LMFDB - Embedded newform 8009.2.a.a.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8009,2,Mod(1,8009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8009 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8009.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:	$63.9521869788$
Analytic rank:	$1$
Dimension:	$306$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50471 q^{2} +3.07842 q^{3} +4.27357 q^{4} +4.01883 q^{5} -7.71055 q^{6} -2.99037 q^{7} -5.69464 q^{8} +6.47666 q^{9} +O(q^{10})$	\(q-2.50471 q^{2} +3.07842 q^{3} +4.27357 q^{4} +4.01883 q^{5} -7.71055 q^{6} -2.99037 q^{7} -5.69464 q^{8} +6.47666 q^{9} -10.0660 q^{10} -2.47324 q^{11} +13.1558 q^{12} -4.56250 q^{13} +7.49000 q^{14} +12.3716 q^{15} +5.71628 q^{16} +3.10076 q^{17} -16.2222 q^{18} -7.61544 q^{19} +17.1748 q^{20} -9.20560 q^{21} +6.19474 q^{22} -6.94903 q^{23} -17.5305 q^{24} +11.1510 q^{25} +11.4277 q^{26} +10.7026 q^{27} -12.7795 q^{28} -6.85859 q^{29} -30.9874 q^{30} +8.64703 q^{31} -2.92834 q^{32} -7.61365 q^{33} -7.76650 q^{34} -12.0178 q^{35} +27.6785 q^{36} -1.50294 q^{37} +19.0745 q^{38} -14.0453 q^{39} -22.8858 q^{40} +5.40661 q^{41} +23.0574 q^{42} -6.67296 q^{43} -10.5696 q^{44} +26.0286 q^{45} +17.4053 q^{46} -7.23723 q^{47} +17.5971 q^{48} +1.94229 q^{49} -27.9300 q^{50} +9.54543 q^{51} -19.4982 q^{52} -10.1870 q^{53} -26.8070 q^{54} -9.93951 q^{55} +17.0291 q^{56} -23.4435 q^{57} +17.1788 q^{58} +6.62252 q^{59} +52.8711 q^{60} +4.43869 q^{61} -21.6583 q^{62} -19.3676 q^{63} -4.09791 q^{64} -18.3359 q^{65} +19.0700 q^{66} -0.789590 q^{67} +13.2513 q^{68} -21.3920 q^{69} +30.1010 q^{70} -11.8008 q^{71} -36.8823 q^{72} -5.10089 q^{73} +3.76443 q^{74} +34.3274 q^{75} -32.5451 q^{76} +7.39588 q^{77} +35.1794 q^{78} -4.28402 q^{79} +22.9727 q^{80} +13.5172 q^{81} -13.5420 q^{82} +12.8936 q^{83} -39.3408 q^{84} +12.4614 q^{85} +16.7138 q^{86} -21.1136 q^{87} +14.0842 q^{88} -11.8019 q^{89} -65.1941 q^{90} +13.6436 q^{91} -29.6972 q^{92} +26.6192 q^{93} +18.1272 q^{94} -30.6051 q^{95} -9.01466 q^{96} +16.9586 q^{97} -4.86488 q^{98} -16.0183 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) $ 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * 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 $

$2$	−2.50471	−1.77110	−0.885549	−	0.464546i	$-0.846217\pi$
$2$	−2.50471	−1.77110	−0.885549	+	0.464546i	$0.846217\pi$
$3$	3.07842	1.77733	0.888663	−	0.458561i	$-0.151635\pi$
$3$	3.07842	1.77733	0.888663	+	0.458561i	$0.151635\pi$
$4$	4.27357	2.13679
$5$	4.01883	1.79727	0.898637	−	0.438693i	$-0.144558\pi$
$5$	4.01883	1.79727	0.898637	+	0.438693i	$0.144558\pi$
$6$	−7.71055	−3.14782
$7$	−2.99037	−1.13025	−0.565126	−	0.825005i	$-0.691173\pi$
$7$	−2.99037	−1.13025	−0.565126	+	0.825005i	$0.691173\pi$
$8$	−5.69464	−2.01336
$9$	6.47666	2.15889
$10$	−10.0660	−3.18315
$11$	−2.47324	−0.745708	−0.372854	−	0.927890i	$-0.621621\pi$
$11$	−2.47324	−0.745708	−0.372854	+	0.927890i	$0.621621\pi$
$12$	13.1558	3.79777
$13$	−4.56250	−1.26541	−0.632705	−	0.774393i	$-0.718056\pi$
$13$	−4.56250	−1.26541	−0.632705	+	0.774393i	$0.718056\pi$
$14$	7.49000	2.00179
$15$	12.3716	3.19434
$16$	5.71628	1.42907
$17$	3.10076	0.752045	0.376022	−	0.926611i	$-0.377292\pi$
$17$	3.10076	0.752045	0.376022	+	0.926611i	$0.377292\pi$
$18$	−16.2222	−3.82360
$19$	−7.61544	−1.74710	−0.873551	−	0.486733i	$-0.838188\pi$
$19$	−7.61544	−1.74710	−0.873551	+	0.486733i	$0.838188\pi$
$20$	17.1748	3.84039
$21$	−9.20560	−2.00883
$22$	6.19474	1.32072
$23$	−6.94903	−1.44897	−0.724486	−	0.689289i	$-0.757923\pi$
$23$	−6.94903	−1.44897	−0.724486	+	0.689289i	$0.757923\pi$
$24$	−17.5305	−3.57840
$25$	11.1510	2.23019
$26$	11.4277	2.24117
$27$	10.7026	2.05972
$28$	−12.7795	−2.41511
$29$	−6.85859	−1.27361	−0.636804	−	0.771026i	$-0.719744\pi$
$29$	−6.85859	−1.27361	−0.636804	+	0.771026i	$0.719744\pi$
$30$	−30.9874	−5.65749
$31$	8.64703	1.55305	0.776527	−	0.630084i	$-0.216980\pi$
$31$	8.64703	1.55305	0.776527	+	0.630084i	$0.216980\pi$
$32$	−2.92834	−0.517662
$33$	−7.61365	−1.32537
$34$	−7.76650	−1.33194
$35$	−12.0178	−2.03137
$36$	27.6785	4.61308
$37$	−1.50294	−0.247082	−0.123541	−	0.992339i	$-0.539425\pi$
$37$	−1.50294	−0.247082	−0.123541	+	0.992339i	$0.539425\pi$
$38$	19.0745	3.09429
$39$	−14.0453	−2.24905
$40$	−22.8858	−3.61856
$41$	5.40661	0.844370	0.422185	−	0.906510i	$-0.361263\pi$
$41$	5.40661	0.844370	0.422185	+	0.906510i	$0.361263\pi$
$42$	23.0574	3.55783
$43$	−6.67296	−1.01762	−0.508808	−	0.860880i	$-0.669914\pi$
$43$	−6.67296	−1.01762	−0.508808	+	0.860880i	$0.669914\pi$
$44$	−10.5696	−1.59342
$45$	26.0286	3.88011
$46$	17.4053	2.56627
$47$	−7.23723	−1.05566	−0.527829	−	0.849350i	$-0.676994\pi$
$47$	−7.23723	−1.05566	−0.527829	+	0.849350i	$0.676994\pi$
$48$	17.5971	2.53992
$49$	1.94229	0.277470
$50$	−27.9300	−3.94989
$51$	9.54543	1.33663
$52$	−19.4982	−2.70391
$53$	−10.1870	−1.39929	−0.699643	−	0.714493i	$-0.746658\pi$
$53$	−10.1870	−1.39929	−0.699643	+	0.714493i	$0.746658\pi$
$54$	−26.8070	−3.64796
$55$	−9.93951	−1.34024
$56$	17.0291	2.27560
$57$	−23.4435	−3.10517
$58$	17.1788	2.25568
$59$	6.62252	0.862179	0.431089	−	0.902309i	$-0.358129\pi$
$59$	6.62252	0.862179	0.431089	+	0.902309i	$0.358129\pi$
$60$	52.8711	6.82563
$61$	4.43869	0.568316	0.284158	−	0.958778i	$-0.408286\pi$
$61$	4.43869	0.568316	0.284158	+	0.958778i	$0.408286\pi$
$62$	−21.6583	−2.75061
$63$	−19.3676	−2.44009
$64$	−4.09791	−0.512239
$65$	−18.3359	−2.27429
$66$	19.0700	2.34735
$67$	−0.789590	−0.0964638	−0.0482319	−	0.998836i	$-0.515359\pi$
$67$	−0.789590	−0.0964638	−0.0482319	+	0.998836i	$0.515359\pi$
$68$	13.2513	1.60696
$69$	−21.3920	−2.57530
$70$	30.1010	3.59776
$71$	−11.8008	−1.40049	−0.700247	−	0.713901i	$-0.746927\pi$
$71$	−11.8008	−1.40049	−0.700247	+	0.713901i	$0.746927\pi$
$72$	−36.8823	−4.34662
$73$	−5.10089	−0.597014	−0.298507	−	0.954407i	$-0.596489\pi$
$73$	−5.10089	−0.597014	−0.298507	+	0.954407i	$0.596489\pi$
$74$	3.76443	0.437606
$75$	34.3274	3.96378
$76$	−32.5451	−3.73318
$77$	7.39588	0.842839
$78$	35.1794	3.98328
$79$	−4.28402	−0.481990	−0.240995	−	0.970526i	$-0.577474\pi$
$79$	−4.28402	−0.481990	−0.240995	+	0.970526i	$0.577474\pi$
$80$	22.9727	2.56843
$81$	13.5172	1.50191
$82$	−13.5420	−1.49546
$83$	12.8936	1.41526	0.707631	−	0.706582i	$-0.249764\pi$
$83$	12.8936	1.41526	0.707631	+	0.706582i	$0.249764\pi$
$84$	−39.3408	−4.29243
$85$	12.4614	1.35163
$86$	16.7138	1.80230
$87$	−21.1136	−2.26362
$88$	14.0842	1.50138
$89$	−11.8019	−1.25100	−0.625501	−	0.780223i	$-0.715105\pi$
$89$	−11.8019	−1.25100	−0.625501	+	0.780223i	$0.715105\pi$
$90$	−65.1941	−6.87206
$91$	13.6436	1.43023
$92$	−29.6972	−3.09614
$93$	26.6192	2.76028
$94$	18.1272	1.86968
$95$	−30.6051	−3.14002
$96$	−9.01466	−0.920055
$97$	16.9586	1.72188	0.860940	−	0.508706i	$-0.169876\pi$
$97$	16.9586	1.72188	0.860940	+	0.508706i	$0.169876\pi$
$98$	−4.86488	−0.491427
$99$	−16.0183	−1.60990
$100$	47.6545	4.76545
$101$	−14.9059	−1.48319	−0.741595	−	0.670848i	$-0.765930\pi$
$101$	−14.9059	−1.48319	−0.741595	+	0.670848i	$0.765930\pi$
$102$	−23.9085	−2.36730
$103$	−2.15505	−0.212343	−0.106171	−	0.994348i	$-0.533859\pi$
$103$	−2.15505	−0.212343	−0.106171	+	0.994348i	$0.533859\pi$
$104$	25.9818	2.54773
$105$	−36.9957	−3.61041
$106$	25.5154	2.47827
$107$	0.932764	0.0901737	0.0450869	−	0.998983i	$-0.485644\pi$
$107$	0.932764	0.0901737	0.0450869	+	0.998983i	$0.485644\pi$
$108$	45.7384	4.40118
$109$	3.01387	0.288676	0.144338	−	0.989528i	$-0.453895\pi$
$109$	3.01387	0.288676	0.144338	+	0.989528i	$0.453895\pi$
$110$	24.8956	2.37370
$111$	−4.62668	−0.439145
$112$	−17.0938	−1.61521
$113$	−3.27747	−0.308319	−0.154159	−	0.988046i	$-0.549267\pi$
$113$	−3.27747	−0.308319	−0.154159	+	0.988046i	$0.549267\pi$
$114$	58.7192	5.49956
$115$	−27.9269	−2.60420
$116$	−29.3107	−2.72143
$117$	−29.5498	−2.73188
$118$	−16.5875	−1.52700
$119$	−9.27241	−0.850000
$120$	−70.4520	−6.43136
$121$	−4.88311	−0.443919
$122$	−11.1176	−1.00654
$123$	16.6438	1.50072
$124$	36.9537	3.31854
$125$	24.7197	2.21100
$126$	48.5102	4.32163
$127$	15.3571	1.36272	0.681360	−	0.731949i	$-0.261389\pi$
$127$	15.3571	1.36272	0.681360	+	0.731949i	$0.261389\pi$
$128$	16.1208	1.42489
$129$	−20.5421	−1.80864
$130$	45.9261	4.02799
$131$	9.85994	0.861467	0.430733	−	0.902479i	$-0.358255\pi$
$131$	9.85994	0.861467	0.430733	+	0.902479i	$0.358255\pi$
$132$	−32.5375	−2.83203
$133$	22.7729	1.97467
$134$	1.97769	0.170847
$135$	43.0120	3.70188
$136$	−17.6577	−1.51414
$137$	2.79860	0.239100	0.119550	−	0.992828i	$-0.461855\pi$
$137$	2.79860	0.239100	0.119550	+	0.992828i	$0.461855\pi$
$138$	53.5808	4.56110
$139$	−10.8571	−0.920890	−0.460445	−	0.887688i	$-0.652310\pi$
$139$	−10.8571	−0.920890	−0.460445	+	0.887688i	$0.652310\pi$
$140$	−51.3588	−4.34061
$141$	−22.2792	−1.87625
$142$	29.5575	2.48041
$143$	11.2841	0.943627
$144$	37.0224	3.08520
$145$	−27.5635	−2.28902
$146$	12.7763	1.05737
$147$	5.97919	0.493155
$148$	−6.42293	−0.527961
$149$	−15.0233	−1.23076	−0.615379	−	0.788231i	$-0.710997\pi$
$149$	−15.0233	−1.23076	−0.615379	+	0.788231i	$0.710997\pi$
$150$	−85.9801	−7.02025
$151$	−0.167783	−0.0136540	−0.00682700	−	0.999977i	$-0.502173\pi$
$151$	−0.167783	−0.0136540	−0.00682700	+	0.999977i	$0.502173\pi$
$152$	43.3672	3.51754
$153$	20.0826	1.62358
$154$	−18.5245	−1.49275
$155$	34.7509	2.79126
$156$	−60.0236	−4.80573
$157$	22.3490	1.78364	0.891820	−	0.452390i	$-0.149428\pi$
$157$	22.3490	1.78364	0.891820	+	0.452390i	$0.149428\pi$
$158$	10.7302	0.853652
$159$	−31.3597	−2.48699
$160$	−11.7685	−0.930381
$161$	20.7801	1.63770
$162$	−33.8566	−2.66002
$163$	−7.14549	−0.559678	−0.279839	−	0.960047i	$-0.590281\pi$
$163$	−7.14549	−0.559678	−0.279839	+	0.960047i	$0.590281\pi$
$164$	23.1055	1.80424
$165$	−30.5980	−2.38205
$166$	−32.2948	−2.50657
$167$	23.8082	1.84233	0.921165	−	0.389171i	$-0.127239\pi$
$167$	23.8082	1.84233	0.921165	+	0.389171i	$0.127239\pi$
$168$	52.4226	4.04449
$169$	7.81643	0.601264
$170$	−31.2122	−2.39387
$171$	−49.3226	−3.77179
$172$	−28.5174	−2.17443
$173$	−2.04369	−0.155379	−0.0776894	−	0.996978i	$-0.524754\pi$
$173$	−2.04369	−0.155379	−0.0776894	+	0.996978i	$0.524754\pi$
$174$	52.8835	4.00909
$175$	−33.3455	−2.52068
$176$	−14.1377	−1.06567
$177$	20.3869	1.53237
$178$	29.5604	2.21565
$179$	2.06915	0.154655	0.0773277	−	0.997006i	$-0.475361\pi$
$179$	2.06915	0.154655	0.0773277	+	0.997006i	$0.475361\pi$
$180$	111.235	8.29097
$181$	−1.80312	−0.134025	−0.0670123	−	0.997752i	$-0.521347\pi$
$181$	−1.80312	−0.134025	−0.0670123	+	0.997752i	$0.521347\pi$
$182$	−34.1732	−2.53308
$183$	13.6641	1.01008
$184$	39.5722	2.91730
$185$	−6.04006	−0.444074
$186$	−66.6734	−4.88873
$187$	−7.66891	−0.560806
$188$	−30.9288	−2.25572
$189$	−32.0047	−2.32800
$190$	76.6570	5.56128
$191$	−19.8842	−1.43877	−0.719384	−	0.694613i	$-0.755576\pi$
$191$	−19.8842	−1.43877	−0.719384	+	0.694613i	$0.755576\pi$
$192$	−12.6151	−0.910416
$193$	5.07551	0.365343	0.182672	−	0.983174i	$-0.441525\pi$
$193$	5.07551	0.365343	0.182672	+	0.983174i	$0.441525\pi$
$194$	−42.4763	−3.04962
$195$	−56.4456	−4.04215
$196$	8.30052	0.592895
$197$	−12.6683	−0.902577	−0.451289	−	0.892378i	$-0.649035\pi$
$197$	−12.6683	−0.902577	−0.451289	+	0.892378i	$0.649035\pi$
$198$	40.1212	2.85129
$199$	−21.6024	−1.53136	−0.765678	−	0.643224i	$-0.777596\pi$
$199$	−21.6024	−1.53136	−0.765678	+	0.643224i	$0.777596\pi$
$200$	−63.5008	−4.49018
$201$	−2.43069	−0.171448
$202$	37.3349	2.62687
$203$	20.5097	1.43950
$204$	40.7931	2.85609
$205$	21.7282	1.51756
$206$	5.39776	0.376080
$207$	−45.0065	−3.12817
$208$	−26.0805	−1.80836
$209$	18.8348	1.30283
$210$	92.6635	6.39439
$211$	−15.1607	−1.04371	−0.521854	−	0.853035i	$-0.674759\pi$
$211$	−15.1607	−1.04371	−0.521854	+	0.853035i	$0.674759\pi$
$212$	−43.5347	−2.98997
$213$	−36.3277	−2.48913
$214$	−2.33630	−0.159706
$215$	−26.8175	−1.82894
$216$	−60.9476	−4.14696
$217$	−25.8578	−1.75534
$218$	−7.54887	−0.511274
$219$	−15.7027	−1.06109
$220$	−42.4772	−2.86381
$221$	−14.1472	−0.951645
$222$	11.5885	0.777769
$223$	−14.4728	−0.969172	−0.484586	−	0.874744i	$-0.661030\pi$
$223$	−14.4728	−0.969172	−0.484586	+	0.874744i	$0.661030\pi$
$224$	8.75681	0.585089
$225$	72.2211	4.81474
$226$	8.20912	0.546063
$227$	−21.5170	−1.42813	−0.714067	−	0.700077i	$-0.753149\pi$
$227$	−21.5170	−1.42813	−0.714067	+	0.700077i	$0.753149\pi$
$228$	−100.188	−6.63508
$229$	13.5749	0.897054	0.448527	−	0.893769i	$-0.351949\pi$
$229$	13.5749	0.897054	0.448527	+	0.893769i	$0.351949\pi$
$230$	69.9489	4.61229
$231$	22.7676	1.49800
$232$	39.0572	2.56423
$233$	14.4466	0.946431	0.473216	−	0.880947i	$-0.343093\pi$
$233$	14.4466	0.946431	0.473216	+	0.880947i	$0.343093\pi$
$234$	74.0136	4.83842
$235$	−29.0852	−1.89731
$236$	28.3018	1.84229
$237$	−13.1880	−0.856654
$238$	23.2247	1.50543
$239$	20.5299	1.32797	0.663985	−	0.747746i	$-0.268864\pi$
$239$	20.5299	1.32797	0.663985	+	0.747746i	$0.268864\pi$
$240$	70.7197	4.56494
$241$	−21.3294	−1.37395	−0.686973	−	0.726683i	$-0.741061\pi$
$241$	−21.3294	−1.37395	−0.686973	+	0.726683i	$0.741061\pi$
$242$	12.2308	0.786224
$243$	9.50360	0.609656
$244$	18.9691	1.21437
$245$	7.80573	0.498690
$246$	−41.6879	−2.65792
$247$	34.7455	2.21080
$248$	−49.2418	−3.12685
$249$	39.6920	2.51538
$250$	−61.9157	−3.91589
$251$	−20.5318	−1.29596	−0.647978	−	0.761659i	$-0.724385\pi$
$251$	−20.5318	−1.29596	−0.647978	+	0.761659i	$0.724385\pi$
$252$	−82.7688	−5.21394
$253$	17.1866	1.08051
$254$	−38.4650	−2.41351
$255$	38.3615	2.40229
$256$	−32.1820	−2.01138
$257$	15.5067	0.967279	0.483639	−	0.875267i	$-0.339315\pi$
$257$	15.5067	0.967279	0.483639	+	0.875267i	$0.339315\pi$
$258$	51.4521	3.20327
$259$	4.49434	0.279265
$260$	−78.3599	−4.85967
$261$	−44.4208	−2.74958
$262$	−24.6963	−1.52574
$263$	13.4750	0.830904	0.415452	−	0.909615i	$-0.363623\pi$
$263$	13.4750	0.830904	0.415452	+	0.909615i	$0.363623\pi$
$264$	43.3570	2.66844
$265$	−40.9396	−2.51490
$266$	−57.0396	−3.49732
$267$	−36.3313	−2.22344
$268$	−3.37437	−0.206122
$269$	−16.2645	−0.991661	−0.495831	−	0.868419i	$-0.665136\pi$
$269$	−16.2645	−0.991661	−0.495831	+	0.868419i	$0.665136\pi$
$270$	−107.733	−6.55639
$271$	−8.35959	−0.507809	−0.253904	−	0.967229i	$-0.581715\pi$
$271$	−8.35959	−0.507809	−0.253904	+	0.967229i	$0.581715\pi$
$272$	17.7248	1.07472
$273$	42.0006	2.54199
$274$	−7.00967	−0.423470
$275$	−27.5790	−1.66308
$276$	−91.4203	−5.50286
$277$	−20.8181	−1.25084	−0.625419	−	0.780289i	$-0.715072\pi$
$277$	−20.8181	−1.25084	−0.625419	+	0.780289i	$0.715072\pi$
$278$	27.1940	1.63099
$279$	56.0039	3.35287
$280$	68.4369	4.08988
$281$	7.03880	0.419900	0.209950	−	0.977712i	$-0.432670\pi$
$281$	7.03880	0.419900	0.209950	+	0.977712i	$0.432670\pi$
$282$	55.8030	3.32302
$283$	15.3413	0.911943	0.455972	−	0.889994i	$-0.349292\pi$
$283$	15.3413	0.911943	0.455972	+	0.889994i	$0.349292\pi$
$284$	−50.4314	−2.99256
$285$	−94.2154	−5.58084
$286$	−28.2635	−1.67126
$287$	−16.1677	−0.954351
$288$	−18.9659	−1.11757
$289$	−7.38529	−0.434429
$290$	69.0386	4.05408
$291$	52.2055	3.06034
$292$	−21.7990	−1.27569
$293$	−24.6791	−1.44177	−0.720885	−	0.693054i	$-0.756265\pi$
$293$	−24.6791	−1.44177	−0.720885	+	0.693054i	$0.756265\pi$
$294$	−14.9761	−0.873425
$295$	26.6148	1.54957
$296$	8.55871	0.497465
$297$	−26.4701	−1.53595
$298$	37.6290	2.17979
$299$	31.7050	1.83355
$300$	146.700	8.46976
$301$	19.9546	1.15016
$302$	0.420248	0.0241826
$303$	−45.8865	−2.63611
$304$	−43.5320	−2.49673
$305$	17.8383	1.02142
$306$	−50.3010	−2.87552
$307$	−8.53609	−0.487181	−0.243590	−	0.969878i	$-0.578325\pi$
$307$	−8.53609	−0.487181	−0.243590	+	0.969878i	$0.578325\pi$
$308$	31.6068	1.80097
$309$	−6.63413	−0.377403
$310$	−87.0410	−4.94360
$311$	−23.0120	−1.30489	−0.652445	−	0.757836i	$-0.726257\pi$
$311$	−23.0120	−1.30489	−0.652445	+	0.757836i	$0.726257\pi$
$312$	79.9829	4.52814
$313$	7.48281	0.422954	0.211477	−	0.977383i	$-0.432173\pi$
$313$	7.48281	0.422954	0.211477	+	0.977383i	$0.432173\pi$
$314$	−55.9776	−3.15900
$315$	−77.8350	−4.38551
$316$	−18.3081	−1.02991
$317$	−25.5387	−1.43439	−0.717197	−	0.696870i	$-0.754575\pi$
$317$	−25.5387	−1.43439	−0.717197	+	0.696870i	$0.754575\pi$
$318$	78.5470	4.40470
$319$	16.9629	0.949741
$320$	−16.4688	−0.920634
$321$	2.87144	0.160268
$322$	−52.0482	−2.90053
$323$	−23.6136	−1.31390
$324$	57.7665	3.20925
$325$	−50.8763	−2.82211
$326$	17.8974	0.991244
$327$	9.27795	0.513072
$328$	−30.7887	−1.70002
$329$	21.6420	1.19316
$330$	76.6390	4.21884
$331$	−2.12376	−0.116732	−0.0583661	−	0.998295i	$-0.518589\pi$
$331$	−2.12376	−0.116732	−0.0583661	+	0.998295i	$0.518589\pi$
$332$	55.1019	3.02411
$333$	−9.73404	−0.533422
$334$	−59.6325	−3.26295
$335$	−3.17323	−0.173372
$336$	−52.6218	−2.87075
$337$	7.75210	0.422284	0.211142	−	0.977455i	$-0.432282\pi$
$337$	7.75210	0.422284	0.211142	+	0.977455i	$0.432282\pi$
$338$	−19.5779	−1.06490
$339$	−10.0894	−0.547983
$340$	53.2548	2.88815
$341$	−21.3862	−1.15812
$342$	123.539	6.68021
$343$	15.1244	0.816641
$344$	38.0001	2.04883
$345$	−85.9708	−4.62851
$346$	5.11885	0.275191
$347$	27.6837	1.48614	0.743070	−	0.669214i	$-0.233369\pi$
$347$	27.6837	1.48614	0.743070	+	0.669214i	$0.233369\pi$
$348$	−90.2306	−4.83687
$349$	−6.42574	−0.343962	−0.171981	−	0.985100i	$-0.555017\pi$
$349$	−6.42574	−0.343962	−0.171981	+	0.985100i	$0.555017\pi$
$350$	83.5208	4.46437
$351$	−48.8307	−2.60639
$352$	7.24247	0.386025
$353$	3.78881	0.201658	0.100829	−	0.994904i	$-0.467851\pi$
$353$	3.78881	0.201658	0.100829	+	0.994904i	$0.467851\pi$
$354$	−51.0632	−2.71398
$355$	−47.4253	−2.51707
$356$	−50.4364	−2.67312
$357$	−28.5443	−1.51073
$358$	−5.18261	−0.273910
$359$	−5.82139	−0.307241	−0.153621	−	0.988130i	$-0.549093\pi$
$359$	−5.82139	−0.307241	−0.153621	+	0.988130i	$0.549093\pi$
$360$	−148.223	−7.81206
$361$	38.9949	2.05236
$362$	4.51628	0.237370
$363$	−15.0322	−0.788989
$364$	58.3067	3.05610
$365$	−20.4996	−1.07300
$366$	−34.2247	−1.78895
$367$	−33.5026	−1.74882	−0.874410	−	0.485188i	$-0.838751\pi$
$367$	−33.5026	−1.74882	−0.874410	+	0.485188i	$0.838751\pi$
$368$	−39.7226	−2.07068
$369$	35.0168	1.82290
$370$	15.1286	0.786498
$371$	30.4627	1.58155
$372$	113.759	5.89813
$373$	17.2875	0.895111	0.447555	−	0.894256i	$-0.352295\pi$
$373$	17.2875	0.895111	0.447555	+	0.894256i	$0.352295\pi$
$374$	19.2084	0.993242
$375$	76.0976	3.92966
$376$	41.2134	2.12542
$377$	31.2923	1.61164
$378$	80.1626	4.12312
$379$	−12.4523	−0.639633	−0.319817	−	0.947479i	$-0.603621\pi$
$379$	−12.4523	−0.639633	−0.319817	+	0.947479i	$0.603621\pi$
$380$	−130.793	−6.70955
$381$	47.2755	2.42200
$382$	49.8041	2.54820
$383$	−15.2380	−0.778625	−0.389313	−	0.921106i	$-0.627287\pi$
$383$	−15.2380	−0.778625	−0.389313	+	0.921106i	$0.627287\pi$
$384$	49.6265	2.53249
$385$	29.7228	1.51481
$386$	−12.7127	−0.647059
$387$	−43.2185	−2.19692
$388$	72.4736	3.67929
$389$	22.1844	1.12479	0.562396	−	0.826868i	$-0.309880\pi$
$389$	22.1844	1.12479	0.562396	+	0.826868i	$0.309880\pi$
$390$	141.380	7.15905
$391$	−21.5473	−1.08969
$392$	−11.0607	−0.558647
$393$	30.3530	1.53111
$394$	31.7304	1.59855
$395$	−17.2168	−0.866269
$396$	−68.4554	−3.44001
$397$	−11.9863	−0.601576	−0.300788	−	0.953691i	$-0.597250\pi$
$397$	−11.9863	−0.601576	−0.300788	+	0.953691i	$0.597250\pi$
$398$	54.1079	2.71218
$399$	70.1047	3.50962
$400$	63.7421	3.18710
$401$	3.74888	0.187210	0.0936052	−	0.995609i	$-0.470161\pi$
$401$	3.74888	0.187210	0.0936052	+	0.995609i	$0.470161\pi$
$402$	6.08817	0.303650
$403$	−39.4521	−1.96525
$404$	−63.7013	−3.16926
$405$	54.3231	2.69934
$406$	−51.3709	−2.54949
$407$	3.71713	0.184251
$408$	−54.3578	−2.69111
$409$	28.4230	1.40543	0.702713	−	0.711473i	$-0.251972\pi$
$409$	28.4230	1.40543	0.702713	+	0.711473i	$0.251972\pi$
$410$	−54.4229	−2.68776
$411$	8.61525	0.424959
$412$	−9.20974	−0.453731
$413$	−19.8038	−0.974479
$414$	112.728	5.54029
$415$	51.8173	2.54361
$416$	13.3606	0.655055
$417$	−33.4228	−1.63672
$418$	−47.1756	−2.30744
$419$	34.7632	1.69829	0.849146	−	0.528158i	$-0.177117\pi$
$419$	34.7632	1.69829	0.849146	+	0.528158i	$0.177117\pi$
$420$	−158.104	−7.71468
$421$	21.5751	1.05151	0.525753	−	0.850638i	$-0.323784\pi$
$421$	21.5751	1.05151	0.525753	+	0.850638i	$0.323784\pi$
$422$	37.9732	1.84851
$423$	−46.8731	−2.27905
$424$	58.0110	2.81727
$425$	34.5765	1.67721
$426$	90.9904	4.40850
$427$	−13.2733	−0.642340
$428$	3.98624	0.192682
$429$	34.7373	1.67713
$430$	67.1699	3.23922
$431$	32.8196	1.58087	0.790433	−	0.612548i	$-0.209856\pi$
$431$	32.8196	1.58087	0.790433	+	0.612548i	$0.209856\pi$
$432$	61.1791	2.94348
$433$	−11.1786	−0.537207	−0.268604	−	0.963251i	$-0.586562\pi$
$433$	−11.1786	−0.537207	−0.268604	+	0.963251i	$0.586562\pi$
$434$	64.7663	3.10888
$435$	−84.8520	−4.06834
$436$	12.8800	0.616840
$437$	52.9199	2.53150
$438$	39.3307	1.87929
$439$	−34.5960	−1.65118	−0.825589	−	0.564271i	$-0.809157\pi$
$439$	−34.5960	−1.65118	−0.825589	+	0.564271i	$0.809157\pi$
$440$	56.6019	2.69839
$441$	12.5796	0.599027
$442$	35.4347	1.68546
$443$	16.4279	0.780513	0.390257	−	0.920706i	$-0.372386\pi$
$443$	16.4279	0.780513	0.390257	+	0.920706i	$0.372386\pi$
$444$	−19.7725	−0.938359
$445$	−47.4299	−2.24839
$446$	36.2502	1.71650
$447$	−46.2481	−2.18746
$448$	12.2543	0.578960
$449$	6.32538	0.298513	0.149257	−	0.988798i	$-0.452312\pi$
$449$	6.32538	0.298513	0.149257	+	0.988798i	$0.452312\pi$
$450$	−180.893	−8.52737
$451$	−13.3718	−0.629654
$452$	−14.0065	−0.658811
$453$	−0.516507	−0.0242676
$454$	53.8939	2.52937
$455$	54.8311	2.57052
$456$	133.502	6.25182
$457$	26.3744	1.23374	0.616870	−	0.787065i	$-0.288400\pi$
$457$	26.3744	1.23374	0.616870	+	0.787065i	$0.288400\pi$
$458$	−34.0011	−1.58877
$459$	33.1862	1.54900
$460$	−119.348	−5.56462
$461$	11.9869	0.558284	0.279142	−	0.960250i	$-0.409950\pi$
$461$	11.9869	0.558284	0.279142	+	0.960250i	$0.409950\pi$
$462$	−57.0263	−2.65310
$463$	−24.0624	−1.11827	−0.559137	−	0.829075i	$-0.688868\pi$
$463$	−24.0624	−1.11827	−0.559137	+	0.829075i	$0.688868\pi$
$464$	−39.2056	−1.82008
$465$	106.978	4.96098
$466$	−36.1847	−1.67622
$467$	19.6583	0.909676	0.454838	−	0.890574i	$-0.349697\pi$
$467$	19.6583	0.909676	0.454838	+	0.890574i	$0.349697\pi$
$468$	−126.283	−5.83744
$469$	2.36116	0.109028
$470$	72.8500	3.36032
$471$	68.7994	3.17011
$472$	−37.7129	−1.73588
$473$	16.5038	0.758845
$474$	33.0322	1.51722
$475$	−84.9195	−3.89638
$476$	−39.6263	−1.81627
$477$	−65.9774	−3.02090
$478$	−51.4215	−2.35196
$479$	2.77955	0.127001	0.0635004	−	0.997982i	$-0.479774\pi$
$479$	2.77955	0.127001	0.0635004	+	0.997982i	$0.479774\pi$
$480$	−36.2284	−1.65359
$481$	6.85717	0.312660
$482$	53.4239	2.43339
$483$	63.9700	2.91073
$484$	−20.8683	−0.948560
$485$	68.1535	3.09469
$486$	−23.8038	−1.07976
$487$	−11.7331	−0.531677	−0.265838	−	0.964018i	$-0.585649\pi$
$487$	−11.7331	−0.531677	−0.265838	+	0.964018i	$0.585649\pi$
$488$	−25.2767	−1.14422
$489$	−21.9968	−0.994729
$490$	−19.5511	−0.883229
$491$	35.6686	1.60970	0.804851	−	0.593476i	$-0.202245\pi$
$491$	35.6686	1.60970	0.804851	+	0.593476i	$0.202245\pi$
$492$	71.1285	3.20672
$493$	−21.2668	−0.957810
$494$	−87.0273	−3.91554
$495$	−64.3748	−2.89343
$496$	49.4289	2.21942
$497$	35.2886	1.58291
$498$	−99.4171	−4.45499
$499$	8.76481	0.392367	0.196183	−	0.980567i	$-0.437145\pi$
$499$	8.76481	0.392367	0.196183	+	0.980567i	$0.437145\pi$
$500$	105.641	4.72443
$501$	73.2915	3.27442
$502$	51.4262	2.29526
$503$	23.7069	1.05704	0.528520	−	0.848921i	$-0.322747\pi$
$503$	23.7069	1.05704	0.528520	+	0.848921i	$0.322747\pi$
$504$	110.291	4.91277
$505$	−59.9041	−2.66570
$506$	−43.0474	−1.91369
$507$	24.0623	1.06864
$508$	65.6296	2.91184
$509$	3.35654	0.148776	0.0743881	−	0.997229i	$-0.476300\pi$
$509$	3.35654	0.148776	0.0743881	+	0.997229i	$0.476300\pi$
$510$	−96.0843	−4.25469
$511$	15.2535	0.674777
$512$	48.3651	2.13746
$513$	−81.5051	−3.59854
$514$	−38.8397	−1.71315
$515$	−8.66076	−0.381638
$516$	−87.7884	−3.86467
$517$	17.8994	0.787214
$518$	−11.2570	−0.494605
$519$	−6.29133	−0.276159
$520$	104.416	4.57896
$521$	−30.8065	−1.34966	−0.674828	−	0.737975i	$-0.735782\pi$
$521$	−30.8065	−1.34966	−0.674828	+	0.737975i	$0.735782\pi$
$522$	111.261	4.86977
$523$	−21.5046	−0.940330	−0.470165	−	0.882578i	$-0.655806\pi$
$523$	−21.5046	−0.940330	−0.470165	+	0.882578i	$0.655806\pi$
$524$	42.1372	1.84077
$525$	−102.651	−4.48007
$526$	−33.7510	−1.47161
$527$	26.8124	1.16797
$528$	−43.5218	−1.89404
$529$	25.2890	1.09952
$530$	102.542	4.45413
$531$	42.8918	1.86135
$532$	97.3219	4.21944
$533$	−24.6677	−1.06848
$534$	90.9993	3.93793
$535$	3.74862	0.162067
$536$	4.49643	0.194216
$537$	6.36970	0.274873
$538$	40.7377	1.75633
$539$	−4.80374	−0.206912
$540$	183.815	7.91013
$541$	30.1447	1.29602	0.648012	−	0.761630i	$-0.275601\pi$
$541$	30.1447	1.29602	0.648012	+	0.761630i	$0.275601\pi$
$542$	20.9383	0.899379
$543$	−5.55074	−0.238205
$544$	−9.08008	−0.389305
$545$	12.1122	0.518831
$546$	−105.199	−4.50211
$547$	7.44812	0.318459	0.159229	−	0.987242i	$-0.449099\pi$
$547$	7.44812	0.318459	0.159229	+	0.987242i	$0.449099\pi$
$548$	11.9600	0.510906
$549$	28.7479	1.22693
$550$	69.0773	2.94547
$551$	52.2312	2.22512
$552$	121.820	5.18500
$553$	12.8108	0.544771
$554$	52.1433	2.21536
$555$	−18.5938	−0.789264
$556$	−46.3987	−1.96774
$557$	22.7920	0.965730	0.482865	−	0.875695i	$-0.339596\pi$
$557$	22.7920	0.965730	0.482865	+	0.875695i	$0.339596\pi$
$558$	−140.274	−5.93825
$559$	30.4454	1.28770
$560$	−68.6969	−2.90297
$561$	−23.6081	−0.996735
$562$	−17.6302	−0.743684
$563$	−44.1755	−1.86178	−0.930888	−	0.365305i	$-0.880965\pi$
$563$	−44.1755	−1.86178	−0.930888	+	0.365305i	$0.880965\pi$
$564$	−95.2119	−4.00915
$565$	−13.1716	−0.554133
$566$	−38.4254	−1.61514
$567$	−40.4212	−1.69753
$568$	67.2011	2.81970
$569$	−7.46970	−0.313146	−0.156573	−	0.987666i	$-0.550045\pi$
$569$	−7.46970	−0.313146	−0.156573	+	0.987666i	$0.550045\pi$
$570$	235.982	9.88421
$571$	−35.9630	−1.50501	−0.752503	−	0.658589i	$-0.771153\pi$
$571$	−35.9630	−1.50501	−0.752503	+	0.658589i	$0.771153\pi$
$572$	48.2236	2.01633
$573$	−61.2118	−2.55716
$574$	40.4955	1.69025
$575$	−77.4884	−3.23149
$576$	−26.5408	−1.10587
$577$	−19.0855	−0.794540	−0.397270	−	0.917702i	$-0.630042\pi$
$577$	−19.0855	−0.794540	−0.397270	+	0.917702i	$0.630042\pi$
$578$	18.4980	0.769416
$579$	15.6245	0.649334
$580$	−117.795	−4.89115
$581$	−38.5567	−1.59960
$582$	−130.760	−5.42016
$583$	25.1947	1.04346
$584$	29.0478	1.20200
$585$	−118.755	−4.90993
$586$	61.8141	2.55352
$587$	17.4070	0.718464	0.359232	−	0.933248i	$-0.383039\pi$
$587$	17.4070	0.718464	0.359232	+	0.933248i	$0.383039\pi$
$588$	25.5525	1.05377
$589$	−65.8510	−2.71334
$590$	−66.6623	−2.74444
$591$	−38.9982	−1.60417
$592$	−8.59123	−0.353097
$593$	−16.2440	−0.667062	−0.333531	−	0.942739i	$-0.608240\pi$
$593$	−16.2440	−0.667062	−0.333531	+	0.942739i	$0.608240\pi$
$594$	66.2999	2.72032
$595$	−37.2642	−1.52768
$596$	−64.2032	−2.62987
$597$	−66.5014	−2.72172
$598$	−79.4117	−3.24739
$599$	−12.6423	−0.516550	−0.258275	−	0.966071i	$-0.583154\pi$
$599$	−12.6423	−0.516550	−0.258275	+	0.966071i	$0.583154\pi$
$600$	−195.482	−7.98052
$601$	−36.8018	−1.50118	−0.750588	−	0.660770i	$-0.770230\pi$
$601$	−36.8018	−1.50118	−0.750588	+	0.660770i	$0.770230\pi$
$602$	−49.9804	−2.03705
$603$	−5.11391	−0.208254
$604$	−0.717034	−0.0291757
$605$	−19.6244	−0.797844
$606$	114.932	4.66881
$607$	−14.3408	−0.582075	−0.291037	−	0.956712i	$-0.594000\pi$
$607$	−14.3408	−0.582075	−0.291037	+	0.956712i	$0.594000\pi$
$608$	22.3006	0.904408
$609$	63.1374	2.55846
$610$	−44.6798	−1.80903
$611$	33.0199	1.33584
$612$	85.8243	3.46924
$613$	45.8509	1.85190	0.925950	−	0.377646i	$-0.123266\pi$
$613$	45.8509	1.85190	0.925950	+	0.377646i	$0.123266\pi$
$614$	21.3804	0.862844
$615$	66.8886	2.69721
$616$	−42.1169	−1.69694
$617$	−7.67197	−0.308862	−0.154431	−	0.988004i	$-0.549354\pi$
$617$	−7.67197	−0.308862	−0.154431	+	0.988004i	$0.549354\pi$
$618$	16.6166	0.668417
$619$	−28.6718	−1.15242	−0.576208	−	0.817303i	$-0.695468\pi$
$619$	−28.6718	−1.15242	−0.576208	+	0.817303i	$0.695468\pi$
$620$	148.511	5.96433
$621$	−74.3728	−2.98448
$622$	57.6383	2.31109
$623$	35.2921	1.41395
$624$	−80.2868	−3.21405
$625$	43.5893	1.74357
$626$	−18.7423	−0.749092
$627$	57.9813	2.31555
$628$	95.5099	3.81126
$629$	−4.66026	−0.185817
$630$	194.954	7.76716
$631$	−34.4089	−1.36980	−0.684899	−	0.728638i	$-0.740154\pi$
$631$	−34.4089	−1.36980	−0.684899	+	0.728638i	$0.740154\pi$
$632$	24.3960	0.970420
$633$	−46.6710	−1.85501
$634$	63.9670	2.54045
$635$	61.7174	2.44918
$636$	−134.018	−5.31416
$637$	−8.86171	−0.351114
$638$	−42.4872	−1.68208
$639$	−76.4296	−3.02351
$640$	64.7866	2.56091
$641$	17.3113	0.683754	0.341877	−	0.939745i	$-0.388937\pi$
$641$	17.3113	0.683754	0.341877	+	0.939745i	$0.388937\pi$
$642$	−7.19212	−0.283850
$643$	44.8412	1.76837	0.884183	−	0.467141i	$-0.154716\pi$
$643$	44.8412	1.76837	0.884183	+	0.467141i	$0.154716\pi$
$644$	88.8054	3.49942
$645$	−82.5553	−3.25061
$646$	59.1453	2.32704
$647$	−1.45453	−0.0571836	−0.0285918	−	0.999591i	$-0.509102\pi$
$647$	−1.45453	−0.0571836	−0.0285918	+	0.999591i	$0.509102\pi$
$648$	−76.9753	−3.02388
$649$	−16.3790	−0.642934
$650$	127.431	4.99824
$651$	−79.6011	−3.11981
$652$	−30.5368	−1.19591
$653$	−26.4113	−1.03355	−0.516777	−	0.856120i	$-0.672868\pi$
$653$	−26.4113	−1.03355	−0.516777	+	0.856120i	$0.672868\pi$
$654$	−23.2386	−0.908701
$655$	39.6254	1.54829
$656$	30.9057	1.20666
$657$	−33.0368	−1.28889
$658$	−54.2069	−2.11320
$659$	−19.0823	−0.743342	−0.371671	−	0.928365i	$-0.621215\pi$
$659$	−19.0823	−0.743342	−0.371671	+	0.928365i	$0.621215\pi$
$660$	−130.763	−5.08993
$661$	12.0579	0.468999	0.234500	−	0.972116i	$-0.424655\pi$
$661$	12.0579	0.468999	0.234500	+	0.972116i	$0.424655\pi$
$662$	5.31940	0.206744
$663$	−43.5511	−1.69138
$664$	−73.4247	−2.84943
$665$	91.5205	3.54901
$666$	24.3809	0.944742
$667$	47.6605	1.84542
$668$	101.746	3.93667
$669$	−44.5534	−1.72253
$670$	7.94801	0.307058
$671$	−10.9779	−0.423798
$672$	26.9571	1.03989
$673$	42.3794	1.63360	0.816802	−	0.576917i	$-0.195745\pi$
$673$	42.3794	1.63360	0.816802	+	0.576917i	$0.195745\pi$
$674$	−19.4168	−0.747906
$675$	119.345	4.59358
$676$	33.4041	1.28477
$677$	19.9519	0.766813	0.383407	−	0.923580i	$-0.374751\pi$
$677$	19.9519	0.766813	0.383407	+	0.923580i	$0.374751\pi$
$678$	25.2711	0.970531
$679$	−50.7123	−1.94616
$680$	−70.9633	−2.72132
$681$	−66.2384	−2.53826
$682$	53.5661	2.05115
$683$	−12.0312	−0.460362	−0.230181	−	0.973148i	$-0.573932\pi$
$683$	−12.0312	−0.460362	−0.230181	+	0.973148i	$0.573932\pi$
$684$	−210.784	−8.05952
$685$	11.2471	0.429729
$686$	−37.8822	−1.44635
$687$	41.7892	1.59436
$688$	−38.1445	−1.45424
$689$	46.4780	1.77067
$690$	215.332	8.19755
$691$	27.5248	1.04709	0.523546	−	0.851997i	$-0.324609\pi$
$691$	27.5248	1.04709	0.523546	+	0.851997i	$0.324609\pi$
$692$	−8.73386	−0.332011
$693$	47.9006	1.81959
$694$	−69.3397	−2.63210
$695$	−43.6329	−1.65509
$696$	120.234	4.55748
$697$	16.7646	0.635004
$698$	16.0946	0.609191
$699$	44.4728	1.68212
$700$	−142.504	−5.38616
$701$	26.8403	1.01375	0.506873	−	0.862021i	$-0.330801\pi$
$701$	26.8403	1.01375	0.506873	+	0.862021i	$0.330801\pi$
$702$	122.307	4.61617
$703$	11.4456	0.431677
$704$	10.1351	0.381981
$705$	−89.5364	−3.37214
$706$	−9.48986	−0.357156
$707$	44.5740	1.67638
$708$	87.1248	3.27435
$709$	20.0774	0.754021	0.377011	−	0.926209i	$-0.376952\pi$
$709$	20.0774	0.754021	0.377011	+	0.926209i	$0.376952\pi$
$710$	118.787	4.45798
$711$	−27.7462	−1.04056
$712$	67.2077	2.51872
$713$	−60.0885	−2.25033
$714$	71.4953	2.67565
$715$	45.3490	1.69596
$716$	8.84265	0.330465
$717$	63.1997	2.36023
$718$	14.5809	0.544154
$719$	20.7500	0.773844	0.386922	−	0.922112i	$-0.373538\pi$
$719$	20.7500	0.773844	0.386922	+	0.922112i	$0.373538\pi$
$720$	148.787	5.54495
$721$	6.44438	0.240001
$722$	−97.6709	−3.63493
$723$	−65.6607	−2.44195
$724$	−7.70575	−0.286382
$725$	−76.4800	−2.84039
$726$	37.6514	1.39738
$727$	32.5991	1.20903	0.604517	−	0.796592i	$-0.293366\pi$
$727$	32.5991	1.20903	0.604517	+	0.796592i	$0.293366\pi$
$728$	−77.6951	−2.87957
$729$	−11.2954	−0.418348
$730$	51.3456	1.90039
$731$	−20.6912	−0.765293
$732$	58.3947	2.15833
$733$	14.9919	0.553737	0.276868	−	0.960908i	$-0.410703\pi$
$733$	14.9919	0.553737	0.276868	+	0.960908i	$0.410703\pi$
$734$	83.9142	3.09733
$735$	24.0293	0.886335
$736$	20.3491	0.750078
$737$	1.95284	0.0719338
$738$	−87.7068	−3.22853
$739$	−17.6475	−0.649175	−0.324587	−	0.945856i	$-0.605225\pi$
$739$	−17.6475	−0.649175	−0.324587	+	0.945856i	$0.605225\pi$
$740$	−25.8126	−0.948891
$741$	106.961	3.92931
$742$	−76.3003	−2.80107
$743$	−25.1599	−0.923026	−0.461513	−	0.887133i	$-0.652693\pi$
$743$	−25.1599	−0.923026	−0.461513	+	0.887133i	$0.652693\pi$
$744$	−151.587	−5.55744
$745$	−60.3761	−2.21201
$746$	−43.3001	−1.58533
$747$	83.5078	3.05539
$748$	−32.7736	−1.19832
$749$	−2.78931	−0.101919
$750$	−190.602	−6.95982
$751$	16.2600	0.593336	0.296668	−	0.954981i	$-0.404125\pi$
$751$	16.2600	0.593336	0.296668	+	0.954981i	$0.404125\pi$
$752$	−41.3700	−1.50861
$753$	−63.2054	−2.30333
$754$	−78.3782	−2.85437
$755$	−0.674292	−0.0245400
$756$	−136.775	−4.97444
$757$	−4.22225	−0.153460	−0.0767301	−	0.997052i	$-0.524448\pi$
$757$	−4.22225	−0.153460	−0.0767301	+	0.997052i	$0.524448\pi$
$758$	31.1895	1.13285
$759$	52.9075	1.92042
$760$	174.285	6.32199
$761$	19.0586	0.690875	0.345438	−	0.938442i	$-0.387730\pi$
$761$	19.0586	0.690875	0.345438	+	0.938442i	$0.387730\pi$
$762$	−118.411	−4.28959
$763$	−9.01258	−0.326277
$764$	−84.9764	−3.07434
$765$	80.7084	2.91802
$766$	38.1668	1.37902
$767$	−30.2153	−1.09101
$768$	−99.0697	−3.57487
$769$	49.1125	1.77104	0.885520	−	0.464600i	$-0.153802\pi$
$769$	49.1125	1.77104	0.885520	+	0.464600i	$0.153802\pi$
$770$	−74.4469	−2.68288
$771$	47.7360	1.71917
$772$	21.6906	0.780661
$773$	42.2413	1.51931	0.759656	−	0.650325i	$-0.225367\pi$
$773$	42.2413	1.51931	0.759656	+	0.650325i	$0.225367\pi$
$774$	108.250	3.89096
$775$	96.4229	3.46361
$776$	−96.5729	−3.46676
$777$	13.8355	0.496345
$778$	−55.5654	−1.99212
$779$	−41.1737	−1.47520
$780$	−241.224	−8.63722
$781$	29.1861	1.04436
$782$	53.9696	1.92995
$783$	−73.4049	−2.62328
$784$	11.1027	0.396524
$785$	89.8166	3.20569
$786$	−76.0255	−2.71174
$787$	14.3469	0.511411	0.255706	−	0.966755i	$-0.417692\pi$
$787$	14.3469	0.511411	0.255706	+	0.966755i	$0.417692\pi$
$788$	−54.1388	−1.92861
$789$	41.4817	1.47679
$790$	43.1230	1.53425
$791$	9.80084	0.348478
$792$	91.2185	3.24131
$793$	−20.2515	−0.719153
$794$	30.0223	1.06545
$795$	−126.029	−4.46980
$796$	−92.3196	−3.27218
$797$	20.1423	0.713476	0.356738	−	0.934205i	$-0.383889\pi$
$797$	20.1423	0.713476	0.356738	+	0.934205i	$0.383889\pi$
$798$	−175.592	−6.21588
$799$	−22.4409	−0.793903
$800$	−32.6538	−1.15449
$801$	−76.4371	−2.70077
$802$	−9.38987	−0.331568
$803$	12.6157	0.445199
$804$	−10.3877	−0.366347
$805$	83.5118	2.94340
$806$	98.8161	3.48065
$807$	−50.0688	−1.76251
$808$	84.8836	2.98619
$809$	6.88833	0.242181	0.121090	−	0.992641i	$-0.461361\pi$
$809$	6.88833	0.242181	0.121090	+	0.992641i	$0.461361\pi$
$810$	−136.064	−4.78079
$811$	−24.3832	−0.856210	−0.428105	−	0.903729i	$-0.640819\pi$
$811$	−24.3832	−0.856210	−0.428105	+	0.903729i	$0.640819\pi$
$812$	87.6497	3.07590
$813$	−25.7343	−0.902542
$814$	−9.31032	−0.326327
$815$	−28.7165	−1.00589
$816$	54.5644	1.91014
$817$	50.8175	1.77788
$818$	−71.1913	−2.48915
$819$	88.3647	3.08771
$820$	92.8571	3.24271
$821$	−6.40603	−0.223572	−0.111786	−	0.993732i	$-0.535657\pi$
$821$	−6.40603	−0.223572	−0.111786	+	0.993732i	$0.535657\pi$
$822$	−21.5787	−0.752644
$823$	2.14576	0.0747964	0.0373982	−	0.999300i	$-0.488093\pi$
$823$	2.14576	0.0747964	0.0373982	+	0.999300i	$0.488093\pi$
$824$	12.2722	0.427523
$825$	−84.8996	−2.95583
$826$	49.6027	1.72590
$827$	−11.8903	−0.413467	−0.206733	−	0.978397i	$-0.566283\pi$
$827$	−11.8903	−0.413467	−0.206733	+	0.978397i	$0.566283\pi$
$828$	−192.339	−6.68423
$829$	15.4295	0.535889	0.267944	−	0.963434i	$-0.413656\pi$
$829$	15.4295	0.535889	0.267944	+	0.963434i	$0.413656\pi$
$830$	−129.787	−4.50499
$831$	−64.0868	−2.22315
$832$	18.6967	0.648193
$833$	6.02258	0.208670
$834$	83.7144	2.89879
$835$	95.6809	3.31117
$836$	80.4918	2.78387
$837$	92.5459	3.19885
$838$	−87.0717	−3.00784
$839$	8.67117	0.299362	0.149681	−	0.988734i	$-0.452175\pi$
$839$	8.67117	0.299362	0.149681	+	0.988734i	$0.452175\pi$
$840$	210.677	7.26906
$841$	18.0403	0.622078
$842$	−54.0393	−1.86232
$843$	21.6684	0.746299
$844$	−64.7905	−2.23018
$845$	31.4129	1.08064
$846$	117.404	4.03642
$847$	14.6023	0.501740
$848$	−58.2315	−1.99968
$849$	47.2268	1.62082
$850$	−86.6041	−2.97050
$851$	10.4440	0.358015
$852$	−155.249	−5.31875
$853$	6.94277	0.237716	0.118858	−	0.992911i	$-0.462077\pi$
$853$	6.94277	0.237716	0.118858	+	0.992911i	$0.462077\pi$
$854$	33.2458	1.13765
$855$	−198.219	−6.77895
$856$	−5.31176	−0.181552
$857$	43.0555	1.47075	0.735375	−	0.677661i	$-0.237006\pi$
$857$	43.0555	1.47075	0.735375	+	0.677661i	$0.237006\pi$
$858$	−87.0069	−2.97037
$859$	0.483348	0.0164916	0.00824581	−	0.999966i	$-0.497375\pi$
$859$	0.483348	0.0164916	0.00824581	+	0.999966i	$0.497375\pi$
$860$	−114.606	−3.90804
$861$	−49.7711	−1.69619
$862$	−82.2037	−2.79987
$863$	52.5598	1.78916	0.894578	−	0.446912i	$-0.147476\pi$
$863$	52.5598	1.78916	0.894578	+	0.446912i	$0.147476\pi$
$864$	−31.3409	−1.06624
$865$	−8.21323	−0.279258
$866$	27.9991	0.951447
$867$	−22.7350	−0.772122
$868$	−110.505	−3.75079
$869$	10.5954	0.359424
$870$	212.530	7.20543
$871$	3.60251	0.122066
$872$	−17.1629	−0.581209
$873$	109.835	3.71734
$874$	−132.549	−4.48354
$875$	−73.9210	−2.49898
$876$	−67.1066	−2.26732
$877$	−6.48557	−0.219002	−0.109501	−	0.993987i	$-0.534925\pi$
$877$	−6.48557	−0.219002	−0.109501	+	0.993987i	$0.534925\pi$
$878$	86.6530	2.92440
$879$	−75.9727	−2.56250
$880$	−56.8170	−1.91530
$881$	−18.5146	−0.623774	−0.311887	−	0.950119i	$-0.600961\pi$
$881$	−18.5146	−0.623774	−0.311887	+	0.950119i	$0.600961\pi$
$882$	−31.5082	−1.06093
$883$	−20.8496	−0.701644	−0.350822	−	0.936442i	$-0.614098\pi$
$883$	−20.8496	−0.701644	−0.350822	+	0.936442i	$0.614098\pi$
$884$	−60.4592	−2.03346
$885$	81.9314	2.75409
$886$	−41.1471	−1.38237
$887$	−5.13909	−0.172554	−0.0862769	−	0.996271i	$-0.527497\pi$
$887$	−5.13909	−0.172554	−0.0862769	+	0.996271i	$0.527497\pi$
$888$	26.3473	0.884157
$889$	−45.9233	−1.54022
$890$	118.798	3.98212
$891$	−33.4311	−1.11998
$892$	−61.8507	−2.07091
$893$	55.1147	1.84434
$894$	115.838	3.87420
$895$	8.31555	0.277958
$896$	−48.2070	−1.61048
$897$	97.6011	3.25881
$898$	−15.8432	−0.528696
$899$	−59.3065	−1.97798
$900$	308.642	10.2881
$901$	−31.5873	−1.05233
$902$	33.4925	1.11518
$903$	61.4286	2.04421
$904$	18.6640	0.620756
$905$	−7.24641	−0.240879
$906$	1.29370	0.0429803
$907$	−54.0907	−1.79605	−0.898026	−	0.439943i	$-0.854999\pi$
$907$	−54.0907	−1.79605	−0.898026	+	0.439943i	$0.854999\pi$
$908$	−91.9545	−3.05162
$909$	−96.5403	−3.20204
$910$	−137.336	−4.55264
$911$	−10.7392	−0.355806	−0.177903	−	0.984048i	$-0.556931\pi$
$911$	−10.7392	−0.355806	−0.177903	+	0.984048i	$0.556931\pi$
$912$	−134.010	−4.43750
$913$	−31.8890	−1.05537
$914$	−66.0601	−2.18508
$915$	54.9138	1.81539
$916$	58.0133	1.91681
$917$	−29.4848	−0.973675
$918$	−83.1219	−2.74343
$919$	−39.3591	−1.29834	−0.649168	−	0.760645i	$-0.724883\pi$
$919$	−39.3591	−1.29834	−0.649168	+	0.760645i	$0.724883\pi$
$920$	159.034	5.24319
$921$	−26.2777	−0.865879
$922$	−30.0236	−0.988776
$923$	53.8410	1.77220
$924$	97.2990	3.20090
$925$	−16.7593	−0.551041
$926$	60.2693	1.98057
$927$	−13.9575	−0.458424
$928$	20.0843	0.659299
$929$	41.2438	1.35317	0.676583	−	0.736366i	$-0.263460\pi$
$929$	41.2438	1.35317	0.676583	+	0.736366i	$0.263460\pi$
$930$	−267.949	−8.78638
$931$	−14.7914	−0.484768
$932$	61.7388	2.02232
$933$	−70.8405	−2.31921
$934$	−49.2382	−1.61112
$935$	−30.8200	−1.00792
$936$	168.275	5.50025
$937$	−41.5599	−1.35770	−0.678851	−	0.734276i	$-0.737522\pi$
$937$	−41.5599	−1.35770	−0.678851	+	0.734276i	$0.737522\pi$
$938$	−5.91403	−0.193100
$939$	23.0352	0.751727
$940$	−124.298	−4.05414
$941$	−33.9989	−1.10833	−0.554166	−	0.832406i	$-0.686963\pi$
$941$	−33.9989	−1.10833	−0.554166	+	0.832406i	$0.686963\pi$
$942$	−172.323	−5.61457
$943$	−37.5707	−1.22347
$944$	37.8562	1.23211
$945$	−128.622	−4.18406
$946$	−41.3372	−1.34399
$947$	−4.35918	−0.141654	−0.0708271	−	0.997489i	$-0.522564\pi$
$947$	−4.35918	−0.141654	−0.0708271	+	0.997489i	$0.522564\pi$
$948$	−56.3600	−1.83049
$949$	23.2728	0.755468
$950$	212.699	6.90086
$951$	−78.6187	−2.54939
$952$	52.8030	1.71136
$953$	−39.3116	−1.27343	−0.636713	−	0.771101i	$-0.719706\pi$
$953$	−39.3116	−1.27343	−0.636713	+	0.771101i	$0.719706\pi$
$954$	165.254	5.35031
$955$	−79.9110	−2.58586
$956$	87.7361	2.83759
$957$	52.2189	1.68800
$958$	−6.96196	−0.224931
$959$	−8.36883	−0.270244
$960$	−50.6979	−1.63627
$961$	43.7712	1.41197
$962$	−17.1752	−0.553751
$963$	6.04120	0.194675
$964$	−91.1526	−2.93583
$965$	20.3976	0.656622
$966$	−160.226	−5.15519
$967$	−55.8132	−1.79483	−0.897416	−	0.441185i	$-0.854558\pi$
$967$	−55.8132	−1.79483	−0.897416	+	0.441185i	$0.854558\pi$
$968$	27.8075	0.893768
$969$	−72.6927	−2.33522
$970$	−170.705	−5.48100
$971$	25.2325	0.809751	0.404876	−	0.914372i	$-0.367315\pi$
$971$	25.2325	0.809751	0.404876	+	0.914372i	$0.367315\pi$
$972$	40.6143	1.30271
$973$	32.4668	1.04084
$974$	29.3880	0.941652
$975$	−156.619	−5.01581
$976$	25.3728	0.812163
$977$	32.4572	1.03840	0.519200	−	0.854653i	$-0.326230\pi$
$977$	32.4572	1.03840	0.519200	+	0.854653i	$0.326230\pi$
$978$	55.0956	1.76176
$979$	29.1889	0.932883
$980$	33.3584	1.06559
$981$	19.5198	0.623220
$982$	−89.3396	−2.85094
$983$	−20.3997	−0.650650	−0.325325	−	0.945602i	$-0.605474\pi$
$983$	−20.3997	−0.650650	−0.325325	+	0.945602i	$0.605474\pi$
$984$	−94.7805	−3.02149
$985$	−50.9116	−1.62218
$986$	53.2673	1.69638
$987$	66.6231	2.12064
$988$	148.487	4.72401
$989$	46.3705	1.47450
$990$	161.240	5.12455
$991$	−28.6270	−0.909368	−0.454684	−	0.890653i	$-0.650248\pi$
$991$	−28.6270	−0.909368	−0.454684	+	0.890653i	$0.650248\pi$
$992$	−25.3215	−0.803957
$993$	−6.53781	−0.207471
$994$	−88.3878	−2.80349
$995$	−86.8165	−2.75227
$996$	169.627	5.37483
$997$	16.0170	0.507264	0.253632	−	0.967301i	$-0.418375\pi$
$997$	16.0170	0.507264	0.253632	+	0.967301i	$0.418375\pi$
$998$	−21.9533	−0.694920
$999$	−16.0854	−0.508919

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.a.1.20	✓	306

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.a.1.20	✓	306	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:	\( N \)	\(=\)	\( 8009 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50471 q^{2} +3.07842 q^{3} +4.27357 q^{4} +4.01883 q^{5} -7.71055 q^{6} -2.99037 q^{7} -5.69464 q^{8} +6.47666 q^{9} +O(q^{10})\)	\(q-2.50471 q^{2} +3.07842 q^{3} +4.27357 q^{4} +4.01883 q^{5} -7.71055 q^{6} -2.99037 q^{7} -5.69464 q^{8} +6.47666 q^{9} -10.0660 q^{10} -2.47324 q^{11} +13.1558 q^{12} -4.56250 q^{13} +7.49000 q^{14} +12.3716 q^{15} +5.71628 q^{16} +3.10076 q^{17} -16.2222 q^{18} -7.61544 q^{19} +17.1748 q^{20} -9.20560 q^{21} +6.19474 q^{22} -6.94903 q^{23} -17.5305 q^{24} +11.1510 q^{25} +11.4277 q^{26} +10.7026 q^{27} -12.7795 q^{28} -6.85859 q^{29} -30.9874 q^{30} +8.64703 q^{31} -2.92834 q^{32} -7.61365 q^{33} -7.76650 q^{34} -12.0178 q^{35} +27.6785 q^{36} -1.50294 q^{37} +19.0745 q^{38} -14.0453 q^{39} -22.8858 q^{40} +5.40661 q^{41} +23.0574 q^{42} -6.67296 q^{43} -10.5696 q^{44} +26.0286 q^{45} +17.4053 q^{46} -7.23723 q^{47} +17.5971 q^{48} +1.94229 q^{49} -27.9300 q^{50} +9.54543 q^{51} -19.4982 q^{52} -10.1870 q^{53} -26.8070 q^{54} -9.93951 q^{55} +17.0291 q^{56} -23.4435 q^{57} +17.1788 q^{58} +6.62252 q^{59} +52.8711 q^{60} +4.43869 q^{61} -21.6583 q^{62} -19.3676 q^{63} -4.09791 q^{64} -18.3359 q^{65} +19.0700 q^{66} -0.789590 q^{67} +13.2513 q^{68} -21.3920 q^{69} +30.1010 q^{70} -11.8008 q^{71} -36.8823 q^{72} -5.10089 q^{73} +3.76443 q^{74} +34.3274 q^{75} -32.5451 q^{76} +7.39588 q^{77} +35.1794 q^{78} -4.28402 q^{79} +22.9727 q^{80} +13.5172 q^{81} -13.5420 q^{82} +12.8936 q^{83} -39.3408 q^{84} +12.4614 q^{85} +16.7138 q^{86} -21.1136 q^{87} +14.0842 q^{88} -11.8019 q^{89} -65.1941 q^{90} +13.6436 q^{91} -29.6972 q^{92} +26.6192 q^{93} +18.1272 q^{94} -30.6051 q^{95} -9.01466 q^{96} +16.9586 q^{97} -4.86488 q^{98} -16.0183 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * q^99

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