LMFDB - Embedded newform 60.2.i.a.53.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,2,Mod(17,60)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("60.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	60.i (of order $4$, degree $2$, 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:	$0.479102412128$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	60.53
Dual form			60.2.i.a.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.61803 - 0.618034i) q^{3} -2.23607 q^{5} +(-1.00000 + 1.00000i) q^{7} +(2.23607 - 2.00000i) q^{9} +O(q^{10})$	\(q+(1.61803 - 0.618034i) q^{3} -2.23607 q^{5} +(-1.00000 + 1.00000i) q^{7} +(2.23607 - 2.00000i) q^{9} +4.47214i q^{11} +(-3.00000 - 3.00000i) q^{13} +(-3.61803 + 1.38197i) q^{15} +(-2.23607 - 2.23607i) q^{17} +2.00000i q^{19} +(-1.00000 + 2.23607i) q^{21} +(2.23607 - 2.23607i) q^{23} +5.00000 q^{25} +(2.38197 - 4.61803i) q^{27} +4.47214 q^{29} +4.00000 q^{31} +(2.76393 + 7.23607i) q^{33} +(2.23607 - 2.23607i) q^{35} +(-3.00000 + 3.00000i) q^{37} +(-6.70820 - 3.00000i) q^{39} -8.94427i q^{41} +(-3.00000 - 3.00000i) q^{43} +(-5.00000 + 4.47214i) q^{45} +(6.70820 + 6.70820i) q^{47} +5.00000i q^{49} +(-5.00000 - 2.23607i) q^{51} +(-2.23607 + 2.23607i) q^{53} -10.0000i q^{55} +(1.23607 + 3.23607i) q^{57} -8.94427 q^{59} -6.00000 q^{61} +(-0.236068 + 4.23607i) q^{63} +(6.70820 + 6.70820i) q^{65} +(-1.00000 + 1.00000i) q^{67} +(2.23607 - 5.00000i) q^{69} -4.47214i q^{71} +(1.00000 + 1.00000i) q^{73} +(8.09017 - 3.09017i) q^{75} +(-4.47214 - 4.47214i) q^{77} +6.00000i q^{79} +(1.00000 - 8.94427i) q^{81} +(-6.70820 + 6.70820i) q^{83} +(5.00000 + 5.00000i) q^{85} +(7.23607 - 2.76393i) q^{87} -4.47214 q^{89} +6.00000 q^{91} +(6.47214 - 2.47214i) q^{93} -4.47214i q^{95} +(9.00000 - 9.00000i) q^{97} +(8.94427 + 10.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^3 - 4 * q^7	$ 4 q + 2 q^{3} - 4 q^{7} - 12 q^{13} - 10 q^{15} - 4 q^{21} + 20 q^{25} + 14 q^{27} + 16 q^{31} + 20 q^{33} - 12 q^{37} - 12 q^{43} - 20 q^{45} - 20 q^{51} - 4 q^{57} - 24 q^{61} + 8 q^{63} - 4 q^{67} + 4 q^{73} + 10 q^{75} + 4 q^{81} + 20 q^{85} + 20 q^{87} + 24 q^{91} + 8 q^{93} + 36 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^7 - 12 * q^13 - 10 * q^15 - 4 * q^21 + 20 * q^25 + 14 * q^27 + 16 * q^31 + 20 * q^33 - 12 * q^37 - 12 * q^43 - 20 * q^45 - 20 * q^51 - 4 * q^57 - 24 * q^61 + 8 * q^63 - 4 * q^67 + 4 * q^73 + 10 * q^75 + 4 * q^81 + 20 * q^85 + 20 * q^87 + 24 * q^91 + 8 * q^93 + 36 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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			0
$2$		0			0
$3$	1.61803	−	0.618034i	0.934172	−	0.356822i
$3$	1.61803	−	0.618034i	0.934172	−	0.356822i
$4$		0			0
$5$	−2.23607			−1.00000
$5$	−2.23607			−1.00000
$6$		0			0
$7$	−1.00000	+	1.00000i	−0.377964	+	0.377964i	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−1.00000	+	1.00000i	−0.377964	+	0.377964i	0.492403	+	0.870367i	$0.336119\pi$
$8$		0			0
$9$	2.23607	−	2.00000i	0.745356	−	0.666667i
$10$		0			0
$11$			4.47214i			1.34840i	0.738549	+	0.674200i	$0.235511\pi$
$11$			4.47214i			1.34840i	−0.738549	+	0.674200i	$0.764489\pi$
$12$		0			0
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	0.155747	−	0.987797i	$-0.450222\pi$
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	−0.987797	+	0.155747i	$0.950222\pi$
$14$		0			0
$15$	−3.61803	+	1.38197i	−0.934172	+	0.356822i
$16$		0			0
$17$	−2.23607	−	2.23607i	−0.542326	−	0.542326i	0.381884	−	0.924210i	$-0.375275\pi$
$17$	−2.23607	−	2.23607i	−0.542326	−	0.542326i	−0.924210	+	0.381884i	$0.875275\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$	−1.00000	+	2.23607i	−0.218218	+	0.487950i
$22$		0			0
$23$	2.23607	−	2.23607i	0.466252	−	0.466252i	−0.434446	−	0.900698i	$-0.643056\pi$
$23$	2.23607	−	2.23607i	0.466252	−	0.466252i	0.900698	+	0.434446i	$0.143056\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	2.38197	−	4.61803i	0.458410	−	0.888741i
$28$		0			0
$29$	4.47214			0.830455			0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214			0.830455			0.415227	+	0.909718i	$0.363702\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$	2.76393	+	7.23607i	0.481139	+	1.25964i
$34$		0			0
$35$	2.23607	−	2.23607i	0.377964	−	0.377964i
$36$		0			0
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	−0.909312	−	0.416115i	$-0.863391\pi$
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	0.416115	+	0.909312i	$0.363391\pi$
$38$		0			0
$39$	−6.70820	−	3.00000i	−1.07417	−	0.480384i
$40$		0			0
$41$		−	8.94427i		−	1.39686i	−0.715678	−	0.698430i	$-0.753882\pi$
$41$		−	8.94427i		−	1.39686i	0.715678	−	0.698430i	$-0.246118\pi$
$42$		0			0
$43$	−3.00000	−	3.00000i	−0.457496	−	0.457496i	0.440337	−	0.897833i	$-0.354859\pi$
$43$	−3.00000	−	3.00000i	−0.457496	−	0.457496i	−0.897833	+	0.440337i	$0.854859\pi$
$44$		0			0
$45$	−5.00000	+	4.47214i	−0.745356	+	0.666667i
$46$		0			0
$47$	6.70820	+	6.70820i	0.978492	+	0.978492i	0.999774	−	0.0212814i	$-0.00677460\pi$
$47$	6.70820	+	6.70820i	0.978492	+	0.978492i	−0.0212814	+	0.999774i	$0.506775\pi$
$48$		0			0
$49$			5.00000i			0.714286i
$50$		0			0
$51$	−5.00000	−	2.23607i	−0.700140	−	0.313112i
$52$		0			0
$53$	−2.23607	+	2.23607i	−0.307148	+	0.307148i	−0.843802	−	0.536655i	$-0.819688\pi$
$53$	−2.23607	+	2.23607i	−0.307148	+	0.307148i	0.536655	+	0.843802i	$0.319688\pi$
$54$		0			0
$55$		−	10.0000i		−	1.34840i
$56$		0			0
$57$	1.23607	+	3.23607i	0.163721	+	0.428628i
$58$		0			0
$59$	−8.94427			−1.16445			−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427			−1.16445			−0.582223	+	0.813029i	$0.697817\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$	−0.236068	+	4.23607i	−0.0297418	+	0.533694i
$64$		0			0
$65$	6.70820	+	6.70820i	0.832050	+	0.832050i
$66$		0			0
$67$	−1.00000	+	1.00000i	−0.122169	+	0.122169i	−0.765548	−	0.643379i	$-0.777532\pi$
$67$	−1.00000	+	1.00000i	−0.122169	+	0.122169i	0.643379	+	0.765548i	$0.277532\pi$
$68$		0			0
$69$	2.23607	−	5.00000i	0.269191	−	0.601929i
$70$		0			0
$71$		−	4.47214i		−	0.530745i	−0.964146	−	0.265372i	$-0.914505\pi$
$71$		−	4.47214i		−	0.530745i	0.964146	−	0.265372i	$-0.0854949\pi$
$72$		0			0
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	0.763202	−	0.646160i	$-0.223626\pi$
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	−0.646160	+	0.763202i	$0.723626\pi$
$74$		0			0
$75$	8.09017	−	3.09017i	0.934172	−	0.356822i
$76$		0			0
$77$	−4.47214	−	4.47214i	−0.509647	−	0.509647i
$78$		0			0
$79$			6.00000i			0.675053i	0.941316	+	0.337526i	$0.109590\pi$
$79$			6.00000i			0.675053i	−0.941316	+	0.337526i	$0.890410\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	−6.70820	+	6.70820i	−0.736321	+	0.736321i	−0.971864	−	0.235543i	$-0.924313\pi$
$83$	−6.70820	+	6.70820i	−0.736321	+	0.736321i	0.235543	+	0.971864i	$0.424313\pi$
$84$		0			0
$85$	5.00000	+	5.00000i	0.542326	+	0.542326i
$86$		0			0
$87$	7.23607	−	2.76393i	0.775788	−	0.296325i
$88$		0			0
$89$	−4.47214			−0.474045			−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214			−0.474045			−0.237023	+	0.971504i	$0.576172\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		0			0
$93$	6.47214	−	2.47214i	0.671129	−	0.256349i
$94$		0			0
$95$		−	4.47214i		−	0.458831i
$96$		0			0
$97$	9.00000	−	9.00000i	0.913812	−	0.913812i	−0.0827581	−	0.996570i	$-0.526373\pi$
$97$	9.00000	−	9.00000i	0.913812	−	0.913812i	0.996570	+	0.0827581i	$0.0263729\pi$
$98$		0			0
$99$	8.94427	+	10.0000i	0.898933	+	1.00504i
$100$		0			0
$101$			8.94427i			0.889988i	0.895533	+	0.444994i	$0.146794\pi$
$101$			8.94427i			0.889988i	−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$	1.00000	+	1.00000i	0.0985329	+	0.0985329i	0.754655	−	0.656122i	$-0.227804\pi$
$103$	1.00000	+	1.00000i	0.0985329	+	0.0985329i	−0.656122	+	0.754655i	$0.727804\pi$
$104$		0			0
$105$	2.23607	−	5.00000i	0.218218	−	0.487950i
$106$		0			0
$107$	−2.23607	−	2.23607i	−0.216169	−	0.216169i	0.590713	−	0.806882i	$-0.298847\pi$
$107$	−2.23607	−	2.23607i	−0.216169	−	0.216169i	−0.806882	+	0.590713i	$0.798847\pi$
$108$		0			0
$109$		−	8.00000i		−	0.766261i	−0.923694	−	0.383131i	$-0.874846\pi$
$109$		−	8.00000i		−	0.766261i	0.923694	−	0.383131i	$-0.125154\pi$
$110$		0			0
$111$	−3.00000	+	6.70820i	−0.284747	+	0.636715i
$112$		0			0
$113$	−2.23607	+	2.23607i	−0.210352	+	0.210352i	−0.804417	−	0.594065i	$-0.797522\pi$
$113$	−2.23607	+	2.23607i	−0.210352	+	0.210352i	0.594065	+	0.804417i	$0.297522\pi$
$114$		0			0
$115$	−5.00000	+	5.00000i	−0.466252	+	0.466252i
$116$		0			0
$117$	−12.7082	−	0.708204i	−1.17487	−	0.0654735i
$118$		0			0
$119$	4.47214			0.409960
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$	−5.52786	−	14.4721i	−0.498431	−	1.30491i
$124$		0			0
$125$	−11.1803			−1.00000
$126$		0			0
$127$	−13.0000	+	13.0000i	−1.15356	+	1.15356i	−0.167731	+	0.985833i	$0.553644\pi$
$127$	−13.0000	+	13.0000i	−1.15356	+	1.15356i	−0.985833	+	0.167731i	$0.946356\pi$
$128$		0			0
$129$	−6.70820	−	3.00000i	−0.590624	−	0.264135i
$130$		0			0
$131$			4.47214i			0.390732i	0.980730	+	0.195366i	$0.0625895\pi$
$131$			4.47214i			0.390732i	−0.980730	+	0.195366i	$0.937410\pi$
$132$		0			0
$133$	−2.00000	−	2.00000i	−0.173422	−	0.173422i
$134$		0			0
$135$	−5.32624	+	10.3262i	−0.458410	+	0.888741i
$136$		0			0
$137$	6.70820	+	6.70820i	0.573121	+	0.573121i	0.932999	−	0.359879i	$-0.117182\pi$
$137$	6.70820	+	6.70820i	0.573121	+	0.573121i	−0.359879	+	0.932999i	$0.617182\pi$
$138$		0			0
$139$		−	14.0000i		−	1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$		−	14.0000i		−	1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$		0			0
$141$	15.0000	+	6.70820i	1.26323	+	0.564933i
$142$		0			0
$143$	13.4164	−	13.4164i	1.12194	−	1.12194i
$144$		0			0
$145$	−10.0000			−0.830455
$146$		0			0
$147$	3.09017	+	8.09017i	0.254873	+	0.667266i
$148$		0			0
$149$	13.4164			1.09911			0.549557	−	0.835456i	$-0.314796\pi$
$149$	13.4164			1.09911			0.549557	+	0.835456i	$0.314796\pi$
$150$		0			0
$151$	4.00000			0.325515			0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000			0.325515			0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$	−9.47214	−	0.527864i	−0.765777	−	0.0426753i
$154$		0			0
$155$	−8.94427			−0.718421
$156$		0			0
$157$	9.00000	−	9.00000i	0.718278	−	0.718278i	−0.249974	−	0.968252i	$-0.580422\pi$
$157$	9.00000	−	9.00000i	0.718278	−	0.718278i	0.968252	+	0.249974i	$0.0804222\pi$
$158$		0			0
$159$	−2.23607	+	5.00000i	−0.177332	+	0.396526i
$160$		0			0
$161$			4.47214i			0.352454i
$162$		0			0
$163$	17.0000	+	17.0000i	1.33154	+	1.33154i	0.903991	+	0.427552i	$0.140624\pi$
$163$	17.0000	+	17.0000i	1.33154	+	1.33154i	0.427552	+	0.903991i	$0.359376\pi$
$164$		0			0
$165$	−6.18034	−	16.1803i	−0.481139	−	1.25964i
$166$		0			0
$167$	−11.1803	−	11.1803i	−0.865161	−	0.865161i	0.126771	−	0.991932i	$-0.459539\pi$
$167$	−11.1803	−	11.1803i	−0.865161	−	0.865161i	−0.991932	+	0.126771i	$0.959539\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	4.00000	+	4.47214i	0.305888	+	0.341993i
$172$		0			0
$173$	15.6525	−	15.6525i	1.19004	−	1.19004i	0.212979	−	0.977057i	$-0.431683\pi$
$173$	15.6525	−	15.6525i	1.19004	−	1.19004i	0.977057	−	0.212979i	$-0.0683166\pi$
$174$		0			0
$175$	−5.00000	+	5.00000i	−0.377964	+	0.377964i
$176$		0			0
$177$	−14.4721	+	5.52786i	−1.08779	+	0.415500i
$178$		0			0
$179$	8.94427			0.668526			0.334263	−	0.942480i	$-0.391513\pi$
$179$	8.94427			0.668526			0.334263	+	0.942480i	$0.391513\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$	−9.70820	+	3.70820i	−0.717651	+	0.274118i
$184$		0			0
$185$	6.70820	−	6.70820i	0.493197	−	0.493197i
$186$		0			0
$187$	10.0000	−	10.0000i	0.731272	−	0.731272i
$188$		0			0
$189$	2.23607	+	7.00000i	0.162650	+	0.509175i
$190$		0			0
$191$		−	4.47214i		−	0.323592i	−0.986824	−	0.161796i	$-0.948271\pi$
$191$		−	4.47214i		−	0.323592i	0.986824	−	0.161796i	$-0.0517287\pi$
$192$		0			0
$193$	−3.00000	−	3.00000i	−0.215945	−	0.215945i	0.590842	−	0.806787i	$-0.298796\pi$
$193$	−3.00000	−	3.00000i	−0.215945	−	0.215945i	−0.806787	+	0.590842i	$0.798796\pi$
$194$		0			0
$195$	15.0000	+	6.70820i	1.07417	+	0.480384i
$196$		0			0
$197$	6.70820	+	6.70820i	0.477940	+	0.477940i	0.904472	−	0.426532i	$-0.140265\pi$
$197$	6.70820	+	6.70820i	0.477940	+	0.477940i	−0.426532	+	0.904472i	$0.640265\pi$
$198$		0			0
$199$		−	18.0000i		−	1.27599i	−0.770042	−	0.637993i	$-0.779765\pi$
$199$		−	18.0000i		−	1.27599i	0.770042	−	0.637993i	$-0.220235\pi$
$200$		0			0
$201$	−1.00000	+	2.23607i	−0.0705346	+	0.157720i
$202$		0			0
$203$	−4.47214	+	4.47214i	−0.313882	+	0.313882i
$204$		0			0
$205$			20.0000i			1.39686i
$206$		0			0
$207$	0.527864	−	9.47214i	0.0366891	−	0.658359i
$208$		0			0
$209$	−8.94427			−0.618688
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$	−2.76393	−	7.23607i	−0.189382	−	0.495807i
$214$		0			0
$215$	6.70820	+	6.70820i	0.457496	+	0.457496i
$216$		0			0
$217$	−4.00000	+	4.00000i	−0.271538	+	0.271538i
$218$		0			0
$219$	2.23607	+	1.00000i	0.151099	+	0.0675737i
$220$		0			0
$221$			13.4164i			0.902485i
$222$		0			0
$223$	−3.00000	−	3.00000i	−0.200895	−	0.200895i	0.599489	−	0.800383i	$-0.295371\pi$
$223$	−3.00000	−	3.00000i	−0.200895	−	0.200895i	−0.800383	+	0.599489i	$0.795371\pi$
$224$		0			0
$225$	11.1803	−	10.0000i	0.745356	−	0.666667i
$226$		0			0
$227$	−11.1803	−	11.1803i	−0.742065	−	0.742065i	0.230910	−	0.972975i	$-0.425830\pi$
$227$	−11.1803	−	11.1803i	−0.742065	−	0.742065i	−0.972975	+	0.230910i	$0.925830\pi$
$228$		0			0
$229$			12.0000i			0.792982i	0.918039	+	0.396491i	$0.129772\pi$
$229$			12.0000i			0.792982i	−0.918039	+	0.396491i	$0.870228\pi$
$230$		0			0
$231$	−10.0000	−	4.47214i	−0.657952	−	0.294245i
$232$		0			0
$233$	−20.1246	+	20.1246i	−1.31841	+	1.31841i	−0.403370	+	0.915037i	$0.632161\pi$
$233$	−20.1246	+	20.1246i	−1.31841	+	1.31841i	−0.915037	+	0.403370i	$0.867839\pi$
$234$		0			0
$235$	−15.0000	−	15.0000i	−0.978492	−	0.978492i
$236$		0			0
$237$	3.70820	+	9.70820i	0.240874	+	0.630616i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−26.0000			−1.67481			−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000			−1.67481			−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$	−3.90983	−	15.0902i	−0.250816	−	0.968035i
$244$		0			0
$245$		−	11.1803i		−	0.714286i
$246$		0			0
$247$	6.00000	−	6.00000i	0.381771	−	0.381771i
$248$		0			0
$249$	−6.70820	+	15.0000i	−0.425115	+	0.950586i
$250$		0			0
$251$		−	13.4164i		−	0.846836i	−0.905934	−	0.423418i	$-0.860830\pi$
$251$		−	13.4164i		−	0.846836i	0.905934	−	0.423418i	$-0.139170\pi$
$252$		0			0
$253$	10.0000	+	10.0000i	0.628695	+	0.628695i
$254$		0			0
$255$	11.1803	+	5.00000i	0.700140	+	0.313112i
$256$		0			0
$257$	15.6525	+	15.6525i	0.976375	+	0.976375i	0.999727	−	0.0233527i	$-0.00743406\pi$
$257$	15.6525	+	15.6525i	0.976375	+	0.976375i	−0.0233527	+	0.999727i	$0.507434\pi$
$258$		0			0
$259$		−	6.00000i		−	0.372822i
$260$		0			0
$261$	10.0000	−	8.94427i	0.618984	−	0.553637i
$262$		0			0
$263$	−15.6525	+	15.6525i	−0.965173	+	0.965173i	−0.999414	−	0.0342406i	$-0.989099\pi$
$263$	−15.6525	+	15.6525i	−0.965173	+	0.965173i	0.0342406	+	0.999414i	$0.489099\pi$
$264$		0			0
$265$	5.00000	−	5.00000i	0.307148	−	0.307148i
$266$		0			0
$267$	−7.23607	+	2.76393i	−0.442840	+	0.169150i
$268$		0			0
$269$	−22.3607			−1.36335			−0.681677	−	0.731653i	$-0.738749\pi$
$269$	−22.3607			−1.36335			−0.681677	+	0.731653i	$0.738749\pi$
$270$		0			0
$271$	12.0000			0.728948			0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000			0.728948			0.364474	+	0.931214i	$0.381249\pi$
$272$		0			0
$273$	9.70820	−	3.70820i	0.587567	−	0.224431i
$274$		0			0
$275$			22.3607i			1.34840i
$276$		0			0
$277$	9.00000	−	9.00000i	0.540758	−	0.540758i	−0.382993	−	0.923751i	$-0.625107\pi$
$277$	9.00000	−	9.00000i	0.540758	−	0.540758i	0.923751	+	0.382993i	$0.125107\pi$
$278$		0			0
$279$	8.94427	−	8.00000i	0.535480	−	0.478947i
$280$		0			0
$281$		−	26.8328i		−	1.60071i	−0.599525	−	0.800356i	$-0.704644\pi$
$281$		−	26.8328i		−	1.60071i	0.599525	−	0.800356i	$-0.295356\pi$
$282$		0			0
$283$	17.0000	+	17.0000i	1.01055	+	1.01055i	0.999944	+	0.0106013i	$0.00337456\pi$
$283$	17.0000	+	17.0000i	1.01055	+	1.01055i	0.0106013	+	0.999944i	$0.496625\pi$
$284$		0			0
$285$	−2.76393	−	7.23607i	−0.163721	−	0.428628i
$286$		0			0
$287$	8.94427	+	8.94427i	0.527964	+	0.527964i
$288$		0			0
$289$		−	7.00000i		−	0.411765i
$290$		0			0
$291$	9.00000	−	20.1246i	0.527589	−	1.17973i
$292$		0			0
$293$	6.70820	−	6.70820i	0.391897	−	0.391897i	−0.483466	−	0.875363i	$-0.660622\pi$
$293$	6.70820	−	6.70820i	0.391897	−	0.391897i	0.875363	+	0.483466i	$0.160622\pi$
$294$		0			0
$295$	20.0000			1.16445
$296$		0			0
$297$	20.6525	+	10.6525i	1.19838	+	0.618119i
$298$		0			0
$299$	−13.4164			−0.775891
$300$		0			0
$301$	6.00000			0.345834
$302$		0			0
$303$	5.52786	+	14.4721i	0.317567	+	0.831402i
$304$		0			0
$305$	13.4164			0.768221
$306$		0			0
$307$	−21.0000	+	21.0000i	−1.19853	+	1.19853i	−0.223928	+	0.974606i	$0.571888\pi$
$307$	−21.0000	+	21.0000i	−1.19853	+	1.19853i	−0.974606	+	0.223928i	$0.928112\pi$
$308$		0			0
$309$	2.23607	+	1.00000i	0.127205	+	0.0568880i
$310$		0			0
$311$		−	4.47214i		−	0.253592i	−0.991929	−	0.126796i	$-0.959531\pi$
$311$		−	4.47214i		−	0.253592i	0.991929	−	0.126796i	$-0.0404693\pi$
$312$		0			0
$313$	−3.00000	−	3.00000i	−0.169570	−	0.169570i	0.617220	−	0.786790i	$-0.288259\pi$
$313$	−3.00000	−	3.00000i	−0.169570	−	0.169570i	−0.786790	+	0.617220i	$0.788259\pi$
$314$		0			0
$315$	0.527864	−	9.47214i	0.0297418	−	0.533694i
$316$		0			0
$317$	−20.1246	−	20.1246i	−1.13031	−	1.13031i	−0.990125	−	0.140186i	$-0.955230\pi$
$317$	−20.1246	−	20.1246i	−1.13031	−	1.13031i	−0.140186	−	0.990125i	$-0.544770\pi$
$318$		0			0
$319$			20.0000i			1.11979i
$320$		0			0
$321$	−5.00000	−	2.23607i	−0.279073	−	0.124805i
$322$		0			0
$323$	4.47214	−	4.47214i	0.248836	−	0.248836i
$324$		0			0
$325$	−15.0000	−	15.0000i	−0.832050	−	0.832050i
$326$		0			0
$327$	−4.94427	−	12.9443i	−0.273419	−	0.715820i
$328$		0			0
$329$	−13.4164			−0.739671
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$	−0.708204	+	12.7082i	−0.0388093	+	0.696405i
$334$		0			0
$335$	2.23607	−	2.23607i	0.122169	−	0.122169i
$336$		0			0
$337$	9.00000	−	9.00000i	0.490261	−	0.490261i	−0.418127	−	0.908388i	$-0.637313\pi$
$337$	9.00000	−	9.00000i	0.490261	−	0.490261i	0.908388	+	0.418127i	$0.137313\pi$
$338$		0			0
$339$	−2.23607	+	5.00000i	−0.121447	+	0.271563i
$340$		0			0
$341$			17.8885i			0.968719i
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$	−5.00000	+	11.1803i	−0.269191	+	0.601929i
$346$		0			0
$347$	24.5967	+	24.5967i	1.32042	+	1.32042i	0.913433	+	0.406990i	$0.133422\pi$
$347$	24.5967	+	24.5967i	1.32042	+	1.32042i	0.406990	+	0.913433i	$0.366578\pi$
$348$		0			0
$349$		−	4.00000i		−	0.214115i	−0.994253	−	0.107058i	$-0.965857\pi$
$349$		−	4.00000i		−	0.214115i	0.994253	−	0.107058i	$-0.0341429\pi$
$350$		0			0
$351$	−21.0000	+	6.70820i	−1.12090	+	0.358057i
$352$		0			0
$353$	6.70820	−	6.70820i	0.357042	−	0.357042i	−0.505680	−	0.862721i	$-0.668758\pi$
$353$	6.70820	−	6.70820i	0.357042	−	0.357042i	0.862721	+	0.505680i	$0.168758\pi$
$354$		0			0
$355$			10.0000i			0.530745i
$356$		0			0
$357$	7.23607	−	2.76393i	0.382973	−	0.146283i
$358$		0			0
$359$	35.7771			1.88824			0.944121	−	0.329598i	$-0.106913\pi$
$359$	35.7771			1.88824			0.944121	+	0.329598i	$0.106913\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	−14.5623	+	5.56231i	−0.764323	+	0.291945i
$364$		0			0
$365$	−2.23607	−	2.23607i	−0.117041	−	0.117041i
$366$		0			0
$367$	−1.00000	+	1.00000i	−0.0521996	+	0.0521996i	−0.732725	−	0.680525i	$-0.761752\pi$
$367$	−1.00000	+	1.00000i	−0.0521996	+	0.0521996i	0.680525	+	0.732725i	$0.261752\pi$
$368$		0			0
$369$	−17.8885	−	20.0000i	−0.931240	−	1.04116i
$370$		0			0
$371$		−	4.47214i		−	0.232182i
$372$		0			0
$373$	−23.0000	−	23.0000i	−1.19089	−	1.19089i	−0.976816	−	0.214078i	$-0.931325\pi$
$373$	−23.0000	−	23.0000i	−1.19089	−	1.19089i	−0.214078	−	0.976816i	$-0.568675\pi$
$374$		0			0
$375$	−18.0902	+	6.90983i	−0.934172	+	0.356822i
$376$		0			0
$377$	−13.4164	−	13.4164i	−0.690980	−	0.690980i
$378$		0			0
$379$			2.00000i			0.102733i	0.998680	+	0.0513665i	$0.0163577\pi$
$379$			2.00000i			0.102733i	−0.998680	+	0.0513665i	$0.983642\pi$
$380$		0			0
$381$	−13.0000	+	29.0689i	−0.666010	+	1.48924i
$382$		0			0
$383$	−6.70820	+	6.70820i	−0.342773	+	0.342773i	−0.857409	−	0.514636i	$-0.827927\pi$
$383$	−6.70820	+	6.70820i	−0.342773	+	0.342773i	0.514636	+	0.857409i	$0.327927\pi$
$384$		0			0
$385$	10.0000	+	10.0000i	0.509647	+	0.509647i
$386$		0			0
$387$	−12.7082	−	0.708204i	−0.645994	−	0.0360000i
$388$		0			0
$389$	13.4164			0.680239			0.340119	−	0.940382i	$-0.389532\pi$
$389$	13.4164			0.680239			0.340119	+	0.940382i	$0.389532\pi$
$390$		0			0
$391$	−10.0000			−0.505722
$392$		0			0
$393$	2.76393	+	7.23607i	0.139422	+	0.365011i
$394$		0			0
$395$		−	13.4164i		−	0.675053i
$396$		0			0
$397$	−11.0000	+	11.0000i	−0.552074	+	0.552074i	−0.927039	−	0.374965i	$-0.877655\pi$
$397$	−11.0000	+	11.0000i	−0.552074	+	0.552074i	0.374965	+	0.927039i	$0.377655\pi$
$398$		0			0
$399$	−4.47214	−	2.00000i	−0.223887	−	0.100125i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−12.0000	−	12.0000i	−0.597763	−	0.597763i
$404$		0			0
$405$	−2.23607	+	20.0000i	−0.111111	+	0.993808i
$406$		0			0
$407$	−13.4164	−	13.4164i	−0.665027	−	0.665027i
$408$		0			0
$409$			12.0000i			0.593362i	0.954977	+	0.296681i	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$		0			0
$411$	15.0000	+	6.70820i	0.739895	+	0.330891i
$412$		0			0
$413$	8.94427	−	8.94427i	0.440119	−	0.440119i
$414$		0			0
$415$	15.0000	−	15.0000i	0.736321	−	0.736321i
$416$		0			0
$417$	−8.65248	−	22.6525i	−0.423714	−	1.10930i
$418$		0			0
$419$	26.8328			1.31087			0.655434	−	0.755252i	$-0.272486\pi$
$419$	26.8328			1.31087			0.655434	+	0.755252i	$0.272486\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$		0			0
$423$	28.4164	+	1.58359i	1.38165	+	0.0769969i
$424$		0			0
$425$	−11.1803	−	11.1803i	−0.542326	−	0.542326i
$426$		0			0
$427$	6.00000	−	6.00000i	0.290360	−	0.290360i
$428$		0			0
$429$	13.4164	−	30.0000i	0.647750	−	1.44841i
$430$		0			0
$431$			31.3050i			1.50791i	0.656928	+	0.753953i	$0.271855\pi$
$431$			31.3050i			1.50791i	−0.656928	+	0.753953i	$0.728145\pi$
$432$		0			0
$433$	17.0000	+	17.0000i	0.816968	+	0.816968i	0.985668	−	0.168700i	$-0.0539568\pi$
$433$	17.0000	+	17.0000i	0.816968	+	0.816968i	−0.168700	+	0.985668i	$0.553957\pi$
$434$		0			0
$435$	−16.1803	+	6.18034i	−0.775788	+	0.296325i
$436$		0			0
$437$	4.47214	+	4.47214i	0.213931	+	0.213931i
$438$		0			0
$439$			22.0000i			1.05000i	0.851101	+	0.525001i	$0.175935\pi$
$439$			22.0000i			1.05000i	−0.851101	+	0.525001i	$0.824065\pi$
$440$		0			0
$441$	10.0000	+	11.1803i	0.476190	+	0.532397i
$442$		0			0
$443$	20.1246	−	20.1246i	0.956149	−	0.956149i	−0.0429290	−	0.999078i	$-0.513669\pi$
$443$	20.1246	−	20.1246i	0.956149	−	0.956149i	0.999078	+	0.0429290i	$0.0136689\pi$
$444$		0			0
$445$	10.0000			0.474045
$446$		0			0
$447$	21.7082	−	8.29180i	1.02676	−	0.392188i
$448$		0			0
$449$	4.47214			0.211053			0.105527	−	0.994416i	$-0.466347\pi$
$449$	4.47214			0.211053			0.105527	+	0.994416i	$0.466347\pi$
$450$		0			0
$451$	40.0000			1.88353
$452$		0			0
$453$	6.47214	−	2.47214i	0.304087	−	0.116151i
$454$		0			0
$455$	−13.4164			−0.628971
$456$		0			0
$457$	−3.00000	+	3.00000i	−0.140334	+	0.140334i	−0.773784	−	0.633450i	$-0.781638\pi$
$457$	−3.00000	+	3.00000i	−0.140334	+	0.140334i	0.633450	+	0.773784i	$0.281638\pi$
$458$		0			0
$459$	−15.6525	+	5.00000i	−0.730595	+	0.233380i
$460$		0			0
$461$		−	8.94427i		−	0.416576i	−0.978068	−	0.208288i	$-0.933211\pi$
$461$		−	8.94427i		−	0.416576i	0.978068	−	0.208288i	$-0.0667892\pi$
$462$		0			0
$463$	−19.0000	−	19.0000i	−0.883005	−	0.883005i	0.110834	−	0.993839i	$-0.464648\pi$
$463$	−19.0000	−	19.0000i	−0.883005	−	0.883005i	−0.993839	+	0.110834i	$0.964648\pi$
$464$		0			0
$465$	−14.4721	+	5.52786i	−0.671129	+	0.256349i
$466$		0			0
$467$	−2.23607	−	2.23607i	−0.103473	−	0.103473i	0.653475	−	0.756948i	$-0.273310\pi$
$467$	−2.23607	−	2.23607i	−0.103473	−	0.103473i	−0.756948	+	0.653475i	$0.773310\pi$
$468$		0			0
$469$		−	2.00000i		−	0.0923514i
$470$		0			0
$471$	9.00000	−	20.1246i	0.414698	−	0.927293i
$472$		0			0
$473$	13.4164	−	13.4164i	0.616887	−	0.616887i
$474$		0			0
$475$			10.0000i			0.458831i
$476$		0			0
$477$	−0.527864	+	9.47214i	−0.0241692	+	0.433699i
$478$		0			0
$479$	−35.7771			−1.63470			−0.817348	−	0.576144i	$-0.804557\pi$
$479$	−35.7771			−1.63470			−0.817348	+	0.576144i	$0.804557\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$	2.76393	+	7.23607i	0.125763	+	0.329252i
$484$		0			0
$485$	−20.1246	+	20.1246i	−0.913812	+	0.913812i
$486$		0			0
$487$	−21.0000	+	21.0000i	−0.951601	+	0.951601i	−0.998882	−	0.0472808i	$-0.984944\pi$
$487$	−21.0000	+	21.0000i	−0.951601	+	0.951601i	0.0472808	+	0.998882i	$0.484944\pi$
$488$		0			0
$489$	38.0132	+	17.0000i	1.71901	+	0.768767i
$490$		0			0
$491$			22.3607i			1.00912i	0.863376	+	0.504562i	$0.168346\pi$
$491$			22.3607i			1.00912i	−0.863376	+	0.504562i	$0.831654\pi$
$492$		0			0
$493$	−10.0000	−	10.0000i	−0.450377	−	0.450377i
$494$		0			0
$495$	−20.0000	−	22.3607i	−0.898933	−	1.00504i
$496$		0			0
$497$	4.47214	+	4.47214i	0.200603	+	0.200603i
$498$		0			0
$499$		−	38.0000i		−	1.70111i	−0.525883	−	0.850557i	$-0.676265\pi$
$499$		−	38.0000i		−	1.70111i	0.525883	−	0.850557i	$-0.323735\pi$
$500$		0			0
$501$	−25.0000	−	11.1803i	−1.11692	−	0.499501i
$502$		0			0
$503$	−6.70820	+	6.70820i	−0.299104	+	0.299104i	−0.840663	−	0.541559i	$-0.817834\pi$
$503$	−6.70820	+	6.70820i	−0.299104	+	0.299104i	0.541559	+	0.840663i	$0.317834\pi$
$504$		0			0
$505$		−	20.0000i		−	0.889988i
$506$		0			0
$507$	3.09017	+	8.09017i	0.137239	+	0.359297i
$508$		0			0
$509$	4.47214			0.198224			0.0991120	−	0.995076i	$-0.468400\pi$
$509$	4.47214			0.198224			0.0991120	+	0.995076i	$0.468400\pi$
$510$		0			0
$511$	−2.00000			−0.0884748
$512$		0			0
$513$	9.23607	+	4.76393i	0.407782	+	0.210333i
$514$		0			0
$515$	−2.23607	−	2.23607i	−0.0985329	−	0.0985329i
$516$		0			0
$517$	−30.0000	+	30.0000i	−1.31940	+	1.31940i
$518$		0			0
$519$	15.6525	−	35.0000i	0.687067	−	1.53633i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−3.00000	−	3.00000i	−0.131181	−	0.131181i	0.638468	−	0.769649i	$-0.279569\pi$
$523$	−3.00000	−	3.00000i	−0.131181	−	0.131181i	−0.769649	+	0.638468i	$0.779569\pi$
$524$		0			0
$525$	−5.00000	+	11.1803i	−0.218218	+	0.487950i
$526$		0			0
$527$	−8.94427	−	8.94427i	−0.389619	−	0.389619i
$528$		0			0
$529$			13.0000i			0.565217i
$530$		0			0
$531$	−20.0000	+	17.8885i	−0.867926	+	0.776297i
$532$		0			0
$533$	−26.8328	+	26.8328i	−1.16226	+	1.16226i
$534$		0			0
$535$	5.00000	+	5.00000i	0.216169	+	0.216169i
$536$		0			0
$537$	14.4721	−	5.52786i	0.624519	−	0.238545i
$538$		0			0
$539$	−22.3607			−0.963143
$540$		0			0
$541$	−26.0000			−1.11783			−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000			−1.11783			−0.558914	+	0.829226i	$0.688782\pi$
$542$		0			0
$543$	22.6525	−	8.65248i	0.972111	−	0.371313i
$544$		0			0
$545$			17.8885i			0.766261i
$546$		0			0
$547$	27.0000	−	27.0000i	1.15444	−	1.15444i	0.168783	−	0.985653i	$-0.446016\pi$
$547$	27.0000	−	27.0000i	1.15444	−	1.15444i	0.985653	−	0.168783i	$-0.0539839\pi$
$548$		0			0
$549$	−13.4164	+	12.0000i	−0.572598	+	0.512148i
$550$		0			0
$551$			8.94427i			0.381039i
$552$		0			0
$553$	−6.00000	−	6.00000i	−0.255146	−	0.255146i
$554$		0			0
$555$	6.70820	−	15.0000i	0.284747	−	0.636715i
$556$		0			0
$557$	6.70820	+	6.70820i	0.284236	+	0.284236i	0.834796	−	0.550560i	$-0.185586\pi$
$557$	6.70820	+	6.70820i	0.284236	+	0.284236i	−0.550560	+	0.834796i	$0.685586\pi$
$558$		0			0
$559$			18.0000i			0.761319i
$560$		0			0
$561$	10.0000	−	22.3607i	0.422200	−	0.944069i
$562$		0			0
$563$	−6.70820	+	6.70820i	−0.282717	+	0.282717i	−0.834192	−	0.551475i	$-0.814065\pi$
$563$	−6.70820	+	6.70820i	−0.282717	+	0.282717i	0.551475	+	0.834192i	$0.314065\pi$
$564$		0			0
$565$	5.00000	−	5.00000i	0.210352	−	0.210352i
$566$		0			0
$567$	7.94427	+	9.94427i	0.333628	+	0.417620i
$568$		0			0
$569$	22.3607			0.937408			0.468704	−	0.883355i	$-0.344721\pi$
$569$	22.3607			0.937408			0.468704	+	0.883355i	$0.344721\pi$
$570$		0			0
$571$	−16.0000			−0.669579			−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000			−0.669579			−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$	−2.76393	−	7.23607i	−0.115465	−	0.302291i
$574$		0			0
$575$	11.1803	−	11.1803i	0.466252	−	0.466252i
$576$		0			0
$577$	−11.0000	+	11.0000i	−0.457936	+	0.457936i	−0.897977	−	0.440041i	$-0.854964\pi$
$577$	−11.0000	+	11.0000i	−0.457936	+	0.457936i	0.440041	+	0.897977i	$0.354964\pi$
$578$		0			0
$579$	−6.70820	−	3.00000i	−0.278783	−	0.124676i
$580$		0			0
$581$		−	13.4164i		−	0.556606i
$582$		0			0
$583$	−10.0000	−	10.0000i	−0.414158	−	0.414158i
$584$		0			0
$585$	28.4164	+	1.58359i	1.17487	+	0.0654735i
$586$		0			0
$587$	6.70820	+	6.70820i	0.276877	+	0.276877i	0.831861	−	0.554984i	$-0.187276\pi$
$587$	6.70820	+	6.70820i	0.276877	+	0.276877i	−0.554984	+	0.831861i	$0.687276\pi$
$588$		0			0
$589$			8.00000i			0.329634i
$590$		0			0
$591$	15.0000	+	6.70820i	0.617018	+	0.275939i
$592$		0			0
$593$	−11.1803	+	11.1803i	−0.459122	+	0.459122i	−0.898367	−	0.439246i	$-0.855246\pi$
$593$	−11.1803	+	11.1803i	−0.459122	+	0.459122i	0.439246	+	0.898367i	$0.355246\pi$
$594$		0			0
$595$	−10.0000			−0.409960
$596$		0			0
$597$	−11.1246	−	29.1246i	−0.455300	−	1.19199i
$598$		0			0
$599$	−35.7771			−1.46181			−0.730906	−	0.682478i	$-0.760902\pi$
$599$	−35.7771			−1.46181			−0.730906	+	0.682478i	$0.760902\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$	−0.236068	+	4.23607i	−0.00961343	+	0.172506i
$604$		0			0
$605$	20.1246			0.818182
$606$		0			0
$607$	27.0000	−	27.0000i	1.09590	−	1.09590i	0.101011	−	0.994885i	$-0.467792\pi$
$607$	27.0000	−	27.0000i	1.09590	−	1.09590i	0.994885	−	0.101011i	$-0.0322077\pi$
$608$		0			0
$609$	−4.47214	+	10.0000i	−0.181220	+	0.405220i
$610$		0			0
$611$		−	40.2492i		−	1.62831i
$612$		0			0
$613$	21.0000	+	21.0000i	0.848182	+	0.848182i	0.989906	−	0.141724i	$-0.0452646\pi$
$613$	21.0000	+	21.0000i	0.848182	+	0.848182i	−0.141724	+	0.989906i	$0.545265\pi$
$614$		0			0
$615$	12.3607	+	32.3607i	0.498431	+	1.30491i
$616$		0			0
$617$	−20.1246	−	20.1246i	−0.810186	−	0.810186i	0.174475	−	0.984662i	$-0.444177\pi$
$617$	−20.1246	−	20.1246i	−0.810186	−	0.810186i	−0.984662	+	0.174475i	$0.944177\pi$
$618$		0			0
$619$			42.0000i			1.68812i	0.536247	+	0.844061i	$0.319842\pi$
$619$			42.0000i			1.68812i	−0.536247	+	0.844061i	$0.680158\pi$
$620$		0			0
$621$	−5.00000	−	15.6525i	−0.200643	−	0.628112i
$622$		0			0
$623$	4.47214	−	4.47214i	0.179172	−	0.179172i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	−14.4721	+	5.52786i	−0.577961	+	0.220762i
$628$		0			0
$629$	13.4164			0.534947
$630$		0			0
$631$	4.00000			0.159237			0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000			0.159237			0.0796187	+	0.996825i	$0.474630\pi$
$632$		0			0
$633$	−25.8885	+	9.88854i	−1.02898	+	0.393034i
$634$		0			0
$635$	29.0689	−	29.0689i	1.15356	−	1.15356i
$636$		0			0
$637$	15.0000	−	15.0000i	0.594322	−	0.594322i
$638$		0			0
$639$	−8.94427	−	10.0000i	−0.353830	−	0.395594i
$640$		0			0
$641$			8.94427i			0.353278i	0.984276	+	0.176639i	$0.0565224\pi$
$641$			8.94427i			0.353278i	−0.984276	+	0.176639i	$0.943478\pi$
$642$		0			0
$643$	21.0000	+	21.0000i	0.828159	+	0.828159i	0.987262	−	0.159103i	$-0.0508601\pi$
$643$	21.0000	+	21.0000i	0.828159	+	0.828159i	−0.159103	+	0.987262i	$0.550860\pi$
$644$		0			0
$645$	15.0000	+	6.70820i	0.590624	+	0.264135i
$646$		0			0
$647$	15.6525	+	15.6525i	0.615362	+	0.615362i	0.944338	−	0.328976i	$-0.106704\pi$
$647$	15.6525	+	15.6525i	0.615362	+	0.615362i	−0.328976	+	0.944338i	$0.606704\pi$
$648$		0			0
$649$		−	40.0000i		−	1.57014i
$650$		0			0
$651$	−4.00000	+	8.94427i	−0.156772	+	0.350554i
$652$		0			0
$653$	−20.1246	+	20.1246i	−0.787537	+	0.787537i	−0.981090	−	0.193553i	$-0.937999\pi$
$653$	−20.1246	+	20.1246i	−0.787537	+	0.787537i	0.193553	+	0.981090i	$0.437999\pi$
$654$		0			0
$655$		−	10.0000i		−	0.390732i
$656$		0			0
$657$	4.23607	+	0.236068i	0.165265	+	0.00920989i
$658$		0			0
$659$	8.94427			0.348419			0.174210	−	0.984709i	$-0.444263\pi$
$659$	8.94427			0.348419			0.174210	+	0.984709i	$0.444263\pi$
$660$		0			0
$661$	2.00000			0.0777910			0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000			0.0777910			0.0388955	+	0.999243i	$0.487616\pi$
$662$		0			0
$663$	8.29180	+	21.7082i	0.322027	+	0.843077i
$664$		0			0
$665$	4.47214	+	4.47214i	0.173422	+	0.173422i
$666$		0			0
$667$	10.0000	−	10.0000i	0.387202	−	0.387202i
$668$		0			0
$669$	−6.70820	−	3.00000i	−0.259354	−	0.115987i
$670$		0			0
$671$		−	26.8328i		−	1.03587i
$672$		0			0
$673$	1.00000	+	1.00000i	0.0385472	+	0.0385472i	0.726118	−	0.687570i	$-0.241323\pi$
$673$	1.00000	+	1.00000i	0.0385472	+	0.0385472i	−0.687570	+	0.726118i	$0.741323\pi$
$674$		0			0
$675$	11.9098	−	23.0902i	0.458410	−	0.888741i
$676$		0			0
$677$	24.5967	+	24.5967i	0.945330	+	0.945330i	0.998581	−	0.0532513i	$-0.0169584\pi$
$677$	24.5967	+	24.5967i	0.945330	+	0.945330i	−0.0532513	+	0.998581i	$0.516958\pi$
$678$		0			0
$679$			18.0000i			0.690777i
$680$		0			0
$681$	−25.0000	−	11.1803i	−0.958002	−	0.428432i
$682$		0			0
$683$	20.1246	−	20.1246i	0.770047	−	0.770047i	−0.208068	−	0.978114i	$-0.566717\pi$
$683$	20.1246	−	20.1246i	0.770047	−	0.770047i	0.978114	+	0.208068i	$0.0667174\pi$
$684$		0			0
$685$	−15.0000	−	15.0000i	−0.573121	−	0.573121i
$686$		0			0
$687$	7.41641	+	19.4164i	0.282954	+	0.740782i
$688$		0			0
$689$	13.4164			0.511124
$690$		0			0
$691$	−48.0000			−1.82601			−0.913003	−	0.407953i	$-0.866243\pi$
$691$	−48.0000			−1.82601			−0.913003	+	0.407953i	$0.866243\pi$
$692$		0			0
$693$	−18.9443	−	1.05573i	−0.719633	−	0.0401038i
$694$		0			0
$695$			31.3050i			1.18746i
$696$		0			0
$697$	−20.0000	+	20.0000i	−0.757554	+	0.757554i
$698$		0			0
$699$	−20.1246	+	45.0000i	−0.761183	+	1.70206i
$700$		0			0
$701$		−	35.7771i		−	1.35128i	−0.737231	−	0.675641i	$-0.763867\pi$
$701$		−	35.7771i		−	1.35128i	0.737231	−	0.675641i	$-0.236133\pi$
$702$		0			0
$703$	−6.00000	−	6.00000i	−0.226294	−	0.226294i
$704$		0			0
$705$	−33.5410	−	15.0000i	−1.26323	−	0.564933i
$706$		0			0
$707$	−8.94427	−	8.94427i	−0.336384	−	0.336384i
$708$		0			0
$709$		−	28.0000i		−	1.05156i	−0.850620	−	0.525781i	$-0.823773\pi$
$709$		−	28.0000i		−	1.05156i	0.850620	−	0.525781i	$-0.176227\pi$
$710$		0			0
$711$	12.0000	+	13.4164i	0.450035	+	0.503155i
$712$		0			0
$713$	8.94427	−	8.94427i	0.334966	−	0.334966i
$714$		0			0
$715$	−30.0000	+	30.0000i	−1.12194	+	1.12194i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−35.7771			−1.33426			−0.667130	−	0.744941i	$-0.732478\pi$
$719$	−35.7771			−1.33426			−0.667130	+	0.744941i	$0.732478\pi$
$720$		0			0
$721$	−2.00000			−0.0744839
$722$		0			0
$723$	−42.0689	+	16.0689i	−1.56456	+	0.597608i
$724$		0			0
$725$	22.3607			0.830455
$726$		0			0
$727$	−21.0000	+	21.0000i	−0.778847	+	0.778847i	−0.979635	−	0.200788i	$-0.935650\pi$
$727$	−21.0000	+	21.0000i	−0.778847	+	0.778847i	0.200788	+	0.979635i	$0.435650\pi$
$728$		0			0
$729$	−15.6525	−	22.0000i	−0.579721	−	0.814815i
$730$		0			0
$731$			13.4164i			0.496224i
$732$		0			0
$733$	1.00000	+	1.00000i	0.0369358	+	0.0369358i	0.725333	−	0.688398i	$-0.241686\pi$
$733$	1.00000	+	1.00000i	0.0369358	+	0.0369358i	−0.688398	+	0.725333i	$0.741686\pi$
$734$		0			0
$735$	−6.90983	−	18.0902i	−0.254873	−	0.667266i
$736$		0			0
$737$	−4.47214	−	4.47214i	−0.164733	−	0.164733i
$738$		0			0
$739$		−	14.0000i		−	0.514998i	−0.966279	−	0.257499i	$-0.917102\pi$
$739$		−	14.0000i		−	0.514998i	0.966279	−	0.257499i	$-0.0828985\pi$
$740$		0			0
$741$	6.00000	−	13.4164i	0.220416	−	0.492864i
$742$		0			0
$743$	11.1803	−	11.1803i	0.410167	−	0.410167i	−0.471630	−	0.881797i	$-0.656334\pi$
$743$	11.1803	−	11.1803i	0.410167	−	0.410167i	0.881797	+	0.471630i	$0.156334\pi$
$744$		0			0
$745$	−30.0000			−1.09911
$746$		0			0
$747$	−1.58359	+	28.4164i	−0.0579406	+	1.03970i
$748$		0			0
$749$	4.47214			0.163408
$750$		0			0
$751$	12.0000			0.437886			0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000			0.437886			0.218943	+	0.975738i	$0.429739\pi$
$752$		0			0
$753$	−8.29180	−	21.7082i	−0.302170	−	0.791091i
$754$		0			0
$755$	−8.94427			−0.325515
$756$		0			0
$757$	9.00000	−	9.00000i	0.327111	−	0.327111i	−0.524376	−	0.851487i	$-0.675701\pi$
$757$	9.00000	−	9.00000i	0.327111	−	0.327111i	0.851487	+	0.524376i	$0.175701\pi$
$758$		0			0
$759$	22.3607	+	10.0000i	0.811641	+	0.362977i
$760$		0			0
$761$			17.8885i			0.648459i	0.945978	+	0.324230i	$0.105105\pi$
$761$			17.8885i			0.648459i	−0.945978	+	0.324230i	$0.894895\pi$
$762$		0			0
$763$	8.00000	+	8.00000i	0.289619	+	0.289619i
$764$		0			0
$765$	21.1803	+	1.18034i	0.765777	+	0.0426753i
$766$		0			0
$767$	26.8328	+	26.8328i	0.968877	+	0.968877i
$768$		0			0
$769$		−	8.00000i		−	0.288487i	−0.989542	−	0.144244i	$-0.953925\pi$
$769$		−	8.00000i		−	0.288487i	0.989542	−	0.144244i	$-0.0460749\pi$
$770$		0			0
$771$	35.0000	+	15.6525i	1.26049	+	0.563710i
$772$		0			0
$773$	6.70820	−	6.70820i	0.241277	−	0.241277i	−0.576101	−	0.817378i	$-0.695427\pi$
$773$	6.70820	−	6.70820i	0.241277	−	0.241277i	0.817378	+	0.576101i	$0.195427\pi$
$774$		0			0
$775$	20.0000			0.718421
$776$		0			0
$777$	−3.70820	−	9.70820i	−0.133031	−	0.348280i
$778$		0			0
$779$	17.8885			0.640924
$780$		0			0
$781$	20.0000			0.715656
$782$		0			0
$783$	10.6525	−	20.6525i	0.380688	−	0.738059i
$784$		0			0
$785$	−20.1246	+	20.1246i	−0.718278	+	0.718278i
$786$		0			0
$787$	19.0000	−	19.0000i	0.677277	−	0.677277i	−0.282106	−	0.959383i	$-0.591033\pi$
$787$	19.0000	−	19.0000i	0.677277	−	0.677277i	0.959383	+	0.282106i	$0.0910331\pi$
$788$		0			0
$789$	−15.6525	+	35.0000i	−0.557243	+	1.24603i
$790$		0			0
$791$		−	4.47214i		−	0.159011i
$792$		0			0
$793$	18.0000	+	18.0000i	0.639199	+	0.639199i
$794$		0			0
$795$	5.00000	−	11.1803i	0.177332	−	0.396526i
$796$		0			0
$797$	−20.1246	−	20.1246i	−0.712850	−	0.712850i	0.254280	−	0.967131i	$-0.418161\pi$
$797$	−20.1246	−	20.1246i	−0.712850	−	0.712850i	−0.967131	+	0.254280i	$0.918161\pi$
$798$		0			0
$799$		−	30.0000i		−	1.06132i
$800$		0			0
$801$	−10.0000	+	8.94427i	−0.353333	+	0.316030i
$802$		0			0
$803$	−4.47214	+	4.47214i	−0.157818	+	0.157818i
$804$		0			0
$805$		−	10.0000i		−	0.352454i
$806$		0			0
$807$	−36.1803	+	13.8197i	−1.27361	+	0.486475i
$808$		0			0
$809$	−4.47214			−0.157232			−0.0786160	−	0.996905i	$-0.525050\pi$
$809$	−4.47214			−0.157232			−0.0786160	+	0.996905i	$0.525050\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$		0			0
$813$	19.4164	−	7.41641i	0.680963	−	0.260105i
$814$		0			0
$815$	−38.0132	−	38.0132i	−1.33154	−	1.33154i
$816$		0			0
$817$	6.00000	−	6.00000i	0.209913	−	0.209913i
$818$		0			0
$819$	13.4164	−	12.0000i	0.468807	−	0.419314i
$820$		0			0
$821$			17.8885i			0.624314i	0.950030	+	0.312157i	$0.101052\pi$
$821$			17.8885i			0.624314i	−0.950030	+	0.312157i	$0.898948\pi$
$822$		0			0
$823$	17.0000	+	17.0000i	0.592583	+	0.592583i	0.938328	−	0.345746i	$-0.112374\pi$
$823$	17.0000	+	17.0000i	0.592583	+	0.592583i	−0.345746	+	0.938328i	$0.612374\pi$
$824$		0			0
$825$	13.8197	+	36.1803i	0.481139	+	1.25964i
$826$		0			0
$827$	−20.1246	−	20.1246i	−0.699801	−	0.699801i	0.264566	−	0.964368i	$-0.414771\pi$
$827$	−20.1246	−	20.1246i	−0.699801	−	0.699801i	−0.964368	+	0.264566i	$0.914771\pi$
$828$		0			0
$829$			16.0000i			0.555703i	0.960624	+	0.277851i	$0.0896223\pi$
$829$			16.0000i			0.555703i	−0.960624	+	0.277851i	$0.910378\pi$
$830$		0			0
$831$	9.00000	−	20.1246i	0.312207	−	0.698115i
$832$		0			0
$833$	11.1803	−	11.1803i	0.387376	−	0.387376i
$834$		0			0
$835$	25.0000	+	25.0000i	0.865161	+	0.865161i
$836$		0			0
$837$	9.52786	−	18.4721i	0.329331	−	0.638490i
$838$		0			0
$839$	35.7771			1.23516			0.617581	−	0.786507i	$-0.288113\pi$
$839$	35.7771			1.23516			0.617581	+	0.786507i	$0.288113\pi$
$840$		0			0
$841$	−9.00000			−0.310345
$842$		0			0
$843$	−16.5836	−	43.4164i	−0.571169	−	1.49534i
$844$		0			0
$845$		−	11.1803i		−	0.384615i
$846$		0			0
$847$	9.00000	−	9.00000i	0.309244	−	0.309244i
$848$		0			0
$849$	38.0132	+	17.0000i	1.30461	+	0.583438i
$850$		0			0
$851$			13.4164i			0.459909i
$852$		0			0
$853$	37.0000	+	37.0000i	1.26686	+	1.26686i	0.947703	+	0.319152i	$0.103398\pi$
$853$	37.0000	+	37.0000i	1.26686	+	1.26686i	0.319152	+	0.947703i	$0.396602\pi$
$854$		0			0
$855$	−8.94427	−	10.0000i	−0.305888	−	0.341993i
$856$		0			0
$857$	6.70820	+	6.70820i	0.229148	+	0.229148i	0.812337	−	0.583189i	$-0.198195\pi$
$857$	6.70820	+	6.70820i	0.229148	+	0.229148i	−0.583189	+	0.812337i	$0.698195\pi$
$858$		0			0
$859$		−	14.0000i		−	0.477674i	−0.971060	−	0.238837i	$-0.923234\pi$
$859$		−	14.0000i		−	0.477674i	0.971060	−	0.238837i	$-0.0767661\pi$
$860$		0			0
$861$	20.0000	+	8.94427i	0.681598	+	0.304820i
$862$		0			0
$863$	−24.5967	+	24.5967i	−0.837283	+	0.837283i	−0.988501	−	0.151217i	$-0.951681\pi$
$863$	−24.5967	+	24.5967i	−0.837283	+	0.837283i	0.151217	+	0.988501i	$0.451681\pi$
$864$		0			0
$865$	−35.0000	+	35.0000i	−1.19004	+	1.19004i
$866$		0			0
$867$	−4.32624	−	11.3262i	−0.146927	−	0.384659i
$868$		0			0
$869$	−26.8328			−0.910241
$870$		0			0
$871$	6.00000			0.203302
$872$		0			0
$873$	2.12461	−	38.1246i	0.0719072	−	1.29032i
$874$		0			0
$875$	11.1803	−	11.1803i	0.377964	−	0.377964i
$876$		0			0
$877$	9.00000	−	9.00000i	0.303908	−	0.303908i	−0.538632	−	0.842541i	$-0.681059\pi$
$877$	9.00000	−	9.00000i	0.303908	−	0.303908i	0.842541	+	0.538632i	$0.181059\pi$
$878$		0			0
$879$	6.70820	−	15.0000i	0.226262	−	0.505937i
$880$		0			0
$881$			44.7214i			1.50670i	0.657619	+	0.753350i	$0.271564\pi$
$881$			44.7214i			1.50670i	−0.657619	+	0.753350i	$0.728436\pi$
$882$		0			0
$883$	−23.0000	−	23.0000i	−0.774012	−	0.774012i	0.204794	−	0.978805i	$-0.434348\pi$
$883$	−23.0000	−	23.0000i	−0.774012	−	0.774012i	−0.978805	+	0.204794i	$0.934348\pi$
$884$		0			0
$885$	32.3607	−	12.3607i	1.08779	−	0.415500i
$886$		0			0
$887$	−20.1246	−	20.1246i	−0.675718	−	0.675718i	0.283310	−	0.959028i	$-0.408567\pi$
$887$	−20.1246	−	20.1246i	−0.675718	−	0.675718i	−0.959028	+	0.283310i	$0.908567\pi$
$888$		0			0
$889$		−	26.0000i		−	0.872012i
$890$		0			0
$891$	40.0000	+	4.47214i	1.34005	+	0.149822i
$892$		0			0
$893$	−13.4164	+	13.4164i	−0.448963	+	0.448963i
$894$		0			0
$895$	−20.0000			−0.668526
$896$		0			0
$897$	−21.7082	+	8.29180i	−0.724816	+	0.276855i
$898$		0			0
$899$	17.8885			0.596616
$900$		0			0
$901$	10.0000			0.333148
$902$		0			0
$903$	9.70820	−	3.70820i	0.323069	−	0.123401i
$904$		0			0
$905$	−31.3050			−1.04061
$906$		0			0
$907$	7.00000	−	7.00000i	0.232431	−	0.232431i	−0.581276	−	0.813707i	$-0.697446\pi$
$907$	7.00000	−	7.00000i	0.232431	−	0.232431i	0.813707	+	0.581276i	$0.197446\pi$
$908$		0			0
$909$	17.8885	+	20.0000i	0.593326	+	0.663358i
$910$		0			0
$911$			49.1935i			1.62985i	0.579564	+	0.814927i	$0.303223\pi$
$911$			49.1935i			1.62985i	−0.579564	+	0.814927i	$0.696777\pi$
$912$		0			0
$913$	−30.0000	−	30.0000i	−0.992855	−	0.992855i
$914$		0			0
$915$	21.7082	−	8.29180i	0.717651	−	0.274118i
$916$		0			0
$917$	−4.47214	−	4.47214i	−0.147683	−	0.147683i
$918$		0			0
$919$			22.0000i			0.725713i	0.931845	+	0.362857i	$0.118198\pi$
$919$			22.0000i			0.725713i	−0.931845	+	0.362857i	$0.881802\pi$
$920$		0			0
$921$	−21.0000	+	46.9574i	−0.691974	+	1.54730i
$922$		0			0
$923$	−13.4164	+	13.4164i	−0.441606	+	0.441606i
$924$		0			0
$925$	−15.0000	+	15.0000i	−0.493197	+	0.493197i
$926$		0			0
$927$	4.23607	+	0.236068i	0.139131	+	0.00775349i
$928$		0			0
$929$	4.47214			0.146726			0.0733630	−	0.997305i	$-0.476627\pi$
$929$	4.47214			0.146726			0.0733630	+	0.997305i	$0.476627\pi$
$930$		0			0
$931$	−10.0000			−0.327737
$932$		0			0
$933$	−2.76393	−	7.23607i	−0.0904871	−	0.236898i
$934$		0			0
$935$	−22.3607	+	22.3607i	−0.731272	+	0.731272i
$936$		0			0
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	−0.863575	−	0.504221i	$-0.831780\pi$
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	0.504221	+	0.863575i	$0.331780\pi$
$938$		0			0
$939$	−6.70820	−	3.00000i	−0.218914	−	0.0979013i
$940$		0			0
$941$		−	26.8328i		−	0.874725i	−0.899285	−	0.437362i	$-0.855913\pi$
$941$		−	26.8328i		−	0.874725i	0.899285	−	0.437362i	$-0.144087\pi$
$942$		0			0
$943$	−20.0000	−	20.0000i	−0.651290	−	0.651290i
$944$		0			0
$945$	−5.00000	−	15.6525i	−0.162650	−	0.509175i
$946$		0			0
$947$	−11.1803	−	11.1803i	−0.363312	−	0.363312i	0.501719	−	0.865031i	$-0.332701\pi$
$947$	−11.1803	−	11.1803i	−0.363312	−	0.363312i	−0.865031	+	0.501719i	$0.832701\pi$
$948$		0			0
$949$		−	6.00000i		−	0.194768i
$950$		0			0
$951$	−45.0000	−	20.1246i	−1.45922	−	0.652585i
$952$		0			0
$953$	33.5410	−	33.5410i	1.08650	−	1.08650i	0.0906141	−	0.995886i	$-0.471117\pi$
$953$	33.5410	−	33.5410i	1.08650	−	1.08650i	0.995886	−	0.0906141i	$-0.0288830\pi$
$954$		0			0
$955$			10.0000i			0.323592i
$956$		0			0
$957$	12.3607	+	32.3607i	0.399564	+	1.04607i
$958$		0			0
$959$	−13.4164			−0.433238
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	−9.47214	−	0.527864i	−0.305235	−	0.0170102i
$964$		0			0
$965$	6.70820	+	6.70820i	0.215945	+	0.215945i
$966$		0			0
$967$	7.00000	−	7.00000i	0.225105	−	0.225105i	−0.585539	−	0.810644i	$-0.699117\pi$
$967$	7.00000	−	7.00000i	0.225105	−	0.225105i	0.810644	+	0.585539i	$0.199117\pi$
$968$		0			0
$969$	4.47214	−	10.0000i	0.143666	−	0.321246i
$970$		0			0
$971$			22.3607i			0.717588i	0.933417	+	0.358794i	$0.116812\pi$
$971$			22.3607i			0.717588i	−0.933417	+	0.358794i	$0.883188\pi$
$972$		0			0
$973$	14.0000	+	14.0000i	0.448819	+	0.448819i
$974$		0			0
$975$	−33.5410	−	15.0000i	−1.07417	−	0.480384i
$976$		0			0
$977$	6.70820	+	6.70820i	0.214614	+	0.214614i	0.806224	−	0.591610i	$-0.201507\pi$
$977$	6.70820	+	6.70820i	0.214614	+	0.214614i	−0.591610	+	0.806224i	$0.701507\pi$
$978$		0			0
$979$		−	20.0000i		−	0.639203i
$980$		0			0
$981$	−16.0000	−	17.8885i	−0.510841	−	0.571137i
$982$		0			0
$983$	11.1803	−	11.1803i	0.356597	−	0.356597i	−0.505960	−	0.862557i	$-0.668862\pi$
$983$	11.1803	−	11.1803i	0.356597	−	0.356597i	0.862557	+	0.505960i	$0.168862\pi$
$984$		0			0
$985$	−15.0000	−	15.0000i	−0.477940	−	0.477940i
$986$		0			0
$987$	−21.7082	+	8.29180i	−0.690980	+	0.263931i
$988$		0			0
$989$	−13.4164			−0.426617
$990$		0			0
$991$	4.00000			0.127064			0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000			0.127064			0.0635321	+	0.997980i	$0.479763\pi$
$992$		0			0
$993$	−12.9443	+	4.94427i	−0.410774	+	0.156902i
$994$		0			0
$995$			40.2492i			1.27599i
$996$		0			0
$997$	−23.0000	+	23.0000i	−0.728417	+	0.728417i	−0.970304	−	0.241887i	$-0.922234\pi$
$997$	−23.0000	+	23.0000i	−0.728417	+	0.728417i	0.241887	+	0.970304i	$0.422234\pi$
$998$		0			0
$999$	6.70820	+	21.0000i	0.212238	+	0.664411i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.2.i.a.53.2	yes	4
3.2	odd	2	inner	60.2.i.a.53.1	yes	4
4.3	odd	2		240.2.v.b.113.1		4
5.2	odd	4	inner	60.2.i.a.17.1	✓	4
5.3	odd	4		300.2.i.a.257.2		4
5.4	even	2		300.2.i.a.293.1		4
8.3	odd	2		960.2.v.h.833.2		4
8.5	even	2		960.2.v.e.833.1		4
9.2	odd	6		1620.2.x.b.53.1		8
9.4	even	3		1620.2.x.b.593.2		8
9.5	odd	6		1620.2.x.b.593.1		8
9.7	even	3		1620.2.x.b.53.2		8
12.11	even	2		240.2.v.b.113.2		4
15.2	even	4	inner	60.2.i.a.17.2	yes	4
15.8	even	4		300.2.i.a.257.1		4
15.14	odd	2		300.2.i.a.293.2		4
20.3	even	4		1200.2.v.i.257.1		4
20.7	even	4		240.2.v.b.17.2		4
20.19	odd	2		1200.2.v.i.593.2		4
24.5	odd	2		960.2.v.e.833.2		4
24.11	even	2		960.2.v.h.833.1		4
40.27	even	4		960.2.v.h.257.1		4
40.37	odd	4		960.2.v.e.257.2		4
45.2	even	12		1620.2.x.b.377.2		8
45.7	odd	12		1620.2.x.b.377.1		8
45.22	odd	12		1620.2.x.b.917.1		8
45.32	even	12		1620.2.x.b.917.2		8
60.23	odd	4		1200.2.v.i.257.2		4
60.47	odd	4		240.2.v.b.17.1		4
60.59	even	2		1200.2.v.i.593.1		4
120.77	even	4		960.2.v.e.257.1		4
120.107	odd	4		960.2.v.h.257.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.i.a.17.1	✓	4	5.2	odd	4	inner
60.2.i.a.17.2	yes	4	15.2	even	4	inner
60.2.i.a.53.1	yes	4	3.2	odd	2	inner
60.2.i.a.53.2	yes	4	1.1	even	1	trivial
240.2.v.b.17.1		4	60.47	odd	4
240.2.v.b.17.2		4	20.7	even	4
240.2.v.b.113.1		4	4.3	odd	2
240.2.v.b.113.2		4	12.11	even	2
300.2.i.a.257.1		4	15.8	even	4
300.2.i.a.257.2		4	5.3	odd	4
300.2.i.a.293.1		4	5.4	even	2
300.2.i.a.293.2		4	15.14	odd	2
960.2.v.e.257.1		4	120.77	even	4
960.2.v.e.257.2		4	40.37	odd	4
960.2.v.e.833.1		4	8.5	even	2
960.2.v.e.833.2		4	24.5	odd	2
960.2.v.h.257.1		4	40.27	even	4
960.2.v.h.257.2		4	120.107	odd	4
960.2.v.h.833.1		4	24.11	even	2
960.2.v.h.833.2		4	8.3	odd	2
1200.2.v.i.257.1		4	20.3	even	4
1200.2.v.i.257.2		4	60.23	odd	4
1200.2.v.i.593.1		4	60.59	even	2
1200.2.v.i.593.2		4	20.19	odd	2
1620.2.x.b.53.1		8	9.2	odd	6
1620.2.x.b.53.2		8	9.7	even	3
1620.2.x.b.377.1		8	45.7	odd	12
1620.2.x.b.377.2		8	45.2	even	12
1620.2.x.b.593.1		8	9.5	odd	6
1620.2.x.b.593.2		8	9.4	even	3
1620.2.x.b.917.1		8	45.22	odd	12
1620.2.x.b.917.2		8	45.32	even	12

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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	60.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.61803 - 0.618034i) q^{3} -2.23607 q^{5} +(-1.00000 + 1.00000i) q^{7} +(2.23607 - 2.00000i) q^{9} +O(q^{10})\)	\(q+(1.61803 - 0.618034i) q^{3} -2.23607 q^{5} +(-1.00000 + 1.00000i) q^{7} +(2.23607 - 2.00000i) q^{9} +4.47214i q^{11} +(-3.00000 - 3.00000i) q^{13} +(-3.61803 + 1.38197i) q^{15} +(-2.23607 - 2.23607i) q^{17} +2.00000i q^{19} +(-1.00000 + 2.23607i) q^{21} +(2.23607 - 2.23607i) q^{23} +5.00000 q^{25} +(2.38197 - 4.61803i) q^{27} +4.47214 q^{29} +4.00000 q^{31} +(2.76393 + 7.23607i) q^{33} +(2.23607 - 2.23607i) q^{35} +(-3.00000 + 3.00000i) q^{37} +(-6.70820 - 3.00000i) q^{39} -8.94427i q^{41} +(-3.00000 - 3.00000i) q^{43} +(-5.00000 + 4.47214i) q^{45} +(6.70820 + 6.70820i) q^{47} +5.00000i q^{49} +(-5.00000 - 2.23607i) q^{51} +(-2.23607 + 2.23607i) q^{53} -10.0000i q^{55} +(1.23607 + 3.23607i) q^{57} -8.94427 q^{59} -6.00000 q^{61} +(-0.236068 + 4.23607i) q^{63} +(6.70820 + 6.70820i) q^{65} +(-1.00000 + 1.00000i) q^{67} +(2.23607 - 5.00000i) q^{69} -4.47214i q^{71} +(1.00000 + 1.00000i) q^{73} +(8.09017 - 3.09017i) q^{75} +(-4.47214 - 4.47214i) q^{77} +6.00000i q^{79} +(1.00000 - 8.94427i) q^{81} +(-6.70820 + 6.70820i) q^{83} +(5.00000 + 5.00000i) q^{85} +(7.23607 - 2.76393i) q^{87} -4.47214 q^{89} +6.00000 q^{91} +(6.47214 - 2.47214i) q^{93} -4.47214i q^{95} +(9.00000 - 9.00000i) q^{97} +(8.94427 + 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^3 - 4 * q^7	\( 4 q + 2 q^{3} - 4 q^{7} - 12 q^{13} - 10 q^{15} - 4 q^{21} + 20 q^{25} + 14 q^{27} + 16 q^{31} + 20 q^{33} - 12 q^{37} - 12 q^{43} - 20 q^{45} - 20 q^{51} - 4 q^{57} - 24 q^{61} + 8 q^{63} - 4 q^{67} + 4 q^{73} + 10 q^{75} + 4 q^{81} + 20 q^{85} + 20 q^{87} + 24 q^{91} + 8 q^{93} + 36 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^7 - 12 * q^13 - 10 * q^15 - 4 * q^21 + 20 * q^25 + 14 * q^27 + 16 * q^31 + 20 * q^33 - 12 * q^37 - 12 * q^43 - 20 * q^45 - 20 * q^51 - 4 * q^57 - 24 * q^61 + 8 * q^63 - 4 * q^67 + 4 * q^73 + 10 * q^75 + 4 * q^81 + 20 * q^85 + 20 * q^87 + 24 * q^91 + 8 * q^93 + 36 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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