LMFDB - Embedded newform 547.3.b.b.546.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [547,3,Mod(546,547)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(547, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("547.546");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 547 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	547.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$14.9046704605$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			546.18
Character	$\chi$	$=$	547.546
Dual form			547.3.b.b.546.71

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76044i q^{2} -1.30646i q^{3} -3.62000 q^{4} +3.69969i q^{5} -3.60639 q^{6} +4.19330i q^{7} -1.04896i q^{8} +7.29317 q^{9} +O(q^{10})$	\(q-2.76044i q^{2} -1.30646i q^{3} -3.62000 q^{4} +3.69969i q^{5} -3.60639 q^{6} +4.19330i q^{7} -1.04896i q^{8} +7.29317 q^{9} +10.2127 q^{10} +2.25936 q^{11} +4.72938i q^{12} -9.40857 q^{13} +11.5753 q^{14} +4.83348 q^{15} -17.3756 q^{16} -20.2826i q^{17} -20.1323i q^{18} +31.8592 q^{19} -13.3929i q^{20} +5.47837 q^{21} -6.23683i q^{22} -34.4234i q^{23} -1.37042 q^{24} +11.3123 q^{25} +25.9718i q^{26} -21.2863i q^{27} -15.1798i q^{28} -14.9273 q^{29} -13.3425i q^{30} -21.7217i q^{31} +43.7684i q^{32} -2.95176i q^{33} -55.9887 q^{34} -15.5139 q^{35} -26.4013 q^{36} +24.2393i q^{37} -87.9453i q^{38} +12.2919i q^{39} +3.88082 q^{40} -44.2412i q^{41} -15.1227i q^{42} -79.6613i q^{43} -8.17890 q^{44} +26.9824i q^{45} -95.0235 q^{46} -31.3411 q^{47} +22.7005i q^{48} +31.4162 q^{49} -31.2269i q^{50} -26.4983 q^{51} +34.0591 q^{52} +53.9572 q^{53} -58.7595 q^{54} +8.35894i q^{55} +4.39861 q^{56} -41.6227i q^{57} +41.2058i q^{58} +8.72851i q^{59} -17.4972 q^{60} +119.629i q^{61} -59.9614 q^{62} +30.5825i q^{63} +51.3173 q^{64} -34.8088i q^{65} -8.14815 q^{66} -1.90568 q^{67} +73.4229i q^{68} -44.9727 q^{69} +42.8251i q^{70} +72.0035i q^{71} -7.65024i q^{72} +46.4732 q^{73} +66.9109 q^{74} -14.7791i q^{75} -115.330 q^{76} +9.47420i q^{77} +33.9310 q^{78} +51.4586i q^{79} -64.2843i q^{80} +37.8288 q^{81} -122.125 q^{82} +122.608i q^{83} -19.8317 q^{84} +75.0391 q^{85} -219.900 q^{86} +19.5019i q^{87} -2.36998i q^{88} -83.1582i q^{89} +74.4833 q^{90} -39.4530i q^{91} +124.613i q^{92} -28.3785 q^{93} +86.5150i q^{94} +117.869i q^{95} +57.1815 q^{96} -7.11450 q^{97} -86.7224i q^{98} +16.4779 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) $ 88 * q - 192 * q^4 - 306 * q^9	$ 88 q - 192 q^{4} - 306 q^{9} - 4 q^{10} - 32 q^{11} + 26 q^{13} - 26 q^{14} + 22 q^{15} + 236 q^{16} - 12 q^{19} - 16 q^{21} - 2 q^{24} - 544 q^{25} - 96 q^{29} + 26 q^{34} + 10 q^{35} + 364 q^{36} + 44 q^{40} + 124 q^{44} - 288 q^{46} - 310 q^{47} - 694 q^{49} + 86 q^{51} - 316 q^{52} + 24 q^{53} - 266 q^{54} + 158 q^{56} - 80 q^{60} + 40 q^{62} - 652 q^{64} + 528 q^{66} + 28 q^{67} + 16 q^{69} + 94 q^{73} - 614 q^{74} - 28 q^{76} - 98 q^{78} + 928 q^{81} - 772 q^{82} + 358 q^{84} + 74 q^{85} - 410 q^{86} - 214 q^{90} + 656 q^{93} - 724 q^{96} + 346 q^{97} + 50 q^{99}+O(q^{100}) $ 88 * q - 192 * q^4 - 306 * q^9 - 4 * q^10 - 32 * q^11 + 26 * q^13 - 26 * q^14 + 22 * q^15 + 236 * q^16 - 12 * q^19 - 16 * q^21 - 2 * q^24 - 544 * q^25 - 96 * q^29 + 26 * q^34 + 10 * q^35 + 364 * q^36 + 44 * q^40 + 124 * q^44 - 288 * q^46 - 310 * q^47 - 694 * q^49 + 86 * q^51 - 316 * q^52 + 24 * q^53 - 266 * q^54 + 158 * q^56 - 80 * q^60 + 40 * q^62 - 652 * q^64 + 528 * q^66 + 28 * q^67 + 16 * q^69 + 94 * q^73 - 614 * q^74 - 28 * q^76 - 98 * q^78 + 928 * q^81 - 772 * q^82 + 358 * q^84 + 74 * q^85 - 410 * q^86 - 214 * q^90 + 656 * q^93 - 724 * q^96 + 346 * q^97 + 50 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/547\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$		−	2.76044i		−	1.38022i	−0.723706	−	0.690109i	$-0.757563\pi$
$2$		−	2.76044i		−	1.38022i	0.723706	−	0.690109i	$-0.242437\pi$
$3$		−	1.30646i		−	0.435486i	−0.976006	−	0.217743i	$-0.930131\pi$
$3$		−	1.30646i		−	0.435486i	0.976006	−	0.217743i	$-0.0698694\pi$
$4$	−3.62000			−0.905000
$5$			3.69969i			0.739937i	0.929044	+	0.369969i	$0.120632\pi$
$5$			3.69969i			0.739937i	−0.929044	+	0.369969i	$0.879368\pi$
$6$	−3.60639			−0.601065
$7$			4.19330i			0.599043i	0.954089	+	0.299522i	$0.0968271\pi$
$7$			4.19330i			0.599043i	−0.954089	+	0.299522i	$0.903173\pi$
$8$		−	1.04896i		−	0.131120i
$9$	7.29317			0.810352
$10$	10.2127			1.02127
$11$	2.25936			0.205397			0.102698	−	0.994713i	$-0.467252\pi$
$11$	2.25936			0.205397			0.102698	+	0.994713i	$0.467252\pi$
$12$			4.72938i			0.394115i
$13$	−9.40857			−0.723737			−0.361868	−	0.932229i	$-0.617861\pi$
$13$	−9.40857			−0.723737			−0.361868	+	0.932229i	$0.617861\pi$
$14$	11.5753			0.826810
$15$	4.83348			0.322232
$16$	−17.3756			−1.08597
$17$		−	20.2826i		−	1.19309i	−0.802579	−	0.596546i	$-0.796539\pi$
$17$		−	20.2826i		−	1.19309i	0.802579	−	0.596546i	$-0.203461\pi$
$18$		−	20.1323i		−	1.11846i
$19$	31.8592			1.67680			0.838401	−	0.545054i	$-0.183491\pi$
$19$	31.8592			1.67680			0.838401	+	0.545054i	$0.183491\pi$
$20$		−	13.3929i		−	0.669644i
$21$	5.47837			0.260875
$22$		−	6.23683i		−	0.283492i
$23$		−	34.4234i		−	1.49667i	−0.663322	−	0.748335i	$-0.730854\pi$
$23$		−	34.4234i		−	1.49667i	0.663322	−	0.748335i	$-0.269146\pi$
$24$	−1.37042			−0.0571009
$25$	11.3123			0.452493
$26$			25.9718i			0.998914i
$27$		−	21.2863i		−	0.788383i
$28$		−	15.1798i		−	0.542135i
$29$	−14.9273			−0.514734			−0.257367	−	0.966314i	$-0.582855\pi$
$29$	−14.9273			−0.514734			−0.257367	+	0.966314i	$0.582855\pi$
$30$		−	13.3425i		−	0.444751i
$31$		−	21.7217i		−	0.700701i	−0.936619	−	0.350351i	$-0.886062\pi$
$31$		−	21.7217i		−	0.700701i	0.936619	−	0.350351i	$-0.113938\pi$
$32$			43.7684i			1.36776i
$33$		−	2.95176i		−	0.0894474i
$34$	−55.9887			−1.64673
$35$	−15.5139			−0.443255
$36$	−26.4013			−0.733369
$37$			24.2393i			0.655115i	0.944831	+	0.327557i	$0.106225\pi$
$37$			24.2393i			0.655115i	−0.944831	+	0.327557i	$0.893775\pi$
$38$		−	87.9453i		−	2.31435i
$39$			12.2919i			0.315177i
$40$	3.88082			0.0970206
$41$		−	44.2412i		−	1.07905i	−0.841968	−	0.539527i	$-0.818603\pi$
$41$		−	44.2412i		−	1.07905i	0.841968	−	0.539527i	$-0.181397\pi$
$42$		−	15.1227i		−	0.360064i
$43$		−	79.6613i		−	1.85259i	−0.376800	−	0.926295i	$-0.622976\pi$
$43$		−	79.6613i		−	1.85259i	0.376800	−	0.926295i	$-0.377024\pi$
$44$	−8.17890			−0.185884
$45$			26.9824i			0.599610i
$46$	−95.0235			−2.06573
$47$	−31.3411			−0.666832			−0.333416	−	0.942780i	$-0.608201\pi$
$47$	−31.3411			−0.666832			−0.333416	+	0.942780i	$0.608201\pi$
$48$			22.7005i			0.472927i
$49$	31.4162			0.641147
$50$		−	31.2269i		−	0.624538i
$51$	−26.4983			−0.519574
$52$	34.0591			0.654982
$53$	53.9572			1.01806			0.509030	−	0.860749i	$-0.330004\pi$
$53$	53.9572			1.01806			0.509030	+	0.860749i	$0.330004\pi$
$54$	−58.7595			−1.08814
$55$			8.35894i			0.151981i
$56$	4.39861			0.0785466
$57$		−	41.6227i		−	0.730223i
$58$			41.2058i			0.710445i
$59$			8.72851i			0.147941i	0.997260	+	0.0739705i	$0.0235671\pi$
$59$			8.72851i			0.147941i	−0.997260	+	0.0739705i	$0.976433\pi$
$60$	−17.4972			−0.291620
$61$			119.629i			1.96113i	0.196199	+	0.980564i	$0.437140\pi$
$61$			119.629i			1.96113i	−0.196199	+	0.980564i	$0.562860\pi$
$62$	−59.9614			−0.967120
$63$			30.5825i			0.485436i
$64$	51.3173			0.801833
$65$		−	34.8088i		−	0.535520i
$66$	−8.14815			−0.123457
$67$	−1.90568			−0.0284429			−0.0142215	−	0.999899i	$-0.504527\pi$
$67$	−1.90568			−0.0284429			−0.0142215	+	0.999899i	$0.504527\pi$
$68$			73.4229i			1.07975i
$69$	−44.9727			−0.651778
$70$			42.8251i			0.611788i
$71$			72.0035i			1.01413i	0.861907	+	0.507067i	$0.169270\pi$
$71$			72.0035i			1.01413i	−0.861907	+	0.507067i	$0.830730\pi$
$72$		−	7.65024i		−	0.106253i
$73$	46.4732			0.636619			0.318309	−	0.947987i	$-0.396885\pi$
$73$	46.4732			0.636619			0.318309	+	0.947987i	$0.396885\pi$
$74$	66.9109			0.904201
$75$		−	14.7791i		−	0.197054i
$76$	−115.330			−1.51751
$77$			9.47420i			0.123042i
$78$	33.9310			0.435013
$79$			51.4586i			0.651375i	0.945478	+	0.325687i	$0.105596\pi$
$79$			51.4586i			0.651375i	−0.945478	+	0.325687i	$0.894404\pi$
$80$		−	64.2843i		−	0.803553i
$81$	37.8288			0.467022
$82$	−122.125			−1.48933
$83$			122.608i			1.47720i	0.674141	+	0.738602i	$0.264514\pi$
$83$			122.608i			1.47720i	−0.674141	+	0.738602i	$0.735486\pi$
$84$	−19.8317			−0.236092
$85$	75.0391			0.882813
$86$	−219.900			−2.55698
$87$			19.5019i			0.224159i
$88$		−	2.36998i		−	0.0269316i
$89$		−	83.1582i		−	0.934361i	−0.884162	−	0.467181i	$-0.845270\pi$
$89$		−	83.1582i		−	0.934361i	0.884162	−	0.467181i	$-0.154730\pi$
$90$	74.4833			0.827592
$91$		−	39.4530i		−	0.433550i
$92$			124.613i			1.35449i
$93$	−28.3785			−0.305145
$94$			86.5150i			0.920373i
$95$			117.869i			1.24073i
$96$	57.1815			0.595641
$97$	−7.11450			−0.0733454			−0.0366727	−	0.999327i	$-0.511676\pi$
$97$	−7.11450			−0.0733454			−0.0366727	+	0.999327i	$0.511676\pi$
$98$		−	86.7224i		−	0.884922i
$99$	16.4779			0.166444
$100$	−40.9506			−0.409506
$101$		−	169.405i		−	1.67728i	−0.544689	−	0.838638i	$-0.683352\pi$
$101$		−	169.405i		−	1.67728i	0.544689	−	0.838638i	$-0.316648\pi$
$102$			73.1468i			0.717126i
$103$		−	142.289i		−	1.38145i	−0.723117	−	0.690725i	$-0.757291\pi$
$103$		−	142.289i		−	1.38145i	0.723117	−	0.690725i	$-0.242709\pi$
$104$			9.86922i			0.0948964i
$105$			20.2683i			0.193031i
$106$		−	148.945i		−	1.40515i
$107$		−	60.6530i		−	0.566851i	−0.958994	−	0.283425i	$-0.908529\pi$
$107$		−	60.6530i		−	0.566851i	0.958994	−	0.283425i	$-0.0914709\pi$
$108$			77.0566i			0.713487i
$109$			37.2964i			0.342168i	0.985256	+	0.171084i	$0.0547270\pi$
$109$			37.2964i			0.342168i	−0.985256	+	0.171084i	$0.945273\pi$
$110$	23.0743			0.209766
$111$	31.6676			0.285293
$112$		−	72.8612i		−	0.650546i
$113$	−157.912			−1.39745			−0.698724	−	0.715392i	$-0.746248\pi$
$113$	−157.912			−1.39745			−0.698724	+	0.715392i	$0.746248\pi$
$114$	−114.897			−1.00787
$115$	127.356			1.10744
$116$	54.0368			0.465835
$117$	−68.6183			−0.586481
$118$	24.0945			0.204191
$119$	85.0509			0.714714
$120$		−	5.07013i		−	0.0422511i
$121$	−115.895			−0.957812
$122$	330.228			2.70678
$123$	−57.7993			−0.469913
$124$			78.6327i			0.634135i
$125$			134.344i			1.07475i
$126$	84.4209			0.670007
$127$	241.014			1.89775			0.948874	−	0.315654i	$-0.102224\pi$
$127$	241.014			1.89775			0.948874	+	0.315654i	$0.102224\pi$
$128$			33.4153i			0.261057i
$129$	−104.074			−0.806776
$130$	−96.0874			−0.739134
$131$	−151.518			−1.15663			−0.578313	−	0.815815i	$-0.696289\pi$
$131$	−151.518			−1.15663			−0.578313	+	0.815815i	$0.696289\pi$
$132$			10.6854i			0.0809499i
$133$			133.595i			1.00448i
$134$			5.26050i			0.0392574i
$135$	78.7528			0.583354
$136$	−21.2756			−0.156438
$137$	198.542			1.44921			0.724606	−	0.689163i	$-0.242022\pi$
$137$	198.542			1.44921			0.724606	+	0.689163i	$0.242022\pi$
$138$			124.144i			0.899596i
$139$	−62.0060			−0.446086			−0.223043	−	0.974809i	$-0.571599\pi$
$139$	−62.0060			−0.446086			−0.223043	+	0.974809i	$0.571599\pi$
$140$	56.1604			0.401146
$141$			40.9458i			0.290396i
$142$	198.761			1.39973
$143$	−21.2574			−0.148653
$144$	−126.723			−0.880022
$145$		−	55.2263i		−	0.380871i
$146$		−	128.286i		−	0.878672i
$147$		−	41.0439i		−	0.279210i
$148$		−	87.7461i		−	0.592879i
$149$	163.056			1.09434			0.547169	−	0.837022i	$-0.315706\pi$
$149$	163.056			1.09434			0.547169	+	0.837022i	$0.315706\pi$
$150$	−40.7967			−0.271978
$151$		−	87.3540i		−	0.578504i	−0.957253	−	0.289252i	$-0.906594\pi$
$151$		−	87.3540i		−	0.578504i	0.957253	−	0.289252i	$-0.0934065\pi$
$152$		−	33.4191i		−	0.219862i
$153$		−	147.924i		−	0.966824i
$154$	26.1529			0.169824
$155$	80.3636			0.518475
$156$		−	44.4967i		−	0.285235i
$157$	−284.545			−1.81239			−0.906196	−	0.422859i	$-0.861027\pi$
$157$	−284.545			−1.81239			−0.906196	+	0.422859i	$0.861027\pi$
$158$	142.048			0.899039
$159$		−	70.4928i		−	0.443351i
$160$	−161.929			−1.01206
$161$	144.348			0.896570
$162$		−	104.424i		−	0.644593i
$163$			232.379i			1.42564i	0.701349	+	0.712818i	$0.252582\pi$
$163$			232.379i			1.42564i	−0.701349	+	0.712818i	$0.747418\pi$
$164$			160.153i			0.976545i
$165$	10.9206			0.0661854
$166$	338.451			2.03886
$167$	−125.381			−0.750785			−0.375393	−	0.926866i	$-0.622492\pi$
$167$	−125.381			−0.750785			−0.375393	+	0.926866i	$0.622492\pi$
$168$		−	5.74660i		−	0.0342059i
$169$	−80.4787			−0.476205
$170$		−	207.141i		−	1.21847i
$171$	232.355			1.35880
$172$			288.374i			1.67659i
$173$			27.8926i			0.161229i	0.996745	+	0.0806143i	$0.0256882\pi$
$173$			27.8926i			0.161229i	−0.996745	+	0.0806143i	$0.974312\pi$
$174$	53.8336			0.309389
$175$			47.4360i			0.271063i
$176$	−39.2578			−0.223056
$177$	11.4034			0.0644262
$178$	−229.553			−1.28962
$179$	208.515			1.16489			0.582446	−	0.812870i	$-0.302096\pi$
$179$	208.515			1.16489			0.582446	+	0.812870i	$0.302096\pi$
$180$		−	97.6765i		−	0.542647i
$181$	105.660			0.583755			0.291878	−	0.956456i	$-0.405720\pi$
$181$	105.660			0.583755			0.291878	+	0.956456i	$0.405720\pi$
$182$	−108.907			−0.598393
$183$	156.290			0.854044
$184$	−36.1088			−0.196243
$185$	−89.6776			−0.484744
$186$			78.3371i			0.421167i
$187$		−	45.8257i		−	0.245057i
$188$	113.455			0.603483
$189$	89.2601			0.472276
$190$	325.370			1.71247
$191$	174.394			0.913059			0.456529	−	0.889708i	$-0.349092\pi$
$191$	174.394			0.913059			0.456529	+	0.889708i	$0.349092\pi$
$192$		−	67.0439i		−	0.349187i
$193$	−265.901			−1.37773			−0.688863	−	0.724891i	$-0.741890\pi$
$193$	−265.901			−1.37773			−0.688863	+	0.724891i	$0.741890\pi$
$194$			19.6391i			0.101233i
$195$	−45.4762			−0.233211
$196$	−113.727			−0.580238
$197$		−	295.751i		−	1.50128i	−0.660714	−	0.750638i	$-0.729746\pi$
$197$		−	295.751i		−	1.50128i	0.660714	−	0.750638i	$-0.270254\pi$
$198$		−	45.4862i		−	0.229728i
$199$	199.914			1.00459			0.502296	−	0.864696i	$-0.332489\pi$
$199$	199.914			1.00459			0.502296	+	0.864696i	$0.332489\pi$
$200$		−	11.8662i		−	0.0593309i
$201$			2.48969i			0.0123865i
$202$	−467.631			−2.31501
$203$		−	62.5947i		−	0.308348i
$204$	95.9239			0.470215
$205$	163.679			0.798433
$206$	−392.781			−1.90670
$207$		−	251.056i		−	1.21283i
$208$	163.480			0.785959
$209$	71.9816			0.344409
$210$	55.9492			0.266425
$211$			194.892i			0.923660i	0.886968	+	0.461830i	$0.152807\pi$
$211$			194.892i			0.923660i	−0.886968	+	0.461830i	$0.847193\pi$
$212$	−195.325			−0.921345
$213$	94.0696			0.441641
$214$	−167.429			−0.782378
$215$	294.722			1.37080
$216$	−22.3285			−0.103373
$217$	91.0858			0.419750
$218$	102.954			0.472267
$219$		−	60.7152i		−	0.277238i
$220$		−	30.2594i		−	0.137543i
$221$			190.830i			0.863484i
$222$		−	87.4162i		−	0.393767i
$223$			177.513i			0.796022i	0.917381	+	0.398011i	$0.130299\pi$
$223$			177.513i			0.796022i	−0.917381	+	0.398011i	$0.869701\pi$
$224$	−183.534			−0.819348
$225$	82.5027			0.366678
$226$			435.905i			1.92878i
$227$	280.322			1.23490			0.617450	−	0.786610i	$-0.288166\pi$
$227$	280.322			1.23490			0.617450	+	0.786610i	$0.288166\pi$
$228$			150.674i			0.660852i
$229$			279.940i			1.22244i	0.791459	+	0.611222i	$0.209322\pi$
$229$			279.940i			1.22244i	−0.791459	+	0.611222i	$0.790678\pi$
$230$		−	351.557i		−	1.52851i
$231$	12.3776			0.0535829
$232$			15.6581i			0.0674920i
$233$	−172.668			−0.741064			−0.370532	−	0.928820i	$-0.620825\pi$
$233$	−172.668			−0.741064			−0.370532	+	0.928820i	$0.620825\pi$
$234$			189.416i			0.809472i
$235$		−	115.952i		−	0.493414i
$236$		−	31.5972i		−	0.133887i
$237$	67.2285			0.283664
$238$		−	234.778i		−	0.986460i
$239$	62.0041			0.259431			0.129716	−	0.991551i	$-0.458594\pi$
$239$	62.0041			0.259431			0.129716	+	0.991551i	$0.458594\pi$
$240$	−83.9847			−0.349936
$241$			293.771i			1.21897i	0.792799	+	0.609483i	$0.208623\pi$
$241$			293.771i			1.21897i	−0.792799	+	0.609483i	$0.791377\pi$
$242$			319.921i			1.32199i
$243$		−	240.999i		−	0.991764i
$244$		−	433.057i		−	1.77482i
$245$			116.230i			0.474409i
$246$			159.551i			0.648582i
$247$	−299.750			−1.21356
$248$	−22.7852			−0.0918760
$249$	160.182			0.643302
$250$	370.848			1.48339
$251$		−	11.7288i		−	0.0467284i	−0.999727	−	0.0233642i	$-0.992562\pi$
$251$		−	11.7288i		−	0.0467284i	0.999727	−	0.0233642i	$-0.00743773\pi$
$252$		−	110.709i		−	0.439320i
$253$		−	77.7750i		−	0.307411i
$254$		−	665.304i		−	2.61931i
$255$		−	98.0354i		−	0.384453i
$256$	297.510			1.16215
$257$			221.128i			0.860419i	0.902729	+	0.430210i	$0.141560\pi$
$257$			221.128i			0.860419i	−0.902729	+	0.430210i	$0.858440\pi$
$258$			287.290i			1.11353i
$259$	−101.643			−0.392442
$260$			126.008i			0.484646i
$261$	−108.867			−0.417116
$262$			418.255i			1.59639i
$263$	73.7094			0.280264			0.140132	−	0.990133i	$-0.455247\pi$
$263$	73.7094			0.280264			0.140132	+	0.990133i	$0.455247\pi$
$264$	−3.09628			−0.0117283
$265$			199.625i			0.753301i
$266$	368.781			1.38640
$267$	−108.643			−0.406901
$268$	6.89855			0.0257409
$269$	−83.5784			−0.310700			−0.155350	−	0.987859i	$-0.549651\pi$
$269$	−83.5784			−0.310700			−0.155350	+	0.987859i	$0.549651\pi$
$270$		−	217.392i		−	0.805155i
$271$			454.441i			1.67690i	0.544977	+	0.838451i	$0.316538\pi$
$271$			454.441i			1.67690i	−0.544977	+	0.838451i	$0.683462\pi$
$272$			352.421i			1.29567i
$273$	−51.5437			−0.188805
$274$		−	548.062i		−	2.00023i
$275$	25.5586			0.0929405
$276$	162.801			0.589860
$277$	−262.401			−0.947296			−0.473648	−	0.880714i	$-0.657063\pi$
$277$	−262.401			−0.947296			−0.473648	+	0.880714i	$0.657063\pi$
$278$			171.164i			0.615696i
$279$		−	158.420i		−	0.567815i
$280$			16.2735i			0.0581196i
$281$		−	11.5643i		−	0.0411541i	−0.999788	−	0.0205770i	$-0.993450\pi$
$281$		−	11.5643i		−	0.0411541i	0.999788	−	0.0205770i	$-0.00655034\pi$
$282$	113.028			0.400809
$283$		−	171.107i		−	0.604620i	−0.953210	−	0.302310i	$-0.902242\pi$
$283$		−	171.107i		−	0.604620i	0.953210	−	0.302310i	$-0.0977578\pi$
$284$		−	260.653i		−	0.917792i
$285$	153.991			0.540319
$286$			58.6797i			0.205174i
$287$	185.517			0.646400
$288$			319.210i			1.10837i
$289$	−122.382			−0.423467
$290$	−152.449			−0.525685
$291$			9.29480i			0.0319409i
$292$	−168.233			−0.576140
$293$	562.091			1.91840			0.959200	−	0.282728i	$-0.0912394\pi$
$293$	562.091			1.91840			0.959200	+	0.282728i	$0.0912394\pi$
$294$	−113.299			−0.385371
$295$	−32.2928			−0.109467
$296$	25.4260			0.0858987
$297$		−	48.0936i		−	0.161931i
$298$		−	450.106i		−	1.51042i
$299$			323.875i			1.08319i
$300$			53.5002i			0.178334i
$301$	334.044			1.10978
$302$	−241.135			−0.798461
$303$	−221.320			−0.730430
$304$	−553.573			−1.82096
$305$	−442.589			−1.45111
$306$	−408.335			−1.33443
$307$			212.866i			0.693373i	0.937981	+	0.346687i	$0.112693\pi$
$307$			212.866i			0.693373i	−0.937981	+	0.346687i	$0.887307\pi$
$308$		−	34.2966i		−	0.111353i
$309$	−185.895			−0.601602
$310$		−	221.839i		−	0.715608i
$311$	−67.7029			−0.217694			−0.108847	−	0.994058i	$-0.534716\pi$
$311$	−67.7029			−0.217694			−0.108847	+	0.994058i	$0.534716\pi$
$312$	12.8937			0.0413260
$313$	−505.615			−1.61538			−0.807691	−	0.589606i	$-0.799283\pi$
$313$	−505.615			−1.61538			−0.807691	+	0.589606i	$0.799283\pi$
$314$			785.469i			2.50149i
$315$	−113.146			−0.359192
$316$		−	186.280i		−	0.589494i
$317$	−581.296			−1.83374			−0.916870	−	0.399185i	$-0.869293\pi$
$317$	−581.296			−1.83374			−0.916870	+	0.399185i	$0.869293\pi$
$318$	−194.591			−0.611921
$319$	−33.7262			−0.105725
$320$			189.858i			0.593306i
$321$	−79.2406			−0.246856
$322$		−	398.463i		−	1.23746i
$323$		−	646.186i		−	2.00058i
$324$	−136.940			−0.422656
$325$	−106.433			−0.327486
$326$	641.467			1.96769
$327$	48.7261			0.149010
$328$	−46.4073			−0.141486
$329$		−	131.423i		−	0.399461i
$330$		−	30.1456i		−	0.0913503i
$331$			108.791i			0.328675i	0.986404	+	0.164337i	$0.0525486\pi$
$331$			108.791i			0.328675i	−0.986404	+	0.164337i	$0.947451\pi$
$332$		−	443.841i		−	1.33687i
$333$			176.781i			0.530874i
$334$			346.106i			1.03625i
$335$		−	7.05041i		−	0.0210460i
$336$	−95.1900			−0.283304
$337$		−	545.930i		−	1.61997i	−0.586450	−	0.809986i	$-0.699475\pi$
$337$		−	545.930i		−	1.61997i	0.586450	−	0.809986i	$-0.300525\pi$
$338$			222.156i			0.657267i
$339$			206.305i			0.608569i
$340$	−271.642			−0.798946
$341$		−	49.0773i		−	0.143922i
$342$		−	641.400i		−	1.87544i
$343$			337.210i			0.983118i
$344$	−83.5616			−0.242912
$345$		−	166.385i		−	0.482275i
$346$	76.9956			0.222531
$347$	244.695			0.705173			0.352587	−	0.935779i	$-0.385302\pi$
$347$	244.695			0.705173			0.352587	+	0.935779i	$0.385302\pi$
$348$		−	70.5968i		−	0.202864i
$349$	−156.094			−0.447260			−0.223630	−	0.974674i	$-0.571791\pi$
$349$	−156.094			−0.447260			−0.223630	+	0.974674i	$0.571791\pi$
$350$	130.944			0.374126
$351$			200.274i			0.570581i
$352$			98.8886i			0.280934i
$353$	−128.751			−0.364734			−0.182367	−	0.983231i	$-0.558376\pi$
$353$	−128.751			−0.364734			−0.182367	+	0.983231i	$0.558376\pi$
$354$		−	31.4784i		−	0.0889221i
$355$	−266.391			−0.750396
$356$			301.033i			0.845597i
$357$		−	111.115i		−	0.311248i
$358$		−	575.593i		−	1.60780i
$359$			385.188i			1.07295i	0.843917	+	0.536473i	$0.180244\pi$
$359$			385.188i			1.07295i	−0.843917	+	0.536473i	$0.819756\pi$
$360$	28.3035			0.0786209
$361$	654.010			1.81166
$362$		−	291.667i		−	0.805709i
$363$			151.412i			0.417114i
$364$			142.820i			0.392363i
$365$			171.936i			0.471058i
$366$		−	431.428i		−	1.17877i
$367$	−134.592			−0.366735			−0.183368	−	0.983044i	$-0.558700\pi$
$367$	−134.592			−0.366735			−0.183368	+	0.983044i	$0.558700\pi$
$368$			598.127i			1.62534i
$369$		−	322.659i		−	0.874414i
$370$			247.549i			0.669052i
$371$			226.259i			0.609863i
$372$	102.730			0.276157
$373$			410.831i			1.10142i	0.834695	+	0.550712i	$0.185644\pi$
$373$			410.831i			1.10142i	−0.834695	+	0.550712i	$0.814356\pi$
$374$	−126.499			−0.338232
$375$	175.515			0.468040
$376$			32.8756i			0.0874350i
$377$	140.445			0.372532
$378$		−	246.397i		−	0.651843i
$379$	−335.415			−0.885000			−0.442500	−	0.896769i	$-0.645908\pi$
$379$	−335.415			−0.885000			−0.442500	+	0.896769i	$0.645908\pi$
$380$		−	426.686i		−	1.12286i
$381$		−	314.875i		−	0.826443i
$382$		−	481.404i		−	1.26022i
$383$	403.421			1.05332			0.526660	−	0.850076i	$-0.323444\pi$
$383$	403.421			1.05332			0.526660	+	0.850076i	$0.323444\pi$
$384$	43.6556			0.113687
$385$	−35.0516			−0.0910430
$386$			734.003i			1.90156i
$387$		−	580.984i		−	1.50125i
$388$	25.7545			0.0663776
$389$			604.322i			1.55353i	0.629792	+	0.776764i	$0.283140\pi$
$389$			604.322i			1.55353i	−0.629792	+	0.776764i	$0.716860\pi$
$390$			125.534i			0.321882i
$391$	−698.194			−1.78566
$392$		−	32.9543i		−	0.0840672i
$393$			197.952i			0.503694i
$394$	−816.403			−2.07209
$395$	−190.381			−0.481976
$396$	−59.6501			−0.150632
$397$			558.630i			1.40713i	0.710632	+	0.703564i	$0.248409\pi$
$397$			558.630i			1.40713i	−0.710632	+	0.703564i	$0.751591\pi$
$398$		−	551.849i		−	1.38656i
$399$	174.537			0.437435
$400$	−196.558			−0.491396
$401$	477.788			1.19149			0.595746	−	0.803173i	$-0.296857\pi$
$401$	477.788			1.19149			0.595746	+	0.803173i	$0.296857\pi$
$402$	6.87262			0.0170961
$403$			204.371i			0.507123i
$404$			613.246i			1.51794i
$405$			139.955i			0.345567i
$406$	−172.789			−0.425587
$407$			54.7653i			0.134558i
$408$			27.7957i			0.0681266i
$409$	−649.839			−1.58885			−0.794425	−	0.607363i	$-0.792228\pi$
$409$	−649.839			−1.58885			−0.794425	+	0.607363i	$0.792228\pi$
$410$		−	451.824i		−	1.10201i
$411$		−	259.387i		−	0.631111i
$412$			515.088i			1.25021i
$413$	−36.6013			−0.0886230
$414$	−693.023			−1.67397
$415$	−453.611			−1.09304
$416$		−	411.798i		−	0.989899i
$417$			81.0082i			0.194264i
$418$		−	198.700i		−	0.475360i
$419$	228.610			0.545608			0.272804	−	0.962070i	$-0.412049\pi$
$419$	228.610			0.545608			0.272804	+	0.962070i	$0.412049\pi$
$420$		−	73.3712i		−	0.174693i
$421$			56.3021i			0.133734i	0.997762	+	0.0668671i	$0.0213003\pi$
$421$			56.3021i			0.133734i	−0.997762	+	0.0668671i	$0.978700\pi$
$422$	537.987			1.27485
$423$	−228.576			−0.540368
$424$		−	56.5990i		−	0.133488i
$425$		−	229.443i		−	0.539865i
$426$		−	259.673i		−	0.609561i
$427$	−501.640			−1.17480
$428$			219.564i			0.513000i
$429$			27.7719i			0.0647363i
$430$		−	813.561i		−	1.89200i
$431$		−	181.031i		−	0.420025i	−0.977699	−	0.210012i	$-0.932650\pi$
$431$		−	181.031i		−	0.420025i	0.977699	−	0.210012i	$-0.0673504\pi$
$432$			369.863i			0.856164i
$433$			419.008i			0.967685i	0.875155	+	0.483843i	$0.160759\pi$
$433$			419.008i			0.967685i	−0.875155	+	0.483843i	$0.839241\pi$
$434$		−	251.437i		−	0.579347i
$435$	−72.1508			−0.165864
$436$		−	135.013i		−	0.309663i
$437$		−	1096.70i		−	2.50962i
$438$	−167.600			−0.382649
$439$	−543.228			−1.23742			−0.618710	−	0.785619i	$-0.712345\pi$
$439$	−543.228			−1.23742			−0.618710	+	0.785619i	$0.712345\pi$
$440$	8.76820			0.0199277
$441$	229.124			0.519555
$442$	526.774			1.19180
$443$	885.338			1.99851			0.999253	−	0.0386498i	$-0.0123057\pi$
$443$	885.338			1.99851			0.999253	+	0.0386498i	$0.0123057\pi$
$444$	−114.637			−0.258191
$445$	307.659			0.691369
$446$	490.013			1.09868
$447$		−	213.026i		−	0.476568i
$448$			215.189i			0.480333i
$449$	−343.135			−0.764221			−0.382110	−	0.924117i	$-0.624803\pi$
$449$	−343.135			−0.764221			−0.382110	+	0.924117i	$0.624803\pi$
$450$		−	227.743i		−	0.506096i
$451$		−	99.9570i		−	0.221634i
$452$	571.640			1.26469
$453$	−114.124			−0.251930
$454$		−	773.811i		−	1.70443i
$455$	145.964			0.320800
$456$	−43.6606			−0.0957469
$457$		−	683.085i		−	1.49472i	−0.664422	−	0.747358i	$-0.731322\pi$
$457$		−	683.085i		−	1.49472i	0.664422	−	0.747358i	$-0.268678\pi$
$458$	772.755			1.68724
$459$	−431.741			−0.940613
$460$	−461.028			−1.00223
$461$			499.587i			1.08370i	0.840474	+	0.541851i	$0.182276\pi$
$461$			499.587i			1.08370i	−0.840474	+	0.541851i	$0.817724\pi$
$462$		−	34.1677i		−	0.0739560i
$463$			416.299i			0.899135i	0.893246	+	0.449567i	$0.148422\pi$
$463$			416.299i			0.899135i	−0.893246	+	0.449567i	$0.851578\pi$
$464$	259.371			0.558988
$465$		−	104.992i		−	0.225788i
$466$			476.639i			1.02283i
$467$	405.203			0.867673			0.433836	−	0.900992i	$-0.357160\pi$
$467$	405.203			0.867673			0.433836	+	0.900992i	$0.357160\pi$
$468$	248.398			0.530766
$469$		−	7.99108i		−	0.0170386i
$470$	−320.078			−0.681018
$471$			371.747i			0.789271i
$472$	9.15587			0.0193980
$473$		−	179.984i		−	0.380516i
$474$		−	185.580i		−	0.391519i
$475$	360.402			0.758740
$476$	−307.884			−0.646816
$477$	393.519			0.824988
$478$		−	171.158i		−	0.358072i
$479$	−77.3436			−0.161469			−0.0807344	−	0.996736i	$-0.525727\pi$
$479$	−77.3436			−0.161469			−0.0807344	+	0.996736i	$0.525727\pi$
$480$			211.554i			0.440737i
$481$		−	228.057i		−	0.474131i
$482$	810.936			1.68244
$483$		−	188.584i		−	0.390444i
$484$	419.541			0.866820
$485$		−	26.3214i		−	0.0542710i
$486$	−665.261			−1.36885
$487$			352.023i			0.722841i	0.932403	+	0.361420i	$0.117708\pi$
$487$			352.023i			0.722841i	−0.932403	+	0.361420i	$0.882292\pi$
$488$	125.486			0.257143
$489$	303.593			0.620845
$490$	320.846			0.654787
$491$			532.523i			1.08457i	0.840195	+	0.542284i	$0.182440\pi$
$491$			532.523i			1.08457i	−0.840195	+	0.542284i	$0.817560\pi$
$492$	209.234			0.425271
$493$			302.764i			0.614125i
$494$			827.440i			1.67498i
$495$			60.9631i			0.123158i
$496$			377.428i			0.760944i
$497$	−301.933			−0.607511
$498$		−	442.172i		−	0.887896i
$499$	890.843			1.78526			0.892628	−	0.450794i	$-0.148859\pi$
$499$	890.843			1.78526			0.892628	+	0.450794i	$0.148859\pi$
$500$		−	486.326i		−	0.972652i
$501$			163.805i			0.326956i
$502$	−32.3767			−0.0644953
$503$		−	407.590i		−	0.810319i	−0.914246	−	0.405159i	$-0.867216\pi$
$503$		−	407.590i		−	0.810319i	0.914246	−	0.405159i	$-0.132784\pi$
$504$	32.0798			0.0636504
$505$	626.745			1.24108
$506$	−214.693			−0.424294
$507$			105.142i			0.207381i
$508$	−872.471			−1.71746
$509$	−907.226			−1.78237			−0.891185	−	0.453641i	$-0.850125\pi$
$509$	−907.226			−1.78237			−0.891185	+	0.453641i	$0.850125\pi$
$510$	−270.620			−0.530628
$511$			194.876i			0.381362i
$512$		−	687.596i		−	1.34296i
$513$		−	678.166i		−	1.32196i
$514$	610.409			1.18757
$515$	526.426			1.02219
$516$	376.749			0.730133
$517$	−70.8109			−0.136965
$518$			280.578i			0.541656i
$519$	36.4405			0.0702128
$520$	−36.5130			−0.0702174
$521$	−176.308			−0.338403			−0.169202	−	0.985581i	$-0.554119\pi$
$521$	−176.308			−0.338403			−0.169202	+	0.985581i	$0.554119\pi$
$522$			300.521i			0.575711i
$523$		−	551.409i		−	1.05432i	−0.849766	−	0.527160i	$-0.823257\pi$
$523$		−	551.409i		−	1.05432i	0.849766	−	0.527160i	$-0.176743\pi$
$524$	548.495			1.04675
$525$	61.9731			0.118044
$526$		−	203.470i		−	0.386825i
$527$	−440.572			−0.836000
$528$			51.2886i			0.0971376i
$529$	−655.970			−1.24002
$530$	551.051			1.03972
$531$			63.6585i			0.119884i
$532$		−	483.616i		−	0.909052i
$533$			416.247i			0.780951i
$534$			299.901i			0.561612i
$535$	224.397			0.419434
$536$			1.99898i			0.00372944i
$537$		−	272.417i		−	0.507294i
$538$			230.713i			0.428834i
$539$	70.9806			0.131689
$540$	−285.085			−0.527935
$541$		−	908.496i		−	1.67929i	−0.543136	−	0.839645i	$-0.682763\pi$
$541$		−	908.496i		−	1.67929i	0.543136	−	0.839645i	$-0.317237\pi$
$542$	1254.45			2.31449
$543$		−	138.040i		−	0.254217i
$544$	887.734			1.63186
$545$	−137.985			−0.253183
$546$			142.283i			0.260592i
$547$	457.475	−	299.875i	0.836335	−	0.548218i
$547$	457.475	−	299.875i	0.836335	−	0.548218i
$548$	−718.722			−1.31154
$549$			872.473i			1.58920i
$550$		−	70.5530i		−	0.128278i
$551$	−475.572			−0.863107
$552$			47.1746i			0.0854612i
$553$	−215.782			−0.390202
$554$			724.341i			1.30747i
$555$			117.160i			0.211099i
$556$	224.462			0.403708
$557$	−597.431			−1.07259			−0.536293	−	0.844032i	$-0.680176\pi$
$557$	−597.431			−1.07259			−0.536293	+	0.844032i	$0.680176\pi$
$558$	−437.309			−0.783708
$559$			749.500i			1.34079i
$560$	269.563			0.481363
$561$	−59.8693			−0.106719
$562$	−31.9225			−0.0568016
$563$	−104.202			−0.185084			−0.0925418	−	0.995709i	$-0.529499\pi$
$563$	−104.202			−0.185084			−0.0925418	+	0.995709i	$0.529499\pi$
$564$		−	148.224i		−	0.262808i
$565$		−	584.223i		−	1.03402i
$566$	−472.331			−0.834507
$567$			158.628i			0.279767i
$568$	75.5289			0.132973
$569$			81.3757i			0.143015i	0.997440	+	0.0715076i	$0.0227810\pi$
$569$			81.3757i			0.143015i	−0.997440	+	0.0715076i	$0.977219\pi$
$570$		−	425.082i		−	0.745758i
$571$	1064.79			1.86478			0.932388	−	0.361458i	$-0.117721\pi$
$571$	1064.79			1.86478			0.932388	+	0.361458i	$0.117721\pi$
$572$	76.9518			0.134531
$573$		−	227.839i		−	0.397624i
$574$		−	512.107i		−	0.892173i
$575$		−	389.408i		−	0.677232i
$576$	374.266			0.649767
$577$			230.445i			0.399386i	0.979859	+	0.199693i	$0.0639944\pi$
$577$			230.445i			0.399386i	−0.979859	+	0.199693i	$0.936006\pi$
$578$			337.828i			0.584477i
$579$			347.389i			0.599980i
$580$			199.919i			0.344688i
$581$	−514.133			−0.884910
$582$	25.6577			0.0440854
$583$	121.909			0.209106
$584$		−	48.7485i		−	0.0834735i
$585$		−	253.866i		−	0.433959i
$586$		−	1551.62i		−	2.64781i
$587$	−809.566			−1.37916			−0.689579	−	0.724211i	$-0.742204\pi$
$587$	−809.566			−1.37916			−0.689579	+	0.724211i	$0.742204\pi$
$588$			148.579i			0.252686i
$589$		−	692.038i		−	1.17494i
$590$			89.1421i			0.151088i
$591$	−386.387			−0.653785
$592$		−	421.171i		−	0.711438i
$593$	534.969			0.902139			0.451070	−	0.892489i	$-0.351043\pi$
$593$	534.969			0.902139			0.451070	+	0.892489i	$0.351043\pi$
$594$	−132.759			−0.223500
$595$			314.662i			0.528843i
$596$	−590.264			−0.990376
$597$		−	261.179i		−	0.437486i
$598$	894.036			1.49504
$599$	−1034.51			−1.72706			−0.863530	−	0.504297i	$-0.831752\pi$
$599$	−1034.51			−1.72706			−0.863530	+	0.504297i	$0.831752\pi$
$600$	−15.5027			−0.0258378
$601$	−466.596			−0.776366			−0.388183	−	0.921582i	$-0.626897\pi$
$601$	−466.596			−0.776366			−0.388183	+	0.921582i	$0.626897\pi$
$602$		−	922.107i		−	1.53174i
$603$	−13.8984			−0.0230488
$604$			316.222i			0.523546i
$605$		−	428.776i		−	0.708721i
$606$			610.941i			1.00815i
$607$	651.797			1.07380			0.536900	−	0.843646i	$-0.319595\pi$
$607$	651.797			1.07380			0.536900	+	0.843646i	$0.319595\pi$
$608$			1394.43i			2.29346i
$609$	−81.7773			−0.134281
$610$			1221.74i			2.00285i
$611$	294.875			0.482610
$612$			535.485i			0.874976i
$613$	−552.703			−0.901636			−0.450818	−	0.892616i	$-0.648868\pi$
$613$	−552.703			−0.901636			−0.450818	+	0.892616i	$0.648868\pi$
$614$	587.601			0.957006
$615$		−	213.839i		−	0.347706i
$616$	9.93806			0.0161332
$617$			747.560i			1.21161i	0.795615	+	0.605803i	$0.207148\pi$
$617$			747.560i			1.21161i	−0.795615	+	0.605803i	$0.792852\pi$
$618$			513.151i			0.830342i
$619$		−	1152.11i		−	1.86124i	−0.365991	−	0.930619i	$-0.619270\pi$
$619$		−	1152.11i		−	1.86124i	0.365991	−	0.930619i	$-0.380730\pi$
$620$	−290.916			−0.469220
$621$	−732.748			−1.17995
$622$			186.890i			0.300466i
$623$	348.707			0.559723
$624$		−	213.579i		−	0.342274i
$625$	−214.223			−0.342757
$626$			1395.72i			2.22958i
$627$		−	94.0409i		−	0.149985i
$628$	1030.05			1.64021
$629$	491.634			0.781612
$630$			312.331i			0.495763i
$631$	869.645			1.37820			0.689101	−	0.724666i	$-0.258006\pi$
$631$	869.645			1.37820			0.689101	+	0.724666i	$0.258006\pi$
$632$	53.9780			0.0854083
$633$	254.618			0.402241
$634$			1604.63i			2.53096i
$635$			891.676i			1.40421i
$636$			255.184i			0.401233i
$637$	−295.582			−0.464021
$638$			93.0989i			0.145923i
$639$			525.134i			0.821806i
$640$	−123.626			−0.193166
$641$			363.632i			0.567288i	0.958930	+	0.283644i	$0.0915434\pi$
$641$			363.632i			0.567288i	−0.958930	+	0.283644i	$0.908457\pi$
$642$			218.739i			0.340714i
$643$	377.275			0.586742			0.293371	−	0.955999i	$-0.405223\pi$
$643$	377.275			0.586742			0.293371	+	0.955999i	$0.405223\pi$
$644$	−522.539			−0.811396
$645$		−	385.042i		−	0.596964i
$646$	−1783.76			−2.76123
$647$	473.881			0.732429			0.366214	−	0.930531i	$-0.380654\pi$
$647$	473.881			0.732429			0.366214	+	0.930531i	$0.380654\pi$
$648$		−	39.6809i		−	0.0612360i
$649$			19.7209i			0.0303866i
$650$			293.801i			0.452001i
$651$		−	119.000i		−	0.182795i
$652$		−	841.212i		−	1.29020i
$653$		−	970.334i		−	1.48596i	−0.669311	−	0.742982i	$-0.733411\pi$
$653$		−	970.334i		−	1.48596i	0.669311	−	0.742982i	$-0.266589\pi$
$654$		−	134.505i		−	0.205666i
$655$		−	560.569i		−	0.855830i
$656$			768.718i			1.17183i
$657$	338.937			0.515885
$658$	−362.784			−0.551343
$659$		−	881.433i		−	1.33753i	−0.743474	−	0.668765i	$-0.766823\pi$
$659$		−	881.433i		−	1.33753i	0.743474	−	0.668765i	$-0.233177\pi$
$660$	−39.5326			−0.0598979
$661$	271.964			0.411444			0.205722	−	0.978611i	$-0.434046\pi$
$661$	271.964			0.411444			0.205722	+	0.978611i	$0.434046\pi$
$662$	300.311			0.453643
$663$	249.311			0.376035
$664$	128.611			0.193691
$665$	−494.261			−0.743250
$666$	487.992			0.732721
$667$			513.848i			0.770387i
$668$	453.880			0.679461
$669$	231.913			0.346656
$670$	−19.4622			−0.0290481
$671$			270.285i			0.402809i
$672$			239.779i			0.356815i
$673$	−509.395			−0.756902			−0.378451	−	0.925621i	$-0.623543\pi$
$673$	−509.395			−0.756902			−0.378451	+	0.925621i	$0.623543\pi$
$674$	−1507.01			−2.23591
$675$		−	240.798i		−	0.356738i
$676$	291.333			0.430966
$677$	123.800			0.182866			0.0914331	−	0.995811i	$-0.470855\pi$
$677$	123.800			0.182866			0.0914331	+	0.995811i	$0.470855\pi$
$678$	569.491			0.839957
$679$		−	29.8333i		−	0.0439371i
$680$		−	78.7130i		−	0.115754i
$681$		−	366.229i		−	0.537781i
$682$	−135.475			−0.198643
$683$	117.481			0.172008			0.0860038	−	0.996295i	$-0.472590\pi$
$683$	117.481			0.172008			0.0860038	+	0.996295i	$0.472590\pi$
$684$	−841.124			−1.22971
$685$			734.543i			1.07233i
$686$	930.845			1.35692
$687$	365.729			0.532357
$688$			1384.16i			2.01186i
$689$	−507.661			−0.736808
$690$	−459.295			−0.665644
$691$	274.166			0.396768			0.198384	−	0.980124i	$-0.436431\pi$
$691$	274.166			0.396768			0.198384	+	0.980124i	$0.436431\pi$
$692$		−	100.971i		−	0.145912i
$693$			69.0969i			0.0997070i
$694$		−	675.465i		−	0.973292i
$695$		−	229.403i		−	0.330076i
$696$	20.4567			0.0293918
$697$	−897.325			−1.28741
$698$			430.887i			0.617316i
$699$			225.583i			0.322723i
$700$		−	171.718i		−	0.245312i
$701$	509.173			0.726352			0.363176	−	0.931721i	$-0.381692\pi$
$701$	509.173			0.726352			0.363176	+	0.931721i	$0.381692\pi$
$702$	552.844			0.787526
$703$			772.244i			1.09850i
$704$	115.945			0.164694
$705$	−151.487			−0.214875
$706$			355.409i			0.503412i
$707$	710.366			1.00476
$708$	−41.2804			−0.0583057
$709$		−	825.256i		−	1.16397i	−0.813199	−	0.581986i	$-0.802276\pi$
$709$		−	825.256i		−	1.16397i	0.813199	−	0.581986i	$-0.197724\pi$
$710$			735.354i			1.03571i
$711$			375.296i			0.527843i
$712$	−87.2296			−0.122513
$713$	−747.736			−1.04872
$714$	−306.727			−0.429590
$715$		−	78.6457i		−	0.109994i
$716$	−754.826			−1.05423
$717$		−	81.0057i		−	0.112979i
$718$	1063.29			1.48090
$719$			66.9545i			0.0931217i	0.998915	+	0.0465608i	$0.0148261\pi$
$719$			66.9545i			0.0931217i	−0.998915	+	0.0465608i	$0.985174\pi$
$720$		−	468.836i		−	0.651161i
$721$	596.663			0.827549
$722$		−	1805.35i		−	2.50049i
$723$	383.799			0.530843
$724$	−382.488			−0.528299
$725$	−168.862			−0.232913
$726$	417.964			0.575708
$727$			798.630i			1.09853i	0.835649	+	0.549264i	$0.185092\pi$
$727$			798.630i			1.09853i	−0.835649	+	0.549264i	$0.814908\pi$
$728$	−41.3847			−0.0568471
$729$	25.6047			0.0351231
$730$	474.619			0.650162
$731$	−1615.74			−2.21031
$732$	−565.770			−0.772910
$733$			302.643i			0.412883i	0.978459	+	0.206441i	$0.0661883\pi$
$733$			302.643i			0.412883i	−0.978459	+	0.206441i	$0.933812\pi$
$734$			371.532i			0.506174i
$735$	151.850			0.206598
$736$	1506.66			2.04709
$737$	−4.30562			−0.00584209
$738$	−890.678			−1.20688
$739$			449.301i			0.607985i	0.952674	+	0.303992i	$0.0983198\pi$
$739$			449.301i			0.607985i	−0.952674	+	0.303992i	$0.901680\pi$
$740$	324.633			0.438693
$741$			391.611i			0.528489i
$742$	624.573			0.841743
$743$	500.889			0.674144			0.337072	−	0.941479i	$-0.390563\pi$
$743$	500.889			0.674144			0.337072	+	0.941479i	$0.390563\pi$
$744$			29.7679i			0.0400107i
$745$			603.257i			0.809741i
$746$	1134.07			1.52021
$747$			894.201i			1.19706i
$748$			165.889i			0.221777i
$749$	254.337			0.339568
$750$		−	484.498i		−	0.645997i
$751$	−177.714			−0.236637			−0.118318	−	0.992976i	$-0.537750\pi$
$751$	−177.714			−0.236637			−0.118318	+	0.992976i	$0.537750\pi$
$752$	544.570			0.724162
$753$	−15.3232			−0.0203496
$754$		−	387.688i		−	0.514175i
$755$	323.183			0.428056
$756$	−323.122			−0.427410
$757$	1195.93			1.57982			0.789912	−	0.613220i	$-0.210126\pi$
$757$	1195.93			1.57982			0.789912	+	0.613220i	$0.210126\pi$
$758$			925.891i			1.22149i
$759$	−101.610			−0.133873
$760$	123.640			0.162684
$761$	10.3723			0.0136299			0.00681493	−	0.999977i	$-0.497831\pi$
$761$	10.3723			0.0136299			0.00681493	+	0.999977i	$0.497831\pi$
$762$	−869.191			−1.14067
$763$	−156.395			−0.204974
$764$	−631.307			−0.826319
$765$	547.273			0.715389
$766$		−	1113.62i		−	1.45381i
$767$		−	82.1229i		−	0.107070i
$768$		−	388.684i		−	0.506099i
$769$			237.734i			0.309147i	0.987981	+	0.154574i	$0.0494004\pi$
$769$			237.734i			0.309147i	−0.987981	+	0.154574i	$0.950600\pi$
$770$			96.7576i			0.125659i
$771$	288.894			0.374701
$772$	962.563			1.24684
$773$			572.025i			0.740006i	0.929031	+	0.370003i	$0.120643\pi$
$773$			572.025i			0.740006i	−0.929031	+	0.370003i	$0.879357\pi$
$774$	−1603.77			−2.07205
$775$		−	245.723i		−	0.317062i
$776$			7.46283i			0.00961705i
$777$			132.792i			0.170903i
$778$	1668.19			2.14421
$779$		−	1409.49i		−	1.80936i
$780$	164.624			0.211056
$781$			162.682i			0.208300i
$782$			1927.32i			2.46460i
$783$			317.747i			0.405807i
$784$	−545.875			−0.696269
$785$		−	1052.73i		−	1.34106i
$786$	546.433			0.695207
$787$	−353.020			−0.448564			−0.224282	−	0.974524i	$-0.572004\pi$
$787$	−353.020			−0.448564			−0.224282	+	0.974524i	$0.572004\pi$
$788$			1070.62i			1.35866i
$789$		−	96.2982i		−	0.122051i
$790$			525.533i			0.665232i
$791$		−	662.171i		−	0.837132i
$792$		−	17.2847i		−	0.0218241i
$793$		−	1125.54i		−	1.41934i
$794$	1542.06			1.94214
$795$	260.801			0.328052
$796$	−723.689			−0.909157
$797$	−1327.25			−1.66531			−0.832653	−	0.553795i	$-0.813179\pi$
$797$	−1327.25			−1.66531			−0.832653	+	0.553795i	$0.813179\pi$
$798$		−	481.797i		−	0.603756i
$799$			635.677i			0.795591i
$800$			495.122i			0.618902i
$801$		−	606.486i		−	0.757162i
$802$		−	1318.90i		−	1.64452i
$803$	105.000			0.130759
$804$		−	9.01267i		−	0.0112098i
$805$			534.041i			0.663405i
$806$	564.152			0.699940
$807$			109.192i			0.135306i
$808$	−177.699			−0.219925
$809$		−	481.689i		−	0.595413i	−0.954657	−	0.297707i	$-0.903778\pi$
$809$		−	481.689i		−	0.595413i	0.954657	−	0.297707i	$-0.0962217\pi$
$810$	386.336			0.476958
$811$	−428.821			−0.528756			−0.264378	−	0.964419i	$-0.585167\pi$
$811$	−428.821			−0.528756			−0.264378	+	0.964419i	$0.585167\pi$
$812$			226.593i			0.279055i
$813$	593.707			0.730267
$814$	151.176			0.185720
$815$	−859.729			−1.05488
$816$	460.424			0.564245
$817$		−	2537.95i		−	3.10642i
$818$			1793.84i			2.19296i
$819$		−	287.737i		−	0.351328i
$820$	−592.517			−0.722582
$821$			127.503i			0.155302i	0.996981	+	0.0776508i	$0.0247419\pi$
$821$			127.503i			0.155302i	−0.996981	+	0.0776508i	$0.975258\pi$
$822$	−716.020			−0.871071
$823$	−848.529			−1.03102			−0.515510	−	0.856884i	$-0.672397\pi$
$823$	−848.529			−1.03102			−0.515510	+	0.856884i	$0.672397\pi$
$824$	−149.256			−0.181136
$825$		−	33.3913i		−	0.0404743i
$826$			101.036i			0.122319i
$827$			1344.68i			1.62598i	0.582278	+	0.812990i	$0.302162\pi$
$827$			1344.68i			1.62598i	−0.582278	+	0.812990i	$0.697838\pi$
$828$			908.822i			1.09761i
$829$	308.030			0.371568			0.185784	−	0.982591i	$-0.440518\pi$
$829$	308.030			0.371568			0.185784	+	0.982591i	$0.440518\pi$
$830$			1252.16i			1.50863i
$831$			342.816i			0.412534i
$832$	−482.823			−0.580316
$833$		−	637.201i		−	0.764947i
$834$	223.618			0.268127
$835$		−	463.871i		−	0.555534i
$836$	−260.573			−0.311691
$837$	−462.376			−0.552421
$838$		−	631.063i		−	0.753058i
$839$	−93.5568			−0.111510			−0.0557550	−	0.998444i	$-0.517757\pi$
$839$	−93.5568			−0.111510			−0.0557550	+	0.998444i	$0.517757\pi$
$840$	21.2606			0.0253103
$841$	−618.176			−0.735049
$842$	155.418			0.184582
$843$	−15.1083			−0.0179220
$844$		−	705.510i		−	0.835913i
$845$		−	297.746i		−	0.352362i
$846$			630.969i			0.745826i
$847$		−	485.984i		−	0.573771i
$848$	−937.539			−1.10559
$849$	−223.545			−0.263303
$850$	−633.362			−0.745131
$851$	834.397			0.980490
$852$	−340.532			−0.399686
$853$	1566.72			1.83672			0.918359	−	0.395748i	$-0.129514\pi$
$853$	1566.72			1.83672			0.918359	+	0.395748i	$0.129514\pi$
$854$			1384.74i			1.62148i
$855$			859.639i			1.00543i
$856$	−63.6226			−0.0743255
$857$		−	860.972i		−	1.00463i	−0.864683	−	0.502317i	$-0.832481\pi$
$857$		−	860.972i		−	1.00463i	0.864683	−	0.502317i	$-0.167519\pi$
$858$	76.6625			0.0893502
$859$	−427.319			−0.497461			−0.248731	−	0.968573i	$-0.580013\pi$
$859$	−427.319			−0.497461			−0.248731	+	0.968573i	$0.580013\pi$
$860$	−1066.89			−1.24057
$861$		−	242.370i		−	0.281498i
$862$	−499.723			−0.579725
$863$		−	1662.60i		−	1.92654i	−0.268539	−	0.963269i	$-0.586541\pi$
$863$		−	1662.60i		−	1.92654i	0.268539	−	0.963269i	$-0.413459\pi$
$864$	931.668			1.07832
$865$	−103.194			−0.119299
$866$	1156.64			1.33562
$867$			159.887i			0.184414i
$868$	−329.731			−0.379874
$869$			116.264i			0.133790i
$870$			199.168i			0.228928i
$871$	17.9297			0.0205852
$872$	39.1224			0.0448651
$873$	−51.8873			−0.0594356
$874$	−3027.38			−3.46382
$875$	−563.346			−0.643824
$876$			219.789i			0.250901i
$877$		−	1286.10i		−	1.46647i	−0.679975	−	0.733236i	$-0.738009\pi$
$877$		−	1286.10i		−	1.46647i	0.679975	−	0.733236i	$-0.261991\pi$
$878$			1499.54i			1.70791i
$879$		−	734.348i		−	0.835436i
$880$		−	145.242i		−	0.165047i
$881$		−	401.116i		−	0.455297i	−0.973743	−	0.227648i	$-0.926896\pi$
$881$		−	401.116i		−	0.455297i	0.973743	−	0.227648i	$-0.0731036\pi$
$882$		−	632.481i		−	0.717099i
$883$	681.810			0.772152			0.386076	−	0.922467i	$-0.373830\pi$
$883$	681.810			0.772152			0.386076	+	0.922467i	$0.373830\pi$
$884$		−	690.805i		−	0.781453i
$885$			42.1891i			0.0476713i
$886$		−	2443.92i		−	2.75837i
$887$	−393.684			−0.443837			−0.221919	−	0.975065i	$-0.571232\pi$
$887$	−393.684			−0.443837			−0.221919	+	0.975065i	$0.571232\pi$
$888$		−	33.2180i		−	0.0374077i
$889$			1010.65i			1.13683i
$890$		−	849.273i		−	0.954239i
$891$	85.4691			0.0959249
$892$		−	642.597i		−	0.720400i
$893$	−998.503			−1.11814
$894$	−588.045			−0.657768
$895$			771.442i			0.861946i
$896$	−140.120			−0.156384
$897$	423.129			0.471716
$898$			947.202i			1.05479i
$899$			324.247i			0.360675i
$900$	−298.660			−0.331844
$901$		−	1094.39i		−	1.21464i
$902$	−275.925			−0.305903
$903$		−	436.415i		−	0.483294i
$904$			165.643i			0.183233i
$905$			390.908i			0.431942i
$906$			315.033i			0.347718i
$907$	−426.676			−0.470426			−0.235213	−	0.971944i	$-0.575579\pi$
$907$	−426.676			−0.470426			−0.235213	+	0.971944i	$0.575579\pi$
$908$	−1014.77			−1.11758
$909$		−	1235.50i		−	1.35918i
$910$		−	402.924i		−	0.442773i
$911$		−	916.296i		−	1.00581i	−0.864341	−	0.502907i	$-0.832264\pi$
$911$		−	916.296i		−	1.00581i	0.864341	−	0.502907i	$-0.167736\pi$
$912$			723.220i			0.793004i
$913$			277.016i			0.303413i
$914$	−1885.61			−2.06303
$915$			578.224i			0.631939i
$916$		−	1013.38i		−	1.10631i
$917$		−	635.361i		−	0.692869i
$918$			1191.79i			1.29825i
$919$	1445.93			1.57337			0.786686	−	0.617353i	$-0.211795\pi$
$919$	1445.93			1.57337			0.786686	+	0.617353i	$0.211795\pi$
$920$		−	133.591i		−	0.145208i
$921$	278.100			0.301954
$922$	1379.08			1.49575
$923$		−	677.451i		−	0.733966i
$924$	−44.8071			−0.0484925
$925$			274.202i			0.296435i
$926$	1149.17			1.24100
$927$		−	1037.74i		−	1.11946i
$928$		−	653.343i		−	0.704033i
$929$		−	648.641i		−	0.698214i	−0.937083	−	0.349107i	$-0.886485\pi$
$929$		−	648.641i		−	0.698214i	0.937083	−	0.349107i	$-0.113515\pi$
$930$	−289.823			−0.311637
$931$	1000.90			1.07508
$932$	625.058			0.670663
$933$			88.4510i			0.0948028i
$934$		−	1118.54i		−	1.19758i
$935$	169.541			0.181327
$936$			71.9779i			0.0768995i
$937$		−	278.605i		−	0.297337i	−0.988887	−	0.148669i	$-0.952501\pi$
$937$		−	278.605i		−	0.297337i	0.988887	−	0.148669i	$-0.0474988\pi$
$938$	−22.0589			−0.0235169
$939$			660.564i			0.703476i
$940$			419.747i			0.446539i
$941$	−1246.35			−1.32450			−0.662248	−	0.749285i	$-0.730397\pi$
$941$	−1246.35			−1.32450			−0.662248	+	0.749285i	$0.730397\pi$
$942$	1026.18			1.08937
$943$	−1522.93			−1.61499
$944$		−	151.663i		−	0.160660i
$945$			330.234i			0.349454i
$946$	−496.834			−0.525195
$947$	63.2163			0.0667543			0.0333771	−	0.999443i	$-0.489374\pi$
$947$	63.2163			0.0667543			0.0333771	+	0.999443i	$0.489374\pi$
$948$	−243.367			−0.256716
$949$	−437.246			−0.460744
$950$		−	994.866i		−	1.04723i
$951$			759.438i			0.798568i
$952$		−	89.2150i		−	0.0937133i
$953$	1453.93			1.52564			0.762818	−	0.646614i	$-0.223815\pi$
$953$	1453.93			1.52564			0.762818	+	0.646614i	$0.223815\pi$
$954$		−	1086.28i		−	1.13866i
$955$			645.204i			0.675606i
$956$	−224.455			−0.234785
$957$			44.0618i			0.0460416i
$958$			213.502i			0.222862i
$959$			832.547i			0.868141i
$960$	248.041			0.258377
$961$	489.166			0.509018
$962$	−629.536			−0.654403
$963$		−	442.353i		−	0.459349i
$964$		−	1063.45i		−	1.10317i
$965$		−	983.751i		−	1.01943i
$966$	−520.574			−0.538897
$967$		−	64.5636i		−	0.0667669i	−0.999443	−	0.0333835i	$-0.989372\pi$
$967$		−	64.5636i		−	0.0667669i	0.999443	−	0.0333835i	$-0.0106283\pi$
$968$			121.570i			0.125588i
$969$	−844.215			−0.871223
$970$	−72.6586			−0.0749058
$971$			1131.96i			1.16576i	0.812557	+	0.582881i	$0.198075\pi$
$971$			1131.96i			1.16576i	−0.812557	+	0.582881i	$0.801925\pi$
$972$			872.416i			0.897547i
$973$		−	260.010i		−	0.267225i
$974$	971.738			0.997678
$975$			139.050i			0.142615i
$976$		−	2078.62i		−	2.12974i
$977$		−	56.5216i		−	0.0578522i	−0.999582	−	0.0289261i	$-0.990791\pi$
$977$		−	56.5216i		−	0.0578522i	0.999582	−	0.0289261i	$-0.00920875\pi$
$978$		−	838.049i		−	0.856901i
$979$		−	187.885i		−	0.191915i
$980$		−	420.753i		−	0.429340i
$981$			272.009i			0.277277i
$982$	1470.00			1.49694
$983$		−	550.816i		−	0.560342i	−0.959950	−	0.280171i	$-0.909609\pi$
$983$		−	550.816i		−	0.560342i	0.959950	−	0.280171i	$-0.0903911\pi$
$984$			60.6292i			0.0616150i
$985$	1094.19			1.11085
$986$	835.759			0.847626
$987$	−171.698			−0.173960
$988$	1085.10			1.09827
$989$	−2742.21			−2.77271
$990$	168.285			0.169985
$991$	−904.112			−0.912323			−0.456162	−	0.889897i	$-0.650776\pi$
$991$	−904.112			−0.912323			−0.456162	+	0.889897i	$0.650776\pi$
$992$	950.725			0.958392
$993$	142.131			0.143133
$994$			833.466i			0.838497i
$995$			739.619i			0.743335i
$996$	−579.860			−0.582188
$997$		−	78.1355i		−	0.0783706i	−0.999232	−	0.0391853i	$-0.987524\pi$
$997$		−	78.1355i		−	0.0783706i	0.999232	−	0.0391853i	$-0.0124763\pi$
$998$		−	2459.11i		−	2.46404i
$999$	515.965			0.516481

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.3.b.b.546.18	✓	88
547.546	odd	2	inner	547.3.b.b.546.71	yes	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.3.b.b.546.18	✓	88	1.1	even	1	trivial
547.3.b.b.546.71	yes	88	547.546	odd	2	inner

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:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	547.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.76044i q^{2} -1.30646i q^{3} -3.62000 q^{4} +3.69969i q^{5} -3.60639 q^{6} +4.19330i q^{7} -1.04896i q^{8} +7.29317 q^{9} +O(q^{10})\)	\(q-2.76044i q^{2} -1.30646i q^{3} -3.62000 q^{4} +3.69969i q^{5} -3.60639 q^{6} +4.19330i q^{7} -1.04896i q^{8} +7.29317 q^{9} +10.2127 q^{10} +2.25936 q^{11} +4.72938i q^{12} -9.40857 q^{13} +11.5753 q^{14} +4.83348 q^{15} -17.3756 q^{16} -20.2826i q^{17} -20.1323i q^{18} +31.8592 q^{19} -13.3929i q^{20} +5.47837 q^{21} -6.23683i q^{22} -34.4234i q^{23} -1.37042 q^{24} +11.3123 q^{25} +25.9718i q^{26} -21.2863i q^{27} -15.1798i q^{28} -14.9273 q^{29} -13.3425i q^{30} -21.7217i q^{31} +43.7684i q^{32} -2.95176i q^{33} -55.9887 q^{34} -15.5139 q^{35} -26.4013 q^{36} +24.2393i q^{37} -87.9453i q^{38} +12.2919i q^{39} +3.88082 q^{40} -44.2412i q^{41} -15.1227i q^{42} -79.6613i q^{43} -8.17890 q^{44} +26.9824i q^{45} -95.0235 q^{46} -31.3411 q^{47} +22.7005i q^{48} +31.4162 q^{49} -31.2269i q^{50} -26.4983 q^{51} +34.0591 q^{52} +53.9572 q^{53} -58.7595 q^{54} +8.35894i q^{55} +4.39861 q^{56} -41.6227i q^{57} +41.2058i q^{58} +8.72851i q^{59} -17.4972 q^{60} +119.629i q^{61} -59.9614 q^{62} +30.5825i q^{63} +51.3173 q^{64} -34.8088i q^{65} -8.14815 q^{66} -1.90568 q^{67} +73.4229i q^{68} -44.9727 q^{69} +42.8251i q^{70} +72.0035i q^{71} -7.65024i q^{72} +46.4732 q^{73} +66.9109 q^{74} -14.7791i q^{75} -115.330 q^{76} +9.47420i q^{77} +33.9310 q^{78} +51.4586i q^{79} -64.2843i q^{80} +37.8288 q^{81} -122.125 q^{82} +122.608i q^{83} -19.8317 q^{84} +75.0391 q^{85} -219.900 q^{86} +19.5019i q^{87} -2.36998i q^{88} -83.1582i q^{89} +74.4833 q^{90} -39.4530i q^{91} +124.613i q^{92} -28.3785 q^{93} +86.5150i q^{94} +117.869i q^{95} +57.1815 q^{96} -7.11450 q^{97} -86.7224i q^{98} +16.4779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) \) 88 * q - 192 * q^4 - 306 * q^9	\( 88 q - 192 q^{4} - 306 q^{9} - 4 q^{10} - 32 q^{11} + 26 q^{13} - 26 q^{14} + 22 q^{15} + 236 q^{16} - 12 q^{19} - 16 q^{21} - 2 q^{24} - 544 q^{25} - 96 q^{29} + 26 q^{34} + 10 q^{35} + 364 q^{36} + 44 q^{40} + 124 q^{44} - 288 q^{46} - 310 q^{47} - 694 q^{49} + 86 q^{51} - 316 q^{52} + 24 q^{53} - 266 q^{54} + 158 q^{56} - 80 q^{60} + 40 q^{62} - 652 q^{64} + 528 q^{66} + 28 q^{67} + 16 q^{69} + 94 q^{73} - 614 q^{74} - 28 q^{76} - 98 q^{78} + 928 q^{81} - 772 q^{82} + 358 q^{84} + 74 q^{85} - 410 q^{86} - 214 q^{90} + 656 q^{93} - 724 q^{96} + 346 q^{97} + 50 q^{99}+O(q^{100}) \) 88 * q - 192 * q^4 - 306 * q^9 - 4 * q^10 - 32 * q^11 + 26 * q^13 - 26 * q^14 + 22 * q^15 + 236 * q^16 - 12 * q^19 - 16 * q^21 - 2 * q^24 - 544 * q^25 - 96 * q^29 + 26 * q^34 + 10 * q^35 + 364 * q^36 + 44 * q^40 + 124 * q^44 - 288 * q^46 - 310 * q^47 - 694 * q^49 + 86 * q^51 - 316 * q^52 + 24 * q^53 - 266 * q^54 + 158 * q^56 - 80 * q^60 + 40 * q^62 - 652 * q^64 + 528 * q^66 + 28 * q^67 + 16 * q^69 + 94 * q^73 - 614 * q^74 - 28 * q^76 - 98 * q^78 + 928 * q^81 - 772 * q^82 + 358 * q^84 + 74 * q^85 - 410 * q^86 - 214 * q^90 + 656 * q^93 - 724 * q^96 + 346 * q^97 + 50 * q^99

Introduction

L-functions

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