LMFDB - Embedded newform 273.2.c.b.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:	$6$
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			64.3
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	273.64
Dual form			273.2.c.b.64.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311108i q^{2} +1.00000 q^{3} +1.90321 q^{4} +1.52543i q^{5} -0.311108i q^{6} -1.00000i q^{7} -1.21432i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.311108i q^{2} +1.00000 q^{3} +1.90321 q^{4} +1.52543i q^{5} -0.311108i q^{6} -1.00000i q^{7} -1.21432i q^{8} +1.00000 q^{9} +0.474572 q^{10} -1.09679i q^{11} +1.90321 q^{12} +(-0.311108 + 3.59210i) q^{13} -0.311108 q^{14} +1.52543i q^{15} +3.42864 q^{16} -4.42864 q^{17} -0.311108i q^{18} -1.80642i q^{19} +2.90321i q^{20} -1.00000i q^{21} -0.341219 q^{22} -3.80642 q^{23} -1.21432i q^{24} +2.67307 q^{25} +(1.11753 + 0.0967881i) q^{26} +1.00000 q^{27} -1.90321i q^{28} -0.755569 q^{29} +0.474572 q^{30} -4.85728i q^{31} -3.49532i q^{32} -1.09679i q^{33} +1.37778i q^{34} +1.52543 q^{35} +1.90321 q^{36} -5.80642i q^{37} -0.561993 q^{38} +(-0.311108 + 3.59210i) q^{39} +1.85236 q^{40} +11.3319i q^{41} -0.311108 q^{42} -5.24443 q^{43} -2.08742i q^{44} +1.52543i q^{45} +1.18421i q^{46} -2.28100i q^{47} +3.42864 q^{48} -1.00000 q^{49} -0.831613i q^{50} -4.42864 q^{51} +(-0.592104 + 6.83654i) q^{52} -6.00000 q^{53} -0.311108i q^{54} +1.67307 q^{55} -1.21432 q^{56} -1.80642i q^{57} +0.235063i q^{58} -0.474572i q^{59} +2.90321i q^{60} -13.0923 q^{61} -1.51114 q^{62} -1.00000i q^{63} +5.76986 q^{64} +(-5.47949 - 0.474572i) q^{65} -0.341219 q^{66} +9.80642i q^{67} -8.42864 q^{68} -3.80642 q^{69} -0.474572i q^{70} -13.0049i q^{71} -1.21432i q^{72} +3.47949i q^{73} -1.80642 q^{74} +2.67307 q^{75} -3.43801i q^{76} -1.09679 q^{77} +(1.11753 + 0.0967881i) q^{78} -5.37778 q^{79} +5.23014i q^{80} +1.00000 q^{81} +3.52543 q^{82} +13.8938i q^{83} -1.90321i q^{84} -6.75557i q^{85} +1.63158i q^{86} -0.755569 q^{87} -1.33185 q^{88} -13.1383i q^{89} +0.474572 q^{90} +(3.59210 + 0.311108i) q^{91} -7.24443 q^{92} -4.85728i q^{93} -0.709636 q^{94} +2.75557 q^{95} -3.49532i q^{96} +4.42864i q^{97} +0.311108i q^{98} -1.09679i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	0.311108i		−	0.219986i	−0.993932	−	0.109993i	$-0.964917\pi$
$2$		−	0.311108i		−	0.219986i	0.993932	−	0.109993i	$-0.0350829\pi$
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	1.90321			0.951606
$5$			1.52543i			0.682192i	0.940028	+	0.341096i	$0.110798\pi$
$5$			1.52543i			0.682192i	−0.940028	+	0.341096i	$0.889202\pi$
$6$		−	0.311108i		−	0.127009i
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		−	1.21432i		−	0.429327i
$9$	1.00000			0.333333
$10$	0.474572			0.150073
$11$		−	1.09679i		−	0.330694i	−0.986235	−	0.165347i	$-0.947126\pi$
$11$		−	1.09679i		−	0.330694i	0.986235	−	0.165347i	$-0.0528744\pi$
$12$	1.90321			0.549410
$13$	−0.311108	+	3.59210i	−0.0862858	+	0.996270i
$13$	−0.311108	+	3.59210i	−0.0862858	+	0.996270i
$14$	−0.311108			−0.0831471
$15$			1.52543i			0.393864i
$16$	3.42864			0.857160
$17$	−4.42864			−1.07410			−0.537051	−	0.843550i	$-0.680462\pi$
$17$	−4.42864			−1.07410			−0.537051	+	0.843550i	$0.680462\pi$
$18$		−	0.311108i		−	0.0733288i
$19$		−	1.80642i		−	0.414422i	−0.978296	−	0.207211i	$-0.933561\pi$
$19$		−	1.80642i		−	0.414422i	0.978296	−	0.207211i	$-0.0664387\pi$
$20$			2.90321i			0.649178i
$21$		−	1.00000i		−	0.218218i
$22$	−0.341219			−0.0727482
$23$	−3.80642			−0.793694			−0.396847	−	0.917885i	$-0.629896\pi$
$23$	−3.80642			−0.793694			−0.396847	+	0.917885i	$0.629896\pi$
$24$		−	1.21432i		−	0.247872i
$25$	2.67307			0.534614
$26$	1.11753	+	0.0967881i	0.219166	+	0.0189817i
$27$	1.00000			0.192450
$28$		−	1.90321i		−	0.359673i
$29$	−0.755569			−0.140306			−0.0701528	−	0.997536i	$-0.522349\pi$
$29$	−0.755569			−0.140306			−0.0701528	+	0.997536i	$0.522349\pi$
$30$	0.474572			0.0866447
$31$		−	4.85728i		−	0.872393i	−0.899851	−	0.436197i	$-0.856325\pi$
$31$		−	4.85728i		−	0.872393i	0.899851	−	0.436197i	$-0.143675\pi$
$32$		−	3.49532i		−	0.617890i
$33$		−	1.09679i		−	0.190926i
$34$			1.37778i			0.236288i
$35$	1.52543			0.257844
$36$	1.90321			0.317202
$37$		−	5.80642i		−	0.954570i	−0.878749	−	0.477285i	$-0.841621\pi$
$37$		−	5.80642i		−	0.954570i	0.878749	−	0.477285i	$-0.158379\pi$
$38$	−0.561993			−0.0911672
$39$	−0.311108	+	3.59210i	−0.0498171	+	0.575197i
$40$	1.85236			0.292883
$41$			11.3319i			1.76974i	0.465840	+	0.884869i	$0.345752\pi$
$41$			11.3319i			1.76974i	−0.465840	+	0.884869i	$0.654248\pi$
$42$	−0.311108			−0.0480050
$43$	−5.24443			−0.799768			−0.399884	−	0.916566i	$-0.630950\pi$
$43$	−5.24443			−0.799768			−0.399884	+	0.916566i	$0.630950\pi$
$44$		−	2.08742i		−	0.314690i
$45$			1.52543i			0.227397i
$46$			1.18421i			0.174602i
$47$		−	2.28100i		−	0.332718i	−0.986065	−	0.166359i	$-0.946799\pi$
$47$		−	2.28100i		−	0.332718i	0.986065	−	0.166359i	$-0.0532010\pi$
$48$	3.42864			0.494881
$49$	−1.00000			−0.142857
$50$		−	0.831613i		−	0.117608i
$51$	−4.42864			−0.620134
$52$	−0.592104	+	6.83654i	−0.0821101	+	0.948057i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		−	0.311108i		−	0.0423364i
$55$	1.67307			0.225597
$56$	−1.21432			−0.162270
$57$		−	1.80642i		−	0.239267i
$58$			0.235063i			0.0308653i
$59$		−	0.474572i		−	0.0617841i	−0.999523	−	0.0308920i	$-0.990165\pi$
$59$		−	0.474572i		−	0.0617841i	0.999523	−	0.0308920i	$-0.00983480\pi$
$60$			2.90321i			0.374803i
$61$	−13.0923			−1.67630			−0.838151	−	0.545438i	$-0.816363\pi$
$61$	−13.0923			−1.67630			−0.838151	+	0.545438i	$0.816363\pi$
$62$	−1.51114			−0.191915
$63$		−	1.00000i		−	0.125988i
$64$	5.76986			0.721232
$65$	−5.47949	−	0.474572i	−0.679648	−	0.0588635i
$66$	−0.341219			−0.0420012
$67$			9.80642i			1.19805i	0.800732	+	0.599023i	$0.204444\pi$
$67$			9.80642i			1.19805i	−0.800732	+	0.599023i	$0.795556\pi$
$68$	−8.42864			−1.02212
$69$	−3.80642			−0.458240
$70$		−	0.474572i		−	0.0567223i
$71$		−	13.0049i		−	1.54340i	−0.635987	−	0.771700i	$-0.719407\pi$
$71$		−	13.0049i		−	1.54340i	0.635987	−	0.771700i	$-0.280593\pi$
$72$		−	1.21432i		−	0.143109i
$73$			3.47949i			0.407244i	0.979050	+	0.203622i	$0.0652714\pi$
$73$			3.47949i			0.407244i	−0.979050	+	0.203622i	$0.934729\pi$
$74$	−1.80642			−0.209993
$75$	2.67307			0.308660
$76$		−	3.43801i		−	0.394366i
$77$	−1.09679			−0.124991
$78$	1.11753	+	0.0967881i	0.126536	+	0.0109591i
$79$	−5.37778			−0.605048			−0.302524	−	0.953142i	$-0.597829\pi$
$79$	−5.37778			−0.605048			−0.302524	+	0.953142i	$0.597829\pi$
$80$			5.23014i			0.584748i
$81$	1.00000			0.111111
$82$	3.52543			0.389318
$83$			13.8938i			1.52505i	0.646960	+	0.762524i	$0.276040\pi$
$83$			13.8938i			1.52505i	−0.646960	+	0.762524i	$0.723960\pi$
$84$		−	1.90321i		−	0.207657i
$85$		−	6.75557i		−	0.732744i
$86$			1.63158i			0.175938i
$87$	−0.755569			−0.0810055
$88$	−1.33185			−0.141976
$89$		−	13.1383i		−	1.39265i	−0.717724	−	0.696327i	$-0.754816\pi$
$89$		−	13.1383i		−	1.39265i	0.717724	−	0.696327i	$-0.245184\pi$
$90$	0.474572			0.0500243
$91$	3.59210	+	0.311108i	0.376555	+	0.0326130i
$92$	−7.24443			−0.755284
$93$		−	4.85728i		−	0.503676i
$94$	−0.709636			−0.0731933
$95$	2.75557			0.282715
$96$		−	3.49532i		−	0.356739i
$97$			4.42864i			0.449660i	0.974398	+	0.224830i	$0.0721827\pi$
$97$			4.42864i			0.449660i	−0.974398	+	0.224830i	$0.927817\pi$
$98$			0.311108i			0.0314266i
$99$		−	1.09679i		−	0.110231i
$100$	5.08742			0.508742
$101$	9.28592			0.923983			0.461992	−	0.886884i	$-0.347135\pi$
$101$	9.28592			0.923983			0.461992	+	0.886884i	$0.347135\pi$
$102$			1.37778i			0.136421i
$103$	0.815792			0.0803824			0.0401912	−	0.999192i	$-0.487203\pi$
$103$	0.815792			0.0803824			0.0401912	+	0.999192i	$0.487203\pi$
$104$	4.36196	+	0.377784i	0.427726	+	0.0370448i
$105$	1.52543			0.148866
$106$			1.86665i			0.181305i
$107$	6.56199			0.634372			0.317186	−	0.948363i	$-0.397262\pi$
$107$	6.56199			0.634372			0.317186	+	0.948363i	$0.397262\pi$
$108$	1.90321			0.183137
$109$			12.2953i			1.17767i	0.808252	+	0.588837i	$0.200414\pi$
$109$			12.2953i			1.17767i	−0.808252	+	0.588837i	$0.799586\pi$
$110$		−	0.520505i		−	0.0496282i
$111$		−	5.80642i		−	0.551121i
$112$		−	3.42864i		−	0.323976i
$113$	3.24443			0.305210			0.152605	−	0.988287i	$-0.451234\pi$
$113$	3.24443			0.305210			0.152605	+	0.988287i	$0.451234\pi$
$114$	−0.561993			−0.0526354
$115$		−	5.80642i		−	0.541452i
$116$	−1.43801			−0.133516
$117$	−0.311108	+	3.59210i	−0.0287619	+	0.332090i
$118$	−0.147643			−0.0135917
$119$			4.42864i			0.405973i
$120$	1.85236			0.169096
$121$	9.79706			0.890641
$122$			4.07313i			0.368764i
$123$			11.3319i			1.02176i
$124$		−	9.24443i		−	0.830174i
$125$			11.7047i			1.04690i
$126$	−0.311108			−0.0277157
$127$	2.62222			0.232684			0.116342	−	0.993209i	$-0.462883\pi$
$127$	2.62222			0.232684			0.116342	+	0.993209i	$0.462883\pi$
$128$		−	8.78568i		−	0.776552i
$129$	−5.24443			−0.461746
$130$	−0.147643	+	1.70471i	−0.0129492	+	0.149513i
$131$	18.3684			1.60486			0.802428	−	0.596749i	$-0.203541\pi$
$131$	18.3684			1.60486			0.802428	+	0.596749i	$0.203541\pi$
$132$		−	2.08742i		−	0.181687i
$133$	−1.80642			−0.156637
$134$	3.05086			0.263554
$135$			1.52543i			0.131288i
$136$			5.37778i			0.461141i
$137$		−	16.3827i		−	1.39967i	−0.714305	−	0.699835i	$-0.753257\pi$
$137$		−	16.3827i		−	1.39967i	0.714305	−	0.699835i	$-0.246743\pi$
$138$			1.18421i			0.100806i
$139$	3.18421			0.270081			0.135041	−	0.990840i	$-0.456884\pi$
$139$	3.18421			0.270081			0.135041	+	0.990840i	$0.456884\pi$
$140$	2.90321			0.245366
$141$		−	2.28100i		−	0.192095i
$142$	−4.04593			−0.339527
$143$	3.93978	+	0.341219i	0.329461	+	0.0285342i
$144$	3.42864			0.285720
$145$		−	1.15257i		−	0.0957153i
$146$	1.08250			0.0895882
$147$	−1.00000			−0.0824786
$148$		−	11.0509i		−	0.908375i
$149$		−	16.0874i		−	1.31793i	−0.752172	−	0.658966i	$-0.770994\pi$
$149$		−	16.0874i		−	1.31793i	0.752172	−	0.658966i	$-0.229006\pi$
$150$		−	0.831613i		−	0.0679009i
$151$			1.51114i			0.122975i	0.998108	+	0.0614873i	$0.0195844\pi$
$151$			1.51114i			0.122975i	−0.998108	+	0.0614873i	$0.980416\pi$
$152$	−2.19358			−0.177923
$153$	−4.42864			−0.358034
$154$			0.341219i			0.0274962i
$155$	7.40943			0.595140
$156$	−0.592104	+	6.83654i	−0.0474063	+	0.547361i
$157$	2.85728			0.228036			0.114018	−	0.993479i	$-0.463628\pi$
$157$	2.85728			0.228036			0.114018	+	0.993479i	$0.463628\pi$
$158$			1.67307i			0.133102i
$159$	−6.00000			−0.475831
$160$	5.33185			0.421520
$161$			3.80642i			0.299988i
$162$		−	0.311108i		−	0.0244429i
$163$		−	13.8064i		−	1.08140i	−0.841215	−	0.540701i	$-0.818159\pi$
$163$		−	13.8064i		−	1.08140i	0.841215	−	0.540701i	$-0.181841\pi$
$164$			21.5669i			1.68409i
$165$	1.67307			0.130248
$166$	4.32248			0.335490
$167$		−	0.769859i		−	0.0595735i	−0.999556	−	0.0297867i	$-0.990517\pi$
$167$		−	0.769859i		−	0.0595735i	0.999556	−	0.0297867i	$-0.00948281\pi$
$168$	−1.21432			−0.0936868
$169$	−12.8064	−	2.23506i	−0.985110	−	0.171928i
$170$	−2.10171			−0.161194
$171$		−	1.80642i		−	0.138141i
$172$	−9.98126			−0.761064
$173$	5.67307			0.431316			0.215658	−	0.976469i	$-0.430810\pi$
$173$	5.67307			0.431316			0.215658	+	0.976469i	$0.430810\pi$
$174$			0.235063i			0.0178201i
$175$		−	2.67307i		−	0.202065i
$176$		−	3.76049i		−	0.283458i
$177$		−	0.474572i		−	0.0356710i
$178$	−4.08742			−0.306365
$179$	21.9081			1.63749			0.818745	−	0.574157i	$-0.194670\pi$
$179$	21.9081			1.63749			0.818745	+	0.574157i	$0.194670\pi$
$180$			2.90321i			0.216393i
$181$	0.488863			0.0363369			0.0181684	−	0.999835i	$-0.494216\pi$
$181$	0.488863			0.0363369			0.0181684	+	0.999835i	$0.494216\pi$
$182$	0.0967881	−	1.11753i	0.00717441	−	0.0828370i
$183$	−13.0923			−0.967814
$184$			4.62222i			0.340754i
$185$	8.85728			0.651200
$186$	−1.51114			−0.110802
$187$			4.85728i			0.355199i
$188$		−	4.34122i		−	0.316616i
$189$		−	1.00000i		−	0.0727393i
$190$		−	0.857279i		−	0.0621936i
$191$	25.1338			1.81862			0.909310	−	0.416119i	$-0.136610\pi$
$191$	25.1338			1.81862			0.909310	+	0.416119i	$0.136610\pi$
$192$	5.76986			0.416404
$193$			6.66370i			0.479664i	0.970814	+	0.239832i	$0.0770923\pi$
$193$			6.66370i			0.479664i	−0.970814	+	0.239832i	$0.922908\pi$
$194$	1.37778			0.0989192
$195$	−5.47949	−	0.474572i	−0.392395	−	0.0339848i
$196$	−1.90321			−0.135944
$197$			18.1891i			1.29592i	0.761674	+	0.647961i	$0.224378\pi$
$197$			18.1891i			1.29592i	−0.761674	+	0.647961i	$0.775622\pi$
$198$	−0.341219			−0.0242494
$199$	−14.1017			−0.999644			−0.499822	−	0.866128i	$-0.666601\pi$
$199$	−14.1017			−0.999644			−0.499822	+	0.866128i	$0.666601\pi$
$200$		−	3.24596i		−	0.229524i
$201$			9.80642i			0.691692i
$202$		−	2.88892i		−	0.203264i
$203$			0.755569i			0.0530305i
$204$	−8.42864			−0.590123
$205$	−17.2859			−1.20730
$206$		−	0.253799i		−	0.0176830i
$207$	−3.80642			−0.264565
$208$	−1.06668	+	12.3160i	−0.0739607	+	0.853963i
$209$	−1.98126			−0.137047
$210$		−	0.474572i		−	0.0327486i
$211$	8.13335			0.559923			0.279962	−	0.960011i	$-0.409678\pi$
$211$	8.13335			0.559923			0.279962	+	0.960011i	$0.409678\pi$
$212$	−11.4193			−0.784279
$213$		−	13.0049i		−	0.891083i
$214$		−	2.04149i		−	0.139553i
$215$		−	8.00000i		−	0.545595i
$216$		−	1.21432i		−	0.0826240i
$217$	−4.85728			−0.329734
$218$	3.82516			0.259073
$219$			3.47949i			0.235122i
$220$	3.18421			0.214679
$221$	1.37778	−	15.9081i	0.0926798	−	1.07010i
$222$	−1.80642			−0.121239
$223$			9.53972i			0.638827i	0.947615	+	0.319413i	$0.103486\pi$
$223$			9.53972i			0.638827i	−0.947615	+	0.319413i	$0.896514\pi$
$224$	−3.49532			−0.233541
$225$	2.67307			0.178205
$226$		−	1.00937i		−	0.0671422i
$227$		−	1.81087i		−	0.120192i	−0.998193	−	0.0600958i	$-0.980859\pi$
$227$		−	1.81087i		−	0.120192i	0.998193	−	0.0600958i	$-0.0191406\pi$
$228$		−	3.43801i		−	0.227688i
$229$			13.9684i			0.923055i	0.887126	+	0.461528i	$0.152698\pi$
$229$			13.9684i			0.923055i	−0.887126	+	0.461528i	$0.847302\pi$
$230$	−1.80642			−0.119112
$231$	−1.09679			−0.0721634
$232$			0.917502i			0.0602370i
$233$	−7.51114			−0.492071			−0.246035	−	0.969261i	$-0.579128\pi$
$233$	−7.51114			−0.492071			−0.246035	+	0.969261i	$0.579128\pi$
$234$	1.11753	+	0.0967881i	0.0730553	+	0.00632723i
$235$	3.47949			0.226977
$236$		−	0.903212i		−	0.0587941i
$237$	−5.37778			−0.349325
$238$	1.37778			0.0893085
$239$			21.9541i			1.42009i	0.704156	+	0.710045i	$0.251326\pi$
$239$			21.9541i			1.42009i	−0.704156	+	0.710045i	$0.748674\pi$
$240$			5.23014i			0.337604i
$241$		−	25.8479i		−	1.66501i	−0.554017	−	0.832505i	$-0.686906\pi$
$241$		−	25.8479i		−	1.66501i	0.554017	−	0.832505i	$-0.313094\pi$
$242$		−	3.04794i		−	0.195929i
$243$	1.00000			0.0641500
$244$	−24.9175			−1.59518
$245$		−	1.52543i		−	0.0974560i
$246$	3.52543			0.224773
$247$	6.48886	+	0.561993i	0.412876	+	0.0357587i
$248$	−5.89829			−0.374542
$249$			13.8938i			0.880487i
$250$	3.64143			0.230304
$251$	23.2257			1.46599			0.732996	−	0.680232i	$-0.238121\pi$
$251$	23.2257			1.46599			0.732996	+	0.680232i	$0.238121\pi$
$252$		−	1.90321i		−	0.119891i
$253$			4.17484i			0.262470i
$254$		−	0.815792i		−	0.0511873i
$255$		−	6.75557i		−	0.423050i
$256$	8.80642			0.550401
$257$	−27.7748			−1.73254			−0.866272	−	0.499573i	$-0.833490\pi$
$257$	−27.7748			−1.73254			−0.866272	+	0.499573i	$0.833490\pi$
$258$			1.63158i			0.101578i
$259$	−5.80642			−0.360794
$260$	−10.4286	−	0.903212i	−0.646757	−	0.0560148i
$261$	−0.755569			−0.0467685
$262$		−	5.71456i		−	0.353047i
$263$	−6.68244			−0.412057			−0.206028	−	0.978546i	$-0.566054\pi$
$263$	−6.68244			−0.412057			−0.206028	+	0.978546i	$0.566054\pi$
$264$	−1.33185			−0.0819698
$265$		−	9.15257i		−	0.562238i
$266$			0.561993i			0.0344580i
$267$		−	13.1383i		−	0.804049i
$268$			18.6637i			1.14007i
$269$	−22.1432			−1.35009			−0.675047	−	0.737774i	$-0.735877\pi$
$269$	−22.1432			−1.35009			−0.675047	+	0.737774i	$0.735877\pi$
$270$	0.474572			0.0288816
$271$			2.19358i			0.133250i	0.997778	+	0.0666251i	$0.0212232\pi$
$271$			2.19358i			0.133250i	−0.997778	+	0.0666251i	$0.978777\pi$
$272$	−15.1842			−0.920678
$273$	3.59210	+	0.311108i	0.217404	+	0.0188291i
$274$	−5.09679			−0.307908
$275$		−	2.93179i		−	0.176794i
$276$	−7.24443			−0.436064
$277$	−27.3274			−1.64194			−0.820972	−	0.570968i	$-0.806568\pi$
$277$	−27.3274			−1.64194			−0.820972	+	0.570968i	$0.806568\pi$
$278$		−	0.990632i		−	0.0594142i
$279$		−	4.85728i		−	0.290798i
$280$		−	1.85236i		−	0.110699i
$281$			22.8430i			1.36270i	0.731958	+	0.681349i	$0.238606\pi$
$281$			22.8430i			1.36270i	−0.731958	+	0.681349i	$0.761394\pi$
$282$	−0.709636			−0.0422582
$283$	21.1240			1.25569			0.627845	−	0.778338i	$-0.283937\pi$
$283$	21.1240			1.25569			0.627845	+	0.778338i	$0.283937\pi$
$284$		−	24.7511i		−	1.46871i
$285$	2.75557			0.163226
$286$	0.106156	−	1.22570i	0.00627714	−	0.0724769i
$287$	11.3319			0.668898
$288$		−	3.49532i		−	0.205963i
$289$	2.61285			0.153697
$290$	−0.358572			−0.0210561
$291$			4.42864i			0.259611i
$292$			6.62222i			0.387536i
$293$			11.6271i			0.679265i	0.940558	+	0.339632i	$0.110303\pi$
$293$			11.6271i			0.679265i	−0.940558	+	0.339632i	$0.889697\pi$
$294$			0.311108i			0.0181442i
$295$	0.723926			0.0421486
$296$	−7.05086			−0.409823
$297$		−	1.09679i		−	0.0636421i
$298$	−5.00492			−0.289927
$299$	1.18421	−	13.6731i	0.0684845	−	0.790734i
$300$	5.08742			0.293722
$301$			5.24443i			0.302284i
$302$	0.470127			0.0270528
$303$	9.28592			0.533462
$304$		−	6.19358i		−	0.355226i
$305$		−	19.9714i		−	1.14356i
$306$			1.37778i			0.0787627i
$307$		−	32.8573i		−	1.87526i	−0.347629	−	0.937632i	$-0.613013\pi$
$307$		−	32.8573i		−	1.87526i	0.347629	−	0.937632i	$-0.386987\pi$
$308$	−2.08742			−0.118942
$309$	0.815792			0.0464088
$310$		−	2.30513i		−	0.130923i
$311$	18.3684			1.04158			0.520789	−	0.853686i	$-0.325638\pi$
$311$	18.3684			1.04158			0.520789	+	0.853686i	$0.325638\pi$
$312$	4.36196	+	0.377784i	0.246948	+	0.0213878i
$313$	7.11108			0.401942			0.200971	−	0.979597i	$-0.435590\pi$
$313$	7.11108			0.401942			0.200971	+	0.979597i	$0.435590\pi$
$314$		−	0.888922i		−	0.0501648i
$315$	1.52543			0.0859481
$316$	−10.2351			−0.575767
$317$			8.47457i			0.475979i	0.971268	+	0.237990i	$0.0764885\pi$
$317$			8.47457i			0.475979i	−0.971268	+	0.237990i	$0.923512\pi$
$318$			1.86665i			0.104676i
$319$			0.828699i			0.0463982i
$320$			8.80150i			0.492019i
$321$	6.56199			0.366255
$322$	1.18421			0.0659933
$323$			8.00000i			0.445132i
$324$	1.90321			0.105734
$325$	−0.831613	+	9.60195i	−0.0461296	+	0.532620i
$326$	−4.29529			−0.237894
$327$			12.2953i			0.679931i
$328$	13.7605			0.759796
$329$	−2.28100			−0.125755
$330$		−	0.520505i		−	0.0286529i
$331$			28.2034i			1.55020i	0.631838	+	0.775100i	$0.282301\pi$
$331$			28.2034i			1.55020i	−0.631838	+	0.775100i	$0.717699\pi$
$332$			26.4429i			1.45124i
$333$		−	5.80642i		−	0.318190i
$334$	−0.239509			−0.0131054
$335$	−14.9590			−0.817297
$336$		−	3.42864i		−	0.187048i
$337$	24.5303			1.33625			0.668127	−	0.744048i	$-0.267096\pi$
$337$	24.5303			1.33625			0.668127	+	0.744048i	$0.267096\pi$
$338$	−0.695346	+	3.98418i	−0.0378218	+	0.216711i
$339$	3.24443			0.176213
$340$		−	12.8573i		−	0.697284i
$341$	−5.32741			−0.288495
$342$	−0.561993			−0.0303891
$343$			1.00000i			0.0539949i
$344$			6.36842i			0.343362i
$345$		−	5.80642i		−	0.312607i
$346$		−	1.76494i		−	0.0948836i
$347$	−9.52098			−0.511113			−0.255557	−	0.966794i	$-0.582259\pi$
$347$	−9.52098			−0.511113			−0.255557	+	0.966794i	$0.582259\pi$
$348$	−1.43801			−0.0770853
$349$		−	9.37778i		−	0.501981i	−0.967990	−	0.250991i	$-0.919244\pi$
$349$		−	9.37778i		−	0.501981i	0.967990	−	0.250991i	$-0.0807563\pi$
$350$	−0.831613			−0.0444516
$351$	−0.311108	+	3.59210i	−0.0166057	+	0.191732i
$352$	−3.83362			−0.204333
$353$		−	11.5067i		−	0.612439i	−0.951961	−	0.306220i	$-0.900936\pi$
$353$		−	11.5067i		−	0.612439i	0.951961	−	0.306220i	$-0.0990642\pi$
$354$	−0.147643			−0.00784715
$355$	19.8381			1.05290
$356$		−	25.0049i		−	1.32526i
$357$			4.42864i			0.234388i
$358$		−	6.81579i		−	0.360226i
$359$			15.3733i			0.811374i	0.914012	+	0.405687i	$0.132968\pi$
$359$			15.3733i			0.811374i	−0.914012	+	0.405687i	$0.867032\pi$
$360$	1.85236			0.0976278
$361$	15.7368			0.828254
$362$		−	0.152089i		−	0.00799362i
$363$	9.79706			0.514212
$364$	6.83654	+	0.592104i	0.358332	+	0.0310347i
$365$	−5.30772			−0.277819
$366$			4.07313i			0.212906i
$367$	−24.8988			−1.29971			−0.649853	−	0.760060i	$-0.725169\pi$
$367$	−24.8988			−1.29971			−0.649853	+	0.760060i	$0.725169\pi$
$368$	−13.0509			−0.680323
$369$			11.3319i			0.589913i
$370$		−	2.75557i		−	0.143255i
$371$			6.00000i			0.311504i
$372$		−	9.24443i		−	0.479301i
$373$	17.1842			0.889765			0.444882	−	0.895589i	$-0.353246\pi$
$373$	17.1842			0.889765			0.444882	+	0.895589i	$0.353246\pi$
$374$	1.51114			0.0781391
$375$			11.7047i			0.604429i
$376$	−2.76986			−0.142845
$377$	0.235063	−	2.71408i	0.0121064	−	0.139782i
$378$	−0.311108			−0.0160017
$379$			13.1240i			0.674134i	0.941481	+	0.337067i	$0.109435\pi$
$379$			13.1240i			0.674134i	−0.941481	+	0.337067i	$0.890565\pi$
$380$	5.24443			0.269034
$381$	2.62222			0.134340
$382$		−	7.81933i		−	0.400072i
$383$		−	30.7511i		−	1.57131i	−0.618666	−	0.785654i	$-0.712326\pi$
$383$		−	30.7511i		−	1.57131i	0.618666	−	0.785654i	$-0.287674\pi$
$384$		−	8.78568i		−	0.448342i
$385$		−	1.67307i		−	0.0852676i
$386$	2.07313			0.105520
$387$	−5.24443			−0.266589
$388$			8.42864i			0.427899i
$389$	−14.8573			−0.753294			−0.376647	−	0.926357i	$-0.622923\pi$
$389$	−14.8573			−0.753294			−0.376647	+	0.926357i	$0.622923\pi$
$390$	−0.147643	+	1.70471i	−0.00747620	+	0.0863215i
$391$	16.8573			0.852509
$392$			1.21432i			0.0613324i
$393$	18.3684			0.926564
$394$	5.65878			0.285085
$395$		−	8.20342i		−	0.412759i
$396$		−	2.08742i		−	0.104897i
$397$			5.93978i			0.298109i	0.988829	+	0.149054i	$0.0476230\pi$
$397$			5.93978i			0.298109i	−0.988829	+	0.149054i	$0.952377\pi$
$398$			4.38715i			0.219908i
$399$	−1.80642			−0.0904343
$400$	9.16500			0.458250
$401$			6.87157i			0.343150i	0.985171	+	0.171575i	$0.0548856\pi$
$401$			6.87157i			0.343150i	−0.985171	+	0.171575i	$0.945114\pi$
$402$	3.05086			0.152163
$403$	17.4479	+	1.51114i	0.869139	+	0.0752751i
$404$	17.6731			0.879268
$405$			1.52543i			0.0757991i
$406$	0.235063			0.0116660
$407$	−6.36842			−0.315671
$408$			5.37778i			0.266240i
$409$		−	31.0005i		−	1.53287i	−0.642319	−	0.766437i	$-0.722028\pi$
$409$		−	31.0005i		−	1.53287i	0.642319	−	0.766437i	$-0.277972\pi$
$410$			5.37778i			0.265590i
$411$		−	16.3827i		−	0.808099i
$412$	1.55262			0.0764923
$413$	−0.474572			−0.0233522
$414$			1.18421i			0.0582007i
$415$	−21.1941			−1.04038
$416$	12.5555	+	1.08742i	0.615586	+	0.0533152i
$417$	3.18421			0.155931
$418$			0.616387i			0.0301485i
$419$	5.32741			0.260261			0.130130	−	0.991497i	$-0.458460\pi$
$419$	5.32741			0.260261			0.130130	+	0.991497i	$0.458460\pi$
$420$	2.90321			0.141662
$421$			11.2257i			0.547107i	0.961857	+	0.273553i	$0.0881990\pi$
$421$			11.2257i			0.547107i	−0.961857	+	0.273553i	$0.911801\pi$
$422$		−	2.53035i		−	0.123175i
$423$		−	2.28100i		−	0.110906i
$424$			7.28592i			0.353835i
$425$	−11.8381			−0.574231
$426$	−4.04593			−0.196026
$427$			13.0923i			0.633583i
$428$	12.4889			0.603672
$429$	3.93978	+	0.341219i	0.190214	+	0.0164742i
$430$	−2.48886			−0.120024
$431$		−	15.1985i		−	0.732086i	−0.930598	−	0.366043i	$-0.880712\pi$
$431$		−	15.1985i		−	0.732086i	0.930598	−	0.366043i	$-0.119288\pi$
$432$	3.42864			0.164960
$433$	−8.82564			−0.424133			−0.212067	−	0.977255i	$-0.568019\pi$
$433$	−8.82564			−0.424133			−0.212067	+	0.977255i	$0.568019\pi$
$434$			1.51114i			0.0725369i
$435$		−	1.15257i		−	0.0552613i
$436$			23.4005i			1.12068i
$437$			6.87601i			0.328924i
$438$	1.08250			0.0517238
$439$	−23.1842			−1.10652			−0.553261	−	0.833008i	$-0.686617\pi$
$439$	−23.1842			−1.10652			−0.553261	+	0.833008i	$0.686617\pi$
$440$		−	2.03164i		−	0.0968548i
$441$	−1.00000			−0.0476190
$442$	−4.94914	−	0.428639i	−0.235407	−	0.0203883i
$443$	10.2953			0.489144			0.244572	−	0.969631i	$-0.421353\pi$
$443$	10.2953			0.489144			0.244572	+	0.969631i	$0.421353\pi$
$444$		−	11.0509i		−	0.524450i
$445$	20.0415			0.950058
$446$	2.96788			0.140533
$447$		−	16.0874i		−	0.760909i
$448$		−	5.76986i		−	0.272600i
$449$			35.6084i			1.68046i	0.542227	+	0.840232i	$0.317581\pi$
$449$			35.6084i			1.68046i	−0.542227	+	0.840232i	$0.682419\pi$
$450$		−	0.831613i		−	0.0392026i
$451$	12.4286			0.585242
$452$	6.17484			0.290440
$453$			1.51114i			0.0709994i
$454$	−0.563376			−0.0264405
$455$	−0.474572	+	5.47949i	−0.0222483	+	0.256883i
$456$	−2.19358			−0.102724
$457$			27.2543i			1.27490i	0.770491	+	0.637451i	$0.220011\pi$
$457$			27.2543i			1.27490i	−0.770491	+	0.637451i	$0.779989\pi$
$458$	4.34567			0.203060
$459$	−4.42864			−0.206711
$460$		−	11.0509i		−	0.515249i
$461$		−	28.7797i		−	1.34040i	−0.742179	−	0.670202i	$-0.766207\pi$
$461$		−	28.7797i		−	1.34040i	0.742179	−	0.670202i	$-0.233793\pi$
$462$			0.341219i			0.0158750i
$463$		−	29.4193i		−	1.36723i	−0.729843	−	0.683615i	$-0.760407\pi$
$463$		−	29.4193i		−	1.36723i	0.729843	−	0.683615i	$-0.239593\pi$
$464$	−2.59057			−0.120264
$465$	7.40943			0.343604
$466$			2.33677i			0.108249i
$467$	−7.14272			−0.330526			−0.165263	−	0.986250i	$-0.552847\pi$
$467$	−7.14272			−0.330526			−0.165263	+	0.986250i	$0.552847\pi$
$468$	−0.592104	+	6.83654i	−0.0273700	+	0.316019i
$469$	9.80642			0.452819
$470$		−	1.08250i		−	0.0499319i
$471$	2.85728			0.131656
$472$	−0.576283			−0.0265256
$473$			5.75203i			0.264479i
$474$			1.67307i			0.0768467i
$475$		−	4.82870i		−	0.221556i
$476$			8.42864i			0.386326i
$477$	−6.00000			−0.274721
$478$	6.83008			0.312401
$479$		−	16.5575i		−	0.756534i	−0.925697	−	0.378267i	$-0.876520\pi$
$479$		−	16.5575i		−	0.756534i	0.925697	−	0.378267i	$-0.123480\pi$
$480$	5.33185			0.243365
$481$	20.8573	+	1.80642i	0.951010	+	0.0823658i
$482$	−8.04149			−0.366280
$483$			3.80642i			0.173198i
$484$	18.6459			0.847540
$485$	−6.75557			−0.306755
$486$		−	0.311108i		−	0.0141121i
$487$			21.7146i			0.983981i	0.870601	+	0.491990i	$0.163730\pi$
$487$			21.7146i			0.983981i	−0.870601	+	0.491990i	$0.836270\pi$
$488$			15.8983i			0.719682i
$489$		−	13.8064i		−	0.624348i
$490$	−0.474572			−0.0214390
$491$	26.1748			1.18125			0.590627	−	0.806945i	$-0.298880\pi$
$491$	26.1748			1.18125			0.590627	+	0.806945i	$0.298880\pi$
$492$			21.5669i			0.972312i
$493$	3.34614			0.150703
$494$	0.174840	−	2.01874i	0.00786644	−	0.0908272i
$495$	1.67307			0.0751989
$496$		−	16.6539i		−	0.747780i
$497$	−13.0049			−0.583350
$498$	4.32248			0.193695
$499$		−	24.9403i		−	1.11648i	−0.829680	−	0.558240i	$-0.811477\pi$
$499$		−	24.9403i		−	1.11648i	0.829680	−	0.558240i	$-0.188523\pi$
$500$			22.2766i			0.996238i
$501$		−	0.769859i		−	0.0343948i
$502$		−	7.22570i		−	0.322499i
$503$	−14.6351			−0.652548			−0.326274	−	0.945275i	$-0.605793\pi$
$503$	−14.6351			−0.652548			−0.326274	+	0.945275i	$0.605793\pi$
$504$	−1.21432			−0.0540901
$505$			14.1650i			0.630334i
$506$	1.29883			0.0577398
$507$	−12.8064	−	2.23506i	−0.568753	−	0.0992626i
$508$	4.99063			0.221423
$509$		−	7.59856i		−	0.336800i	−0.985719	−	0.168400i	$-0.946140\pi$
$509$		−	7.59856i		−	0.336800i	0.985719	−	0.168400i	$-0.0538600\pi$
$510$	−2.10171			−0.0930653
$511$	3.47949			0.153924
$512$		−	20.3111i		−	0.897633i
$513$		−	1.80642i		−	0.0797556i
$514$			8.64095i			0.381136i
$515$			1.24443i			0.0548362i
$516$	−9.98126			−0.439401
$517$	−2.50177			−0.110028
$518$			1.80642i			0.0793697i
$519$	5.67307			0.249020
$520$	−0.576283	+	6.65386i	−0.0252717	+	0.291791i
$521$	−25.7560			−1.12839			−0.564196	−	0.825641i	$-0.690814\pi$
$521$	−25.7560			−1.12839			−0.564196	+	0.825641i	$0.690814\pi$
$522$			0.235063i			0.0102884i
$523$	−19.5081			−0.853029			−0.426514	−	0.904481i	$-0.640259\pi$
$523$	−19.5081			−0.853029			−0.426514	+	0.904481i	$0.640259\pi$
$524$	34.9590			1.52719
$525$		−	2.67307i		−	0.116662i
$526$			2.07896i			0.0906469i
$527$			21.5111i			0.937040i
$528$		−	3.76049i		−	0.163654i
$529$	−8.51114			−0.370049
$530$	−2.84743			−0.123685
$531$		−	0.474572i		−	0.0205947i
$532$	−3.43801			−0.149057
$533$	−40.7052	−	3.52543i	−1.76314	−	0.152703i
$534$	−4.08742			−0.176880
$535$			10.0098i			0.432763i
$536$	11.9081			0.514353
$537$	21.9081			0.945406
$538$			6.88892i			0.297003i
$539$			1.09679i			0.0472420i
$540$			2.90321i			0.124934i
$541$		−	43.8163i		−	1.88381i	−0.335881	−	0.941904i	$-0.609034\pi$
$541$		−	43.8163i		−	1.88381i	0.335881	−	0.941904i	$-0.390966\pi$
$542$	0.682439			0.0293133
$543$	0.488863			0.0209791
$544$			15.4795i			0.663678i
$545$	−18.7556			−0.803400
$546$	0.0967881	−	1.11753i	0.00414215	−	0.0478259i
$547$	−43.2257			−1.84820			−0.924099	−	0.382154i	$-0.875182\pi$
$547$	−43.2257			−1.84820			−0.924099	+	0.382154i	$0.875182\pi$
$548$		−	31.1798i		−	1.33193i
$549$	−13.0923			−0.558768
$550$	−0.912103			−0.0388922
$551$			1.36488i			0.0581457i
$552$			4.62222i			0.196735i
$553$			5.37778i			0.228687i
$554$			8.50177i			0.361206i
$555$	8.85728			0.375971
$556$	6.06022			0.257011
$557$		−	11.5254i		−	0.488348i	−0.969731	−	0.244174i	$-0.921483\pi$
$557$		−	11.5254i		−	0.488348i	0.969731	−	0.244174i	$-0.0785168\pi$
$558$	−1.51114			−0.0639715
$559$	1.63158	−	18.8385i	0.0690086	−	0.796785i
$560$	5.23014			0.221014
$561$			4.85728i			0.205074i
$562$	7.10663			0.299775
$563$	−44.2034			−1.86295			−0.931476	−	0.363803i	$-0.881478\pi$
$563$	−44.2034			−1.86295			−0.931476	+	0.363803i	$0.881478\pi$
$564$		−	4.34122i		−	0.182798i
$565$			4.94914i			0.208212i
$566$		−	6.57184i		−	0.276235i
$567$		−	1.00000i		−	0.0419961i
$568$	−15.7921			−0.662623
$569$	−33.4924			−1.40407			−0.702037	−	0.712140i	$-0.747726\pi$
$569$	−33.4924			−1.40407			−0.702037	+	0.712140i	$0.747726\pi$
$570$		−	0.857279i		−	0.0359075i
$571$	2.23506			0.0935345			0.0467672	−	0.998906i	$-0.485108\pi$
$571$	2.23506			0.0935345			0.0467672	+	0.998906i	$0.485108\pi$
$572$	7.49823	+	0.649413i	0.313517	+	0.0271533i
$573$	25.1338			1.04998
$574$		−	3.52543i		−	0.147149i
$575$	−10.1748			−0.424320
$576$	5.76986			0.240411
$577$			20.0415i			0.834338i	0.908829	+	0.417169i	$0.136978\pi$
$577$			20.0415i			0.834338i	−0.908829	+	0.417169i	$0.863022\pi$
$578$		−	0.812877i		−	0.0338112i
$579$			6.66370i			0.276934i
$580$		−	2.19358i		−	0.0910833i
$581$	13.8938			0.576414
$582$	1.37778			0.0571110
$583$			6.58073i			0.272546i
$584$	4.22522			0.174841
$585$	−5.47949	−	0.474572i	−0.226549	−	0.0196212i
$586$	3.61729			0.149429
$587$		−	14.1891i		−	0.585648i	−0.956166	−	0.292824i	$-0.905405\pi$
$587$		−	14.1891i		−	0.585648i	0.956166	−	0.292824i	$-0.0945950\pi$
$588$	−1.90321			−0.0784871
$589$	−8.77430			−0.361539
$590$		−	0.225219i		−	0.00927212i
$591$			18.1891i			0.748201i
$592$		−	19.9081i		−	0.818219i
$593$			20.6593i			0.848374i	0.905575	+	0.424187i	$0.139440\pi$
$593$			20.6593i			0.848374i	−0.905575	+	0.424187i	$0.860560\pi$
$594$	−0.341219			−0.0140004
$595$	−6.75557			−0.276951
$596$		−	30.6178i		−	1.25415i
$597$	−14.1017			−0.577145
$598$	−4.25380	−	0.368416i	−0.173951	−	0.0150657i
$599$	−36.1561			−1.47730			−0.738649	−	0.674090i	$-0.764536\pi$
$599$	−36.1561			−1.47730			−0.738649	+	0.674090i	$0.764536\pi$
$600$		−	3.24596i		−	0.132516i
$601$	−45.6829			−1.86344			−0.931722	−	0.363171i	$-0.881694\pi$
$601$	−45.6829			−1.86344			−0.931722	+	0.363171i	$0.881694\pi$
$602$	1.63158			0.0664984
$603$			9.80642i			0.399348i
$604$			2.87601i			0.117023i
$605$			14.9447i			0.607588i
$606$		−	2.88892i		−	0.117354i
$607$	−11.8381			−0.480492			−0.240246	−	0.970712i	$-0.577228\pi$
$607$	−11.8381			−0.480492			−0.240246	+	0.970712i	$0.577228\pi$
$608$	−6.31402			−0.256067
$609$			0.755569i			0.0306172i
$610$	−6.21326			−0.251568
$611$	8.19358	+	0.709636i	0.331477	+	0.0287088i
$612$	−8.42864			−0.340708
$613$		−	14.9590i		−	0.604188i	−0.953278	−	0.302094i	$-0.902314\pi$
$613$		−	14.9590i		−	0.604188i	0.953278	−	0.302094i	$-0.0976856\pi$
$614$	−10.2222			−0.412533
$615$	−17.2859			−0.697036
$616$			1.33185i			0.0536618i
$617$		−	2.49331i		−	0.100377i	−0.998740	−	0.0501884i	$-0.984018\pi$
$617$		−	2.49331i		−	0.100377i	0.998740	−	0.0501884i	$-0.0159822\pi$
$618$		−	0.253799i		−	0.0102093i
$619$		−	25.1526i		−	1.01097i	−0.862836	−	0.505483i	$-0.831314\pi$
$619$		−	25.1526i		−	1.01097i	0.862836	−	0.505483i	$-0.168686\pi$
$620$	14.1017			0.566338
$621$	−3.80642			−0.152747
$622$		−	5.71456i		−	0.229133i
$623$	−13.1383			−0.526374
$624$	−1.06668	+	12.3160i	−0.0427012	+	0.493036i
$625$	−4.48934			−0.179574
$626$		−	2.21231i		−	0.0884218i
$627$	−1.98126			−0.0791241
$628$	5.43801			0.217000
$629$			25.7146i			1.02531i
$630$		−	0.474572i		−	0.0189074i
$631$			40.1116i			1.59682i	0.602117	+	0.798408i	$0.294324\pi$
$631$			40.1116i			1.59682i	−0.602117	+	0.798408i	$0.705676\pi$
$632$			6.53035i			0.259763i
$633$	8.13335			0.323272
$634$	2.63651			0.104709
$635$			4.00000i			0.158735i
$636$	−11.4193			−0.452804
$637$	0.311108	−	3.59210i	0.0123265	−	0.142324i
$638$	0.257815			0.0102070
$639$		−	13.0049i		−	0.514467i
$640$	13.4019			0.529757
$641$	35.5308			1.40338			0.701692	−	0.712481i	$-0.252428\pi$
$641$	35.5308			1.40338			0.701692	+	0.712481i	$0.252428\pi$
$642$		−	2.04149i		−	0.0805711i
$643$			13.2444i			0.522309i	0.965297	+	0.261155i	$0.0841033\pi$
$643$			13.2444i			0.522309i	−0.965297	+	0.261155i	$0.915897\pi$
$644$			7.24443i			0.285471i
$645$		−	8.00000i		−	0.315000i
$646$	2.48886			0.0979230
$647$	0.120446			0.00473523			0.00236761	−	0.999997i	$-0.499246\pi$
$647$	0.120446			0.00473523			0.00236761	+	0.999997i	$0.499246\pi$
$648$		−	1.21432i		−	0.0477030i
$649$	−0.520505			−0.0204316
$650$	2.98724	+	0.258721i	0.117169	+	0.0101479i
$651$	−4.85728			−0.190372
$652$		−	26.2766i		−	1.02907i
$653$	−15.7146			−0.614958			−0.307479	−	0.951555i	$-0.599485\pi$
$653$	−15.7146			−0.614958			−0.307479	+	0.951555i	$0.599485\pi$
$654$	3.82516			0.149576
$655$			28.0197i			1.09482i
$656$			38.8528i			1.51695i
$657$			3.47949i			0.135748i
$658$			0.709636i			0.0276645i
$659$	14.1748			0.552173			0.276087	−	0.961133i	$-0.410962\pi$
$659$	14.1748			0.552173			0.276087	+	0.961133i	$0.410962\pi$
$660$	3.18421			0.123945
$661$			7.21279i			0.280545i	0.990113	+	0.140272i	$0.0447979\pi$
$661$			7.21279i			0.280545i	−0.990113	+	0.140272i	$0.955202\pi$
$662$	8.77430			0.341023
$663$	1.37778	−	15.9081i	0.0535087	−	0.617821i
$664$	16.8716			0.654744
$665$		−	2.75557i		−	0.106856i
$666$	−1.80642			−0.0699975
$667$	2.87601			0.111360
$668$		−	1.46520i		−	0.0566905i
$669$			9.53972i			0.368827i
$670$			4.65386i			0.179794i
$671$			14.3595i			0.554343i
$672$	−3.49532			−0.134835
$673$	39.3274			1.51596			0.757980	−	0.652278i	$-0.226186\pi$
$673$	39.3274			1.51596			0.757980	+	0.652278i	$0.226186\pi$
$674$		−	7.63158i		−	0.293958i
$675$	2.67307			0.102887
$676$	−24.3733	−	4.25380i	−0.937436	−	0.163608i
$677$	27.2672			1.04796			0.523981	−	0.851730i	$-0.324446\pi$
$677$	27.2672			1.04796			0.523981	+	0.851730i	$0.324446\pi$
$678$		−	1.00937i		−	0.0387645i
$679$	4.42864			0.169956
$680$	−8.20342			−0.314587
$681$		−	1.81087i		−	0.0693927i
$682$			1.65740i			0.0634650i
$683$			16.6811i			0.638283i	0.947707	+	0.319141i	$0.103394\pi$
$683$			16.6811i			0.638283i	−0.947707	+	0.319141i	$0.896606\pi$
$684$		−	3.43801i		−	0.131455i
$685$	24.9906			0.954843
$686$	0.311108			0.0118782
$687$			13.9684i			0.532926i
$688$	−17.9813			−0.685529
$689$	1.86665	−	21.5526i	0.0711136	−	0.821090i
$690$	−1.80642			−0.0687694
$691$		−	15.2543i		−	0.580300i	−0.956981	−	0.290150i	$-0.906295\pi$
$691$		−	15.2543i		−	0.580300i	0.956981	−	0.290150i	$-0.0937052\pi$
$692$	10.7971			0.410442
$693$	−1.09679			−0.0416635
$694$			2.96205i			0.112438i
$695$			4.85728i			0.184247i
$696$			0.917502i			0.0347778i
$697$		−	50.1847i		−	1.90088i
$698$	−2.91750			−0.110429
$699$	−7.51114			−0.284097
$700$		−	5.08742i		−	0.192286i
$701$	34.5906			1.30647			0.653234	−	0.757156i	$-0.273412\pi$
$701$	34.5906			1.30647			0.653234	+	0.757156i	$0.273412\pi$
$702$	1.11753	+	0.0967881i	0.0421785	+	0.00365303i
$703$	−10.4889			−0.395595
$704$		−	6.32831i		−	0.238507i
$705$	3.47949			0.131045
$706$	−3.57982			−0.134728
$707$		−	9.28592i		−	0.349233i
$708$		−	0.903212i		−	0.0339448i
$709$		−	14.5433i		−	0.546183i	−0.961988	−	0.273092i	$-0.911954\pi$
$709$		−	14.5433i		−	0.546183i	0.961988	−	0.273092i	$-0.0880463\pi$
$710$		−	6.17178i		−	0.231623i
$711$	−5.37778			−0.201683
$712$	−15.9541			−0.597904
$713$			18.4889i			0.692413i
$714$	1.37778			0.0515623
$715$	−0.520505	+	6.00984i	−0.0194658	+	0.224755i
$716$	41.6958			1.55825
$717$			21.9541i			0.819890i
$718$	4.78277			0.178491
$719$	−3.34614			−0.124790			−0.0623950	−	0.998052i	$-0.519874\pi$
$719$	−3.34614			−0.124790			−0.0623950	+	0.998052i	$0.519874\pi$
$720$			5.23014i			0.194916i
$721$		−	0.815792i		−	0.0303817i
$722$		−	4.89585i		−	0.182205i
$723$		−	25.8479i		−	0.961294i
$724$	0.930409			0.0345784
$725$	−2.01969			−0.0750094
$726$		−	3.04794i		−	0.113120i
$727$	−23.2257			−0.861393			−0.430697	−	0.902497i	$-0.641732\pi$
$727$	−23.2257			−0.861393			−0.430697	+	0.902497i	$0.641732\pi$
$728$	0.377784	−	4.36196i	0.0140016	−	0.161665i
$729$	1.00000			0.0370370
$730$			1.65127i			0.0611163i
$731$	23.2257			0.859033
$732$	−24.9175			−0.920977
$733$			10.8889i			0.402192i	0.979572	+	0.201096i	$0.0644502\pi$
$733$			10.8889i			0.402192i	−0.979572	+	0.201096i	$0.935550\pi$
$734$			7.74620i			0.285917i
$735$		−	1.52543i		−	0.0562662i
$736$			13.3047i			0.490416i
$737$	10.7556			0.396186
$738$	3.52543			0.129773
$739$			33.8064i			1.24359i	0.783180	+	0.621795i	$0.213596\pi$
$739$			33.8064i			1.24359i	−0.783180	+	0.621795i	$0.786404\pi$
$740$	16.8573			0.619686
$741$	6.48886	+	0.561993i	0.238374	+	0.0206453i
$742$	1.86665			0.0685268
$743$		−	46.5990i		−	1.70955i	−0.518996	−	0.854776i	$-0.673694\pi$
$743$		−	46.5990i		−	1.70955i	0.518996	−	0.854776i	$-0.326306\pi$
$744$	−5.89829			−0.216242
$745$	24.5402			0.899083
$746$		−	5.34614i		−	0.195736i
$747$			13.8938i			0.508349i
$748$			9.24443i			0.338010i
$749$		−	6.56199i		−	0.239770i
$750$	3.64143			0.132966
$751$	−17.2444			−0.629258			−0.314629	−	0.949215i	$-0.601880\pi$
$751$	−17.2444			−0.629258			−0.314629	+	0.949215i	$0.601880\pi$
$752$		−	7.82071i		−	0.285192i
$753$	23.2257			0.846391
$754$	−0.844372	−	0.0731300i	−0.0307502	−	0.00266324i
$755$	−2.30513			−0.0838923
$756$		−	1.90321i		−	0.0692191i
$757$	19.8193			0.720346			0.360173	−	0.932886i	$-0.382718\pi$
$757$	19.8193			0.720346			0.360173	+	0.932886i	$0.382718\pi$
$758$	4.08297			0.148300
$759$			4.17484i			0.151537i
$760$		−	3.34614i		−	0.121377i
$761$			24.2810i			0.880185i	0.897952	+	0.440093i	$0.145054\pi$
$761$			24.2810i			0.880185i	−0.897952	+	0.440093i	$0.854946\pi$
$762$		−	0.815792i		−	0.0295530i
$763$	12.2953			0.445119
$764$	47.8350			1.73061
$765$		−	6.75557i		−	0.244248i
$766$	−9.56691			−0.345667
$767$	1.70471	+	0.147643i	0.0615536	+	0.00533109i
$768$	8.80642			0.317774
$769$			8.78721i			0.316875i	0.987369	+	0.158437i	$0.0506456\pi$
$769$			8.78721i			0.316875i	−0.987369	+	0.158437i	$0.949354\pi$
$770$	−0.520505			−0.0187577
$771$	−27.7748			−1.00028
$772$			12.6824i			0.456451i
$773$		−	43.6271i		−	1.56916i	−0.620028	−	0.784580i	$-0.712879\pi$
$773$		−	43.6271i		−	1.56916i	0.620028	−	0.784580i	$-0.287121\pi$
$774$			1.63158i			0.0586461i
$775$		−	12.9839i		−	0.466394i
$776$	5.37778			0.193051
$777$	−5.80642			−0.208304
$778$			4.62222i			0.165714i
$779$	20.4701			0.733418
$780$	−10.4286	−	0.903212i	−0.373405	−	0.0323402i
$781$	−14.2636			−0.510393
$782$		−	5.24443i		−	0.187540i
$783$	−0.755569			−0.0270018
$784$	−3.42864			−0.122451
$785$			4.35857i			0.155564i
$786$		−	5.71456i

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)

\(f(q)\)	\(=\)	\(q-0.311108i q^{2} +1.00000 q^{3} +1.90321 q^{4} +1.52543i q^{5} -0.311108i q^{6} -1.00000i q^{7} -1.21432i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.311108i q^{2} +1.00000 q^{3} +1.90321 q^{4} +1.52543i q^{5} -0.311108i q^{6} -1.00000i q^{7} -1.21432i q^{8} +1.00000 q^{9} +0.474572 q^{10} -1.09679i q^{11} +1.90321 q^{12} +(-0.311108 + 3.59210i) q^{13} -0.311108 q^{14} +1.52543i q^{15} +3.42864 q^{16} -4.42864 q^{17} -0.311108i q^{18} -1.80642i q^{19} +2.90321i q^{20} -1.00000i q^{21} -0.341219 q^{22} -3.80642 q^{23} -1.21432i q^{24} +2.67307 q^{25} +(1.11753 + 0.0967881i) q^{26} +1.00000 q^{27} -1.90321i q^{28} -0.755569 q^{29} +0.474572 q^{30} -4.85728i q^{31} -3.49532i q^{32} -1.09679i q^{33} +1.37778i q^{34} +1.52543 q^{35} +1.90321 q^{36} -5.80642i q^{37} -0.561993 q^{38} +(-0.311108 + 3.59210i) q^{39} +1.85236 q^{40} +11.3319i q^{41} -0.311108 q^{42} -5.24443 q^{43} -2.08742i q^{44} +1.52543i q^{45} +1.18421i q^{46} -2.28100i q^{47} +3.42864 q^{48} -1.00000 q^{49} -0.831613i q^{50} -4.42864 q^{51} +(-0.592104 + 6.83654i) q^{52} -6.00000 q^{53} -0.311108i q^{54} +1.67307 q^{55} -1.21432 q^{56} -1.80642i q^{57} +0.235063i q^{58} -0.474572i q^{59} +2.90321i q^{60} -13.0923 q^{61} -1.51114 q^{62} -1.00000i q^{63} +5.76986 q^{64} +(-5.47949 - 0.474572i) q^{65} -0.341219 q^{66} +9.80642i q^{67} -8.42864 q^{68} -3.80642 q^{69} -0.474572i q^{70} -13.0049i q^{71} -1.21432i q^{72} +3.47949i q^{73} -1.80642 q^{74} +2.67307 q^{75} -3.43801i q^{76} -1.09679 q^{77} +(1.11753 + 0.0967881i) q^{78} -5.37778 q^{79} +5.23014i q^{80} +1.00000 q^{81} +3.52543 q^{82} +13.8938i q^{83} -1.90321i q^{84} -6.75557i q^{85} +1.63158i q^{86} -0.755569 q^{87} -1.33185 q^{88} -13.1383i q^{89} +0.474572 q^{90} +(3.59210 + 0.311108i) q^{91} -7.24443 q^{92} -4.85728i q^{93} -0.709636 q^{94} +2.75557 q^{95} -3.49532i q^{96} +4.42864i q^{97} +0.311108i q^{98} -1.09679i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9}+O(q^{10}) \) 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