LMFDB - Embedded newform 585.2.b.b.181.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(181,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, 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("585.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	585.b (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:	$4.67124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	585.181
Dual form			585.2.b.b.181.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{5} -4.00000i q^{7} -3.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{5} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{10} +(3.00000 + 2.00000i) q^{13} -4.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} +4.00000i q^{19} -1.00000i q^{20} -8.00000 q^{23} -1.00000 q^{25} +(2.00000 - 3.00000i) q^{26} -4.00000i q^{28} +8.00000 q^{29} -4.00000i q^{31} -5.00000i q^{32} +4.00000i q^{34} -4.00000 q^{35} -4.00000i q^{37} +4.00000 q^{38} -3.00000 q^{40} +2.00000i q^{41} +8.00000 q^{43} +8.00000i q^{46} -8.00000i q^{47} -9.00000 q^{49} +1.00000i q^{50} +(3.00000 + 2.00000i) q^{52} +12.0000 q^{53} -12.0000 q^{56} -8.00000i q^{58} -2.00000 q^{61} -4.00000 q^{62} -7.00000 q^{64} +(2.00000 - 3.00000i) q^{65} +12.0000i q^{67} -4.00000 q^{68} +4.00000i q^{70} +4.00000i q^{71} -4.00000i q^{73} -4.00000 q^{74} +4.00000i q^{76} -8.00000 q^{79} +1.00000i q^{80} +2.00000 q^{82} +12.0000i q^{83} +4.00000i q^{85} -8.00000i q^{86} +18.0000i q^{89} +(8.00000 - 12.0000i) q^{91} -8.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} -4.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} - 2 q^{10} + 6 q^{13} - 8 q^{14} - 2 q^{16} - 8 q^{17} - 16 q^{23} - 2 q^{25} + 4 q^{26} + 16 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} - 18 q^{49} + 6 q^{52} + 24 q^{53} - 24 q^{56} - 4 q^{61} - 8 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{68} - 8 q^{74} - 16 q^{79} + 4 q^{82} + 16 q^{91} - 16 q^{92} - 16 q^{94} + 8 q^{95}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^10 + 6 * q^13 - 8 * q^14 - 2 * q^16 - 8 * q^17 - 16 * q^23 - 2 * q^25 + 4 * q^26 + 16 * q^29 - 8 * q^35 + 8 * q^38 - 6 * q^40 + 16 * q^43 - 18 * q^49 + 6 * q^52 + 24 * q^53 - 24 * q^56 - 4 * q^61 - 8 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^68 - 8 * q^74 - 16 * q^79 + 4 * q^82 + 16 * q^91 - 16 * q^92 - 16 * q^94 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$496$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.00000i		−	0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$		−	1.00000i		−	0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		−	3.00000i		−	1.06066i
$9$		0			0
$10$	−1.00000			−0.316228
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$14$	−4.00000			−1.06904
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−4.00000			−0.970143			−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000			−0.970143			−0.485071	+	0.874475i	$0.661206\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		−	1.00000i		−	0.223607i
$21$		0			0
$22$		0			0
$23$	−8.00000			−1.66812			−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000			−1.66812			−0.834058	+	0.551677i	$0.813988\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	2.00000	−	3.00000i	0.392232	−	0.588348i
$27$		0			0
$28$		−	4.00000i		−	0.755929i
$29$	8.00000			1.48556			0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000			1.48556			0.742781	+	0.669534i	$0.233506\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$		−	5.00000i		−	0.883883i
$33$		0			0
$34$			4.00000i			0.685994i
$35$	−4.00000			−0.676123
$36$		0			0
$37$		−	4.00000i		−	0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$		−	4.00000i		−	0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	4.00000			0.648886
$39$		0			0
$40$	−3.00000			−0.474342
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$			8.00000i			1.17954i
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$			1.00000i			0.141421i
$51$		0			0
$52$	3.00000	+	2.00000i	0.416025	+	0.277350i
$53$	12.0000			1.64833			0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000			1.64833			0.824163	+	0.566352i	$0.191646\pi$
$54$		0			0
$55$		0			0
$56$	−12.0000			−1.60357
$57$		0			0
$58$		−	8.00000i		−	1.05045i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000			−0.508001
$63$		0			0
$64$	−7.00000			−0.875000
$65$	2.00000	−	3.00000i	0.248069	−	0.372104i
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	−4.00000			−0.485071
$69$		0			0
$70$			4.00000i			0.478091i
$71$			4.00000i			0.474713i	0.971423	+	0.237356i	$0.0762809\pi$
$71$			4.00000i			0.474713i	−0.971423	+	0.237356i	$0.923719\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−4.00000			−0.464991
$75$		0			0
$76$			4.00000i			0.458831i
$77$		0			0
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$			1.00000i			0.111803i
$81$		0			0
$82$	2.00000			0.220863
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$			4.00000i			0.433861i
$86$		−	8.00000i		−	0.862662i
$87$		0			0
$88$		0			0
$89$			18.0000i			1.90800i	0.299813	+	0.953998i	$0.403076\pi$
$89$			18.0000i			1.90800i	−0.299813	+	0.953998i	$0.596924\pi$
$90$		0			0
$91$	8.00000	−	12.0000i	0.838628	−	1.25794i
$92$	−8.00000			−0.834058
$93$		0			0
$94$	−8.00000			−0.825137
$95$	4.00000			0.410391
$96$		0			0
$97$		−	4.00000i		−	0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$		−	4.00000i		−	0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$			9.00000i			0.909137i
$99$		0			0
$100$	−1.00000			−0.100000
$101$	16.0000			1.59206			0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000			1.59206			0.796030	+	0.605257i	$0.206930\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$	6.00000	−	9.00000i	0.588348	−	0.882523i
$105$		0			0
$106$		−	12.0000i		−	1.16554i
$107$	8.00000			0.773389			0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000			0.773389			0.386695	+	0.922208i	$0.373617\pi$
$108$		0			0
$109$			8.00000i			0.766261i	0.923694	+	0.383131i	$0.125154\pi$
$109$			8.00000i			0.766261i	−0.923694	+	0.383131i	$0.874846\pi$
$110$		0			0
$111$		0			0
$112$			4.00000i			0.377964i
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$			8.00000i			0.746004i
$116$	8.00000			0.742781
$117$		0			0
$118$		0			0
$119$			16.0000i			1.46672i
$120$		0			0
$121$	11.0000			1.00000
$122$			2.00000i			0.181071i
$123$		0			0
$124$		−	4.00000i		−	0.359211i
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		−	3.00000i		−	0.265165i
$129$		0			0
$130$	−3.00000	−	2.00000i	−0.263117	−	0.175412i
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	16.0000			1.38738
$134$	12.0000			1.03664
$135$		0			0
$136$			12.0000i			1.02899i
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$	−4.00000			−0.338062
$141$		0			0
$142$	4.00000			0.335673
$143$		0			0
$144$		0			0
$145$		−	8.00000i		−	0.664364i
$146$	−4.00000			−0.331042
$147$		0			0
$148$		−	4.00000i		−	0.328798i
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$		0			0
$151$		−	20.0000i		−	1.62758i	−0.581161	−	0.813788i	$-0.697401\pi$
$151$		−	20.0000i		−	1.62758i	0.581161	−	0.813788i	$-0.302599\pi$
$152$	12.0000			0.973329
$153$		0			0
$154$		0			0
$155$	−4.00000			−0.321288
$156$		0			0
$157$	−22.0000			−1.75579			−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000			−1.75579			−0.877896	+	0.478852i	$0.841053\pi$
$158$			8.00000i			0.636446i
$159$		0			0
$160$	−5.00000			−0.395285
$161$			32.0000i			2.52195i
$162$		0			0
$163$		−	12.0000i		−	0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$		−	12.0000i		−	0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$			2.00000i			0.156174i
$165$		0			0
$166$	12.0000			0.931381
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$	4.00000			0.306786
$171$		0			0
$172$	8.00000			0.609994
$173$	4.00000			0.304114			0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000			0.304114			0.152057	+	0.988372i	$0.451410\pi$
$174$		0			0
$175$			4.00000i			0.302372i
$176$		0			0
$177$		0			0
$178$	18.0000			1.34916
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$	−12.0000	−	8.00000i	−0.889499	−	0.592999i
$183$		0			0
$184$			24.0000i			1.76930i
$185$	−4.00000			−0.294086
$186$		0			0
$187$		0			0
$188$		−	8.00000i		−	0.583460i
$189$		0			0
$190$		−	4.00000i		−	0.290191i
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	−4.00000			−0.287183
$195$		0			0
$196$	−9.00000			−0.642857
$197$			26.0000i			1.85242i	0.377004	+	0.926212i	$0.376954\pi$
$197$			26.0000i			1.85242i	−0.377004	+	0.926212i	$0.623046\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$			3.00000i			0.212132i
$201$		0			0
$202$		−	16.0000i		−	1.12576i
$203$		−	32.0000i		−	2.24596i
$204$		0			0
$205$	2.00000			0.139686
$206$		−	4.00000i		−	0.278693i
$207$		0			0
$208$	−3.00000	−	2.00000i	−0.208013	−	0.138675i
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	12.0000			0.824163
$213$		0			0
$214$		−	8.00000i		−	0.546869i
$215$		−	8.00000i		−	0.545595i
$216$		0			0
$217$	−16.0000			−1.08615
$218$	8.00000			0.541828
$219$		0			0
$220$		0			0
$221$	−12.0000	−	8.00000i	−0.807207	−	0.538138i
$222$		0			0
$223$			4.00000i			0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$			4.00000i			0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	−20.0000			−1.33631
$225$		0			0
$226$		−	4.00000i		−	0.266076i
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$			16.0000i			1.05731i	0.848837	+	0.528655i	$0.177303\pi$
$229$			16.0000i			1.05731i	−0.848837	+	0.528655i	$0.822697\pi$
$230$	8.00000			0.527504
$231$		0			0
$232$		−	24.0000i		−	1.57568i
$233$	−28.0000			−1.83434			−0.917170	−	0.398495i	$-0.869533\pi$
$233$	−28.0000			−1.83434			−0.917170	+	0.398495i	$0.869533\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$	16.0000			1.03713
$239$		−	20.0000i		−	1.29369i	−0.762620	−	0.646846i	$-0.776088\pi$
$239$		−	20.0000i		−	1.29369i	0.762620	−	0.646846i	$-0.223912\pi$
$240$		0			0
$241$			24.0000i			1.54598i	0.634421	+	0.772988i	$0.281239\pi$
$241$			24.0000i			1.54598i	−0.634421	+	0.772988i	$0.718761\pi$
$242$		−	11.0000i		−	0.707107i
$243$		0			0
$244$	−2.00000			−0.128037
$245$			9.00000i			0.574989i
$246$		0			0
$247$	−8.00000	+	12.0000i	−0.509028	+	0.763542i
$248$	−12.0000			−0.762001
$249$		0			0
$250$	1.00000			0.0632456
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$		0			0
$254$		−	12.0000i		−	0.752947i
$255$		0			0
$256$	−17.0000			−1.06250
$257$	12.0000			0.748539			0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000			0.748539			0.374270	+	0.927320i	$0.377893\pi$
$258$		0			0
$259$	−16.0000			−0.994192
$260$	2.00000	−	3.00000i	0.124035	−	0.186052i
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		−	12.0000i		−	0.737154i
$266$		−	16.0000i		−	0.981023i
$267$		0			0
$268$			12.0000i			0.733017i
$269$	16.0000			0.975537			0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000			0.975537			0.487769	+	0.872973i	$0.337811\pi$
$270$		0			0
$271$			20.0000i			1.21491i	0.794353	+	0.607457i	$0.207810\pi$
$271$			20.0000i			1.21491i	−0.794353	+	0.607457i	$0.792190\pi$
$272$	4.00000			0.242536
$273$		0			0
$274$	6.00000			0.362473
$275$		0			0
$276$		0			0
$277$	−14.0000			−0.841178			−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000			−0.841178			−0.420589	+	0.907251i	$0.638177\pi$
$278$			20.0000i			1.19952i
$279$		0			0
$280$			12.0000i			0.717137i
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$	−16.0000			−0.951101			−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000			−0.951101			−0.475551	+	0.879688i	$0.657751\pi$
$284$			4.00000i			0.237356i
$285$		0			0
$286$		0			0
$287$	8.00000			0.472225
$288$		0			0
$289$	−1.00000			−0.0588235
$290$	−8.00000			−0.469776
$291$		0			0
$292$		−	4.00000i		−	0.234082i
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$		0			0
$296$	−12.0000			−0.697486
$297$		0			0
$298$	−10.0000			−0.579284
$299$	−24.0000	−	16.0000i	−1.38796	−	0.925304i
$300$		0			0
$301$		−	32.0000i		−	1.84445i
$302$	−20.0000			−1.15087
$303$		0			0
$304$		−	4.00000i		−	0.229416i
$305$			2.00000i			0.114520i
$306$		0			0
$307$			28.0000i			1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$			28.0000i			1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$		0			0
$309$		0			0
$310$			4.00000i			0.227185i
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$			22.0000i			1.24153i
$315$		0			0
$316$	−8.00000			−0.450035
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$		0			0
$319$		0			0
$320$			7.00000i			0.391312i
$321$		0			0
$322$	32.0000			1.78329
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$	−3.00000	−	2.00000i	−0.166410	−	0.110940i
$326$	−12.0000			−0.664619
$327$		0			0
$328$	6.00000			0.331295
$329$	−32.0000			−1.76422
$330$		0			0
$331$		−	4.00000i		−	0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$		−	4.00000i		−	0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$			12.0000i			0.658586i
$333$		0			0
$334$	8.00000			0.437741
$335$	12.0000			0.655630
$336$		0			0
$337$	18.0000			0.980522			0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000			0.980522			0.490261	+	0.871576i	$0.336901\pi$
$338$	12.0000	−	5.00000i	0.652714	−	0.271964i
$339$		0			0
$340$			4.00000i			0.216930i
$341$		0			0
$342$		0			0
$343$			8.00000i			0.431959i
$344$		−	24.0000i		−	1.29399i
$345$		0			0
$346$		−	4.00000i		−	0.215041i
$347$	−32.0000			−1.71785			−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000			−1.71785			−0.858925	+	0.512101i	$0.828867\pi$
$348$		0			0
$349$		−	16.0000i		−	0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	4.00000			0.213809
$351$		0			0
$352$		0			0
$353$			18.0000i			0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$			18.0000i			0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$		0			0
$355$	4.00000			0.212298
$356$			18.0000i			0.953998i
$357$		0			0
$358$		0			0
$359$		−	28.0000i		−	1.47778i	−0.673824	−	0.738892i	$-0.735349\pi$
$359$		−	28.0000i		−	1.47778i	0.673824	−	0.738892i	$-0.264651\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		−	6.00000i		−	0.315353i
$363$		0			0
$364$	8.00000	−	12.0000i	0.419314	−	0.628971i
$365$	−4.00000			−0.209370
$366$		0			0
$367$	12.0000			0.626395			0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000			0.626395			0.313197	+	0.949688i	$0.398600\pi$
$368$	8.00000			0.417029
$369$		0			0
$370$			4.00000i			0.207950i
$371$		−	48.0000i		−	2.49204i
$372$		0			0
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$		0			0
$376$	−24.0000			−1.23771
$377$	24.0000	+	16.0000i	1.23606	+	0.824042i
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$	4.00000			0.205196
$381$		0			0
$382$			8.00000i			0.409316i
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$		0			0
$385$		0			0
$386$	−4.00000			−0.203595
$387$		0			0
$388$		−	4.00000i		−	0.203069i
$389$	16.0000			0.811232			0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000			0.811232			0.405616	+	0.914044i	$0.367057\pi$
$390$		0			0
$391$	32.0000			1.61831
$392$			27.0000i			1.36371i
$393$		0			0
$394$	26.0000			1.30986
$395$			8.00000i			0.402524i
$396$		0			0
$397$		−	20.0000i		−	1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$		−	20.0000i		−	1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			0.0500000
$401$			10.0000i			0.499376i	0.968326	+	0.249688i	$0.0803281\pi$
$401$			10.0000i			0.499376i	−0.968326	+	0.249688i	$0.919672\pi$
$402$		0			0
$403$	8.00000	−	12.0000i	0.398508	−	0.597763i
$404$	16.0000			0.796030
$405$		0			0
$406$	−32.0000			−1.58813
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		−	2.00000i		−	0.0987730i
$411$		0			0
$412$	4.00000			0.197066
$413$		0			0
$414$		0			0
$415$	12.0000			0.589057
$416$	10.0000	−	15.0000i	0.490290	−	0.735436i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$			16.0000i			0.779792i	0.920859	+	0.389896i	$0.127489\pi$
$421$			16.0000i			0.779792i	−0.920859	+	0.389896i	$0.872511\pi$
$422$		−	12.0000i		−	0.584151i
$423$		0			0
$424$		−	36.0000i		−	1.74831i
$425$	4.00000			0.194029
$426$		0			0
$427$			8.00000i			0.387147i
$428$	8.00000			0.386695
$429$		0			0
$430$	−8.00000			−0.385794
$431$			12.0000i			0.578020i	0.957326	+	0.289010i	$0.0933260\pi$
$431$			12.0000i			0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$			16.0000i			0.768025i
$435$		0			0
$436$			8.00000i			0.383131i
$437$		−	32.0000i		−	1.53077i
$438$		0			0
$439$	−32.0000			−1.52728			−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000			−1.52728			−0.763638	+	0.645644i	$0.776589\pi$
$440$		0			0
$441$		0			0
$442$	−8.00000	+	12.0000i	−0.380521	+	0.570782i
$443$	−24.0000			−1.14027			−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000			−1.14027			−0.570137	+	0.821549i	$0.693110\pi$
$444$		0			0
$445$	18.0000			0.853282
$446$	4.00000			0.189405
$447$		0			0
$448$			28.0000i			1.32288i
$449$		−	6.00000i		−	0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$		−	6.00000i		−	0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$		0			0
$451$		0			0
$452$	4.00000			0.188144
$453$		0			0
$454$	−12.0000			−0.563188
$455$	−12.0000	−	8.00000i	−0.562569	−	0.375046i
$456$		0			0
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	16.0000			0.747631
$459$		0			0
$460$			8.00000i			0.373002i
$461$			34.0000i			1.58354i	0.610821	+	0.791769i	$0.290840\pi$
$461$			34.0000i			1.58354i	−0.610821	+	0.791769i	$0.709160\pi$
$462$		0			0
$463$		−	20.0000i		−	0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$		−	20.0000i		−	0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	−8.00000			−0.371391
$465$		0			0
$466$			28.0000i			1.29707i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	48.0000			2.21643
$470$			8.00000i			0.369012i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	4.00000i		−	0.183533i
$476$			16.0000i			0.733359i
$477$		0			0
$478$	−20.0000			−0.914779
$479$			12.0000i			0.548294i	0.961688	+	0.274147i	$0.0883955\pi$
$479$			12.0000i			0.548294i	−0.961688	+	0.274147i	$0.911605\pi$
$480$		0			0
$481$	8.00000	−	12.0000i	0.364769	−	0.547153i
$482$	24.0000			1.09317
$483$		0			0
$484$	11.0000			0.500000
$485$	−4.00000			−0.181631
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$			6.00000i			0.271607i
$489$		0			0
$490$	9.00000			0.406579
$491$	−40.0000			−1.80517			−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000			−1.80517			−0.902587	+	0.430507i	$0.858335\pi$
$492$		0			0
$493$	−32.0000			−1.44121
$494$	12.0000	+	8.00000i	0.539906	+	0.359937i
$495$		0			0
$496$			4.00000i			0.179605i
$497$	16.0000			0.717698
$498$		0			0
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$			1.00000i			0.0447214i
$501$		0			0
$502$		−	24.0000i		−	1.07117i
$503$	−16.0000			−0.713405			−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000			−0.713405			−0.356702	+	0.934218i	$0.616099\pi$
$504$		0			0
$505$		−	16.0000i		−	0.711991i
$506$		0			0
$507$		0			0
$508$	12.0000			0.532414
$509$			18.0000i			0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$509$			18.0000i			0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$		0			0
$511$	−16.0000			−0.707798
$512$			11.0000i			0.486136i
$513$		0			0
$514$		−	12.0000i		−	0.529297i
$515$		−	4.00000i		−	0.176261i
$516$		0			0
$517$		0			0
$518$			16.0000i			0.703000i
$519$		0			0
$520$	−9.00000	−	6.00000i	−0.394676	−	0.263117i
$521$	16.0000			0.700973			0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000			0.700973			0.350486	+	0.936568i	$0.386016\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	41.0000			1.78261
$530$	−12.0000			−0.521247
$531$		0			0
$532$	16.0000			0.693688
$533$	−4.00000	+	6.00000i	−0.173259	+	0.259889i
$534$		0			0
$535$		−	8.00000i		−	0.345870i
$536$	36.0000			1.55496
$537$		0			0
$538$		−	16.0000i		−	0.689809i
$539$		0			0
$540$		0			0
$541$			8.00000i			0.343947i	0.985102	+	0.171973i	$0.0550143\pi$
$541$			8.00000i			0.343947i	−0.985102	+	0.171973i	$0.944986\pi$
$542$	20.0000			0.859074
$543$		0			0
$544$			20.0000i			0.857493i
$545$	8.00000			0.342682
$546$		0			0
$547$	−16.0000			−0.684111			−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000			−0.684111			−0.342055	+	0.939680i	$0.611123\pi$
$548$			6.00000i			0.256307i
$549$		0			0
$550$		0			0
$551$			32.0000i			1.36325i
$552$		0			0
$553$			32.0000i			1.36078i
$554$			14.0000i			0.594803i
$555$		0			0
$556$	−20.0000			−0.848189
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$		0			0
$559$	24.0000	+	16.0000i	1.01509	+	0.676728i
$560$	4.00000			0.169031
$561$		0			0
$562$	−18.0000			−0.759284
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$		−	4.00000i		−	0.168281i
$566$			16.0000i			0.672530i
$567$		0			0
$568$	12.0000			0.503509
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		0			0
$574$		−	8.00000i		−	0.333914i
$575$	8.00000			0.333623
$576$		0			0
$577$		−	4.00000i		−	0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$		−	4.00000i		−	0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$			1.00000i			0.0415945i
$579$		0			0
$580$		−	8.00000i		−	0.332182i
$581$	48.0000			1.99138
$582$		0			0
$583$		0			0
$584$	−12.0000			−0.496564
$585$		0			0
$586$	−6.00000			−0.247858
$587$			20.0000i			0.825488i	0.910847	+	0.412744i	$0.135430\pi$
$587$			20.0000i			0.825488i	−0.910847	+	0.412744i	$0.864570\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$		0			0
$592$			4.00000i			0.164399i
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$		0			0
$595$	16.0000			0.655936
$596$		−	10.0000i		−	0.409616i
$597$		0			0
$598$	−16.0000	+	24.0000i	−0.654289	+	0.981433i
$599$	16.0000			0.653742			0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000			0.653742			0.326871	+	0.945069i	$0.394006\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$	−32.0000			−1.30422
$603$		0			0
$604$		−	20.0000i		−	0.813788i
$605$		−	11.0000i		−	0.447214i
$606$		0			0
$607$	−20.0000			−0.811775			−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000			−0.811775			−0.405887	+	0.913923i	$0.633038\pi$
$608$	20.0000			0.811107
$609$		0			0
$610$	2.00000			0.0809776
$611$	16.0000	−	24.0000i	0.647291	−	0.970936i
$612$		0			0
$613$			44.0000i			1.77714i	0.458738	+	0.888572i	$0.348302\pi$
$613$			44.0000i			1.77714i	−0.458738	+	0.888572i	$0.651698\pi$
$614$	28.0000			1.12999
$615$		0			0
$616$		0			0
$617$		−	38.0000i		−	1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$		−	38.0000i		−	1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$		0			0
$619$			28.0000i			1.12542i	0.826656	+	0.562708i	$0.190240\pi$
$619$			28.0000i			1.12542i	−0.826656	+	0.562708i	$0.809760\pi$
$620$	−4.00000			−0.160644
$621$		0			0
$622$			8.00000i			0.320771i
$623$	72.0000			2.88462
$624$		0			0
$625$	1.00000			0.0400000
$626$			10.0000i			0.399680i
$627$		0			0
$628$	−22.0000			−0.877896
$629$			16.0000i			0.637962i
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$			24.0000i			0.954669i
$633$		0			0
$634$	2.00000			0.0794301
$635$		−	12.0000i		−	0.476205i
$636$		0			0
$637$	−27.0000	−	18.0000i	−1.06978	−	0.713186i
$638$		0			0
$639$		0			0
$640$	−3.00000			−0.118585
$641$	−24.0000			−0.947943			−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000			−0.947943			−0.473972	+	0.880540i	$0.657180\pi$
$642$		0			0
$643$		−	4.00000i		−	0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$		−	4.00000i		−	0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$			32.0000i			1.26098i
$645$		0			0
$646$	−16.0000			−0.629512
$647$	−32.0000			−1.25805			−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000			−1.25805			−0.629025	+	0.777385i	$0.716546\pi$
$648$		0			0
$649$		0			0
$650$	−2.00000	+	3.00000i	−0.0784465	+	0.117670i
$651$		0			0
$652$		−	12.0000i		−	0.469956i
$653$	4.00000			0.156532			0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000			0.156532			0.0782660	+	0.996933i	$0.475062\pi$
$654$		0			0
$655$		0			0
$656$		−	2.00000i		−	0.0780869i
$657$		0			0
$658$			32.0000i			1.24749i
$659$	−24.0000			−0.934907			−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000			−0.934907			−0.467454	+	0.884018i	$0.654829\pi$
$660$		0			0
$661$		−	32.0000i		−	1.24466i	−0.782757	−	0.622328i	$-0.786187\pi$
$661$		−	32.0000i		−	1.24466i	0.782757	−	0.622328i	$-0.213813\pi$
$662$	−4.00000			−0.155464
$663$		0			0
$664$	36.0000			1.39707
$665$		−	16.0000i		−	0.620453i
$666$		0			0
$667$	−64.0000			−2.47809
$668$			8.00000i			0.309529i
$669$		0			0
$670$		−	12.0000i		−	0.463600i
$671$		0			0
$672$		0			0
$673$	−50.0000			−1.92736			−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000			−1.92736			−0.963679	+	0.267063i	$0.913947\pi$
$674$		−	18.0000i		−	0.693334i
$675$		0			0
$676$	5.00000	+	12.0000i	0.192308	+	0.461538i
$677$	−36.0000			−1.38359			−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000			−1.38359			−0.691796	+	0.722093i	$0.743180\pi$
$678$		0			0
$679$	−16.0000			−0.614024
$680$	12.0000			0.460179
$681$		0			0
$682$		0			0
$683$		−	20.0000i		−	0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$		−	20.0000i		−	0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$	8.00000			0.305441
$687$		0			0
$688$	−8.00000			−0.304997
$689$	36.0000	+	24.0000i	1.37149	+	0.914327i
$690$		0			0
$691$			28.0000i			1.06517i	0.846376	+	0.532585i	$0.178779\pi$
$691$			28.0000i			1.06517i	−0.846376	+	0.532585i	$0.821221\pi$
$692$	4.00000			0.152057
$693$		0			0
$694$			32.0000i			1.21470i
$695$			20.0000i			0.758643i
$696$		0			0
$697$		−	8.00000i		−	0.303022i
$698$	−16.0000			−0.605609
$699$		0			0
$700$			4.00000i			0.151186i
$701$	−24.0000			−0.906467			−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000			−0.906467			−0.453234	+	0.891392i	$0.649730\pi$
$702$		0			0
$703$	16.0000			0.603451
$704$		0			0
$705$		0			0
$706$	18.0000			0.677439
$707$		−	64.0000i		−	2.40697i
$708$		0			0
$709$		−	16.0000i		−	0.600893i	−0.953799	−	0.300446i	$-0.902864\pi$
$709$		−	16.0000i		−	0.600893i	0.953799	−	0.300446i	$-0.0971356\pi$
$710$		−	4.00000i		−	0.150117i
$711$		0			0
$712$	54.0000			2.02374
$713$			32.0000i			1.19841i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−28.0000			−1.04495
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$		−	16.0000i		−	0.595871i
$722$		−	3.00000i		−	0.111648i
$723$		0			0
$724$	6.00000			0.222988
$725$	−8.00000			−0.297113
$726$		0			0
$727$	44.0000			1.63187			0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000			1.63187			0.815935	+	0.578144i	$0.196223\pi$
$728$	−36.0000	−	24.0000i	−1.33425	−	0.889499i
$729$		0			0
$730$			4.00000i			0.148047i
$731$	−32.0000			−1.18356
$732$		0			0
$733$			36.0000i			1.32969i	0.746981	+	0.664845i	$0.231502\pi$
$733$			36.0000i			1.32969i	−0.746981	+	0.664845i	$0.768498\pi$
$734$		−	12.0000i		−	0.442928i
$735$		0			0
$736$			40.0000i			1.47442i
$737$		0			0
$738$		0			0
$739$		−	36.0000i		−	1.32428i	−0.749380	−	0.662141i	$-0.769648\pi$
$739$		−	36.0000i		−	1.32428i	0.749380	−	0.662141i	$-0.230352\pi$
$740$	−4.00000			−0.147043
$741$		0			0
$742$	−48.0000			−1.76214
$743$			32.0000i			1.17397i	0.809599	+	0.586983i	$0.199684\pi$
$743$			32.0000i			1.17397i	−0.809599	+	0.586983i	$0.800316\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$			14.0000i			0.512576i
$747$		0			0
$748$		0			0
$749$		−	32.0000i		−	1.16925i
$750$		0			0
$751$	16.0000			0.583848			0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000			0.583848			0.291924	+	0.956441i	$0.405705\pi$
$752$			8.00000i			0.291730i
$753$		0			0
$754$	16.0000	−	24.0000i	0.582686	−	0.874028i
$755$	−20.0000			−0.727875
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	−12.0000			−0.435860
$759$		0			0
$760$		−	12.0000i		−	0.435286i
$761$		−	34.0000i		−	1.23250i	−0.787551	−	0.616250i	$-0.788651\pi$
$761$		−	34.0000i		−	1.23250i	0.787551	−	0.616250i	$-0.211349\pi$
$762$		0			0
$763$	32.0000			1.15848
$764$	−8.00000			−0.289430
$765$		0			0
$766$	−16.0000			−0.578103
$767$		0			0
$768$		0			0
$769$		−	40.0000i		−	1.44244i	−0.692708	−	0.721218i	$-0.743582\pi$
$769$		−	40.0000i		−	1.44244i	0.692708	−	0.721218i	$-0.256418\pi$
$770$		0			0
$771$		0			0
$772$		−	4.00000i		−	0.143963i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$		0			0
$775$			4.00000i			0.143684i
$776$	−12.0000			−0.430775
$777$		0			0
$778$		−	16.0000i		−	0.573628i
$779$	−8.00000			−0.286630
$780$		0			0
$781$		0			0
$782$		−	32.0000i		−	1.14432i
$783$		0			0
$784$	9.00000			0.321429
$785$			22.0000i			0.785214i
$786$		0			0
$787$			44.0000i			1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$			44.0000i			1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$			26.0000i			0.926212i
$789$		0			0
$790$	8.00000			0.284627
$791$		−	16.0000i		−	0.568895i
$792$		0			0
$793$	−6.00000	−	4.00000i	−0.213066	−	0.142044i
$794$	−20.0000			−0.709773
$795$		0			0
$796$		0			0
$797$	28.0000			0.991811			0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000			0.991811			0.495905	+	0.868377i	$0.334836\pi$
$798$		0			0
$799$			32.0000i			1.13208i
$800$			5.00000i			0.176777i
$801$		0			0
$802$	10.0000			0.353112
$803$		0			0
$804$		0			0
$805$	32.0000			1.12785
$806$	−12.0000	−	8.00000i	−0.422682	−	0.281788i
$807$		0			0
$808$		−	48.0000i		−	1.68863i
$809$	−24.0000			−0.843795			−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000			−0.843795			−0.421898	+	0.906644i	$0.638636\pi$
$810$		0			0
$811$		−	28.0000i		−	0.983213i	−0.870817	−	0.491606i	$-0.836410\pi$
$811$		−	28.0000i		−	0.983213i	0.870817	−	0.491606i	$-0.163590\pi$
$812$		−	32.0000i		−	1.12298i
$813$		0			0
$814$		0			0
$815$	−12.0000			−0.420342
$816$		0			0
$817$			32.0000i			1.11954i
$818$		0			0
$819$		0			0
$820$	2.00000			0.0698430
$821$		−	42.0000i		−	1.46581i	−0.680331	−	0.732905i	$-0.738164\pi$
$821$		−	42.0000i		−	1.46581i	0.680331	−	0.732905i	$-0.261836\pi$
$822$		0			0
$823$	−12.0000			−0.418294			−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000			−0.418294			−0.209147	+	0.977884i	$0.567069\pi$
$824$		−	12.0000i		−	0.418040i
$825$		0			0
$826$		0			0
$827$		−	44.0000i		−	1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$		−	44.0000i		−	1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		−	12.0000i		−	0.416526i
$831$		0			0
$832$	−21.0000	−	14.0000i	−0.728044	−	0.485363i
$833$	36.0000			1.24733
$834$		0			0
$835$	8.00000			0.276851
$836$		0			0
$837$		0			0
$838$		0			0
$839$			36.0000i			1.24286i	0.783470	+	0.621429i	$0.213448\pi$
$839$			36.0000i			1.24286i	−0.783470	+	0.621429i	$0.786552\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$	16.0000			0.551396
$843$		0			0
$844$	12.0000			0.413057
$845$	12.0000	−	5.00000i	0.412813	−	0.172005i
$846$		0			0
$847$		−	44.0000i		−	1.51186i
$848$	−12.0000			−0.412082
$849$		0			0
$850$		−	4.00000i		−	0.137199i
$851$			32.0000i			1.09695i
$852$		0			0
$853$		−	12.0000i		−	0.410872i	−0.978671	−	0.205436i	$-0.934139\pi$
$853$		−	12.0000i		−	0.410872i	0.978671	−	0.205436i	$-0.0658613\pi$
$854$	8.00000			0.273754
$855$		0			0
$856$		−	24.0000i		−	0.820303i
$857$	−28.0000			−0.956462			−0.478231	−	0.878234i	$-0.658722\pi$
$857$	−28.0000			−0.956462			−0.478231	+	0.878234i	$0.658722\pi$
$858$		0			0
$859$	44.0000			1.50126			0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000			1.50126			0.750630	+	0.660722i	$0.229750\pi$
$860$		−	8.00000i		−	0.272798i
$861$		0			0
$862$	12.0000			0.408722
$863$		−	48.0000i		−	1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$		−	48.0000i		−	1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$		0			0
$865$		−	4.00000i		−	0.136004i
$866$		−	2.00000i		−	0.0679628i
$867$		0			0
$868$	−16.0000			−0.543075
$869$		0			0
$870$		0			0
$871$	−24.0000	+	36.0000i	−0.813209	+	1.21981i
$872$	24.0000			0.812743
$873$		0			0
$874$	−32.0000			−1.08242
$875$	4.00000			0.135225
$876$		0			0
$877$			52.0000i			1.75592i	0.478738	+	0.877958i	$0.341094\pi$
$877$			52.0000i			1.75592i	−0.478738	+	0.877958i	$0.658906\pi$
$878$			32.0000i			1.07995i
$879$		0			0
$880$		0			0
$881$	24.0000			0.808581			0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000			0.808581			0.404290	+	0.914631i	$0.367519\pi$
$882$		0			0
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$	−12.0000	−	8.00000i	−0.403604	−	0.269069i
$885$		0			0
$886$			24.0000i			0.806296i
$887$	−40.0000			−1.34307			−0.671534	−	0.740973i	$-0.734364\pi$
$887$	−40.0000			−1.34307			−0.671534	+	0.740973i	$0.734364\pi$
$888$		0			0
$889$		−	48.0000i		−	1.60987i
$890$		−	18.0000i		−	0.603361i
$891$		0			0
$892$			4.00000i			0.133930i
$893$	32.0000			1.07084
$894$		0			0
$895$		0			0
$896$	−12.0000			−0.400892
$897$		0			0
$898$	−6.00000			−0.200223
$899$		−	32.0000i		−	1.06726i
$900$		0			0
$901$	−48.0000			−1.59911
$902$		0			0
$903$		0			0
$904$		−	12.0000i		−	0.399114i
$905$		−	6.00000i		−	0.199447i
$906$		0			0
$907$	40.0000			1.32818			0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000			1.32818			0.664089	+	0.747653i	$0.268820\pi$
$908$		−	12.0000i		−	0.398234i
$909$		0			0
$910$	−8.00000	+	12.0000i	−0.265197	+	0.397796i
$911$	8.00000			0.265052			0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000			0.265052			0.132526	+	0.991180i	$0.457691\pi$
$912$		0			0
$913$		0			0
$914$	28.0000			0.926158
$915$		0			0
$916$			16.0000i			0.528655i
$917$		0			0
$918$		0			0
$919$	−8.00000			−0.263896			−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000			−0.263896			−0.131948	+	0.991257i	$0.542123\pi$
$920$	24.0000			0.791257
$921$		0			0
$922$	34.0000			1.11973
$923$	−8.00000	+	12.0000i	−0.263323	+	0.394985i
$924$		0			0
$925$			4.00000i			0.131519i
$926$	−20.0000			−0.657241
$927$		0			0
$928$		−	40.0000i		−	1.31306i
$929$			26.0000i			0.853032i	0.904480	+	0.426516i	$0.140259\pi$
$929$			26.0000i			0.853032i	−0.904480	+	0.426516i	$0.859741\pi$
$930$		0			0
$931$		−	36.0000i		−	1.17985i
$932$	−28.0000			−0.917170
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$		−	48.0000i		−	1.56726i
$939$		0			0
$940$	−8.00000			−0.260931
$941$		−	2.00000i		−	0.0651981i	−0.999469	−	0.0325991i	$-0.989622\pi$
$941$		−	2.00000i		−	0.0651981i	0.999469	−	0.0325991i	$-0.0103784\pi$
$942$		0			0
$943$		−	16.0000i		−	0.521032i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	8.00000	−	12.0000i	0.259691	−	0.389536i
$950$	−4.00000			−0.129777
$951$		0			0
$952$	48.0000			1.55569
$953$	12.0000			0.388718			0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000			0.388718			0.194359	+	0.980930i	$0.437737\pi$
$954$		0			0
$955$			8.00000i			0.258874i
$956$		−	20.0000i		−	0.646846i
$957$		0			0
$958$	12.0000			0.387702
$959$	24.0000			0.775000
$960$		0			0
$961$	15.0000			0.483871
$962$	−12.0000	−	8.00000i	−0.386896	−	0.257930i
$963$		0			0
$964$			24.0000i			0.772988i
$965$	−4.00000			−0.128765
$966$		0			0
$967$			20.0000i			0.643157i	0.946883	+	0.321578i	$0.104213\pi$
$967$			20.0000i			0.643157i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		−	33.0000i		−	1.06066i
$969$		0			0
$970$			4.00000i			0.128432i
$971$	−24.0000			−0.770197			−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000			−0.770197			−0.385098	+	0.922876i	$0.625832\pi$
$972$		0			0
$973$			80.0000i			2.56468i
$974$	−12.0000			−0.384505
$975$		0			0
$976$	2.00000			0.0640184
$977$			14.0000i			0.447900i	0.974601	+	0.223950i	$0.0718952\pi$
$977$			14.0000i			0.447900i	−0.974601	+	0.223950i	$0.928105\pi$
$978$		0			0
$979$		0			0
$980$			9.00000i			0.287494i
$981$		0			0
$982$			40.0000i			1.27645i
$983$			40.0000i			1.27580i	0.770118	+	0.637901i	$0.220197\pi$
$983$			40.0000i			1.27580i	−0.770118	+	0.637901i	$0.779803\pi$
$984$		0			0
$985$	26.0000			0.828429
$986$			32.0000i			1.01909i
$987$		0			0
$988$	−8.00000	+	12.0000i	−0.254514	+	0.381771i
$989$	−64.0000			−2.03508
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$	−20.0000			−0.635001
$993$		0			0
$994$		−	16.0000i		−	0.507489i
$995$		0			0
$996$		0			0
$997$	30.0000			0.950110			0.475055	−	0.879956i	$-0.342428\pi$
$997$	30.0000			0.950110			0.475055	+	0.879956i	$0.342428\pi$
$998$	−20.0000			−0.633089
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.b.b.181.1	✓	2
3.2	odd	2		585.2.b.c.181.2	yes	2
13.5	odd	4		7605.2.a.e.1.1		1
13.8	odd	4		7605.2.a.o.1.1		1
13.12	even	2	inner	585.2.b.b.181.2	yes	2
39.5	even	4		7605.2.a.r.1.1		1
39.8	even	4		7605.2.a.b.1.1		1
39.38	odd	2		585.2.b.c.181.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.b.b.181.1	✓	2	1.1	even	1	trivial
585.2.b.b.181.2	yes	2	13.12	even	2	inner
585.2.b.c.181.1	yes	2	39.38	odd	2
585.2.b.c.181.2	yes	2	3.2	odd	2
7605.2.a.b.1.1		1	39.8	even	4
7605.2.a.e.1.1		1	13.5	odd	4
7605.2.a.o.1.1		1	13.8	odd	4
7605.2.a.r.1.1		1	39.5	even	4

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:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{5} -4.00000i q^{7} -3.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{5} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{10} +(3.00000 + 2.00000i) q^{13} -4.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} +4.00000i q^{19} -1.00000i q^{20} -8.00000 q^{23} -1.00000 q^{25} +(2.00000 - 3.00000i) q^{26} -4.00000i q^{28} +8.00000 q^{29} -4.00000i q^{31} -5.00000i q^{32} +4.00000i q^{34} -4.00000 q^{35} -4.00000i q^{37} +4.00000 q^{38} -3.00000 q^{40} +2.00000i q^{41} +8.00000 q^{43} +8.00000i q^{46} -8.00000i q^{47} -9.00000 q^{49} +1.00000i q^{50} +(3.00000 + 2.00000i) q^{52} +12.0000 q^{53} -12.0000 q^{56} -8.00000i q^{58} -2.00000 q^{61} -4.00000 q^{62} -7.00000 q^{64} +(2.00000 - 3.00000i) q^{65} +12.0000i q^{67} -4.00000 q^{68} +4.00000i q^{70} +4.00000i q^{71} -4.00000i q^{73} -4.00000 q^{74} +4.00000i q^{76} -8.00000 q^{79} +1.00000i q^{80} +2.00000 q^{82} +12.0000i q^{83} +4.00000i q^{85} -8.00000i q^{86} +18.0000i q^{89} +(8.00000 - 12.0000i) q^{91} -8.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} -4.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} - 2 q^{10} + 6 q^{13} - 8 q^{14} - 2 q^{16} - 8 q^{17} - 16 q^{23} - 2 q^{25} + 4 q^{26} + 16 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} - 18 q^{49} + 6 q^{52} + 24 q^{53} - 24 q^{56} - 4 q^{61} - 8 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{68} - 8 q^{74} - 16 q^{79} + 4 q^{82} + 16 q^{91} - 16 q^{92} - 16 q^{94} + 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^10 + 6 * q^13 - 8 * q^14 - 2 * q^16 - 8 * q^17 - 16 * q^23 - 2 * q^25 + 4 * q^26 + 16 * q^29 - 8 * q^35 + 8 * q^38 - 6 * q^40 + 16 * q^43 - 18 * q^49 + 6 * q^52 + 24 * q^53 - 24 * q^56 - 4 * q^61 - 8 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^68 - 8 * q^74 - 16 * q^79 + 4 * q^82 + 16 * q^91 - 16 * q^92 - 16 * q^94 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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