LMFDB - Embedded newform 8009.2.a.a.1.9

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			1.9
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)

$f(q)$	$=$	$q-2.63809 q^{2} -1.56073 q^{3} +4.95951 q^{4} +2.27175 q^{5} +4.11735 q^{6} -0.895063 q^{7} -7.80745 q^{8} -0.564115 q^{9} +O(q^{10})$	\(q-2.63809 q^{2} -1.56073 q^{3} +4.95951 q^{4} +2.27175 q^{5} +4.11735 q^{6} -0.895063 q^{7} -7.80745 q^{8} -0.564115 q^{9} -5.99307 q^{10} +2.40193 q^{11} -7.74047 q^{12} +3.50691 q^{13} +2.36126 q^{14} -3.54559 q^{15} +10.6777 q^{16} +3.29820 q^{17} +1.48818 q^{18} +4.26866 q^{19} +11.2668 q^{20} +1.39695 q^{21} -6.33650 q^{22} -0.226836 q^{23} +12.1853 q^{24} +0.160843 q^{25} -9.25153 q^{26} +5.56263 q^{27} -4.43908 q^{28} +4.21438 q^{29} +9.35359 q^{30} -7.66261 q^{31} -12.5539 q^{32} -3.74876 q^{33} -8.70095 q^{34} -2.03336 q^{35} -2.79773 q^{36} +9.29199 q^{37} -11.2611 q^{38} -5.47334 q^{39} -17.7366 q^{40} -10.6191 q^{41} -3.68529 q^{42} -10.3829 q^{43} +11.9124 q^{44} -1.28153 q^{45} +0.598414 q^{46} -4.77734 q^{47} -16.6651 q^{48} -6.19886 q^{49} -0.424318 q^{50} -5.14761 q^{51} +17.3925 q^{52} +1.00730 q^{53} -14.6747 q^{54} +5.45657 q^{55} +6.98816 q^{56} -6.66224 q^{57} -11.1179 q^{58} -2.66744 q^{59} -17.5844 q^{60} +2.53691 q^{61} +20.2146 q^{62} +0.504918 q^{63} +11.7628 q^{64} +7.96681 q^{65} +9.88957 q^{66} -11.2835 q^{67} +16.3575 q^{68} +0.354030 q^{69} +5.36418 q^{70} -10.1832 q^{71} +4.40430 q^{72} +7.99291 q^{73} -24.5131 q^{74} -0.251033 q^{75} +21.1705 q^{76} -2.14988 q^{77} +14.4392 q^{78} +2.03694 q^{79} +24.2571 q^{80} -6.98943 q^{81} +28.0141 q^{82} +4.26918 q^{83} +6.92821 q^{84} +7.49269 q^{85} +27.3911 q^{86} -6.57753 q^{87} -18.7529 q^{88} -2.06999 q^{89} +3.38078 q^{90} -3.13890 q^{91} -1.12500 q^{92} +11.9593 q^{93} +12.6031 q^{94} +9.69733 q^{95} +19.5932 q^{96} -8.20065 q^{97} +16.3531 q^{98} -1.35496 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.63809	−1.86541	−0.932705	−	0.360640i	$-0.882558\pi$
$2$	−2.63809	−1.86541	−0.932705	+	0.360640i	$0.882558\pi$
$3$	−1.56073	−0.901089	−0.450545	−	0.892754i	$-0.648770\pi$
$3$	−1.56073	−0.901089	−0.450545	+	0.892754i	$0.648770\pi$
$4$	4.95951	2.47976
$5$	2.27175	1.01596	0.507979	−	0.861370i	$-0.330393\pi$
$5$	2.27175	1.01596	0.507979	+	0.861370i	$0.330393\pi$
$6$	4.11735	1.68090
$7$	−0.895063	−0.338302	−0.169151	−	0.985590i	$-0.554103\pi$
$7$	−0.895063	−0.338302	−0.169151	+	0.985590i	$0.554103\pi$
$8$	−7.80745	−2.76035
$9$	−0.564115	−0.188038
$10$	−5.99307	−1.89518
$11$	2.40193	0.724208	0.362104	−	0.932138i	$-0.382058\pi$
$11$	2.40193	0.724208	0.362104	+	0.932138i	$0.382058\pi$
$12$	−7.74047	−2.23448
$13$	3.50691	0.972641	0.486320	−	0.873781i	$-0.338339\pi$
$13$	3.50691	0.972641	0.486320	+	0.873781i	$0.338339\pi$
$14$	2.36126	0.631072
$15$	−3.54559	−0.915468
$16$	10.6777	2.66943
$17$	3.29820	0.799932	0.399966	−	0.916530i	$-0.369022\pi$
$17$	3.29820	0.799932	0.399966	+	0.916530i	$0.369022\pi$
$18$	1.48818	0.350768
$19$	4.26866	0.979298	0.489649	−	0.871920i	$-0.337125\pi$
$19$	4.26866	0.979298	0.489649	+	0.871920i	$0.337125\pi$
$20$	11.2668	2.51932
$21$	1.39695	0.304840
$22$	−6.33650	−1.35095
$23$	−0.226836	−0.0472986	−0.0236493	−	0.999720i	$-0.507529\pi$
$23$	−0.226836	−0.0472986	−0.0236493	+	0.999720i	$0.507529\pi$
$24$	12.1853	2.48732
$25$	0.160843	0.0321686
$26$	−9.25153	−1.81437
$27$	5.56263	1.07053
$28$	−4.43908	−0.838906
$29$	4.21438	0.782591	0.391296	−	0.920265i	$-0.372027\pi$
$29$	4.21438	0.782591	0.391296	+	0.920265i	$0.372027\pi$
$30$	9.35359	1.70772
$31$	−7.66261	−1.37624	−0.688122	−	0.725595i	$-0.741565\pi$
$31$	−7.66261	−1.37624	−0.688122	+	0.725595i	$0.741565\pi$
$32$	−12.5539	−2.21923
$33$	−3.74876	−0.652576
$34$	−8.70095	−1.49220
$35$	−2.03336	−0.343700
$36$	−2.79773	−0.466289
$37$	9.29199	1.52759	0.763797	−	0.645457i	$-0.223333\pi$
$37$	9.29199	1.52759	0.763797	+	0.645457i	$0.223333\pi$
$38$	−11.2611	−1.82679
$39$	−5.47334	−0.876436
$40$	−17.7366	−2.80440
$41$	−10.6191	−1.65842	−0.829212	−	0.558935i	$-0.811210\pi$
$41$	−10.6191	−1.65842	−0.829212	+	0.558935i	$0.811210\pi$
$42$	−3.68529	−0.568652
$43$	−10.3829	−1.58338	−0.791692	−	0.610920i	$-0.790800\pi$
$43$	−10.3829	−1.58338	−0.791692	+	0.610920i	$0.790800\pi$
$44$	11.9124	1.79586
$45$	−1.28153	−0.191039
$46$	0.598414	0.0882313
$47$	−4.77734	−0.696847	−0.348424	−	0.937337i	$-0.613283\pi$
$47$	−4.77734	−0.696847	−0.348424	+	0.937337i	$0.613283\pi$
$48$	−16.6651	−2.40540
$49$	−6.19886	−0.885552
$50$	−0.424318	−0.0600077
$51$	−5.14761	−0.720810
$52$	17.3925	2.41191
$53$	1.00730	0.138363	0.0691814	−	0.997604i	$-0.477961\pi$
$53$	1.00730	0.138363	0.0691814	+	0.997604i	$0.477961\pi$
$54$	−14.6747	−1.99697
$55$	5.45657	0.735764
$56$	6.98816	0.933832
$57$	−6.66224	−0.882435
$58$	−11.1179	−1.45985
$59$	−2.66744	−0.347271	−0.173636	−	0.984810i	$-0.555551\pi$
$59$	−2.66744	−0.347271	−0.173636	+	0.984810i	$0.555551\pi$
$60$	−17.5844	−2.27014
$61$	2.53691	0.324818	0.162409	−	0.986724i	$-0.448074\pi$
$61$	2.53691	0.324818	0.162409	+	0.986724i	$0.448074\pi$
$62$	20.2146	2.56726
$63$	0.504918	0.0636137
$64$	11.7628	1.47035
$65$	7.96681	0.988161
$66$	9.88957	1.21732
$67$	−11.2835	−1.37850	−0.689252	−	0.724522i	$-0.742061\pi$
$67$	−11.2835	−1.37850	−0.689252	+	0.724522i	$0.742061\pi$
$68$	16.3575	1.98363
$69$	0.354030	0.0426203
$70$	5.36418	0.641142
$71$	−10.1832	−1.20852	−0.604260	−	0.796787i	$-0.706531\pi$
$71$	−10.1832	−1.20852	−0.604260	+	0.796787i	$0.706531\pi$
$72$	4.40430	0.519051
$73$	7.99291	0.935499	0.467750	−	0.883861i	$-0.345065\pi$
$73$	7.99291	0.935499	0.467750	+	0.883861i	$0.345065\pi$
$74$	−24.5131	−2.84959
$75$	−0.251033	−0.0289868
$76$	21.1705	2.42842
$77$	−2.14988	−0.245001
$78$	14.4392	1.63491
$79$	2.03694	0.229174	0.114587	−	0.993413i	$-0.463446\pi$
$79$	2.03694	0.229174	0.114587	+	0.993413i	$0.463446\pi$
$80$	24.2571	2.71203
$81$	−6.98943	−0.776603
$82$	28.0141	3.09364
$83$	4.26918	0.468603	0.234302	−	0.972164i	$-0.424720\pi$
$83$	4.26918	0.468603	0.234302	+	0.972164i	$0.424720\pi$
$84$	6.92821	0.755929
$85$	7.49269	0.812696
$86$	27.3911	2.95366
$87$	−6.57753	−0.705185
$88$	−18.7529	−1.99907
$89$	−2.06999	−0.219418	−0.109709	−	0.993964i	$-0.534992\pi$
$89$	−2.06999	−0.219418	−0.109709	+	0.993964i	$0.534992\pi$
$90$	3.38078	0.356366
$91$	−3.13890	−0.329046
$92$	−1.12500	−0.117289
$93$	11.9593	1.24012
$94$	12.6031	1.29991
$95$	9.69733	0.994925
$96$	19.5932	1.99973
$97$	−8.20065	−0.832650	−0.416325	−	0.909216i	$-0.636682\pi$
$97$	−8.20065	−0.832650	−0.416325	+	0.909216i	$0.636682\pi$
$98$	16.3531	1.65192
$99$	−1.35496	−0.136179
$100$	0.797703	0.0797703
$101$	−9.72761	−0.967934	−0.483967	−	0.875086i	$-0.660805\pi$
$101$	−9.72761	−0.967934	−0.483967	+	0.875086i	$0.660805\pi$
$102$	13.5799	1.34461
$103$	1.06317	0.104757	0.0523784	−	0.998627i	$-0.483320\pi$
$103$	1.06317	0.104757	0.0523784	+	0.998627i	$0.483320\pi$
$104$	−27.3800	−2.68483
$105$	3.17353	0.309705
$106$	−2.65734	−0.258103
$107$	7.24129	0.700042	0.350021	−	0.936742i	$-0.386174\pi$
$107$	7.24129	0.700042	0.350021	+	0.936742i	$0.386174\pi$
$108$	27.5879	2.65465
$109$	−1.67129	−0.160081	−0.0800405	−	0.996792i	$-0.525505\pi$
$109$	−1.67129	−0.160081	−0.0800405	+	0.996792i	$0.525505\pi$
$110$	−14.3949	−1.37250
$111$	−14.5023	−1.37650
$112$	−9.55724	−0.903074
$113$	−11.1676	−1.05056	−0.525281	−	0.850929i	$-0.676040\pi$
$113$	−11.1676	−1.05056	−0.525281	+	0.850929i	$0.676040\pi$
$114$	17.5756	1.64610
$115$	−0.515315	−0.0480533
$116$	20.9013	1.94064
$117$	−1.97830	−0.182894
$118$	7.03694	0.647803
$119$	−2.95210	−0.270619
$120$	27.6820	2.52701
$121$	−5.23075	−0.475523
$122$	−6.69260	−0.605919
$123$	16.5736	1.49439
$124$	−38.0028	−3.41275
$125$	−10.9933	−0.983275
$126$	−1.33202	−0.118666
$127$	−9.97490	−0.885130	−0.442565	−	0.896737i	$-0.645931\pi$
$127$	−9.97490	−0.885130	−0.442565	+	0.896737i	$0.645931\pi$
$128$	−5.92353	−0.523571
$129$	16.2050	1.42677
$130$	−21.0172	−1.84333
$131$	−8.64818	−0.755595	−0.377798	−	0.925888i	$-0.623319\pi$
$131$	−8.64818	−0.755595	−0.377798	+	0.925888i	$0.623319\pi$
$132$	−18.5920	−1.61823
$133$	−3.82072	−0.331299
$134$	29.7670	2.57148
$135$	12.6369	1.08761
$136$	−25.7506	−2.20809
$137$	4.65302	0.397534	0.198767	−	0.980047i	$-0.436306\pi$
$137$	4.65302	0.397534	0.198767	+	0.980047i	$0.436306\pi$
$138$	−0.933964	−0.0795043
$139$	−14.4068	−1.22196	−0.610982	−	0.791644i	$-0.709225\pi$
$139$	−14.4068	−1.22196	−0.610982	+	0.791644i	$0.709225\pi$
$140$	−10.0845	−0.852293
$141$	7.45615	0.627922
$142$	26.8641	2.25438
$143$	8.42333	0.704395
$144$	−6.02346	−0.501955
$145$	9.57402	0.795079
$146$	−21.0860	−1.74509
$147$	9.67476	0.797961
$148$	46.0837	3.78806
$149$	−1.39976	−0.114673	−0.0573363	−	0.998355i	$-0.518261\pi$
$149$	−1.39976	−0.114673	−0.0573363	+	0.998355i	$0.518261\pi$
$150$	0.662248	0.0540723
$151$	−4.27782	−0.348124	−0.174062	−	0.984735i	$-0.555689\pi$
$151$	−4.27782	−0.348124	−0.174062	+	0.984735i	$0.555689\pi$
$152$	−33.3274	−2.70321
$153$	−1.86056	−0.150418
$154$	5.67156	0.457028
$155$	−17.4075	−1.39821
$156$	−27.1451	−2.17335
$157$	−0.961986	−0.0767749	−0.0383874	−	0.999263i	$-0.512222\pi$
$157$	−0.961986	−0.0767749	−0.0383874	+	0.999263i	$0.512222\pi$
$158$	−5.37363	−0.427503
$159$	−1.57212	−0.124677
$160$	−28.5193	−2.25465
$161$	0.203033	0.0160012
$162$	18.4387	1.44868
$163$	−18.3875	−1.44022	−0.720111	−	0.693859i	$-0.755909\pi$
$163$	−18.3875	−1.44022	−0.720111	+	0.693859i	$0.755909\pi$
$164$	−52.6655	−4.11248
$165$	−8.51625	−0.662989
$166$	−11.2625	−0.874137
$167$	−1.59107	−0.123121	−0.0615605	−	0.998103i	$-0.519608\pi$
$167$	−1.59107	−0.123121	−0.0615605	+	0.998103i	$0.519608\pi$
$168$	−10.9066	−0.841466
$169$	−0.701605	−0.0539696
$170$	−19.7664	−1.51601
$171$	−2.40801	−0.184145
$172$	−51.4943	−3.92641
$173$	7.23883	0.550358	0.275179	−	0.961393i	$-0.411263\pi$
$173$	7.23883	0.550358	0.275179	+	0.961393i	$0.411263\pi$
$174$	17.3521	1.31546
$175$	−0.143965	−0.0108827
$176$	25.6471	1.93322
$177$	4.16316	0.312922
$178$	5.46081	0.409305
$179$	3.65352	0.273077	0.136538	−	0.990635i	$-0.456402\pi$
$179$	3.65352	0.273077	0.136538	+	0.990635i	$0.456402\pi$
$180$	−6.35574	−0.473729
$181$	−6.13247	−0.455823	−0.227911	−	0.973682i	$-0.573190\pi$
$181$	−6.13247	−0.455823	−0.227911	+	0.973682i	$0.573190\pi$
$182$	8.28070	0.613807
$183$	−3.95944	−0.292690
$184$	1.77101	0.130561
$185$	21.1091	1.55197
$186$	−31.5496	−2.31333
$187$	7.92204	0.579317
$188$	−23.6933	−1.72801
$189$	−4.97890	−0.362162
$190$	−25.5824	−1.85594
$191$	−0.155763	−0.0112706	−0.00563532	−	0.999984i	$-0.501794\pi$
$191$	−0.155763	−0.0112706	−0.00563532	+	0.999984i	$0.501794\pi$
$192$	−18.3586	−1.32492
$193$	−11.2340	−0.808639	−0.404319	−	0.914618i	$-0.632492\pi$
$193$	−11.2340	−0.808639	−0.404319	+	0.914618i	$0.632492\pi$
$194$	21.6341	1.55323
$195$	−12.4341	−0.890422
$196$	−30.7433	−2.19595
$197$	−1.34988	−0.0961750	−0.0480875	−	0.998843i	$-0.515313\pi$
$197$	−1.34988	−0.0961750	−0.0480875	+	0.998843i	$0.515313\pi$
$198$	3.57451	0.254029
$199$	18.1267	1.28497	0.642483	−	0.766300i	$-0.277904\pi$
$199$	18.1267	1.28497	0.642483	+	0.766300i	$0.277904\pi$
$200$	−1.25577	−0.0887967
$201$	17.6106	1.24216
$202$	25.6623	1.80559
$203$	−3.77214	−0.264752
$204$	−25.5296	−1.78743
$205$	−24.1239	−1.68489
$206$	−2.80473	−0.195415
$207$	0.127962	0.00889394
$208$	37.4458	2.59640
$209$	10.2530	0.709216
$210$	−8.37205	−0.577726
$211$	−16.4493	−1.13242	−0.566209	−	0.824262i	$-0.691590\pi$
$211$	−16.4493	−1.13242	−0.566209	+	0.824262i	$0.691590\pi$
$212$	4.99570	0.343106
$213$	15.8932	1.08898
$214$	−19.1032	−1.30587
$215$	−23.5874	−1.60865
$216$	−43.4299	−2.95503
$217$	6.85852	0.465587
$218$	4.40902	0.298617
$219$	−12.4748	−0.842968
$220$	27.0619	1.82452
$221$	11.5665	0.778046
$222$	38.2584	2.56773
$223$	12.8404	0.859856	0.429928	−	0.902863i	$-0.358539\pi$
$223$	12.8404	0.859856	0.429928	+	0.902863i	$0.358539\pi$
$224$	11.2365	0.750771
$225$	−0.0907340	−0.00604893
$226$	29.4612	1.95973
$227$	13.0782	0.868032	0.434016	−	0.900905i	$-0.357096\pi$
$227$	13.0782	0.868032	0.434016	+	0.900905i	$0.357096\pi$
$228$	−33.0414	−2.18822
$229$	10.6343	0.702731	0.351366	−	0.936238i	$-0.385717\pi$
$229$	10.6343	0.702731	0.351366	+	0.936238i	$0.385717\pi$
$230$	1.35945	0.0896392
$231$	3.35538	0.220768
$232$	−32.9036	−2.16023
$233$	0.256056	0.0167748	0.00838739	−	0.999965i	$-0.497330\pi$
$233$	0.256056	0.0167748	0.00838739	+	0.999965i	$0.497330\pi$
$234$	5.21892	0.341172
$235$	−10.8529	−0.707967
$236$	−13.2292	−0.861147
$237$	−3.17912	−0.206506
$238$	7.78790	0.504815
$239$	−5.47095	−0.353886	−0.176943	−	0.984221i	$-0.556621\pi$
$239$	−5.47095	−0.353886	−0.176943	+	0.984221i	$0.556621\pi$
$240$	−37.8588	−2.44378
$241$	−12.0164	−0.774046	−0.387023	−	0.922070i	$-0.626497\pi$
$241$	−12.0164	−0.774046	−0.387023	+	0.922070i	$0.626497\pi$
$242$	13.7992	0.887045
$243$	−5.77926	−0.370739
$244$	12.5818	0.805470
$245$	−14.0823	−0.899682
$246$	−43.7225	−2.78765
$247$	14.9698	0.952506
$248$	59.8254	3.79892
$249$	−6.66304	−0.422253
$250$	29.0014	1.83421
$251$	−20.7328	−1.30864	−0.654321	−	0.756217i	$-0.727045\pi$
$251$	−20.7328	−1.30864	−0.654321	+	0.756217i	$0.727045\pi$
$252$	2.50415	0.157746
$253$	−0.544844	−0.0342540
$254$	26.3147	1.65113
$255$	−11.6941	−0.732312
$256$	−7.89879	−0.493674
$257$	2.57561	0.160662	0.0803310	−	0.996768i	$-0.474402\pi$
$257$	2.57561	0.160662	0.0803310	+	0.996768i	$0.474402\pi$
$258$	−42.7502	−2.66151
$259$	−8.31692	−0.516788
$260$	39.5115	2.45040
$261$	−2.37740	−0.147157
$262$	22.8147	1.40950
$263$	17.0052	1.04859	0.524293	−	0.851538i	$-0.324330\pi$
$263$	17.0052	1.04859	0.524293	+	0.851538i	$0.324330\pi$
$264$	29.2683	1.80134
$265$	2.28832	0.140571
$266$	10.0794	0.618008
$267$	3.23069	0.197715
$268$	−55.9608	−3.41835
$269$	26.0223	1.58661	0.793304	−	0.608826i	$-0.208359\pi$
$269$	26.0223	1.58661	0.793304	+	0.608826i	$0.208359\pi$
$270$	−33.3372	−2.02884
$271$	−7.08658	−0.430479	−0.215240	−	0.976561i	$-0.569053\pi$
$271$	−7.08658	−0.430479	−0.215240	+	0.976561i	$0.569053\pi$
$272$	35.2173	2.13536
$273$	4.89899	0.296500
$274$	−12.2751	−0.741564
$275$	0.386333	0.0232968
$276$	1.75582	0.105688
$277$	8.32710	0.500327	0.250164	−	0.968204i	$-0.419516\pi$
$277$	8.32710	0.500327	0.250164	+	0.968204i	$0.419516\pi$
$278$	38.0063	2.27947
$279$	4.32259	0.258787
$280$	15.8753	0.948733
$281$	24.4647	1.45944	0.729720	−	0.683746i	$-0.239650\pi$
$281$	24.4647	1.45944	0.729720	+	0.683746i	$0.239650\pi$
$282$	−19.6700	−1.17133
$283$	−1.88512	−0.112059	−0.0560293	−	0.998429i	$-0.517844\pi$
$283$	−1.88512	−0.112059	−0.0560293	+	0.998429i	$0.517844\pi$
$284$	−50.5035	−2.99683
$285$	−15.1349	−0.896516
$286$	−22.2215	−1.31398
$287$	9.50476	0.561048
$288$	7.08182	0.417301
$289$	−6.12186	−0.360109
$290$	−25.2571	−1.48315
$291$	12.7990	0.750292
$292$	39.6409	2.31981
$293$	6.59667	0.385382	0.192691	−	0.981260i	$-0.438279\pi$
$293$	6.59667	0.385382	0.192691	+	0.981260i	$0.438279\pi$
$294$	−25.5229	−1.48852
$295$	−6.05975	−0.352813
$296$	−72.5467	−4.21669
$297$	13.3610	0.775285
$298$	3.69269	0.213912
$299$	−0.795493	−0.0460046
$300$	−1.24500	−0.0718802
$301$	9.29339	0.535662
$302$	11.2853	0.649394
$303$	15.1822	0.872195
$304$	45.5796	2.61417
$305$	5.76322	0.330001
$306$	4.90833	0.280591
$307$	−27.5239	−1.57087	−0.785437	−	0.618941i	$-0.787562\pi$
$307$	−27.5239	−1.57087	−0.785437	+	0.618941i	$0.787562\pi$
$308$	−10.6623	−0.607543
$309$	−1.65932	−0.0943953
$310$	45.9226	2.60823
$311$	−12.5158	−0.709706	−0.354853	−	0.934922i	$-0.615469\pi$
$311$	−12.5158	−0.709706	−0.354853	+	0.934922i	$0.615469\pi$
$312$	42.7329	2.41927
$313$	−16.2060	−0.916015	−0.458007	−	0.888948i	$-0.651437\pi$
$313$	−16.2060	−0.916015	−0.458007	+	0.888948i	$0.651437\pi$
$314$	2.53780	0.143217
$315$	1.14705	0.0646288
$316$	10.1022	0.568295
$317$	−1.79796	−0.100984	−0.0504918	−	0.998724i	$-0.516079\pi$
$317$	−1.79796	−0.100984	−0.0504918	+	0.998724i	$0.516079\pi$
$318$	4.14739	0.232574
$319$	10.1226	0.566759
$320$	26.7221	1.49381
$321$	−11.3017	−0.630800
$322$	−0.535618	−0.0298488
$323$	14.0789	0.783372
$324$	−34.6642	−1.92579
$325$	0.564062	0.0312885
$326$	48.5079	2.68660
$327$	2.60844	0.144247
$328$	82.9080	4.57783
$329$	4.27602	0.235745
$330$	22.4666	1.23675
$331$	−16.4630	−0.904890	−0.452445	−	0.891792i	$-0.649448\pi$
$331$	−16.4630	−0.904890	−0.452445	+	0.891792i	$0.649448\pi$
$332$	21.1730	1.16202
$333$	−5.24175	−0.287246
$334$	4.19739	0.229671
$335$	−25.6334	−1.40050
$336$	14.9163	0.813750
$337$	5.00082	0.272412	0.136206	−	0.990681i	$-0.456509\pi$
$337$	5.00082	0.272412	0.136206	+	0.990681i	$0.456509\pi$
$338$	1.85090	0.100675
$339$	17.4297	0.946650
$340$	37.1601	2.01529
$341$	−18.4050	−0.996688
$342$	6.35255	0.343507
$343$	11.8138	0.637886
$344$	81.0643	4.37070
$345$	0.804268	0.0433004
$346$	−19.0967	−1.02664
$347$	6.96585	0.373947	0.186973	−	0.982365i	$-0.440132\pi$
$347$	6.96585	0.373947	0.186973	+	0.982365i	$0.440132\pi$
$348$	−32.6213	−1.74869
$349$	−16.9351	−0.906514	−0.453257	−	0.891380i	$-0.649738\pi$
$349$	−16.9351	−0.906514	−0.453257	+	0.891380i	$0.649738\pi$
$350$	0.379792	0.0203007
$351$	19.5076	1.04124
$352$	−30.1535	−1.60719
$353$	2.64189	0.140614	0.0703068	−	0.997525i	$-0.477602\pi$
$353$	2.64189	0.140614	0.0703068	+	0.997525i	$0.477602\pi$
$354$	−10.9828	−0.583728
$355$	−23.1336	−1.22780
$356$	−10.2661	−0.544103
$357$	4.60744	0.243851
$358$	−9.63830	−0.509400
$359$	−1.61975	−0.0854874	−0.0427437	−	0.999086i	$-0.513610\pi$
$359$	−1.61975	−0.0854874	−0.0427437	+	0.999086i	$0.513610\pi$
$360$	10.0055	0.527334
$361$	−0.778525	−0.0409750
$362$	16.1780	0.850297
$363$	8.16380	0.428488
$364$	−15.5674	−0.815955
$365$	18.1579	0.950427
$366$	10.4453	0.545987
$367$	13.2057	0.689331	0.344666	−	0.938726i	$-0.387992\pi$
$367$	13.2057	0.689331	0.344666	+	0.938726i	$0.387992\pi$
$368$	−2.42209	−0.126260
$369$	5.99038	0.311847
$370$	−55.6876	−2.89506
$371$	−0.901594	−0.0468084
$372$	59.3122	3.07519
$373$	18.6773	0.967073	0.483536	−	0.875324i	$-0.339352\pi$
$373$	18.6773	0.967073	0.483536	+	0.875324i	$0.339352\pi$
$374$	−20.8990	−1.08066
$375$	17.1577	0.886019
$376$	37.2989	1.92354
$377$	14.7795	0.761181
$378$	13.1348	0.675581
$379$	−7.15343	−0.367447	−0.183724	−	0.982978i	$-0.558815\pi$
$379$	−7.15343	−0.367447	−0.183724	+	0.982978i	$0.558815\pi$
$380$	48.0940	2.46717
$381$	15.5682	0.797581
$382$	0.410917	0.0210244
$383$	22.3617	1.14263	0.571314	−	0.820732i	$-0.306434\pi$
$383$	22.3617	1.14263	0.571314	+	0.820732i	$0.306434\pi$
$384$	9.24504	0.471784
$385$	−4.88398	−0.248911
$386$	29.6362	1.50844
$387$	5.85717	0.297737
$388$	−40.6712	−2.06477
$389$	−11.5179	−0.583983	−0.291991	−	0.956421i	$-0.594318\pi$
$389$	−11.5179	−0.583983	−0.291991	+	0.956421i	$0.594318\pi$
$390$	32.8022	1.66100
$391$	−0.748152	−0.0378357
$392$	48.3973	2.44443
$393$	13.4975	0.680859
$394$	3.56111	0.179406
$395$	4.62742	0.232831
$396$	−6.71995	−0.337690
$397$	−3.90136	−0.195804	−0.0979019	−	0.995196i	$-0.531213\pi$
$397$	−3.90136	−0.195804	−0.0979019	+	0.995196i	$0.531213\pi$
$398$	−47.8198	−2.39699
$399$	5.96312	0.298530
$400$	1.71744	0.0858719
$401$	14.3777	0.717989	0.358994	−	0.933340i	$-0.383120\pi$
$401$	14.3777	0.717989	0.358994	+	0.933340i	$0.383120\pi$
$402$	−46.4583	−2.31713
$403$	−26.8720	−1.33859
$404$	−48.2442	−2.40024
$405$	−15.8782	−0.788996
$406$	9.95124	0.493872
$407$	22.3187	1.10630
$408$	40.1897	1.98969
$409$	−1.65654	−0.0819106	−0.0409553	−	0.999161i	$-0.513040\pi$
$409$	−1.65654	−0.0819106	−0.0409553	+	0.999161i	$0.513040\pi$
$410$	63.6410	3.14301
$411$	−7.26211	−0.358214
$412$	5.27278	0.259771
$413$	2.38753	0.117483
$414$	−0.337574	−0.0165909
$415$	9.69850	0.476081
$416$	−44.0253	−2.15852
$417$	22.4851	1.10110
$418$	−27.0484	−1.32298
$419$	6.63313	0.324049	0.162025	−	0.986787i	$-0.448198\pi$
$419$	6.63313	0.324049	0.162025	+	0.986787i	$0.448198\pi$
$420$	15.7391	0.767992
$421$	38.0158	1.85277	0.926387	−	0.376572i	$-0.122897\pi$
$421$	38.0158	1.85277	0.926387	+	0.376572i	$0.122897\pi$
$422$	43.3948	2.11242
$423$	2.69497	0.131034
$424$	−7.86441	−0.381930
$425$	0.530493	0.0257327
$426$	−41.9277	−2.03140
$427$	−2.27070	−0.109887
$428$	35.9133	1.73593
$429$	−13.1466	−0.634722
$430$	62.2258	3.00079
$431$	−20.1174	−0.969023	−0.484512	−	0.874785i	$-0.661003\pi$
$431$	−20.1174	−0.969023	−0.484512	+	0.874785i	$0.661003\pi$
$432$	59.3962	2.85770
$433$	3.44025	0.165328	0.0826639	−	0.996577i	$-0.473657\pi$
$433$	3.44025	0.165328	0.0826639	+	0.996577i	$0.473657\pi$
$434$	−18.0934	−0.868510
$435$	−14.9425	−0.716437
$436$	−8.28880	−0.396961
$437$	−0.968287	−0.0463194
$438$	32.9096	1.57248
$439$	−28.1149	−1.34185	−0.670925	−	0.741525i	$-0.734103\pi$
$439$	−28.1149	−1.34185	−0.670925	+	0.741525i	$0.734103\pi$
$440$	−42.6019	−2.03097
$441$	3.49687	0.166518
$442$	−30.5134	−1.45138
$443$	−41.2048	−1.95770	−0.978849	−	0.204582i	$-0.934416\pi$
$443$	−41.2048	−1.95770	−0.978849	+	0.204582i	$0.934416\pi$
$444$	−71.9243	−3.41338
$445$	−4.70249	−0.222919
$446$	−33.8741	−1.60398
$447$	2.18465	0.103330
$448$	−10.5284	−0.497422
$449$	30.9698	1.46156	0.730778	−	0.682615i	$-0.239157\pi$
$449$	30.9698	1.46156	0.730778	+	0.682615i	$0.239157\pi$
$450$	0.239364	0.0112837
$451$	−25.5063	−1.20104
$452$	−55.3859	−2.60514
$453$	6.67653	0.313691
$454$	−34.5015	−1.61924
$455$	−7.13080	−0.334297
$456$	52.0151	2.43583
$457$	20.4028	0.954405	0.477202	−	0.878793i	$-0.341651\pi$
$457$	20.4028	0.954405	0.477202	+	0.878793i	$0.341651\pi$
$458$	−28.0541	−1.31088
$459$	18.3467	0.856350
$460$	−2.55571	−0.119161
$461$	−23.3626	−1.08811	−0.544054	−	0.839051i	$-0.683111\pi$
$461$	−23.3626	−1.08811	−0.544054	+	0.839051i	$0.683111\pi$
$462$	−8.85179	−0.411823
$463$	13.4369	0.624463	0.312232	−	0.950006i	$-0.398923\pi$
$463$	13.4369	0.624463	0.312232	+	0.950006i	$0.398923\pi$
$464$	45.0000	2.08907
$465$	27.1685	1.25991
$466$	−0.675498	−0.0312919
$467$	−2.58547	−0.119641	−0.0598206	−	0.998209i	$-0.519053\pi$
$467$	−2.58547	−0.119641	−0.0598206	+	0.998209i	$0.519053\pi$
$468$	−9.81139	−0.453531
$469$	10.0995	0.466351
$470$	28.6310	1.32065
$471$	1.50140	0.0691810
$472$	20.8259	0.958590
$473$	−24.9391	−1.14670
$474$	8.38679	0.385218
$475$	0.686585	0.0315027
$476$	−14.6410	−0.671068
$477$	−0.568230	−0.0260175
$478$	14.4329	0.660143
$479$	33.3158	1.52224	0.761120	−	0.648612i	$-0.224650\pi$
$479$	33.3158	1.52224	0.761120	+	0.648612i	$0.224650\pi$
$480$	44.5109	2.03164
$481$	32.5861	1.48580
$482$	31.7004	1.44391
$483$	−0.316880	−0.0144185
$484$	−25.9419	−1.17918
$485$	−18.6298	−0.845937
$486$	15.2462	0.691581
$487$	8.33397	0.377648	0.188824	−	0.982011i	$-0.439532\pi$
$487$	8.33397	0.377648	0.188824	+	0.982011i	$0.439532\pi$
$488$	−19.8068	−0.896612
$489$	28.6980	1.29777
$490$	37.1502	1.67828
$491$	35.9921	1.62430	0.812150	−	0.583449i	$-0.198297\pi$
$491$	35.9921	1.62430	0.812150	+	0.583449i	$0.198297\pi$
$492$	82.1967	3.70572
$493$	13.8999	0.626020
$494$	−39.4917	−1.77681
$495$	−3.07813	−0.138352
$496$	−81.8192	−3.67379
$497$	9.11458	0.408845
$498$	17.5777	0.787675
$499$	−7.79564	−0.348981	−0.174490	−	0.984659i	$-0.555828\pi$
$499$	−7.79564	−0.348981	−0.174490	+	0.984659i	$0.555828\pi$
$500$	−54.5216	−2.43828
$501$	2.48324	0.110943
$502$	54.6949	2.44115
$503$	22.4978	1.00313	0.501564	−	0.865121i	$-0.332758\pi$
$503$	22.4978	1.00313	0.501564	+	0.865121i	$0.332758\pi$
$504$	−3.94212	−0.175596
$505$	−22.0987	−0.983379
$506$	1.43735	0.0638978
$507$	1.09502	0.0486315
$508$	−49.4706	−2.19490
$509$	−4.18513	−0.185503	−0.0927513	−	0.995689i	$-0.529566\pi$
$509$	−4.18513	−0.185503	−0.0927513	+	0.995689i	$0.529566\pi$
$510$	30.8500	1.36606
$511$	−7.15416	−0.316481
$512$	32.6848	1.44448
$513$	23.7450	1.04837
$514$	−6.79468	−0.299700
$515$	2.41525	0.106428
$516$	80.3689	3.53804
$517$	−11.4748	−0.504662
$518$	21.9408	0.964022
$519$	−11.2979	−0.495922
$520$	−62.2005	−2.72767
$521$	−20.6221	−0.903470	−0.451735	−	0.892152i	$-0.649195\pi$
$521$	−20.6221	−0.903470	−0.451735	+	0.892152i	$0.649195\pi$
$522$	6.27178	0.274508
$523$	11.3596	0.496721	0.248361	−	0.968668i	$-0.420108\pi$
$523$	11.3596	0.496721	0.248361	+	0.968668i	$0.420108\pi$
$524$	−42.8908	−1.87369
$525$	0.224690	0.00980630
$526$	−44.8612	−1.95604
$527$	−25.2728	−1.10090
$528$	−40.0283	−1.74201
$529$	−22.9485	−0.997763
$530$	−6.03680	−0.262222
$531$	1.50474	0.0653002
$532$	−18.9489	−0.821540
$533$	−37.2402	−1.61305
$534$	−8.52286	−0.368820
$535$	16.4504	0.711213
$536$	88.0957	3.80515
$537$	−5.70216	−0.246066
$538$	−68.6491	−2.95967
$539$	−14.8892	−0.641324
$540$	62.6728	2.69701
$541$	31.5518	1.35652	0.678259	−	0.734823i	$-0.262735\pi$
$541$	31.5518	1.35652	0.678259	+	0.734823i	$0.262735\pi$
$542$	18.6950	0.803020
$543$	9.57114	0.410737
$544$	−41.4052	−1.77523
$545$	−3.79676	−0.162635
$546$	−12.9240	−0.553095
$547$	−12.7608	−0.545610	−0.272805	−	0.962069i	$-0.587951\pi$
$547$	−12.7608	−0.545610	−0.272805	+	0.962069i	$0.587951\pi$
$548$	23.0767	0.985787
$549$	−1.43111	−0.0610782
$550$	−1.01918	−0.0434581
$551$	17.9898	0.766390
$552$	−2.76408	−0.117647
$553$	−1.82319	−0.0775299
$554$	−21.9676	−0.933315
$555$	−32.9456	−1.39846
$556$	−71.4504	−3.03017
$557$	−19.0787	−0.808392	−0.404196	−	0.914672i	$-0.632449\pi$
$557$	−19.0787	−0.808392	−0.404196	+	0.914672i	$0.632449\pi$
$558$	−11.4034	−0.482743
$559$	−36.4120	−1.54006
$560$	−21.7116	−0.917484
$561$	−12.3642	−0.522016
$562$	−64.5400	−2.72245
$563$	46.2041	1.94727	0.973635	−	0.228110i	$-0.0732545\pi$
$563$	46.2041	1.94727	0.973635	+	0.228110i	$0.0732545\pi$
$564$	36.9789	1.55709
$565$	−25.3700	−1.06733
$566$	4.97310	0.209035
$567$	6.25598	0.262727
$568$	79.5046	3.33594
$569$	−13.2978	−0.557472	−0.278736	−	0.960368i	$-0.589915\pi$
$569$	−13.2978	−0.557472	−0.278736	+	0.960368i	$0.589915\pi$
$570$	39.9273	1.67237
$571$	−24.2173	−1.01346	−0.506732	−	0.862104i	$-0.669147\pi$
$571$	−24.2173	−1.01346	−0.506732	+	0.862104i	$0.669147\pi$
$572$	41.7756	1.74673
$573$	0.243105	0.0101559
$574$	−25.0744	−1.04658
$575$	−0.0364850	−0.00152153
$576$	−6.63556	−0.276482
$577$	−10.6045	−0.441469	−0.220735	−	0.975334i	$-0.570845\pi$
$577$	−10.6045	−0.441469	−0.220735	+	0.975334i	$0.570845\pi$
$578$	16.1500	0.671752
$579$	17.5332	0.728656
$580$	47.4825	1.97160
$581$	−3.82118	−0.158529
$582$	−33.7650	−1.39960
$583$	2.41945	0.100203
$584$	−62.4042	−2.58231
$585$	−4.49419	−0.185812
$586$	−17.4026	−0.718895
$587$	4.54819	0.187724	0.0938618	−	0.995585i	$-0.470079\pi$
$587$	4.54819	0.187724	0.0938618	+	0.995585i	$0.470079\pi$
$588$	47.9821	1.97875
$589$	−32.7091	−1.34775
$590$	15.9862	0.658140
$591$	2.10680	0.0866623
$592$	99.2173	4.07780
$593$	−11.8846	−0.488040	−0.244020	−	0.969770i	$-0.578466\pi$
$593$	−11.8846	−0.488040	−0.244020	+	0.969770i	$0.578466\pi$
$594$	−35.2476	−1.44623
$595$	−6.70643	−0.274937
$596$	−6.94212	−0.284360
$597$	−28.2909	−1.15787
$598$	2.09858	0.0858174
$599$	1.04134	0.0425478	0.0212739	−	0.999774i	$-0.493228\pi$
$599$	1.04134	0.0425478	0.0212739	+	0.999774i	$0.493228\pi$
$600$	1.95993	0.0800137
$601$	−2.17051	−0.0885369	−0.0442685	−	0.999020i	$-0.514096\pi$
$601$	−2.17051	−0.0885369	−0.0442685	+	0.999020i	$0.514096\pi$
$602$	−24.5168	−0.999230
$603$	6.36521	0.259211
$604$	−21.2159	−0.863262
$605$	−11.8829	−0.483110
$606$	−40.0520	−1.62700
$607$	−19.2485	−0.781272	−0.390636	−	0.920545i	$-0.627745\pi$
$607$	−19.2485	−0.781272	−0.390636	+	0.920545i	$0.627745\pi$
$608$	−53.5883	−2.17329
$609$	5.88730	0.238565
$610$	−15.2039	−0.615588
$611$	−16.7537	−0.677782
$612$	−9.22749	−0.372999
$613$	−0.272738	−0.0110158	−0.00550789	−	0.999985i	$-0.501753\pi$
$613$	−0.272738	−0.0110158	−0.00550789	+	0.999985i	$0.501753\pi$
$614$	72.6106	2.93033
$615$	37.6510	1.51823
$616$	16.7851	0.676289
$617$	17.2491	0.694421	0.347210	−	0.937787i	$-0.387129\pi$
$617$	17.2491	0.694421	0.347210	+	0.937787i	$0.387129\pi$
$618$	4.37743	0.176086
$619$	18.8017	0.755706	0.377853	−	0.925866i	$-0.376662\pi$
$619$	18.8017	0.755706	0.377853	+	0.925866i	$0.376662\pi$
$620$	−86.3328	−3.46721
$621$	−1.26181	−0.0506345
$622$	33.0178	1.32389
$623$	1.85277	0.0742296
$624$	−58.4428	−2.33959
$625$	−25.7783	−1.03113
$626$	42.7527	1.70874
$627$	−16.0022	−0.639067
$628$	−4.77098	−0.190383
$629$	30.6469	1.22197
$630$	−3.02601	−0.120559
$631$	4.16674	0.165875	0.0829377	−	0.996555i	$-0.473570\pi$
$631$	4.16674	0.165875	0.0829377	+	0.996555i	$0.473570\pi$
$632$	−15.9033	−0.632600
$633$	25.6730	1.02041
$634$	4.74318	0.188376
$635$	−22.6605	−0.899254
$636$	−7.79694	−0.309169
$637$	−21.7388	−0.861324
$638$	−26.7044	−1.05724
$639$	5.74447	0.227248
$640$	−13.4568	−0.531925
$641$	36.1243	1.42682	0.713412	−	0.700745i	$-0.247149\pi$
$641$	36.1243	1.42682	0.713412	+	0.700745i	$0.247149\pi$
$642$	29.8149	1.17670
$643$	10.6517	0.420063	0.210031	−	0.977695i	$-0.432643\pi$
$643$	10.6517	0.420063	0.210031	+	0.977695i	$0.432643\pi$
$644$	1.00694	0.0396791
$645$	36.8137	1.44954
$646$	−37.1414	−1.46131
$647$	13.6622	0.537116	0.268558	−	0.963263i	$-0.413453\pi$
$647$	13.6622	0.537116	0.268558	+	0.963263i	$0.413453\pi$
$648$	54.5696	2.14370
$649$	−6.40700	−0.251497
$650$	−1.48805	−0.0583659
$651$	−10.7043	−0.419535
$652$	−91.1931	−3.57140
$653$	−9.36591	−0.366517	−0.183258	−	0.983065i	$-0.558664\pi$
$653$	−9.36591	−0.366517	−0.183258	+	0.983065i	$0.558664\pi$
$654$	−6.88130	−0.269080
$655$	−19.6465	−0.767652
$656$	−113.388	−4.42705
$657$	−4.50892	−0.175910
$658$	−11.2805	−0.439761
$659$	−10.2965	−0.401095	−0.200547	−	0.979684i	$-0.564272\pi$
$659$	−10.2965	−0.401095	−0.200547	+	0.979684i	$0.564272\pi$
$660$	−42.2364	−1.64405
$661$	3.31039	0.128759	0.0643797	−	0.997925i	$-0.479493\pi$
$661$	3.31039	0.128759	0.0643797	+	0.997925i	$0.479493\pi$
$662$	43.4309	1.68799
$663$	−18.0522	−0.701089
$664$	−33.3314	−1.29351
$665$	−8.67972	−0.336585
$666$	13.8282	0.535831
$667$	−0.955975	−0.0370155
$668$	−7.89095	−0.305310
$669$	−20.0404	−0.774807
$670$	67.6231	2.61251
$671$	6.09347	0.235236
$672$	−17.5372	−0.676512
$673$	−3.06827	−0.118273	−0.0591365	−	0.998250i	$-0.518835\pi$
$673$	−3.06827	−0.118273	−0.0591365	+	0.998250i	$0.518835\pi$
$674$	−13.1926	−0.508160
$675$	0.894711	0.0344374
$676$	−3.47962	−0.133831
$677$	17.7836	0.683481	0.341740	−	0.939794i	$-0.388984\pi$
$677$	17.7836	0.683481	0.341740	+	0.939794i	$0.388984\pi$
$678$	−45.9810	−1.76589
$679$	7.34010	0.281687
$680$	−58.4988	−2.24333
$681$	−20.4116	−0.782175
$682$	48.5541	1.85923
$683$	−0.966003	−0.0369631	−0.0184815	−	0.999829i	$-0.505883\pi$
$683$	−0.966003	−0.0369631	−0.0184815	+	0.999829i	$0.505883\pi$
$684$	−11.9426	−0.456636
$685$	10.5705	0.403877
$686$	−31.1659	−1.18992
$687$	−16.5972	−0.633223
$688$	−110.866	−4.22673
$689$	3.53249	0.134577
$690$	−2.12173	−0.0807729
$691$	−37.2480	−1.41698	−0.708491	−	0.705720i	$-0.750624\pi$
$691$	−37.2480	−1.41698	−0.708491	+	0.705720i	$0.750624\pi$
$692$	35.9011	1.36475
$693$	1.21278	0.0460696
$694$	−18.3765	−0.697564
$695$	−32.7285	−1.24146
$696$	51.3537	1.94656
$697$	−35.0239	−1.32663
$698$	44.6762	1.69102
$699$	−0.399635	−0.0151156
$700$	−0.713995	−0.0269865
$701$	−31.3311	−1.18336	−0.591681	−	0.806173i	$-0.701535\pi$
$701$	−31.3311	−1.18336	−0.591681	+	0.806173i	$0.701535\pi$
$702$	−51.4628	−1.94234
$703$	39.6644	1.49597
$704$	28.2534	1.06484
$705$	16.9385	0.637941
$706$	−6.96954	−0.262302
$707$	8.70683	0.327454
$708$	20.6472	0.775971
$709$	−8.87977	−0.333487	−0.166743	−	0.986000i	$-0.553325\pi$
$709$	−8.87977	−0.333487	−0.166743	+	0.986000i	$0.553325\pi$
$710$	61.0285	2.29036
$711$	−1.14907	−0.0430934
$712$	16.1613	0.605671
$713$	1.73816	0.0650945
$714$	−12.1548	−0.454883
$715$	19.1357	0.715635
$716$	18.1196	0.677163
$717$	8.53869	0.318883
$718$	4.27306	0.159469
$719$	−36.6051	−1.36514	−0.682570	−	0.730820i	$-0.739138\pi$
$719$	−36.6051	−1.36514	−0.682570	+	0.730820i	$0.739138\pi$
$720$	−13.6838	−0.509965
$721$	−0.951601	−0.0354395
$722$	2.05382	0.0764352
$723$	18.7544	0.697485
$724$	−30.4140	−1.13033
$725$	0.677855	0.0251749
$726$	−21.5368	−0.799306
$727$	10.6957	0.396681	0.198340	−	0.980133i	$-0.436445\pi$
$727$	10.6957	0.396681	0.198340	+	0.980133i	$0.436445\pi$
$728$	24.5068	0.908284
$729$	29.9882	1.11067
$730$	−47.9021	−1.77294
$731$	−34.2451	−1.26660
$732$	−19.6369	−0.725800
$733$	−35.9851	−1.32914	−0.664570	−	0.747226i	$-0.731385\pi$
$733$	−35.9851	−1.32914	−0.664570	+	0.747226i	$0.731385\pi$
$734$	−34.8378	−1.28589
$735$	21.9786	0.810694
$736$	2.84767	0.104967
$737$	−27.1022	−0.998324
$738$	−15.8032	−0.581722
$739$	−7.94288	−0.292184	−0.146092	−	0.989271i	$-0.546670\pi$
$739$	−7.94288	−0.292184	−0.146092	+	0.989271i	$0.546670\pi$
$740$	104.691	3.84850
$741$	−23.3638	−0.858292
$742$	2.37848	0.0873169
$743$	−4.00221	−0.146827	−0.0734134	−	0.997302i	$-0.523389\pi$
$743$	−4.00221	−0.146827	−0.0734134	+	0.997302i	$0.523389\pi$
$744$	−93.3715	−3.42316
$745$	−3.17990	−0.116503
$746$	−49.2723	−1.80399
$747$	−2.40830	−0.0881153
$748$	39.2894	1.43656
$749$	−6.48141	−0.236826
$750$	−45.2635	−1.65279
$751$	15.5480	0.567355	0.283678	−	0.958920i	$-0.408445\pi$
$751$	15.5480	0.567355	0.283678	+	0.958920i	$0.408445\pi$
$752$	−51.0112	−1.86019
$753$	32.3583	1.17920
$754$	−38.9895	−1.41991
$755$	−9.71813	−0.353679
$756$	−24.6929	−0.898073
$757$	17.1063	0.621739	0.310869	−	0.950453i	$-0.399380\pi$
$757$	17.1063	0.621739	0.310869	+	0.950453i	$0.399380\pi$
$758$	18.8714	0.685440
$759$	0.850355	0.0308659
$760$	−75.7114	−2.74634
$761$	22.7383	0.824261	0.412130	−	0.911125i	$-0.364785\pi$
$761$	22.7383	0.824261	0.412130	+	0.911125i	$0.364785\pi$
$762$	−41.0702	−1.48782
$763$	1.49591	0.0541557
$764$	−0.772510	−0.0279484
$765$	−4.22673	−0.152818
$766$	−58.9920	−2.13147
$767$	−9.35446	−0.337770
$768$	12.3279	0.444845
$769$	−37.7665	−1.36189	−0.680947	−	0.732332i	$-0.738432\pi$
$769$	−37.7665	−1.36189	−0.680947	+	0.732332i	$0.738432\pi$
$770$	12.8844	0.464320
$771$	−4.01983	−0.144771
$772$	−55.7150	−2.00523
$773$	−45.8069	−1.64756	−0.823780	−	0.566910i	$-0.808139\pi$
$773$	−45.8069	−1.64756	−0.823780	+	0.566910i	$0.808139\pi$
$774$	−15.4517	−0.555401
$775$	−1.23248	−0.0442719
$776$	64.0262	2.29841
$777$	12.9805	0.465672
$778$	30.3853	1.08937
$779$	−45.3293	−1.62409
$780$	−61.6669	−2.20803
$781$	−24.4592	−0.875220
$782$	1.97369	0.0705790
$783$	23.4431	0.837786
$784$	−66.1897	−2.36392
$785$	−2.18539	−0.0780000
$786$	−35.6076	−1.27008
$787$	−38.1830	−1.36108	−0.680538	−	0.732713i	$-0.738254\pi$
$787$	−38.1830	−1.36108	−0.680538	+	0.732713i	$0.738254\pi$
$788$	−6.69475	−0.238491
$789$	−26.5406	−0.944870
$790$	−12.2075	−0.434325
$791$	9.99573	0.355407
$792$	10.5788	0.375901
$793$	8.89671	0.315931
$794$	10.2921	0.365254
$795$	−3.57146	−0.126667
$796$	89.8994	3.18640
$797$	49.4597	1.75195	0.875976	−	0.482354i	$-0.160218\pi$
$797$	49.4597	1.75195	0.875976	+	0.482354i	$0.160218\pi$
$798$	−15.7313	−0.556880
$799$	−15.7566	−0.557430
$800$	−2.01921	−0.0713897
$801$	1.16771	0.0412590
$802$	−37.9297	−1.33934
$803$	19.1984	0.677496
$804$	87.3399	3.08024
$805$	0.461239	0.0162565
$806$	70.8908	2.49702
$807$	−40.6138	−1.42967
$808$	75.9479	2.67184
$809$	−13.5459	−0.476248	−0.238124	−	0.971235i	$-0.576532\pi$
$809$	−13.5459	−0.476248	−0.238124	+	0.971235i	$0.576532\pi$
$810$	41.8882	1.47180
$811$	−31.3290	−1.10011	−0.550056	−	0.835128i	$-0.685394\pi$
$811$	−31.3290	−1.10011	−0.550056	+	0.835128i	$0.685394\pi$
$812$	−18.7080	−0.656521
$813$	11.0603	0.387900
$814$	−58.8786	−2.06370
$815$	−41.7718	−1.46320
$816$	−54.9648	−1.92415
$817$	−44.3213	−1.55061
$818$	4.37010	0.152797
$819$	1.77070	0.0618733
$820$	−119.643	−4.17811
$821$	36.2594	1.26546	0.632732	−	0.774371i	$-0.281934\pi$
$821$	36.2594	1.26546	0.632732	+	0.774371i	$0.281934\pi$
$822$	19.1581	0.668215
$823$	−9.27225	−0.323210	−0.161605	−	0.986856i	$-0.551667\pi$
$823$	−9.27225	−0.323210	−0.161605	+	0.986856i	$0.551667\pi$
$824$	−8.30062	−0.289166
$825$	−0.602963	−0.0209925
$826$	−6.29851	−0.219153
$827$	−39.6936	−1.38028	−0.690140	−	0.723676i	$-0.742451\pi$
$827$	−39.6936	−1.38028	−0.690140	+	0.723676i	$0.742451\pi$
$828$	0.634627	0.0220548
$829$	47.4878	1.64932	0.824659	−	0.565630i	$-0.191367\pi$
$829$	47.4878	1.64932	0.824659	+	0.565630i	$0.191367\pi$
$830$	−25.5855	−0.888085
$831$	−12.9964	−0.450839
$832$	41.2510	1.43012
$833$	−20.4451	−0.708381
$834$	−59.3176	−2.05400
$835$	−3.61452	−0.125086
$836$	50.8499	1.75868
$837$	−42.6242	−1.47331
$838$	−17.4988	−0.604485
$839$	46.5430	1.60684	0.803421	−	0.595412i	$-0.203011\pi$
$839$	46.5430	1.60684	0.803421	+	0.595412i	$0.203011\pi$
$840$	−24.7772	−0.854894
$841$	−11.2390	−0.387551
$842$	−100.289	−3.45618
$843$	−38.1828	−1.31509
$844$	−81.5806	−2.80812
$845$	−1.59387	−0.0548308
$846$	−7.10957	−0.244432
$847$	4.68185	0.160870
$848$	10.7556	0.369350
$849$	2.94216	0.100975
$850$	−1.39949	−0.0480021
$851$	−2.10776	−0.0722530
$852$	78.8225	2.70041
$853$	45.0603	1.54284	0.771418	−	0.636328i	$-0.219548\pi$
$853$	45.0603	1.54284	0.771418	+	0.636328i	$0.219548\pi$
$854$	5.99030	0.204984
$855$	−5.47040	−0.187084
$856$	−56.5360	−1.93236
$857$	14.7509	0.503879	0.251940	−	0.967743i	$-0.418932\pi$
$857$	14.7509	0.503879	0.251940	+	0.967743i	$0.418932\pi$
$858$	34.6818	1.18402
$859$	−57.2530	−1.95345	−0.976724	−	0.214499i	$-0.931188\pi$
$859$	−57.2530	−1.95345	−0.976724	+	0.214499i	$0.931188\pi$
$860$	−116.982	−3.98906
$861$	−14.8344	−0.505554
$862$	53.0716	1.80763
$863$	−23.2806	−0.792482	−0.396241	−	0.918147i	$-0.629686\pi$
$863$	−23.2806	−0.792482	−0.396241	+	0.918147i	$0.629686\pi$
$864$	−69.8326	−2.37575
$865$	16.4448	0.559140
$866$	−9.07567	−0.308404
$867$	9.55458	0.324491
$868$	34.0149	1.15454
$869$	4.89258	0.165969
$870$	39.4196	1.33645
$871$	−39.5703	−1.34079
$872$	13.0485	0.441879
$873$	4.62611	0.156570
$874$	2.55443	0.0864048
$875$	9.83974	0.332644
$876$	−61.8688	−2.09035
$877$	7.33600	0.247719	0.123860	−	0.992300i	$-0.460473\pi$
$877$	7.33600	0.247719	0.123860	+	0.992300i	$0.460473\pi$
$878$	74.1695	2.50310
$879$	−10.2956	−0.347263
$880$	58.2638	1.96407
$881$	−32.9813	−1.11117	−0.555584	−	0.831460i	$-0.687505\pi$
$881$	−32.9813	−1.11117	−0.555584	+	0.831460i	$0.687505\pi$
$882$	−9.22505	−0.310624
$883$	−28.5738	−0.961584	−0.480792	−	0.876835i	$-0.659651\pi$
$883$	−28.5738	−0.961584	−0.480792	+	0.876835i	$0.659651\pi$
$884$	57.3641	1.92936
$885$	9.45765	0.317916
$886$	108.702	3.65191
$887$	35.4558	1.19049	0.595244	−	0.803545i	$-0.297055\pi$
$887$	35.4558	1.19049	0.595244	+	0.803545i	$0.297055\pi$
$888$	113.226	3.79962
$889$	8.92817	0.299441
$890$	12.4056	0.415836
$891$	−16.7881	−0.562423
$892$	63.6820	2.13223
$893$	−20.3929	−0.682421
$894$	−5.76330	−0.192753
$895$	8.29987	0.277434
$896$	5.30193	0.177125
$897$	1.24155	0.0414542
$898$	−81.7011	−2.72640
$899$	−32.2932	−1.07704
$900$	−0.449996	−0.0149999
$901$	3.32227	0.110681
$902$	67.2878	2.24044
$903$	−14.5045	−0.482679
$904$	87.1907	2.89992
$905$	−13.9314	−0.463096
$906$	−17.6133	−0.585162
$907$	−25.8840	−0.859465	−0.429733	−	0.902956i	$-0.641392\pi$
$907$	−25.8840	−0.859465	−0.429733	+	0.902956i	$0.641392\pi$
$908$	64.8616	2.15251
$909$	5.48749	0.182009
$910$	18.8117	0.623601
$911$	51.0144	1.69018	0.845092	−	0.534622i	$-0.179546\pi$
$911$	51.0144	1.69018	0.845092	+	0.534622i	$0.179546\pi$
$912$	−71.1375	−2.35560
$913$	10.2542	0.339366
$914$	−53.8245	−1.78036
$915$	−8.99485	−0.297361
$916$	52.7407	1.74260
$917$	7.74067	0.255619
$918$	−48.4002	−1.59744
$919$	−11.9753	−0.395030	−0.197515	−	0.980300i	$-0.563287\pi$
$919$	−11.9753	−0.395030	−0.197515	+	0.980300i	$0.563287\pi$
$920$	4.02329	0.132644
$921$	42.9575	1.41550
$922$	61.6327	2.02977
$923$	−35.7114	−1.17546
$924$	16.6410	0.547450
$925$	1.49455	0.0491406
$926$	−35.4476	−1.16488
$927$	−0.599748	−0.0196983
$928$	−52.9069	−1.73675
$929$	−31.3341	−1.02804	−0.514020	−	0.857778i	$-0.671844\pi$
$929$	−31.3341	−1.02804	−0.514020	+	0.857778i	$0.671844\pi$
$930$	−71.6728	−2.35025
$931$	−26.4608	−0.867219
$932$	1.26991	0.0415974
$933$	19.5338	0.639508
$934$	6.82070	0.223180
$935$	17.9969	0.588561
$936$	15.4455	0.504851
$937$	−18.7842	−0.613652	−0.306826	−	0.951766i	$-0.599267\pi$
$937$	−18.7842	−0.613652	−0.306826	+	0.951766i	$0.599267\pi$
$938$	−26.6433	−0.869936
$939$	25.2932	0.825411
$940$	−53.8252	−1.75558
$941$	−42.1883	−1.37530	−0.687650	−	0.726042i	$-0.741358\pi$
$941$	−42.1883	−1.37530	−0.687650	+	0.726042i	$0.741358\pi$
$942$	−3.96083	−0.129051
$943$	2.40879	0.0784411
$944$	−28.4822	−0.927016
$945$	−11.3108	−0.367941
$946$	65.7915	2.13907
$947$	−36.5646	−1.18819	−0.594095	−	0.804395i	$-0.702490\pi$
$947$	−36.5646	−1.18819	−0.594095	+	0.804395i	$0.702490\pi$
$948$	−15.7669	−0.512084
$949$	28.0304	0.909905
$950$	−1.81127	−0.0587654
$951$	2.80614	0.0909953
$952$	23.0484	0.747002
$953$	−9.78324	−0.316910	−0.158455	−	0.987366i	$-0.550651\pi$
$953$	−9.78324	−0.316910	−0.158455	+	0.987366i	$0.550651\pi$
$954$	1.49904	0.0485333
$955$	−0.353855	−0.0114505
$956$	−27.1332	−0.877552
$957$	−15.7987	−0.510701
$958$	−87.8901	−2.83960
$959$	−4.16474	−0.134487
$960$	−41.7061	−1.34606
$961$	27.7155	0.894050
$962$	−85.9651	−2.77163
$963$	−4.08492	−0.131635
$964$	−59.5956	−1.91945
$965$	−25.5208	−0.821542
$966$	0.835957	0.0268965
$967$	−43.4848	−1.39838	−0.699189	−	0.714937i	$-0.746455\pi$
$967$	−43.4848	−1.39838	−0.699189	+	0.714937i	$0.746455\pi$
$968$	40.8388	1.31261
$969$	−21.9734	−0.705888
$970$	49.1471	1.57802
$971$	39.2696	1.26022	0.630110	−	0.776506i	$-0.283010\pi$
$971$	39.2696	1.26022	0.630110	+	0.776506i	$0.283010\pi$
$972$	−28.6623	−0.919343
$973$	12.8950	0.413393
$974$	−21.9857	−0.704469
$975$	−0.880350	−0.0281938
$976$	27.0884	0.867080
$977$	−1.90421	−0.0609210	−0.0304605	−	0.999536i	$-0.509697\pi$
$977$	−1.90421	−0.0609210	−0.0304605	+	0.999536i	$0.509697\pi$
$978$	−75.7078	−2.42087
$979$	−4.97195	−0.158904
$980$	−69.8411	−2.23099
$981$	0.942801	0.0301013
$982$	−94.9503	−3.02999
$983$	−6.91942	−0.220695	−0.110348	−	0.993893i	$-0.535196\pi$
$983$	−6.91942	−0.220695	−0.110348	+	0.993893i	$0.535196\pi$
$984$	−129.397	−4.12503
$985$	−3.06659	−0.0977097
$986$	−36.6691	−1.16778
$987$	−6.67373	−0.212427
$988$	74.2429	2.36198
$989$	2.35523	0.0748919
$990$	8.12039	0.258083
$991$	18.0567	0.573591	0.286796	−	0.957992i	$-0.407410\pi$
$991$	18.0567	0.573591	0.286796	+	0.957992i	$0.407410\pi$
$992$	96.1954	3.05421
$993$	25.6944	0.815386
$994$	−24.0451	−0.762663
$995$	41.1793	1.30547
$996$	−33.0454	−1.04708
$997$	−0.379446	−0.0120172	−0.00600859	−	0.999982i	$-0.501913\pi$
$997$	−0.379446	−0.0120172	−0.00600859	+	0.999982i	$0.501913\pi$
$998$	20.5656	0.650992
$999$	51.6879	1.63533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.a.1.9	✓	306

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.a.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-2.63809 q^{2} -1.56073 q^{3} +4.95951 q^{4} +2.27175 q^{5} +4.11735 q^{6} -0.895063 q^{7} -7.80745 q^{8} -0.564115 q^{9} +O(q^{10})\)	\(q-2.63809 q^{2} -1.56073 q^{3} +4.95951 q^{4} +2.27175 q^{5} +4.11735 q^{6} -0.895063 q^{7} -7.80745 q^{8} -0.564115 q^{9} -5.99307 q^{10} +2.40193 q^{11} -7.74047 q^{12} +3.50691 q^{13} +2.36126 q^{14} -3.54559 q^{15} +10.6777 q^{16} +3.29820 q^{17} +1.48818 q^{18} +4.26866 q^{19} +11.2668 q^{20} +1.39695 q^{21} -6.33650 q^{22} -0.226836 q^{23} +12.1853 q^{24} +0.160843 q^{25} -9.25153 q^{26} +5.56263 q^{27} -4.43908 q^{28} +4.21438 q^{29} +9.35359 q^{30} -7.66261 q^{31} -12.5539 q^{32} -3.74876 q^{33} -8.70095 q^{34} -2.03336 q^{35} -2.79773 q^{36} +9.29199 q^{37} -11.2611 q^{38} -5.47334 q^{39} -17.7366 q^{40} -10.6191 q^{41} -3.68529 q^{42} -10.3829 q^{43} +11.9124 q^{44} -1.28153 q^{45} +0.598414 q^{46} -4.77734 q^{47} -16.6651 q^{48} -6.19886 q^{49} -0.424318 q^{50} -5.14761 q^{51} +17.3925 q^{52} +1.00730 q^{53} -14.6747 q^{54} +5.45657 q^{55} +6.98816 q^{56} -6.66224 q^{57} -11.1179 q^{58} -2.66744 q^{59} -17.5844 q^{60} +2.53691 q^{61} +20.2146 q^{62} +0.504918 q^{63} +11.7628 q^{64} +7.96681 q^{65} +9.88957 q^{66} -11.2835 q^{67} +16.3575 q^{68} +0.354030 q^{69} +5.36418 q^{70} -10.1832 q^{71} +4.40430 q^{72} +7.99291 q^{73} -24.5131 q^{74} -0.251033 q^{75} +21.1705 q^{76} -2.14988 q^{77} +14.4392 q^{78} +2.03694 q^{79} +24.2571 q^{80} -6.98943 q^{81} +28.0141 q^{82} +4.26918 q^{83} +6.92821 q^{84} +7.49269 q^{85} +27.3911 q^{86} -6.57753 q^{87} -18.7529 q^{88} -2.06999 q^{89} +3.38078 q^{90} -3.13890 q^{91} -1.12500 q^{92} +11.9593 q^{93} +12.6031 q^{94} +9.69733 q^{95} +19.5932 q^{96} -8.20065 q^{97} +16.3531 q^{98} -1.35496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) \) 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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