LMFDB - Embedded newform 8009.2.a.b.1.6

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:	$0$
Dimension:	$361$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.76744 q^{2} +2.44821 q^{3} +5.65873 q^{4} -3.94684 q^{5} -6.77527 q^{6} -1.82730 q^{7} -10.1253 q^{8} +2.99372 q^{9} +O(q^{10})$	\(q-2.76744 q^{2} +2.44821 q^{3} +5.65873 q^{4} -3.94684 q^{5} -6.77527 q^{6} -1.82730 q^{7} -10.1253 q^{8} +2.99372 q^{9} +10.9226 q^{10} -3.61880 q^{11} +13.8537 q^{12} -1.49651 q^{13} +5.05695 q^{14} -9.66267 q^{15} +16.7038 q^{16} -4.03926 q^{17} -8.28494 q^{18} -0.0611975 q^{19} -22.3341 q^{20} -4.47361 q^{21} +10.0148 q^{22} -3.25870 q^{23} -24.7889 q^{24} +10.5775 q^{25} +4.14151 q^{26} -0.0153816 q^{27} -10.3402 q^{28} -0.751461 q^{29} +26.7409 q^{30} -5.67932 q^{31} -25.9761 q^{32} -8.85956 q^{33} +11.1784 q^{34} +7.21206 q^{35} +16.9406 q^{36} -1.59940 q^{37} +0.169360 q^{38} -3.66377 q^{39} +39.9630 q^{40} -11.4590 q^{41} +12.3805 q^{42} -7.11990 q^{43} -20.4778 q^{44} -11.8157 q^{45} +9.01825 q^{46} -10.9162 q^{47} +40.8943 q^{48} -3.66097 q^{49} -29.2726 q^{50} -9.88894 q^{51} -8.46837 q^{52} +0.857981 q^{53} +0.0425675 q^{54} +14.2828 q^{55} +18.5020 q^{56} -0.149824 q^{57} +2.07962 q^{58} +10.0274 q^{59} -54.6785 q^{60} +13.2736 q^{61} +15.7172 q^{62} -5.47042 q^{63} +38.4797 q^{64} +5.90649 q^{65} +24.5183 q^{66} -10.6245 q^{67} -22.8571 q^{68} -7.97796 q^{69} -19.9589 q^{70} -5.17815 q^{71} -30.3124 q^{72} -12.0980 q^{73} +4.42626 q^{74} +25.8959 q^{75} -0.346300 q^{76} +6.61263 q^{77} +10.1393 q^{78} -1.42884 q^{79} -65.9270 q^{80} -9.01881 q^{81} +31.7121 q^{82} +2.55106 q^{83} -25.3150 q^{84} +15.9423 q^{85} +19.7039 q^{86} -1.83973 q^{87} +36.6415 q^{88} +2.25876 q^{89} +32.6993 q^{90} +2.73458 q^{91} -18.4401 q^{92} -13.9041 q^{93} +30.2100 q^{94} +0.241536 q^{95} -63.5948 q^{96} +4.81279 q^{97} +10.1315 q^{98} -10.8337 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) $ 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * 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 $

$2$	−2.76744	−1.95688	−0.978438	−	0.206540i	$-0.933780\pi$
$2$	−2.76744	−1.95688	−0.978438	+	0.206540i	$0.933780\pi$
$3$	2.44821	1.41347	0.706736	−	0.707477i	$-0.250167\pi$
$3$	2.44821	1.41347	0.706736	+	0.707477i	$0.250167\pi$
$4$	5.65873	2.82937
$5$	−3.94684	−1.76508	−0.882539	−	0.470239i	$-0.844168\pi$
$5$	−3.94684	−1.76508	−0.882539	+	0.470239i	$0.844168\pi$
$6$	−6.77527	−2.76599
$7$	−1.82730	−0.690655	−0.345327	−	0.938482i	$-0.612232\pi$
$7$	−1.82730	−0.690655	−0.345327	+	0.938482i	$0.612232\pi$
$8$	−10.1253	−3.57984
$9$	2.99372	0.997906
$10$	10.9226	3.45404
$11$	−3.61880	−1.09111	−0.545554	−	0.838076i	$-0.683681\pi$
$11$	−3.61880	−1.09111	−0.545554	+	0.838076i	$0.683681\pi$
$12$	13.8537	3.99923
$13$	−1.49651	−0.415058	−0.207529	−	0.978229i	$-0.566542\pi$
$13$	−1.49651	−0.415058	−0.207529	+	0.978229i	$0.566542\pi$
$14$	5.05695	1.35153
$15$	−9.66267	−2.49489
$16$	16.7038	4.17594
$17$	−4.03926	−0.979664	−0.489832	−	0.871817i	$-0.662942\pi$
$17$	−4.03926	−0.979664	−0.489832	+	0.871817i	$0.662942\pi$
$18$	−8.28494	−1.95278
$19$	−0.0611975	−0.0140397	−0.00701983	−	0.999975i	$-0.502234\pi$
$19$	−0.0611975	−0.0140397	−0.00701983	+	0.999975i	$0.502234\pi$
$20$	−22.3341	−4.99405
$21$	−4.47361	−0.976222
$22$	10.0148	2.13516
$23$	−3.25870	−0.679485	−0.339742	−	0.940519i	$-0.610340\pi$
$23$	−3.25870	−0.679485	−0.339742	+	0.940519i	$0.610340\pi$
$24$	−24.7889	−5.06001
$25$	10.5775	2.11550
$26$	4.14151	0.812217
$27$	−0.0153816	−0.00296018
$28$	−10.3402	−1.95412
$29$	−0.751461	−0.139543	−0.0697714	−	0.997563i	$-0.522227\pi$
$29$	−0.751461	−0.139543	−0.0697714	+	0.997563i	$0.522227\pi$
$30$	26.7409	4.88219
$31$	−5.67932	−1.02004	−0.510018	−	0.860164i	$-0.670361\pi$
$31$	−5.67932	−1.02004	−0.510018	+	0.860164i	$0.670361\pi$
$32$	−25.9761	−4.59196
$33$	−8.85956	−1.54225
$34$	11.1784	1.91708
$35$	7.21206	1.21906
$36$	16.9406	2.82344
$37$	−1.59940	−0.262940	−0.131470	−	0.991320i	$-0.541970\pi$
$37$	−1.59940	−0.262940	−0.131470	+	0.991320i	$0.541970\pi$
$38$	0.169360	0.0274739
$39$	−3.66377	−0.586673
$40$	39.9630	6.31870
$41$	−11.4590	−1.78959	−0.894797	−	0.446472i	$-0.852680\pi$
$41$	−11.4590	−1.78959	−0.894797	+	0.446472i	$0.852680\pi$
$42$	12.3805	1.91035
$43$	−7.11990	−1.08578	−0.542888	−	0.839805i	$-0.682669\pi$
$43$	−7.11990	−1.08578	−0.542888	+	0.839805i	$0.682669\pi$
$44$	−20.4778	−3.08714
$45$	−11.8157	−1.76138
$46$	9.01825	1.32967
$47$	−10.9162	−1.59230	−0.796148	−	0.605102i	$-0.793133\pi$
$47$	−10.9162	−1.59230	−0.796148	+	0.605102i	$0.793133\pi$
$48$	40.8943	5.90258
$49$	−3.66097	−0.522996
$50$	−29.2726	−4.13978
$51$	−9.88894	−1.38473
$52$	−8.46837	−1.17435
$53$	0.857981	0.117853	0.0589264	−	0.998262i	$-0.481232\pi$
$53$	0.857981	0.117853	0.0589264	+	0.998262i	$0.481232\pi$
$54$	0.0425675	0.00579271
$55$	14.2828	1.92589
$56$	18.5020	2.47244
$57$	−0.149824	−0.0198447
$58$	2.07962	0.273068
$59$	10.0274	1.30546	0.652730	−	0.757590i	$-0.273624\pi$
$59$	10.0274	1.30546	0.652730	+	0.757590i	$0.273624\pi$
$60$	−54.6785	−7.05896
$61$	13.2736	1.69951	0.849755	−	0.527178i	$-0.176750\pi$
$61$	13.2736	1.69951	0.849755	+	0.527178i	$0.176750\pi$
$62$	15.7172	1.99608
$63$	−5.47042	−0.689208
$64$	38.4797	4.80996
$65$	5.90649	0.732610
$66$	24.5183	3.01800
$67$	−10.6245	−1.29799	−0.648994	−	0.760793i	$-0.724810\pi$
$67$	−10.6245	−1.29799	−0.648994	+	0.760793i	$0.724810\pi$
$68$	−22.8571	−2.77183
$69$	−7.97796	−0.960434
$70$	−19.9589	−2.38555
$71$	−5.17815	−0.614533	−0.307267	−	0.951623i	$-0.599414\pi$
$71$	−5.17815	−0.614533	−0.307267	+	0.951623i	$0.599414\pi$
$72$	−30.3124	−3.57234
$73$	−12.0980	−1.41597	−0.707983	−	0.706229i	$-0.750395\pi$
$73$	−12.0980	−1.41597	−0.707983	+	0.706229i	$0.750395\pi$
$74$	4.42626	0.514542
$75$	25.8959	2.99021
$76$	−0.346300	−0.0397233
$77$	6.61263	0.753579
$78$	10.1393	1.14805
$79$	−1.42884	−0.160757	−0.0803783	−	0.996764i	$-0.525613\pi$
$79$	−1.42884	−0.160757	−0.0803783	+	0.996764i	$0.525613\pi$
$80$	−65.9270	−7.37087
$81$	−9.01881	−1.00209
$82$	31.7121	3.50202
$83$	2.55106	0.280015	0.140008	−	0.990150i	$-0.455287\pi$
$83$	2.55106	0.280015	0.140008	+	0.990150i	$0.455287\pi$
$84$	−25.3150	−2.76209
$85$	15.9423	1.72918
$86$	19.7039	2.12473
$87$	−1.83973	−0.197240
$88$	36.6415	3.90600
$89$	2.25876	0.239428	0.119714	−	0.992808i	$-0.461802\pi$
$89$	2.25876	0.239428	0.119714	+	0.992808i	$0.461802\pi$
$90$	32.6993	3.44681
$91$	2.73458	0.286662
$92$	−18.4401	−1.92251
$93$	−13.9041	−1.44179
$94$	30.2100	3.11593
$95$	0.241536	0.0247811
$96$	−63.5948	−6.49062
$97$	4.81279	0.488665	0.244333	−	0.969692i	$-0.421431\pi$
$97$	4.81279	0.488665	0.244333	+	0.969692i	$0.421431\pi$
$98$	10.1315	1.02344
$99$	−10.8337	−1.08882
$100$	59.8553	5.98553
$101$	11.5776	1.15202	0.576009	−	0.817444i	$-0.304609\pi$
$101$	11.5776	1.15202	0.576009	+	0.817444i	$0.304609\pi$
$102$	27.3670	2.70974
$103$	0.680261	0.0670281	0.0335141	−	0.999438i	$-0.489330\pi$
$103$	0.680261	0.0670281	0.0335141	+	0.999438i	$0.489330\pi$
$104$	15.1527	1.48584
$105$	17.6566	1.72311
$106$	−2.37441	−0.230623
$107$	7.13810	0.690066	0.345033	−	0.938591i	$-0.387868\pi$
$107$	7.13810	0.690066	0.345033	+	0.938591i	$0.387868\pi$
$108$	−0.0870401	−0.00837543
$109$	8.33925	0.798755	0.399378	−	0.916787i	$-0.369226\pi$
$109$	8.33925	0.798755	0.399378	+	0.916787i	$0.369226\pi$
$110$	−39.5268	−3.76873
$111$	−3.91567	−0.371659
$112$	−30.5228	−2.88414
$113$	1.37813	0.129644	0.0648218	−	0.997897i	$-0.479352\pi$
$113$	1.37813	0.129644	0.0648218	+	0.997897i	$0.479352\pi$
$114$	0.414629	0.0388336
$115$	12.8615	1.19934
$116$	−4.25231	−0.394817
$117$	−4.48014	−0.414189
$118$	−27.7503	−2.55462
$119$	7.38094	0.676609
$120$	97.8377	8.93132
$121$	2.09569	0.190518
$122$	−36.7339	−3.32573
$123$	−28.0540	−2.52954
$124$	−32.1377	−2.88605
$125$	−22.0135	−1.96895
$126$	15.1391	1.34870
$127$	−5.65371	−0.501685	−0.250843	−	0.968028i	$-0.580708\pi$
$127$	−5.65371	−0.501685	−0.250843	+	0.968028i	$0.580708\pi$
$128$	−54.5381	−4.82054
$129$	−17.4310	−1.53471
$130$	−16.3459	−1.43363
$131$	−9.11789	−0.796633	−0.398317	−	0.917248i	$-0.630406\pi$
$131$	−9.11789	−0.796633	−0.398317	+	0.917248i	$0.630406\pi$
$132$	−50.1339	−4.36360
$133$	0.111826	0.00969656
$134$	29.4027	2.54000
$135$	0.0607085	0.00522495
$136$	40.8988	3.50704
$137$	−13.5862	−1.16075	−0.580375	−	0.814349i	$-0.697094\pi$
$137$	−13.5862	−1.16075	−0.580375	+	0.814349i	$0.697094\pi$
$138$	22.0785	1.87945
$139$	10.7338	0.910431	0.455215	−	0.890381i	$-0.349562\pi$
$139$	10.7338	0.910431	0.455215	+	0.890381i	$0.349562\pi$
$140$	40.8111	3.44917
$141$	−26.7252	−2.25067
$142$	14.3302	1.20257
$143$	5.41558	0.452873
$144$	50.0064	4.16720
$145$	2.96589	0.246304
$146$	33.4806	2.77087
$147$	−8.96281	−0.739240
$148$	−9.05059	−0.743954
$149$	9.65471	0.790945	0.395472	−	0.918478i	$-0.370581\pi$
$149$	9.65471	0.790945	0.395472	+	0.918478i	$0.370581\pi$
$150$	−71.6655	−5.85146
$151$	7.77811	0.632973	0.316487	−	0.948597i	$-0.397497\pi$
$151$	7.77811	0.632973	0.316487	+	0.948597i	$0.397497\pi$
$152$	0.619644	0.0502598
$153$	−12.0924	−0.977612
$154$	−18.3001	−1.47466
$155$	22.4153	1.80044
$156$	−20.7323	−1.65991
$157$	−14.8893	−1.18830	−0.594148	−	0.804356i	$-0.702511\pi$
$157$	−14.8893	−1.18830	−0.594148	+	0.804356i	$0.702511\pi$
$158$	3.95422	0.314581
$159$	2.10052	0.166582
$160$	102.523	8.10518
$161$	5.95462	0.469290
$162$	24.9590	1.96097
$163$	−19.2525	−1.50798	−0.753988	−	0.656888i	$-0.771872\pi$
$163$	−19.2525	−1.50798	−0.753988	+	0.656888i	$0.771872\pi$
$164$	−64.8434	−5.06342
$165$	34.9672	2.72220
$166$	−7.05991	−0.547955
$167$	10.3103	0.797832	0.398916	−	0.916988i	$-0.369387\pi$
$167$	10.3103	0.797832	0.398916	+	0.916988i	$0.369387\pi$
$168$	45.2968	3.49472
$169$	−10.7604	−0.827727
$170$	−44.1193	−3.38380
$171$	−0.183208	−0.0140103
$172$	−40.2896	−3.07206
$173$	−0.536805	−0.0408125	−0.0204062	−	0.999792i	$-0.506496\pi$
$173$	−0.536805	−0.0408125	−0.0204062	+	0.999792i	$0.506496\pi$
$174$	5.09135	0.385974
$175$	−19.3283	−1.46108
$176$	−60.4476	−4.55641
$177$	24.5492	1.84523
$178$	−6.25099	−0.468531
$179$	−21.3672	−1.59706	−0.798530	−	0.601955i	$-0.794388\pi$
$179$	−21.3672	−1.59706	−0.798530	+	0.601955i	$0.794388\pi$
$180$	−66.8619	−4.98359
$181$	−17.8584	−1.32740	−0.663701	−	0.747998i	$-0.731015\pi$
$181$	−17.8584	−1.32740	−0.663701	+	0.747998i	$0.731015\pi$
$182$	−7.56779	−0.560962
$183$	32.4965	2.40221
$184$	32.9953	2.43245
$185$	6.31258	0.464110
$186$	38.4789	2.82141
$187$	14.6172	1.06892
$188$	−61.7720	−4.50519
$189$	0.0281067	0.00204446
$190$	−0.668438	−0.0484936
$191$	−22.4726	−1.62606	−0.813031	−	0.582221i	$-0.802184\pi$
$191$	−22.4726	−1.62606	−0.813031	+	0.582221i	$0.802184\pi$
$192$	94.2062	6.79875
$193$	−14.6201	−1.05238	−0.526189	−	0.850368i	$-0.676380\pi$
$193$	−14.6201	−1.05238	−0.526189	+	0.850368i	$0.676380\pi$
$194$	−13.3191	−0.956257
$195$	14.4603	1.03552
$196$	−20.7164	−1.47975
$197$	3.36008	0.239396	0.119698	−	0.992810i	$-0.461807\pi$
$197$	3.36008	0.239396	0.119698	+	0.992810i	$0.461807\pi$
$198$	29.9815	2.13069
$199$	−24.3043	−1.72289	−0.861444	−	0.507853i	$-0.830439\pi$
$199$	−24.3043	−1.72289	−0.861444	+	0.507853i	$0.830439\pi$
$200$	−107.101	−7.57317
$201$	−26.0110	−1.83467
$202$	−32.0404	−2.25436
$203$	1.37314	0.0963759
$204$	−55.9588	−3.91790
$205$	45.2268	3.15878
$206$	−1.88258	−0.131166
$207$	−9.75561	−0.678062
$208$	−24.9974	−1.73326
$209$	0.221461	0.0153188
$210$	−48.8636	−3.37191
$211$	1.37929	0.0949543	0.0474772	−	0.998872i	$-0.484882\pi$
$211$	1.37929	0.0949543	0.0474772	+	0.998872i	$0.484882\pi$
$212$	4.85509	0.333449
$213$	−12.6772	−0.868626
$214$	−19.7543	−1.35037
$215$	28.1011	1.91648
$216$	0.155743	0.0105970
$217$	10.3778	0.704493
$218$	−23.0784	−1.56307
$219$	−29.6185	−2.00143
$220$	80.8225	5.44905
$221$	6.04480	0.406617
$222$	10.8364	0.727291
$223$	24.4853	1.63966	0.819829	−	0.572609i	$-0.194069\pi$
$223$	24.4853	1.63966	0.819829	+	0.572609i	$0.194069\pi$
$224$	47.4661	3.17146
$225$	31.6661	2.11107
$226$	−3.81390	−0.253697
$227$	−8.39090	−0.556924	−0.278462	−	0.960447i	$-0.589825\pi$
$227$	−8.39090	−0.556924	−0.278462	+	0.960447i	$0.589825\pi$
$228$	−0.847814	−0.0561479
$229$	−13.6849	−0.904326	−0.452163	−	0.891935i	$-0.649348\pi$
$229$	−13.6849	−0.904326	−0.452163	+	0.891935i	$0.649348\pi$
$230$	−35.5935	−2.34697
$231$	16.1891	1.06516
$232$	7.60878	0.499541
$233$	13.8725	0.908817	0.454409	−	0.890793i	$-0.349851\pi$
$233$	13.8725	0.908817	0.454409	+	0.890793i	$0.349851\pi$
$234$	12.3985	0.810516
$235$	43.0846	2.81053
$236$	56.7425	3.69362
$237$	−3.49808	−0.227225
$238$	−20.4263	−1.32404
$239$	19.7204	1.27561	0.637804	−	0.770199i	$-0.279843\pi$
$239$	19.7204	1.27561	0.637804	+	0.770199i	$0.279843\pi$
$240$	−161.403	−10.4185
$241$	20.5375	1.32294	0.661469	−	0.749973i	$-0.269933\pi$
$241$	20.5375	1.32294	0.661469	+	0.749973i	$0.269933\pi$
$242$	−5.79971	−0.372819
$243$	−22.0338	−1.41347
$244$	75.1117	4.80854
$245$	14.4492	0.923129
$246$	77.6378	4.95000
$247$	0.0915828	0.00582728
$248$	57.5049	3.65157
$249$	6.24552	0.395794
$250$	60.9211	3.85299
$251$	21.9682	1.38662	0.693311	−	0.720639i	$-0.256151\pi$
$251$	21.9682	1.38662	0.693311	+	0.720639i	$0.256151\pi$
$252$	−30.9557	−1.95002
$253$	11.7926	0.741392
$254$	15.6463	0.981736
$255$	39.0300	2.44415
$256$	73.9717	4.62323
$257$	−0.487736	−0.0304242	−0.0152121	−	0.999884i	$-0.504842\pi$
$257$	−0.487736	−0.0304242	−0.0152121	+	0.999884i	$0.504842\pi$
$258$	48.2393	3.00325
$259$	2.92259	0.181601
$260$	33.4233	2.07282
$261$	−2.24966	−0.139251
$262$	25.2332	1.55891
$263$	−9.19689	−0.567104	−0.283552	−	0.958957i	$-0.591513\pi$
$263$	−9.19689	−0.567104	−0.283552	+	0.958957i	$0.591513\pi$
$264$	89.7059	5.52102
$265$	−3.38631	−0.208019
$266$	−0.309472	−0.0189750
$267$	5.52991	0.338425
$268$	−60.1211	−3.67248
$269$	−30.8111	−1.87858	−0.939292	−	0.343119i	$-0.888516\pi$
$269$	−30.8111	−1.87858	−0.939292	+	0.343119i	$0.888516\pi$
$270$	−0.168007	−0.0102246
$271$	14.8572	0.902510	0.451255	−	0.892395i	$-0.350977\pi$
$271$	14.8572	0.902510	0.451255	+	0.892395i	$0.350977\pi$
$272$	−67.4708	−4.09102
$273$	6.69482	0.405189
$274$	37.5991	2.27145
$275$	−38.2779	−2.30824
$276$	−45.1451	−2.71742
$277$	20.3735	1.22412	0.612062	−	0.790809i	$-0.290340\pi$
$277$	20.3735	1.22412	0.612062	+	0.790809i	$0.290340\pi$
$278$	−29.7052	−1.78160
$279$	−17.0023	−1.01790
$280$	−73.0244	−4.36404
$281$	13.1839	0.786488	0.393244	−	0.919434i	$-0.371353\pi$
$281$	13.1839	0.786488	0.393244	+	0.919434i	$0.371353\pi$
$282$	73.9604	4.40428
$283$	12.5273	0.744673	0.372337	−	0.928098i	$-0.378557\pi$
$283$	12.5273	0.744673	0.372337	+	0.928098i	$0.378557\pi$
$284$	−29.3017	−1.73874
$285$	0.591331	0.0350274
$286$	−14.9873	−0.886217
$287$	20.9390	1.23599
$288$	−77.7650	−4.58235
$289$	−0.684407	−0.0402593
$290$	−8.20793	−0.481986
$291$	11.7827	0.690715
$292$	−68.4594	−4.00629
$293$	17.5215	1.02362	0.511809	−	0.859099i	$-0.328975\pi$
$293$	17.5215	1.02362	0.511809	+	0.859099i	$0.328975\pi$
$294$	24.8041	1.44660
$295$	−39.5766	−2.30424
$296$	16.1945	0.941285
$297$	0.0556627	0.00322988
$298$	−26.7188	−1.54778
$299$	4.87668	0.282026
$300$	146.538	8.46038
$301$	13.0102	0.749896
$302$	−21.5254	−1.23865
$303$	28.3444	1.62834
$304$	−1.02223	−0.0586288
$305$	−52.3887	−2.99977
$306$	33.4650	1.91307
$307$	9.10738	0.519786	0.259893	−	0.965637i	$-0.416313\pi$
$307$	9.10738	0.519786	0.259893	+	0.965637i	$0.416313\pi$
$308$	37.4191	2.13215
$309$	1.66542	0.0947424
$310$	−62.0331	−3.52324
$311$	6.30629	0.357597	0.178798	−	0.983886i	$-0.442779\pi$
$311$	6.30629	0.357597	0.178798	+	0.983886i	$0.442779\pi$
$312$	37.0969	2.10020
$313$	−15.3469	−0.867458	−0.433729	−	0.901043i	$-0.642803\pi$
$313$	−15.3469	−0.867458	−0.433729	+	0.901043i	$0.642803\pi$
$314$	41.2053	2.32535
$315$	21.5909	1.21651
$316$	−8.08539	−0.454839
$317$	29.5676	1.66068	0.830342	−	0.557255i	$-0.188145\pi$
$317$	29.5676	1.66068	0.830342	+	0.557255i	$0.188145\pi$
$318$	−5.81305	−0.325980
$319$	2.71938	0.152256
$320$	−151.873	−8.48996
$321$	17.4755	0.975390
$322$	−16.4791	−0.918342
$323$	0.247192	0.0137541
$324$	−51.0350	−2.83528
$325$	−15.8294	−0.878057
$326$	53.2803	2.95092
$327$	20.4162	1.12902
$328$	116.026	6.40647
$329$	19.9472	1.09973
$330$	−96.7698	−5.32700
$331$	25.2404	1.38734	0.693668	−	0.720295i	$-0.255994\pi$
$331$	25.2404	1.38734	0.693668	+	0.720295i	$0.255994\pi$
$332$	14.4358	0.792265
$333$	−4.78816	−0.262390
$334$	−28.5330	−1.56126
$335$	41.9331	2.29105
$336$	−74.7262	−4.07665
$337$	−19.0279	−1.03652	−0.518258	−	0.855225i	$-0.673419\pi$
$337$	−19.0279	−1.03652	−0.518258	+	0.855225i	$0.673419\pi$
$338$	29.7789	1.61976
$339$	3.37395	0.183248
$340$	90.2131	4.89249
$341$	20.5523	1.11297
$342$	0.507017	0.0274163
$343$	19.4808	1.05186
$344$	72.0913	3.88690
$345$	31.4877	1.69524
$346$	1.48558	0.0798650
$347$	−18.0144	−0.967064	−0.483532	−	0.875327i	$-0.660646\pi$
$347$	−18.0144	−0.967064	−0.483532	+	0.875327i	$0.660646\pi$
$348$	−10.4105	−0.558064
$349$	−13.0591	−0.699036	−0.349518	−	0.936930i	$-0.613655\pi$
$349$	−13.0591	−0.699036	−0.349518	+	0.936930i	$0.613655\pi$
$350$	53.4899	2.85916
$351$	0.0230187	0.00122865
$352$	94.0021	5.01033
$353$	−12.7867	−0.680568	−0.340284	−	0.940323i	$-0.610523\pi$
$353$	−12.7867	−0.680568	−0.340284	+	0.940323i	$0.610523\pi$
$354$	−67.9385	−3.61089
$355$	20.4373	1.08470
$356$	12.7817	0.677430
$357$	18.0701	0.956369
$358$	59.1325	3.12525
$359$	−28.0956	−1.48283	−0.741415	−	0.671047i	$-0.765845\pi$
$359$	−28.0956	−1.48283	−0.741415	+	0.671047i	$0.765845\pi$
$360$	119.638	6.30547
$361$	−18.9963	−0.999803
$362$	49.4220	2.59756
$363$	5.13069	0.269291
$364$	15.4743	0.811071
$365$	47.7489	2.49929
$366$	−89.9322	−4.70083
$367$	12.2638	0.640165	0.320083	−	0.947390i	$-0.396289\pi$
$367$	12.2638	0.640165	0.320083	+	0.947390i	$0.396289\pi$
$368$	−54.4325	−2.83749
$369$	−34.3050	−1.78585
$370$	−17.4697	−0.908207
$371$	−1.56779	−0.0813956
$372$	−78.6798	−4.07936
$373$	−9.04345	−0.468252	−0.234126	−	0.972206i	$-0.575223\pi$
$373$	−9.04345	−0.468252	−0.234126	+	0.972206i	$0.575223\pi$
$374$	−40.4524	−2.09174
$375$	−53.8937	−2.78306
$376$	110.530	5.70017
$377$	1.12457	0.0579184
$378$	−0.0777837	−0.00400076
$379$	−17.4521	−0.896452	−0.448226	−	0.893920i	$-0.647944\pi$
$379$	−17.4521	−0.896452	−0.448226	+	0.893920i	$0.647944\pi$
$380$	1.36679	0.0701148
$381$	−13.8414	−0.709119
$382$	62.1917	3.18200
$383$	24.9617	1.27548	0.637742	−	0.770250i	$-0.279869\pi$
$383$	24.9617	1.27548	0.637742	+	0.770250i	$0.279869\pi$
$384$	−133.521	−6.81370
$385$	−26.0990	−1.33013
$386$	40.4603	2.05937
$387$	−21.3150	−1.08350
$388$	27.2343	1.38261
$389$	26.6619	1.35181	0.675907	−	0.736987i	$-0.263752\pi$
$389$	26.6619	1.35181	0.675907	+	0.736987i	$0.263752\pi$
$390$	−40.0181	−2.02639
$391$	13.1627	0.665667
$392$	37.0685	1.87224
$393$	−22.3225	−1.12602
$394$	−9.29883	−0.468468
$395$	5.63938	0.283748
$396$	−61.3047	−3.08068
$397$	−11.4737	−0.575847	−0.287924	−	0.957653i	$-0.592965\pi$
$397$	−11.4737	−0.575847	−0.287924	+	0.957653i	$0.592965\pi$
$398$	67.2608	3.37148
$399$	0.273774	0.0137058
$400$	176.684	8.83422
$401$	−25.4146	−1.26915	−0.634573	−	0.772863i	$-0.718824\pi$
$401$	−25.4146	−1.26915	−0.634573	+	0.772863i	$0.718824\pi$
$402$	71.9838	3.59023
$403$	8.49918	0.423374
$404$	65.5147	3.25948
$405$	35.5958	1.76877
$406$	−3.80010	−0.188596
$407$	5.78792	0.286896
$408$	100.129	4.95711
$409$	9.12692	0.451297	0.225648	−	0.974209i	$-0.427550\pi$
$409$	9.12692	0.451297	0.225648	+	0.974209i	$0.427550\pi$
$410$	−125.162	−6.18133
$411$	−33.2619	−1.64069
$412$	3.84941	0.189647
$413$	−18.3231	−0.901623
$414$	26.9981	1.32688
$415$	−10.0686	−0.494249
$416$	38.8735	1.90593
$417$	26.2786	1.28687
$418$	−0.612881	−0.0299770
$419$	−9.53612	−0.465870	−0.232935	−	0.972492i	$-0.574833\pi$
$419$	−9.53612	−0.465870	−0.232935	+	0.972492i	$0.574833\pi$
$420$	99.9140	4.87530
$421$	−5.75628	−0.280544	−0.140272	−	0.990113i	$-0.544798\pi$
$421$	−5.75628	−0.280544	−0.140272	+	0.990113i	$0.544798\pi$
$422$	−3.81711	−0.185814
$423$	−32.6801	−1.58896
$424$	−8.68734	−0.421894
$425$	−42.7253	−2.07248
$426$	35.0833	1.69979
$427$	−24.2549	−1.17378
$428$	40.3926	1.95245
$429$	13.2585	0.640124
$430$	−77.7681	−3.75031
$431$	28.1396	1.35544	0.677718	−	0.735321i	$-0.262969\pi$
$431$	28.1396	1.35544	0.677718	+	0.735321i	$0.262969\pi$
$432$	−0.256930	−0.0123615
$433$	−31.8838	−1.53224	−0.766119	−	0.642699i	$-0.777815\pi$
$433$	−31.8838	−1.53224	−0.766119	+	0.642699i	$0.777815\pi$
$434$	−28.7200	−1.37860
$435$	7.26112	0.348144
$436$	47.1896	2.25997
$437$	0.199424	0.00953974
$438$	81.9673	3.91655
$439$	−14.1600	−0.675821	−0.337911	−	0.941178i	$-0.609720\pi$
$439$	−14.1600	−0.675821	−0.337911	+	0.941178i	$0.609720\pi$
$440$	−144.618	−6.89439
$441$	−10.9599	−0.521901
$442$	−16.7286	−0.795700
$443$	−12.1947	−0.579388	−0.289694	−	0.957119i	$-0.593554\pi$
$443$	−12.1947	−0.579388	−0.289694	+	0.957119i	$0.593554\pi$
$444$	−22.1577	−1.05156
$445$	−8.91496	−0.422610
$446$	−67.7616	−3.20861
$447$	23.6367	1.11798
$448$	−70.3140	−3.32202
$449$	12.7956	0.603860	0.301930	−	0.953330i	$-0.402369\pi$
$449$	12.7956	0.603860	0.301930	+	0.953330i	$0.402369\pi$
$450$	−87.6340	−4.13111
$451$	41.4678	1.95264
$452$	7.79847	0.366809
$453$	19.0424	0.894691
$454$	23.2213	1.08983
$455$	−10.7929	−0.505981
$456$	1.51702	0.0710408
$457$	−5.79244	−0.270959	−0.135479	−	0.990780i	$-0.543257\pi$
$457$	−5.79244	−0.270959	−0.135479	+	0.990780i	$0.543257\pi$
$458$	37.8723	1.76965
$459$	0.0621300	0.00289998
$460$	72.7800	3.39338
$461$	31.0450	1.44591	0.722956	−	0.690894i	$-0.242783\pi$
$461$	31.0450	1.44591	0.722956	+	0.690894i	$0.242783\pi$
$462$	−44.8024	−2.08439
$463$	8.91911	0.414506	0.207253	−	0.978287i	$-0.433548\pi$
$463$	8.91911	0.414506	0.207253	+	0.978287i	$0.433548\pi$
$464$	−12.5522	−0.582723
$465$	54.8774	2.54488
$466$	−38.3913	−1.77844
$467$	10.4260	0.482456	0.241228	−	0.970468i	$-0.422450\pi$
$467$	10.4260	0.482456	0.241228	+	0.970468i	$0.422450\pi$
$468$	−25.3519	−1.17189
$469$	19.4141	0.896462
$470$	−119.234	−5.49986
$471$	−36.4521	−1.67962
$472$	−101.531	−4.67334
$473$	25.7655	1.18470
$474$	9.68074	0.444651
$475$	−0.647317	−0.0297009
$476$	41.7667	1.91438
$477$	2.56855	0.117606
$478$	−54.5751	−2.49621
$479$	−28.2340	−1.29004	−0.645022	−	0.764164i	$-0.723152\pi$
$479$	−28.2340	−1.29004	−0.645022	+	0.764164i	$0.723152\pi$
$480$	250.998	11.4564
$481$	2.39353	0.109136
$482$	−56.8364	−2.58883
$483$	14.5781	0.663328
$484$	11.8590	0.539044
$485$	−18.9953	−0.862532
$486$	60.9772	2.76598
$487$	39.3620	1.78366	0.891830	−	0.452370i	$-0.149422\pi$
$487$	39.3620	1.78366	0.891830	+	0.452370i	$0.149422\pi$
$488$	−134.399	−6.08398
$489$	−47.1342	−2.13148
$490$	−39.9874	−1.80645
$491$	27.1906	1.22709	0.613546	−	0.789659i	$-0.289742\pi$
$491$	27.1906	1.22709	0.613546	+	0.789659i	$0.289742\pi$
$492$	−158.750	−7.15700
$493$	3.03534	0.136705
$494$	−0.253450	−0.0114033
$495$	42.7587	1.92186
$496$	−94.8660	−4.25961
$497$	9.46204	0.424430
$498$	−17.2841	−0.774520
$499$	24.9347	1.11623	0.558115	−	0.829763i	$-0.311525\pi$
$499$	24.9347	1.11623	0.558115	+	0.829763i	$0.311525\pi$
$500$	−124.569	−5.57088
$501$	25.2416	1.12771
$502$	−60.7957	−2.71345
$503$	−24.0170	−1.07087	−0.535433	−	0.844578i	$-0.679852\pi$
$503$	−24.0170	−1.07087	−0.535433	+	0.844578i	$0.679852\pi$
$504$	55.3898	2.46726
$505$	−45.6950	−2.03340
$506$	−32.6352	−1.45081
$507$	−26.3438	−1.16997
$508$	−31.9928	−1.41945
$509$	−17.2418	−0.764229	−0.382114	−	0.924115i	$-0.624804\pi$
$509$	−17.2418	−0.764229	−0.382114	+	0.924115i	$0.624804\pi$
$510$	−108.013	−4.78291
$511$	22.1067	0.977944
$512$	−95.6361	−4.22656
$513$	0.000941312 0	4.15599e−5 0
$514$	1.34978	0.0595363
$515$	−2.68488	−0.118310
$516$	−98.6373	−4.34227
$517$	39.5036	1.73737
$518$	−8.08810	−0.355371
$519$	−1.31421	−0.0576874
$520$	−59.8051	−2.62263
$521$	35.7059	1.56430	0.782152	−	0.623088i	$-0.214122\pi$
$521$	35.7059	1.56430	0.782152	+	0.623088i	$0.214122\pi$
$522$	6.22580	0.272496
$523$	−38.6962	−1.69207	−0.846034	−	0.533129i	$-0.821016\pi$
$523$	−38.6962	−1.69207	−0.846034	+	0.533129i	$0.821016\pi$
$524$	−51.5957	−2.25397
$525$	−47.3197	−2.06520
$526$	25.4518	1.10975
$527$	22.9402	0.999292
$528$	−147.988	−6.44036
$529$	−12.3809	−0.538300
$530$	9.37142	0.407068
$531$	30.0193	1.30273
$532$	0.632794	0.0274351
$533$	17.1485	0.742786
$534$	−15.3037	−0.662256
$535$	−28.1729	−1.21802
$536$	107.576	4.64659
$537$	−52.3113	−2.25740
$538$	85.2678	3.67616
$539$	13.2483	0.570645
$540$	0.343533	0.0147833
$541$	23.7359	1.02049	0.510244	−	0.860030i	$-0.329555\pi$
$541$	23.7359	1.02049	0.510244	+	0.860030i	$0.329555\pi$
$542$	−41.1164	−1.76610
$543$	−43.7210	−1.87625
$544$	104.924	4.49858
$545$	−32.9136	−1.40987
$546$	−18.5275	−0.792905
$547$	24.8278	1.06156	0.530780	−	0.847510i	$-0.321899\pi$
$547$	24.8278	1.06156	0.530780	+	0.847510i	$0.321899\pi$
$548$	−76.8809	−3.28419
$549$	39.7374	1.69595
$550$	105.932	4.51695
$551$	0.0459875	0.00195913
$552$	80.7794	3.43820
$553$	2.61091	0.111027
$554$	−56.3825	−2.39546
$555$	15.4545	0.656007
$556$	60.7398	2.57594
$557$	−18.0211	−0.763577	−0.381789	−	0.924250i	$-0.624692\pi$
$557$	−18.0211	−0.763577	−0.381789	+	0.924250i	$0.624692\pi$
$558$	47.0528	1.99190
$559$	10.6550	0.450660
$560$	120.469	5.09073
$561$	35.7861	1.51089
$562$	−36.4858	−1.53906
$563$	−1.23949	−0.0522383	−0.0261192	−	0.999659i	$-0.508315\pi$
$563$	−1.23949	−0.0522383	−0.0261192	+	0.999659i	$0.508315\pi$
$564$	−151.231	−6.36796
$565$	−5.43926	−0.228831
$566$	−34.6687	−1.45723
$567$	16.4801	0.692098
$568$	52.4304	2.19993
$569$	−28.2168	−1.18291	−0.591455	−	0.806338i	$-0.701446\pi$
$569$	−28.2168	−1.18291	−0.591455	+	0.806338i	$0.701446\pi$
$570$	−1.63647	−0.0685443
$571$	−1.43862	−0.0602042	−0.0301021	−	0.999547i	$-0.509583\pi$
$571$	−1.43862	−0.0602042	−0.0301021	+	0.999547i	$0.509583\pi$
$572$	30.6453	1.28134
$573$	−55.0176	−2.29839
$574$	−57.9476	−2.41868
$575$	−34.4689	−1.43745
$576$	115.197	4.79989
$577$	11.2851	0.469805	0.234903	−	0.972019i	$-0.424523\pi$
$577$	11.2851	0.469805	0.234903	+	0.972019i	$0.424523\pi$
$578$	1.89406	0.0787824
$579$	−35.7931	−1.48751
$580$	16.7832	0.696884
$581$	−4.66155	−0.193394
$582$	−32.6080	−1.35164
$583$	−3.10486	−0.128590
$584$	122.496	5.06894
$585$	17.6824	0.731076
$586$	−48.4898	−2.00310
$587$	−36.9000	−1.52303	−0.761513	−	0.648150i	$-0.775543\pi$
$587$	−36.9000	−1.52303	−0.761513	+	0.648150i	$0.775543\pi$
$588$	−50.7181	−2.09158
$589$	0.347560	0.0143210
$590$	109.526	4.50911
$591$	8.22617	0.338380
$592$	−26.7161	−1.09802
$593$	−12.1389	−0.498487	−0.249243	−	0.968441i	$-0.580182\pi$
$593$	−12.1389	−0.498487	−0.249243	+	0.968441i	$0.580182\pi$
$594$	−0.154043	−0.00632047
$595$	−29.1313	−1.19427
$596$	54.6334	2.23787
$597$	−59.5020	−2.43525
$598$	−13.4959	−0.551890
$599$	−47.3585	−1.93502	−0.967508	−	0.252839i	$-0.918636\pi$
$599$	−47.3585	−1.93502	−0.967508	+	0.252839i	$0.918636\pi$
$600$	−262.205	−10.7045
$601$	−44.8302	−1.82866	−0.914330	−	0.404969i	$-0.867282\pi$
$601$	−44.8302	−1.82866	−0.914330	+	0.404969i	$0.867282\pi$
$602$	−36.0050	−1.46745
$603$	−31.8067	−1.29527
$604$	44.0142	1.79091
$605$	−8.27136	−0.336278
$606$	−78.4415	−3.18647
$607$	−0.409094	−0.0166046	−0.00830231	−	0.999966i	$-0.502643\pi$
$607$	−0.409094	−0.0166046	−0.00830231	+	0.999966i	$0.502643\pi$
$608$	1.58967	0.0644696
$609$	3.36174	0.136225
$610$	144.983	5.87018
$611$	16.3363	0.660896
$612$	−68.4276	−2.76602
$613$	12.9626	0.523556	0.261778	−	0.965128i	$-0.415691\pi$
$613$	12.9626	0.523556	0.261778	+	0.965128i	$0.415691\pi$
$614$	−25.2041	−1.01716
$615$	110.725	4.46484
$616$	−66.9550	−2.69770
$617$	−6.56344	−0.264234	−0.132117	−	0.991234i	$-0.542177\pi$
$617$	−6.56344	−0.264234	−0.132117	+	0.991234i	$0.542177\pi$
$618$	−4.60895	−0.185399
$619$	7.22017	0.290203	0.145102	−	0.989417i	$-0.453649\pi$
$619$	7.22017	0.290203	0.145102	+	0.989417i	$0.453649\pi$
$620$	126.842	5.09411
$621$	0.0501238	0.00201140
$622$	−17.4523	−0.699773
$623$	−4.12744	−0.165362
$624$	−61.1989	−2.44992
$625$	33.9962	1.35985
$626$	42.4716	1.69751
$627$	0.542183	0.0216527
$628$	−84.2545	−3.36212
$629$	6.46040	0.257593
$630$	−59.7514	−2.38055
$631$	12.2641	0.488227	0.244114	−	0.969747i	$-0.421503\pi$
$631$	12.2641	0.488227	0.244114	+	0.969747i	$0.421503\pi$
$632$	14.4674	0.575483
$633$	3.37679	0.134215
$634$	−81.8267	−3.24975
$635$	22.3143	0.885514
$636$	11.8863	0.471321
$637$	5.47869	0.217074
$638$	−7.52573	−0.297947
$639$	−15.5019	−0.613246
$640$	215.253	8.50863
$641$	19.2920	0.761989	0.380994	−	0.924577i	$-0.375582\pi$
$641$	19.2920	0.761989	0.380994	+	0.924577i	$0.375582\pi$
$642$	−48.3625	−1.90872
$643$	10.9824	0.433104	0.216552	−	0.976271i	$-0.430519\pi$
$643$	10.9824	0.433104	0.216552	+	0.976271i	$0.430519\pi$
$644$	33.6956	1.32779
$645$	68.7973	2.70889
$646$	−0.684090	−0.0269152
$647$	−14.1323	−0.555596	−0.277798	−	0.960639i	$-0.589605\pi$
$647$	−14.1323	−0.555596	−0.277798	+	0.960639i	$0.589605\pi$
$648$	91.3183	3.58732
$649$	−36.2872	−1.42440
$650$	43.8069	1.71825
$651$	25.4071	0.995781
$652$	−108.945	−4.26661
$653$	−1.79489	−0.0702395	−0.0351197	−	0.999383i	$-0.511181\pi$
$653$	−1.79489	−0.0702395	−0.0351197	+	0.999383i	$0.511181\pi$
$654$	−56.5006	−2.20935
$655$	35.9868	1.40612
$656$	−191.408	−7.47325
$657$	−36.2181	−1.41300
$658$	−55.2028	−2.15203
$659$	44.8575	1.74740	0.873700	−	0.486465i	$-0.161714\pi$
$659$	44.8575	1.74740	0.873700	+	0.486465i	$0.161714\pi$
$660$	197.870	7.70209
$661$	3.06163	0.119083	0.0595417	−	0.998226i	$-0.481036\pi$
$661$	3.06163	0.119083	0.0595417	+	0.998226i	$0.481036\pi$
$662$	−69.8512	−2.71484
$663$	14.7989	0.574743
$664$	−25.8303	−1.00241
$665$	−0.441360	−0.0171152
$666$	13.2510	0.513464
$667$	2.44878	0.0948172
$668$	58.3430	2.25736
$669$	59.9451	2.31761
$670$	−116.047	−4.48330
$671$	−48.0345	−1.85435
$672$	116.207	4.48278
$673$	10.6844	0.411852	0.205926	−	0.978568i	$-0.433979\pi$
$673$	10.6844	0.411852	0.205926	+	0.978568i	$0.433979\pi$
$674$	52.6586	2.02833
$675$	−0.162699	−0.00626227
$676$	−60.8905	−2.34194
$677$	3.10274	0.119248	0.0596241	−	0.998221i	$-0.481010\pi$
$677$	3.10274	0.119248	0.0596241	+	0.998221i	$0.481010\pi$
$678$	−9.33721	−0.358593
$679$	−8.79442	−0.337499
$680$	−161.421	−6.19020
$681$	−20.5427	−0.787197
$682$	−56.8773	−2.17794
$683$	−2.17072	−0.0830602	−0.0415301	−	0.999137i	$-0.513223\pi$
$683$	−2.17072	−0.0830602	−0.0415301	+	0.999137i	$0.513223\pi$
$684$	−1.03672	−0.0396401
$685$	53.6227	2.04882
$686$	−53.9120	−2.05837
$687$	−33.5036	−1.27824
$688$	−118.929	−4.53414
$689$	−1.28398	−0.0489158
$690$	−87.1404	−3.31738
$691$	38.9539	1.48188	0.740939	−	0.671573i	$-0.234381\pi$
$691$	38.9539	1.48188	0.740939	+	0.671573i	$0.234381\pi$
$692$	−3.03763	−0.115473
$693$	19.7964	0.752001
$694$	49.8538	1.89243
$695$	−42.3646	−1.60698
$696$	18.6279	0.706088
$697$	46.2858	1.75320
$698$	36.1402	1.36793
$699$	33.9627	1.28459
$700$	−109.374	−4.13394
$701$	11.8271	0.446703	0.223352	−	0.974738i	$-0.428300\pi$
$701$	11.8271	0.446703	0.223352	+	0.974738i	$0.428300\pi$
$702$	−0.0637029	−0.00240431
$703$	0.0978794	0.00369159
$704$	−139.250	−5.24819
$705$	105.480	3.97261
$706$	35.3865	1.33179
$707$	−21.1558	−0.795646
$708$	138.917	5.22084
$709$	−8.96442	−0.336666	−0.168333	−	0.985730i	$-0.553838\pi$
$709$	−8.96442	−0.336666	−0.168333	+	0.985730i	$0.553838\pi$
$710$	−56.5590	−2.12262
$711$	−4.27753	−0.160420
$712$	−22.8707	−0.857115
$713$	18.5072	0.693099
$714$	−50.0078	−1.87150
$715$	−21.3744	−0.799357
$716$	−120.911	−4.51866
$717$	48.2797	1.80304
$718$	77.7530	2.90172
$719$	21.7823	0.812342	0.406171	−	0.913797i	$-0.366864\pi$
$719$	21.7823	0.812342	0.406171	+	0.913797i	$0.366864\pi$
$720$	−197.367	−7.35543
$721$	−1.24304	−0.0462933
$722$	52.5710	1.95649
$723$	50.2801	1.86994
$724$	−101.056	−3.75571
$725$	−7.94858	−0.295203
$726$	−14.1989	−0.526970
$727$	4.29805	0.159406	0.0797028	−	0.996819i	$-0.474603\pi$
$727$	4.29805	0.159406	0.0797028	+	0.996819i	$0.474603\pi$
$728$	−27.6885	−1.02620
$729$	−26.8868	−0.995807
$730$	−132.142	−4.89081
$731$	28.7591	1.06369
$732$	183.889	6.79673
$733$	−32.5542	−1.20242	−0.601208	−	0.799092i	$-0.705314\pi$
$733$	−32.5542	−1.20242	−0.601208	+	0.799092i	$0.705314\pi$
$734$	−33.9393	−1.25272
$735$	35.3748	1.30482
$736$	84.6481	3.12017
$737$	38.4479	1.41625
$738$	94.9371	3.49468
$739$	10.9746	0.403707	0.201853	−	0.979416i	$-0.435304\pi$
$739$	10.9746	0.403707	0.201853	+	0.979416i	$0.435304\pi$
$740$	35.7212	1.31314
$741$	0.224214	0.00823670
$742$	4.33877	0.159281
$743$	44.2040	1.62169	0.810844	−	0.585262i	$-0.199008\pi$
$743$	44.2040	1.62169	0.810844	+	0.585262i	$0.199008\pi$
$744$	140.784	5.16139
$745$	−38.1056	−1.39608
$746$	25.0272	0.916311
$747$	7.63715	0.279429
$748$	82.7151	3.02436
$749$	−13.0435	−0.476598
$750$	149.148	5.44610
$751$	30.3718	1.10828	0.554141	−	0.832423i	$-0.313047\pi$
$751$	30.3718	1.10828	0.554141	+	0.832423i	$0.313047\pi$
$752$	−182.342	−6.64934
$753$	53.7827	1.95995
$754$	−3.11218	−0.113339
$755$	−30.6989	−1.11725
$756$	0.159048	0.00578453
$757$	−1.71323	−0.0622684	−0.0311342	−	0.999515i	$-0.509912\pi$
$757$	−1.71323	−0.0622684	−0.0311342	+	0.999515i	$0.509912\pi$
$758$	48.2976	1.75425
$759$	28.8706	1.04794
$760$	−2.44563	−0.0887125
$761$	40.2601	1.45943	0.729713	−	0.683753i	$-0.239654\pi$
$761$	40.2601	1.45943	0.729713	+	0.683753i	$0.239654\pi$
$762$	38.3054	1.38766
$763$	−15.2383	−0.551664
$764$	−127.167	−4.60072
$765$	47.7267	1.72556
$766$	−69.0801	−2.49597
$767$	−15.0062	−0.541842
$768$	181.098	6.53481
$769$	−33.0863	−1.19312	−0.596562	−	0.802567i	$-0.703467\pi$
$769$	−33.0863	−1.19312	−0.596562	+	0.802567i	$0.703467\pi$
$770$	72.2274	2.60289
$771$	−1.19408	−0.0430037
$772$	−82.7313	−2.97756
$773$	−35.8250	−1.28853	−0.644267	−	0.764800i	$-0.722838\pi$
$773$	−35.8250	−1.28853	−0.644267	+	0.764800i	$0.722838\pi$
$774$	58.9880	2.12028
$775$	−60.0731	−2.15789
$776$	−48.7311	−1.74934
$777$	7.15511	0.256688
$778$	−73.7853	−2.64533
$779$	0.701262	0.0251253
$780$	81.8270	2.92988
$781$	18.7387	0.670522
$782$	−36.4270	−1.30263
$783$	0.0115586	0.000413072 0
$784$	−61.1520	−2.18400
$785$	58.7656	2.09743
$786$	61.7761	2.20348
$787$	−27.2836	−0.972557	−0.486278	−	0.873804i	$-0.661646\pi$
$787$	−27.2836	−0.972557	−0.486278	+	0.873804i	$0.661646\pi$
$788$	19.0138	0.677338
$789$	−22.5159	−0.801587
$790$	−15.6066	−0.555260
$791$	−2.51826	−0.0895390
$792$	109.694	3.89782
$793$	−19.8641	−0.705396
$794$	31.7527	1.12686
$795$	−8.29039	−0.294030
$796$	−137.532	−4.87468
$797$	42.2845	1.49780	0.748898	−	0.662686i	$-0.230583\pi$
$797$	42.2845	1.49780	0.748898	+	0.662686i	$0.230583\pi$
$798$	−0.757653	−0.0268206
$799$	44.0935	1.55992
$800$	−274.762	−9.71431
$801$	6.76209	0.238927
$802$	70.3335	2.48356
$803$	43.7803	1.54497
$804$	−147.189	−5.19096
$805$	−23.5019	−0.828333
$806$	−23.5210	−0.828491
$807$	−75.4319	−2.65533
$808$	−117.227	−4.12404
$809$	40.3667	1.41922	0.709609	−	0.704596i	$-0.248872\pi$
$809$	40.3667	1.41922	0.709609	+	0.704596i	$0.248872\pi$
$810$	−98.5092	−3.46126
$811$	2.08490	0.0732106	0.0366053	−	0.999330i	$-0.488346\pi$
$811$	2.08490	0.0732106	0.0366053	+	0.999330i	$0.488346\pi$
$812$	7.77026	0.272683
$813$	36.3735	1.27567
$814$	−16.0177	−0.561421
$815$	75.9866	2.66170
$816$	−165.183	−5.78255
$817$	0.435720	0.0152439
$818$	−25.2582	−0.883132
$819$	8.18656	0.286062
$820$	255.926	8.93733
$821$	−21.1700	−0.738839	−0.369419	−	0.929263i	$-0.620443\pi$
$821$	−21.1700	−0.738839	−0.369419	+	0.929263i	$0.620443\pi$
$822$	92.0504	3.21063
$823$	28.5212	0.994188	0.497094	−	0.867697i	$-0.334400\pi$
$823$	28.5212	0.994188	0.497094	+	0.867697i	$0.334400\pi$
$824$	−6.88786	−0.239950
$825$	−93.7122	−3.26264
$826$	50.7082	1.76436
$827$	32.0677	1.11510	0.557551	−	0.830143i	$-0.311741\pi$
$827$	32.0677	1.11510	0.557551	+	0.830143i	$0.311741\pi$
$828$	−55.2044	−1.91848
$829$	2.48098	0.0861680	0.0430840	−	0.999071i	$-0.486282\pi$
$829$	2.48098	0.0861680	0.0430840	+	0.999071i	$0.486282\pi$
$830$	27.8643	0.967184
$831$	49.8785	1.73027
$832$	−57.5854	−1.99641
$833$	14.7876	0.512360
$834$	−72.7245	−2.51824
$835$	−40.6929	−1.40824
$836$	1.25319	0.0433425
$837$	0.0873567	0.00301949
$838$	26.3906	0.911650
$839$	15.5929	0.538325	0.269163	−	0.963095i	$-0.413253\pi$
$839$	15.5929	0.538325	0.269163	+	0.963095i	$0.413253\pi$
$840$	−178.779	−6.16846
$841$	−28.4353	−0.980528
$842$	15.9302	0.548989
$843$	32.2770	1.11168
$844$	7.80504	0.268661
$845$	42.4697	1.46100
$846$	90.4403	3.10940
$847$	−3.82946	−0.131582
$848$	14.3315	0.492147
$849$	30.6695	1.05258
$850$	118.240	4.05559
$851$	5.21197	0.178664
$852$	−71.7367	−2.45766
$853$	−45.0518	−1.54254	−0.771272	−	0.636506i	$-0.780379\pi$
$853$	−45.0518	−1.54254	−0.771272	+	0.636506i	$0.780379\pi$
$854$	67.1239	2.29693
$855$	0.723092	0.0247292
$856$	−72.2756	−2.47033
$857$	−49.0873	−1.67679	−0.838395	−	0.545063i	$-0.816506\pi$
$857$	−49.0873	−1.67679	−0.838395	+	0.545063i	$0.816506\pi$
$858$	−36.6920	−1.25264
$859$	35.2483	1.20266	0.601328	−	0.799002i	$-0.294639\pi$
$859$	35.2483	1.20266	0.601328	+	0.799002i	$0.294639\pi$
$860$	159.017	5.42242
$861$	51.2631	1.74704
$862$	−77.8747	−2.65242
$863$	−13.5816	−0.462323	−0.231162	−	0.972915i	$-0.574253\pi$
$863$	−13.5816	−0.462323	−0.231162	+	0.972915i	$0.574253\pi$
$864$	0.399552	0.0135930
$865$	2.11868	0.0720373
$866$	88.2365	2.99840
$867$	−1.67557	−0.0569054
$868$	58.7253	1.99327
$869$	5.17066	0.175403
$870$	−20.0947	−0.681275
$871$	15.8997	0.538741
$872$	−84.4376	−2.85942
$873$	14.4081	0.487642
$874$	−0.551894	−0.0186681
$875$	40.2253	1.35986
$876$	−167.603	−5.66278
$877$	7.53426	0.254414	0.127207	−	0.991876i	$-0.459399\pi$
$877$	7.53426	0.254414	0.127207	+	0.991876i	$0.459399\pi$
$878$	39.1871	1.32250
$879$	42.8963	1.44686
$880$	238.577	8.04242
$881$	−46.3488	−1.56153	−0.780765	−	0.624825i	$-0.785170\pi$
$881$	−46.3488	−1.56153	−0.780765	+	0.624825i	$0.785170\pi$
$882$	30.3309	1.02129
$883$	43.6517	1.46900	0.734499	−	0.678610i	$-0.237417\pi$
$883$	43.6517	1.46900	0.734499	+	0.678610i	$0.237417\pi$
$884$	34.2059	1.15047
$885$	−96.8918	−3.25698
$886$	33.7481	1.13379
$887$	37.5011	1.25916	0.629582	−	0.776934i	$-0.283226\pi$
$887$	37.5011	1.25916	0.629582	+	0.776934i	$0.283226\pi$
$888$	39.6474	1.33048
$889$	10.3310	0.346491
$890$	24.6716	0.826995
$891$	32.6372	1.09339
$892$	138.556	4.63919
$893$	0.668046	0.0223553
$894$	−65.4132	−2.18775
$895$	84.3328	2.81894
$896$	99.6576	3.32933
$897$	11.9391	0.398636
$898$	−35.4110	−1.18168
$899$	4.26778	0.142339
$900$	179.190	5.97299
$901$	−3.46561	−0.115456
$902$	−114.760	−3.82108
$903$	31.8517	1.05996
$904$	−13.9540	−0.464104
$905$	70.4840	2.34297
$906$	−52.6988	−1.75080
$907$	36.1244	1.19949	0.599746	−	0.800191i	$-0.295268\pi$
$907$	36.1244	1.19949	0.599746	+	0.800191i	$0.295268\pi$
$908$	−47.4819	−1.57574
$909$	34.6601	1.14960
$910$	29.8688	0.990142
$911$	26.5693	0.880280	0.440140	−	0.897929i	$-0.354929\pi$
$911$	26.5693	0.880280	0.440140	+	0.897929i	$0.354929\pi$
$912$	−2.50263	−0.0828703
$913$	−9.23177	−0.305527
$914$	16.0302	0.530233
$915$	−128.258	−4.24009
$916$	−77.4394	−2.55867
$917$	16.6611	0.550199
$918$	−0.171941	−0.00567491
$919$	−28.1676	−0.929162	−0.464581	−	0.885531i	$-0.653795\pi$
$919$	−28.1676	−0.929162	−0.464581	+	0.885531i	$0.653795\pi$
$920$	−130.227	−4.29346
$921$	22.2968	0.734703
$922$	−85.9153	−2.82947
$923$	7.74917	0.255067
$924$	91.6097	3.01374
$925$	−16.9177	−0.556251
$926$	−24.6831	−0.811137
$927$	2.03651	0.0668877
$928$	19.5200	0.640775
$929$	−11.9958	−0.393571	−0.196785	−	0.980447i	$-0.563050\pi$
$929$	−11.9958	−0.393571	−0.196785	+	0.980447i	$0.563050\pi$
$930$	−151.870	−4.98001
$931$	0.224042	0.00734268
$932$	78.5007	2.57138
$933$	15.4391	0.505453
$934$	−28.8532	−0.944107
$935$	−57.6919	−1.88673
$936$	45.3628	1.48273
$937$	15.1644	0.495401	0.247700	−	0.968837i	$-0.420325\pi$
$937$	15.1644	0.495401	0.247700	+	0.968837i	$0.420325\pi$
$938$	−53.7275	−1.75427
$939$	−37.5724	−1.22613
$940$	243.804	7.95201
$941$	−34.9958	−1.14083	−0.570415	−	0.821357i	$-0.693218\pi$
$941$	−34.9958	−1.14083	−0.570415	+	0.821357i	$0.693218\pi$
$942$	100.879	3.28682
$943$	37.3414	1.21600
$944$	167.496	5.45153
$945$	−0.110933	−0.00360864
$946$	−71.3045	−2.31831
$947$	−39.4244	−1.28112	−0.640560	−	0.767908i	$-0.721298\pi$
$947$	−39.4244	−1.28112	−0.640560	+	0.767908i	$0.721298\pi$
$948$	−19.7947	−0.642903
$949$	18.1049	0.587708
$950$	1.79141	0.0581211
$951$	72.3877	2.34733
$952$	−74.7344	−2.42216
$953$	−21.2125	−0.687140	−0.343570	−	0.939127i	$-0.611636\pi$
$953$	−21.2125	−0.687140	−0.343570	+	0.939127i	$0.611636\pi$
$954$	−7.10832	−0.230140
$955$	88.6957	2.87013
$956$	111.593	3.60916
$957$	6.65761	0.215210
$958$	78.1360	2.52446
$959$	24.8262	0.801678
$960$	−371.817	−12.0003
$961$	1.25465	0.0404727
$962$	−6.62395	−0.213565
$963$	21.3695	0.688621
$964$	116.216	3.74307
$965$	57.7032	1.85753
$966$	−40.3441	−1.29805
$967$	−58.3417	−1.87614	−0.938071	−	0.346443i	$-0.887389\pi$
$967$	−58.3417	−1.87614	−0.938071	+	0.346443i	$0.887389\pi$
$968$	−21.2196	−0.682023
$969$	0.605178	0.0194411
$970$	52.5684	1.68787
$971$	−52.6381	−1.68924	−0.844618	−	0.535369i	$-0.820173\pi$
$971$	−52.6381	−1.68924	−0.844618	+	0.535369i	$0.820173\pi$
$972$	−124.683	−3.99921
$973$	−19.6139	−0.628793
$974$	−108.932	−3.49040
$975$	−38.7536	−1.24111
$976$	221.719	7.09706
$977$	6.41975	0.205386	0.102693	−	0.994713i	$-0.467254\pi$
$977$	6.41975	0.205386	0.102693	+	0.994713i	$0.467254\pi$
$978$	130.441	4.17105
$979$	−8.17400	−0.261242
$980$	81.7644	2.61187
$981$	24.9653	0.797082
$982$	−75.2482	−2.40127
$983$	−32.3973	−1.03331	−0.516656	−	0.856193i	$-0.672823\pi$
$983$	−32.3973	−1.03331	−0.516656	+	0.856193i	$0.672823\pi$
$984$	284.056	9.05537
$985$	−13.2617	−0.422552
$986$	−8.40013	−0.267515
$987$	48.8350	1.55444
$988$	0.518243	0.0164875
$989$	23.2016	0.737768
$990$	−118.332	−3.76084
$991$	29.5471	0.938595	0.469298	−	0.883040i	$-0.344507\pi$
$991$	29.5471	0.938595	0.469298	+	0.883040i	$0.344507\pi$
$992$	147.526	4.68397
$993$	61.7936	1.96096
$994$	−26.1856	−0.830558
$995$	95.9251	3.04103
$996$	35.3417	1.11985
$997$	12.4299	0.393658	0.196829	−	0.980438i	$-0.436936\pi$
$997$	12.4299	0.393658	0.196829	+	0.980438i	$0.436936\pi$
$998$	−69.0053	−2.18433
$999$	0.0246013	0.000778351 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.b.1.6	✓	361

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.b.1.6	✓	361	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)

\(f(q)\)	\(=\)	\(q-2.76744 q^{2} +2.44821 q^{3} +5.65873 q^{4} -3.94684 q^{5} -6.77527 q^{6} -1.82730 q^{7} -10.1253 q^{8} +2.99372 q^{9} +O(q^{10})\)	\(q-2.76744 q^{2} +2.44821 q^{3} +5.65873 q^{4} -3.94684 q^{5} -6.77527 q^{6} -1.82730 q^{7} -10.1253 q^{8} +2.99372 q^{9} +10.9226 q^{10} -3.61880 q^{11} +13.8537 q^{12} -1.49651 q^{13} +5.05695 q^{14} -9.66267 q^{15} +16.7038 q^{16} -4.03926 q^{17} -8.28494 q^{18} -0.0611975 q^{19} -22.3341 q^{20} -4.47361 q^{21} +10.0148 q^{22} -3.25870 q^{23} -24.7889 q^{24} +10.5775 q^{25} +4.14151 q^{26} -0.0153816 q^{27} -10.3402 q^{28} -0.751461 q^{29} +26.7409 q^{30} -5.67932 q^{31} -25.9761 q^{32} -8.85956 q^{33} +11.1784 q^{34} +7.21206 q^{35} +16.9406 q^{36} -1.59940 q^{37} +0.169360 q^{38} -3.66377 q^{39} +39.9630 q^{40} -11.4590 q^{41} +12.3805 q^{42} -7.11990 q^{43} -20.4778 q^{44} -11.8157 q^{45} +9.01825 q^{46} -10.9162 q^{47} +40.8943 q^{48} -3.66097 q^{49} -29.2726 q^{50} -9.88894 q^{51} -8.46837 q^{52} +0.857981 q^{53} +0.0425675 q^{54} +14.2828 q^{55} +18.5020 q^{56} -0.149824 q^{57} +2.07962 q^{58} +10.0274 q^{59} -54.6785 q^{60} +13.2736 q^{61} +15.7172 q^{62} -5.47042 q^{63} +38.4797 q^{64} +5.90649 q^{65} +24.5183 q^{66} -10.6245 q^{67} -22.8571 q^{68} -7.97796 q^{69} -19.9589 q^{70} -5.17815 q^{71} -30.3124 q^{72} -12.0980 q^{73} +4.42626 q^{74} +25.8959 q^{75} -0.346300 q^{76} +6.61263 q^{77} +10.1393 q^{78} -1.42884 q^{79} -65.9270 q^{80} -9.01881 q^{81} +31.7121 q^{82} +2.55106 q^{83} -25.3150 q^{84} +15.9423 q^{85} +19.7039 q^{86} -1.83973 q^{87} +36.6415 q^{88} +2.25876 q^{89} +32.6993 q^{90} +2.73458 q^{91} -18.4401 q^{92} -13.9041 q^{93} +30.2100 q^{94} +0.241536 q^{95} -63.5948 q^{96} +4.81279 q^{97} +10.1315 q^{98} -10.8337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * q^99

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