LMFDB - Embedded newform 273.2.c.c.64.3

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
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			64.3
Root			$-1.29051i$ of defining polynomial
Character	$\chi$	$=$	273.64
Dual form			273.2.c.c.64.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29051i q^{2} -1.00000 q^{3} +0.334573 q^{4} +1.33457i q^{5} +1.29051i q^{6} -1.00000i q^{7} -3.01280i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.29051i q^{2} -1.00000 q^{3} +0.334573 q^{4} +1.33457i q^{5} +1.29051i q^{6} -1.00000i q^{7} -3.01280i q^{8} +1.00000 q^{9} +1.72229 q^{10} -4.88434i q^{11} -0.334573 q^{12} +(3.59383 + 0.290514i) q^{13} -1.29051 q^{14} -1.33457i q^{15} -3.21892 q^{16} +4.58103 q^{17} -1.29051i q^{18} -2.96874i q^{19} +0.446512i q^{20} +1.00000i q^{21} -6.30331 q^{22} -6.13080 q^{23} +3.01280i q^{24} +3.21892 q^{25} +(0.374913 - 4.63789i) q^{26} -1.00000 q^{27} -0.334573i q^{28} -7.63789 q^{29} -1.72229 q^{30} +5.63789i q^{31} -1.87154i q^{32} +4.88434i q^{33} -5.91188i q^{34} +1.33457 q^{35} +0.334573 q^{36} +9.76869i q^{37} -3.83120 q^{38} +(-3.59383 - 0.290514i) q^{39} +4.02080 q^{40} -3.69669i q^{41} +1.29051 q^{42} +9.63789 q^{43} -1.63417i q^{44} +1.33457i q^{45} +7.91188i q^{46} +5.27206i q^{47} +3.21892 q^{48} -1.00000 q^{49} -4.15406i q^{50} -4.58103 q^{51} +(1.20240 + 0.0971982i) q^{52} +10.7374 q^{53} +1.29051i q^{54} +6.51851 q^{55} -3.01280 q^{56} +2.96874i q^{57} +9.85680i q^{58} +10.3033i q^{59} -0.446512i q^{60} -11.1877 q^{61} +7.27577 q^{62} -1.00000i q^{63} -8.85308 q^{64} +(-0.387712 + 4.79623i) q^{65} +6.30331 q^{66} -5.50709i q^{67} +1.53269 q^{68} +6.13080 q^{69} -1.72229i q^{70} +4.88434i q^{71} -3.01280i q^{72} -4.45023i q^{73} +12.6066 q^{74} -3.21892 q^{75} -0.993260i q^{76} -4.88434 q^{77} +(-0.374913 + 4.63789i) q^{78} -3.54977 q^{79} -4.29588i q^{80} +1.00000 q^{81} -4.77063 q^{82} -5.27206i q^{83} +0.334573i q^{84} +6.11372i q^{85} -12.4378i q^{86} +7.63789 q^{87} -14.7155 q^{88} +9.94120i q^{89} +1.72229 q^{90} +(0.290514 - 3.59383i) q^{91} -2.05120 q^{92} -5.63789i q^{93} +6.80366 q^{94} +3.96200 q^{95} +1.87154i q^{96} +18.2189i q^{97} +1.29051i q^{98} -4.88434i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9	$ 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9} + 4 q^{10} + 14 q^{12} - 6 q^{13} - 2 q^{14} + 34 q^{16} + 20 q^{17} - 24 q^{22} - 6 q^{23} - 34 q^{25} + 28 q^{26} - 8 q^{27} - 18 q^{29} - 4 q^{30} - 6 q^{35} - 14 q^{36} + 36 q^{38} + 6 q^{39} - 8 q^{40} + 2 q^{42} + 34 q^{43} - 34 q^{48} - 8 q^{49} - 20 q^{51} + 18 q^{52} - 10 q^{53} + 16 q^{55} - 6 q^{56} - 20 q^{61} - 28 q^{62} - 18 q^{64} - 10 q^{65} + 24 q^{66} - 24 q^{68} + 6 q^{69} + 48 q^{74} + 34 q^{75} + 4 q^{77} - 28 q^{78} - 2 q^{79} + 8 q^{81} - 48 q^{82} + 18 q^{87} - 8 q^{88} + 4 q^{90} - 6 q^{91} + 56 q^{92} - 72 q^{94} - 78 q^{95}+O(q^{100}) $ 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9 + 4 * q^10 + 14 * q^12 - 6 * q^13 - 2 * q^14 + 34 * q^16 + 20 * q^17 - 24 * q^22 - 6 * q^23 - 34 * q^25 + 28 * q^26 - 8 * q^27 - 18 * q^29 - 4 * q^30 - 6 * q^35 - 14 * q^36 + 36 * q^38 + 6 * q^39 - 8 * q^40 + 2 * q^42 + 34 * q^43 - 34 * q^48 - 8 * q^49 - 20 * q^51 + 18 * q^52 - 10 * q^53 + 16 * q^55 - 6 * q^56 - 20 * q^61 - 28 * q^62 - 18 * q^64 - 10 * q^65 + 24 * q^66 - 24 * q^68 + 6 * q^69 + 48 * q^74 + 34 * q^75 + 4 * q^77 - 28 * q^78 - 2 * q^79 + 8 * q^81 - 48 * q^82 + 18 * q^87 - 8 * q^88 + 4 * q^90 - 6 * q^91 + 56 * q^92 - 72 * q^94 - 78 * 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 $

$2$		−	1.29051i		−	0.912531i	−0.889844	−	0.456266i	$-0.849187\pi$
$2$		−	1.29051i		−	0.912531i	0.889844	−	0.456266i	$-0.150813\pi$
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	0.334573			0.167286
$5$			1.33457i			0.596839i	0.954435	+	0.298420i	$0.0964595\pi$
$5$			1.33457i			0.596839i	−0.954435	+	0.298420i	$0.903541\pi$
$6$			1.29051i			0.526850i
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		−	3.01280i		−	1.06519i
$9$	1.00000			0.333333
$10$	1.72229			0.544634
$11$		−	4.88434i		−	1.47268i	−0.676609	−	0.736342i	$-0.736551\pi$
$11$		−	4.88434i		−	1.47268i	0.676609	−	0.736342i	$-0.263449\pi$
$12$	−0.334573			−0.0965829
$13$	3.59383	+	0.290514i	0.996749	+	0.0805742i
$13$	3.59383	+	0.290514i	0.996749	+	0.0805742i
$14$	−1.29051			−0.344904
$15$		−	1.33457i		−	0.344585i
$16$	−3.21892			−0.804729
$17$	4.58103			1.11106			0.555531	−	0.831496i	$-0.312515\pi$
$17$	4.58103			1.11106			0.555531	+	0.831496i	$0.312515\pi$
$18$		−	1.29051i		−	0.304177i
$19$		−	2.96874i		−	0.681076i	−0.940231	−	0.340538i	$-0.889391\pi$
$19$		−	2.96874i		−	0.681076i	0.940231	−	0.340538i	$-0.110609\pi$
$20$			0.446512i			0.0998431i
$21$			1.00000i			0.218218i
$22$	−6.30331			−1.34387
$23$	−6.13080			−1.27836			−0.639180	−	0.769057i	$-0.720726\pi$
$23$	−6.13080			−1.27836			−0.639180	+	0.769057i	$0.720726\pi$
$24$			3.01280i			0.614985i
$25$	3.21892			0.643783
$26$	0.374913	−	4.63789i	0.0735265	−	0.909564i
$27$	−1.00000			−0.192450
$28$		−	0.334573i		−	0.0632283i
$29$	−7.63789			−1.41832			−0.709160	−	0.705048i	$-0.750926\pi$
$29$	−7.63789			−1.41832			−0.709160	+	0.705048i	$0.750926\pi$
$30$	−1.72229			−0.314445
$31$			5.63789i			1.01259i	0.862359	+	0.506297i	$0.168986\pi$
$31$			5.63789i			1.01259i	−0.862359	+	0.506297i	$0.831014\pi$
$32$		−	1.87154i		−	0.330845i
$33$			4.88434i			0.850255i
$34$		−	5.91188i		−	1.01388i
$35$	1.33457			0.225584
$36$	0.334573			0.0557621
$37$			9.76869i			1.60596i	0.596005	+	0.802981i	$0.296754\pi$
$37$			9.76869i			1.60596i	−0.596005	+	0.802981i	$0.703246\pi$
$38$	−3.83120			−0.621503
$39$	−3.59383	−	0.290514i	−0.575473	−	0.0465195i
$40$	4.02080			0.635744
$41$		−	3.69669i		−	0.577325i	−0.957431	−	0.288663i	$-0.906789\pi$
$41$		−	3.69669i		−	0.577325i	0.957431	−	0.288663i	$-0.0932106\pi$
$42$	1.29051			0.199131
$43$	9.63789			1.46976			0.734882	−	0.678195i	$-0.237238\pi$
$43$	9.63789			1.46976			0.734882	+	0.678195i	$0.237238\pi$
$44$		−	1.63417i		−	0.246360i
$45$			1.33457i			0.198946i
$46$			7.91188i			1.16654i
$47$			5.27206i			0.769008i	0.923123	+	0.384504i	$0.125628\pi$
$47$			5.27206i			0.769008i	−0.923123	+	0.384504i	$0.874372\pi$
$48$	3.21892			0.464610
$49$	−1.00000			−0.142857
$50$		−	4.15406i		−	0.587472i
$51$	−4.58103			−0.641472
$52$	1.20240	+	0.0971982i	0.166743	+	0.0134790i
$53$	10.7374			1.47490			0.737449	−	0.675402i	$-0.236030\pi$
$53$	10.7374			1.47490			0.737449	+	0.675402i	$0.236030\pi$
$54$			1.29051i			0.175617i
$55$	6.51851			0.878956
$56$	−3.01280			−0.402602
$57$			2.96874i			0.393219i
$58$			9.85680i			1.29426i
$59$			10.3033i			1.34138i	0.741739	+	0.670689i	$0.234001\pi$
$59$			10.3033i			1.34138i	−0.741739	+	0.670689i	$0.765999\pi$
$60$		−	0.446512i		−	0.0576444i
$61$	−11.1877			−1.43243			−0.716216	−	0.697878i	$-0.754128\pi$
$61$	−11.1877			−1.43243			−0.716216	+	0.697878i	$0.754128\pi$
$62$	7.27577			0.924024
$63$		−	1.00000i		−	0.125988i
$64$	−8.85308			−1.10664
$65$	−0.387712	+	4.79623i	−0.0480898	+	0.594899i
$66$	6.30331			0.775884
$67$		−	5.50709i		−	0.672798i	−0.941720	−	0.336399i	$-0.890791\pi$
$67$		−	5.50709i		−	0.672798i	0.941720	−	0.336399i	$-0.109209\pi$
$68$	1.53269			0.185866
$69$	6.13080			0.738061
$70$		−	1.72229i		−	0.205852i
$71$			4.88434i			0.579665i	0.957077	+	0.289832i	$0.0935996\pi$
$71$			4.88434i			0.579665i	−0.957077	+	0.289832i	$0.906400\pi$
$72$		−	3.01280i		−	0.355062i
$73$		−	4.45023i		−	0.520860i	−0.965493	−	0.260430i	$-0.916136\pi$
$73$		−	4.45023i		−	0.520860i	0.965493	−	0.260430i	$-0.0838643\pi$
$74$	12.6066			1.46549
$75$	−3.21892			−0.371688
$76$		−	0.993260i		−	0.113935i
$77$	−4.88434			−0.556622
$78$	−0.374913	+	4.63789i	−0.0424505	+	0.525137i
$79$	−3.54977			−0.399380			−0.199690	−	0.979859i	$-0.563994\pi$
$79$	−3.54977			−0.399380			−0.199690	+	0.979859i	$0.563994\pi$
$80$		−	4.29588i		−	0.480294i
$81$	1.00000			0.111111
$82$	−4.77063			−0.526828
$83$		−	5.27206i		−	0.578683i	−0.957226	−	0.289342i	$-0.906564\pi$
$83$		−	5.27206i		−	0.578683i	0.957226	−	0.289342i	$-0.0934364\pi$
$84$			0.334573i			0.0365049i
$85$			6.11372i			0.663126i
$86$		−	12.4378i		−	1.34121i
$87$	7.63789			0.818867
$88$	−14.7155			−1.56868
$89$			9.94120i			1.05377i	0.849938	+	0.526883i	$0.176639\pi$
$89$			9.94120i			1.05377i	−0.849938	+	0.526883i	$0.823361\pi$
$90$	1.72229			0.181545
$91$	0.290514	−	3.59383i	0.0304542	−	0.376736i
$92$	−2.05120			−0.213852
$93$		−	5.63789i		−	0.584622i
$94$	6.80366			0.701744
$95$	3.96200			0.406493
$96$			1.87154i			0.191014i
$97$			18.2189i			1.84985i	0.380149	+	0.924925i	$0.375873\pi$
$97$			18.2189i			1.84985i	−0.380149	+	0.924925i	$0.624127\pi$
$98$			1.29051i			0.130362i
$99$		−	4.88434i		−	0.490895i
$100$	1.07696			0.107696
$101$	−10.6947			−1.06417			−0.532083	−	0.846692i	$-0.678591\pi$
$101$	−10.6947			−1.06417			−0.532083	+	0.846692i	$0.678591\pi$
$102$			5.91188i			0.585364i
$103$	1.48149			0.145975			0.0729877	−	0.997333i	$-0.476747\pi$
$103$	1.48149			0.145975			0.0729877	+	0.997333i	$0.476747\pi$
$104$	0.875261	−	10.8275i	0.0858264	−	1.06172i
$105$	−1.33457			−0.130241
$106$		−	13.8568i		−	1.34589i
$107$	4.49291			0.434346			0.217173	−	0.976133i	$-0.430316\pi$
$107$	4.49291			0.434346			0.217173	+	0.976133i	$0.430316\pi$
$108$	−0.334573			−0.0321943
$109$		−	7.83120i		−	0.750093i	−0.927006	−	0.375047i	$-0.877627\pi$
$109$		−	7.83120i		−	0.750093i	0.927006	−	0.375047i	$-0.122373\pi$
$110$		−	8.41223i		−	0.802075i
$111$		−	9.76869i		−	0.927203i
$112$			3.21892i			0.304159i
$113$	7.63789			0.718512			0.359256	−	0.933239i	$-0.383031\pi$
$113$	7.63789			0.718512			0.359256	+	0.933239i	$0.383031\pi$
$114$	3.83120			0.358825
$115$		−	8.18200i		−	0.762975i
$116$	−2.55543			−0.237266
$117$	3.59383	+	0.290514i	0.332250	+	0.0268581i
$118$	13.2966			1.22405
$119$		−	4.58103i		−	0.419942i
$120$	−4.02080			−0.367047
$121$	−12.8568			−1.16880
$122$			14.4378i			1.30714i
$123$			3.69669i			0.333319i
$124$			1.88628i			0.169393i
$125$			10.9687i			0.981074i
$126$	−1.29051			−0.114968
$127$	−16.4122			−1.45635			−0.728175	−	0.685391i	$-0.759631\pi$
$127$	−16.4122			−1.45635			−0.728175	+	0.685391i	$0.759631\pi$
$128$			7.68195i			0.678994i
$129$	−9.63789			−0.848569
$130$	6.18960	+	0.500349i	0.542864	+	0.0438835i
$131$	−8.43783			−0.737217			−0.368608	−	0.929585i	$-0.620166\pi$
$131$	−8.43783			−0.737217			−0.368608	+	0.929585i	$0.620166\pi$
$132$			1.63417i			0.142236i
$133$	−2.96874			−0.257423
$134$	−7.10698			−0.613949
$135$		−	1.33457i		−	0.114862i
$136$		−	13.8017i		−	1.18349i
$137$			16.8037i			1.43563i	0.696232	+	0.717817i	$0.254859\pi$
$137$			16.8037i			1.43563i	−0.696232	+	0.717817i	$0.745141\pi$
$138$		−	7.91188i		−	0.673504i
$139$	2.51851			0.213617			0.106809	−	0.994280i	$-0.465937\pi$
$139$	2.51851			0.213617			0.106809	+	0.994280i	$0.465937\pi$
$140$	0.446512			0.0377371
$141$		−	5.27206i		−	0.443987i
$142$	6.30331			0.528962
$143$	1.41897	−	17.5535i	0.118660	−	1.46790i
$144$	−3.21892			−0.268243
$145$		−	10.1933i		−	0.846509i
$146$	−5.74309			−0.475301
$147$	1.00000			0.0824786
$148$			3.26834i			0.268656i
$149$		−	11.6416i		−	0.953717i	−0.878980	−	0.476859i	$-0.841775\pi$
$149$		−	11.6416i		−	0.953717i	0.878980	−	0.476859i	$-0.158225\pi$
$150$			4.15406i			0.339177i
$151$		−	15.5374i		−	1.26441i	−0.774800	−	0.632207i	$-0.782149\pi$
$151$		−	15.5374i		−	1.26441i	0.774800	−	0.632207i	$-0.217851\pi$
$152$	−8.94422			−0.725472
$153$	4.58103			0.370354
$154$			6.30331i			0.507936i
$155$	−7.52417			−0.604356
$156$	−1.20240	−	0.0971982i	−0.0962688	−	0.00778208i
$157$	−7.16206			−0.571594			−0.285797	−	0.958290i	$-0.592258\pi$
$157$	−7.16206			−0.571594			−0.285797	+	0.958290i	$0.592258\pi$
$158$			4.58103i			0.364447i
$159$	−10.7374			−0.851533
$160$	2.49771			0.197461
$161$			6.13080i			0.483175i
$162$		−	1.29051i		−	0.101392i
$163$		−	7.70617i		−	0.603594i	−0.953372	−	0.301797i	$-0.902414\pi$
$163$		−	7.70617i		−	0.603594i	0.953372	−	0.301797i	$-0.0975864\pi$
$164$		−	1.23681i		−	0.0965787i
$165$	−6.51851			−0.507465
$166$	−6.80366			−0.528067
$167$			12.5478i			0.970980i	0.874242	+	0.485490i	$0.161359\pi$
$167$			12.5478i			0.970980i	−0.874242	+	0.485490i	$0.838641\pi$
$168$	3.01280			0.232443
$169$	12.8312	+	2.08812i	0.987016	+	0.160624i
$170$	7.88984			0.605123
$171$		−	2.96874i		−	0.227025i
$172$	3.22458			0.245872
$173$	14.5185			1.10382			0.551911	−	0.833903i	$-0.313899\pi$
$173$	14.5185			1.10382			0.551911	+	0.833903i	$0.313899\pi$
$174$		−	9.85680i		−	0.747242i
$175$		−	3.21892i		−	0.243327i
$176$			15.7223i			1.18511i
$177$		−	10.3033i		−	0.774444i
$178$	12.8293			0.961594
$179$	16.0683			1.20100			0.600500	−	0.799625i	$-0.294968\pi$
$179$	16.0683			1.20100			0.600500	+	0.799625i	$0.294968\pi$
$180$			0.446512i			0.0332810i
$181$	17.2758			1.28410			0.642049	−	0.766664i	$-0.278085\pi$
$181$	17.2758			1.28410			0.642049	+	0.766664i	$0.278085\pi$
$182$	−4.63789	−	0.374913i	−0.343783	−	0.0277904i
$183$	11.1877			0.827015
$184$			18.4709i			1.36169i
$185$	−13.0370			−0.958501
$186$	−7.27577			−0.533486
$187$		−	22.3753i		−	1.63624i
$188$			1.76389i			0.128645i
$189$			1.00000i			0.0727393i
$190$		−	5.11302i		−	0.370937i
$191$	−4.66915			−0.337848			−0.168924	−	0.985629i	$-0.554029\pi$
$191$	−4.66915			−0.337848			−0.168924	+	0.985629i	$0.554029\pi$
$192$	8.85308			0.638916
$193$			2.66915i			0.192129i	0.995375	+	0.0960647i	$0.0306256\pi$
$193$			2.66915i			0.192129i	−0.995375	+	0.0960647i	$0.969374\pi$
$194$	23.5118			1.68805
$195$	0.387712	−	4.79623i	0.0277647	−	0.343465i
$196$	−0.334573			−0.0238981
$197$		−	13.6967i		−	0.975848i	−0.872886	−	0.487924i	$-0.837754\pi$
$197$		−	13.6967i		−	0.975848i	0.872886	−	0.487924i	$-0.162246\pi$
$198$	−6.30331			−0.447957
$199$	−5.93748			−0.420897			−0.210448	−	0.977605i	$-0.567492\pi$
$199$	−5.93748			−0.420897			−0.210448	+	0.977605i	$0.567492\pi$
$200$		−	9.69795i		−	0.685748i
$201$			5.50709i			0.388440i
$202$			13.8017i			0.971086i
$203$			7.63789i			0.536075i
$204$	−1.53269			−0.107310
$205$	4.93350			0.344570
$206$		−	1.91188i		−	0.133207i
$207$	−6.13080			−0.426120
$208$	−11.5682	−	0.935141i	−0.802112	−	0.0648404i
$209$	−14.5003			−1.00301
$210$			1.72229i			0.118849i
$211$	18.6493			1.28387			0.641936	−	0.766758i	$-0.278132\pi$
$211$	18.6493			1.28387			0.641936	+	0.766758i	$0.278132\pi$
$212$	3.59245			0.246731
$213$		−	4.88434i		−	0.334670i
$214$		−	5.79817i		−	0.396355i
$215$			12.8625i			0.877213i
$216$			3.01280i			0.204995i
$217$	5.63789			0.382725
$218$	−10.1063			−0.684484
$219$			4.45023i			0.300719i
$220$	2.18092			0.147037
$221$	16.4634	+	1.33085i	1.10745	+	0.0895230i
$222$	−12.6066			−0.846101
$223$		−	15.4066i		−	1.03170i	−0.856679	−	0.515850i	$-0.827476\pi$
$223$		−	15.4066i		−	1.03170i	0.856679	−	0.515850i	$-0.172524\pi$
$224$	−1.87154			−0.125048
$225$	3.21892			0.214594
$226$		−	9.85680i		−	0.655665i
$227$			7.89576i			0.524060i	0.965060	+	0.262030i	$0.0843920\pi$
$227$			7.89576i			0.524060i	−0.965060	+	0.262030i	$0.915608\pi$
$228$			0.993260i			0.0657803i
$229$			9.01142i			0.595492i	0.954645	+	0.297746i	$0.0962348\pi$
$229$			9.01142i			0.595492i	−0.954645	+	0.297746i	$0.903765\pi$
$230$	−10.5590			−0.696239
$231$	4.88434			0.321366
$232$			23.0114i			1.51077i
$233$	−10.7374			−0.703432			−0.351716	−	0.936107i	$-0.614402\pi$
$233$	−10.7374			−0.703432			−0.351716	+	0.936107i	$0.614402\pi$
$234$	0.374913	−	4.63789i	0.0245088	−	0.303188i
$235$	−7.03594			−0.458974
$236$			3.44721i			0.224394i
$237$	3.54977			0.230582
$238$	−5.91188			−0.383210
$239$			15.9839i			1.03391i	0.856012	+	0.516956i	$0.172935\pi$
$239$			15.9839i			1.03391i	−0.856012	+	0.516956i	$0.827065\pi$
$240$			4.29588i			0.277298i
$241$		−	21.6635i		−	1.39547i	−0.716357	−	0.697734i	$-0.754192\pi$
$241$		−	21.6635i		−	1.39547i	0.716357	−	0.697734i	$-0.245808\pi$
$242$			16.5919i			1.06657i
$243$	−1.00000			−0.0641500
$244$	−3.74309			−0.239627
$245$		−	1.33457i		−	0.0852627i
$246$	4.77063			0.304164
$247$	0.862462	−	10.6691i	0.0548771	−	0.678861i
$248$	16.9858			1.07860
$249$			5.27206i			0.334103i
$250$	14.1553			0.895261
$251$	−3.87496			−0.244586			−0.122293	−	0.992494i	$-0.539025\pi$
$251$	−3.87496			−0.244586			−0.122293	+	0.992494i	$0.539025\pi$
$252$		−	0.334573i		−	0.0210761i
$253$			29.9449i			1.88262i
$254$			21.1802i			1.32897i
$255$		−	6.11372i		−	0.382856i
$256$	−7.79251			−0.487032
$257$	16.8938			1.05381			0.526904	−	0.849925i	$-0.323353\pi$
$257$	16.8938			1.05381			0.526904	+	0.849925i	$0.323353\pi$
$258$			12.4378i			0.774346i
$259$	9.76869			0.606997
$260$	−0.129718	+	1.60469i	−0.00804477	+	0.0995184i
$261$	−7.63789			−0.472773
$262$			10.8891i			0.672733i
$263$	9.40657			0.580034			0.290017	−	0.957021i	$-0.406339\pi$
$263$	9.40657			0.580034			0.290017	+	0.957021i	$0.406339\pi$
$264$	14.7155			0.905679
$265$			14.3299i			0.880277i
$266$			3.83120i			0.234906i
$267$		−	9.94120i		−	0.608392i
$268$		−	1.84252i		−	0.112550i
$269$	−17.6181			−1.07419			−0.537096	−	0.843521i	$-0.680479\pi$
$269$	−17.6181			−1.07419			−0.537096	+	0.843521i	$0.680479\pi$
$270$	−1.72229			−0.104815
$271$			23.1070i			1.40365i	0.712350	+	0.701824i	$0.247631\pi$
$271$			23.1070i			1.40365i	−0.712350	+	0.701824i	$0.752369\pi$
$272$	−14.7459			−0.894104
$273$	−0.290514	+	3.59383i	−0.0175827	+	0.217508i
$274$	21.6854			1.31006
$275$		−	15.7223i		−	0.948089i
$276$	2.05120			0.123468
$277$	−20.0757			−1.20623			−0.603116	−	0.797653i	$-0.706075\pi$
$277$	−20.0757			−1.20623			−0.603116	+	0.797653i	$0.706075\pi$
$278$		−	3.25017i		−	0.194932i
$279$			5.63789i			0.337531i
$280$		−	4.02080i		−	0.240289i
$281$			11.2038i			0.668361i	0.942509	+	0.334181i	$0.108460\pi$
$281$			11.2038i			0.668361i	−0.942509	+	0.334181i	$0.891540\pi$
$282$	−6.80366			−0.405152
$283$	−1.33829			−0.0795532			−0.0397766	−	0.999209i	$-0.512665\pi$
$283$	−1.33829			−0.0795532			−0.0397766	+	0.999209i	$0.512665\pi$
$284$			1.63417i			0.0969700i
$285$	−3.96200			−0.234689
$286$	−22.6530	−	1.83120i	−1.33950	−	0.108281i
$287$	−3.69669			−0.218208
$288$		−	1.87154i		−	0.110282i
$289$	3.98582			0.234460
$290$	−13.1546			−0.772466
$291$		−	18.2189i		−	1.06801i
$292$		−	1.48893i		−	0.0871328i
$293$		−	8.60291i		−	0.502587i	−0.967911	−	0.251294i	$-0.919144\pi$
$293$		−	8.60291i		−	0.502587i	0.967911	−	0.251294i	$-0.0808560\pi$
$294$		−	1.29051i		−	0.0752643i
$295$	−13.7505			−0.800586
$296$	29.4311			1.71065
$297$			4.88434i			0.283418i
$298$	−15.0237			−0.870297
$299$	−22.0330	−	1.78108i	−1.27420	−	0.103003i
$300$	−1.07696			−0.0621784
$301$		−	9.63789i		−	0.555519i
$302$	−20.0512			−1.15382
$303$	10.6947			0.614397
$304$			9.55613i			0.548081i
$305$		−	14.9307i		−	0.854932i
$306$		−	5.91188i		−	0.337960i
$307$		−	7.57537i		−	0.432349i	−0.976355	−	0.216175i	$-0.930642\pi$
$307$		−	7.57537i		−	0.432349i	0.976355	−	0.216175i	$-0.0693580\pi$
$308$	−1.63417			−0.0931154
$309$	−1.48149			−0.0842790
$310$			9.71005i			0.551494i
$311$	6.11372			0.346677			0.173339	−	0.984862i	$-0.444545\pi$
$311$	6.11372			0.346677			0.173339	+	0.984862i	$0.444545\pi$
$312$	−0.875261	+	10.8275i	−0.0495519	+	0.612986i
$313$	−7.31269			−0.413338			−0.206669	−	0.978411i	$-0.566262\pi$
$313$	−7.31269			−0.413338			−0.206669	+	0.978411i	$0.566262\pi$
$314$			9.24274i			0.521598i
$315$	1.33457			0.0751947
$316$	−1.18766			−0.0668109
$317$			1.70412i			0.0957131i	0.998854	+	0.0478565i	$0.0152390\pi$
$317$			1.70412i			0.0957131i	−0.998854	+	0.0478565i	$0.984761\pi$
$318$			13.8568i			0.777051i
$319$			37.3061i			2.08874i
$320$		−	11.8151i		−	0.660483i
$321$	−4.49291			−0.250770
$322$	7.91188			0.440912
$323$		−	13.5999i		−	0.756718i
$324$	0.334573			0.0185874
$325$	11.5682	+	0.935141i	0.641690	+	0.0518723i
$326$	−9.94492			−0.550798
$327$			7.83120i			0.433067i
$328$	−11.1374			−0.614959
$329$	5.27206			0.290658
$330$			8.41223i			0.463078i
$331$		−	16.5232i		−	0.908197i	−0.890952	−	0.454098i	$-0.849961\pi$
$331$		−	16.5232i		−	0.908197i	0.890952	−	0.454098i	$-0.150039\pi$
$332$		−	1.76389i		−	0.0968058i
$333$			9.76869i			0.535321i
$334$	16.1932			0.886050
$335$	7.34961			0.401552
$336$		−	3.21892i		−	0.175606i
$337$	12.3043			0.670257			0.335128	−	0.942172i	$-0.391220\pi$
$337$	12.3043			0.670257			0.335128	+	0.942172i	$0.391220\pi$
$338$	2.69474	−	16.5589i	0.146575	−	0.900683i
$339$	−7.63789			−0.414833
$340$			2.04548i			0.110932i
$341$	27.5374			1.49123
$342$	−3.83120			−0.207168
$343$			1.00000i			0.0539949i
$344$		−	29.0370i		−	1.56557i
$345$			8.18200i			0.440504i
$346$		−	18.7363i		−	1.00727i
$347$	9.88240			0.530515			0.265258	−	0.964178i	$-0.414543\pi$
$347$	9.88240			0.530515			0.265258	+	0.964178i	$0.414543\pi$
$348$	2.55543			0.136985
$349$		−	29.1497i		−	1.56035i	−0.625564	−	0.780173i	$-0.715131\pi$
$349$		−	29.1497i		−	1.56035i	0.625564	−	0.780173i	$-0.284869\pi$
$350$	−4.15406			−0.222044
$351$	−3.59383	−	0.290514i	−0.191824	−	0.0155065i
$352$	−9.14126			−0.487231
$353$		−	8.90994i		−	0.474228i	−0.971482	−	0.237114i	$-0.923798\pi$
$353$		−	8.90994i		−	0.474228i	0.971482	−	0.237114i	$-0.0762016\pi$
$354$	−13.2966			−0.706705
$355$	−6.51851			−0.345967
$356$			3.32606i			0.176281i
$357$			4.58103i			0.242454i
$358$		−	20.7363i		−	1.09595i
$359$			29.8776i			1.57688i	0.615112	+	0.788440i	$0.289111\pi$
$359$			29.8776i			1.57688i	−0.615112	+	0.788440i	$0.710889\pi$
$360$	4.02080			0.211915
$361$	10.1866			0.536136
$362$		−	22.2946i		−	1.17178i
$363$	12.8568			0.674807
$364$	0.0971982	−	1.20240i	0.00509457	−	0.0630227i
$365$	5.93916			0.310870
$366$		−	14.4378i		−	0.754678i
$367$	−5.91932			−0.308986			−0.154493	−	0.987994i	$-0.549374\pi$
$367$	−5.91932			−0.308986			−0.154493	+	0.987994i	$0.549374\pi$
$368$	19.7345			1.02873
$369$		−	3.69669i		−	0.192442i
$370$			16.8245i			0.874662i
$371$		−	10.7374i		−	0.557459i
$372$		−	1.88628i		−	0.0977993i
$373$	7.35645			0.380903			0.190451	−	0.981697i	$-0.439005\pi$
$373$	7.35645			0.380903			0.190451	+	0.981697i	$0.439005\pi$
$374$	−28.8757			−1.49312
$375$		−	10.9687i		−	0.566423i
$376$	15.8836			0.819136
$377$	−27.4493	−	2.21892i	−1.41371	−	0.114280i
$378$	1.29051			0.0663769
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$	1.32558			0.0680007
$381$	16.4122			0.840824
$382$			6.02560i			0.308296i
$383$		−	15.2891i		−	0.781238i	−0.920552	−	0.390619i	$-0.872261\pi$
$383$		−	15.2891i		−	0.781238i	0.920552	−	0.390619i	$-0.127739\pi$
$384$		−	7.68195i		−	0.392018i
$385$		−	6.51851i		−	0.332214i
$386$	3.44457			0.175324
$387$	9.63789			0.489921
$388$			6.09555i			0.309455i
$389$	−7.51453			−0.381002			−0.190501	−	0.981687i	$-0.561011\pi$
$389$	−7.51453			−0.381002			−0.190501	+	0.981687i	$0.561011\pi$
$390$	−6.18960	−	0.500349i	−0.313422	−	0.0253361i
$391$	−28.0854			−1.42034
$392$			3.01280i			0.152169i
$393$	8.43783			0.425632
$394$	−17.6758			−0.890492
$395$		−	4.73743i		−	0.238366i
$396$		−	1.63417i		−	0.0821200i
$397$		−	10.3951i		−	0.521718i	−0.965377	−	0.260859i	$-0.915994\pi$
$397$		−	10.3951i		−	0.521718i	0.965377	−	0.260859i	$-0.0840057\pi$
$398$			7.66241i			0.384082i
$399$	2.96874			0.148623
$400$	−10.3614			−0.518071
$401$		−	13.4029i		−	0.669307i	−0.942341	−	0.334653i	$-0.891381\pi$
$401$		−	13.4029i		−	0.669307i	0.942341	−	0.334653i	$-0.108619\pi$
$402$	7.10698			0.354464
$403$	−1.63789	+	20.2616i	−0.0815890	+	1.00930i
$404$	−3.57817			−0.178021
$405$			1.33457i			0.0663155i
$406$	9.85680			0.489185
$407$	47.7136			2.36508
$408$			13.8017i			0.683287i
$409$			34.7421i			1.71789i	0.512071	+	0.858943i	$0.328879\pi$
$409$			34.7421i			1.71789i	−0.512071	+	0.858943i	$0.671121\pi$
$410$		−	6.36675i		−	0.314431i
$411$		−	16.8037i		−	0.828864i
$412$	0.495666			0.0244197
$413$	10.3033			0.506993
$414$			7.91188i			0.388848i
$415$	7.03594			0.345381
$416$	0.543710	−	6.72600i	0.0266576	−	0.329769i
$417$	−2.51851			−0.123332
$418$			18.7129i			0.915278i
$419$	3.01418			0.147252			0.0736261	−	0.997286i	$-0.476543\pi$
$419$	3.01418			0.147252			0.0736261	+	0.997286i	$0.476543\pi$
$420$	−0.446512			−0.0217875
$421$		−	32.1366i		−	1.56624i	−0.621871	−	0.783120i	$-0.713627\pi$
$421$		−	32.1366i		−	1.56624i	0.621871	−	0.783120i	$-0.286373\pi$
$422$		−	24.0672i		−	1.17157i
$423$			5.27206i			0.256336i
$424$		−	32.3497i		−	1.57104i
$425$	14.7459			0.715283
$426$	−6.30331			−0.305397
$427$			11.1877i			0.541409i
$428$	1.50321			0.0726602
$429$	−1.41897	+	17.5535i	−0.0685086	+	0.847490i
$430$	16.5992			0.800484
$431$		−	9.00938i		−	0.433966i	−0.976175	−	0.216983i	$-0.930378\pi$
$431$		−	9.00938i		−	0.433966i	0.976175	−	0.216983i	$-0.0696217\pi$
$432$	3.21892			0.154870
$433$	−38.6397			−1.85690			−0.928452	−	0.371453i	$-0.878860\pi$
$433$	−38.6397			−1.85690			−0.928452	+	0.371453i	$0.878860\pi$
$434$		−	7.27577i		−	0.349248i
$435$			10.1933i			0.488732i
$436$		−	2.62011i		−	0.125480i
$437$			18.2008i			0.870660i
$438$	5.74309			0.274415
$439$	1.48149			0.0707076			0.0353538	−	0.999375i	$-0.488744\pi$
$439$	1.48149			0.0707076			0.0353538	+	0.999375i	$0.488744\pi$
$440$		−	19.6390i		−	0.936251i
$441$	−1.00000			−0.0476190
$442$	1.71749	−	21.2463i	0.0816925	−	1.01058i
$443$	−12.0683			−0.573381			−0.286691	−	0.958023i	$-0.592555\pi$
$443$	−12.0683			−0.573381			−0.286691	+	0.958023i	$0.592555\pi$
$444$		−	3.26834i		−	0.155108i
$445$	−13.2673			−0.628928
$446$	−19.8824			−0.941459
$447$			11.6416i			0.550629i
$448$			8.85308i			0.418269i
$449$		−	24.0794i		−	1.13638i	−0.822898	−	0.568189i	$-0.807644\pi$
$449$		−	24.0794i		−	1.13638i	0.822898	−	0.568189i	$-0.192356\pi$
$450$		−	4.15406i		−	0.195824i
$451$	−18.0559			−0.850218
$452$	2.55543			0.120197
$453$			15.5374i			0.730009i
$454$	10.1896			0.478222
$455$	4.79623	+	0.387712i	0.224851	+	0.0181762i
$456$	8.94422			0.418852
$457$			32.9933i			1.54336i	0.636011	+	0.771680i	$0.280583\pi$
$457$			32.9933i			1.54336i	−0.636011	+	0.771680i	$0.719417\pi$
$458$	11.6294			0.543405
$459$	−4.58103			−0.213824
$460$		−	2.73747i		−	0.127635i
$461$		−	13.4654i		−	0.627145i	−0.949564	−	0.313572i	$-0.898474\pi$
$461$		−	13.4654i		−	0.627145i	0.949564	−	0.313572i	$-0.101526\pi$
$462$		−	6.30331i		−	0.293257i
$463$			22.3679i			1.03952i	0.854311	+	0.519762i	$0.173979\pi$
$463$			22.3679i			1.03952i	−0.854311	+	0.519762i	$0.826021\pi$
$464$	24.5857			1.14136
$465$	7.52417			0.348925
$466$			13.8568i			0.641904i
$467$	16.5629			0.766438			0.383219	−	0.923658i	$-0.374815\pi$
$467$	16.5629			0.766438			0.383219	+	0.923658i	$0.374815\pi$
$468$	1.20240	+	0.0971982i	0.0555808	+	0.00449299i
$469$	−5.50709			−0.254294
$470$			9.07998i			0.418828i
$471$	7.16206			0.330010
$472$	31.0418			1.42882
$473$		−	47.0747i		−	2.16450i
$474$		−	4.58103i		−	0.210414i
$475$		−	9.55613i		−	0.438465i
$476$		−	1.53269i		−	0.0702506i
$477$	10.7374			0.491633
$478$	20.6274			0.943477
$479$			10.1403i			0.463321i	0.972797	+	0.231661i	$0.0744159\pi$
$479$			10.1403i			0.463321i	−0.972797	+	0.231661i	$0.925584\pi$
$480$	−2.49771			−0.114004
$481$	−2.83794	+	35.1070i	−0.129399	+	1.60074i
$482$	−27.9570			−1.27341
$483$		−	6.13080i		−	0.278961i
$484$	−4.30154			−0.195524
$485$	−24.3145			−1.10406
$486$			1.29051i			0.0585389i
$487$		−	11.6624i		−	0.528474i	−0.964458	−	0.264237i	$-0.914880\pi$
$487$		−	11.6624i		−	0.528474i	0.964458	−	0.264237i	$-0.0851201\pi$
$488$			33.7062i			1.52581i
$489$			7.70617i			0.348485i
$490$	−1.72229			−0.0778049
$491$	−24.8454			−1.12126			−0.560628	−	0.828068i	$-0.689440\pi$
$491$	−24.8454			−1.12126			−0.560628	+	0.828068i	$0.689440\pi$
$492$			1.23681i			0.0557597i
$493$	−34.9894			−1.57584
$494$	−13.7687	−	1.11302i	−0.619482	−	0.0500771i
$495$	6.51851			0.292985
$496$		−	18.1479i		−	0.814864i
$497$	4.88434			0.219093
$498$	6.80366			0.304879
$499$		−	7.40081i		−	0.331306i	−0.986184	−	0.165653i	$-0.947027\pi$
$499$		−	7.40081i		−	0.331306i	0.986184	−	0.165653i	$-0.0529731\pi$
$500$			3.66984i			0.164120i
$501$		−	12.5478i		−	0.560596i
$502$			5.00070i			0.223192i
$503$	−35.8898			−1.60025			−0.800124	−	0.599834i	$-0.795233\pi$
$503$	−35.8898			−1.60025			−0.800124	+	0.599834i	$0.795233\pi$
$504$	−3.01280			−0.134201
$505$		−	14.2729i		−	0.635136i
$506$	38.6443			1.71795
$507$	−12.8312	−	2.08812i	−0.569854	−	0.0927365i
$508$	−5.49109			−0.243628
$509$			3.99628i			0.177132i	0.996070	+	0.0885660i	$0.0282284\pi$
$509$			3.99628i			0.177132i	−0.996070	+	0.0885660i	$0.971772\pi$
$510$	−7.88984			−0.349368
$511$	−4.45023			−0.196867
$512$			25.4202i			1.12343i
$513$			2.96874i			0.131073i
$514$		−	21.8017i		−	0.961633i
$515$			1.97716i			0.0871239i
$516$	−3.22458			−0.141954
$517$	25.7505			1.13251
$518$		−	12.6066i		−	0.553903i
$519$	−14.5185			−0.637292
$520$	14.4501	+	1.16810i	0.633677	+	0.0512246i
$521$	−28.5562			−1.25107			−0.625536	−	0.780196i	$-0.715119\pi$
$521$	−28.5562			−1.25107			−0.625536	+	0.780196i	$0.715119\pi$
$522$			9.85680i			0.431421i
$523$	−23.5555			−1.03001			−0.515006	−	0.857187i	$-0.672210\pi$
$523$	−23.5555			−1.03001			−0.515006	+	0.857187i	$0.672210\pi$
$524$	−2.82307			−0.123326
$525$			3.21892i			0.140485i
$526$		−	12.1393i		−	0.529299i
$527$			25.8273i			1.12506i
$528$		−	15.7223i		−	0.684225i
$529$	14.5867			0.634204
$530$	18.4929			0.803281
$531$			10.3033i			0.447126i
$532$	−0.993260			−0.0430633
$533$	1.07394	−	13.2853i	0.0465175	−	0.575448i
$534$	−12.8293			−0.555176
$535$			5.99612i			0.259235i
$536$	−16.5918			−0.716655
$537$	−16.0683			−0.693397
$538$			22.7363i			0.980233i
$539$			4.88434i			0.210384i
$540$		−	0.446512i		−	0.0192148i
$541$			6.32411i			0.271895i	0.990716	+	0.135947i	$0.0434078\pi$
$541$			6.32411i			0.271895i	−0.990716	+	0.135947i	$0.956592\pi$
$542$	29.8199			1.28087
$543$	−17.2758			−0.741374
$544$		−	8.57359i		−	0.367590i
$545$	10.4513			0.447685
$546$	4.63789	+	0.374913i	0.198483	+	0.0160448i
$547$	17.8141			0.761677			0.380838	−	0.924642i	$-0.375635\pi$
$547$	17.8141			0.761677			0.380838	+	0.924642i	$0.375635\pi$
$548$			5.62205i			0.240162i
$549$	−11.1877			−0.477478
$550$	−20.2898			−0.865161
$551$			22.6749i			0.965984i
$552$		−	18.4709i		−	0.786172i
$553$			3.54977i			0.150952i
$554$			25.9080i			1.10073i
$555$	13.0370			0.553391
$556$	0.842625			0.0357353
$557$		−	2.04915i		−	0.0868255i	−0.999057	−	0.0434127i	$-0.986177\pi$
$557$		−	2.04915i		−	0.0868255i	0.999057	−	0.0434127i	$-0.0138230\pi$
$558$	7.27577			0.308008
$559$	34.6369	+	2.79994i	1.46499	+	0.118425i
$560$	−4.29588			−0.181534
$561$			22.3753i			0.944686i
$562$	14.4586			0.609901
$563$	−29.5999			−1.24749			−0.623743	−	0.781629i	$-0.714389\pi$
$563$	−29.5999			−1.24749			−0.623743	+	0.781629i	$0.714389\pi$
$564$		−	1.76389i		−	0.0742730i
$565$			10.1933i			0.428836i
$566$			1.72708i			0.0725948i
$567$		−	1.00000i		−	0.0419961i
$568$	14.7155			0.617451
$569$	7.63789			0.320197			0.160098	−	0.987101i	$-0.448819\pi$
$569$	7.63789			0.320197			0.160098	+	0.987101i	$0.448819\pi$
$570$			5.11302i			0.214161i
$571$	−11.9364			−0.499523			−0.249761	−	0.968307i	$-0.580352\pi$
$571$	−11.9364			−0.499523			−0.249761	+	0.968307i	$0.580352\pi$
$572$	0.474749	−	5.87292i	0.0198503	−	0.245559i
$573$	4.66915			0.195056
$574$			4.77063i			0.199122i
$575$	−19.7345			−0.822986
$576$	−8.85308			−0.368878
$577$			47.8320i			1.99127i	0.0933200	+	0.995636i	$0.470252\pi$
$577$			47.8320i			1.99127i	−0.0933200	+	0.995636i	$0.529748\pi$
$578$		−	5.14376i		−	0.213952i
$579$		−	2.66915i		−	0.110926i
$580$		−	3.41041i		−	0.141609i
$581$	−5.27206			−0.218722
$582$	−23.5118			−0.974594
$583$		−	52.4453i		−	2.17206i
$584$	−13.4077			−0.554813
$585$	−0.387712	+	4.79623i	−0.0160299	+	0.198300i
$586$	−11.1022			−0.458627
$587$		−	12.2350i		−	0.504994i	−0.967598	−	0.252497i	$-0.918748\pi$
$587$		−	12.2350i		−	0.504994i	0.967598	−	0.252497i	$-0.0812518\pi$
$588$	0.334573			0.0137976
$589$	16.7374			0.689654
$590$			17.7452i			0.730560i
$591$			13.6967i			0.563406i
$592$		−	31.4446i		−	1.29236i
$593$		−	33.5714i		−	1.37861i	−0.724471	−	0.689305i	$-0.757916\pi$
$593$		−	33.5714i		−	1.37861i	0.724471	−	0.689305i	$-0.242084\pi$
$594$	6.30331			0.258628
$595$	6.11372			0.250638
$596$		−	3.89496i		−	0.159544i
$597$	5.93748			0.243005
$598$	−2.29852	+	28.4339i	−0.0939933	+	1.16275i
$599$	20.4549			0.835765			0.417883	−	0.908501i	$-0.362772\pi$
$599$	20.4549			0.835765			0.417883	+	0.908501i	$0.362772\pi$
$600$			9.69795i			0.395917i
$601$	25.3014			1.03206			0.516032	−	0.856569i	$-0.327408\pi$
$601$	25.3014			1.03206			0.516032	+	0.856569i	$0.327408\pi$
$602$	−12.4378			−0.506928
$603$		−	5.50709i		−	0.224266i
$604$		−	5.19838i		−	0.211519i
$605$		−	17.1583i		−	0.697586i
$606$		−	13.8017i		−	0.560657i
$607$	−16.0330			−0.650761			−0.325380	−	0.945583i	$-0.605492\pi$
$607$	−16.0330			−0.650761			−0.325380	+	0.945583i	$0.605492\pi$
$608$	−5.55613			−0.225331
$609$		−	7.63789i		−	0.309503i
$610$	−19.2683			−0.780152
$611$	−1.53161	+	18.9469i	−0.0619622	+	0.766508i
$612$	1.53269			0.0619552
$613$			39.9752i			1.61458i	0.590153	+	0.807292i	$0.299067\pi$
$613$			39.9752i			1.61458i	−0.590153	+	0.807292i	$0.700933\pi$
$614$	−9.77612			−0.394532
$615$	−4.93350			−0.198938
$616$			14.7155i			0.592906i
$617$			1.05782i			0.0425863i	0.999773	+	0.0212932i	$0.00677834\pi$
$617$			1.05782i			0.0425863i	−0.999773	+	0.0212932i	$0.993222\pi$
$618$			1.91188i			0.0769072i
$619$			41.8199i			1.68088i	0.541902	+	0.840442i	$0.317704\pi$
$619$			41.8199i			1.68088i	−0.541902	+	0.840442i	$0.682296\pi$
$620$	−2.51738			−0.101101
$621$	6.13080			0.246020
$622$		−	7.88984i		−	0.316354i
$623$	9.94120			0.398286
$624$	11.5682	+	0.935141i	0.463100	+	0.0374356i
$625$	1.45599			0.0582397
$626$			9.43713i			0.377184i
$627$	14.5003			0.579088
$628$	−2.39623			−0.0956200
$629$			44.7506i			1.78432i
$630$		−	1.72229i		−	0.0686175i
$631$			20.0928i			0.799882i	0.916541	+	0.399941i	$0.130969\pi$
$631$			20.0928i			0.799882i	−0.916541	+	0.399941i	$0.869031\pi$
$632$			10.6947i			0.425414i
$633$	−18.6493			−0.741243
$634$	2.19920			0.0873412
$635$		−	21.9033i		−	0.869207i
$636$	−3.59245			−0.142450
$637$	−3.59383	−	0.290514i	−0.142393	−	0.0115106i
$638$	48.1440			1.90604
$639$			4.88434i			0.193222i
$640$	−10.2521			−0.405250
$641$	12.2008			0.481901			0.240950	−	0.970537i	$-0.422541\pi$
$641$	12.2008			0.481901			0.240950	+	0.970537i	$0.422541\pi$
$642$			5.79817i			0.228835i
$643$		−	0.0511985i		−	0.00201907i	−0.999999	−	0.00100954i	$-0.999679\pi$
$643$		−	0.0511985i		−	0.00201907i	0.999999	−	0.00100954i	$-0.000321345\pi$
$644$			2.05120i			0.0808285i
$645$		−	12.8625i		−	0.506459i
$646$	−17.5508			−0.690529
$647$	6.63691			0.260924			0.130462	−	0.991453i	$-0.458354\pi$
$647$	6.63691			0.260924			0.130462	+	0.991453i	$0.458354\pi$
$648$		−	3.01280i		−	0.118354i
$649$	50.3249			1.97543
$650$	1.20681	−	14.9290i	0.0473351	−	0.585562i
$651$	−5.63789			−0.220966
$652$		−	2.57827i		−	0.100973i
$653$	14.1250			0.552755			0.276378	−	0.961049i	$-0.410866\pi$
$653$	14.1250			0.552755			0.276378	+	0.961049i	$0.410866\pi$
$654$	10.1063			0.395187
$655$		−	11.2609i		−	0.440000i
$656$			11.8993i			0.464590i
$657$		−	4.45023i		−	0.173620i
$658$		−	6.80366i		−	0.265234i
$659$	−9.95456			−0.387775			−0.193887	−	0.981024i	$-0.562110\pi$
$659$	−9.95456			−0.387775			−0.193887	+	0.981024i	$0.562110\pi$
$660$	−2.18092			−0.0848921
$661$			11.0359i			0.429248i	0.976697	+	0.214624i	$0.0688527\pi$
$661$			11.0359i			0.429248i	−0.976697	+	0.214624i	$0.931147\pi$
$662$	−21.3234			−0.828758
$663$	−16.4634	−	1.33085i	−0.639387	−	0.0516861i
$664$	−15.8836			−0.616405
$665$		−	3.96200i		−	0.153640i
$666$	12.6066			0.488497
$667$	46.8263			1.81312
$668$			4.19816i			0.162432i
$669$			15.4066i			0.595652i
$670$		−	9.48478i		−	0.366429i
$671$			54.6443i			2.10952i
$672$	1.87154			0.0721963
$673$	−9.64921			−0.371950			−0.185975	−	0.982555i	$-0.559544\pi$
$673$	−9.64921			−0.371950			−0.185975	+	0.982555i	$0.559544\pi$
$674$		−	15.8788i		−	0.611630i
$675$	−3.21892			−0.123896
$676$	4.29297	+	0.698627i	0.165114	+	0.0268703i
$677$	27.8320			1.06967			0.534835	−	0.844956i	$-0.320374\pi$
$677$	27.8320			1.06967			0.534835	+	0.844956i	$0.320374\pi$
$678$			9.85680i			0.378548i
$679$	18.2189			0.699178
$680$	18.4194			0.706352
$681$		−	7.89576i		−	0.302566i
$682$		−	35.5374i		−	1.36080i
$683$			43.8151i			1.67654i	0.545257	+	0.838269i	$0.316432\pi$
$683$			43.8151i			1.67654i	−0.545257	+	0.838269i	$0.683568\pi$
$684$		−	0.993260i		−	0.0379782i
$685$	−22.4257			−0.856842
$686$	1.29051			0.0492721
$687$		−	9.01142i		−	0.343807i
$688$	−31.0235			−1.18276
$689$	38.5885	+	3.11938i	1.47010	+	0.118839i
$690$	10.5590			0.401974
$691$		−	41.6057i		−	1.58275i	−0.611329	−	0.791377i	$-0.709365\pi$
$691$		−	41.6057i		−	1.58275i	0.611329	−	0.791377i	$-0.290635\pi$
$692$	4.85750			0.184654
$693$	−4.88434			−0.185541
$694$		−	12.7534i		−	0.484112i
$695$			3.36114i			0.127495i
$696$		−	23.0114i		−	0.872246i
$697$		−	16.9346i		−	0.641445i
$698$	−37.6181			−1.42386
$699$	10.7374			0.406127
$700$		−	1.07696i		−	0.0407053i
$701$	−35.4765			−1.33993			−0.669965	−	0.742393i	$-0.733691\pi$
$701$	−35.4765			−1.33993			−0.669965	+	0.742393i	$0.733691\pi$
$702$	−0.374913	+	4.63789i	−0.0141502	+	0.175046i
$703$	29.0007			1.09378
$704$			43.2415i			1.62973i
$705$	7.03594			0.264989
$706$	−11.4984			−0.432748
$707$			10.6947i			0.402217i
$708$		−	3.44721i		−	0.129554i
$709$		−	17.8937i		−	0.672013i	−0.941860	−	0.336006i	$-0.890924\pi$
$709$		−	17.8937i		−	0.672013i	0.941860	−	0.336006i	$-0.109076\pi$
$710$			8.41223i			0.315705i
$711$	−3.54977			−0.133127
$712$	29.9508			1.12246
$713$		−	34.5647i		−	1.29446i
$714$	5.91188			0.221247
$715$	23.4264	+	1.89372i	0.876098	+	0.0708211i
$716$	5.37601			0.200911
$717$		−	15.9839i		−	0.596929i
$718$	38.5575			1.43895
$719$	−34.1137			−1.27223			−0.636113	−	0.771596i	$-0.719459\pi$
$719$	−34.1137			−1.27223			−0.636113	+	0.771596i	$0.719459\pi$
$720$		−	4.29588i		−	0.160098i
$721$		−	1.48149i		−	0.0551735i
$722$		−	13.1459i		−	0.489241i
$723$			21.6635i			0.805674i
$724$	5.78000			0.214812
$725$	−24.5857			−0.913090
$726$		−	16.5919i		−	0.615783i
$727$	−0.599191			−0.0222228			−0.0111114	−	0.999938i	$-0.503537\pi$
$727$	−0.599191			−0.0222228			−0.0111114	+	0.999938i	$0.503537\pi$
$728$	−10.8275	−	0.875261i	−0.401293	−	0.0324393i
$729$	1.00000			0.0370370
$730$		−	7.66457i		−	0.283678i
$731$	44.1514			1.63300
$732$	3.74309			0.138348
$733$		−	41.7125i		−	1.54069i	−0.637629	−	0.770344i	$-0.720085\pi$
$733$		−	41.7125i		−	1.54069i	0.637629	−	0.770344i	$-0.279915\pi$
$734$			7.63897i			0.281959i
$735$			1.33457i			0.0492265i
$736$			11.4741i			0.422939i
$737$	−26.8985			−0.990819
$738$	−4.77063			−0.175609
$739$			19.4446i			0.715280i	0.933860	+	0.357640i	$0.116419\pi$
$739$			19.4446i			0.715280i	−0.933860	+	0.357640i	$0.883581\pi$
$740$	−4.36183			−0.160344
$741$	−0.862462	+	10.6691i	−0.0316833	+	0.391941i
$742$	−13.8568			−0.508699
$743$			9.44721i			0.346584i	0.984870	+	0.173292i	$0.0554405\pi$
$743$			9.44721i			0.346584i	−0.984870	+	0.173292i	$0.944559\pi$
$744$	−16.9858			−0.622730
$745$	15.5366			0.569216
$746$		−	9.49361i		−	0.347586i
$747$		−	5.27206i		−	0.192894i
$748$		−	7.48617i		−	0.273722i
$749$		−	4.49291i		−	0.164167i
$750$	−14.1553			−0.516879
$751$	−8.91366			−0.325264			−0.162632	−	0.986687i	$-0.551998\pi$
$751$	−8.91366			−0.325264			−0.162632	+	0.986687i	$0.551998\pi$
$752$		−	16.9703i		−	0.618843i
$753$	3.87496			0.141212
$754$	−2.86354	+	35.4237i	−0.104284	+	1.29005i
$755$	20.7358			0.754651
$756$			0.334573i			0.0121683i
$757$	−43.8416			−1.59345			−0.796726	−	0.604341i	$-0.793437\pi$
$757$	−43.8416			−1.59345			−0.796726	+	0.604341i	$0.793437\pi$
$758$	−15.4862			−0.562483
$759$		−	29.9449i		−	1.08693i
$760$		−	11.9367i		−	0.432990i
$761$		−	11.2283i		−	0.407025i	−0.979072	−	0.203513i	$-0.934764\pi$
$761$		−	11.2283i		−	0.407025i	0.979072	−	0.203513i	$-0.0652358\pi$
$762$		−	21.1802i		−	0.767278i
$763$	−7.83120			−0.283509
$764$	−1.56217			−0.0565173
$765$			6.11372i			0.221042i
$766$	−19.7309			−0.712905
$767$	−2.99326	+	37.0283i	−0.108080	+	1.33702i
$768$	7.79251			0.281188
$769$		−	13.1860i		−	0.475499i	−0.971327	−	0.237749i	$-0.923590\pi$
$769$		−	13.1860i		−	0.475499i	0.971327	−	0.237749i	$-0.0764097\pi$
$770$	−8.41223			−0.303156
$771$	−16.8938			−0.608416
$772$			0.893024i			0.0321406i
$773$		−	25.6094i		−	0.921105i	−0.887632	−	0.460552i	$-0.847651\pi$
$773$		−	25.6094i		−	0.921105i	0.887632	−	0.460552i	$-0.152349\pi$
$774$		−	12.4378i		−	0.447069i
$775$			18.1479i			0.651891i
$776$	54.8899			1.97043
$777$	−9.76869			−0.350450
$778$			9.69760i			0.347676i
$779$	−10.9745			−0.393202
$780$	0.129718	−	1.60469i	0.00464465	−	0.0574570i
$781$	23.8568			0.853663
$782$			36.2446i			1.29610i
$783$	7.63789			0.272956
$784$	3.21892			0.114961

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)

\(f(q)\)	\(=\)	\(q-1.29051i q^{2} -1.00000 q^{3} +0.334573 q^{4} +1.33457i q^{5} +1.29051i q^{6} -1.00000i q^{7} -3.01280i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.29051i q^{2} -1.00000 q^{3} +0.334573 q^{4} +1.33457i q^{5} +1.29051i q^{6} -1.00000i q^{7} -3.01280i q^{8} +1.00000 q^{9} +1.72229 q^{10} -4.88434i q^{11} -0.334573 q^{12} +(3.59383 + 0.290514i) q^{13} -1.29051 q^{14} -1.33457i q^{15} -3.21892 q^{16} +4.58103 q^{17} -1.29051i q^{18} -2.96874i q^{19} +0.446512i q^{20} +1.00000i q^{21} -6.30331 q^{22} -6.13080 q^{23} +3.01280i q^{24} +3.21892 q^{25} +(0.374913 - 4.63789i) q^{26} -1.00000 q^{27} -0.334573i q^{28} -7.63789 q^{29} -1.72229 q^{30} +5.63789i q^{31} -1.87154i q^{32} +4.88434i q^{33} -5.91188i q^{34} +1.33457 q^{35} +0.334573 q^{36} +9.76869i q^{37} -3.83120 q^{38} +(-3.59383 - 0.290514i) q^{39} +4.02080 q^{40} -3.69669i q^{41} +1.29051 q^{42} +9.63789 q^{43} -1.63417i q^{44} +1.33457i q^{45} +7.91188i q^{46} +5.27206i q^{47} +3.21892 q^{48} -1.00000 q^{49} -4.15406i q^{50} -4.58103 q^{51} +(1.20240 + 0.0971982i) q^{52} +10.7374 q^{53} +1.29051i q^{54} +6.51851 q^{55} -3.01280 q^{56} +2.96874i q^{57} +9.85680i q^{58} +10.3033i q^{59} -0.446512i q^{60} -11.1877 q^{61} +7.27577 q^{62} -1.00000i q^{63} -8.85308 q^{64} +(-0.387712 + 4.79623i) q^{65} +6.30331 q^{66} -5.50709i q^{67} +1.53269 q^{68} +6.13080 q^{69} -1.72229i q^{70} +4.88434i q^{71} -3.01280i q^{72} -4.45023i q^{73} +12.6066 q^{74} -3.21892 q^{75} -0.993260i q^{76} -4.88434 q^{77} +(-0.374913 + 4.63789i) q^{78} -3.54977 q^{79} -4.29588i q^{80} +1.00000 q^{81} -4.77063 q^{82} -5.27206i q^{83} +0.334573i q^{84} +6.11372i q^{85} -12.4378i q^{86} +7.63789 q^{87} -14.7155 q^{88} +9.94120i q^{89} +1.72229 q^{90} +(0.290514 - 3.59383i) q^{91} -2.05120 q^{92} -5.63789i q^{93} +6.80366 q^{94} +3.96200 q^{95} +1.87154i q^{96} +18.2189i q^{97} +1.29051i q^{98} -4.88434i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9	\( 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9} + 4 q^{10} + 14 q^{12} - 6 q^{13} - 2 q^{14} + 34 q^{16} + 20 q^{17} - 24 q^{22} - 6 q^{23} - 34 q^{25} + 28 q^{26} - 8 q^{27} - 18 q^{29} - 4 q^{30} - 6 q^{35} - 14 q^{36} + 36 q^{38} + 6 q^{39} - 8 q^{40} + 2 q^{42} + 34 q^{43} - 34 q^{48} - 8 q^{49} - 20 q^{51} + 18 q^{52} - 10 q^{53} + 16 q^{55} - 6 q^{56} - 20 q^{61} - 28 q^{62} - 18 q^{64} - 10 q^{65} + 24 q^{66} - 24 q^{68} + 6 q^{69} + 48 q^{74} + 34 q^{75} + 4 q^{77} - 28 q^{78} - 2 q^{79} + 8 q^{81} - 48 q^{82} + 18 q^{87} - 8 q^{88} + 4 q^{90} - 6 q^{91} + 56 q^{92} - 72 q^{94} - 78 q^{95}+O(q^{100}) \) 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9 + 4 * q^10 + 14 * q^12 - 6 * q^13 - 2 * q^14 + 34 * q^16 + 20 * q^17 - 24 * q^22 - 6 * q^23 - 34 * q^25 + 28 * q^26 - 8 * q^27 - 18 * q^29 - 4 * q^30 - 6 * q^35 - 14 * q^36 + 36 * q^38 + 6 * q^39 - 8 * q^40 + 2 * q^42 + 34 * q^43 - 34 * q^48 - 8 * q^49 - 20 * q^51 + 18 * q^52 - 10 * q^53 + 16 * q^55 - 6 * q^56 - 20 * q^61 - 28 * q^62 - 18 * q^64 - 10 * q^65 + 24 * q^66 - 24 * q^68 + 6 * q^69 + 48 * q^74 + 34 * q^75 + 4 * q^77 - 28 * q^78 - 2 * q^79 + 8 * q^81 - 48 * q^82 + 18 * q^87 - 8 * q^88 + 4 * q^90 - 6 * q^91 + 56 * q^92 - 72 * q^94 - 78 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more