LMFDB - Embedded newform 8009.2.a.a.1.2

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			1.2
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)

$f(q)$	$=$	$q-2.76073 q^{2} +2.52985 q^{3} +5.62166 q^{4} +0.0311099 q^{5} -6.98425 q^{6} +0.584590 q^{7} -9.99844 q^{8} +3.40015 q^{9} +O(q^{10})$	\(q-2.76073 q^{2} +2.52985 q^{3} +5.62166 q^{4} +0.0311099 q^{5} -6.98425 q^{6} +0.584590 q^{7} -9.99844 q^{8} +3.40015 q^{9} -0.0858861 q^{10} -3.58892 q^{11} +14.2220 q^{12} -4.55481 q^{13} -1.61390 q^{14} +0.0787034 q^{15} +16.3597 q^{16} +0.794243 q^{17} -9.38691 q^{18} -0.270503 q^{19} +0.174889 q^{20} +1.47893 q^{21} +9.90805 q^{22} +6.64741 q^{23} -25.2946 q^{24} -4.99903 q^{25} +12.5746 q^{26} +1.01232 q^{27} +3.28636 q^{28} +3.53068 q^{29} -0.217279 q^{30} +1.78343 q^{31} -25.1680 q^{32} -9.07943 q^{33} -2.19269 q^{34} +0.0181865 q^{35} +19.1145 q^{36} +5.16878 q^{37} +0.746786 q^{38} -11.5230 q^{39} -0.311050 q^{40} +10.8098 q^{41} -4.08292 q^{42} -10.2253 q^{43} -20.1757 q^{44} +0.105778 q^{45} -18.3517 q^{46} +4.94294 q^{47} +41.3877 q^{48} -6.65825 q^{49} +13.8010 q^{50} +2.00932 q^{51} -25.6056 q^{52} -2.54049 q^{53} -2.79475 q^{54} -0.111651 q^{55} -5.84499 q^{56} -0.684332 q^{57} -9.74728 q^{58} -6.97711 q^{59} +0.442444 q^{60} -9.66589 q^{61} -4.92357 q^{62} +1.98769 q^{63} +36.7627 q^{64} -0.141700 q^{65} +25.0659 q^{66} +13.1930 q^{67} +4.46496 q^{68} +16.8170 q^{69} -0.0502082 q^{70} +3.53603 q^{71} -33.9962 q^{72} -7.33620 q^{73} -14.2696 q^{74} -12.6468 q^{75} -1.52067 q^{76} -2.09804 q^{77} +31.8120 q^{78} -15.2307 q^{79} +0.508949 q^{80} -7.63943 q^{81} -29.8431 q^{82} +5.45781 q^{83} +8.31402 q^{84} +0.0247088 q^{85} +28.2295 q^{86} +8.93210 q^{87} +35.8836 q^{88} -9.55626 q^{89} -0.292026 q^{90} -2.66270 q^{91} +37.3695 q^{92} +4.51181 q^{93} -13.6461 q^{94} -0.00841531 q^{95} -63.6712 q^{96} -8.48384 q^{97} +18.3817 q^{98} -12.2029 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.76073	−1.95213	−0.976067	−	0.217469i	$-0.930220\pi$
$2$	−2.76073	−1.95213	−0.976067	+	0.217469i	$0.930220\pi$
$3$	2.52985	1.46061	0.730305	−	0.683121i	$-0.239378\pi$
$3$	2.52985	1.46061	0.730305	+	0.683121i	$0.239378\pi$
$4$	5.62166	2.81083
$5$	0.0311099	0.0139128	0.00695638	−	0.999976i	$-0.497786\pi$
$5$	0.0311099	0.0139128	0.00695638	+	0.999976i	$0.497786\pi$
$6$	−6.98425	−2.85131
$7$	0.584590	0.220954	0.110477	−	0.993879i	$-0.464762\pi$
$7$	0.584590	0.220954	0.110477	+	0.993879i	$0.464762\pi$
$8$	−9.99844	−3.53498
$9$	3.40015	1.13338
$10$	−0.0858861	−0.0271596
$11$	−3.58892	−1.08210	−0.541050	−	0.840991i	$-0.681973\pi$
$11$	−3.58892	−1.08210	−0.541050	+	0.840991i	$0.681973\pi$
$12$	14.2220	4.10553
$13$	−4.55481	−1.26328	−0.631639	−	0.775263i	$-0.717617\pi$
$13$	−4.55481	−1.26328	−0.631639	+	0.775263i	$0.717617\pi$
$14$	−1.61390	−0.431332
$15$	0.0787034	0.0203211
$16$	16.3597	4.08993
$17$	0.794243	0.192632	0.0963161	−	0.995351i	$-0.469294\pi$
$17$	0.794243	0.192632	0.0963161	+	0.995351i	$0.469294\pi$
$18$	−9.38691	−2.21252
$19$	−0.270503	−0.0620576	−0.0310288	−	0.999518i	$-0.509878\pi$
$19$	−0.270503	−0.0620576	−0.0310288	+	0.999518i	$0.509878\pi$
$20$	0.174889	0.0391064
$21$	1.47893	0.322728
$22$	9.90805	2.11240
$23$	6.64741	1.38608	0.693040	−	0.720899i	$-0.256271\pi$
$23$	6.64741	1.38608	0.693040	+	0.720899i	$0.256271\pi$
$24$	−25.2946	−5.16323
$25$	−4.99903	−0.999806
$26$	12.5746	2.46609
$27$	1.01232	0.194821
$28$	3.28636	0.621065
$29$	3.53068	0.655631	0.327816	−	0.944742i	$-0.393688\pi$
$29$	3.53068	0.655631	0.327816	+	0.944742i	$0.393688\pi$
$30$	−0.217279	−0.0396696
$31$	1.78343	0.320313	0.160157	−	0.987092i	$-0.448800\pi$
$31$	1.78343	0.320313	0.160157	+	0.987092i	$0.448800\pi$
$32$	−25.1680	−4.44911
$33$	−9.07943	−1.58053
$34$	−2.19269	−0.376044
$35$	0.0181865	0.00307408
$36$	19.1145	3.18575
$37$	5.16878	0.849742	0.424871	−	0.905254i	$-0.360319\pi$
$37$	5.16878	0.849742	0.424871	+	0.905254i	$0.360319\pi$
$38$	0.746786	0.121145
$39$	−11.5230	−1.84516
$40$	−0.311050	−0.0491814
$41$	10.8098	1.68821	0.844106	−	0.536176i	$-0.180132\pi$
$41$	10.8098	1.68821	0.844106	+	0.536176i	$0.180132\pi$
$42$	−4.08292	−0.630009
$43$	−10.2253	−1.55935	−0.779675	−	0.626184i	$-0.784616\pi$
$43$	−10.2253	−1.55935	−0.779675	+	0.626184i	$0.784616\pi$
$44$	−20.1757	−3.04160
$45$	0.105778	0.0157685
$46$	−18.3517	−2.70582
$47$	4.94294	0.721002	0.360501	−	0.932759i	$-0.382606\pi$
$47$	4.94294	0.721002	0.360501	+	0.932759i	$0.382606\pi$
$48$	41.3877	5.97379
$49$	−6.65825	−0.951179
$50$	13.8010	1.95176
$51$	2.00932	0.281361
$52$	−25.6056	−3.55086
$53$	−2.54049	−0.348963	−0.174481	−	0.984660i	$-0.555825\pi$
$53$	−2.54049	−0.348963	−0.174481	+	0.984660i	$0.555825\pi$
$54$	−2.79475	−0.380317
$55$	−0.111651	−0.0150550
$56$	−5.84499	−0.781069
$57$	−0.684332	−0.0906419
$58$	−9.74728	−1.27988
$59$	−6.97711	−0.908342	−0.454171	−	0.890914i	$-0.650065\pi$
$59$	−6.97711	−0.908342	−0.454171	+	0.890914i	$0.650065\pi$
$60$	0.442444	0.0571192
$61$	−9.66589	−1.23759	−0.618795	−	0.785552i	$-0.712379\pi$
$61$	−9.66589	−1.23759	−0.618795	+	0.785552i	$0.712379\pi$
$62$	−4.92357	−0.625294
$63$	1.98769	0.250426
$64$	36.7627	4.59533
$65$	−0.141700	−0.0175757
$66$	25.0659	3.08540
$67$	13.1930	1.61178	0.805890	−	0.592065i	$-0.201687\pi$
$67$	13.1930	1.61178	0.805890	+	0.592065i	$0.201687\pi$
$68$	4.46496	0.541456
$69$	16.8170	2.02452
$70$	−0.0502082	−0.00600103
$71$	3.53603	0.419649	0.209825	−	0.977739i	$-0.432711\pi$
$71$	3.53603	0.419649	0.209825	+	0.977739i	$0.432711\pi$
$72$	−33.9962	−4.00649
$73$	−7.33620	−0.858637	−0.429319	−	0.903153i	$-0.641246\pi$
$73$	−7.33620	−0.858637	−0.429319	+	0.903153i	$0.641246\pi$
$74$	−14.2696	−1.65881
$75$	−12.6468	−1.46033
$76$	−1.52067	−0.174433
$77$	−2.09804	−0.239094
$78$	31.8120	3.60199
$79$	−15.2307	−1.71358	−0.856792	−	0.515662i	$-0.827546\pi$
$79$	−15.2307	−1.71358	−0.856792	+	0.515662i	$0.827546\pi$
$80$	0.508949	0.0569022
$81$	−7.63943	−0.848825
$82$	−29.8431	−3.29562
$83$	5.45781	0.599072	0.299536	−	0.954085i	$-0.403168\pi$
$83$	5.45781	0.599072	0.299536	+	0.954085i	$0.403168\pi$
$84$	8.31402	0.907133
$85$	0.0247088	0.00268005
$86$	28.2295	3.04406
$87$	8.93210	0.957622
$88$	35.8836	3.82520
$89$	−9.55626	−1.01296	−0.506481	−	0.862251i	$-0.669054\pi$
$89$	−9.55626	−1.01296	−0.506481	+	0.862251i	$0.669054\pi$
$90$	−0.292026	−0.0307822
$91$	−2.66270	−0.279127
$92$	37.3695	3.89604
$93$	4.51181	0.467853
$94$	−13.6461	−1.40749
$95$	−0.00841531	−0.000863392 0
$96$	−63.6712	−6.49842
$97$	−8.48384	−0.861404	−0.430702	−	0.902494i	$-0.641734\pi$
$97$	−8.48384	−0.861404	−0.430702	+	0.902494i	$0.641734\pi$
$98$	18.3817	1.85683
$99$	−12.2029	−1.22643
$100$	−28.1028	−2.81028
$101$	18.7434	1.86503	0.932517	−	0.361127i	$-0.117608\pi$
$101$	18.7434	1.86503	0.932517	+	0.361127i	$0.117608\pi$
$102$	−5.54719	−0.549254
$103$	1.32324	0.130382	0.0651912	−	0.997873i	$-0.479234\pi$
$103$	1.32324	0.130382	0.0651912	+	0.997873i	$0.479234\pi$
$104$	45.5410	4.46566
$105$	0.0460092	0.00449004
$106$	7.01361	0.681222
$107$	−1.92258	−0.185863	−0.0929315	−	0.995673i	$-0.529624\pi$
$107$	−1.92258	−0.185863	−0.0929315	+	0.995673i	$0.529624\pi$
$108$	5.69092	0.547609
$109$	13.5223	1.29520	0.647599	−	0.761981i	$-0.275773\pi$
$109$	13.5223	1.29520	0.647599	+	0.761981i	$0.275773\pi$
$110$	0.308238	0.0293894
$111$	13.0762	1.24114
$112$	9.56373	0.903687
$113$	−13.0036	−1.22327	−0.611637	−	0.791138i	$-0.709489\pi$
$113$	−13.0036	−1.22327	−0.611637	+	0.791138i	$0.709489\pi$
$114$	1.88926	0.176945
$115$	0.206800	0.0192842
$116$	19.8483	1.84287
$117$	−15.4870	−1.43178
$118$	19.2620	1.77321
$119$	0.464306	0.0425629
$120$	−0.786911	−0.0718348
$121$	1.88032	0.170938
$122$	26.6850	2.41594
$123$	27.3473	2.46582
$124$	10.0258	0.900346
$125$	−0.311069	−0.0278228
$126$	−5.48750	−0.488865
$127$	−9.65112	−0.856398	−0.428199	−	0.903684i	$-0.640852\pi$
$127$	−9.65112	−0.856398	−0.428199	+	0.903684i	$0.640852\pi$
$128$	−51.1560	−4.52160
$129$	−25.8686	−2.27760
$130$	0.391195	0.0343101
$131$	−5.80826	−0.507470	−0.253735	−	0.967274i	$-0.581659\pi$
$131$	−5.80826	−0.507470	−0.253735	+	0.967274i	$0.581659\pi$
$132$	−51.0414	−4.44259
$133$	−0.158133	−0.0137119
$134$	−36.4223	−3.14641
$135$	0.0314932	0.00271050
$136$	−7.94119	−0.680951
$137$	3.55221	0.303486	0.151743	−	0.988420i	$-0.451511\pi$
$137$	3.55221	0.303486	0.151743	+	0.988420i	$0.451511\pi$
$138$	−46.4272	−3.95214
$139$	−18.1670	−1.54090	−0.770451	−	0.637499i	$-0.779969\pi$
$139$	−18.1670	−1.54090	−0.770451	+	0.637499i	$0.779969\pi$
$140$	0.102238	0.00864072
$141$	12.5049	1.05310
$142$	−9.76203	−0.819212
$143$	16.3468	1.36699
$144$	55.6255	4.63546
$145$	0.109839	0.00912164
$146$	20.2533	1.67618
$147$	−16.8444	−1.38930
$148$	29.0571	2.38848
$149$	−14.5112	−1.18880	−0.594402	−	0.804168i	$-0.702611\pi$
$149$	−14.5112	−1.18880	−0.594402	+	0.804168i	$0.702611\pi$
$150$	34.9145	2.85076
$151$	17.9411	1.46003	0.730013	−	0.683434i	$-0.239514\pi$
$151$	17.9411	1.46003	0.730013	+	0.683434i	$0.239514\pi$
$152$	2.70460	0.219372
$153$	2.70055	0.218326
$154$	5.79214	0.466744
$155$	0.0554823	0.00445644
$156$	−64.7784	−5.18642
$157$	1.35405	0.108065	0.0540324	−	0.998539i	$-0.482793\pi$
$157$	1.35405	0.108065	0.0540324	+	0.998539i	$0.482793\pi$
$158$	42.0479	3.34515
$159$	−6.42706	−0.509699
$160$	−0.782973	−0.0618994
$161$	3.88601	0.306260
$162$	21.0904	1.65702
$163$	7.46533	0.584729	0.292365	−	0.956307i	$-0.405558\pi$
$163$	7.46533	0.584729	0.292365	+	0.956307i	$0.405558\pi$
$164$	60.7692	4.74528
$165$	−0.282460	−0.0219895
$166$	−15.0676	−1.16947
$167$	−5.58076	−0.431852	−0.215926	−	0.976410i	$-0.569277\pi$
$167$	−5.58076	−0.431852	−0.215926	+	0.976410i	$0.569277\pi$
$168$	−14.7869	−1.14084
$169$	7.74632	0.595871
$170$	−0.0682145	−0.00523181
$171$	−0.919750	−0.0703350
$172$	−57.4834	−4.38307
$173$	1.50379	0.114331	0.0571654	−	0.998365i	$-0.481794\pi$
$173$	1.50379	0.114331	0.0571654	+	0.998365i	$0.481794\pi$
$174$	−24.6592	−1.86941
$175$	−2.92238	−0.220911
$176$	−58.7137	−4.42571
$177$	−17.6511	−1.32673
$178$	26.3823	1.97744
$179$	−12.3393	−0.922286	−0.461143	−	0.887326i	$-0.652560\pi$
$179$	−12.3393	−0.922286	−0.461143	+	0.887326i	$0.652560\pi$
$180$	0.594649	0.0443225
$181$	1.18923	0.0883947	0.0441974	−	0.999023i	$-0.485927\pi$
$181$	1.18923	0.0883947	0.0441974	+	0.999023i	$0.485927\pi$
$182$	7.35100	0.544893
$183$	−24.4533	−1.80764
$184$	−66.4637	−4.89977
$185$	0.160800	0.0118223
$186$	−12.4559	−0.913312
$187$	−2.85047	−0.208447
$188$	27.7875	2.02661
$189$	0.591793	0.0430466
$190$	0.0232324	0.00168546
$191$	8.45651	0.611891	0.305946	−	0.952049i	$-0.401027\pi$
$191$	8.45651	0.611891	0.305946	+	0.952049i	$0.401027\pi$
$192$	93.0041	6.71199
$193$	−16.3892	−1.17972	−0.589861	−	0.807505i	$-0.700818\pi$
$193$	−16.3892	−1.17972	−0.589861	+	0.807505i	$0.700818\pi$
$194$	23.4216	1.68158
$195$	−0.358479	−0.0256712
$196$	−37.4304	−2.67360
$197$	17.8196	1.26960	0.634798	−	0.772678i	$-0.281083\pi$
$197$	17.8196	1.26960	0.634798	+	0.772678i	$0.281083\pi$
$198$	33.6889	2.39416
$199$	10.0614	0.713231	0.356616	−	0.934251i	$-0.383931\pi$
$199$	10.0614	0.713231	0.356616	+	0.934251i	$0.383931\pi$
$200$	49.9825	3.53430
$201$	33.3763	2.35418
$202$	−51.7454	−3.64080
$203$	2.06400	0.144864
$204$	11.2957	0.790857
$205$	0.336293	0.0234877
$206$	−3.65311	−0.254524
$207$	22.6022	1.57096
$208$	−74.5154	−5.16672
$209$	0.970812	0.0671524
$210$	−0.127019	−0.00876516
$211$	−21.8757	−1.50598	−0.752992	−	0.658030i	$-0.771390\pi$
$211$	−21.8757	−1.50598	−0.752992	+	0.658030i	$0.771390\pi$
$212$	−14.2817	−0.980875
$213$	8.94563	0.612944
$214$	5.30774	0.362830
$215$	−0.318109	−0.0216949
$216$	−10.1216	−0.688690
$217$	1.04257	0.0707746
$218$	−37.3314	−2.52840
$219$	−18.5595	−1.25413
$220$	−0.627662	−0.0423170
$221$	−3.61763	−0.243348
$222$	−36.1000	−2.42288
$223$	−4.43083	−0.296710	−0.148355	−	0.988934i	$-0.547398\pi$
$223$	−4.43083	−0.296710	−0.148355	+	0.988934i	$0.547398\pi$
$224$	−14.7129	−0.983050
$225$	−16.9975	−1.13316
$226$	35.8995	2.38800
$227$	−5.40467	−0.358720	−0.179360	−	0.983783i	$-0.557403\pi$
$227$	−5.40467	−0.358720	−0.179360	+	0.983783i	$0.557403\pi$
$228$	−3.84708	−0.254779
$229$	0.717354	0.0474041	0.0237020	−	0.999719i	$-0.492455\pi$
$229$	0.717354	0.0474041	0.0237020	+	0.999719i	$0.492455\pi$
$230$	−0.570920	−0.0376454
$231$	−5.30774	−0.349224
$232$	−35.3013	−2.31764
$233$	3.97895	0.260670	0.130335	−	0.991470i	$-0.458395\pi$
$233$	3.97895	0.260670	0.130335	+	0.991470i	$0.458395\pi$
$234$	42.7556	2.79502
$235$	0.153774	0.0100311
$236$	−39.2229	−2.55319
$237$	−38.5313	−2.50288
$238$	−1.28183	−0.0830885
$239$	−30.8671	−1.99663	−0.998313	−	0.0580658i	$-0.981507\pi$
$239$	−30.8671	−1.99663	−0.998313	+	0.0580658i	$0.981507\pi$
$240$	1.28757	0.0831120
$241$	−22.9083	−1.47566	−0.737828	−	0.674989i	$-0.764148\pi$
$241$	−22.9083	−1.47566	−0.737828	+	0.674989i	$0.764148\pi$
$242$	−5.19107	−0.333695
$243$	−22.3636	−1.43462
$244$	−54.3383	−3.47865
$245$	−0.207138	−0.0132335
$246$	−75.4986	−4.81361
$247$	1.23209	0.0783959
$248$	−17.8315	−1.13230
$249$	13.8074	0.875011
$250$	0.858778	0.0543139
$251$	2.53210	0.159825	0.0799125	−	0.996802i	$-0.474536\pi$
$251$	2.53210	0.159825	0.0799125	+	0.996802i	$0.474536\pi$
$252$	11.1741	0.703904
$253$	−23.8570	−1.49988
$254$	26.6442	1.67180
$255$	0.0625096	0.00391450
$256$	67.7029	4.23143
$257$	−12.1298	−0.756639	−0.378320	−	0.925675i	$-0.623498\pi$
$257$	−12.1298	−0.756639	−0.378320	+	0.925675i	$0.623498\pi$
$258$	71.4164	4.44619
$259$	3.02162	0.187754
$260$	−0.796587	−0.0494022
$261$	12.0048	0.743081
$262$	16.0351	0.990650
$263$	13.1222	0.809152	0.404576	−	0.914504i	$-0.367419\pi$
$263$	13.1222	0.809152	0.404576	+	0.914504i	$0.367419\pi$
$264$	90.7801	5.58713
$265$	−0.0790343	−0.00485504
$266$	0.436564	0.0267674
$267$	−24.1759	−1.47954
$268$	74.1664	4.53044
$269$	−20.7937	−1.26781	−0.633906	−	0.773410i	$-0.718550\pi$
$269$	−20.7937	−1.26781	−0.633906	+	0.773410i	$0.718550\pi$
$270$	−0.0869444	−0.00529126
$271$	−30.5140	−1.85359	−0.926796	−	0.375564i	$-0.877449\pi$
$271$	−30.5140	−1.85359	−0.926796	+	0.375564i	$0.877449\pi$
$272$	12.9936	0.787852
$273$	−6.73623	−0.407695
$274$	−9.80670	−0.592445
$275$	17.9411	1.08189
$276$	94.5392	5.69059
$277$	−15.2194	−0.914447	−0.457224	−	0.889352i	$-0.651156\pi$
$277$	−15.2194	−0.914447	−0.457224	+	0.889352i	$0.651156\pi$
$278$	50.1542	3.00805
$279$	6.06392	0.363038
$280$	−0.181837	−0.0108668
$281$	6.21715	0.370884	0.185442	−	0.982655i	$-0.440628\pi$
$281$	6.21715	0.370884	0.185442	+	0.982655i	$0.440628\pi$
$282$	−34.5227	−2.05580
$283$	−19.1104	−1.13600	−0.567999	−	0.823029i	$-0.692282\pi$
$283$	−19.1104	−1.13600	−0.567999	+	0.823029i	$0.692282\pi$
$284$	19.8783	1.17956
$285$	−0.0212895	−0.00126108
$286$	−45.1293	−2.66855
$287$	6.31932	0.373018
$288$	−85.5749	−5.04255
$289$	−16.3692	−0.962893
$290$	−0.303237	−0.0178067
$291$	−21.4629	−1.25818
$292$	−41.2416	−2.41348
$293$	13.0709	0.763611	0.381806	−	0.924243i	$-0.375302\pi$
$293$	13.0709	0.763611	0.381806	+	0.924243i	$0.375302\pi$
$294$	46.5029	2.71211
$295$	−0.217057	−0.0126376
$296$	−51.6797	−3.00382
$297$	−3.63314	−0.210816
$298$	40.0616	2.32070
$299$	−30.2777	−1.75100
$300$	−71.0960	−4.10473
$301$	−5.97763	−0.344545
$302$	−49.5306	−2.85017
$303$	47.4179	2.72409
$304$	−4.42535	−0.253811
$305$	−0.300705	−0.0172183
$306$	−7.45549	−0.426202
$307$	13.1708	0.751698	0.375849	−	0.926681i	$-0.377351\pi$
$307$	13.1708	0.751698	0.375849	+	0.926681i	$0.377351\pi$
$308$	−11.7945	−0.672053
$309$	3.34760	0.190438
$310$	−0.153172	−0.00869957
$311$	10.4112	0.590365	0.295182	−	0.955441i	$-0.404620\pi$
$311$	10.4112	0.590365	0.295182	+	0.955441i	$0.404620\pi$
$312$	115.212	6.52259
$313$	−2.94688	−0.166568	−0.0832838	−	0.996526i	$-0.526541\pi$
$313$	−2.94688	−0.166568	−0.0832838	+	0.996526i	$0.526541\pi$
$314$	−3.73817	−0.210957
$315$	0.0618369	0.00348412
$316$	−85.6216	−4.81659
$317$	−16.7500	−0.940776	−0.470388	−	0.882460i	$-0.655886\pi$
$317$	−16.7500	−0.940776	−0.470388	+	0.882460i	$0.655886\pi$
$318$	17.7434	0.995000
$319$	−12.6713	−0.709458
$320$	1.14368	0.0639338
$321$	−4.86385	−0.271473
$322$	−10.7282	−0.597861
$323$	−0.214845	−0.0119543
$324$	−42.9463	−2.38590
$325$	22.7697	1.26303
$326$	−20.6098	−1.14147
$327$	34.2093	1.89178
$328$	−108.081	−5.96780
$329$	2.88959	0.159308
$330$	0.779797	0.0429264
$331$	−2.18739	−0.120230	−0.0601149	−	0.998191i	$-0.519147\pi$
$331$	−2.18739	−0.120230	−0.0601149	+	0.998191i	$0.519147\pi$
$332$	30.6819	1.68389
$333$	17.5746	0.963084
$334$	15.4070	0.843033
$335$	0.410432	0.0224243
$336$	24.1948	1.31994
$337$	−5.32424	−0.290030	−0.145015	−	0.989429i	$-0.546323\pi$
$337$	−5.32424	−0.290030	−0.145015	+	0.989429i	$0.546323\pi$
$338$	−21.3855	−1.16322
$339$	−32.8972	−1.78673
$340$	0.138904	0.00753315
$341$	−6.40058	−0.346611
$342$	2.53919	0.137303
$343$	−7.98448	−0.431121
$344$	102.237	5.51227
$345$	0.523174	0.0281667
$346$	−4.15156	−0.223189
$347$	−15.9261	−0.854960	−0.427480	−	0.904025i	$-0.640599\pi$
$347$	−15.9261	−0.854960	−0.427480	+	0.904025i	$0.640599\pi$
$348$	50.2132	2.69171
$349$	−28.1539	−1.50704	−0.753522	−	0.657423i	$-0.771647\pi$
$349$	−28.1539	−1.50704	−0.753522	+	0.657423i	$0.771647\pi$
$350$	8.06793	0.431249
$351$	−4.61093	−0.246113
$352$	90.3257	4.81438
$353$	8.43996	0.449214	0.224607	−	0.974449i	$-0.427890\pi$
$353$	8.43996	0.449214	0.224607	+	0.974449i	$0.427890\pi$
$354$	48.7299	2.58996
$355$	0.110005	0.00583848
$356$	−53.7220	−2.84726
$357$	1.17463	0.0621678
$358$	34.0657	1.80043
$359$	−11.5124	−0.607601	−0.303800	−	0.952736i	$-0.598256\pi$
$359$	−11.5124	−0.607601	−0.303800	+	0.952736i	$0.598256\pi$
$360$	−1.05762	−0.0557413
$361$	−18.9268	−0.996149
$362$	−3.28315	−0.172558
$363$	4.75694	0.249674
$364$	−14.9688	−0.784577
$365$	−0.228228	−0.0119460
$366$	67.5090	3.52875
$367$	−10.1004	−0.527235	−0.263618	−	0.964627i	$-0.584916\pi$
$367$	−10.1004	−0.527235	−0.263618	+	0.964627i	$0.584916\pi$
$368$	108.750	5.66897
$369$	36.7551	1.91339
$370$	−0.443927	−0.0230786
$371$	−1.48514	−0.0771048
$372$	25.3638	1.31505
$373$	−26.3273	−1.36318	−0.681588	−	0.731736i	$-0.738710\pi$
$373$	−26.3273	−1.36318	−0.681588	+	0.731736i	$0.738710\pi$
$374$	7.86940	0.406917
$375$	−0.786958	−0.0406383
$376$	−49.4217	−2.54873
$377$	−16.0816	−0.828244
$378$	−1.63378	−0.0840327
$379$	31.9365	1.64047	0.820234	−	0.572029i	$-0.193843\pi$
$379$	31.9365	1.64047	0.820234	+	0.572029i	$0.193843\pi$
$380$	−0.0473080	−0.00242685
$381$	−24.4159	−1.25086
$382$	−23.3462	−1.19449
$383$	−29.7308	−1.51917	−0.759587	−	0.650405i	$-0.774599\pi$
$383$	−29.7308	−1.51917	−0.759587	+	0.650405i	$0.774599\pi$
$384$	−129.417	−6.60429
$385$	−0.0652699	−0.00332646
$386$	45.2463	2.30298
$387$	−34.7677	−1.76734
$388$	−47.6932	−2.42126
$389$	−17.2767	−0.875964	−0.437982	−	0.898984i	$-0.644307\pi$
$389$	−17.2767	−0.875964	−0.437982	+	0.898984i	$0.644307\pi$
$390$	0.989666	0.0501137
$391$	5.27966	0.267004
$392$	66.5721	3.36240
$393$	−14.6940	−0.741217
$394$	−49.1953	−2.47842
$395$	−0.473825	−0.0238407
$396$	−68.6003	−3.44729
$397$	24.3580	1.22249	0.611246	−	0.791440i	$-0.290668\pi$
$397$	24.3580	1.22249	0.611246	+	0.791440i	$0.290668\pi$
$398$	−27.7768	−1.39232
$399$	−0.400053	−0.0200277
$400$	−81.7828	−4.08914
$401$	19.2063	0.959116	0.479558	−	0.877510i	$-0.340797\pi$
$401$	19.2063	0.959116	0.479558	+	0.877510i	$0.340797\pi$
$402$	−92.1431	−4.59568
$403$	−8.12318	−0.404645
$404$	105.369	5.24229
$405$	−0.237662	−0.0118095
$406$	−5.69816	−0.282795
$407$	−18.5503	−0.919505
$408$	−20.0900	−0.994605
$409$	22.3094	1.10313	0.551565	−	0.834132i	$-0.314031\pi$
$409$	22.3094	1.10313	0.551565	+	0.834132i	$0.314031\pi$
$410$	−0.928415	−0.0458511
$411$	8.98656	0.443274
$412$	7.43879	0.366483
$413$	−4.07875	−0.200702
$414$	−62.3987	−3.06673
$415$	0.169792	0.00833475
$416$	114.635	5.62046
$417$	−45.9597	−2.25066
$418$	−2.68015	−0.131091
$419$	28.8963	1.41168	0.705839	−	0.708372i	$-0.250570\pi$
$419$	28.8963	1.41168	0.705839	+	0.708372i	$0.250570\pi$
$420$	0.258648	0.0126207
$421$	5.49326	0.267725	0.133863	−	0.991000i	$-0.457262\pi$
$421$	5.49326	0.267725	0.133863	+	0.991000i	$0.457262\pi$
$422$	60.3930	2.93988
$423$	16.8067	0.817172
$424$	25.4009	1.23358
$425$	−3.97045	−0.192595
$426$	−24.6965	−1.19655
$427$	−5.65058	−0.273451
$428$	−10.8081	−0.522429
$429$	41.3551	1.99664
$430$	0.878215	0.0423513
$431$	−18.8904	−0.909917	−0.454958	−	0.890513i	$-0.650346\pi$
$431$	−18.8904	−0.909917	−0.454958	+	0.890513i	$0.650346\pi$
$432$	16.5613	0.796805
$433$	−5.92335	−0.284658	−0.142329	−	0.989819i	$-0.545459\pi$
$433$	−5.92335	−0.284658	−0.142329	+	0.989819i	$0.545459\pi$
$434$	−2.87827	−0.138161
$435$	0.277877	0.0133232
$436$	76.0176	3.64058
$437$	−1.79814	−0.0860168
$438$	51.2379	2.44824
$439$	32.1973	1.53669	0.768346	−	0.640035i	$-0.221080\pi$
$439$	32.1973	1.53669	0.768346	+	0.640035i	$0.221080\pi$
$440$	1.11633	0.0532191
$441$	−22.6391	−1.07805
$442$	9.98731	0.475048
$443$	17.6415	0.838173	0.419087	−	0.907946i	$-0.362350\pi$
$443$	17.6415	0.838173	0.419087	+	0.907946i	$0.362350\pi$
$444$	73.5102	3.48864
$445$	−0.297294	−0.0140931
$446$	12.2324	0.579219
$447$	−36.7112	−1.73638
$448$	21.4911	1.01536
$449$	−12.8846	−0.608061	−0.304030	−	0.952662i	$-0.598332\pi$
$449$	−12.8846	−0.608061	−0.304030	+	0.952662i	$0.598332\pi$
$450$	46.9255	2.21209
$451$	−38.7956	−1.82681
$452$	−73.1017	−3.43842
$453$	45.3883	2.13253
$454$	14.9209	0.700270
$455$	−0.0828362	−0.00388342
$456$	6.84225	0.320418
$457$	34.0717	1.59381	0.796904	−	0.604106i	$-0.206469\pi$
$457$	34.0717	1.59381	0.796904	+	0.604106i	$0.206469\pi$
$458$	−1.98042	−0.0925391
$459$	0.804029	0.0375289
$460$	1.16256	0.0542046
$461$	−21.2202	−0.988324	−0.494162	−	0.869370i	$-0.664525\pi$
$461$	−21.2202	−0.988324	−0.494162	+	0.869370i	$0.664525\pi$
$462$	14.6533	0.681732
$463$	25.2505	1.17349	0.586745	−	0.809771i	$-0.300409\pi$
$463$	25.2505	1.17349	0.586745	+	0.809771i	$0.300409\pi$
$464$	57.7610	2.68148
$465$	0.140362	0.00650913
$466$	−10.9848	−0.508862
$467$	10.5010	0.485930	0.242965	−	0.970035i	$-0.421880\pi$
$467$	10.5010	0.485930	0.242965	+	0.970035i	$0.421880\pi$
$468$	−87.0629	−4.02448
$469$	7.71249	0.356130
$470$	−0.424530	−0.0195821
$471$	3.42554	0.157841
$472$	69.7602	3.21097
$473$	36.6979	1.68737
$474$	106.375	4.88596
$475$	1.35225	0.0620456
$476$	2.61017	0.119637
$477$	−8.63804	−0.395509
$478$	85.2158	3.89768
$479$	−37.4106	−1.70933	−0.854667	−	0.519177i	$-0.826239\pi$
$479$	−37.4106	−1.70933	−0.854667	+	0.519177i	$0.826239\pi$
$480$	−1.98080	−0.0904110
$481$	−23.5428	−1.07346
$482$	63.2438	2.88068
$483$	9.83103	0.447327
$484$	10.5705	0.480479
$485$	−0.263931	−0.0119845
$486$	61.7399	2.80058
$487$	26.9424	1.22088	0.610438	−	0.792064i	$-0.290994\pi$
$487$	26.9424	1.22088	0.610438	+	0.792064i	$0.290994\pi$
$488$	96.6438	4.37486
$489$	18.8862	0.854062
$490$	0.571852	0.0258336
$491$	15.8434	0.715004	0.357502	−	0.933912i	$-0.383628\pi$
$491$	15.8434	0.715004	0.357502	+	0.933912i	$0.383628\pi$
$492$	153.737	6.93100
$493$	2.80422	0.126296
$494$	−3.40147	−0.153039
$495$	−0.379629	−0.0170631
$496$	29.1764	1.31006
$497$	2.06713	0.0927233
$498$	−38.1187	−1.70814
$499$	−1.51851	−0.0679776	−0.0339888	−	0.999422i	$-0.510821\pi$
$499$	−1.51851	−0.0679776	−0.0339888	+	0.999422i	$0.510821\pi$
$500$	−1.74872	−0.0782052
$501$	−14.1185	−0.630767
$502$	−6.99047	−0.312000
$503$	−18.7723	−0.837014	−0.418507	−	0.908213i	$-0.637447\pi$
$503$	−18.7723	−0.837014	−0.418507	+	0.908213i	$0.637447\pi$
$504$	−19.8738	−0.885251
$505$	0.583104	0.0259478
$506$	65.8628	2.92796
$507$	19.5970	0.870335
$508$	−54.2553	−2.40719
$509$	−19.5349	−0.865871	−0.432936	−	0.901425i	$-0.642522\pi$
$509$	−19.5349	−0.865871	−0.432936	+	0.901425i	$0.642522\pi$
$510$	−0.172573	−0.00764164
$511$	−4.28867	−0.189720
$512$	−84.5977	−3.73873
$513$	−0.273836	−0.0120901
$514$	33.4873	1.47706
$515$	0.0411658	0.00181398
$516$	−145.424	−6.40195
$517$	−17.7398	−0.780196
$518$	−8.34188	−0.366521
$519$	3.80436	0.166993
$520$	1.41678	0.0621297
$521$	−13.1629	−0.576676	−0.288338	−	0.957529i	$-0.593103\pi$
$521$	−13.1629	−0.576676	−0.288338	+	0.957529i	$0.593103\pi$
$522$	−33.1422	−1.45059
$523$	−25.6688	−1.12242	−0.561209	−	0.827674i	$-0.689663\pi$
$523$	−25.6688	−1.12242	−0.561209	+	0.827674i	$0.689663\pi$
$524$	−32.6521	−1.42641
$525$	−7.39320	−0.322666
$526$	−36.2270	−1.57957
$527$	1.41648	0.0617026
$528$	−148.537	−6.46424
$529$	21.1880	0.921219
$530$	0.218193	0.00947768
$531$	−23.7232	−1.02950
$532$	−0.888970	−0.0385418
$533$	−49.2368	−2.13268
$534$	66.7433	2.88827
$535$	−0.0598113	−0.00258587
$536$	−131.909	−5.69761
$537$	−31.2167	−1.34710
$538$	57.4058	2.47494
$539$	23.8959	1.02927
$540$	0.177044	0.00761876
$541$	25.8362	1.11078	0.555392	−	0.831589i	$-0.312568\pi$
$541$	25.8362	1.11078	0.555392	+	0.831589i	$0.312568\pi$
$542$	84.2410	3.61846
$543$	3.00857	0.129110
$544$	−19.9895	−0.857042
$545$	0.420676	0.0180198
$546$	18.5969	0.795876
$547$	25.9791	1.11078	0.555392	−	0.831589i	$-0.312568\pi$
$547$	25.9791	1.11078	0.555392	+	0.831589i	$0.312568\pi$
$548$	19.9693	0.853046
$549$	−32.8655	−1.40266
$550$	−49.5306	−2.11199
$551$	−0.955059	−0.0406869
$552$	−168.143	−7.15665
$553$	−8.90370	−0.378624
$554$	42.0168	1.78512
$555$	0.406800	0.0172677
$556$	−102.128	−4.33121
$557$	−11.1225	−0.471274	−0.235637	−	0.971841i	$-0.575718\pi$
$557$	−11.1225	−0.471274	−0.235637	+	0.971841i	$0.575718\pi$
$558$	−16.7409	−0.708698
$559$	46.5745	1.96989
$560$	0.297526	0.0125728
$561$	−7.21127	−0.304460
$562$	−17.1639	−0.724016
$563$	8.61248	0.362973	0.181486	−	0.983393i	$-0.441909\pi$
$563$	8.61248	0.362973	0.181486	+	0.983393i	$0.441909\pi$
$564$	70.2983	2.96009
$565$	−0.404540	−0.0170191
$566$	52.7589	2.21762
$567$	−4.46593	−0.187552
$568$	−35.3547	−1.48345
$569$	6.29877	0.264058	0.132029	−	0.991246i	$-0.457851\pi$
$569$	6.29877	0.264058	0.132029	+	0.991246i	$0.457851\pi$
$570$	0.0587746	0.00246180
$571$	25.3882	1.06246	0.531231	−	0.847227i	$-0.321730\pi$
$571$	25.3882	1.06246	0.531231	+	0.847227i	$0.321730\pi$
$572$	91.8964	3.84238
$573$	21.3937	0.893735
$574$	−17.4460	−0.728181
$575$	−33.2306	−1.38581
$576$	124.999	5.20827
$577$	15.2648	0.635482	0.317741	−	0.948178i	$-0.397076\pi$
$577$	15.2648	0.635482	0.317741	+	0.948178i	$0.397076\pi$
$578$	45.1910	1.87970
$579$	−41.4623	−1.72312
$580$	0.617478	0.0256394
$581$	3.19058	0.132368
$582$	59.2533	2.45613
$583$	9.11760	0.377612
$584$	73.3505	3.03527
$585$	−0.481800	−0.0199200
$586$	−36.0853	−1.49067
$587$	−22.3978	−0.924455	−0.462228	−	0.886761i	$-0.652950\pi$
$587$	−22.3978	−0.924455	−0.462228	+	0.886761i	$0.652950\pi$
$588$	−94.6934	−3.90509
$589$	−0.482422	−0.0198779
$590$	0.599237	0.0246702
$591$	45.0810	1.85438
$592$	84.5598	3.47539
$593$	−7.76673	−0.318941	−0.159471	−	0.987203i	$-0.550979\pi$
$593$	−7.76673	−0.318941	−0.159471	+	0.987203i	$0.550979\pi$
$594$	10.0301	0.411541
$595$	0.0144445	0.000592168 0
$596$	−81.5770	−3.34152
$597$	25.4538	1.04175
$598$	83.5887	3.41820
$599$	−28.2634	−1.15481	−0.577406	−	0.816457i	$-0.695935\pi$
$599$	−28.2634	−1.15481	−0.577406	+	0.816457i	$0.695935\pi$
$600$	126.448	5.16223
$601$	27.9528	1.14022	0.570110	−	0.821569i	$-0.306901\pi$
$601$	27.9528	1.14022	0.570110	+	0.821569i	$0.306901\pi$
$602$	16.5027	0.672598
$603$	44.8581	1.82676
$604$	100.859	4.10388
$605$	0.0584966	0.00237823
$606$	−130.908	−5.31779
$607$	27.5948	1.12004	0.560018	−	0.828480i	$-0.310794\pi$
$607$	27.5948	1.12004	0.560018	+	0.828480i	$0.310794\pi$
$608$	6.80800	0.276101
$609$	5.22162	0.211591
$610$	0.830166	0.0336124
$611$	−22.5142	−0.910826
$612$	15.1815	0.613677
$613$	−0.705984	−0.0285144	−0.0142572	−	0.999898i	$-0.504538\pi$
$613$	−0.705984	−0.0285144	−0.0142572	+	0.999898i	$0.504538\pi$
$614$	−36.3611	−1.46742
$615$	0.850771	0.0343064
$616$	20.9772	0.845194
$617$	7.68456	0.309369	0.154684	−	0.987964i	$-0.450564\pi$
$617$	7.68456	0.309369	0.154684	+	0.987964i	$0.450564\pi$
$618$	−9.24182	−0.371761
$619$	42.8453	1.72210	0.861049	−	0.508522i	$-0.169808\pi$
$619$	42.8453	1.72210	0.861049	+	0.508522i	$0.169808\pi$
$620$	0.311902	0.0125263
$621$	6.72931	0.270038
$622$	−28.7426	−1.15247
$623$	−5.58649	−0.223818
$624$	−188.513	−7.54656
$625$	24.9855	0.999419
$626$	8.13556	0.325162
$627$	2.45601	0.0980836
$628$	7.61199	0.303752
$629$	4.10527	0.163688
$630$	−0.170715	−0.00680146
$631$	−38.9298	−1.54977	−0.774885	−	0.632102i	$-0.782193\pi$
$631$	−38.9298	−1.54977	−0.774885	+	0.632102i	$0.782193\pi$
$632$	152.283	6.05749
$633$	−55.3422	−2.19966
$634$	46.2424	1.83652
$635$	−0.300245	−0.0119149
$636$	−36.1307	−1.43268
$637$	30.3271	1.20160
$638$	34.9822	1.38496
$639$	12.0230	0.475624
$640$	−1.59146	−0.0629079
$641$	20.5400	0.811282	0.405641	−	0.914032i	$-0.367048\pi$
$641$	20.5400	0.811282	0.405641	+	0.914032i	$0.367048\pi$
$642$	13.4278	0.529953
$643$	−10.6676	−0.420688	−0.210344	−	0.977627i	$-0.567458\pi$
$643$	−10.6676	−0.420688	−0.210344	+	0.977627i	$0.567458\pi$
$644$	21.8458	0.860845
$645$	−0.804769	−0.0316878
$646$	0.593130	0.0233364
$647$	41.6687	1.63817	0.819083	−	0.573675i	$-0.194483\pi$
$647$	41.6687	1.63817	0.819083	+	0.573675i	$0.194483\pi$
$648$	76.3823	3.00058
$649$	25.0403	0.982916
$650$	−62.8610	−2.46561
$651$	2.63756	0.103374
$652$	41.9675	1.64357
$653$	29.2092	1.14304	0.571522	−	0.820587i	$-0.306353\pi$
$653$	29.2092	1.14304	0.571522	+	0.820587i	$0.306353\pi$
$654$	−94.4429	−3.69301
$655$	−0.180694	−0.00706032
$656$	176.846	6.90467
$657$	−24.9442	−0.973165
$658$	−7.97740	−0.310991
$659$	30.0866	1.17201	0.586003	−	0.810309i	$-0.300701\pi$
$659$	30.0866	1.17201	0.586003	+	0.810309i	$0.300701\pi$
$660$	−1.58789	−0.0618087
$661$	−39.3272	−1.52965	−0.764826	−	0.644237i	$-0.777175\pi$
$661$	−39.3272	−1.52965	−0.764826	+	0.644237i	$0.777175\pi$
$662$	6.03880	0.234705
$663$	−9.15206	−0.355437
$664$	−54.5696	−2.11771
$665$	−0.00491950	−0.000190770 0
$666$	−48.5189	−1.88007
$667$	23.4699	0.908758
$668$	−31.3731	−1.21386
$669$	−11.2093	−0.433378
$670$	−1.13309	−0.0437753
$671$	34.6901	1.33920
$672$	−37.2216	−1.43585
$673$	−27.5800	−1.06313	−0.531565	−	0.847017i	$-0.678396\pi$
$673$	−27.5800	−1.06313	−0.531565	+	0.847017i	$0.678396\pi$
$674$	14.6988	0.566177
$675$	−5.06063	−0.194784
$676$	43.5471	1.67489
$677$	−48.7735	−1.87452	−0.937259	−	0.348634i	$-0.886646\pi$
$677$	−48.7735	−1.87452	−0.937259	+	0.348634i	$0.886646\pi$
$678$	90.8203	3.48793
$679$	−4.95957	−0.190331
$680$	−0.247049	−0.00947391
$681$	−13.6730	−0.523951
$682$	17.6703	0.676630
$683$	24.0058	0.918555	0.459277	−	0.888293i	$-0.348108\pi$
$683$	24.0058	0.918555	0.459277	+	0.888293i	$0.348108\pi$
$684$	−5.17052	−0.197700
$685$	0.110509	0.00422232
$686$	22.0430	0.841607
$687$	1.81480	0.0692389
$688$	−167.284	−6.37763
$689$	11.5714	0.440837
$690$	−1.44434	−0.0549852
$691$	−10.7625	−0.409425	−0.204713	−	0.978822i	$-0.565626\pi$
$691$	−10.7625	−0.409425	−0.204713	+	0.978822i	$0.565626\pi$
$692$	8.45378	0.321365
$693$	−7.13367	−0.270986
$694$	43.9678	1.66900
$695$	−0.565172	−0.0214382
$696$	−89.3070	−3.38517
$697$	8.58563	0.325204
$698$	77.7255	2.94195
$699$	10.0661	0.380737
$700$	−16.4286	−0.620944
$701$	−7.17135	−0.270858	−0.135429	−	0.990787i	$-0.543241\pi$
$701$	−7.17135	−0.270858	−0.135429	+	0.990787i	$0.543241\pi$
$702$	12.7296	0.480446
$703$	−1.39817	−0.0527329
$704$	−131.938	−4.97261
$705$	0.389026	0.0146516
$706$	−23.3005	−0.876925
$707$	10.9572	0.412087
$708$	−99.2282	−3.72922
$709$	4.88932	0.183622	0.0918111	−	0.995776i	$-0.470734\pi$
$709$	4.88932	0.183622	0.0918111	+	0.995776i	$0.470734\pi$
$710$	−0.303696	−0.0113975
$711$	−51.7866	−1.94215
$712$	95.5477	3.58080
$713$	11.8552	0.443980
$714$	−3.24283	−0.121360
$715$	0.508548	0.0190186
$716$	−69.3676	−2.59239
$717$	−78.0891	−2.91629
$718$	31.7827	1.18612
$719$	−3.04245	−0.113464	−0.0567322	−	0.998389i	$-0.518068\pi$
$719$	−3.04245	−0.113464	−0.0567322	+	0.998389i	$0.518068\pi$
$720$	1.73050	0.0644920
$721$	0.773551	0.0288086
$722$	52.2520	1.94462
$723$	−57.9547	−2.15536
$724$	6.68544	0.248462
$725$	−17.6500	−0.655504
$726$	−13.1326	−0.487398
$727$	42.0945	1.56120	0.780599	−	0.625032i	$-0.214914\pi$
$727$	42.0945	1.56120	0.780599	+	0.625032i	$0.214914\pi$
$728$	26.6228	0.986707
$729$	−33.6583	−1.24660
$730$	0.630078	0.0233202
$731$	−8.12141	−0.300381
$732$	−137.468	−5.08096
$733$	−24.2768	−0.896685	−0.448343	−	0.893862i	$-0.647986\pi$
$733$	−24.2768	−0.896685	−0.448343	+	0.893862i	$0.647986\pi$
$734$	27.8845	1.02923
$735$	−0.524027	−0.0193290
$736$	−167.302	−6.16683
$737$	−47.3485	−1.74411
$738$	−101.471	−3.73520
$739$	9.24622	0.340128	0.170064	−	0.985433i	$-0.445603\pi$
$739$	9.24622	0.340128	0.170064	+	0.985433i	$0.445603\pi$
$740$	0.903963	0.0332303
$741$	3.11700	0.114506
$742$	4.10009	0.150519
$743$	−45.1249	−1.65547	−0.827736	−	0.561118i	$-0.810371\pi$
$743$	−45.1249	−1.65547	−0.827736	+	0.561118i	$0.810371\pi$
$744$	−45.1110	−1.65385
$745$	−0.451442	−0.0165395
$746$	72.6827	2.66110
$747$	18.5574	0.678979
$748$	−16.0244	−0.585909
$749$	−1.12392	−0.0410672
$750$	2.17258	0.0793315
$751$	−54.2564	−1.97984	−0.989922	−	0.141615i	$-0.954771\pi$
$751$	−54.2564	−1.97984	−0.989922	+	0.141615i	$0.954771\pi$
$752$	80.8651	2.94885
$753$	6.40585	0.233442
$754$	44.3970	1.61684
$755$	0.558145	0.0203130
$756$	3.32686	0.120997
$757$	−40.1288	−1.45851	−0.729254	−	0.684243i	$-0.760133\pi$
$757$	−40.1288	−1.45851	−0.729254	+	0.684243i	$0.760133\pi$
$758$	−88.1682	−3.20241
$759$	−60.3547	−2.19074
$760$	0.0841399	0.00305208
$761$	33.4516	1.21262	0.606309	−	0.795229i	$-0.292649\pi$
$761$	33.4516	1.21262	0.606309	+	0.795229i	$0.292649\pi$
$762$	67.4058	2.44186
$763$	7.90498	0.286180
$764$	47.5396	1.71992
$765$	0.0840137	0.00303752
$766$	82.0790	2.96563
$767$	31.7794	1.14749
$768$	171.278	6.18047
$769$	−24.1579	−0.871154	−0.435577	−	0.900151i	$-0.643456\pi$
$769$	−24.1579	−0.871154	−0.435577	+	0.900151i	$0.643456\pi$
$770$	0.180193	0.00649370
$771$	−30.6867	−1.10516
$772$	−92.1346	−3.31600
$773$	47.0249	1.69137	0.845684	−	0.533685i	$-0.179193\pi$
$773$	47.0249	1.69137	0.845684	+	0.533685i	$0.179193\pi$
$774$	95.9844	3.45009
$775$	−8.91542	−0.320251
$776$	84.8251	3.04505
$777$	7.64424	0.274236
$778$	47.6964	1.71000
$779$	−2.92409	−0.104766
$780$	−2.01525	−0.0721574
$781$	−12.6905	−0.454102
$782$	−14.5757	−0.521227
$783$	3.57418	0.127731
$784$	−108.927	−3.89026
$785$	0.0421243	0.00150348
$786$	40.5664	1.44695
$787$	−27.2227	−0.970386	−0.485193	−	0.874407i	$-0.661251\pi$
$787$	−27.2227	−0.970386	−0.485193	+	0.874407i	$0.661251\pi$
$788$	100.176	3.56862
$789$	33.1973	1.18186
$790$	1.30810	0.0465402
$791$	−7.60177	−0.270288
$792$	122.009	4.33542
$793$	44.0263	1.56342
$794$	−67.2460	−2.38647
$795$	−0.199945	−0.00709132
$796$	56.5616	2.00477
$797$	20.5513	0.727965	0.363983	−	0.931406i	$-0.381417\pi$
$797$	20.5513	0.727965	0.363983	+	0.931406i	$0.381417\pi$
$798$	1.10444	0.0390968
$799$	3.92590	0.138888
$800$	125.815	4.44825
$801$	−32.4927	−1.14807
$802$	−53.0235	−1.87232
$803$	26.3290	0.929131
$804$	187.630	6.61720
$805$	0.120893	0.00426093
$806$	22.4260	0.789920
$807$	−52.6049	−1.85178
$808$	−187.404	−6.59286
$809$	−25.1741	−0.885073	−0.442537	−	0.896750i	$-0.645921\pi$
$809$	−25.1741	−0.885073	−0.442537	+	0.896750i	$0.645921\pi$
$810$	0.656121	0.0230537
$811$	−53.6582	−1.88419	−0.942097	−	0.335340i	$-0.891149\pi$
$811$	−53.6582	−1.88419	−0.942097	+	0.335340i	$0.891149\pi$
$812$	11.6031	0.407189
$813$	−77.1959	−2.70738
$814$	51.2125	1.79500
$815$	0.232245	0.00813520
$816$	32.8719	1.15075
$817$	2.76598	0.0967695
$818$	−61.5904	−2.15346
$819$	−9.05357	−0.316357
$820$	1.89052	0.0660199
$821$	12.1552	0.424219	0.212109	−	0.977246i	$-0.431967\pi$
$821$	12.1552	0.424219	0.212109	+	0.977246i	$0.431967\pi$
$822$	−24.8095	−0.865331
$823$	36.9874	1.28930	0.644650	−	0.764478i	$-0.277003\pi$
$823$	36.9874	1.28930	0.644650	+	0.764478i	$0.277003\pi$
$824$	−13.2303	−0.460900
$825$	45.3883	1.58022
$826$	11.2603	0.391797
$827$	−21.7715	−0.757069	−0.378534	−	0.925587i	$-0.623572\pi$
$827$	−21.7715	−0.757069	−0.378534	+	0.925587i	$0.623572\pi$
$828$	127.062	4.41570
$829$	37.0897	1.28818	0.644089	−	0.764950i	$-0.277236\pi$
$829$	37.0897	1.28818	0.644089	+	0.764950i	$0.277236\pi$
$830$	−0.468750	−0.0162706
$831$	−38.5029	−1.33565
$832$	−167.447	−5.80518
$833$	−5.28827	−0.183228
$834$	126.883	4.39359
$835$	−0.173617	−0.00600825
$836$	5.45757	0.188754
$837$	1.80540	0.0624038
$838$	−79.7752	−2.75579
$839$	39.0040	1.34657	0.673285	−	0.739383i	$-0.264883\pi$
$839$	39.0040	1.34657	0.673285	+	0.739383i	$0.264883\pi$
$840$	−0.460020	−0.0158722
$841$	−16.5343	−0.570148
$842$	−15.1654	−0.522635
$843$	15.7285	0.541718
$844$	−122.978	−4.23306
$845$	0.240987	0.00829021
$846$	−46.3990	−1.59523
$847$	1.09922	0.0377696
$848$	−41.5617	−1.42723
$849$	−48.3466	−1.65925
$850$	10.9614	0.375971
$851$	34.3590	1.17781
$852$	50.2892	1.72288
$853$	46.1420	1.57987	0.789936	−	0.613189i	$-0.210114\pi$
$853$	46.1420	1.57987	0.789936	+	0.613189i	$0.210114\pi$
$854$	15.5998	0.533813
$855$	−0.0286133	−0.000978555 0
$856$	19.2228	0.657022
$857$	26.2997	0.898380	0.449190	−	0.893436i	$-0.351713\pi$
$857$	26.2997	0.898380	0.449190	+	0.893436i	$0.351713\pi$
$858$	−114.170	−3.89771
$859$	−8.87209	−0.302712	−0.151356	−	0.988479i	$-0.548364\pi$
$859$	−8.87209	−0.302712	−0.151356	+	0.988479i	$0.548364\pi$
$860$	−1.78830	−0.0609806
$861$	15.9869	0.544834
$862$	52.1513	1.77628
$863$	−29.5893	−1.00723	−0.503617	−	0.863927i	$-0.667998\pi$
$863$	−29.5893	−1.00723	−0.503617	+	0.863927i	$0.667998\pi$
$864$	−25.4781	−0.866781
$865$	0.0467827	0.00159066
$866$	16.3528	0.555691
$867$	−41.4116	−1.40641
$868$	5.86100	0.198935
$869$	54.6616	1.85427
$870$	−0.767144	−0.0260086
$871$	−60.0916	−2.03613
$872$	−135.202	−4.57850
$873$	−28.8463	−0.976301
$874$	4.96419	0.167916
$875$	−0.181848	−0.00614757
$876$	−104.335	−3.52516
$877$	39.8662	1.34619	0.673094	−	0.739557i	$-0.264965\pi$
$877$	39.8662	1.34619	0.673094	+	0.739557i	$0.264965\pi$
$878$	−88.8881	−2.99983
$879$	33.0675	1.11534
$880$	−1.82658	−0.0615738
$881$	−6.25572	−0.210761	−0.105380	−	0.994432i	$-0.533606\pi$
$881$	−6.25572	−0.210761	−0.105380	+	0.994432i	$0.533606\pi$
$882$	62.5005	2.10450
$883$	−47.1700	−1.58740	−0.793699	−	0.608311i	$-0.791847\pi$
$883$	−47.1700	−1.58740	−0.793699	+	0.608311i	$0.791847\pi$
$884$	−20.3371	−0.684010
$885$	−0.549122	−0.0184585
$886$	−48.7035	−1.63623
$887$	−8.00801	−0.268883	−0.134441	−	0.990922i	$-0.542924\pi$
$887$	−8.00801	−0.268883	−0.134441	+	0.990922i	$0.542924\pi$
$888$	−130.742	−4.38741
$889$	−5.64195	−0.189225
$890$	0.820750	0.0275116
$891$	27.4173	0.918513
$892$	−24.9086	−0.834002
$893$	−1.33708	−0.0447436
$894$	101.350	3.38965
$895$	−0.383875	−0.0128315
$896$	−29.9053	−0.999066
$897$	−76.5981	−2.55754
$898$	35.5709	1.18702
$899$	6.29672	0.210007
$900$	−95.5539	−3.18513
$901$	−2.01776	−0.0672215
$902$	107.104	3.56618
$903$	−15.1225	−0.503246
$904$	130.016	4.32425
$905$	0.0369968	0.00122981
$906$	−125.305	−4.16298
$907$	40.7718	1.35381	0.676903	−	0.736073i	$-0.263322\pi$
$907$	40.7718	1.35381	0.676903	+	0.736073i	$0.263322\pi$
$908$	−30.3832	−1.00830
$909$	63.7302	2.11380
$910$	0.228689	0.00758096
$911$	35.0600	1.16159	0.580795	−	0.814050i	$-0.302742\pi$
$911$	35.0600	1.16159	0.580795	+	0.814050i	$0.302742\pi$
$912$	−11.1955	−0.370719
$913$	−19.5876	−0.648256
$914$	−94.0630	−3.11133
$915$	−0.760738	−0.0251492
$916$	4.03272	0.133245
$917$	−3.39545	−0.112128
$918$	−2.21971	−0.0732614
$919$	−55.4624	−1.82954	−0.914768	−	0.403979i	$-0.867627\pi$
$919$	−55.4624	−1.82954	−0.914768	+	0.403979i	$0.867627\pi$
$920$	−2.06768	−0.0681693
$921$	33.3202	1.09794
$922$	58.5834	1.92934
$923$	−16.1059	−0.530134
$924$	−29.8383	−0.981608
$925$	−25.8389	−0.849578
$926$	−69.7100	−2.29081
$927$	4.49921	0.147773
$928$	−88.8601	−2.91698
$929$	−16.9560	−0.556308	−0.278154	−	0.960537i	$-0.589723\pi$
$929$	−16.9560	−0.556308	−0.278154	+	0.960537i	$0.589723\pi$
$930$	−0.387502	−0.0127067
$931$	1.80108	0.0590279
$932$	22.3683	0.732698
$933$	26.3388	0.862293
$934$	−28.9906	−0.948600
$935$	−0.0886779	−0.00290008
$936$	154.846	5.06131
$937$	8.05148	0.263031	0.131515	−	0.991314i	$-0.458016\pi$
$937$	8.05148	0.263031	0.131515	+	0.991314i	$0.458016\pi$
$938$	−21.2921	−0.695213
$939$	−7.45517	−0.243290
$940$	0.864467	0.0281958
$941$	−8.33597	−0.271745	−0.135872	−	0.990726i	$-0.543384\pi$
$941$	−8.33597	−0.271745	−0.135872	+	0.990726i	$0.543384\pi$
$942$	−9.45701	−0.308126
$943$	71.8574	2.34000
$944$	−114.144	−3.71506
$945$	0.0184106	0.000598897 0
$946$	−101.313	−3.29398
$947$	−24.5444	−0.797587	−0.398794	−	0.917041i	$-0.630571\pi$
$947$	−24.5444	−0.797587	−0.398794	+	0.917041i	$0.630571\pi$
$948$	−216.610	−7.03517
$949$	33.4150	1.08470
$950$	−3.73321	−0.121121
$951$	−42.3751	−1.37411
$952$	−4.64234	−0.150459
$953$	27.0376	0.875832	0.437916	−	0.899016i	$-0.355717\pi$
$953$	27.0376	0.875832	0.437916	+	0.899016i	$0.355717\pi$
$954$	23.8473	0.772086
$955$	0.263081	0.00851310
$956$	−173.524	−5.61217
$957$	−32.0566	−1.03624
$958$	103.281	3.33685
$959$	2.07658	0.0670564
$960$	2.89335	0.0933824
$961$	−27.8194	−0.897399
$962$	64.9955	2.09554
$963$	−6.53707	−0.210654
$964$	−128.783	−4.14781
$965$	−0.509867	−0.0164132
$966$	−27.1409	−0.873243
$967$	16.3509	0.525808	0.262904	−	0.964822i	$-0.415320\pi$
$967$	16.3509	0.525808	0.262904	+	0.964822i	$0.415320\pi$
$968$	−18.8003	−0.604264
$969$	−0.543526	−0.0174606
$970$	0.728644	0.0233954
$971$	−49.0632	−1.57451	−0.787256	−	0.616626i	$-0.788499\pi$
$971$	−49.0632	−1.57451	−0.787256	+	0.616626i	$0.788499\pi$
$972$	−125.720	−4.03248
$973$	−10.6202	−0.340469
$974$	−74.3807	−2.38331
$975$	57.6039	1.84480
$976$	−158.131	−5.06166
$977$	−32.8155	−1.04986	−0.524931	−	0.851145i	$-0.675909\pi$
$977$	−32.8155	−1.04986	−0.524931	+	0.851145i	$0.675909\pi$
$978$	−52.1397	−1.66724
$979$	34.2966	1.09612
$980$	−1.16446	−0.0371972
$981$	45.9778	1.46796
$982$	−43.7395	−1.39578
$983$	15.1453	0.483059	0.241529	−	0.970394i	$-0.422351\pi$
$983$	15.1453	0.483059	0.241529	+	0.970394i	$0.422351\pi$
$984$	−273.430	−8.71663
$985$	0.554366	0.0176636
$986$	−7.74171	−0.246546
$987$	7.31024	0.232688
$988$	6.92638	0.220358
$989$	−67.9721	−2.16139
$990$	1.04806	0.0333094
$991$	13.5435	0.430223	0.215112	−	0.976589i	$-0.430988\pi$
$991$	13.5435	0.430223	0.215112	+	0.976589i	$0.430988\pi$
$992$	−44.8853	−1.42511
$993$	−5.53377	−0.175609
$994$	−5.70679	−0.181008
$995$	0.313008	0.00992302
$996$	77.6207	2.45951
$997$	−45.3568	−1.43646	−0.718232	−	0.695804i	$-0.755048\pi$
$997$	−45.3568	−1.43646	−0.718232	+	0.695804i	$0.755048\pi$
$998$	4.19219	0.132701
$999$	5.23246	0.165548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.a.1.2	✓	306

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.a.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-2.76073 q^{2} +2.52985 q^{3} +5.62166 q^{4} +0.0311099 q^{5} -6.98425 q^{6} +0.584590 q^{7} -9.99844 q^{8} +3.40015 q^{9} +O(q^{10})\)	\(q-2.76073 q^{2} +2.52985 q^{3} +5.62166 q^{4} +0.0311099 q^{5} -6.98425 q^{6} +0.584590 q^{7} -9.99844 q^{8} +3.40015 q^{9} -0.0858861 q^{10} -3.58892 q^{11} +14.2220 q^{12} -4.55481 q^{13} -1.61390 q^{14} +0.0787034 q^{15} +16.3597 q^{16} +0.794243 q^{17} -9.38691 q^{18} -0.270503 q^{19} +0.174889 q^{20} +1.47893 q^{21} +9.90805 q^{22} +6.64741 q^{23} -25.2946 q^{24} -4.99903 q^{25} +12.5746 q^{26} +1.01232 q^{27} +3.28636 q^{28} +3.53068 q^{29} -0.217279 q^{30} +1.78343 q^{31} -25.1680 q^{32} -9.07943 q^{33} -2.19269 q^{34} +0.0181865 q^{35} +19.1145 q^{36} +5.16878 q^{37} +0.746786 q^{38} -11.5230 q^{39} -0.311050 q^{40} +10.8098 q^{41} -4.08292 q^{42} -10.2253 q^{43} -20.1757 q^{44} +0.105778 q^{45} -18.3517 q^{46} +4.94294 q^{47} +41.3877 q^{48} -6.65825 q^{49} +13.8010 q^{50} +2.00932 q^{51} -25.6056 q^{52} -2.54049 q^{53} -2.79475 q^{54} -0.111651 q^{55} -5.84499 q^{56} -0.684332 q^{57} -9.74728 q^{58} -6.97711 q^{59} +0.442444 q^{60} -9.66589 q^{61} -4.92357 q^{62} +1.98769 q^{63} +36.7627 q^{64} -0.141700 q^{65} +25.0659 q^{66} +13.1930 q^{67} +4.46496 q^{68} +16.8170 q^{69} -0.0502082 q^{70} +3.53603 q^{71} -33.9962 q^{72} -7.33620 q^{73} -14.2696 q^{74} -12.6468 q^{75} -1.52067 q^{76} -2.09804 q^{77} +31.8120 q^{78} -15.2307 q^{79} +0.508949 q^{80} -7.63943 q^{81} -29.8431 q^{82} +5.45781 q^{83} +8.31402 q^{84} +0.0247088 q^{85} +28.2295 q^{86} +8.93210 q^{87} +35.8836 q^{88} -9.55626 q^{89} -0.292026 q^{90} -2.66270 q^{91} +37.3695 q^{92} +4.51181 q^{93} -13.6461 q^{94} -0.00841531 q^{95} -63.6712 q^{96} -8.48384 q^{97} +18.3817 q^{98} -12.2029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) \) 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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