LMFDB - Embedded newform 273.2.c.b.64.4

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			64.4
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	273.64
Dual form			273.2.c.b.64.3

$q$-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	$0.0350829\pi$
$2$			0.311108i			0.219986i	−0.993932	+	0.109993i	$0.964917\pi$
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	1.90321			0.951606
$5$		−	1.52543i		−	0.682192i	−0.940028	−	0.341096i	$-0.889202\pi$
$5$		−	1.52543i		−	0.682192i	0.940028	−	0.341096i	$-0.110798\pi$
$6$			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	$0.0528744\pi$
$11$			1.09679i			0.330694i	−0.986235	+	0.165347i	$0.947126\pi$
$12$	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	$0.0664387\pi$
$19$			1.80642i			0.414422i	−0.978296	+	0.207211i	$0.933561\pi$
$20$		−	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	$0.143675\pi$
$31$			4.85728i			0.872393i	−0.899851	+	0.436197i	$0.856325\pi$
$32$			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	$0.158379\pi$
$37$			5.80642i			0.954570i	−0.878749	+	0.477285i	$0.841621\pi$
$38$	−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	$-0.654248\pi$
$41$		−	11.3319i		−	1.76974i	0.465840	−	0.884869i	$-0.345752\pi$
$42$	−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	$0.0532010\pi$
$47$			2.28100i			0.332718i	−0.986065	+	0.166359i	$0.946799\pi$
$48$	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	$0.00983480\pi$
$59$			0.474572i			0.0617841i	−0.999523	+	0.0308920i	$0.990165\pi$
$60$		−	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	$-0.795556\pi$
$67$		−	9.80642i		−	1.19805i	0.800732	−	0.599023i	$-0.204444\pi$
$68$	−8.42864			−1.02212
$69$	−3.80642			−0.458240
$70$			0.474572i			0.0567223i
$71$			13.0049i			1.54340i	0.635987	+	0.771700i	$0.280593\pi$
$71$			13.0049i			1.54340i	−0.635987	+	0.771700i	$0.719407\pi$
$72$			1.21432i			0.143109i
$73$		−	3.47949i		−	0.407244i	−0.979050	−	0.203622i	$-0.934729\pi$
$73$		−	3.47949i		−	0.407244i	0.979050	−	0.203622i	$-0.0652714\pi$
$74$	−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	$-0.723960\pi$
$83$		−	13.8938i		−	1.52505i	0.646960	−	0.762524i	$-0.276040\pi$
$84$			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	$0.245184\pi$
$89$			13.1383i			1.39265i	−0.717724	+	0.696327i	$0.754816\pi$
$90$	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	$-0.927817\pi$
$97$		−	4.42864i		−	0.449660i	0.974398	−	0.224830i	$-0.0721827\pi$
$98$		−	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	$-0.799586\pi$
$109$		−	12.2953i		−	1.17767i	0.808252	−	0.588837i	$-0.200414\pi$
$110$			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	$0.246743\pi$
$137$			16.3827i			1.39967i	−0.714305	+	0.699835i	$0.753257\pi$
$138$		−	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	$0.229006\pi$
$149$			16.0874i			1.31793i	−0.752172	+	0.658966i	$0.770994\pi$
$150$			0.831613i			0.0679009i
$151$		−	1.51114i		−	0.122975i	−0.998108	−	0.0614873i	$-0.980416\pi$
$151$		−	1.51114i		−	0.122975i	0.998108	−	0.0614873i	$-0.0195844\pi$
$152$	−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	$0.181841\pi$
$163$			13.8064i			1.08140i	−0.841215	+	0.540701i	$0.818159\pi$
$164$		−	21.5669i		−	1.68409i
$165$	1.67307			0.130248
$166$	4.32248			0.335490
$167$			0.769859i			0.0595735i	0.999556	+	0.0297867i	$0.00948281\pi$
$167$			0.769859i			0.0595735i	−0.999556	+	0.0297867i	$0.990517\pi$
$168$	−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	$-0.922908\pi$
$193$		−	6.66370i		−	0.479664i	0.970814	−	0.239832i	$-0.0770923\pi$
$194$	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	$-0.775622\pi$
$197$		−	18.1891i		−	1.29592i	0.761674	−	0.647961i	$-0.224378\pi$
$198$	−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	$-0.896514\pi$
$223$		−	9.53972i		−	0.638827i	0.947615	−	0.319413i	$-0.103486\pi$
$224$	−3.49532			−0.233541
$225$	2.67307			0.178205
$226$			1.00937i			0.0671422i
$227$			1.81087i			0.120192i	0.998193	+	0.0600958i	$0.0191406\pi$
$227$			1.81087i			0.120192i	−0.998193	+	0.0600958i	$0.980859\pi$
$228$			3.43801i			0.227688i
$229$		−	13.9684i		−	0.923055i	−0.887126	−	0.461528i	$-0.847302\pi$
$229$		−	13.9684i		−	0.923055i	0.887126	−	0.461528i	$-0.152698\pi$
$230$	−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	$-0.748674\pi$
$239$		−	21.9541i		−	1.42009i	0.704156	−	0.710045i	$-0.251326\pi$
$240$		−	5.23014i		−	0.337604i
$241$			25.8479i			1.66501i	0.554017	+	0.832505i	$0.313094\pi$
$241$			25.8479i			1.66501i	−0.554017	+	0.832505i	$0.686906\pi$
$242$			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	$-0.978777\pi$
$271$		−	2.19358i		−	0.133250i	0.997778	−	0.0666251i	$-0.0212232\pi$
$272$	−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	$-0.761394\pi$
$281$		−	22.8430i		−	1.36270i	0.731958	−	0.681349i	$-0.238606\pi$
$282$	−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	$-0.889697\pi$
$293$		−	11.6271i		−	0.679265i	0.940558	−	0.339632i	$-0.110303\pi$
$294$		−	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	$0.386987\pi$
$307$			32.8573i			1.87526i	−0.347629	+	0.937632i	$0.613013\pi$
$308$	−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	$-0.923512\pi$
$317$		−	8.47457i		−	0.475979i	0.971268	−	0.237990i	$-0.0764885\pi$
$318$		−	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	$-0.717699\pi$
$331$		−	28.2034i		−	1.55020i	0.631838	−	0.775100i	$-0.282301\pi$
$332$		−	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	$0.0807563\pi$
$349$			9.37778i			0.501981i	−0.967990	+	0.250991i	$0.919244\pi$
$350$	−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	$0.0990642\pi$
$353$			11.5067i			0.612439i	−0.951961	+	0.306220i	$0.900936\pi$
$354$	−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	$-0.867032\pi$
$359$		−	15.3733i		−	0.811374i	0.914012	−	0.405687i	$-0.132968\pi$
$360$	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	$-0.890565\pi$
$379$		−	13.1240i		−	0.674134i	0.941481	−	0.337067i	$-0.109435\pi$
$380$	5.24443			0.269034
$381$	2.62222			0.134340
$382$			7.81933i			0.400072i
$383$			30.7511i			1.57131i	0.618666	+	0.785654i	$0.287674\pi$
$383$			30.7511i			1.57131i	−0.618666	+	0.785654i	$0.712326\pi$
$384$			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	$-0.952377\pi$
$397$		−	5.93978i		−	0.298109i	0.988829	−	0.149054i	$-0.0476230\pi$
$398$		−	4.38715i		−	0.219908i
$399$	−1.80642			−0.0904343
$400$	9.16500			0.458250
$401$		−	6.87157i		−	0.343150i	−0.985171	−	0.171575i	$-0.945114\pi$
$401$		−	6.87157i		−	0.343150i	0.985171	−	0.171575i	$-0.0548856\pi$
$402$	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	$0.277972\pi$
$409$			31.0005i			1.53287i	−0.642319	+	0.766437i	$0.722028\pi$
$410$		−	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	$-0.911801\pi$
$421$		−	11.2257i		−	0.547107i	0.961857	−	0.273553i	$-0.0881990\pi$
$422$			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	$0.119288\pi$
$431$			15.1985i			0.732086i	−0.930598	+	0.366043i	$0.880712\pi$
$432$	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	$-0.682419\pi$
$449$		−	35.6084i		−	1.68046i	0.542227	−	0.840232i	$-0.317581\pi$
$450$			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	$-0.779989\pi$
$457$		−	27.2543i		−	1.27490i	0.770491	−	0.637451i	$-0.220011\pi$
$458$	4.34567			0.203060
$459$	−4.42864			−0.206711
$460$			11.0509i			0.515249i
$461$			28.7797i			1.34040i	0.742179	+	0.670202i	$0.233793\pi$
$461$			28.7797i			1.34040i	−0.742179	+	0.670202i	$0.766207\pi$
$462$		−	0.341219i		−	0.0158750i
$463$			29.4193i			1.36723i	0.729843	+	0.683615i	$0.239593\pi$
$463$			29.4193i			1.36723i	−0.729843	+	0.683615i	$0.760407\pi$
$464$	−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	$0.123480\pi$
$479$			16.5575i			0.756534i	−0.925697	+	0.378267i	$0.876520\pi$
$480$	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	$-0.836270\pi$
$487$		−	21.7146i		−	0.983981i	0.870601	−	0.491990i	$-0.163730\pi$
$488$		−	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	$0.188523\pi$
$499$			24.9403i			1.11648i	−0.829680	+	0.558240i	$0.811477\pi$
$500$		−	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	$0.0538600\pi$
$509$			7.59856i			0.336800i	−0.985719	+	0.168400i	$0.946140\pi$
$510$	−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	$0.390966\pi$
$541$			43.8163i			1.88381i	−0.335881	+	0.941904i	$0.609034\pi$
$542$	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	$0.0785168\pi$
$557$			11.5254i			0.488348i	−0.969731	+	0.244174i	$0.921483\pi$
$558$	−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	$-0.863022\pi$
$577$		−	20.0415i		−	0.834338i	0.908829	−	0.417169i	$-0.136978\pi$
$578$			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	$0.0945950\pi$
$587$			14.1891i			0.585648i	−0.956166	+	0.292824i	$0.905405\pi$
$588$	−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	$-0.860560\pi$
$593$		−	20.6593i		−	0.848374i	0.905575	−	0.424187i	$-0.139440\pi$
$594$	−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	$0.0976856\pi$
$613$			14.9590i			0.604188i	−0.953278	+	0.302094i	$0.902314\pi$
$614$	−10.2222			−0.412533
$615$	−17.2859			−0.697036
$616$		−	1.33185i		−	0.0536618i
$617$			2.49331i			0.100377i	0.998740	+	0.0501884i	$0.0159822\pi$
$617$			2.49331i			0.100377i	−0.998740	+	0.0501884i	$0.984018\pi$
$618$			0.253799i			0.0102093i
$619$			25.1526i			1.01097i	0.862836	+	0.505483i	$0.168686\pi$
$619$			25.1526i			1.01097i	−0.862836	+	0.505483i	$0.831314\pi$
$620$	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	$-0.705676\pi$
$631$		−	40.1116i		−	1.59682i	0.602117	−	0.798408i	$-0.294324\pi$
$632$		−	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	$-0.915897\pi$
$643$		−	13.2444i		−	0.522309i	0.965297	−	0.261155i	$-0.0841033\pi$
$644$		−	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	$-0.955202\pi$
$661$		−	7.21279i		−	0.280545i	0.990113	−	0.140272i	$-0.0447979\pi$
$662$	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	$-0.896606\pi$
$683$		−	16.6811i		−	0.638283i	0.947707	−	0.319141i	$-0.103394\pi$
$684$			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	$0.0937052\pi$
$691$			15.2543i			0.580300i	−0.956981	+	0.290150i	$0.906295\pi$
$692$	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	$0.0880463\pi$
$709$			14.5433i			0.546183i	−0.961988	+	0.273092i	$0.911954\pi$
$710$			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	$-0.935550\pi$
$733$		−	10.8889i		−	0.402192i	0.979572	−	0.201096i	$-0.0644502\pi$
$734$		−	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	$-0.786404\pi$
$739$		−	33.8064i		−	1.24359i	0.783180	−	0.621795i	$-0.213596\pi$
$740$	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	$0.326306\pi$
$743$			46.5990i			1.70955i	−0.518996	+	0.854776i	$0.673694\pi$
$744$	−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	$-0.854946\pi$
$761$		−	24.2810i		−	0.880185i	0.897952	−	0.440093i	$-0.145054\pi$
$762$			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	$-0.949354\pi$
$769$		−	8.78721i		−	0.316875i	0.987369	−	0.158437i	$-0.0506456\pi$
$770$	−0.520505			−0.0187577
$771$	−27.7748			−1.00028
$772$		−	12.6824i		−	0.456451i
$773$			43.6271i			1.56916i	0.620028	+	0.784580i	$0.287121\pi$
$773$			43.6271i			1.56916i	−0.620028	+	0.784580i	$0.712879\pi$
$774$		−	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

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