LMFDB - Embedded newform 273.2.c.a.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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	273.64
Dual form			273.2.c.a.64.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} +1.00000 q^{3} -2.00000 q^{4} -3.00000i q^{5} -2.00000i q^{6} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} +1.00000 q^{3} -2.00000 q^{4} -3.00000i q^{5} -2.00000i q^{6} +1.00000i q^{7} +1.00000 q^{9} -6.00000 q^{10} -2.00000 q^{12} +(2.00000 + 3.00000i) q^{13} +2.00000 q^{14} -3.00000i q^{15} -4.00000 q^{16} -2.00000 q^{17} -2.00000i q^{18} -1.00000i q^{19} +6.00000i q^{20} +1.00000i q^{21} -1.00000 q^{23} -4.00000 q^{25} +(6.00000 - 4.00000i) q^{26} +1.00000 q^{27} -2.00000i q^{28} +5.00000 q^{29} -6.00000 q^{30} -5.00000i q^{31} +8.00000i q^{32} +4.00000i q^{34} +3.00000 q^{35} -2.00000 q^{36} +8.00000i q^{37} -2.00000 q^{38} +(2.00000 + 3.00000i) q^{39} +10.0000i q^{41} +2.00000 q^{42} +9.00000 q^{43} -3.00000i q^{45} +2.00000i q^{46} -7.00000i q^{47} -4.00000 q^{48} -1.00000 q^{49} +8.00000i q^{50} -2.00000 q^{51} +(-4.00000 - 6.00000i) q^{52} +9.00000 q^{53} -2.00000i q^{54} -1.00000i q^{57} -10.0000i q^{58} +4.00000i q^{59} +6.00000i q^{60} -8.00000 q^{61} -10.0000 q^{62} +1.00000i q^{63} +8.00000 q^{64} +(9.00000 - 6.00000i) q^{65} -2.00000i q^{67} +4.00000 q^{68} -1.00000 q^{69} -6.00000i q^{70} -9.00000i q^{73} +16.0000 q^{74} -4.00000 q^{75} +2.00000i q^{76} +(6.00000 - 4.00000i) q^{78} +15.0000 q^{79} +12.0000i q^{80} +1.00000 q^{81} +20.0000 q^{82} -9.00000i q^{83} -2.00000i q^{84} +6.00000i q^{85} -18.0000i q^{86} +5.00000 q^{87} +9.00000i q^{89} -6.00000 q^{90} +(-3.00000 + 2.00000i) q^{91} +2.00000 q^{92} -5.00000i q^{93} -14.0000 q^{94} -3.00000 q^{95} +8.00000i q^{96} +13.0000i q^{97} +2.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{4} + 2 q^{9} - 12 q^{10} - 4 q^{12} + 4 q^{13} + 4 q^{14} - 8 q^{16} - 4 q^{17} - 2 q^{23} - 8 q^{25} + 12 q^{26} + 2 q^{27} + 10 q^{29} - 12 q^{30} + 6 q^{35} - 4 q^{36} - 4 q^{38} + 4 q^{39} + 4 q^{42} + 18 q^{43} - 8 q^{48} - 2 q^{49} - 4 q^{51} - 8 q^{52} + 18 q^{53} - 16 q^{61} - 20 q^{62} + 16 q^{64} + 18 q^{65} + 8 q^{68} - 2 q^{69} + 32 q^{74} - 8 q^{75} + 12 q^{78} + 30 q^{79} + 2 q^{81} + 40 q^{82} + 10 q^{87} - 12 q^{90} - 6 q^{91} + 4 q^{92} - 28 q^{94} - 6 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9 - 12 * q^10 - 4 * q^12 + 4 * q^13 + 4 * q^14 - 8 * q^16 - 4 * q^17 - 2 * q^23 - 8 * q^25 + 12 * q^26 + 2 * q^27 + 10 * q^29 - 12 * q^30 + 6 * q^35 - 4 * q^36 - 4 * q^38 + 4 * q^39 + 4 * q^42 + 18 * q^43 - 8 * q^48 - 2 * q^49 - 4 * q^51 - 8 * q^52 + 18 * q^53 - 16 * q^61 - 20 * q^62 + 16 * q^64 + 18 * q^65 + 8 * q^68 - 2 * q^69 + 32 * q^74 - 8 * q^75 + 12 * q^78 + 30 * q^79 + 2 * q^81 + 40 * q^82 + 10 * q^87 - 12 * q^90 - 6 * q^91 + 4 * q^92 - 28 * q^94 - 6 * 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 $

$2$		−	2.00000i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$		−	2.00000i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−2.00000			−1.00000
$5$		−	3.00000i		−	1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$		−	3.00000i		−	1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$		−	2.00000i		−	0.816497i
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$	−6.00000			−1.89737
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−2.00000			−0.577350
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$	2.00000			0.534522
$15$		−	3.00000i		−	0.774597i
$16$	−4.00000			−1.00000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		−	2.00000i		−	0.471405i
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$			6.00000i			1.34164i
$21$			1.00000i			0.218218i
$22$		0			0
$23$	−1.00000			−0.208514			−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000			−0.208514			−0.104257	+	0.994550i	$0.533247\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$	6.00000	−	4.00000i	1.17670	−	0.784465i
$27$	1.00000			0.192450
$28$		−	2.00000i		−	0.377964i
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$	−6.00000			−1.09545
$31$		−	5.00000i		−	0.898027i	−0.893525	−	0.449013i	$-0.851776\pi$
$31$		−	5.00000i		−	0.898027i	0.893525	−	0.449013i	$-0.148224\pi$
$32$			8.00000i			1.41421i
$33$		0			0
$34$			4.00000i			0.685994i
$35$	3.00000			0.507093
$36$	−2.00000			−0.333333
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	−2.00000			−0.324443
$39$	2.00000	+	3.00000i	0.320256	+	0.480384i
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$	2.00000			0.308607
$43$	9.00000			1.37249			0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000			1.37249			0.686244	+	0.727372i	$0.259258\pi$
$44$		0			0
$45$		−	3.00000i		−	0.447214i
$46$			2.00000i			0.294884i
$47$		−	7.00000i		−	1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$		−	7.00000i		−	1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	−4.00000			−0.577350
$49$	−1.00000			−0.142857
$50$			8.00000i			1.13137i
$51$	−2.00000			−0.280056
$52$	−4.00000	−	6.00000i	−0.554700	−	0.832050i
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		−	2.00000i		−	0.272166i
$55$		0			0
$56$		0			0
$57$		−	1.00000i		−	0.132453i
$58$		−	10.0000i		−	1.31306i
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$			6.00000i			0.774597i
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$	−10.0000			−1.27000
$63$			1.00000i			0.125988i
$64$	8.00000			1.00000
$65$	9.00000	−	6.00000i	1.11631	−	0.744208i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	4.00000			0.485071
$69$	−1.00000			−0.120386
$70$		−	6.00000i		−	0.717137i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	9.00000i		−	1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$		−	9.00000i		−	1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$	16.0000			1.85996
$75$	−4.00000			−0.461880
$76$			2.00000i			0.229416i
$77$		0			0
$78$	6.00000	−	4.00000i	0.679366	−	0.452911i
$79$	15.0000			1.68763			0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000			1.68763			0.843816	+	0.536633i	$0.180304\pi$
$80$			12.0000i			1.34164i
$81$	1.00000			0.111111
$82$	20.0000			2.20863
$83$		−	9.00000i		−	0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$		−	9.00000i		−	0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$		−	2.00000i		−	0.218218i
$85$			6.00000i			0.650791i
$86$		−	18.0000i		−	1.94099i
$87$	5.00000			0.536056
$88$		0			0
$89$			9.00000i			0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$			9.00000i			0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$	−6.00000			−0.632456
$91$	−3.00000	+	2.00000i	−0.314485	+	0.209657i
$92$	2.00000			0.208514
$93$		−	5.00000i		−	0.518476i
$94$	−14.0000			−1.44399
$95$	−3.00000			−0.307794
$96$			8.00000i			0.816497i
$97$			13.0000i			1.31995i	0.751288	+	0.659975i	$0.229433\pi$
$97$			13.0000i			1.31995i	−0.751288	+	0.659975i	$0.770567\pi$
$98$			2.00000i			0.202031i
$99$		0			0
$100$	8.00000			0.800000
$101$	−8.00000			−0.796030			−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000			−0.796030			−0.398015	+	0.917379i	$0.630301\pi$
$102$			4.00000i			0.396059i
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$		0			0
$105$	3.00000			0.292770
$106$		−	18.0000i		−	1.74831i
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$	−2.00000			−0.192450
$109$			14.0000i			1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$			14.0000i			1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$		0			0
$111$			8.00000i			0.759326i
$112$		−	4.00000i		−	0.377964i
$113$	−21.0000			−1.97551			−0.987757	−	0.156001i	$-0.950140\pi$
$113$	−21.0000			−1.97551			−0.987757	+	0.156001i	$0.950140\pi$
$114$	−2.00000			−0.187317
$115$			3.00000i			0.279751i
$116$	−10.0000			−0.928477
$117$	2.00000	+	3.00000i	0.184900	+	0.277350i
$118$	8.00000			0.736460
$119$		−	2.00000i		−	0.183340i
$120$		0			0
$121$	11.0000			1.00000
$122$			16.0000i			1.44857i
$123$			10.0000i			0.901670i
$124$			10.0000i			0.898027i
$125$		−	3.00000i		−	0.268328i
$126$	2.00000			0.178174
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	9.00000			0.792406
$130$	−12.0000	−	18.0000i	−1.05247	−	1.57870i
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$	1.00000			0.0867110
$134$	−4.00000			−0.345547
$135$		−	3.00000i		−	0.258199i
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$			2.00000i			0.170251i
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$	−6.00000			−0.507093
$141$		−	7.00000i		−	0.589506i
$142$		0			0
$143$		0			0
$144$	−4.00000			−0.333333
$145$		−	15.0000i		−	1.24568i
$146$	−18.0000			−1.48969
$147$	−1.00000			−0.0824786
$148$		−	16.0000i		−	1.31519i
$149$			4.00000i			0.327693i	0.986486	+	0.163846i	$0.0523901\pi$
$149$			4.00000i			0.327693i	−0.986486	+	0.163846i	$0.947610\pi$
$150$			8.00000i			0.653197i
$151$			20.0000i			1.62758i	0.581161	+	0.813788i	$0.302599\pi$
$151$			20.0000i			1.62758i	−0.581161	+	0.813788i	$0.697401\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$	−15.0000			−1.20483
$156$	−4.00000	−	6.00000i	−0.320256	−	0.480384i
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		−	30.0000i		−	2.38667i
$159$	9.00000			0.713746
$160$	24.0000			1.89737
$161$		−	1.00000i		−	0.0788110i
$162$		−	2.00000i		−	0.157135i
$163$			6.00000i			0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$			6.00000i			0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$		−	20.0000i		−	1.56174i
$165$		0			0
$166$	−18.0000			−1.39707
$167$			3.00000i			0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$			3.00000i			0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$	12.0000			0.920358
$171$		−	1.00000i		−	0.0764719i
$172$	−18.0000			−1.37249
$173$	24.0000			1.82469			0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000			1.82469			0.912343	+	0.409426i	$0.134271\pi$
$174$		−	10.0000i		−	0.758098i
$175$		−	4.00000i		−	0.302372i
$176$		0			0
$177$			4.00000i			0.300658i
$178$	18.0000			1.34916
$179$	−5.00000			−0.373718			−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000			−0.373718			−0.186859	+	0.982387i	$0.559831\pi$
$180$			6.00000i			0.447214i
$181$	−8.00000			−0.594635			−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000			−0.594635			−0.297318	+	0.954779i	$0.596092\pi$
$182$	4.00000	+	6.00000i	0.296500	+	0.444750i
$183$	−8.00000			−0.591377
$184$		0			0
$185$	24.0000			1.76452
$186$	−10.0000			−0.733236
$187$		0			0
$188$			14.0000i			1.02105i
$189$			1.00000i			0.0727393i
$190$			6.00000i			0.435286i
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$	8.00000			0.577350
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	26.0000			1.86669
$195$	9.00000	−	6.00000i	0.644503	−	0.429669i
$196$	2.00000			0.142857
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$		−	2.00000i		−	0.141069i
$202$			16.0000i			1.12576i
$203$			5.00000i			0.350931i
$204$	4.00000			0.280056
$205$	30.0000			2.09529
$206$			32.0000i			2.22955i
$207$	−1.00000			−0.0695048
$208$	−8.00000	−	12.0000i	−0.554700	−	0.832050i
$209$		0			0
$210$		−	6.00000i		−	0.414039i
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$	−18.0000			−1.23625
$213$		0			0
$214$			24.0000i			1.64061i
$215$		−	27.0000i		−	1.84138i
$216$		0			0
$217$	5.00000			0.339422
$218$	28.0000			1.89640
$219$		−	9.00000i		−	0.608164i
$220$		0			0
$221$	−4.00000	−	6.00000i	−0.269069	−	0.403604i
$222$	16.0000			1.07385
$223$		−	19.0000i		−	1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$		−	19.0000i		−	1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	−8.00000			−0.534522
$225$	−4.00000			−0.266667
$226$			42.0000i			2.79380i
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$			2.00000i			0.132453i
$229$			14.0000i			0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$			14.0000i			0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$	6.00000			0.395628
$231$		0			0
$232$		0			0
$233$	−1.00000			−0.0655122			−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000			−0.0655122			−0.0327561	+	0.999463i	$0.510428\pi$
$234$	6.00000	−	4.00000i	0.392232	−	0.261488i
$235$	−21.0000			−1.36989
$236$		−	8.00000i		−	0.520756i
$237$	15.0000			0.974355
$238$	−4.00000			−0.259281
$239$		−	6.00000i		−	0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$		−	6.00000i		−	0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$			12.0000i			0.774597i
$241$			15.0000i			0.966235i	0.875556	+	0.483117i	$0.160496\pi$
$241$			15.0000i			0.966235i	−0.875556	+	0.483117i	$0.839504\pi$
$242$		−	22.0000i		−	1.41421i
$243$	1.00000			0.0641500
$244$	16.0000			1.02430
$245$			3.00000i			0.191663i
$246$	20.0000			1.27515
$247$	3.00000	−	2.00000i	0.190885	−	0.127257i
$248$		0			0
$249$		−	9.00000i		−	0.570352i
$250$	−6.00000			−0.379473
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$		−	2.00000i		−	0.125988i
$253$		0			0
$254$		−	16.0000i		−	1.00393i
$255$			6.00000i			0.375735i
$256$	16.0000			1.00000
$257$	−12.0000			−0.748539			−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000			−0.748539			−0.374270	+	0.927320i	$0.622107\pi$
$258$		−	18.0000i		−	1.12063i
$259$	−8.00000			−0.497096
$260$	−18.0000	+	12.0000i	−1.11631	+	0.744208i
$261$	5.00000			0.309492
$262$			36.0000i			2.22409i
$263$	19.0000			1.17159			0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000			1.17159			0.585795	+	0.810459i	$0.300782\pi$
$264$		0			0
$265$		−	27.0000i		−	1.65860i
$266$		−	2.00000i		−	0.122628i
$267$			9.00000i			0.550791i
$268$			4.00000i			0.244339i
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$	−6.00000			−0.365148
$271$		−	20.0000i		−	1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$		−	20.0000i		−	1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$	8.00000			0.485071
$273$	−3.00000	+	2.00000i	−0.181568	+	0.121046i
$274$	−24.0000			−1.44989
$275$		0			0
$276$	2.00000			0.120386
$277$	3.00000			0.180253			0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000			0.180253			0.0901263	+	0.995930i	$0.471273\pi$
$278$			40.0000i			2.39904i
$279$		−	5.00000i		−	0.299342i
$280$		0			0
$281$		−	10.0000i		−	0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$		−	10.0000i		−	0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$	−14.0000			−0.833688
$283$	24.0000			1.42665			0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000			1.42665			0.713326	+	0.700832i	$0.247188\pi$
$284$		0			0
$285$	−3.00000			−0.177705
$286$		0			0
$287$	−10.0000			−0.590281
$288$			8.00000i			0.471405i
$289$	−13.0000			−0.764706
$290$	−30.0000			−1.76166
$291$			13.0000i			0.762073i
$292$			18.0000i			1.05337i
$293$			31.0000i			1.81104i	0.424304	+	0.905520i	$0.360519\pi$
$293$			31.0000i			1.81104i	−0.424304	+	0.905520i	$0.639481\pi$
$294$			2.00000i			0.116642i
$295$	12.0000			0.698667
$296$		0			0
$297$		0			0
$298$	8.00000			0.463428
$299$	−2.00000	−	3.00000i	−0.115663	−	0.173494i
$300$	8.00000			0.461880
$301$			9.00000i			0.518751i
$302$	40.0000			2.30174
$303$	−8.00000			−0.459588
$304$			4.00000i			0.229416i
$305$			24.0000i			1.37424i
$306$			4.00000i			0.228665i
$307$		−	17.0000i		−	0.970241i	−0.874447	−	0.485121i	$-0.838776\pi$
$307$		−	17.0000i		−	0.970241i	0.874447	−	0.485121i	$-0.161224\pi$
$308$		0			0
$309$	−16.0000			−0.910208
$310$			30.0000i			1.70389i
$311$	2.00000			0.113410			0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000			0.113410			0.0567048	+	0.998391i	$0.481941\pi$
$312$		0			0
$313$	4.00000			0.226093			0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000			0.226093			0.113047	+	0.993590i	$0.463939\pi$
$314$			4.00000i			0.225733i
$315$	3.00000			0.169031
$316$	−30.0000			−1.68763
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$		−	18.0000i		−	1.00939i
$319$		0			0
$320$		−	24.0000i		−	1.34164i
$321$	−12.0000			−0.669775
$322$	−2.00000			−0.111456
$323$			2.00000i			0.111283i
$324$	−2.00000			−0.111111
$325$	−8.00000	−	12.0000i	−0.443760	−	0.665640i
$326$	12.0000			0.664619
$327$			14.0000i			0.774202i
$328$		0			0
$329$	7.00000			0.385922
$330$		0			0
$331$		−	20.0000i		−	1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$		−	20.0000i		−	1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$			18.0000i			0.987878i
$333$			8.00000i			0.438397i
$334$	6.00000			0.328305
$335$	−6.00000			−0.327815
$336$		−	4.00000i		−	0.218218i
$337$	23.0000			1.25289			0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000			1.25289			0.626445	+	0.779466i	$0.284509\pi$
$338$	24.0000	+	10.0000i	1.30543	+	0.543928i
$339$	−21.0000			−1.14056
$340$		−	12.0000i		−	0.650791i
$341$		0			0
$342$	−2.00000			−0.108148
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			3.00000i			0.161515i
$346$		−	48.0000i		−	2.58050i
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$	−10.0000			−0.536056
$349$		−	1.00000i		−	0.0535288i	−0.999642	−	0.0267644i	$-0.991480\pi$
$349$		−	1.00000i		−	0.0535288i	0.999642	−	0.0267644i	$-0.00852039\pi$
$350$	−8.00000			−0.427618
$351$	2.00000	+	3.00000i	0.106752	+	0.160128i
$352$		0			0
$353$			6.00000i			0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$			6.00000i			0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	8.00000			0.425195
$355$		0			0
$356$		−	18.0000i		−	0.953998i
$357$		−	2.00000i		−	0.105851i
$358$			10.0000i			0.528516i
$359$			14.0000i			0.738892i	0.929252	+	0.369446i	$0.120452\pi$
$359$			14.0000i			0.738892i	−0.929252	+	0.369446i	$0.879548\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$			16.0000i			0.840941i
$363$	11.0000			0.577350
$364$	6.00000	−	4.00000i	0.314485	−	0.209657i
$365$	−27.0000			−1.41324
$366$			16.0000i			0.836333i
$367$	18.0000			0.939592			0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000			0.939592			0.469796	+	0.882775i	$0.344327\pi$
$368$	4.00000			0.208514
$369$			10.0000i			0.520579i
$370$		−	48.0000i		−	2.49540i
$371$			9.00000i			0.467257i
$372$			10.0000i			0.518476i
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$		−	3.00000i		−	0.154919i
$376$		0			0
$377$	10.0000	+	15.0000i	0.515026	+	0.772539i
$378$	2.00000			0.102869
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	6.00000			0.307794
$381$	8.00000			0.409852
$382$			16.0000i			0.818631i
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$		0			0
$385$		0			0
$386$	−28.0000			−1.42516
$387$	9.00000			0.457496
$388$		−	26.0000i		−	1.31995i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$	−12.0000	−	18.0000i	−0.607644	−	0.911465i
$391$	2.00000			0.101144
$392$		0			0
$393$	−18.0000			−0.907980
$394$	−24.0000			−1.20910
$395$		−	45.0000i		−	2.26420i
$396$		0			0
$397$		−	17.0000i		−	0.853206i	−0.904439	−	0.426603i	$-0.859710\pi$
$397$		−	17.0000i		−	0.853206i	0.904439	−	0.426603i	$-0.140290\pi$
$398$		−	20.0000i		−	1.00251i
$399$	1.00000			0.0500626
$400$	16.0000			0.800000
$401$			10.0000i			0.499376i	0.968326	+	0.249688i	$0.0803281\pi$
$401$			10.0000i			0.499376i	−0.968326	+	0.249688i	$0.919672\pi$
$402$	−4.00000			−0.199502
$403$	15.0000	−	10.0000i	0.747203	−	0.498135i
$404$	16.0000			0.796030
$405$		−	3.00000i		−	0.149071i
$406$	10.0000			0.496292
$407$		0			0
$408$		0			0
$409$		−	31.0000i		−	1.53285i	−0.642333	−	0.766426i	$-0.722033\pi$
$409$		−	31.0000i		−	1.53285i	0.642333	−	0.766426i	$-0.277967\pi$
$410$		−	60.0000i		−	2.96319i
$411$		−	12.0000i		−	0.591916i
$412$	32.0000			1.57653
$413$	−4.00000			−0.196827
$414$			2.00000i			0.0982946i
$415$	−27.0000			−1.32538
$416$	−24.0000	+	16.0000i	−1.17670	+	0.784465i
$417$	−20.0000			−0.979404
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$	−6.00000			−0.292770
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$			26.0000i			1.26566i
$423$		−	7.00000i		−	0.340352i
$424$		0			0
$425$	8.00000			0.388057
$426$		0			0
$427$		−	8.00000i		−	0.387147i
$428$	24.0000			1.16008
$429$		0			0
$430$	−54.0000			−2.60411
$431$		−	20.0000i		−	0.963366i	−0.876346	−	0.481683i	$-0.840026\pi$
$431$		−	20.0000i		−	0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$	−4.00000			−0.192450
$433$	−6.00000			−0.288342			−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000			−0.288342			−0.144171	+	0.989553i	$0.546051\pi$
$434$		−	10.0000i		−	0.480015i
$435$		−	15.0000i		−	0.719195i
$436$		−	28.0000i		−	1.34096i
$437$			1.00000i			0.0478365i
$438$	−18.0000			−0.860073
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$	−12.0000	+	8.00000i	−0.570782	+	0.380521i
$443$	−31.0000			−1.47285			−0.736427	−	0.676517i	$-0.763489\pi$
$443$	−31.0000			−1.47285			−0.736427	+	0.676517i	$0.763489\pi$
$444$		−	16.0000i		−	0.759326i
$445$	27.0000			1.27992
$446$	−38.0000			−1.79935
$447$			4.00000i			0.189194i
$448$			8.00000i			0.377964i
$449$			34.0000i			1.60456i	0.596948	+	0.802280i	$0.296380\pi$
$449$			34.0000i			1.60456i	−0.596948	+	0.802280i	$0.703620\pi$
$450$			8.00000i			0.377124i
$451$		0			0
$452$	42.0000			1.97551
$453$			20.0000i			0.939682i
$454$	−24.0000			−1.12638
$455$	6.00000	+	9.00000i	0.281284	+	0.421927i
$456$		0			0
$457$		−	42.0000i		−	1.96468i	−0.187112	−	0.982339i	$-0.559913\pi$
$457$		−	42.0000i		−	1.96468i	0.187112	−	0.982339i	$-0.440087\pi$
$458$	28.0000			1.30835
$459$	−2.00000			−0.0933520
$460$		−	6.00000i		−	0.279751i
$461$			10.0000i			0.465746i	0.972507	+	0.232873i	$0.0748127\pi$
$461$			10.0000i			0.465746i	−0.972507	+	0.232873i	$0.925187\pi$
$462$		0			0
$463$			26.0000i			1.20832i	0.796862	+	0.604161i	$0.206492\pi$
$463$			26.0000i			1.20832i	−0.796862	+	0.604161i	$0.793508\pi$
$464$	−20.0000			−0.928477
$465$	−15.0000			−0.695608
$466$			2.00000i			0.0926482i
$467$	18.0000			0.832941			0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000			0.832941			0.416470	+	0.909149i	$0.363267\pi$
$468$	−4.00000	−	6.00000i	−0.184900	−	0.277350i
$469$	2.00000			0.0923514
$470$			42.0000i			1.93732i
$471$	−2.00000			−0.0921551
$472$		0			0
$473$		0			0
$474$		−	30.0000i		−	1.37795i
$475$			4.00000i			0.183533i
$476$			4.00000i			0.183340i
$477$	9.00000			0.412082
$478$	−12.0000			−0.548867
$479$		−	21.0000i		−	0.959514i	−0.877401	−	0.479757i	$-0.840725\pi$
$479$		−	21.0000i		−	0.959514i	0.877401	−	0.479757i	$-0.159275\pi$
$480$	24.0000			1.09545
$481$	−24.0000	+	16.0000i	−1.09431	+	0.729537i
$482$	30.0000			1.36646
$483$		−	1.00000i		−	0.0455016i
$484$	−22.0000			−1.00000
$485$	39.0000			1.77090
$486$		−	2.00000i		−	0.0907218i
$487$			8.00000i			0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$			8.00000i			0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$		0			0
$489$			6.00000i			0.271329i
$490$	6.00000			0.271052
$491$	32.0000			1.44414			0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000			1.44414			0.722070	+	0.691820i	$0.243191\pi$
$492$		−	20.0000i		−	0.901670i
$493$	−10.0000			−0.450377
$494$	−4.00000	−	6.00000i	−0.179969	−	0.269953i
$495$		0			0
$496$			20.0000i			0.898027i
$497$		0			0
$498$	−18.0000			−0.806599
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$			6.00000i			0.268328i
$501$			3.00000i			0.134030i
$502$			56.0000i			2.49940i
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$			24.0000i			1.06799i
$506$		0			0
$507$	−5.00000	+	12.0000i	−0.222058	+	0.532939i
$508$	−16.0000			−0.709885
$509$			9.00000i			0.398918i	0.979906	+	0.199459i	$0.0639185\pi$
$509$			9.00000i			0.398918i	−0.979906	+	0.199459i	$0.936082\pi$
$510$	12.0000			0.531369
$511$	9.00000			0.398137
$512$		−	32.0000i		−	1.41421i
$513$		−	1.00000i		−	0.0441511i
$514$			24.0000i			1.05859i
$515$			48.0000i			2.11513i
$516$	−18.0000			−0.792406
$517$		0			0
$518$			16.0000i			0.703000i
$519$	24.0000			1.05348
$520$		0			0
$521$	12.0000			0.525730			0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000			0.525730			0.262865	+	0.964833i	$0.415333\pi$
$522$		−	10.0000i		−	0.437688i
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$	36.0000			1.57267
$525$		−	4.00000i		−	0.174574i
$526$		−	38.0000i		−	1.65688i
$527$			10.0000i			0.435607i
$528$		0			0
$529$	−22.0000			−0.956522
$530$	−54.0000			−2.34561
$531$			4.00000i			0.173585i
$532$	−2.00000			−0.0867110
$533$	−30.0000	+	20.0000i	−1.29944	+	0.866296i
$534$	18.0000			0.778936
$535$			36.0000i			1.55642i
$536$		0			0
$537$	−5.00000			−0.215766
$538$		−	20.0000i		−	0.862261i
$539$		0			0
$540$			6.00000i			0.258199i
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	−40.0000			−1.71815
$543$	−8.00000			−0.343313
$544$		−	16.0000i		−	0.685994i
$545$	42.0000			1.79908
$546$	4.00000	+	6.00000i	0.171184	+	0.256776i
$547$	13.0000			0.555840			0.277920	−	0.960604i	$-0.410355\pi$
$547$	13.0000			0.555840			0.277920	+	0.960604i	$0.410355\pi$
$548$			24.0000i			1.02523i
$549$	−8.00000			−0.341432
$550$		0			0
$551$		−	5.00000i		−	0.213007i
$552$		0			0
$553$			15.0000i			0.637865i
$554$		−	6.00000i		−	0.254916i
$555$	24.0000			1.01874
$556$	40.0000			1.69638
$557$		−	32.0000i		−	1.35588i	−0.735116	−	0.677942i	$-0.762872\pi$
$557$		−	32.0000i		−	1.35588i	0.735116	−	0.677942i	$-0.237128\pi$
$558$	−10.0000			−0.423334
$559$	18.0000	+	27.0000i	0.761319	+	1.14198i
$560$	−12.0000			−0.507093
$561$		0			0
$562$	−20.0000			−0.843649
$563$	−26.0000			−1.09577			−0.547885	−	0.836554i	$-0.684567\pi$
$563$	−26.0000			−1.09577			−0.547885	+	0.836554i	$0.684567\pi$
$564$			14.0000i			0.589506i
$565$			63.0000i			2.65043i
$566$		−	48.0000i		−	2.01759i
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	−5.00000			−0.209611			−0.104805	−	0.994493i	$-0.533422\pi$
$569$	−5.00000			−0.209611			−0.104805	+	0.994493i	$0.533422\pi$
$570$			6.00000i			0.251312i
$571$	−23.0000			−0.962520			−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000			−0.962520			−0.481260	+	0.876578i	$0.659821\pi$
$572$		0			0
$573$	−8.00000			−0.334205
$574$			20.0000i			0.834784i
$575$	4.00000			0.166812
$576$	8.00000			0.333333
$577$		−	22.0000i		−	0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$		−	22.0000i		−	0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$			26.0000i			1.08146i
$579$		−	14.0000i		−	0.581820i
$580$			30.0000i			1.24568i
$581$	9.00000			0.373383
$582$	26.0000			1.07773
$583$		0			0
$584$		0			0
$585$	9.00000	−	6.00000i	0.372104	−	0.248069i
$586$	62.0000			2.56120
$587$			3.00000i			0.123823i	0.998082	+	0.0619116i	$0.0197197\pi$
$587$			3.00000i			0.123823i	−0.998082	+	0.0619116i	$0.980280\pi$
$588$	2.00000			0.0824786
$589$	−5.00000			−0.206021
$590$		−	24.0000i		−	0.988064i
$591$		−	12.0000i		−	0.493614i
$592$		−	32.0000i		−	1.31519i
$593$		−	9.00000i		−	0.369586i	−0.982777	−	0.184793i	$-0.940839\pi$
$593$		−	9.00000i		−	0.369586i	0.982777	−	0.184793i	$-0.0591614\pi$
$594$		0			0
$595$	−6.00000			−0.245976
$596$		−	8.00000i		−	0.327693i
$597$	10.0000			0.409273
$598$	−6.00000	+	4.00000i	−0.245358	+	0.163572i
$599$	−35.0000			−1.43006			−0.715031	−	0.699093i	$-0.753587\pi$
$599$	−35.0000			−1.43006			−0.715031	+	0.699093i	$0.753587\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$	18.0000			0.733625
$603$		−	2.00000i		−	0.0814463i
$604$		−	40.0000i		−	1.62758i
$605$		−	33.0000i		−	1.34164i
$606$			16.0000i			0.649956i
$607$	28.0000			1.13648			0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000			1.13648			0.568242	+	0.822861i	$0.307624\pi$
$608$	8.00000			0.324443
$609$			5.00000i			0.202610i
$610$	48.0000			1.94346
$611$	21.0000	−	14.0000i	0.849569	−	0.566379i
$612$	4.00000			0.161690
$613$			26.0000i			1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$			26.0000i			1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−34.0000			−1.37213
$615$	30.0000			1.20972
$616$		0			0
$617$			48.0000i			1.93241i	0.257780	+	0.966204i	$0.417009\pi$
$617$			48.0000i			1.93241i	−0.257780	+	0.966204i	$0.582991\pi$
$618$			32.0000i			1.28723i
$619$			44.0000i			1.76851i	0.467005	+	0.884255i	$0.345333\pi$
$619$			44.0000i			1.76851i	−0.467005	+	0.884255i	$0.654667\pi$
$620$	30.0000			1.20483
$621$	−1.00000			−0.0401286
$622$		−	4.00000i		−	0.160385i
$623$	−9.00000			−0.360577
$624$	−8.00000	−	12.0000i	−0.320256	−	0.480384i
$625$	−29.0000			−1.16000
$626$		−	8.00000i		−	0.319744i
$627$		0			0
$628$	4.00000			0.159617
$629$		−	16.0000i		−	0.637962i
$630$		−	6.00000i		−	0.239046i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−13.0000			−0.516704
$634$	−4.00000			−0.158860
$635$		−	24.0000i		−	0.952411i
$636$	−18.0000			−0.713746
$637$	−2.00000	−	3.00000i	−0.0792429	−	0.118864i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.0000			0.671460			0.335730	−	0.941958i	$-0.391017\pi$
$641$	17.0000			0.671460			0.335730	+	0.941958i	$0.391017\pi$
$642$			24.0000i			0.947204i
$643$		−	44.0000i		−	1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$		−	44.0000i		−	1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$			2.00000i			0.0788110i
$645$		−	27.0000i		−	1.06312i
$646$	4.00000			0.157378
$647$	−2.00000			−0.0786281			−0.0393141	−	0.999227i	$-0.512517\pi$
$647$	−2.00000			−0.0786281			−0.0393141	+	0.999227i	$0.512517\pi$
$648$		0			0
$649$		0			0
$650$	−24.0000	+	16.0000i	−0.941357	+	0.627572i
$651$	5.00000			0.195965
$652$		−	12.0000i		−	0.469956i
$653$	14.0000			0.547862			0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000			0.547862			0.273931	+	0.961749i	$0.411676\pi$
$654$	28.0000			1.09489
$655$			54.0000i			2.10995i
$656$		−	40.0000i		−	1.56174i
$657$		−	9.00000i		−	0.351123i
$658$		−	14.0000i		−	0.545777i
$659$	15.0000			0.584317			0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000			0.584317			0.292159	+	0.956370i	$0.405627\pi$
$660$		0			0
$661$			15.0000i			0.583432i	0.956505	+	0.291716i	$0.0942263\pi$
$661$			15.0000i			0.583432i	−0.956505	+	0.291716i	$0.905774\pi$
$662$	−40.0000			−1.55464
$663$	−4.00000	−	6.00000i	−0.155347	−	0.233021i
$664$		0			0
$665$		−	3.00000i		−	0.116335i
$666$	16.0000			0.619987
$667$	−5.00000			−0.193601
$668$		−	6.00000i		−	0.232147i
$669$		−	19.0000i		−	0.734582i
$670$			12.0000i			0.463600i
$671$		0			0
$672$	−8.00000			−0.308607
$673$	19.0000			0.732396			0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000			0.732396			0.366198	+	0.930537i	$0.380659\pi$
$674$		−	46.0000i		−	1.77185i
$675$	−4.00000			−0.153960
$676$	10.0000	−	24.0000i	0.384615	−	0.923077i
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$			42.0000i			1.61300i
$679$	−13.0000			−0.498894
$680$		0			0
$681$		−	12.0000i		−	0.459841i
$682$		0			0
$683$		−	24.0000i		−	0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$		−	24.0000i		−	0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$			2.00000i			0.0764719i
$685$	−36.0000			−1.37549
$686$	−2.00000			−0.0763604
$687$			14.0000i			0.534133i
$688$	−36.0000			−1.37249
$689$	18.0000	+	27.0000i	0.685745	+	1.02862i
$690$	6.00000			0.228416
$691$		−	35.0000i		−	1.33146i	−0.746191	−	0.665731i	$-0.768120\pi$
$691$		−	35.0000i		−	1.33146i	0.746191	−	0.665731i	$-0.231880\pi$
$692$	−48.0000			−1.82469
$693$		0			0
$694$			24.0000i			0.911028i
$695$			60.0000i			2.27593i
$696$		0			0
$697$		−	20.0000i		−	0.757554i
$698$	−2.00000			−0.0757011
$699$	−1.00000			−0.0378235
$700$			8.00000i			0.302372i
$701$	−23.0000			−0.868698			−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000			−0.868698			−0.434349	+	0.900745i	$0.643022\pi$
$702$	6.00000	−	4.00000i	0.226455	−	0.150970i
$703$	8.00000			0.301726
$704$		0			0
$705$	−21.0000			−0.790906
$706$	12.0000			0.451626
$707$		−	8.00000i		−	0.300871i
$708$		−	8.00000i		−	0.300658i
$709$		−	16.0000i		−	0.600893i	−0.953799	−	0.300446i	$-0.902864\pi$
$709$		−	16.0000i		−	0.600893i	0.953799	−	0.300446i	$-0.0971356\pi$
$710$		0			0
$711$	15.0000			0.562544
$712$		0			0
$713$			5.00000i			0.187251i
$714$	−4.00000			−0.149696
$715$		0			0
$716$	10.0000			0.373718
$717$		−	6.00000i		−	0.224074i
$718$	28.0000			1.04495
$719$	50.0000			1.86469			0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000			1.86469			0.932343	+	0.361576i	$0.117761\pi$
$720$			12.0000i			0.447214i
$721$		−	16.0000i		−	0.595871i
$722$		−	36.0000i		−	1.33978i
$723$			15.0000i			0.557856i
$724$	16.0000			0.594635
$725$	−20.0000			−0.742781
$726$		−	22.0000i		−	0.816497i
$727$	−2.00000			−0.0741759			−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000			−0.0741759			−0.0370879	+	0.999312i	$0.511808\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$			54.0000i			1.99863i
$731$	−18.0000			−0.665754
$732$	16.0000			0.591377
$733$		−	19.0000i		−	0.701781i	−0.936416	−	0.350891i	$-0.885879\pi$
$733$		−	19.0000i		−	0.701781i	0.936416	−	0.350891i	$-0.114121\pi$
$734$		−	36.0000i		−	1.32878i
$735$			3.00000i			0.110657i
$736$		−	8.00000i		−	0.294884i
$737$		0			0
$738$	20.0000			0.736210
$739$			4.00000i			0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$			4.00000i			0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$	−48.0000			−1.76452
$741$	3.00000	−	2.00000i	0.110208	−	0.0734718i
$742$	18.0000			0.660801
$743$			26.0000i			0.953847i	0.878945	+	0.476924i	$0.158248\pi$
$743$			26.0000i			0.953847i	−0.878945	+	0.476924i	$0.841752\pi$
$744$		0			0
$745$	12.0000			0.439646
$746$			52.0000i			1.90386i
$747$		−	9.00000i		−	0.329293i
$748$		0			0
$749$		−	12.0000i		−	0.438470i
$750$	−6.00000			−0.219089
$751$	−23.0000			−0.839282			−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000			−0.839282			−0.419641	+	0.907690i	$0.637844\pi$
$752$			28.0000i			1.02105i
$753$	−28.0000			−1.02038
$754$	30.0000	−	20.0000i	1.09254	−	0.728357i
$755$	60.0000			2.18362
$756$		−	2.00000i		−	0.0727393i
$757$	13.0000			0.472493			0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000			0.472493			0.236247	+	0.971693i	$0.424083\pi$
$758$	8.00000			0.290573
$759$		0			0
$760$		0			0
$761$		−	45.0000i		−	1.63125i	−0.578582	−	0.815624i	$-0.696394\pi$
$761$		−	45.0000i		−	1.63125i	0.578582	−	0.815624i	$-0.303606\pi$
$762$		−	16.0000i		−	0.579619i
$763$	−14.0000			−0.506834
$764$	16.0000			0.578860
$765$			6.00000i			0.216930i
$766$	32.0000			1.15621
$767$	−12.0000	+	8.00000i	−0.433295	+	0.288863i
$768$	16.0000			0.577350
$769$		−	1.00000i		−	0.0360609i	−0.999837	−	0.0180305i	$-0.994260\pi$
$769$		−	1.00000i		−	0.0360609i	0.999837	−	0.0180305i	$-0.00573959\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$			28.0000i			1.00774i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$		−	18.0000i		−	0.646997i
$775$			20.0000i			0.718421i
$776$		0			0
$777$	−8.00000			−0.286998
$778$		−	60.0000i		−	2.15110i
$779$	10.0000			0.358287
$780$	−18.0000	+	12.0000i	−0.644503	+	0.429669i
$781$		0			0
$782$		−	4.00000i		−	0.143040i
$783$	5.00000			0.178685
$784$	4.00000			0.142857
$785$			6.00000i			0.214149i
$786$			36.0000i			1.28408i
$787$		−	7.00000i		−	0.249523i	−0.992187	−	0.124762i	$-0.960183\pi$
$787$		−	7.00000i		−	0.249523i	0.992187	−	0.124762i	$-0.0398166\pi$
$788$			24.0000i			0.854965i

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)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +1.00000 q^{3} -2.00000 q^{4} -3.00000i q^{5} -2.00000i q^{6} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} +1.00000 q^{3} -2.00000 q^{4} -3.00000i q^{5} -2.00000i q^{6} +1.00000i q^{7} +1.00000 q^{9} -6.00000 q^{10} -2.00000 q^{12} +(2.00000 + 3.00000i) q^{13} +2.00000 q^{14} -3.00000i q^{15} -4.00000 q^{16} -2.00000 q^{17} -2.00000i q^{18} -1.00000i q^{19} +6.00000i q^{20} +1.00000i q^{21} -1.00000 q^{23} -4.00000 q^{25} +(6.00000 - 4.00000i) q^{26} +1.00000 q^{27} -2.00000i q^{28} +5.00000 q^{29} -6.00000 q^{30} -5.00000i q^{31} +8.00000i q^{32} +4.00000i q^{34} +3.00000 q^{35} -2.00000 q^{36} +8.00000i q^{37} -2.00000 q^{38} +(2.00000 + 3.00000i) q^{39} +10.0000i q^{41} +2.00000 q^{42} +9.00000 q^{43} -3.00000i q^{45} +2.00000i q^{46} -7.00000i q^{47} -4.00000 q^{48} -1.00000 q^{49} +8.00000i q^{50} -2.00000 q^{51} +(-4.00000 - 6.00000i) q^{52} +9.00000 q^{53} -2.00000i q^{54} -1.00000i q^{57} -10.0000i q^{58} +4.00000i q^{59} +6.00000i q^{60} -8.00000 q^{61} -10.0000 q^{62} +1.00000i q^{63} +8.00000 q^{64} +(9.00000 - 6.00000i) q^{65} -2.00000i q^{67} +4.00000 q^{68} -1.00000 q^{69} -6.00000i q^{70} -9.00000i q^{73} +16.0000 q^{74} -4.00000 q^{75} +2.00000i q^{76} +(6.00000 - 4.00000i) q^{78} +15.0000 q^{79} +12.0000i q^{80} +1.00000 q^{81} +20.0000 q^{82} -9.00000i q^{83} -2.00000i q^{84} +6.00000i q^{85} -18.0000i q^{86} +5.00000 q^{87} +9.00000i q^{89} -6.00000 q^{90} +(-3.00000 + 2.00000i) q^{91} +2.00000 q^{92} -5.00000i q^{93} -14.0000 q^{94} -3.00000 q^{95} +8.00000i q^{96} +13.0000i q^{97} +2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{4} + 2 q^{9} - 12 q^{10} - 4 q^{12} + 4 q^{13} + 4 q^{14} - 8 q^{16} - 4 q^{17} - 2 q^{23} - 8 q^{25} + 12 q^{26} + 2 q^{27} + 10 q^{29} - 12 q^{30} + 6 q^{35} - 4 q^{36} - 4 q^{38} + 4 q^{39} + 4 q^{42} + 18 q^{43} - 8 q^{48} - 2 q^{49} - 4 q^{51} - 8 q^{52} + 18 q^{53} - 16 q^{61} - 20 q^{62} + 16 q^{64} + 18 q^{65} + 8 q^{68} - 2 q^{69} + 32 q^{74} - 8 q^{75} + 12 q^{78} + 30 q^{79} + 2 q^{81} + 40 q^{82} + 10 q^{87} - 12 q^{90} - 6 q^{91} + 4 q^{92} - 28 q^{94} - 6 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9 - 12 * q^10 - 4 * q^12 + 4 * q^13 + 4 * q^14 - 8 * q^16 - 4 * q^17 - 2 * q^23 - 8 * q^25 + 12 * q^26 + 2 * q^27 + 10 * q^29 - 12 * q^30 + 6 * q^35 - 4 * q^36 - 4 * q^38 + 4 * q^39 + 4 * q^42 + 18 * q^43 - 8 * q^48 - 2 * q^49 - 4 * q^51 - 8 * q^52 + 18 * q^53 - 16 * q^61 - 20 * q^62 + 16 * q^64 + 18 * q^65 + 8 * q^68 - 2 * q^69 + 32 * q^74 - 8 * q^75 + 12 * q^78 + 30 * q^79 + 2 * q^81 + 40 * q^82 + 10 * q^87 - 12 * q^90 - 6 * q^91 + 4 * q^92 - 28 * q^94 - 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more