LMFDB - Embedded newform 273.2.c.b.64.2

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			64.2
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	273.64
Dual form			273.2.c.b.64.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.48119i q^{2} +1.00000 q^{3} -0.193937 q^{4} +4.15633i q^{5} -1.48119i q^{6} +1.00000i q^{7} -2.67513i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.48119i q^{2} +1.00000 q^{3} -0.193937 q^{4} +4.15633i q^{5} -1.48119i q^{6} +1.00000i q^{7} -2.67513i q^{8} +1.00000 q^{9} +6.15633 q^{10} +3.19394i q^{11} -0.193937 q^{12} +(1.48119 - 3.28726i) q^{13} +1.48119 q^{14} +4.15633i q^{15} -4.35026 q^{16} +3.35026 q^{17} -1.48119i q^{18} -2.38787i q^{19} -0.806063i q^{20} +1.00000i q^{21} +4.73084 q^{22} +0.387873 q^{23} -2.67513i q^{24} -12.2750 q^{25} +(-4.86907 - 2.19394i) q^{26} +1.00000 q^{27} -0.193937i q^{28} -7.92478 q^{29} +6.15633 q^{30} -10.7005i q^{31} +1.09332i q^{32} +3.19394i q^{33} -4.96239i q^{34} -4.15633 q^{35} -0.193937 q^{36} +1.61213i q^{37} -3.53690 q^{38} +(1.48119 - 3.28726i) q^{39} +11.1187 q^{40} -1.45580i q^{41} +1.48119 q^{42} +1.92478 q^{43} -0.619421i q^{44} +4.15633i q^{45} -0.574515i q^{46} +3.76845i q^{47} -4.35026 q^{48} -1.00000 q^{49} +18.1817i q^{50} +3.35026 q^{51} +(-0.287258 + 0.637519i) q^{52} -6.00000 q^{53} -1.48119i q^{54} -13.2750 q^{55} +2.67513 q^{56} -2.38787i q^{57} +11.7381i q^{58} +6.15633i q^{59} -0.806063i q^{60} +14.4387 q^{61} -15.8496 q^{62} +1.00000i q^{63} -7.08110 q^{64} +(13.6629 + 6.15633i) q^{65} +4.73084 q^{66} -5.61213i q^{67} -0.649738 q^{68} +0.387873 q^{69} +6.15633i q^{70} -11.8192i q^{71} -2.67513i q^{72} +15.6629i q^{73} +2.38787 q^{74} -12.2750 q^{75} +0.463096i q^{76} -3.19394 q^{77} +(-4.86907 - 2.19394i) q^{78} -8.96239 q^{79} -18.0811i q^{80} +1.00000 q^{81} -2.15633 q^{82} -6.99271i q^{83} -0.193937i q^{84} +13.9248i q^{85} -2.85097i q^{86} -7.92478 q^{87} +8.54420 q^{88} -0.932071i q^{89} +6.15633 q^{90} +(3.28726 + 1.48119i) q^{91} -0.0752228 q^{92} -10.7005i q^{93} +5.58181 q^{94} +9.92478 q^{95} +1.09332i q^{96} +3.35026i q^{97} +1.48119i q^{98} +3.19394i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	1.48119i		−	1.04736i	−0.851914	−	0.523681i	$-0.824558\pi$
$2$		−	1.48119i		−	1.04736i	0.851914	−	0.523681i	$-0.175442\pi$
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−0.193937			−0.0969683
$5$			4.15633i			1.85877i	0.369118	+	0.929383i	$0.379660\pi$
$5$			4.15633i			1.85877i	−0.369118	+	0.929383i	$0.620340\pi$
$6$		−	1.48119i		−	0.604695i
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		−	2.67513i		−	0.945802i
$9$	1.00000			0.333333
$10$	6.15633			1.94680
$11$			3.19394i			0.963008i	0.876444	+	0.481504i	$0.159909\pi$
$11$			3.19394i			0.963008i	−0.876444	+	0.481504i	$0.840091\pi$
$12$	−0.193937			−0.0559847
$13$	1.48119	−	3.28726i	0.410809	−	0.911721i
$13$	1.48119	−	3.28726i	0.410809	−	0.911721i
$14$	1.48119			0.395866
$15$			4.15633i			1.07316i
$16$	−4.35026			−1.08757
$17$	3.35026			0.812558			0.406279	−	0.913749i	$-0.366826\pi$
$17$	3.35026			0.812558			0.406279	+	0.913749i	$0.366826\pi$
$18$		−	1.48119i		−	0.349121i
$19$		−	2.38787i		−	0.547816i	−0.961756	−	0.273908i	$-0.911684\pi$
$19$		−	2.38787i		−	0.547816i	0.961756	−	0.273908i	$-0.0883163\pi$
$20$		−	0.806063i		−	0.180241i
$21$			1.00000i			0.218218i
$22$	4.73084			1.00862
$23$	0.387873			0.0808771			0.0404386	−	0.999182i	$-0.487124\pi$
$23$	0.387873			0.0808771			0.0404386	+	0.999182i	$0.487124\pi$
$24$		−	2.67513i		−	0.546059i
$25$	−12.2750			−2.45501
$26$	−4.86907	−	2.19394i	−0.954903	−	0.430266i
$27$	1.00000			0.192450
$28$		−	0.193937i		−	0.0366506i
$29$	−7.92478			−1.47159			−0.735797	−	0.677202i	$-0.763192\pi$
$29$	−7.92478			−1.47159			−0.735797	+	0.677202i	$0.763192\pi$
$30$	6.15633			1.12399
$31$		−	10.7005i		−	1.92187i	−0.276773	−	0.960935i	$-0.589265\pi$
$31$		−	10.7005i		−	1.92187i	0.276773	−	0.960935i	$-0.410735\pi$
$32$			1.09332i			0.193274i
$33$			3.19394i			0.555993i
$34$		−	4.96239i		−	0.851043i
$35$	−4.15633			−0.702547
$36$	−0.193937			−0.0323228
$37$			1.61213i			0.265032i	0.991181	+	0.132516i	$0.0423056\pi$
$37$			1.61213i			0.265032i	−0.991181	+	0.132516i	$0.957694\pi$
$38$	−3.53690			−0.573762
$39$	1.48119	−	3.28726i	0.237181	−	0.526383i
$40$	11.1187			1.75802
$41$		−	1.45580i		−	0.227358i	−0.993518	−	0.113679i	$-0.963736\pi$
$41$		−	1.45580i		−	0.227358i	0.993518	−	0.113679i	$-0.0362635\pi$
$42$	1.48119			0.228553
$43$	1.92478			0.293526			0.146763	−	0.989172i	$-0.453115\pi$
$43$	1.92478			0.293526			0.146763	+	0.989172i	$0.453115\pi$
$44$		−	0.619421i		−	0.0933812i
$45$			4.15633i			0.619588i
$46$		−	0.574515i		−	0.0847077i
$47$			3.76845i			0.549685i	0.961489	+	0.274843i	$0.0886258\pi$
$47$			3.76845i			0.549685i	−0.961489	+	0.274843i	$0.911374\pi$
$48$	−4.35026			−0.627906
$49$	−1.00000			−0.142857
$50$			18.1817i			2.57128i
$51$	3.35026			0.469130
$52$	−0.287258	+	0.637519i	−0.0398355	+	0.0884080i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		−	1.48119i		−	0.201565i
$55$	−13.2750			−1.79001
$56$	2.67513			0.357479
$57$		−	2.38787i		−	0.316282i
$58$			11.7381i			1.54129i
$59$			6.15633i			0.801485i	0.916191	+	0.400743i	$0.131248\pi$
$59$			6.15633i			0.801485i	−0.916191	+	0.400743i	$0.868752\pi$
$60$		−	0.806063i		−	0.104062i
$61$	14.4387			1.84868			0.924340	−	0.381569i	$-0.124616\pi$
$61$	14.4387			1.84868			0.924340	+	0.381569i	$0.124616\pi$
$62$	−15.8496			−2.01290
$63$			1.00000i			0.125988i
$64$	−7.08110			−0.885138
$65$	13.6629	+	6.15633i	1.69468	+	0.763598i
$66$	4.73084			0.582326
$67$		−	5.61213i		−	0.685630i	−0.939403	−	0.342815i	$-0.888620\pi$
$67$		−	5.61213i		−	0.685630i	0.939403	−	0.342815i	$-0.111380\pi$
$68$	−0.649738			−0.0787923
$69$	0.387873			0.0466944
$70$			6.15633i			0.735822i
$71$		−	11.8192i		−	1.40269i	−0.712824	−	0.701343i	$-0.752584\pi$
$71$		−	11.8192i		−	1.40269i	0.712824	−	0.701343i	$-0.247416\pi$
$72$		−	2.67513i		−	0.315267i
$73$			15.6629i			1.83321i	0.399800	+	0.916603i	$0.369080\pi$
$73$			15.6629i			1.83321i	−0.399800	+	0.916603i	$0.630920\pi$
$74$	2.38787			0.277585
$75$	−12.2750			−1.41740
$76$			0.463096i			0.0531207i
$77$	−3.19394			−0.363983
$78$	−4.86907	−	2.19394i	−0.551313	−	0.248414i
$79$	−8.96239			−1.00835			−0.504174	−	0.863602i	$-0.668203\pi$
$79$	−8.96239			−1.00835			−0.504174	+	0.863602i	$0.668203\pi$
$80$		−	18.0811i		−	2.02153i
$81$	1.00000			0.111111
$82$	−2.15633			−0.238126
$83$		−	6.99271i		−	0.767549i	−0.923427	−	0.383775i	$-0.874624\pi$
$83$		−	6.99271i		−	0.767549i	0.923427	−	0.383775i	$-0.125376\pi$
$84$		−	0.193937i		−	0.0211602i
$85$			13.9248i			1.51035i
$86$		−	2.85097i		−	0.307428i
$87$	−7.92478			−0.849625
$88$	8.54420			0.910815
$89$		−	0.932071i		−	0.0987994i	−0.998779	−	0.0493997i	$-0.984269\pi$
$89$		−	0.932071i		−	0.0987994i	0.998779	−	0.0493997i	$-0.0157308\pi$
$90$	6.15633			0.648934
$91$	3.28726	+	1.48119i	0.344598	+	0.155271i
$92$	−0.0752228			−0.00784252
$93$		−	10.7005i		−	1.10959i
$94$	5.58181			0.575720
$95$	9.92478			1.01826
$96$			1.09332i			0.111587i
$97$			3.35026i			0.340168i	0.985430	+	0.170084i	$0.0544038\pi$
$97$			3.35026i			0.340168i	−0.985430	+	0.170084i	$0.945596\pi$
$98$			1.48119i			0.149623i
$99$			3.19394i			0.321003i
$100$	2.38058			0.238058
$101$	−14.0508			−1.39811			−0.699053	−	0.715070i	$-0.746395\pi$
$101$	−14.0508			−1.39811			−0.699053	+	0.715070i	$0.746395\pi$
$102$		−	4.96239i		−	0.491350i
$103$	1.42548			0.140457			0.0702286	−	0.997531i	$-0.477627\pi$
$103$	1.42548			0.140457			0.0702286	+	0.997531i	$0.477627\pi$
$104$	−8.79384	−	3.96239i	−0.862307	−	0.388544i
$105$	−4.15633			−0.405616
$106$			8.88717i			0.863198i
$107$	9.53690			0.921967			0.460984	−	0.887409i	$-0.347497\pi$
$107$	9.53690			0.921967			0.460984	+	0.887409i	$0.347497\pi$
$108$	−0.193937			−0.0186616
$109$			6.23743i			0.597437i	0.954341	+	0.298719i	$0.0965592\pi$
$109$			6.23743i			0.597437i	−0.954341	+	0.298719i	$0.903441\pi$
$110$			19.6629i			1.87479i
$111$			1.61213i			0.153016i
$112$		−	4.35026i		−	0.411061i
$113$	−3.92478			−0.369212			−0.184606	−	0.982813i	$-0.559101\pi$
$113$	−3.92478			−0.369212			−0.184606	+	0.982813i	$0.559101\pi$
$114$	−3.53690			−0.331261
$115$			1.61213i			0.150332i
$116$	1.53690			0.142698
$117$	1.48119	−	3.28726i	0.136936	−	0.303907i
$118$	9.11871			0.839446
$119$			3.35026i			0.307118i
$120$	11.1187			1.01500
$121$	0.798769			0.0726154
$122$		−	21.3865i		−	1.93624i
$123$		−	1.45580i		−	0.131265i
$124$			2.07522i			0.186361i
$125$		−	30.2374i		−	2.70452i
$126$	1.48119			0.131955
$127$	−0.962389			−0.0853982			−0.0426991	−	0.999088i	$-0.513596\pi$
$127$	−0.962389			−0.0853982			−0.0426991	+	0.999088i	$0.513596\pi$
$128$			12.6751i			1.12033i
$129$	1.92478			0.169467
$130$	9.11871	−	20.2374i	0.799764	−	1.77494i
$131$	17.1490			1.49832			0.749159	−	0.662390i	$-0.230458\pi$
$131$	17.1490			1.49832			0.749159	+	0.662390i	$0.230458\pi$
$132$		−	0.619421i		−	0.0539137i
$133$	2.38787			0.207055
$134$	−8.31265			−0.718104
$135$			4.15633i			0.357720i
$136$		−	8.96239i		−	0.768518i
$137$		−	4.85685i		−	0.414949i	−0.978240	−	0.207474i	$-0.933476\pi$
$137$		−	4.85685i		−	0.414949i	0.978240	−	0.207474i	$-0.0665243\pi$
$138$		−	0.574515i		−	0.0489060i
$139$	2.57452			0.218368			0.109184	−	0.994022i	$-0.465176\pi$
$139$	2.57452			0.218368			0.109184	+	0.994022i	$0.465176\pi$
$140$	0.806063			0.0681248
$141$			3.76845i			0.317361i
$142$	−17.5066			−1.46912
$143$	10.4993	+	4.73084i	0.877995	+	0.395613i
$144$	−4.35026			−0.362522
$145$		−	32.9380i		−	2.73535i
$146$	23.1998			1.92003
$147$	−1.00000			−0.0824786
$148$		−	0.312650i		−	0.0256997i
$149$			13.3806i			1.09618i	0.836419	+	0.548090i	$0.184645\pi$
$149$			13.3806i			1.09618i	−0.836419	+	0.548090i	$0.815355\pi$
$150$			18.1817i			1.48453i
$151$		−	15.8496i		−	1.28982i	−0.764259	−	0.644909i	$-0.776895\pi$
$151$		−	15.8496i		−	1.28982i	0.764259	−	0.644909i	$-0.223105\pi$
$152$	−6.38787			−0.518125
$153$	3.35026			0.270853
$154$			4.73084i			0.381222i
$155$	44.4749			3.57231
$156$	−0.287258	+	0.637519i	−0.0229990	+	0.0510424i
$157$	−12.7005			−1.01361			−0.506806	−	0.862060i	$-0.669174\pi$
$157$	−12.7005			−1.01361			−0.506806	+	0.862060i	$0.669174\pi$
$158$			13.2750i			1.05611i
$159$	−6.00000			−0.475831
$160$	−4.54420			−0.359250
$161$			0.387873i			0.0305687i
$162$		−	1.48119i		−	0.116374i
$163$			9.61213i			0.752880i	0.926441	+	0.376440i	$0.122852\pi$
$163$			9.61213i			0.752880i	−0.926441	+	0.376440i	$0.877148\pi$
$164$			0.282333i			0.0220465i
$165$	−13.2750			−1.03346
$166$	−10.3576			−0.803902
$167$		−	12.0811i		−	0.934864i	−0.884029	−	0.467432i	$-0.845179\pi$
$167$		−	12.0811i		−	0.934864i	0.884029	−	0.467432i	$-0.154821\pi$
$168$	2.67513			0.206391
$169$	−8.61213	−	9.73813i	−0.662471	−	0.749087i
$170$	20.6253			1.58189
$171$		−	2.38787i		−	0.182605i
$172$	−0.373285			−0.0284627
$173$	−9.27504			−0.705168			−0.352584	−	0.935780i	$-0.614697\pi$
$173$	−9.27504			−0.705168			−0.352584	+	0.935780i	$0.614697\pi$
$174$			11.7381i			0.889866i
$175$		−	12.2750i		−	0.927906i
$176$		−	13.8945i		−	1.04733i
$177$			6.15633i			0.462738i
$178$	−1.38058			−0.103479
$179$	−5.01317			−0.374702			−0.187351	−	0.982293i	$-0.559990\pi$
$179$	−5.01317			−0.374702			−0.187351	+	0.982293i	$0.559990\pi$
$180$		−	0.806063i		−	0.0600804i
$181$	−13.8496			−1.02943			−0.514715	−	0.857362i	$-0.672102\pi$
$181$	−13.8496			−1.02943			−0.514715	+	0.857362i	$0.672102\pi$
$182$	2.19394	−	4.86907i	0.162625	−	0.360919i
$183$	14.4387			1.06734
$184$		−	1.03761i		−	0.0764937i
$185$	−6.70052			−0.492632
$186$	−15.8496			−1.16215
$187$			10.7005i			0.782500i
$188$		−	0.730841i		−	0.0533020i
$189$			1.00000i			0.0727393i
$190$		−	14.7005i		−	1.06649i
$191$	−18.5647			−1.34329			−0.671646	−	0.740872i	$-0.734412\pi$
$191$	−18.5647			−1.34329			−0.671646	+	0.740872i	$0.734412\pi$
$192$	−7.08110			−0.511035
$193$			13.0884i			0.942123i	0.882100	+	0.471062i	$0.156129\pi$
$193$			13.0884i			0.942123i	−0.882100	+	0.471062i	$0.843871\pi$
$194$	4.96239			0.356279
$195$	13.6629	+	6.15633i	0.978421	+	0.440864i
$196$	0.193937			0.0138526
$197$			7.24472i			0.516165i	0.966123	+	0.258083i	$0.0830906\pi$
$197$			7.24472i			0.516165i	−0.966123	+	0.258083i	$0.916909\pi$
$198$	4.73084			0.336206
$199$	8.62530			0.611431			0.305716	−	0.952123i	$-0.401104\pi$
$199$	8.62530			0.611431			0.305716	+	0.952123i	$0.401104\pi$
$200$			32.8373i			2.32195i
$201$		−	5.61213i		−	0.395849i
$202$			20.8119i			1.46432i
$203$		−	7.92478i		−	0.556210i
$204$	−0.649738			−0.0454908
$205$	6.05079			0.422605
$206$		−	2.11142i		−	0.147110i
$207$	0.387873			0.0269590
$208$	−6.44358	+	14.3004i	−0.446782	+	0.991557i
$209$	7.62672			0.527551
$210$			6.15633i			0.424827i
$211$	18.8872			1.30025			0.650123	−	0.759829i	$-0.274717\pi$
$211$	18.8872			1.30025			0.650123	+	0.759829i	$0.274717\pi$
$212$	1.16362			0.0799177
$213$		−	11.8192i		−	0.809841i
$214$		−	14.1260i		−	0.965634i
$215$			8.00000i			0.545595i
$216$		−	2.67513i		−	0.182020i
$217$	10.7005			0.726399
$218$	9.23884			0.625733
$219$			15.6629i			1.05840i
$220$	2.57452			0.173574
$221$	4.96239	−	11.0132i	0.333806	−	0.740826i
$222$	2.38787			0.160264
$223$			16.1622i			1.08230i	0.840926	+	0.541151i	$0.182011\pi$
$223$			16.1622i			1.08230i	−0.840926	+	0.541151i	$0.817989\pi$
$224$	−1.09332			−0.0730506
$225$	−12.2750			−0.818336
$226$			5.81336i			0.386699i
$227$			27.2447i			1.80830i	0.427220	+	0.904148i	$0.359493\pi$
$227$			27.2447i			1.80830i	−0.427220	+	0.904148i	$0.640507\pi$
$228$			0.463096i			0.0306693i
$229$			19.5125i			1.28942i	0.764427	+	0.644710i	$0.223022\pi$
$229$			19.5125i			1.28942i	−0.764427	+	0.644710i	$0.776978\pi$
$230$	2.38787			0.157452
$231$	−3.19394			−0.210146
$232$			21.1998i			1.39184i
$233$	−21.8496			−1.43141			−0.715706	−	0.698402i	$-0.753895\pi$
$233$	−21.8496			−1.43141			−0.715706	+	0.698402i	$0.753895\pi$
$234$	−4.86907	−	2.19394i	−0.318301	−	0.143422i
$235$	−15.6629			−1.02174
$236$		−	1.19394i		−	0.0777187i
$237$	−8.96239			−0.582170
$238$	4.96239			0.321664
$239$		−	8.49341i		−	0.549393i	−0.961531	−	0.274697i	$-0.911423\pi$
$239$		−	8.49341i		−	0.549393i	0.961531	−	0.274697i	$-0.0885774\pi$
$240$		−	18.0811i		−	1.16713i
$241$			5.48612i			0.353392i	0.984265	+	0.176696i	$0.0565409\pi$
$241$			5.48612i			0.353392i	−0.984265	+	0.176696i	$0.943459\pi$
$242$		−	1.18313i		−	0.0760546i
$243$	1.00000			0.0641500
$244$	−2.80018			−0.179263
$245$		−	4.15633i		−	0.265538i
$246$	−2.15633			−0.137482
$247$	−7.84955	−	3.53690i	−0.499455	−	0.225048i
$248$	−28.6253			−1.81771
$249$		−	6.99271i		−	0.443145i
$250$	−44.7875			−2.83261
$251$	6.44851			0.407026			0.203513	−	0.979072i	$-0.434764\pi$
$251$	6.44851			0.407026			0.203513	+	0.979072i	$0.434764\pi$
$252$		−	0.193937i		−	0.0122169i
$253$			1.23884i			0.0778853i
$254$			1.42548i			0.0894429i
$255$			13.9248i			0.872003i
$256$	4.61213			0.288258
$257$	9.90034			0.617566			0.308783	−	0.951132i	$-0.400078\pi$
$257$	9.90034			0.617566			0.308783	+	0.951132i	$0.400078\pi$
$258$		−	2.85097i		−	0.177494i
$259$	−1.61213			−0.100173
$260$	−2.64974	−	1.19394i	−0.164330	−	0.0740448i
$261$	−7.92478			−0.490531
$262$		−	25.4010i		−	1.56928i
$263$	3.46168			0.213456			0.106728	−	0.994288i	$-0.465963\pi$
$263$	3.46168			0.213456			0.106728	+	0.994288i	$0.465963\pi$
$264$	8.54420			0.525859
$265$		−	24.9380i		−	1.53193i
$266$		−	3.53690i		−	0.216862i
$267$		−	0.932071i		−	0.0570418i
$268$			1.08840i			0.0664844i
$269$	16.7513			1.02135			0.510673	−	0.859775i	$-0.329396\pi$
$269$	16.7513			1.02135			0.510673	+	0.859775i	$0.329396\pi$
$270$	6.15633			0.374662
$271$		−	6.38787i		−	0.388036i	−0.980998	−	0.194018i	$-0.937848\pi$
$271$		−	6.38787i		−	0.388036i	0.980998	−	0.194018i	$-0.0621520\pi$
$272$	−14.5745			−0.883710
$273$	3.28726	+	1.48119i	0.198954	+	0.0896460i
$274$	−7.19394			−0.434602
$275$		−	39.2057i		−	2.36419i
$276$	−0.0752228			−0.00452788
$277$	12.1768			0.731633			0.365816	−	0.930687i	$-0.380790\pi$
$277$	12.1768			0.731633			0.365816	+	0.930687i	$0.380790\pi$
$278$		−	3.81336i		−	0.228710i
$279$		−	10.7005i		−	0.640624i
$280$			11.1187i			0.664470i
$281$		−	27.3054i		−	1.62890i	−0.580233	−	0.814450i	$-0.697039\pi$
$281$		−	27.3054i		−	1.62890i	0.580233	−	0.814450i	$-0.302961\pi$
$282$	5.58181			0.332392
$283$	27.0738			1.60937			0.804685	−	0.593701i	$-0.202334\pi$
$283$	27.0738			1.60937			0.804685	+	0.593701i	$0.202334\pi$
$284$			2.29218i			0.136016i
$285$	9.92478			0.587893
$286$	7.00729	−	15.5515i	0.414350	−	0.919579i
$287$	1.45580			0.0859333
$288$			1.09332i			0.0644246i
$289$	−5.77575			−0.339750
$290$	−48.7875			−2.86490
$291$			3.35026i			0.196396i
$292$		−	3.03761i		−	0.177763i
$293$			16.7816i			0.980393i	0.871612	+	0.490197i	$0.163075\pi$
$293$			16.7816i			0.980393i	−0.871612	+	0.490197i	$0.836925\pi$
$294$			1.48119i			0.0863850i
$295$	−25.5877			−1.48977
$296$	4.31265			0.250668
$297$			3.19394i			0.185331i
$298$	19.8192			1.14810
$299$	0.574515	−	1.27504i	0.0332251	−	0.0737374i
$300$	2.38058			0.137443
$301$			1.92478i			0.110942i
$302$	−23.4763			−1.35091
$303$	−14.0508			−0.807197
$304$			10.3879i			0.595785i
$305$			60.0118i			3.43626i
$306$		−	4.96239i		−	0.283681i
$307$			17.2995i			0.987333i	0.869651	+	0.493667i	$0.164344\pi$
$307$			17.2995i			0.987333i	−0.869651	+	0.493667i	$0.835656\pi$
$308$	0.619421			0.0352948
$309$	1.42548			0.0810930
$310$		−	65.8759i		−	3.74150i
$311$	17.1490			0.972432			0.486216	−	0.873839i	$-0.338377\pi$
$311$	17.1490			0.972432			0.486216	+	0.873839i	$0.338377\pi$
$312$	−8.79384	−	3.96239i	−0.497853	−	0.224326i
$313$	−10.8119			−0.611127			−0.305564	−	0.952172i	$-0.598845\pi$
$313$	−10.8119			−0.611127			−0.305564	+	0.952172i	$0.598845\pi$
$314$			18.8119i			1.06162i
$315$	−4.15633			−0.234182
$316$	1.73813			0.0977777
$317$		−	14.1563i		−	0.795098i	−0.917581	−	0.397549i	$-0.869861\pi$
$317$		−	14.1563i		−	0.795098i	0.917581	−	0.397549i	$-0.130139\pi$
$318$			8.88717i			0.498368i
$319$		−	25.3112i		−	1.41716i
$320$		−	29.4314i		−	1.64526i
$321$	9.53690			0.532298
$322$	0.574515			0.0320165
$323$		−	8.00000i		−	0.445132i
$324$	−0.193937			−0.0107743
$325$	−18.1817	+	40.3512i	−1.00854	+	2.23828i
$326$	14.2374			0.788538
$327$			6.23743i			0.344931i
$328$	−3.89446			−0.215036
$329$	−3.76845			−0.207761
$330$			19.6629i			1.08241i
$331$			17.2506i			0.948179i	0.880477	+	0.474089i	$0.157223\pi$
$331$			17.2506i			0.948179i	−0.880477	+	0.474089i	$0.842777\pi$
$332$			1.35614i			0.0744279i
$333$			1.61213i			0.0883440i
$334$	−17.8945			−0.979141
$335$	23.3258			1.27443
$336$		−	4.35026i		−	0.237326i
$337$	−5.97556			−0.325510			−0.162755	−	0.986667i	$-0.552038\pi$
$337$	−5.97556			−0.325510			−0.162755	+	0.986667i	$0.552038\pi$
$338$	−14.4241	+	12.7562i	−0.784566	+	0.693848i
$339$	−3.92478			−0.213165
$340$		−	2.70052i		−	0.146456i
$341$	34.1768			1.85078
$342$	−3.53690			−0.191254
$343$		−	1.00000i		−	0.0539949i
$344$		−	5.14903i		−	0.277617i
$345$			1.61213i			0.0867940i
$346$			13.7381i			0.738567i
$347$	25.7889			1.38442			0.692211	−	0.721695i	$-0.256637\pi$
$347$	25.7889			1.38442			0.692211	+	0.721695i	$0.256637\pi$
$348$	1.53690			0.0823867
$349$			12.9624i			0.693861i	0.937891	+	0.346930i	$0.112776\pi$
$349$			12.9624i			0.693861i	−0.937891	+	0.346930i	$0.887224\pi$
$350$	−18.1817			−0.971854
$351$	1.48119	−	3.28726i	0.0790603	−	0.175461i
$352$	−3.49200			−0.186124
$353$		−	3.78304i		−	0.201351i	−0.994919	−	0.100675i	$-0.967900\pi$
$353$		−	3.78304i		−	0.201351i	0.994919	−	0.100675i	$-0.0321004\pi$
$354$	9.11871			0.484654
$355$	49.1246			2.60726
$356$			0.180763i			0.00958041i
$357$			3.35026i			0.177315i
$358$			7.42548i			0.392449i
$359$			10.6702i			0.563152i	0.959539	+	0.281576i	$0.0908571\pi$
$359$			10.6702i			0.563152i	−0.959539	+	0.281576i	$0.909143\pi$
$360$	11.1187			0.586008
$361$	13.2981			0.699898
$362$			20.5139i			1.07819i
$363$	0.798769			0.0419245
$364$	−0.637519	−	0.287258i	−0.0334151	−	0.0150564i
$365$	−65.1002			−3.40750
$366$		−	21.3865i		−	1.11789i
$367$	6.82653			0.356342			0.178171	−	0.984000i	$-0.442982\pi$
$367$	6.82653			0.356342			0.178171	+	0.984000i	$0.442982\pi$
$368$	−1.68735			−0.0879592
$369$		−	1.45580i		−	0.0757860i
$370$			9.92478i			0.515965i
$371$		−	6.00000i		−	0.311504i
$372$			2.07522i			0.107595i
$373$	16.5745			0.858196			0.429098	−	0.903258i	$-0.358832\pi$
$373$	16.5745			0.858196			0.429098	+	0.903258i	$0.358832\pi$
$374$	15.8496			0.819561
$375$		−	30.2374i		−	1.56145i
$376$	10.0811			0.519893
$377$	−11.7381	+	26.0508i	−0.604545	+	1.34168i
$378$	1.48119			0.0761844
$379$		−	19.0738i		−	0.979756i	−0.871791	−	0.489878i	$-0.837041\pi$
$379$		−	19.0738i		−	0.979756i	0.871791	−	0.489878i	$-0.162959\pi$
$380$	−1.92478			−0.0987390
$381$	−0.962389			−0.0493047
$382$			27.4979i			1.40691i
$383$			8.29218i			0.423711i	0.977301	+	0.211855i	$0.0679506\pi$
$383$			8.29218i			0.423711i	−0.977301	+	0.211855i	$0.932049\pi$
$384$			12.6751i			0.646825i
$385$		−	13.2750i		−	0.676559i
$386$	19.3865			0.986745
$387$	1.92478			0.0978419
$388$		−	0.649738i		−	0.0329855i
$389$	0.700523			0.0355180			0.0177590	−	0.999842i	$-0.494347\pi$
$389$	0.700523			0.0355180			0.0177590	+	0.999842i	$0.494347\pi$
$390$	9.11871	−	20.2374i	0.461744	−	1.02476i
$391$	1.29948			0.0657174
$392$			2.67513i			0.135115i
$393$	17.1490			0.865054
$394$	10.7308			0.540612
$395$		−	37.2506i		−	1.87428i
$396$		−	0.619421i		−	0.0311271i
$397$		−	12.4993i		−	0.627322i	−0.949535	−	0.313661i	$-0.898445\pi$
$397$		−	12.4993i		−	0.627322i	0.949535	−	0.313661i	$-0.101555\pi$
$398$		−	12.7757i		−	0.640390i
$399$	2.38787			0.119543
$400$	53.3996			2.66998
$401$			28.7064i			1.43353i	0.697315	+	0.716765i	$0.254378\pi$
$401$			28.7064i			1.43353i	−0.697315	+	0.716765i	$0.745622\pi$
$402$	−8.31265			−0.414597
$403$	−35.1754	−	15.8496i	−1.75221	−	0.789523i
$404$	2.72496			0.135572
$405$			4.15633i			0.206529i
$406$	−11.7381			−0.582554
$407$	−5.14903			−0.255228
$408$		−	8.96239i		−	0.443704i
$409$		−	23.4518i		−	1.15962i	−0.814752	−	0.579809i	$-0.803127\pi$
$409$		−	23.4518i		−	1.15962i	0.814752	−	0.579809i	$-0.196873\pi$
$410$		−	8.96239i		−	0.442621i
$411$		−	4.85685i		−	0.239571i
$412$	−0.276454			−0.0136199
$413$	−6.15633			−0.302933
$414$		−	0.574515i		−	0.0282359i
$415$	29.0640			1.42669
$416$	3.59403	+	1.61942i	0.176212	+	0.0793987i
$417$	2.57452			0.126075
$418$		−	11.2966i		−	0.552537i
$419$	−34.1768			−1.66965			−0.834823	−	0.550519i	$-0.814430\pi$
$419$	−34.1768			−1.66965			−0.834823	+	0.550519i	$0.814430\pi$
$420$	0.806063			0.0393319
$421$			5.55149i			0.270563i	0.990807	+	0.135282i	$0.0431939\pi$
$421$			5.55149i			0.270563i	−0.990807	+	0.135282i	$0.956806\pi$
$422$		−	27.9756i		−	1.36183i
$423$			3.76845i			0.183228i
$424$			16.0508i			0.779495i
$425$	−41.1246			−1.99484
$426$	−17.5066			−0.848197
$427$			14.4387i			0.698736i
$428$	−1.84955			−0.0894016
$429$	10.4993	+	4.73084i	0.506911	+	0.228407i
$430$	11.8496			0.571436
$431$		−	5.43136i		−	0.261620i	−0.991407	−	0.130810i	$-0.958242\pi$
$431$		−	5.43136i		−	0.261620i	0.991407	−	0.130810i	$-0.0417577\pi$
$432$	−4.35026			−0.209302
$433$	40.2130			1.93251			0.966256	−	0.257582i	$-0.0829257\pi$
$433$	40.2130			1.93251			0.966256	+	0.257582i	$0.0829257\pi$
$434$		−	15.8496i		−	0.760803i
$435$		−	32.9380i		−	1.57925i
$436$		−	1.20967i		−	0.0579325i
$437$		−	0.926192i		−	0.0443058i
$438$	23.1998			1.10853
$439$	−22.5745			−1.07742			−0.538711	−	0.842490i	$-0.681089\pi$
$439$	−22.5745			−1.07742			−0.538711	+	0.842490i	$0.681089\pi$
$440$			35.5125i			1.69299i
$441$	−1.00000			−0.0476190
$442$	−16.3127	−	7.35026i	−0.775914	−	0.349616i
$443$	−8.23743			−0.391372			−0.195686	−	0.980667i	$-0.562693\pi$
$443$	−8.23743			−0.391372			−0.195686	+	0.980667i	$0.562693\pi$
$444$		−	0.312650i		−	0.0148377i
$445$	3.87399			0.183645
$446$	23.9394			1.13356
$447$			13.3806i			0.632880i
$448$		−	7.08110i		−	0.334551i
$449$			2.40834i			0.113657i	0.998384	+	0.0568283i	$0.0180988\pi$
$449$			2.40834i			0.113657i	−0.998384	+	0.0568283i	$0.981901\pi$
$450$			18.1817i			0.857094i
$451$	4.64974			0.218948
$452$	0.761158			0.0358019
$453$		−	15.8496i		−	0.744677i
$454$	40.3547			1.89394
$455$	−6.15633	+	13.6629i	−0.288613	+	0.640527i
$456$	−6.38787			−0.299140
$457$			29.5633i			1.38291i	0.722419	+	0.691455i	$0.243030\pi$
$457$			29.5633i			1.38291i	−0.722419	+	0.691455i	$0.756970\pi$
$458$	28.9018			1.35049
$459$	3.35026			0.156377
$460$		−	0.312650i		−	0.0145774i
$461$		−	33.7196i		−	1.57048i	−0.619193	−	0.785239i	$-0.712540\pi$
$461$		−	33.7196i		−	1.57048i	0.619193	−	0.785239i	$-0.287460\pi$
$462$			4.73084i			0.220099i
$463$			16.8364i			0.782453i	0.920294	+	0.391226i	$0.127949\pi$
$463$			16.8364i			0.782453i	−0.920294	+	0.391226i	$0.872051\pi$
$464$	34.4749			1.60045
$465$	44.4749			2.06247
$466$			32.3634i			1.49921i
$467$	−22.7005			−1.05045			−0.525227	−	0.850962i	$-0.676020\pi$
$467$	−22.7005			−1.05045			−0.525227	+	0.850962i	$0.676020\pi$
$468$	−0.287258	+	0.637519i	−0.0132785	+	0.0294693i
$469$	5.61213			0.259144
$470$			23.1998i			1.07013i
$471$	−12.7005			−0.585209
$472$	16.4690			0.758046
$473$			6.14762i			0.282668i
$474$			13.2750i			0.609743i
$475$			29.3112i			1.34489i
$476$		−	0.649738i		−	0.0297807i
$477$	−6.00000			−0.274721
$478$	−12.5804			−0.575414
$479$		−	10.0957i		−	0.461284i	−0.973039	−	0.230642i	$-0.925917\pi$
$479$		−	10.0957i		−	0.461284i	0.973039	−	0.230642i	$-0.0740826\pi$
$480$	−4.54420			−0.207413
$481$	5.29948	+	2.38787i	0.241635	+	0.108878i
$482$	8.12601			0.370130
$483$			0.387873i			0.0176488i
$484$	−0.154911			−0.00704139
$485$	−13.9248			−0.632292
$486$		−	1.48119i		−	0.0671883i
$487$			9.40105i			0.426002i	0.977052	+	0.213001i	$0.0683238\pi$
$487$			9.40105i			0.426002i	−0.977052	+	0.213001i	$0.931676\pi$
$488$		−	38.6253i		−	1.74849i
$489$			9.61213i			0.434675i
$490$	−6.15633			−0.278114
$491$	20.7612			0.936938			0.468469	−	0.883480i	$-0.344806\pi$
$491$	20.7612			0.936938			0.468469	+	0.883480i	$0.344806\pi$
$492$			0.282333i			0.0127286i
$493$	−26.5501			−1.19576
$494$	−5.23884	+	11.6267i	−0.235707	+	0.523111i
$495$	−13.2750			−0.596669
$496$			46.5501i			2.09016i
$497$	11.8192			0.530165
$498$	−10.3576			−0.464133
$499$		−	22.9525i		−	1.02750i	−0.857941	−	0.513748i	$-0.828256\pi$
$499$		−	22.9525i		−	1.02750i	0.857941	−	0.513748i	$-0.171744\pi$
$500$			5.86414i			0.262252i
$501$		−	12.0811i		−	0.539744i
$502$		−	9.55149i		−	0.426304i
$503$	−34.9234			−1.55716			−0.778578	−	0.627548i	$-0.784059\pi$
$503$	−34.9234			−1.55716			−0.778578	+	0.627548i	$0.784059\pi$
$504$	2.67513			0.119160
$505$		−	58.3996i		−	2.59875i
$506$	1.83497			0.0815742
$507$	−8.61213	−	9.73813i	−0.382478	−	0.432486i
$508$	0.186642			0.00828091
$509$			19.2301i			0.852361i	0.904638	+	0.426180i	$0.140141\pi$
$509$			19.2301i			0.852361i	−0.904638	+	0.426180i	$0.859859\pi$
$510$	20.6253			0.913304
$511$	−15.6629			−0.692886
$512$			18.5188i			0.818423i
$513$		−	2.38787i		−	0.105427i
$514$		−	14.6643i		−	0.646816i
$515$			5.92478i			0.261077i
$516$	−0.373285			−0.0164329
$517$	−12.0362			−0.529351
$518$			2.38787i			0.104917i
$519$	−9.27504			−0.407129
$520$	16.4690	−	36.5501i	0.722212	−	1.60283i
$521$	21.5271			0.943117			0.471559	−	0.881835i	$-0.343692\pi$
$521$	21.5271			0.943117			0.471559	+	0.881835i	$0.343692\pi$
$522$			11.7381i			0.513764i
$523$	39.6747			1.73485			0.867426	−	0.497566i	$-0.165773\pi$
$523$	39.6747			1.73485			0.867426	+	0.497566i	$0.165773\pi$
$524$	−3.32582			−0.145289
$525$		−	12.2750i		−	0.535727i
$526$		−	5.12742i		−	0.223566i
$527$		−	35.8496i		−	1.56163i
$528$		−	13.8945i		−	0.604679i
$529$	−22.8496			−0.993459
$530$	−36.9380			−1.60448
$531$			6.15633i			0.267162i
$532$	−0.463096			−0.0200778
$533$	−4.78560	−	2.15633i	−0.207287	−	0.0934008i
$534$	−1.38058			−0.0597435
$535$			39.6385i			1.71372i
$536$	−15.0132			−0.648470
$537$	−5.01317			−0.216334
$538$		−	24.8119i		−	1.06972i
$539$		−	3.19394i		−	0.137573i
$540$		−	0.806063i		−	0.0346874i
$541$		−	10.0263i		−	0.431066i	−0.976497	−	0.215533i	$-0.930851\pi$
$541$		−	10.0263i		−	0.431066i	0.976497	−	0.215533i	$-0.0691489\pi$
$542$	−9.46168			−0.406414
$543$	−13.8496			−0.594341
$544$			3.66291i			0.157046i
$545$	−25.9248			−1.11050
$546$	2.19394	−	4.86907i	0.0938918	−	0.208377i
$547$	−26.4485			−1.13086			−0.565428	−	0.824797i	$-0.691289\pi$
$547$	−26.4485			−1.13086			−0.565428	+	0.824797i	$0.691289\pi$
$548$			0.941921i			0.0402369i
$549$	14.4387			0.616227
$550$	−58.0713			−2.47617
$551$			18.9234i			0.806162i
$552$		−	1.03761i		−	0.0441637i
$553$		−	8.96239i		−	0.381120i
$554$		−	18.0362i		−	0.766285i
$555$	−6.70052			−0.284421
$556$	−0.499293			−0.0211747
$557$			5.84367i			0.247604i	0.992307	+	0.123802i	$0.0395088\pi$
$557$			5.84367i			0.247604i	−0.992307	+	0.123802i	$0.960491\pi$
$558$	−15.8496			−0.670965
$559$	2.85097	−	6.32724i	0.120583	−	0.267614i
$560$	18.0811			0.764066
$561$			10.7005i			0.451776i
$562$	−40.4445			−1.70605
$563$	1.25060			0.0527066			0.0263533	−	0.999653i	$-0.491611\pi$
$563$	1.25060			0.0527066			0.0263533	+	0.999653i	$0.491611\pi$
$564$		−	0.730841i		−	0.0307739i
$565$		−	16.3127i		−	0.686278i
$566$		−	40.1016i		−	1.68559i
$567$			1.00000i			0.0419961i
$568$	−31.6180			−1.32666
$569$	−38.2228			−1.60238			−0.801192	−	0.598407i	$-0.795801\pi$
$569$	−38.2228			−1.60238			−0.801192	+	0.598407i	$0.795801\pi$
$570$		−	14.7005i		−	0.615737i
$571$	−9.73813			−0.407528			−0.203764	−	0.979020i	$-0.565318\pi$
$571$	−9.73813			−0.407528			−0.203764	+	0.979020i	$0.565318\pi$
$572$	−2.03620	−	0.917483i	−0.0851377	−	0.0383619i
$573$	−18.5647			−0.775550
$574$		−	2.15633i		−	0.0900033i
$575$	−4.76116			−0.198554
$576$	−7.08110			−0.295046
$577$		−	3.87399i		−	0.161276i	−0.996743	−	0.0806382i	$-0.974304\pi$
$577$		−	3.87399i		−	0.161276i	0.996743	−	0.0806382i	$-0.0256958\pi$
$578$			8.55500i			0.355841i
$579$			13.0884i			0.543935i
$580$			6.38787i			0.265242i
$581$	6.99271			0.290106
$582$	4.96239			0.205698
$583$		−	19.1636i		−	0.793676i
$584$	41.9003			1.73385
$585$	13.6629	+	6.15633i	0.564892	+	0.254533i
$586$	24.8568			1.02683
$587$		−	11.2447i		−	0.464119i	−0.972702	−	0.232060i	$-0.925454\pi$
$587$		−	11.2447i		−	0.464119i	0.972702	−	0.232060i	$-0.0745465\pi$
$588$	0.193937			0.00799781
$589$	−25.5515			−1.05283
$590$			37.9003i			1.56033i
$591$			7.24472i			0.298008i
$592$		−	7.01317i		−	0.288240i
$593$			28.7210i			1.17943i	0.807612	+	0.589715i	$0.200760\pi$
$593$			28.7210i			1.17943i	−0.807612	+	0.589715i	$0.799240\pi$
$594$	4.73084			0.194109
$595$	−13.9248			−0.570860
$596$		−	2.59498i		−	0.106295i
$597$	8.62530			0.353010
$598$	−1.88858	−	0.850969i	−0.0772298	−	0.0347987i
$599$	−21.1344			−0.863530			−0.431765	−	0.901986i	$-0.642109\pi$
$599$	−21.1344			−0.863530			−0.431765	+	0.901986i	$0.642109\pi$
$600$			32.8373i			1.34058i
$601$	18.9135			0.771498			0.385749	−	0.922604i	$-0.373943\pi$
$601$	18.9135			0.771498			0.385749	+	0.922604i	$0.373943\pi$
$602$	2.85097			0.116197
$603$		−	5.61213i		−	0.228543i
$604$			3.07381i			0.125071i
$605$			3.31994i			0.134975i
$606$			20.8119i			0.845427i
$607$	−41.1246			−1.66920			−0.834598	−	0.550860i	$-0.814300\pi$
$607$	−41.1246			−1.66920			−0.834598	+	0.550860i	$0.814300\pi$
$608$	2.61071			0.105878
$609$		−	7.92478i		−	0.321128i
$610$	88.8891			3.59901
$611$	12.3879	+	5.58181i	0.501160	+	0.225816i
$612$	−0.649738			−0.0262641
$613$		−	23.3258i		−	0.942121i	−0.882101	−	0.471061i	$-0.843871\pi$
$613$		−	23.3258i		−	0.942121i	0.882101	−	0.471061i	$-0.156129\pi$
$614$	25.6239			1.03410
$615$	6.05079			0.243991
$616$			8.54420i			0.344256i
$617$			17.7830i			0.715918i	0.933737	+	0.357959i	$0.116527\pi$
$617$			17.7830i			0.715918i	−0.933737	+	0.357959i	$0.883473\pi$
$618$		−	2.11142i		−	0.0849338i
$619$		−	8.93795i		−	0.359247i	−0.983735	−	0.179623i	$-0.942512\pi$
$619$		−	8.93795i		−	0.359247i	0.983735	−	0.179623i	$-0.0574879\pi$
$620$	−8.62530			−0.346400
$621$	0.387873			0.0155648
$622$		−	25.4010i		−	1.01849i
$623$	0.932071			0.0373427
$624$	−6.44358	+	14.3004i	−0.257950	+	0.572475i
$625$	64.3014			2.57206
$626$			16.0146i			0.640072i
$627$	7.62672			0.304582
$628$	2.46310			0.0982882
$629$			5.40105i			0.215354i
$630$			6.15633i			0.245274i
$631$			32.2638i			1.28440i	0.766537	+	0.642200i	$0.221978\pi$
$631$			32.2638i			1.28440i	−0.766537	+	0.642200i	$0.778022\pi$
$632$			23.9756i			0.953697i
$633$	18.8872			0.750697
$634$	−20.9683			−0.832756
$635$		−	4.00000i		−	0.158735i
$636$	1.16362			0.0461405
$637$	−1.48119	+	3.28726i	−0.0586871	+	0.130246i
$638$	−37.4909			−1.48428
$639$		−	11.8192i		−	0.467562i
$640$	−52.6820			−2.08244
$641$	−49.4274			−1.95226			−0.976132	−	0.217176i	$-0.930315\pi$
$641$	−49.4274			−1.95226			−0.976132	+	0.217176i	$0.930315\pi$
$642$		−	14.1260i		−	0.557509i
$643$		−	6.07522i		−	0.239583i	−0.992799	−	0.119792i	$-0.961777\pi$
$643$		−	6.07522i		−	0.239583i	0.992799	−	0.119792i	$-0.0382227\pi$
$644$		−	0.0752228i		−	0.00296419i
$645$			8.00000i			0.315000i
$646$	−11.8496			−0.466214
$647$	−12.9986			−0.511027			−0.255514	−	0.966805i	$-0.582245\pi$
$647$	−12.9986			−0.511027			−0.255514	+	0.966805i	$0.582245\pi$
$648$		−	2.67513i		−	0.105089i
$649$	−19.6629			−0.771837
$650$	59.7680	+	26.9307i	2.34429	+	1.05631i
$651$	10.7005			0.419387
$652$		−	1.86414i		−	0.0730055i
$653$	15.4010			0.602690			0.301345	−	0.953515i	$-0.402564\pi$
$653$	15.4010			0.602690			0.301345	+	0.953515i	$0.402564\pi$
$654$	9.23884			0.361267
$655$			71.2769i			2.78502i
$656$			6.33312i			0.247267i
$657$			15.6629i			0.611068i
$658$			5.58181i			0.217602i
$659$	8.76116			0.341286			0.170643	−	0.985333i	$-0.445415\pi$
$659$	8.76116			0.341286			0.170643	+	0.985333i	$0.445415\pi$
$660$	2.57452			0.100213
$661$			33.4372i			1.30056i	0.759695	+	0.650279i	$0.225348\pi$
$661$			33.4372i			1.30056i	−0.759695	+	0.650279i	$0.774652\pi$
$662$	25.5515			0.993087
$663$	4.96239	−	11.0132i	0.192723	−	0.427716i
$664$	−18.7064			−0.725949
$665$			9.92478i			0.384866i
$666$	2.38787			0.0925282
$667$	−3.07381			−0.119018
$668$			2.34297i			0.0906521i
$669$			16.1622i			0.624867i
$670$		−	34.5501i		−	1.33479i
$671$			46.1162i			1.78029i
$672$	−1.09332			−0.0421758
$673$	−0.176793			−0.00681488			−0.00340744	−	0.999994i	$-0.501085\pi$
$673$	−0.176793			−0.00681488			−0.00340744	+	0.999994i	$0.501085\pi$
$674$			8.85097i			0.340927i
$675$	−12.2750			−0.472466
$676$	1.67021	+	1.88858i	0.0642387	+	0.0726377i
$677$	−5.67750			−0.218204			−0.109102	−	0.994031i	$-0.534798\pi$
$677$	−5.67750			−0.218204			−0.109102	+	0.994031i	$0.534798\pi$
$678$			5.81336i			0.223261i
$679$	−3.35026			−0.128571
$680$	37.2506			1.42850
$681$			27.2447i			1.04402i
$682$		−	50.6225i		−	1.93843i
$683$		−	50.4299i		−	1.92965i	−0.262896	−	0.964824i	$-0.584678\pi$
$683$		−	50.4299i		−	1.92965i	0.262896	−	0.964824i	$-0.415322\pi$
$684$			0.463096i			0.0177069i
$685$	20.1866			0.771292
$686$	−1.48119			−0.0565523
$687$			19.5125i			0.744447i
$688$	−8.37328			−0.319228
$689$	−8.88717	+	19.7235i	−0.338574	+	0.751407i
$690$	2.38787			0.0909048
$691$		−	41.5633i		−	1.58114i	−0.612371	−	0.790570i	$-0.709784\pi$
$691$		−	41.5633i		−	1.58114i	0.612371	−	0.790570i	$-0.290216\pi$
$692$	1.79877			0.0683789
$693$	−3.19394			−0.121328
$694$		−	38.1984i		−	1.44999i
$695$			10.7005i			0.405894i
$696$			21.1998i			0.803577i
$697$		−	4.87732i		−	0.184742i
$698$	19.1998			0.726724
$699$	−21.8496			−0.826426
$700$			2.38058i			0.0899774i
$701$	−2.47486			−0.0934740			−0.0467370	−	0.998907i	$-0.514882\pi$
$701$	−2.47486			−0.0934740			−0.0467370	+	0.998907i	$0.514882\pi$
$702$	−4.86907	−	2.19394i	−0.183771	−	0.0828048i
$703$	3.84955			0.145189
$704$		−	22.6166i		−	0.852395i
$705$	−15.6629			−0.589899
$706$	−5.60342			−0.210887
$707$		−	14.0508i		−	0.528434i
$708$		−	1.19394i		−	0.0448709i
$709$			7.91019i			0.297073i	0.988907	+	0.148537i	$0.0474563\pi$
$709$			7.91019i			0.297073i	−0.988907	+	0.148537i	$0.952544\pi$
$710$		−	72.7631i		−	2.73075i
$711$	−8.96239			−0.336116
$712$	−2.49341			−0.0934446
$713$		−	4.15045i		−	0.155435i
$714$	4.96239			0.185713
$715$	−19.6629	+	43.6385i	−0.735351	+	1.63199i
$716$	0.972238			0.0363342
$717$		−	8.49341i		−	0.317192i
$718$	15.8046			0.589824
$719$	26.5501			0.990151			0.495075	−	0.868850i	$-0.335140\pi$
$719$	26.5501			0.990151			0.495075	+	0.868850i	$0.335140\pi$
$720$		−	18.0811i		−	0.673843i
$721$			1.42548i			0.0530878i
$722$		−	19.6970i		−	0.733047i
$723$			5.48612i			0.204031i
$724$	2.68594			0.0998220
$725$	97.2769			3.61278
$726$		−	1.18313i		−	0.0439102i
$727$	−6.44851			−0.239162			−0.119581	−	0.992824i	$-0.538155\pi$
$727$	−6.44851			−0.239162			−0.119581	+	0.992824i	$0.538155\pi$
$728$	3.96239	−	8.79384i	0.146856	−	0.325922i
$729$	1.00000			0.0370370
$730$			96.4260i			3.56889i
$731$	6.44851			0.238507
$732$	−2.80018			−0.103498
$733$		−	28.8119i		−	1.06419i	−0.846684	−	0.532097i	$-0.821404\pi$
$733$		−	28.8119i		−	1.06419i	0.846684	−	0.532097i	$-0.178596\pi$
$734$		−	10.1114i		−	0.373219i
$735$		−	4.15633i		−	0.153308i
$736$			0.424070i			0.0156314i
$737$	17.9248			0.660268
$738$	−2.15633			−0.0793754
$739$		−	29.6121i		−	1.08930i	−0.838664	−	0.544650i	$-0.816663\pi$
$739$		−	29.6121i		−	1.08930i	0.838664	−	0.544650i	$-0.183337\pi$
$740$	1.29948			0.0477697
$741$	−7.84955	−	3.53690i	−0.288361	−	0.129931i
$742$	−8.88717			−0.326258
$743$			3.77830i			0.138612i	0.997595	+	0.0693062i	$0.0220786\pi$
$743$			3.77830i			0.138612i	−0.997595	+	0.0693062i	$0.977921\pi$
$744$	−28.6253			−1.04945
$745$	−55.6140			−2.03754
$746$		−	24.5501i		−	0.898842i
$747$		−	6.99271i		−	0.255850i
$748$		−	2.07522i		−	0.0758777i
$749$			9.53690i			0.348471i
$750$	−44.7875			−1.63541
$751$	−10.0752			−0.367650			−0.183825	−	0.982959i	$-0.558848\pi$
$751$	−10.0752			−0.367650			−0.183825	+	0.982959i	$0.558848\pi$
$752$		−	16.3938i		−	0.597819i
$753$	6.44851			0.234997
$754$	38.5863	+	17.3865i	1.40523	+	0.633177i
$755$	65.8759			2.39747
$756$		−	0.193937i		−	0.00705340i
$757$	39.4979			1.43557			0.717787	−	0.696262i	$-0.245155\pi$
$757$	39.4979			1.43557			0.717787	+	0.696262i	$0.245155\pi$
$758$	−28.2520			−1.02616
$759$			1.23884i			0.0449671i
$760$		−	26.5501i		−	0.963073i
$761$		−	25.7685i		−	0.934106i	−0.884229	−	0.467053i	$-0.845316\pi$
$761$		−	25.7685i		−	0.934106i	0.884229	−	0.467053i	$-0.154684\pi$
$762$			1.42548i			0.0516399i
$763$	−6.23743			−0.225810
$764$	3.60037			0.130257
$765$			13.9248i			0.503451i
$766$	12.2823			0.443779
$767$	20.2374	+	9.11871i	0.730731	+	0.329258i
$768$	4.61213			0.166426
$769$		−	49.4372i		−	1.78275i	−0.453264	−	0.891376i	$-0.649741\pi$
$769$		−	49.4372i		−	1.78275i	0.453264	−	0.891376i	$-0.350259\pi$
$770$	−19.6629			−0.708602
$771$	9.90034			0.356552
$772$		−	2.53832i		−	0.0913561i
$773$			15.2184i			0.547367i	0.961820	+	0.273683i	$0.0882421\pi$
$773$			15.2184i			0.547367i	−0.961820	+	0.273683i	$0.911758\pi$
$774$		−	2.85097i		−	0.102476i
$775$			131.349i			4.71821i
$776$	8.96239			0.321731
$777$	−1.61213			−0.0578347
$778$		−	1.03761i		−	0.0372002i
$779$	−3.47627			−0.124550
$780$	−2.64974	−	1.19394i	−0.0948758	−	0.0427498i
$781$	37.7499			1.35080
$782$		−	1.92478i		−	0.0688299i
$783$	−7.92478			−0.283208
$784$	4.35026			0.155366
$785$		−	52.7875i

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)

\(f(q)\)	\(=\)	\(q-1.48119i q^{2} +1.00000 q^{3} -0.193937 q^{4} +4.15633i q^{5} -1.48119i q^{6} +1.00000i q^{7} -2.67513i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.48119i q^{2} +1.00000 q^{3} -0.193937 q^{4} +4.15633i q^{5} -1.48119i q^{6} +1.00000i q^{7} -2.67513i q^{8} +1.00000 q^{9} +6.15633 q^{10} +3.19394i q^{11} -0.193937 q^{12} +(1.48119 - 3.28726i) q^{13} +1.48119 q^{14} +4.15633i q^{15} -4.35026 q^{16} +3.35026 q^{17} -1.48119i q^{18} -2.38787i q^{19} -0.806063i q^{20} +1.00000i q^{21} +4.73084 q^{22} +0.387873 q^{23} -2.67513i q^{24} -12.2750 q^{25} +(-4.86907 - 2.19394i) q^{26} +1.00000 q^{27} -0.193937i q^{28} -7.92478 q^{29} +6.15633 q^{30} -10.7005i q^{31} +1.09332i q^{32} +3.19394i q^{33} -4.96239i q^{34} -4.15633 q^{35} -0.193937 q^{36} +1.61213i q^{37} -3.53690 q^{38} +(1.48119 - 3.28726i) q^{39} +11.1187 q^{40} -1.45580i q^{41} +1.48119 q^{42} +1.92478 q^{43} -0.619421i q^{44} +4.15633i q^{45} -0.574515i q^{46} +3.76845i q^{47} -4.35026 q^{48} -1.00000 q^{49} +18.1817i q^{50} +3.35026 q^{51} +(-0.287258 + 0.637519i) q^{52} -6.00000 q^{53} -1.48119i q^{54} -13.2750 q^{55} +2.67513 q^{56} -2.38787i q^{57} +11.7381i q^{58} +6.15633i q^{59} -0.806063i q^{60} +14.4387 q^{61} -15.8496 q^{62} +1.00000i q^{63} -7.08110 q^{64} +(13.6629 + 6.15633i) q^{65} +4.73084 q^{66} -5.61213i q^{67} -0.649738 q^{68} +0.387873 q^{69} +6.15633i q^{70} -11.8192i q^{71} -2.67513i q^{72} +15.6629i q^{73} +2.38787 q^{74} -12.2750 q^{75} +0.463096i q^{76} -3.19394 q^{77} +(-4.86907 - 2.19394i) q^{78} -8.96239 q^{79} -18.0811i q^{80} +1.00000 q^{81} -2.15633 q^{82} -6.99271i q^{83} -0.193937i q^{84} +13.9248i q^{85} -2.85097i q^{86} -7.92478 q^{87} +8.54420 q^{88} -0.932071i q^{89} +6.15633 q^{90} +(3.28726 + 1.48119i) q^{91} -0.0752228 q^{92} -10.7005i q^{93} +5.58181 q^{94} +9.92478 q^{95} +1.09332i q^{96} +3.35026i q^{97} +1.48119i q^{98} +3.19394i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more