LMFDB - Embedded newform 273.2.c.b.64.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(64,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("273.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	273.c (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:	$2.17991597518$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.1
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	273.64
Dual form			273.2.c.b.64.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17009i q^{2} +1.00000 q^{3} -2.70928 q^{4} +0.630898i q^{5} -2.17009i q^{6} -1.00000i q^{7} +1.53919i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.17009i q^{2} +1.00000 q^{3} -2.70928 q^{4} +0.630898i q^{5} -2.17009i q^{6} -1.00000i q^{7} +1.53919i q^{8} +1.00000 q^{9} +1.36910 q^{10} -5.70928i q^{11} -2.70928 q^{12} +(-2.17009 - 2.87936i) q^{13} -2.17009 q^{14} +0.630898i q^{15} -2.07838 q^{16} +1.07838 q^{17} -2.17009i q^{18} +7.41855i q^{19} -1.70928i q^{20} -1.00000i q^{21} -12.3896 q^{22} +5.41855 q^{23} +1.53919i q^{24} +4.60197 q^{25} +(-6.24846 + 4.70928i) q^{26} +1.00000 q^{27} +2.70928i q^{28} +6.68035 q^{29} +1.36910 q^{30} +6.15676i q^{31} +7.58864i q^{32} -5.70928i q^{33} -2.34017i q^{34} +0.630898 q^{35} -2.70928 q^{36} +3.41855i q^{37} +16.0989 q^{38} +(-2.17009 - 2.87936i) q^{39} -0.971071 q^{40} +1.21235i q^{41} -2.17009 q^{42} -12.6803 q^{43} +15.4680i q^{44} +0.630898i q^{45} -11.7587i q^{46} +6.04945i q^{47} -2.07838 q^{48} -1.00000 q^{49} -9.98667i q^{50} +1.07838 q^{51} +(5.87936 + 7.80098i) q^{52} -6.00000 q^{53} -2.17009i q^{54} +3.60197 q^{55} +1.53919 q^{56} +7.41855i q^{57} -14.4969i q^{58} -1.36910i q^{59} -1.70928i q^{60} +12.6537 q^{61} +13.3607 q^{62} -1.00000i q^{63} +12.3112 q^{64} +(1.81658 - 1.36910i) q^{65} -12.3896 q^{66} +0.581449i q^{67} -2.92162 q^{68} +5.41855 q^{69} -1.36910i q^{70} -4.81432i q^{71} +1.53919i q^{72} -3.81658i q^{73} +7.41855 q^{74} +4.60197 q^{75} -20.0989i q^{76} -5.70928 q^{77} +(-6.24846 + 4.70928i) q^{78} -1.65983 q^{79} -1.31124i q^{80} +1.00000 q^{81} +2.63090 q^{82} -12.8865i q^{83} +2.70928i q^{84} +0.680346i q^{85} +27.5174i q^{86} +6.68035 q^{87} +8.78765 q^{88} +6.20620i q^{89} +1.36910 q^{90} +(-2.87936 + 2.17009i) q^{91} -14.6803 q^{92} +6.15676i q^{93} +13.1278 q^{94} -4.68035 q^{95} +7.58864i q^{96} -1.07838i q^{97} +2.17009i q^{98} -5.70928i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 2 * q^4 + 6 * q^9	$ 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9} + 16 q^{10} - 2 q^{12} - 2 q^{13} - 2 q^{14} - 6 q^{16} - 16 q^{22} + 4 q^{23} - 10 q^{25} - 20 q^{26} + 6 q^{27} - 4 q^{29} + 16 q^{30} - 4 q^{35} - 2 q^{36} + 24 q^{38} - 2 q^{39} + 24 q^{40} - 2 q^{42} - 32 q^{43} - 6 q^{48} - 6 q^{49} + 10 q^{52} - 36 q^{53} - 16 q^{55} + 6 q^{56} + 28 q^{61} - 8 q^{62} + 22 q^{64} + 20 q^{65} - 16 q^{66} - 24 q^{68} + 4 q^{69} + 16 q^{74} - 10 q^{75} - 20 q^{77} - 20 q^{78} - 32 q^{79} + 6 q^{81} + 8 q^{82} - 4 q^{87} + 32 q^{88} + 16 q^{90} + 8 q^{91} - 44 q^{92} + 36 q^{94} + 16 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 2 * q^4 + 6 * q^9 + 16 * q^10 - 2 * q^12 - 2 * q^13 - 2 * q^14 - 6 * q^16 - 16 * q^22 + 4 * q^23 - 10 * q^25 - 20 * q^26 + 6 * q^27 - 4 * q^29 + 16 * q^30 - 4 * q^35 - 2 * q^36 + 24 * q^38 - 2 * q^39 + 24 * q^40 - 2 * q^42 - 32 * q^43 - 6 * q^48 - 6 * q^49 + 10 * q^52 - 36 * q^53 - 16 * q^55 + 6 * q^56 + 28 * q^61 - 8 * q^62 + 22 * q^64 + 20 * q^65 - 16 * q^66 - 24 * q^68 + 4 * q^69 + 16 * q^74 - 10 * q^75 - 20 * q^77 - 20 * q^78 - 32 * q^79 + 6 * q^81 + 8 * q^82 - 4 * q^87 + 32 * q^88 + 16 * q^90 + 8 * q^91 - 44 * q^92 + 36 * q^94 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$.

$n$	$92$	$106$	$157$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		−	2.17009i		−	1.53448i	−0.641358	−	0.767241i	$-0.721629\pi$
$2$		−	2.17009i		−	1.53448i	0.641358	−	0.767241i	$-0.278371\pi$
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−2.70928			−1.35464
$5$			0.630898i			0.282146i	0.989999	+	0.141073i	$0.0450552\pi$
$5$			0.630898i			0.282146i	−0.989999	+	0.141073i	$0.954945\pi$
$6$		−	2.17009i		−	0.885934i
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			1.53919i			0.544185i
$9$	1.00000			0.333333
$10$	1.36910			0.432948
$11$		−	5.70928i		−	1.72141i	−0.509103	−	0.860706i	$-0.670023\pi$
$11$		−	5.70928i		−	1.72141i	0.509103	−	0.860706i	$-0.329977\pi$
$12$	−2.70928			−0.782100
$13$	−2.17009	−	2.87936i	−0.601874	−	0.798591i
$13$	−2.17009	−	2.87936i	−0.601874	−	0.798591i
$14$	−2.17009			−0.579980
$15$			0.630898i			0.162897i
$16$	−2.07838			−0.519594
$17$	1.07838			0.261545			0.130773	−	0.991412i	$-0.458254\pi$
$17$	1.07838			0.261545			0.130773	+	0.991412i	$0.458254\pi$
$18$		−	2.17009i		−	0.511494i
$19$			7.41855i			1.70193i	0.525221	+	0.850966i	$0.323983\pi$
$19$			7.41855i			1.70193i	−0.525221	+	0.850966i	$0.676017\pi$
$20$		−	1.70928i		−	0.382206i
$21$		−	1.00000i		−	0.218218i
$22$	−12.3896			−2.64148
$23$	5.41855			1.12985			0.564923	−	0.825144i	$-0.308906\pi$
$23$	5.41855			1.12985			0.564923	+	0.825144i	$0.308906\pi$
$24$			1.53919i			0.314186i
$25$	4.60197			0.920394
$26$	−6.24846	+	4.70928i	−1.22542	+	0.923565i
$27$	1.00000			0.192450
$28$			2.70928i			0.512005i
$29$	6.68035			1.24051			0.620255	−	0.784401i	$-0.287029\pi$
$29$	6.68035			1.24051			0.620255	+	0.784401i	$0.287029\pi$
$30$	1.36910			0.249963
$31$			6.15676i			1.10579i	0.833252	+	0.552893i	$0.186476\pi$
$31$			6.15676i			1.10579i	−0.833252	+	0.552893i	$0.813524\pi$
$32$			7.58864i			1.34149i
$33$		−	5.70928i		−	0.993857i
$34$		−	2.34017i		−	0.401336i
$35$	0.630898			0.106641
$36$	−2.70928			−0.451546
$37$			3.41855i			0.562006i	0.959707	+	0.281003i	$0.0906671\pi$
$37$			3.41855i			0.562006i	−0.959707	+	0.281003i	$0.909333\pi$
$38$	16.0989			2.61159
$39$	−2.17009	−	2.87936i	−0.347492	−	0.461067i
$40$	−0.971071			−0.153540
$41$			1.21235i			0.189337i	0.995509	+	0.0946684i	$0.0301791\pi$
$41$			1.21235i			0.189337i	−0.995509	+	0.0946684i	$0.969821\pi$
$42$	−2.17009			−0.334852
$43$	−12.6803			−1.93373			−0.966867	−	0.255279i	$-0.917833\pi$
$43$	−12.6803			−1.93373			−0.966867	+	0.255279i	$0.917833\pi$
$44$			15.4680i			2.33189i
$45$			0.630898i			0.0940487i
$46$		−	11.7587i		−	1.73373i
$47$			6.04945i			0.882403i	0.897408	+	0.441201i	$0.145448\pi$
$47$			6.04945i			0.882403i	−0.897408	+	0.441201i	$0.854552\pi$
$48$	−2.07838			−0.299988
$49$	−1.00000			−0.142857
$50$		−	9.98667i		−	1.41233i
$51$	1.07838			0.151003
$52$	5.87936	+	7.80098i	0.815321	+	1.08180i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		−	2.17009i		−	0.295311i
$55$	3.60197			0.485689
$56$	1.53919			0.205683
$57$			7.41855i			0.982611i
$58$		−	14.4969i		−	1.90354i
$59$		−	1.36910i		−	0.178242i	−0.996021	−	0.0891210i	$-0.971594\pi$
$59$		−	1.36910i		−	0.178242i	0.996021	−	0.0891210i	$-0.0284058\pi$
$60$		−	1.70928i		−	0.220667i
$61$	12.6537			1.62014			0.810069	−	0.586334i	$-0.199430\pi$
$61$	12.6537			1.62014			0.810069	+	0.586334i	$0.199430\pi$
$62$	13.3607			1.69681
$63$		−	1.00000i		−	0.125988i
$64$	12.3112			1.53891
$65$	1.81658	−	1.36910i	0.225319	−	0.169816i
$66$	−12.3896			−1.52506
$67$			0.581449i			0.0710353i	0.999369	+	0.0355177i	$0.0113080\pi$
$67$			0.581449i			0.0710353i	−0.999369	+	0.0355177i	$0.988692\pi$
$68$	−2.92162			−0.354299
$69$	5.41855			0.652317
$70$		−	1.36910i		−	0.163639i
$71$		−	4.81432i		−	0.571354i	−0.958326	−	0.285677i	$-0.907782\pi$
$71$		−	4.81432i		−	0.571354i	0.958326	−	0.285677i	$-0.0922185\pi$
$72$			1.53919i			0.181395i
$73$		−	3.81658i		−	0.446697i	−0.974739	−	0.223349i	$-0.928301\pi$
$73$		−	3.81658i		−	0.446697i	0.974739	−	0.223349i	$-0.0716988\pi$
$74$	7.41855			0.862389
$75$	4.60197			0.531390
$76$		−	20.0989i		−	2.30550i
$77$	−5.70928			−0.650632
$78$	−6.24846	+	4.70928i	−0.707499	+	0.533220i
$79$	−1.65983			−0.186745			−0.0933726	−	0.995631i	$-0.529765\pi$
$79$	−1.65983			−0.186745			−0.0933726	+	0.995631i	$0.529765\pi$
$80$		−	1.31124i		−	0.146601i
$81$	1.00000			0.111111
$82$	2.63090			0.290534
$83$		−	12.8865i		−	1.41448i	−0.706972	−	0.707241i	$-0.749939\pi$
$83$		−	12.8865i		−	1.41448i	0.706972	−	0.707241i	$-0.250061\pi$
$84$			2.70928i			0.295606i
$85$			0.680346i			0.0737939i
$86$			27.5174i			2.96728i
$87$	6.68035			0.716208
$88$	8.78765			0.936767
$89$			6.20620i			0.657856i	0.944355	+	0.328928i	$0.106687\pi$
$89$			6.20620i			0.657856i	−0.944355	+	0.328928i	$0.893313\pi$
$90$	1.36910			0.144316
$91$	−2.87936	+	2.17009i	−0.301839	+	0.227487i
$92$	−14.6803			−1.53053
$93$			6.15676i			0.638426i
$94$	13.1278			1.35403
$95$	−4.68035			−0.480193
$96$			7.58864i			0.774512i
$97$		−	1.07838i		−	0.109493i	−0.998500	−	0.0547463i	$-0.982565\pi$
$97$		−	1.07838i		−	0.109493i	0.998500	−	0.0547463i	$-0.0174350\pi$
$98$			2.17009i			0.219212i
$99$		−	5.70928i		−	0.573804i
$100$	−12.4680			−1.24680
$101$	−7.23513			−0.719923			−0.359961	−	0.932967i	$-0.617210\pi$
$101$	−7.23513			−0.719923			−0.359961	+	0.932967i	$0.617210\pi$
$102$		−	2.34017i		−	0.231712i
$103$	13.7587			1.35569			0.677844	−	0.735206i	$-0.262915\pi$
$103$	13.7587			1.35569			0.677844	+	0.735206i	$0.262915\pi$
$104$	4.43188	−	3.34017i	0.434582	−	0.327531i
$105$	0.630898			0.0615693
$106$			13.0205i			1.26466i
$107$	−10.0989			−0.976297			−0.488149	−	0.872761i	$-0.662328\pi$
$107$	−10.0989			−0.976297			−0.488149	+	0.872761i	$0.662328\pi$
$108$	−2.70928			−0.260700
$109$			17.9421i			1.71855i	0.511518	+	0.859273i	$0.329083\pi$
$109$			17.9421i			1.71855i	−0.511518	+	0.859273i	$0.670917\pi$
$110$		−	7.81658i		−	0.745282i
$111$			3.41855i			0.324474i
$112$			2.07838i			0.196388i
$113$	10.6803			1.00472			0.502361	−	0.864658i	$-0.332465\pi$
$113$	10.6803			1.00472			0.502361	+	0.864658i	$0.332465\pi$
$114$	16.0989			1.50780
$115$			3.41855i			0.318781i
$116$	−18.0989			−1.68044
$117$	−2.17009	−	2.87936i	−0.200625	−	0.266197i
$118$	−2.97107			−0.273509
$119$		−	1.07838i		−	0.0988547i
$120$	−0.971071			−0.0886462
$121$	−21.5958			−1.96326
$122$		−	27.4596i		−	2.48607i
$123$			1.21235i			0.109314i
$124$		−	16.6803i		−	1.49794i
$125$			6.05786i			0.541831i
$126$	−2.17009			−0.193327
$127$	6.34017			0.562599			0.281300	−	0.959620i	$-0.409235\pi$
$127$	6.34017			0.562599			0.281300	+	0.959620i	$0.409235\pi$
$128$		−	11.5392i		−	1.01993i
$129$	−12.6803			−1.11644
$130$	−2.97107	−	3.94214i	−0.260580	−	0.345749i
$131$	−7.51745			−0.656802			−0.328401	−	0.944538i	$-0.606510\pi$
$131$	−7.51745			−0.656802			−0.328401	+	0.944538i	$0.606510\pi$
$132$			15.4680i			1.34632i
$133$	7.41855			0.643270
$134$	1.26180			0.109003
$135$			0.630898i			0.0542990i
$136$			1.65983i			0.142329i
$137$		−	4.47414i		−	0.382252i	−0.981566	−	0.191126i	$-0.938786\pi$
$137$		−	4.47414i		−	0.382252i	0.981566	−	0.191126i	$-0.0612139\pi$
$138$		−	11.7587i		−	1.00097i
$139$	−9.75872			−0.827724			−0.413862	−	0.910340i	$-0.635820\pi$
$139$	−9.75872			−0.827724			−0.413862	+	0.910340i	$0.635820\pi$
$140$	−1.70928			−0.144460
$141$			6.04945i			0.509455i
$142$	−10.4475			−0.876733
$143$	−16.4391	+	12.3896i	−1.37470	+	1.03607i
$144$	−2.07838			−0.173198
$145$			4.21461i			0.350005i
$146$	−8.28231			−0.685449
$147$	−1.00000			−0.0824786
$148$		−	9.26180i		−	0.761315i
$149$			1.46800i			0.120263i	0.998190	+	0.0601316i	$0.0191520\pi$
$149$			1.46800i			0.120263i	−0.998190	+	0.0601316i	$0.980848\pi$
$150$		−	9.98667i		−	0.815408i
$151$		−	13.3607i		−	1.08728i	−0.839319	−	0.543639i	$-0.817046\pi$
$151$		−	13.3607i		−	1.08728i	0.839319	−	0.543639i	$-0.182954\pi$
$152$	−11.4186			−0.926167
$153$	1.07838			0.0871817
$154$			12.3896i			0.998384i
$155$	−3.88428			−0.311993
$156$	5.87936	+	7.80098i	0.470726	+	0.624579i
$157$	−8.15676			−0.650980			−0.325490	−	0.945545i	$-0.605529\pi$
$157$	−8.15676			−0.650980			−0.325490	+	0.945545i	$0.605529\pi$
$158$			3.60197i			0.286557i
$159$	−6.00000			−0.475831
$160$	−4.78765			−0.378497
$161$		−	5.41855i		−	0.427042i
$162$		−	2.17009i		−	0.170498i
$163$		−	4.58145i		−	0.358847i	−0.983772	−	0.179423i	$-0.942577\pi$
$163$		−	4.58145i		−	0.358847i	0.983772	−	0.179423i	$-0.0574232\pi$
$164$		−	3.28458i		−	0.256483i
$165$	3.60197			0.280413
$166$	−27.9649			−2.17050
$167$		−	7.31124i		−	0.565761i	−0.959155	−	0.282881i	$-0.908710\pi$
$167$		−	7.31124i		−	0.565761i	0.959155	−	0.282881i	$-0.0912900\pi$
$168$	1.53919			0.118751
$169$	−3.58145	+	12.4969i	−0.275496	+	0.961302i
$170$	1.47641			0.113235
$171$			7.41855i			0.567311i
$172$	34.3545			2.61951
$173$	7.60197			0.577967			0.288983	−	0.957334i	$-0.406683\pi$
$173$	7.60197			0.577967			0.288983	+	0.957334i	$0.406683\pi$
$174$		−	14.4969i		−	1.09901i
$175$		−	4.60197i		−	0.347876i
$176$			11.8660i			0.894436i
$177$		−	1.36910i		−	0.102908i
$178$	13.4680			1.00947
$179$	9.10504			0.680543			0.340271	−	0.940327i	$-0.389481\pi$
$179$	9.10504			0.680543			0.340271	+	0.940327i	$0.389481\pi$
$180$		−	1.70928i		−	0.127402i
$181$	15.3607			1.14175			0.570876	−	0.821037i	$-0.306604\pi$
$181$	15.3607			1.14175			0.570876	+	0.821037i	$0.306604\pi$
$182$	4.70928	+	6.24846i	0.349075	+	0.463167i
$183$	12.6537			0.935387
$184$			8.34017i			0.614846i
$185$	−2.15676			−0.158568
$186$	13.3607			0.979653
$187$		−	6.15676i		−	0.450227i
$188$		−	16.3896i		−	1.19534i
$189$		−	1.00000i		−	0.0727393i
$190$			10.1568i			0.736848i
$191$	−24.5692			−1.77776			−0.888881	−	0.458138i	$-0.848517\pi$
$191$	−24.5692			−1.77776			−0.888881	+	0.458138i	$0.848517\pi$
$192$	12.3112			0.888487
$193$		−	13.5753i		−	0.977172i	−0.872516	−	0.488586i	$-0.837513\pi$
$193$		−	13.5753i		−	0.977172i	0.872516	−	0.488586i	$-0.162487\pi$
$194$	−2.34017			−0.168015
$195$	1.81658	−	1.36910i	0.130088	−	0.0980435i
$196$	2.70928			0.193520
$197$		−	2.94441i		−	0.209780i	−0.994484	−	0.104890i	$-0.966551\pi$
$197$		−	2.94441i		−	0.209780i	0.994484	−	0.104890i	$-0.0334491\pi$
$198$	−12.3896			−0.880492
$199$	−10.5236			−0.745998			−0.372999	−	0.927832i	$-0.621670\pi$
$199$	−10.5236			−0.745998			−0.372999	+	0.927832i	$0.621670\pi$
$200$			7.08330i			0.500865i
$201$			0.581449i			0.0410123i
$202$			15.7009i			1.10471i
$203$		−	6.68035i		−	0.468868i
$204$	−2.92162			−0.204554
$205$	−0.764867			−0.0534206
$206$		−	29.8576i		−	2.08028i
$207$	5.41855			0.376615
$208$	4.51026	+	5.98440i	0.312730	+	0.414944i
$209$	42.3545			2.92973
$210$		−	1.36910i		−	0.0944770i
$211$	−3.02052			−0.207941			−0.103971	−	0.994580i	$-0.533155\pi$
$211$	−3.02052			−0.207941			−0.103971	+	0.994580i	$0.533155\pi$
$212$	16.2557			1.11644
$213$		−	4.81432i		−	0.329871i
$214$			21.9155i			1.49811i
$215$		−	8.00000i		−	0.545595i
$216$			1.53919i			0.104729i
$217$	6.15676			0.417948
$218$	38.9360			2.63708
$219$		−	3.81658i		−	0.257901i
$220$	−9.75872			−0.657933
$221$	−2.34017	−	3.10504i	−0.157417	−	0.208868i
$222$	7.41855			0.497901
$223$			22.6225i			1.51491i	0.652885	+	0.757457i	$0.273558\pi$
$223$			22.6225i			1.51491i	−0.652885	+	0.757457i	$0.726442\pi$
$224$	7.58864			0.507037
$225$	4.60197			0.306798
$226$		−	23.1773i		−	1.54173i
$227$		−	22.9444i		−	1.52287i	−0.648239	−	0.761437i	$-0.724494\pi$
$227$		−	22.9444i		−	1.52287i	0.648239	−	0.761437i	$-0.275506\pi$
$228$		−	20.0989i		−	1.33108i
$229$			21.5441i			1.42367i	0.702344	+	0.711837i	$0.252137\pi$
$229$			21.5441i			1.42367i	−0.702344	+	0.711837i	$0.747863\pi$
$230$	7.41855			0.489165
$231$	−5.70928			−0.375643
$232$			10.2823i			0.675067i
$233$	7.36069			0.482215			0.241107	−	0.970498i	$-0.422489\pi$
$233$	7.36069			0.482215			0.241107	+	0.970498i	$0.422489\pi$
$234$	−6.24846	+	4.70928i	−0.408475	+	0.307855i
$235$	−3.81658			−0.248966
$236$			3.70928i			0.241453i
$237$	−1.65983			−0.107817
$238$	−2.34017			−0.151691
$239$			15.5525i			1.00601i	0.864284	+	0.503004i	$0.167772\pi$
$239$			15.5525i			1.00601i	−0.864284	+	0.503004i	$0.832228\pi$
$240$		−	1.31124i		−	0.0846404i
$241$			7.33403i			0.472426i	0.971701	+	0.236213i	$0.0759064\pi$
$241$			7.33403i			0.472426i	−0.971701	+	0.236213i	$0.924094\pi$
$242$			46.8648i			3.01258i
$243$	1.00000			0.0641500
$244$	−34.2823			−2.19470
$245$		−	0.630898i		−	0.0403066i
$246$	2.63090			0.167740
$247$	21.3607	−	16.0989i	1.35915	−	1.02435i
$248$	−9.47641			−0.601753
$249$		−	12.8865i		−	0.816652i
$250$	13.1461			0.831431
$251$	−13.6742			−0.863108			−0.431554	−	0.902087i	$-0.642035\pi$
$251$	−13.6742			−0.863108			−0.431554	+	0.902087i	$0.642035\pi$
$252$			2.70928i			0.170668i
$253$		−	30.9360i		−	1.94493i
$254$		−	13.7587i		−	0.863299i
$255$			0.680346i			0.0426049i
$256$	−0.418551			−0.0261594
$257$	−26.1256			−1.62967			−0.814834	−	0.579695i	$-0.803172\pi$
$257$	−26.1256			−1.62967			−0.814834	+	0.579695i	$0.803172\pi$
$258$			27.5174i			1.71316i
$259$	3.41855			0.212418
$260$	−4.92162	+	3.70928i	−0.305226	+	0.230039i
$261$	6.68035			0.413503
$262$			16.3135i			1.00785i
$263$	−30.7792			−1.89793			−0.948965	−	0.315382i	$-0.897867\pi$
$263$	−30.7792			−1.89793			−0.948965	+	0.315382i	$0.897867\pi$
$264$	8.78765			0.540843
$265$		−	3.78539i		−	0.232534i
$266$		−	16.0989i		−	0.987087i
$267$			6.20620i			0.379814i
$268$		−	1.57531i		−	0.0962271i
$269$	5.39189			0.328749			0.164375	−	0.986398i	$-0.447439\pi$
$269$	5.39189			0.328749			0.164375	+	0.986398i	$0.447439\pi$
$270$	1.36910			0.0833209
$271$			11.4186i			0.693628i	0.937934	+	0.346814i	$0.112736\pi$
$271$			11.4186i			0.693628i	−0.937934	+	0.346814i	$0.887264\pi$
$272$	−2.24128			−0.135897
$273$	−2.87936	+	2.17009i	−0.174267	+	0.131340i
$274$	−9.70928			−0.586559
$275$		−	26.2739i		−	1.58438i
$276$	−14.6803			−0.883653
$277$	13.1506			0.790144			0.395072	−	0.918650i	$-0.370720\pi$
$277$	13.1506			0.790144			0.395072	+	0.918650i	$0.370720\pi$
$278$			21.1773i			1.27013i
$279$			6.15676i			0.368595i
$280$			0.971071i			0.0580326i
$281$		−	2.14834i		−	0.128160i	−0.997945	−	0.0640798i	$-0.979589\pi$
$281$		−	2.14834i		−	0.128160i	0.997945	−	0.0640798i	$-0.0204112\pi$
$282$	13.1278			0.781751
$283$	−12.1978			−0.725084			−0.362542	−	0.931968i	$-0.618091\pi$
$283$	−12.1978			−0.725084			−0.362542	+	0.931968i	$0.618091\pi$
$284$			13.0433i			0.773978i
$285$	−4.68035			−0.277240
$286$	26.8865	+	35.6742i	1.58984	+	2.10946i
$287$	1.21235			0.0715626
$288$			7.58864i			0.447165i
$289$	−15.8371			−0.931594
$290$	9.14608			0.537076
$291$		−	1.07838i		−	0.0632156i
$292$			10.3402i			0.605113i
$293$			7.15449i			0.417970i	0.977919	+	0.208985i	$0.0670159\pi$
$293$			7.15449i			0.417970i	−0.977919	+	0.208985i	$0.932984\pi$
$294$			2.17009i			0.126562i
$295$	0.863763			0.0502903
$296$	−5.26180			−0.305836
$297$		−	5.70928i		−	0.331286i
$298$	3.18568			0.184542
$299$	−11.7587	−	15.6020i	−0.680025	−	0.902285i
$300$	−12.4680			−0.719840
$301$			12.6803i			0.730883i
$302$	−28.9939			−1.66841
$303$	−7.23513			−0.415648
$304$		−	15.4186i		−	0.884315i
$305$			7.98318i			0.457116i
$306$		−	2.34017i		−	0.133779i
$307$		−	21.8432i		−	1.24666i	−0.781959	−	0.623330i	$-0.785779\pi$
$307$		−	21.8432i		−	1.24666i	0.781959	−	0.623330i	$-0.214221\pi$
$308$	15.4680			0.881371
$309$	13.7587			0.782706
$310$			8.42923i			0.478748i
$311$	−7.51745			−0.426275			−0.213138	−	0.977022i	$-0.568368\pi$
$311$	−7.51745			−0.426275			−0.213138	+	0.977022i	$0.568368\pi$
$312$	4.43188	−	3.34017i	0.250906	−	0.189100i
$313$	25.7009			1.45270			0.726349	−	0.687326i	$-0.241215\pi$
$313$	25.7009			1.45270			0.726349	+	0.687326i	$0.241215\pi$
$314$			17.7009i			0.998918i
$315$	0.630898			0.0355471
$316$	4.49693			0.252972
$317$			9.36910i			0.526221i	0.964766	+	0.263111i	$0.0847484\pi$
$317$			9.36910i			0.526221i	−0.964766	+	0.263111i	$0.915252\pi$
$318$			13.0205i			0.730154i
$319$		−	38.1399i		−	2.13543i
$320$			7.76713i			0.434196i
$321$	−10.0989			−0.563665
$322$	−11.7587			−0.655288
$323$			8.00000i			0.445132i
$324$	−2.70928			−0.150515
$325$	−9.98667	−	13.2507i	−0.553961	−	0.735018i
$326$	−9.94214			−0.550644
$327$			17.9421i			0.992203i
$328$	−1.86603			−0.103034
$329$	6.04945			0.333517
$330$		−	7.81658i		−	0.430289i
$331$			21.0472i			1.15686i	0.815733	+	0.578429i	$0.196334\pi$
$331$			21.0472i			1.15686i	−0.815733	+	0.578429i	$0.803666\pi$
$332$			34.9132i			1.91611i
$333$			3.41855i			0.187335i
$334$	−15.8660			−0.868151
$335$	−0.366835			−0.0200423
$336$			2.07838i			0.113385i
$337$	15.4452			0.841354			0.420677	−	0.907210i	$-0.361793\pi$
$337$	15.4452			0.841354			0.420677	+	0.907210i	$0.361793\pi$
$338$	27.1194	+	7.77205i	1.47510	+	0.422744i
$339$	10.6803			0.580077
$340$		−	1.84324i		−	0.0999640i
$341$	35.1506			1.90351
$342$	16.0989			0.870529
$343$			1.00000i			0.0539949i
$344$		−	19.5174i		−	1.05231i
$345$			3.41855i			0.184049i
$346$		−	16.4969i		−	0.886880i
$347$	21.7321			1.16664			0.583319	−	0.812243i	$-0.301754\pi$
$347$	21.7321			1.16664			0.583319	+	0.812243i	$0.301754\pi$
$348$	−18.0989			−0.970203
$349$		−	5.65983i		−	0.302964i	−0.988460	−	0.151482i	$-0.951596\pi$
$349$		−	5.65983i		−	0.302964i	0.988460	−	0.151482i	$-0.0484045\pi$
$350$	−9.98667			−0.533810
$351$	−2.17009	−	2.87936i	−0.115831	−	0.153689i
$352$	43.3256			2.30926
$353$			33.7237i			1.79493i	0.441087	+	0.897464i	$0.354593\pi$
$353$			33.7237i			1.79493i	−0.441087	+	0.897464i	$0.645407\pi$
$354$	−2.97107			−0.157911
$355$	3.03734			0.161205
$356$		−	16.8143i		−	0.891157i
$357$		−	1.07838i		−	0.0570738i
$358$		−	19.7587i		−	1.04428i
$359$		−	18.7031i		−	0.987114i	−0.869714	−	0.493557i	$-0.835697\pi$
$359$		−	18.7031i		−	0.987114i	0.869714	−	0.493557i	$-0.164303\pi$
$360$	−0.971071			−0.0511799
$361$	−36.0349			−1.89657
$362$		−	33.3340i		−	1.75200i
$363$	−21.5958			−1.13349
$364$	7.80098	−	5.87936i	0.408883	−	0.308162i
$365$	2.40787			0.126034
$366$		−	27.4596i		−	1.43534i
$367$	10.0722			0.525766			0.262883	−	0.964828i	$-0.415327\pi$
$367$	10.0722			0.525766			0.262883	+	0.964828i	$0.415327\pi$
$368$	−11.2618			−0.587062
$369$			1.21235i			0.0631123i
$370$			4.68035i			0.243320i
$371$			6.00000i			0.311504i
$372$		−	16.6803i		−	0.864836i
$373$	4.24128			0.219605			0.109802	−	0.993953i	$-0.464978\pi$
$373$	4.24128			0.219605			0.109802	+	0.993953i	$0.464978\pi$
$374$	−13.3607			−0.690865
$375$			6.05786i			0.312826i
$376$	−9.31124			−0.480191
$377$	−14.4969	−	19.2351i	−0.746630	−	0.990660i
$378$	−2.17009			−0.111617
$379$		−	20.1978i		−	1.03749i	−0.854929	−	0.518745i	$-0.826399\pi$
$379$		−	20.1978i		−	1.03749i	0.854929	−	0.518745i	$-0.173601\pi$
$380$	12.6803			0.650488
$381$	6.34017			0.324817
$382$			53.3172i			2.72795i
$383$			7.04331i			0.359896i	0.983676	+	0.179948i	$0.0575930\pi$
$383$			7.04331i			0.359896i	−0.983676	+	0.179948i	$0.942407\pi$
$384$		−	11.5392i		−	0.588857i
$385$		−	3.60197i		−	0.183573i
$386$	−29.4596			−1.49945
$387$	−12.6803			−0.644578
$388$			2.92162i			0.148323i
$389$	−3.84324			−0.194860			−0.0974301	−	0.995242i	$-0.531062\pi$
$389$	−3.84324			−0.194860			−0.0974301	+	0.995242i	$0.531062\pi$
$390$	−2.97107	−	3.94214i	−0.150446	−	0.199618i
$391$	5.84324			0.295506
$392$		−	1.53919i		−	0.0777408i
$393$	−7.51745			−0.379205
$394$	−6.38962			−0.321904
$395$		−	1.04718i		−	0.0526894i
$396$			15.4680i			0.777296i
$397$		−	14.4391i		−	0.724676i	−0.932047	−	0.362338i	$-0.881979\pi$
$397$		−	14.4391i		−	0.724676i	0.932047	−	0.362338i	$-0.118021\pi$
$398$			22.8371i			1.14472i
$399$	7.41855			0.371392
$400$	−9.56463			−0.478231
$401$			9.83483i			0.491128i	0.969380	+	0.245564i	$0.0789732\pi$
$401$			9.83483i			0.491128i	−0.969380	+	0.245564i	$0.921027\pi$
$402$	1.26180			0.0629326
$403$	17.7275	−	13.3607i	0.883071	−	0.665543i
$404$	19.6020			0.975234
$405$			0.630898i			0.0313496i
$406$	−14.4969			−0.719470
$407$	19.5174			0.967444
$408$			1.65983i			0.0821737i
$409$			7.54864i			0.373256i	0.982431	+	0.186628i	$0.0597560\pi$
$409$			7.54864i			0.373256i	−0.982431	+	0.186628i	$0.940244\pi$
$410$			1.65983i			0.0819730i
$411$		−	4.47414i		−	0.220693i
$412$	−37.2762			−1.83647
$413$	−1.36910			−0.0673691
$414$		−	11.7587i		−	0.577910i
$415$	8.13009			0.399091
$416$	21.8504	−	16.4680i	1.07131	−	0.807410i
$417$	−9.75872			−0.477887
$418$		−	91.9130i		−	4.49561i
$419$	−35.1506			−1.71722			−0.858610	−	0.512630i	$-0.828671\pi$
$419$	−35.1506			−1.71722			−0.858610	+	0.512630i	$0.828671\pi$
$420$	−1.70928			−0.0834041
$421$		−	25.6742i		−	1.25128i	−0.780110	−	0.625642i	$-0.784837\pi$
$421$		−	25.6742i		−	1.25128i	0.780110	−	0.625642i	$-0.215163\pi$
$422$			6.55479i			0.319082i
$423$			6.04945i			0.294134i
$424$		−	9.23513i		−	0.448498i
$425$	4.96266			0.240724
$426$	−10.4475			−0.506182
$427$		−	12.6537i		−	0.612355i
$428$	27.3607			1.32253
$429$	−16.4391	+	12.3896i	−0.793686	+	0.598177i
$430$	−17.3607			−0.837207
$431$		−	16.2329i		−	0.781910i	−0.920410	−	0.390955i	$-0.872145\pi$
$431$		−	16.2329i		−	0.781910i	0.920410	−	0.390955i	$-0.127855\pi$
$432$	−2.07838			−0.0999960
$433$	−5.38735			−0.258900			−0.129450	−	0.991586i	$-0.541321\pi$
$433$	−5.38735			−0.258900			−0.129450	+	0.991586i	$0.541321\pi$
$434$		−	13.3607i		−	0.641334i
$435$			4.21461i			0.202075i
$436$		−	48.6102i		−	2.32801i
$437$			40.1978i			1.92292i
$438$	−8.28231			−0.395744
$439$	−10.2413			−0.488789			−0.244395	−	0.969676i	$-0.578589\pi$
$439$	−10.2413			−0.488789			−0.244395	+	0.969676i	$0.578589\pi$
$440$			5.54411i			0.264305i
$441$	−1.00000			−0.0476190
$442$	−6.73820	+	5.07838i	−0.320504	+	0.241554i
$443$	15.9421			0.757434			0.378717	−	0.925513i	$-0.376365\pi$
$443$	15.9421			0.757434			0.378717	+	0.925513i	$0.376365\pi$
$444$		−	9.26180i		−	0.439545i
$445$	−3.91548			−0.185612
$446$	49.0928			2.32461
$447$			1.46800i			0.0694340i
$448$		−	12.3112i		−	0.581652i
$449$		−	13.2001i		−	0.622949i	−0.950254	−	0.311475i	$-0.899177\pi$
$449$		−	13.2001i		−	0.622949i	0.950254	−	0.311475i	$-0.100823\pi$
$450$		−	9.98667i		−	0.470776i
$451$	6.92162			0.325926
$452$	−28.9360			−1.36103
$453$		−	13.3607i		−	0.627740i
$454$	−49.7914			−2.33682
$455$	−1.36910	−	1.81658i	−0.0641845	−	0.0851627i
$456$	−11.4186			−0.534723
$457$			18.3090i			0.856458i	0.903670	+	0.428229i	$0.140862\pi$
$457$			18.3090i			0.856458i	−0.903670	+	0.428229i	$0.859138\pi$
$458$	46.7526			2.18460
$459$	1.07838			0.0503344
$460$		−	9.26180i		−	0.431833i
$461$		−	18.9399i		−	0.882118i	−0.897478	−	0.441059i	$-0.854603\pi$
$461$		−	18.9399i		−	0.882118i	0.897478	−	0.441059i	$-0.145397\pi$
$462$			12.3896i			0.576417i
$463$		−	1.74435i		−	0.0810667i	−0.999178	−	0.0405334i	$-0.987094\pi$
$463$		−	1.74435i		−	0.0810667i	0.999178	−	0.0405334i	$-0.0129057\pi$
$464$	−13.8843			−0.644562
$465$	−3.88428			−0.180129
$466$		−	15.9733i		−	0.739951i
$467$	−18.1568			−0.840194			−0.420097	−	0.907479i	$-0.638004\pi$
$467$	−18.1568			−0.840194			−0.420097	+	0.907479i	$0.638004\pi$
$468$	5.87936	+	7.80098i	0.271774	+	0.360601i
$469$	0.581449			0.0268488
$470$			8.28231i			0.382035i
$471$	−8.15676			−0.375843
$472$	2.10731			0.0969967
$473$			72.3956i			3.32875i
$474$			3.60197i			0.165444i
$475$			34.1399i			1.56645i
$476$			2.92162i			0.133912i
$477$	−6.00000			−0.274721
$478$	33.7503			1.54370
$479$			30.4619i			1.39184i	0.718120	+	0.695919i	$0.245003\pi$
$479$			30.4619i			1.39184i	−0.718120	+	0.695919i	$0.754997\pi$
$480$	−4.78765			−0.218525
$481$	9.84324	−	7.41855i	0.448813	−	0.338257i
$482$	15.9155			0.724930
$483$		−	5.41855i		−	0.246553i
$484$	58.5090			2.65950
$485$	0.680346			0.0308929
$486$		−	2.17009i		−	0.0984371i
$487$		−	0.313511i		−	0.0142065i	−0.999975	−	0.00710327i	$-0.997739\pi$
$487$		−	0.313511i		−	0.0142065i	0.999975	−	0.00710327i	$-0.00226106\pi$
$488$			19.4764i			0.881656i
$489$		−	4.58145i		−	0.207180i
$490$	−1.36910			−0.0618497
$491$	−8.93600			−0.403276			−0.201638	−	0.979460i	$-0.564626\pi$
$491$	−8.93600			−0.403276			−0.201638	+	0.979460i	$0.564626\pi$
$492$		−	3.28458i		−	0.148080i
$493$	7.20394			0.324449
$494$	−34.9360	−	46.3545i	−1.57184	−	2.08559i
$495$	3.60197			0.161896
$496$		−	12.7961i		−	0.574560i
$497$	−4.81432			−0.215952
$498$	−27.9649			−1.25314
$499$			33.9877i			1.52150i	0.649046	+	0.760750i	$0.275168\pi$
$499$			33.9877i			1.52150i	−0.649046	+	0.760750i	$0.724832\pi$
$500$		−	16.4124i		−	0.733985i
$501$		−	7.31124i		−	0.326642i
$502$			29.6742i			1.32442i
$503$	33.5585			1.49630			0.748149	−	0.663530i	$-0.230943\pi$
$503$	33.5585			1.49630			0.748149	+	0.663530i	$0.230943\pi$
$504$	1.53919			0.0685609
$505$		−	4.56463i		−	0.203123i
$506$	−67.1338			−2.98446
$507$	−3.58145	+	12.4969i	−0.159058	+	0.555008i
$508$	−17.1773			−0.762118
$509$			24.8287i			1.10051i	0.834996	+	0.550256i	$0.185470\pi$
$509$			24.8287i			1.10051i	−0.834996	+	0.550256i	$0.814530\pi$
$510$	1.47641			0.0653765
$511$	−3.81658			−0.168836
$512$		−	22.1701i		−	0.979789i
$513$			7.41855i			0.327537i
$514$			56.6947i			2.50070i
$515$			8.68035i			0.382502i
$516$	34.3545			1.51237
$517$	34.5380			1.51898
$518$		−	7.41855i		−	0.325952i
$519$	7.60197			0.333689
$520$	2.10731	+	2.79606i	0.0924115	+	0.122616i
$521$	20.2290			0.886248			0.443124	−	0.896460i	$-0.353870\pi$
$521$	20.2290			0.886248			0.443124	+	0.896460i	$0.353870\pi$
$522$		−	14.4969i		−	0.634513i
$523$	−40.1666			−1.75636			−0.878181	−	0.478328i	$-0.841243\pi$
$523$	−40.1666			−1.75636			−0.878181	+	0.478328i	$0.841243\pi$
$524$	20.3668			0.889729
$525$		−	4.60197i		−	0.200846i
$526$			66.7936i			2.91234i
$527$			6.63931i			0.289213i
$528$			11.8660i			0.516403i
$529$	6.36069			0.276552
$530$	−8.21461			−0.356820
$531$		−	1.36910i		−	0.0594140i
$532$	−20.0989			−0.871398
$533$	3.49079	−	2.63090i	0.151203	−	0.113957i
$534$	13.4680			0.582817
$535$		−	6.37137i		−	0.275458i
$536$	−0.894960			−0.0386564
$537$	9.10504			0.392911
$538$		−	11.7009i		−	0.504460i
$539$			5.70928i			0.245916i
$540$		−	1.70928i		−	0.0735555i
$541$		−	18.2101i		−	0.782912i	−0.920197	−	0.391456i	$-0.871971\pi$
$541$		−	18.2101i		−	0.782912i	0.920197	−	0.391456i	$-0.128029\pi$
$542$	24.7792			1.06436
$543$	15.3607			0.659190
$544$			8.18342i			0.350861i
$545$	−11.3197			−0.484881
$546$	4.70928	+	6.24846i	0.201538	+	0.267410i
$547$	−6.32580			−0.270472			−0.135236	−	0.990813i	$-0.543179\pi$
$547$	−6.32580			−0.270472			−0.135236	+	0.990813i	$0.543179\pi$
$548$			12.1217i			0.517813i
$549$	12.6537			0.540046
$550$	−57.0166			−2.43120
$551$			49.5585i			2.11126i
$552$			8.34017i			0.354981i
$553$			1.65983i			0.0705830i
$554$		−	28.5380i		−	1.21246i
$555$	−2.15676			−0.0915492
$556$	26.4391			1.12127
$557$		−	10.6309i		−	0.450446i	−0.974307	−	0.225223i	$-0.927689\pi$
$557$		−	10.6309i		−	0.450446i	0.974307	−	0.225223i	$-0.0723110\pi$
$558$	13.3607			0.565603
$559$	27.5174	+	36.5113i	1.16386	+	1.54426i
$560$	−1.31124			−0.0554102
$561$		−	6.15676i		−	0.259938i
$562$	−4.66209			−0.196659
$563$	−37.0472			−1.56135			−0.780676	−	0.624936i	$-0.785125\pi$
$563$	−37.0472			−1.56135			−0.780676	+	0.624936i	$0.785125\pi$
$564$		−	16.3896i		−	0.690128i
$565$			6.73820i			0.283478i
$566$			26.4703i			1.11263i
$567$		−	1.00000i		−	0.0419961i
$568$	7.41014			0.310923
$569$	25.7152			1.07804			0.539019	−	0.842293i	$-0.318795\pi$
$569$	25.7152			1.07804			0.539019	+	0.842293i	$0.318795\pi$
$570$			10.1568i			0.425420i
$571$	−12.4969			−0.522980			−0.261490	−	0.965206i	$-0.584214\pi$
$571$	−12.4969			−0.522980			−0.261490	+	0.965206i	$0.584214\pi$
$572$	44.5380	−	33.5669i	1.86223	−	1.40350i
$573$	−24.5692			−1.02639
$574$		−	2.63090i		−	0.109812i
$575$	24.9360			1.03990
$576$	12.3112			0.512968
$577$		−	3.91548i		−	0.163004i	−0.996673	−	0.0815018i	$-0.974028\pi$
$577$		−	3.91548i		−	0.163004i	0.996673	−	0.0815018i	$-0.0259716\pi$
$578$			34.3679i			1.42952i
$579$		−	13.5753i		−	0.564170i
$580$		−	11.4186i		−	0.474130i
$581$	−12.8865			−0.534624
$582$	−2.34017			−0.0970033
$583$			34.2557i			1.41872i
$584$	5.87444			0.243086
$585$	1.81658	−	1.36910i	0.0751064	−	0.0566054i
$586$	15.5259			0.641367
$587$			6.94441i			0.286626i	0.989677	+	0.143313i	$0.0457756\pi$
$587$			6.94441i			0.286626i	−0.989677	+	0.143313i	$0.954224\pi$
$588$	2.70928			0.111729
$589$	−45.6742			−1.88197
$590$		−	1.87444i		−	0.0771695i
$591$		−	2.94441i		−	0.121117i
$592$		−	7.10504i		−	0.292015i
$593$		−	29.9383i		−	1.22942i	−0.788754	−	0.614709i	$-0.789274\pi$
$593$		−	29.9383i		−	1.22942i	0.788754	−	0.614709i	$-0.210726\pi$
$594$	−12.3896			−0.508352
$595$	0.680346			0.0278915
$596$		−	3.97721i		−	0.162913i
$597$	−10.5236			−0.430702
$598$	−33.8576	+	25.5174i	−1.38454	+	1.04349i
$599$	43.2905			1.76880			0.884402	−	0.466726i	$-0.154567\pi$
$599$	43.2905			1.76880			0.884402	+	0.466726i	$0.154567\pi$
$600$			7.08330i			0.289174i
$601$	−31.2306			−1.27392			−0.636961	−	0.770896i	$-0.719809\pi$
$601$	−31.2306			−1.27392			−0.636961	+	0.770896i	$0.719809\pi$
$602$	27.5174			1.12153
$603$			0.581449i			0.0236784i
$604$			36.1978i			1.47287i
$605$		−	13.6248i		−	0.553925i
$606$			15.7009i			0.637804i
$607$	4.96266			0.201428			0.100714	−	0.994915i	$-0.467887\pi$
$607$	4.96266			0.201428			0.100714	+	0.994915i	$0.467887\pi$
$608$	−56.2967			−2.28313
$609$		−	6.68035i		−	0.270701i
$610$	17.3242			0.701436
$611$	17.4186	−	13.1278i	0.704679	−	0.531095i
$612$	−2.92162			−0.118100
$613$		−	0.366835i		−	0.0148163i	−0.999973	−	0.00740816i	$-0.997642\pi$
$613$		−	0.366835i		−	0.0148163i	0.999973	−	0.00740816i	$-0.00235811\pi$
$614$	−47.4017			−1.91298
$615$	−0.764867			−0.0308424
$616$		−	8.78765i		−	0.354065i
$617$		−	47.7237i		−	1.92128i	−0.277793	−	0.960641i	$-0.589603\pi$
$617$		−	47.7237i		−	1.92128i	0.277793	−	0.960641i	$-0.410397\pi$
$618$		−	29.8576i		−	1.20105i
$619$		−	19.7854i		−	0.795242i	−0.917550	−	0.397621i	$-0.869836\pi$
$619$		−	19.7854i		−	0.795242i	0.917550	−	0.397621i	$-0.130164\pi$
$620$	10.5236			0.422638
$621$	5.41855			0.217439
$622$			16.3135i			0.654112i
$623$	6.20620			0.248646
$624$	4.51026	+	5.98440i	0.180555	+	0.239568i
$625$	19.1880			0.767518
$626$		−	55.7731i		−	2.22914i
$627$	42.3545			1.69148
$628$	22.0989			0.881842
$629$			3.68649i			0.146990i
$630$		−	1.36910i		−	0.0545463i
$631$			20.1522i			0.802247i	0.916024	+	0.401124i	$0.131380\pi$
$631$			20.1522i			0.802247i	−0.916024	+	0.401124i	$0.868620\pi$
$632$		−	2.55479i		−	0.101624i
$633$	−3.02052			−0.120055
$634$	20.3318			0.807477
$635$			4.00000i			0.158735i
$636$	16.2557			0.644579
$637$	2.17009	+	2.87936i	0.0859820	+	0.114084i
$638$	−82.7670			−3.27678
$639$		−	4.81432i		−	0.190451i
$640$	7.28005			0.287769
$641$	−12.1034			−0.478057			−0.239028	−	0.971013i	$-0.576829\pi$
$641$	−12.1034			−0.478057			−0.239028	+	0.971013i	$0.576829\pi$
$642$			21.9155i			0.864935i
$643$			20.6803i			0.815553i	0.913082	+	0.407777i	$0.133696\pi$
$643$			20.6803i			0.815553i	−0.913082	+	0.407777i	$0.866304\pi$
$644$			14.6803i			0.578487i
$645$		−	8.00000i		−	0.315000i
$646$	17.3607			0.683047
$647$	40.8781			1.60709			0.803543	−	0.595247i	$-0.202946\pi$
$647$	40.8781			1.60709			0.803543	+	0.595247i	$0.202946\pi$
$648$			1.53919i			0.0604650i
$649$	−7.81658			−0.306828
$650$	−28.7552	+	21.6719i	−1.12787	+	0.850043i
$651$	6.15676			0.241302
$652$			12.4124i			0.486107i
$653$	6.31351			0.247067			0.123533	−	0.992340i	$-0.460577\pi$
$653$	6.31351			0.247067			0.123533	+	0.992340i	$0.460577\pi$
$654$	38.9360			1.52252
$655$		−	4.74274i		−	0.185314i
$656$		−	2.51971i		−	0.0983783i
$657$		−	3.81658i		−	0.148899i
$658$		−	13.1278i		−	0.511776i
$659$	−20.9360			−0.815551			−0.407775	−	0.913082i	$-0.633695\pi$
$659$	−20.9360			−0.815551			−0.407775	+	0.913082i	$0.633695\pi$
$660$	−9.75872			−0.379858
$661$			22.2245i			0.864431i	0.901770	+	0.432216i	$0.142268\pi$
$661$			22.2245i			0.864431i	−0.901770	+	0.432216i	$0.857732\pi$
$662$	45.6742			1.77518
$663$	−2.34017	−	3.10504i	−0.0908848	−	0.120590i
$664$	19.8348			0.769741
$665$			4.68035i			0.181496i
$666$	7.41855			0.287463
$667$	36.1978			1.40158
$668$			19.8082i			0.766401i
$669$			22.6225i			0.874636i
$670$			0.796064i			0.0307546i
$671$		−	72.2434i		−	2.78892i
$672$	7.58864			0.292738
$673$	−1.15061			−0.0443528			−0.0221764	−	0.999754i	$-0.507060\pi$
$673$	−1.15061			−0.0443528			−0.0221764	+	0.999754i	$0.507060\pi$
$674$		−	33.5174i		−	1.29104i
$675$	4.60197			0.177130
$676$	9.70313	−	33.8576i	0.373197	−	1.30222i
$677$	−33.5897			−1.29096			−0.645478	−	0.763779i	$-0.723342\pi$
$677$	−33.5897			−1.29096			−0.645478	+	0.763779i	$0.723342\pi$
$678$		−	23.1773i		−	0.890118i
$679$	−1.07838			−0.0413843
$680$	−1.04718			−0.0401576
$681$		−	22.9444i		−	0.879232i
$682$		−	76.2799i		−	2.92091i
$683$		−	25.1110i		−	0.960846i	−0.877037	−	0.480423i	$-0.840483\pi$
$683$		−	25.1110i		−	0.960846i	0.877037	−	0.480423i	$-0.159517\pi$
$684$		−	20.0989i		−	0.768501i
$685$	2.82273			0.107851
$686$	2.17009			0.0828543
$687$			21.5441i			0.821959i
$688$	26.3545			1.00476
$689$	13.0205	+	17.2762i	0.496042	+	0.658170i
$690$	7.41855			0.282419
$691$		−	6.30898i		−	0.240005i	−0.992774	−	0.120002i	$-0.961710\pi$
$691$		−	6.30898i		−	0.240005i	0.992774	−	0.120002i	$-0.0382902\pi$
$692$	−20.5958			−0.782936
$693$	−5.70928			−0.216877
$694$		−	47.1605i		−	1.79019i
$695$		−	6.15676i		−	0.233539i
$696$			10.2823i			0.389750i
$697$			1.30737i			0.0495201i
$698$	−12.2823			−0.464892
$699$	7.36069			0.278407
$700$			12.4680i			0.471246i
$701$	45.8843			1.73303			0.866513	−	0.499155i	$-0.166356\pi$
$701$	45.8843			1.73303			0.866513	+	0.499155i	$0.166356\pi$
$702$	−6.24846	+	4.70928i	−0.235833	+	0.177740i
$703$	−25.3607			−0.956497
$704$		−	70.2883i		−	2.64909i
$705$	−3.81658			−0.143741
$706$	73.1832			2.75429
$707$			7.23513i			0.272105i
$708$			3.70928i			0.139403i
$709$			46.4534i			1.74460i	0.488975	+	0.872298i	$0.337371\pi$
$709$			46.4534i			1.74460i	−0.488975	+	0.872298i	$0.662629\pi$
$710$		−	6.59129i		−	0.247367i
$711$	−1.65983			−0.0622484
$712$	−9.55252			−0.357996
$713$			33.3607i			1.24937i
$714$	−2.34017			−0.0875788
$715$	−7.81658	−	10.3714i	−0.292324	−	0.387867i
$716$	−24.6681			−0.921889
$717$			15.5525i			0.580819i
$718$	−40.5874			−1.51471
$719$	−7.20394			−0.268661			−0.134331	−	0.990937i	$-0.542888\pi$
$719$	−7.20394			−0.268661			−0.134331	+	0.990937i	$0.542888\pi$
$720$		−	1.31124i		−	0.0488672i
$721$		−	13.7587i		−	0.512402i
$722$			78.1988i			2.91026i
$723$			7.33403i			0.272756i
$724$	−41.6163			−1.54666
$725$	30.7427			1.14176
$726$			46.8648i			1.73932i
$727$	13.6742			0.507148			0.253574	−	0.967316i	$-0.418394\pi$
$727$	13.6742			0.507148			0.253574	+	0.967316i	$0.418394\pi$
$728$	−3.34017	−	4.43188i	−0.123795	−	0.164256i
$729$	1.00000			0.0370370
$730$		−	5.22529i		−	0.193397i
$731$	−13.6742			−0.505759
$732$	−34.2823			−1.26711
$733$		−	7.70086i		−	0.284438i	−0.989835	−	0.142219i	$-0.954576\pi$
$733$		−	7.70086i		−	0.284438i	0.989835	−	0.142219i	$-0.0454237\pi$
$734$		−	21.8576i		−	0.806779i
$735$		−	0.630898i		−	0.0232710i
$736$			41.1194i			1.51568i
$737$	3.31965			0.122281
$738$	2.63090			0.0968447
$739$			24.5814i			0.904243i	0.891956	+	0.452122i	$0.149333\pi$
$739$			24.5814i			0.904243i	−0.891956	+	0.452122i	$0.850667\pi$
$740$	5.84324			0.214802
$741$	21.3607	−	16.0989i	0.784705	−	0.591408i
$742$	13.0205			0.477998
$743$			24.3773i			0.894318i	0.894455	+	0.447159i	$0.147564\pi$
$743$			24.3773i			0.894318i	−0.894455	+	0.447159i	$0.852436\pi$
$744$	−9.47641			−0.347422
$745$	−0.926157			−0.0339318
$746$		−	9.20394i		−	0.336980i
$747$		−	12.8865i		−	0.471494i
$748$			16.6803i			0.609894i
$749$			10.0989i			0.369006i
$750$	13.1461			0.480027
$751$	−24.6803			−0.900599			−0.450299	−	0.892878i	$-0.648683\pi$
$751$	−24.6803			−0.900599			−0.450299	+	0.892878i	$0.648683\pi$
$752$		−	12.5730i		−	0.458492i
$753$	−13.6742			−0.498316
$754$	−41.7419	+	31.4596i	−1.52015	+	1.14569i
$755$	8.42923			0.306771
$756$			2.70928i			0.0985354i
$757$	−41.3172			−1.50170			−0.750850	−	0.660473i	$-0.770356\pi$
$757$	−41.3172			−1.50170			−0.750850	+	0.660473i	$0.770356\pi$
$758$	−43.8310			−1.59201
$759$		−	30.9360i		−	1.12291i
$760$		−	7.20394i		−	0.261314i
$761$			15.9506i			0.578207i	0.957298	+	0.289104i	$0.0933572\pi$
$761$			15.9506i			0.578207i	−0.957298	+	0.289104i	$0.906643\pi$
$762$		−	13.7587i		−	0.498426i
$763$	17.9421			0.649549
$764$	66.5646			2.40822
$765$			0.680346i			0.0245980i
$766$	15.2846			0.552254
$767$	−3.94214	+	2.97107i	−0.142342	+	0.107279i
$768$	−0.418551			−0.0151031
$769$		−	6.22446i		−	0.224460i	−0.993682	−	0.112230i	$-0.964201\pi$
$769$		−	6.22446i		−	0.224460i	0.993682	−	0.112230i	$-0.0357993\pi$
$770$	−7.81658			−0.281690
$771$	−26.1256			−0.940889
$772$			36.7792i			1.32371i
$773$		−	39.1545i		−	1.40829i	−0.710057	−	0.704145i	$-0.751331\pi$
$773$		−	39.1545i		−	1.40829i	0.710057	−	0.704145i	$-0.248669\pi$
$774$			27.5174i			0.989094i
$775$			28.3332i			1.01776i
$776$	1.65983			0.0595843
$777$	3.41855			0.122640
$778$			8.34017i			0.299010i
$779$	−8.99386			−0.322238
$780$	−4.92162	+	3.70928i	−0.176222	+	0.132813i
$781$	−27.4863			−0.983535
$782$		−	12.6803i		−	0.453448i
$783$	6.68035			0.238736
$784$	2.07838			0.0742278
$785$		−	5.14608i		−	0.183671i
$786$			16.3135i			0.581884i

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:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.17009i q^{2} +1.00000 q^{3} -2.70928 q^{4} +0.630898i q^{5} -2.17009i q^{6} -1.00000i q^{7} +1.53919i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.17009i q^{2} +1.00000 q^{3} -2.70928 q^{4} +0.630898i q^{5} -2.17009i q^{6} -1.00000i q^{7} +1.53919i q^{8} +1.00000 q^{9} +1.36910 q^{10} -5.70928i q^{11} -2.70928 q^{12} +(-2.17009 - 2.87936i) q^{13} -2.17009 q^{14} +0.630898i q^{15} -2.07838 q^{16} +1.07838 q^{17} -2.17009i q^{18} +7.41855i q^{19} -1.70928i q^{20} -1.00000i q^{21} -12.3896 q^{22} +5.41855 q^{23} +1.53919i q^{24} +4.60197 q^{25} +(-6.24846 + 4.70928i) q^{26} +1.00000 q^{27} +2.70928i q^{28} +6.68035 q^{29} +1.36910 q^{30} +6.15676i q^{31} +7.58864i q^{32} -5.70928i q^{33} -2.34017i q^{34} +0.630898 q^{35} -2.70928 q^{36} +3.41855i q^{37} +16.0989 q^{38} +(-2.17009 - 2.87936i) q^{39} -0.971071 q^{40} +1.21235i q^{41} -2.17009 q^{42} -12.6803 q^{43} +15.4680i q^{44} +0.630898i q^{45} -11.7587i q^{46} +6.04945i q^{47} -2.07838 q^{48} -1.00000 q^{49} -9.98667i q^{50} +1.07838 q^{51} +(5.87936 + 7.80098i) q^{52} -6.00000 q^{53} -2.17009i q^{54} +3.60197 q^{55} +1.53919 q^{56} +7.41855i q^{57} -14.4969i q^{58} -1.36910i q^{59} -1.70928i q^{60} +12.6537 q^{61} +13.3607 q^{62} -1.00000i q^{63} +12.3112 q^{64} +(1.81658 - 1.36910i) q^{65} -12.3896 q^{66} +0.581449i q^{67} -2.92162 q^{68} +5.41855 q^{69} -1.36910i q^{70} -4.81432i q^{71} +1.53919i q^{72} -3.81658i q^{73} +7.41855 q^{74} +4.60197 q^{75} -20.0989i q^{76} -5.70928 q^{77} +(-6.24846 + 4.70928i) q^{78} -1.65983 q^{79} -1.31124i q^{80} +1.00000 q^{81} +2.63090 q^{82} -12.8865i q^{83} +2.70928i q^{84} +0.680346i q^{85} +27.5174i q^{86} +6.68035 q^{87} +8.78765 q^{88} +6.20620i q^{89} +1.36910 q^{90} +(-2.87936 + 2.17009i) q^{91} -14.6803 q^{92} +6.15676i q^{93} +13.1278 q^{94} -4.68035 q^{95} +7.58864i q^{96} -1.07838i q^{97} +2.17009i q^{98} -5.70928i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 2 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9} + 16 q^{10} - 2 q^{12} - 2 q^{13} - 2 q^{14} - 6 q^{16} - 16 q^{22} + 4 q^{23} - 10 q^{25} - 20 q^{26} + 6 q^{27} - 4 q^{29} + 16 q^{30} - 4 q^{35} - 2 q^{36} + 24 q^{38} - 2 q^{39} + 24 q^{40} - 2 q^{42} - 32 q^{43} - 6 q^{48} - 6 q^{49} + 10 q^{52} - 36 q^{53} - 16 q^{55} + 6 q^{56} + 28 q^{61} - 8 q^{62} + 22 q^{64} + 20 q^{65} - 16 q^{66} - 24 q^{68} + 4 q^{69} + 16 q^{74} - 10 q^{75} - 20 q^{77} - 20 q^{78} - 32 q^{79} + 6 q^{81} + 8 q^{82} - 4 q^{87} + 32 q^{88} + 16 q^{90} + 8 q^{91} - 44 q^{92} + 36 q^{94} + 16 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 2 * q^4 + 6 * q^9 + 16 * q^10 - 2 * q^12 - 2 * q^13 - 2 * q^14 - 6 * q^16 - 16 * q^22 + 4 * q^23 - 10 * q^25 - 20 * q^26 + 6 * q^27 - 4 * q^29 + 16 * q^30 - 4 * q^35 - 2 * q^36 + 24 * q^38 - 2 * q^39 + 24 * q^40 - 2 * q^42 - 32 * q^43 - 6 * q^48 - 6 * q^49 + 10 * q^52 - 36 * q^53 - 16 * q^55 + 6 * q^56 + 28 * q^61 - 8 * q^62 + 22 * q^64 + 20 * q^65 - 16 * q^66 - 24 * q^68 + 4 * q^69 + 16 * q^74 - 10 * q^75 - 20 * q^77 - 20 * q^78 - 32 * q^79 + 6 * q^81 + 8 * q^82 - 4 * q^87 + 32 * q^88 + 16 * q^90 + 8 * q^91 - 44 * q^92 + 36 * q^94 + 16 * q^95

Introduction

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