LMFDB - Embedded newform 585.2.b.a.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:	no (minimal twist has level 195)
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.a.181.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{5} -2.00000i q^{7} -3.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{5} -2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{10} +(-3.00000 - 2.00000i) q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +2.00000i q^{19} -1.00000i q^{20} +8.00000 q^{23} -1.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -2.00000i q^{28} -2.00000 q^{29} -2.00000i q^{31} -5.00000i q^{32} +2.00000i q^{34} -2.00000 q^{35} -8.00000i q^{37} +2.00000 q^{38} -3.00000 q^{40} +2.00000i q^{41} -4.00000 q^{43} -8.00000i q^{46} +4.00000i q^{47} +3.00000 q^{49} +1.00000i q^{50} +(-3.00000 - 2.00000i) q^{52} +6.00000 q^{53} -6.00000 q^{56} +2.00000i q^{58} +12.0000i q^{59} +10.0000 q^{61} -2.00000 q^{62} -7.00000 q^{64} +(-2.00000 + 3.00000i) q^{65} -6.00000i q^{67} -2.00000 q^{68} +2.00000i q^{70} -8.00000i q^{71} +16.0000i q^{73} -8.00000 q^{74} +2.00000i q^{76} -8.00000 q^{79} +1.00000i q^{80} +2.00000 q^{82} -12.0000i q^{83} +2.00000i q^{85} +4.00000i q^{86} -6.00000i q^{89} +(-4.00000 + 6.00000i) q^{91} +8.00000 q^{92} +4.00000 q^{94} +2.00000 q^{95} +16.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} - 2 q^{10} - 6 q^{13} - 4 q^{14} - 2 q^{16} - 4 q^{17} + 16 q^{23} - 2 q^{25} - 4 q^{26} - 4 q^{29} - 4 q^{35} + 4 q^{38} - 6 q^{40} - 8 q^{43} + 6 q^{49} - 6 q^{52} + 12 q^{53} - 12 q^{56} + 20 q^{61} - 4 q^{62} - 14 q^{64} - 4 q^{65} - 4 q^{68} - 16 q^{74} - 16 q^{79} + 4 q^{82} - 8 q^{91} + 16 q^{92} + 8 q^{94} + 4 q^{95}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^10 - 6 * q^13 - 4 * q^14 - 2 * q^16 - 4 * q^17 + 16 * q^23 - 2 * q^25 - 4 * q^26 - 4 * q^29 - 4 * q^35 + 4 * q^38 - 6 * q^40 - 8 * q^43 + 6 * q^49 - 6 * q^52 + 12 * q^53 - 12 * q^56 + 20 * q^61 - 4 * q^62 - 14 * q^64 - 4 * q^65 - 4 * q^68 - 16 * q^74 - 16 * q^79 + 4 * q^82 - 8 * q^91 + 16 * q^92 + 8 * q^94 + 4 * 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
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		−	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
$14$	−2.00000			−0.534522
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		−	1.00000i		−	0.223607i
$21$		0			0
$22$		0			0
$23$	8.00000			1.66812			0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000			1.66812			0.834058	+	0.551677i	$0.186012\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	−2.00000	+	3.00000i	−0.392232	+	0.588348i
$27$		0			0
$28$		−	2.00000i		−	0.377964i
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$		−	5.00000i		−	0.883883i
$33$		0			0
$34$			2.00000i			0.342997i
$35$	−2.00000			−0.338062
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	2.00000			0.324443
$39$		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
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		−	8.00000i		−	1.17954i
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$			1.00000i			0.141421i
$51$		0			0
$52$	−3.00000	−	2.00000i	−0.416025	−	0.277350i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$	−6.00000			−0.801784
$57$		0			0
$58$			2.00000i			0.262613i
$59$			12.0000i			1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$			12.0000i			1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$	−2.00000			−0.254000
$63$		0			0
$64$	−7.00000			−0.875000
$65$	−2.00000	+	3.00000i	−0.248069	+	0.372104i
$66$		0			0
$67$		−	6.00000i		−	0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$		−	6.00000i		−	0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	−2.00000			−0.242536
$69$		0			0
$70$			2.00000i			0.239046i
$71$		−	8.00000i		−	0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$		−	8.00000i		−	0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$		0			0
$73$			16.0000i			1.87266i	0.351123	+	0.936329i	$0.385800\pi$
$73$			16.0000i			1.87266i	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−8.00000			−0.929981
$75$		0			0
$76$			2.00000i			0.229416i
$77$		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	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$			2.00000i			0.216930i
$86$			4.00000i			0.431331i
$87$		0			0
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$	−4.00000	+	6.00000i	−0.419314	+	0.628971i
$92$	8.00000			0.834058
$93$		0			0
$94$	4.00000			0.412568
$95$	2.00000			0.205196
$96$		0			0
$97$			16.0000i			1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$			16.0000i			1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$		−	3.00000i		−	0.303046i
$99$		0			0
$100$	−1.00000			−0.100000
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$	−6.00000	+	9.00000i	−0.588348	+	0.882523i
$105$		0			0
$106$		−	6.00000i		−	0.582772i
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$		0			0
$112$			2.00000i			0.188982i
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		−	8.00000i		−	0.746004i
$116$	−2.00000			−0.185695
$117$		0			0
$118$	12.0000			1.10469
$119$			4.00000i			0.366679i
$120$		0			0
$121$	11.0000			1.00000
$122$		−	10.0000i		−	0.905357i
$123$		0			0
$124$		−	2.00000i		−	0.179605i
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		−	3.00000i		−	0.265165i
$129$		0			0
$130$	3.00000	+	2.00000i	0.263117	+	0.175412i
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	4.00000			0.346844
$134$	−6.00000			−0.518321
$135$		0			0
$136$			6.00000i			0.514496i
$137$			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
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$	−2.00000			−0.169031
$141$		0			0
$142$	−8.00000			−0.671345
$143$		0			0
$144$		0			0
$145$			2.00000i			0.166091i
$146$	16.0000			1.32417
$147$		0			0
$148$		−	8.00000i		−	0.657596i
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$			14.0000i			1.13930i	0.821886	+	0.569652i	$0.192922\pi$
$151$			14.0000i			1.13930i	−0.821886	+	0.569652i	$0.807078\pi$
$152$	6.00000			0.486664
$153$		0			0
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$			8.00000i			0.636446i
$159$		0			0
$160$	−5.00000			−0.395285
$161$		−	16.0000i		−	1.26098i
$162$		0			0
$163$		−	18.0000i		−	1.40987i	−0.709273	−	0.704934i	$-0.750976\pi$
$163$		−	18.0000i		−	1.40987i	0.709273	−	0.704934i	$-0.249024\pi$
$164$			2.00000i			0.156174i
$165$		0			0
$166$	−12.0000			−0.931381
$167$			20.0000i			1.54765i	0.633402	+	0.773823i	$0.281658\pi$
$167$			20.0000i			1.54765i	−0.633402	+	0.773823i	$0.718342\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$	2.00000			0.153393
$171$		0			0
$172$	−4.00000			−0.304997
$173$	−10.0000			−0.760286			−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000			−0.760286			−0.380143	+	0.924928i	$0.624125\pi$
$174$		0			0
$175$			2.00000i			0.151186i
$176$		0			0
$177$		0			0
$178$	−6.00000			−0.449719
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$	6.00000	+	4.00000i	0.444750	+	0.296500i
$183$		0			0
$184$		−	24.0000i		−	1.76930i
$185$	−8.00000			−0.588172
$186$		0			0
$187$		0			0
$188$			4.00000i			0.291730i
$189$		0			0
$190$		−	2.00000i		−	0.145095i
$191$	20.0000			1.44715			0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000			1.44715			0.723575	+	0.690246i	$0.242498\pi$
$192$		0			0
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	16.0000			1.14873
$195$		0			0
$196$	3.00000			0.214286
$197$		−	22.0000i		−	1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$		−	22.0000i		−	1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$			3.00000i			0.212132i
$201$		0			0
$202$		−	2.00000i		−	0.140720i
$203$			4.00000i			0.280745i
$204$		0			0
$205$	2.00000			0.139686
$206$		−	16.0000i		−	1.11477i
$207$		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	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000			0.412082
$213$		0			0
$214$			8.00000i			0.546869i
$215$			4.00000i			0.272798i
$216$		0			0
$217$	−4.00000			−0.271538
$218$	16.0000			1.08366
$219$		0			0
$220$		0			0
$221$	6.00000	+	4.00000i	0.403604	+	0.269069i
$222$		0			0
$223$		−	22.0000i		−	1.47323i	−0.676313	−	0.736614i	$-0.736423\pi$
$223$		−	22.0000i		−	1.47323i	0.676313	−	0.736614i	$-0.263577\pi$
$224$	−10.0000			−0.668153
$225$		0			0
$226$		−	14.0000i		−	0.931266i
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$			8.00000i			0.528655i	0.964433	+	0.264327i	$0.0851500\pi$
$229$			8.00000i			0.528655i	−0.964433	+	0.264327i	$0.914850\pi$
$230$	−8.00000			−0.527504
$231$		0			0
$232$			6.00000i			0.393919i
$233$	−14.0000			−0.917170			−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$		0			0
$235$	4.00000			0.260931
$236$			12.0000i			0.781133i
$237$		0			0
$238$	4.00000			0.259281
$239$		−	8.00000i		−	0.517477i	−0.965947	−	0.258738i	$-0.916693\pi$
$239$		−	8.00000i		−	0.517477i	0.965947	−	0.258738i	$-0.0833068\pi$
$240$		0			0
$241$			12.0000i			0.772988i	0.922292	+	0.386494i	$0.126314\pi$
$241$			12.0000i			0.772988i	−0.922292	+	0.386494i	$0.873686\pi$
$242$		−	11.0000i		−	0.707107i
$243$		0			0
$244$	10.0000			0.640184
$245$		−	3.00000i		−	0.191663i
$246$		0			0
$247$	4.00000	−	6.00000i	0.254514	−	0.381771i
$248$	−6.00000			−0.381000
$249$		0			0
$250$	1.00000			0.0632456
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−17.0000			−1.06250
$257$	−30.0000			−1.87135			−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000			−1.87135			−0.935674	+	0.352865i	$0.885208\pi$
$258$		0			0
$259$	−16.0000			−0.994192
$260$	−2.00000	+	3.00000i	−0.124035	+	0.186052i
$261$		0			0
$262$		−	12.0000i		−	0.741362i
$263$	−12.0000			−0.739952			−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000			−0.739952			−0.369976	+	0.929041i	$0.620634\pi$
$264$		0			0
$265$		−	6.00000i		−	0.368577i
$266$		−	4.00000i		−	0.245256i
$267$		0			0
$268$		−	6.00000i		−	0.366508i
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$		−	14.0000i		−	0.850439i	−0.905090	−	0.425220i	$-0.860197\pi$
$271$		−	14.0000i		−	0.850439i	0.905090	−	0.425220i	$-0.139803\pi$
$272$	2.00000			0.121268
$273$		0			0
$274$	6.00000			0.362473
$275$		0			0
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		−	4.00000i		−	0.239904i
$279$		0			0
$280$			6.00000i			0.358569i
$281$			6.00000i			0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$			6.00000i			0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		−	8.00000i		−	0.474713i
$285$		0			0
$286$		0			0
$287$	4.00000			0.236113
$288$		0			0
$289$	−13.0000			−0.764706
$290$	2.00000			0.117444
$291$		0			0
$292$			16.0000i			0.936329i
$293$			18.0000i			1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$			18.0000i			1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$	−24.0000			−1.39497
$297$		0			0
$298$	14.0000			0.810998
$299$	−24.0000	−	16.0000i	−1.38796	−	0.925304i
$300$		0			0
$301$			8.00000i			0.461112i
$302$	14.0000			0.805609
$303$		0			0
$304$		−	2.00000i		−	0.114708i
$305$		−	10.0000i		−	0.572598i
$306$		0			0
$307$			26.0000i			1.48390i	0.670456	+	0.741949i	$0.266098\pi$
$307$			26.0000i			1.48390i	−0.670456	+	0.741949i	$0.733902\pi$
$308$		0			0
$309$		0			0
$310$			2.00000i			0.113592i
$311$	−4.00000			−0.226819			−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000			−0.226819			−0.113410	+	0.993548i	$0.536177\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		−	14.0000i		−	0.790066i
$315$		0			0
$316$	−8.00000			−0.450035
$317$			26.0000i			1.46031i	0.683284	+	0.730153i	$0.260551\pi$
$317$			26.0000i			1.46031i	−0.683284	+	0.730153i	$0.739449\pi$
$318$		0			0
$319$		0			0
$320$			7.00000i			0.391312i
$321$		0			0
$322$	−16.0000			−0.891645
$323$		−	4.00000i		−	0.222566i
$324$		0			0
$325$	3.00000	+	2.00000i	0.166410	+	0.110940i
$326$	−18.0000			−0.996928
$327$		0			0
$328$	6.00000			0.331295
$329$	8.00000			0.441054
$330$		0			0
$331$		−	26.0000i		−	1.42909i	−0.699590	−	0.714545i	$-0.746634\pi$
$331$		−	26.0000i		−	1.42909i	0.699590	−	0.714545i	$-0.253366\pi$
$332$		−	12.0000i		−	0.658586i
$333$		0			0
$334$	20.0000			1.09435
$335$	−6.00000			−0.327815
$336$		0			0
$337$	6.00000			0.326841			0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000			0.326841			0.163420	+	0.986557i	$0.447747\pi$
$338$	12.0000	−	5.00000i	0.652714	−	0.271964i
$339$		0			0
$340$			2.00000i			0.108465i
$341$		0			0
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$			12.0000i			0.646997i
$345$		0			0
$346$			10.0000i			0.537603i
$347$	20.0000			1.07366			0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000			1.07366			0.536828	+	0.843692i	$0.319622\pi$
$348$		0			0
$349$		−	20.0000i		−	1.07058i	−0.844670	−	0.535288i	$-0.820203\pi$
$349$		−	20.0000i		−	1.07058i	0.844670	−	0.535288i	$-0.179797\pi$
$350$	2.00000			0.106904
$351$		0			0
$352$		0			0
$353$		−	30.0000i		−	1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$		−	30.0000i		−	1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$		0			0
$355$	−8.00000			−0.424596
$356$		−	6.00000i		−	0.317999i
$357$		0			0
$358$			12.0000i			0.634220i
$359$		−	4.00000i		−	0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$		−	4.00000i		−	0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		−	6.00000i		−	0.315353i
$363$		0			0
$364$	−4.00000	+	6.00000i	−0.209657	+	0.314485i
$365$	16.0000			0.837478
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−8.00000			−0.417029
$369$		0			0
$370$			8.00000i			0.415900i
$371$		−	12.0000i		−	0.623009i
$372$		0			0
$373$	−38.0000			−1.96757			−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000			−1.96757			−0.983783	+	0.179364i	$0.942596\pi$
$374$		0			0
$375$		0			0
$376$	12.0000			0.618853
$377$	6.00000	+	4.00000i	0.309016	+	0.206010i
$378$		0			0
$379$		−	30.0000i		−	1.54100i	−0.637442	−	0.770498i	$-0.720007\pi$
$379$		−	30.0000i		−	1.54100i	0.637442	−	0.770498i	$-0.279993\pi$
$380$	2.00000			0.102598
$381$		0			0
$382$		−	20.0000i		−	1.02329i
$383$		−	4.00000i		−	0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$		−	4.00000i		−	0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$		0			0
$385$		0			0
$386$	4.00000			0.203595
$387$		0			0
$388$			16.0000i			0.812277i
$389$	38.0000			1.92668			0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000			1.92668			0.963338	+	0.268290i	$0.0864585\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		−	9.00000i		−	0.454569i
$393$		0			0
$394$	−22.0000			−1.10834
$395$			8.00000i			0.402524i
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$			24.0000i			1.20301i
$399$		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
$403$	−4.00000	+	6.00000i	−0.199254	+	0.298881i
$404$	2.00000			0.0995037
$405$		0			0
$406$	4.00000			0.198517
$407$		0			0
$408$		0			0
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		−	2.00000i		−	0.0987730i
$411$		0			0
$412$	16.0000			0.788263
$413$	24.0000			1.18096
$414$		0			0
$415$	−12.0000			−0.589057
$416$	−10.0000	+	15.0000i	−0.490290	+	0.735436i
$417$		0			0
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$		−	40.0000i		−	1.94948i	−0.223341	−	0.974740i	$-0.571696\pi$
$421$		−	40.0000i		−	1.94948i	0.223341	−	0.974740i	$-0.428304\pi$
$422$			12.0000i			0.584151i
$423$		0			0
$424$		−	18.0000i		−	0.874157i
$425$	2.00000			0.0970143
$426$		0			0
$427$		−	20.0000i		−	0.967868i
$428$	−8.00000			−0.386695
$429$		0			0
$430$	4.00000			0.192897
$431$		−	12.0000i		−	0.578020i	−0.957326	−	0.289010i	$-0.906674\pi$
$431$		−	12.0000i		−	0.578020i	0.957326	−	0.289010i	$-0.0933260\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$			4.00000i			0.192006i
$435$		0			0
$436$			16.0000i			0.766261i
$437$			16.0000i			0.765384i
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$		0			0
$442$	4.00000	−	6.00000i	0.190261	−	0.285391i
$443$	−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
$445$	−6.00000			−0.284427
$446$	−22.0000			−1.04173
$447$		0			0
$448$			14.0000i			0.661438i
$449$		−	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
$452$	14.0000			0.658505
$453$		0			0
$454$		0			0
$455$	6.00000	+	4.00000i	0.281284	+	0.187523i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	8.00000			0.373815
$459$		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
$463$			14.0000i			0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$			14.0000i			0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$	2.00000			0.0928477
$465$		0			0
$466$			14.0000i			0.648537i
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$		0			0
$469$	−12.0000			−0.554109
$470$		−	4.00000i		−	0.184506i
$471$		0			0
$472$	36.0000			1.65703
$473$		0			0
$474$		0			0
$475$		−	2.00000i		−	0.0917663i
$476$			4.00000i			0.183340i
$477$		0			0
$478$	−8.00000			−0.365911
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	−16.0000	+	24.0000i	−0.729537	+	1.09431i
$482$	12.0000			0.546585
$483$		0			0
$484$	11.0000			0.500000
$485$	16.0000			0.726523
$486$		0			0
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$		−	30.0000i		−	1.35804i
$489$		0			0
$490$	−3.00000			−0.135526
$491$	40.0000			1.80517			0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000			1.80517			0.902587	+	0.430507i	$0.141665\pi$
$492$		0			0
$493$	4.00000			0.180151
$494$	−6.00000	−	4.00000i	−0.269953	−	0.179969i
$495$		0			0
$496$			2.00000i			0.0898027i
$497$	−16.0000			−0.717698
$498$		0			0
$499$		−	10.0000i		−	0.447661i	−0.974628	−	0.223831i	$-0.928144\pi$
$499$		−	10.0000i		−	0.447661i	0.974628	−	0.223831i	$-0.0718563\pi$
$500$			1.00000i			0.0447214i
$501$		0			0
$502$		0			0
$503$	4.00000			0.178351			0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000			0.178351			0.0891756	+	0.996016i	$0.471577\pi$
$504$		0			0
$505$		−	2.00000i		−	0.0889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	6.00000i		−	0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$		−	6.00000i		−	0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$		0			0
$511$	32.0000			1.41560
$512$			11.0000i			0.486136i
$513$		0			0
$514$			30.0000i			1.32324i
$515$		−	16.0000i		−	0.705044i
$516$		0			0
$517$		0			0
$518$			16.0000i			0.703000i
$519$		0			0
$520$	9.00000	+	6.00000i	0.394676	+	0.263117i
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$	−28.0000			−1.22435			−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000			−1.22435			−0.612177	+	0.790721i	$0.709706\pi$
$524$	12.0000			0.524222
$525$		0			0
$526$			12.0000i			0.523225i
$527$			4.00000i			0.174243i
$528$		0			0
$529$	41.0000			1.78261
$530$	−6.00000			−0.260623
$531$		0			0
$532$	4.00000			0.173422
$533$	4.00000	−	6.00000i	0.173259	−	0.259889i
$534$		0			0
$535$			8.00000i			0.345870i
$536$	−18.0000			−0.777482
$537$		0			0
$538$			10.0000i			0.431131i
$539$		0			0
$540$		0			0
$541$		−	8.00000i		−	0.343947i	−0.985102	−	0.171973i	$-0.944986\pi$
$541$		−	8.00000i		−	0.343947i	0.985102	−	0.171973i	$-0.0550143\pi$
$542$	−14.0000			−0.601351
$543$		0			0
$544$			10.0000i			0.428746i
$545$	16.0000			0.685365
$546$		0			0
$547$	−4.00000			−0.171028			−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000			−0.171028			−0.0855138	+	0.996337i	$0.527253\pi$
$548$			6.00000i			0.256307i
$549$		0			0
$550$		0			0
$551$		−	4.00000i		−	0.170406i
$552$		0			0
$553$			16.0000i			0.680389i
$554$			2.00000i			0.0849719i
$555$		0			0
$556$	4.00000			0.169638
$557$		−	6.00000i		−	0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$		−	6.00000i		−	0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$		0			0
$559$	12.0000	+	8.00000i	0.507546	+	0.338364i
$560$	2.00000			0.0845154
$561$		0			0
$562$	6.00000			0.253095
$563$	−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
$565$		−	14.0000i		−	0.588984i
$566$			4.00000i			0.168133i
$567$		0			0
$568$	−24.0000			−1.00702
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$		0			0
$574$		−	4.00000i		−	0.166957i
$575$	−8.00000			−0.333623
$576$		0			0
$577$			4.00000i			0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$			4.00000i			0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$			13.0000i			0.540729i
$579$		0			0
$580$			2.00000i			0.0830455i
$581$	−24.0000			−0.995688
$582$		0			0
$583$		0			0
$584$	48.0000			1.98625
$585$		0			0
$586$	18.0000			0.743573
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$		−	12.0000i		−	0.494032i
$591$		0			0
$592$			8.00000i			0.328798i
$593$			10.0000i			0.410651i	0.978694	+	0.205325i	$0.0658253\pi$
$593$			10.0000i			0.410651i	−0.978694	+	0.205325i	$0.934175\pi$
$594$		0			0
$595$	4.00000			0.163984
$596$			14.0000i			0.573462i
$597$		0			0
$598$	−16.0000	+	24.0000i	−0.654289	+	0.981433i
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$	8.00000			0.326056
$603$		0			0
$604$			14.0000i			0.569652i
$605$		−	11.0000i		−	0.447214i
$606$		0			0
$607$	−8.00000			−0.324710			−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000			−0.324710			−0.162355	+	0.986732i	$0.551909\pi$
$608$	10.0000			0.405554
$609$		0			0
$610$	−10.0000			−0.404888
$611$	8.00000	−	12.0000i	0.323645	−	0.485468i
$612$		0			0
$613$			16.0000i			0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$			16.0000i			0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	26.0000			1.04927
$615$		0			0
$616$		0			0
$617$			10.0000i			0.402585i	0.979531	+	0.201292i	$0.0645141\pi$
$617$			10.0000i			0.402585i	−0.979531	+	0.201292i	$0.935486\pi$
$618$		0			0
$619$			38.0000i			1.52735i	0.645601	+	0.763674i	$0.276607\pi$
$619$			38.0000i			1.52735i	−0.645601	+	0.763674i	$0.723393\pi$
$620$	−2.00000			−0.0803219
$621$		0			0
$622$			4.00000i			0.160385i
$623$	−12.0000			−0.480770
$624$		0			0
$625$	1.00000			0.0400000
$626$			10.0000i			0.399680i
$627$		0			0
$628$	14.0000			0.558661
$629$			16.0000i			0.637962i
$630$		0			0
$631$			38.0000i			1.51276i	0.654135	+	0.756378i	$0.273033\pi$
$631$			38.0000i			1.51276i	−0.654135	+	0.756378i	$0.726967\pi$
$632$			24.0000i			0.954669i
$633$		0			0
$634$	26.0000			1.03259
$635$		0			0
$636$		0			0
$637$	−9.00000	−	6.00000i	−0.356593	−	0.237729i
$638$		0			0
$639$		0			0
$640$	−3.00000			−0.118585
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$		−	14.0000i		−	0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$		−	14.0000i		−	0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$		−	16.0000i		−	0.630488i
$645$		0			0
$646$	−4.00000			−0.157378
$647$	−28.0000			−1.10079			−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000			−1.10079			−0.550397	+	0.834903i	$0.685524\pi$
$648$		0			0
$649$		0			0
$650$	2.00000	−	3.00000i	0.0784465	−	0.117670i
$651$		0			0
$652$		−	18.0000i		−	0.704934i
$653$	2.00000			0.0782660			0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000			0.0782660			0.0391330	+	0.999234i	$0.487540\pi$
$654$		0			0
$655$		−	12.0000i		−	0.468879i
$656$		−	2.00000i		−	0.0780869i
$657$		0			0
$658$		−	8.00000i		−	0.311872i
$659$	−48.0000			−1.86981			−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000			−1.86981			−0.934907	+	0.354892i	$0.884518\pi$
$660$		0			0
$661$		−	4.00000i		−	0.155582i	−0.996970	−	0.0777910i	$-0.975213\pi$
$661$		−	4.00000i		−	0.155582i	0.996970	−	0.0777910i	$-0.0247867\pi$
$662$	−26.0000			−1.01052
$663$		0			0
$664$	−36.0000			−1.39707
$665$		−	4.00000i		−	0.155113i
$666$		0			0
$667$	−16.0000			−0.619522
$668$			20.0000i			0.773823i
$669$		0			0
$670$			6.00000i			0.231800i
$671$		0			0
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$		−	6.00000i		−	0.231111i
$675$		0			0
$676$	5.00000	+	12.0000i	0.192308	+	0.461538i
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$	32.0000			1.22805
$680$	6.00000			0.230089
$681$		0			0
$682$		0			0
$683$			16.0000i			0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$			16.0000i			0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$	−20.0000			−0.763604
$687$		0			0
$688$	4.00000			0.152499
$689$	−18.0000	−	12.0000i	−0.685745	−	0.457164i
$690$		0			0
$691$		−	10.0000i		−	0.380418i	−0.981744	−	0.190209i	$-0.939083\pi$
$691$		−	10.0000i		−	0.380418i	0.981744	−	0.190209i	$-0.0609166\pi$
$692$	−10.0000			−0.380143
$693$		0			0
$694$		−	20.0000i		−	0.759190i
$695$		−	4.00000i		−	0.151729i
$696$		0			0
$697$		−	4.00000i		−	0.151511i
$698$	−20.0000			−0.757011
$699$		0			0
$700$			2.00000i			0.0755929i
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	16.0000			0.603451
$704$		0			0
$705$		0			0
$706$	−30.0000			−1.12906
$707$		−	4.00000i		−	0.150435i
$708$		0			0
$709$		−	32.0000i		−	1.20179i	−0.799330	−	0.600893i	$-0.794812\pi$
$709$		−	32.0000i		−	1.20179i	0.799330	−	0.600893i	$-0.205188\pi$
$710$			8.00000i			0.300235i
$711$		0			0
$712$	−18.0000			−0.674579
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$		0			0
$716$	−12.0000			−0.448461
$717$		0			0
$718$	−4.00000			−0.149279
$719$	36.0000			1.34257			0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000			1.34257			0.671287	+	0.741198i	$0.265742\pi$
$720$		0			0
$721$		−	32.0000i		−	1.19174i
$722$		−	15.0000i		−	0.558242i
$723$		0			0
$724$	6.00000			0.222988
$725$	2.00000			0.0742781
$726$		0			0
$727$	−40.0000			−1.48352			−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000			−1.48352			−0.741759	+	0.670667i	$0.766008\pi$
$728$	18.0000	+	12.0000i	0.667124	+	0.444750i
$729$		0			0
$730$		−	16.0000i		−	0.592187i
$731$	8.00000			0.295891
$732$		0			0
$733$			12.0000i			0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$			12.0000i			0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$		0			0
$735$		0			0
$736$		−	40.0000i		−	1.47442i
$737$		0			0
$738$		0			0
$739$			6.00000i			0.220714i	0.993892	+	0.110357i	$0.0351994\pi$
$739$			6.00000i			0.220714i	−0.993892	+	0.110357i	$0.964801\pi$
$740$	−8.00000			−0.294086
$741$		0			0
$742$	−12.0000			−0.440534
$743$			8.00000i			0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$			8.00000i			0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$		0			0
$745$	14.0000			0.512920
$746$			38.0000i			1.39128i
$747$		0			0
$748$		0			0
$749$			16.0000i			0.584627i
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		−	4.00000i		−	0.145865i
$753$		0			0
$754$	4.00000	−	6.00000i	0.145671	−	0.218507i
$755$	14.0000			0.509512
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$	−30.0000			−1.08965
$759$		0			0
$760$		−	6.00000i		−	0.217643i
$761$			38.0000i			1.37750i	0.724999	+	0.688749i	$0.241840\pi$
$761$			38.0000i			1.37750i	−0.724999	+	0.688749i	$0.758160\pi$
$762$		0			0
$763$	32.0000			1.15848
$764$	20.0000			0.723575
$765$		0			0
$766$	−4.00000			−0.144526
$767$	24.0000	−	36.0000i	0.866590	−	1.29988i
$768$		0			0
$769$			40.0000i			1.44244i	0.692708	+	0.721218i	$0.256418\pi$
$769$			40.0000i			1.44244i	−0.692708	+	0.721218i	$0.743582\pi$
$770$		0			0
$771$		0			0
$772$			4.00000i			0.143963i
$773$			54.0000i			1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$			54.0000i			1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$		0			0
$775$			2.00000i			0.0718421i
$776$	48.0000			1.72310
$777$		0			0
$778$		−	38.0000i		−	1.36237i
$779$	−4.00000			−0.143315
$780$		0			0
$781$		0			0
$782$			16.0000i			0.572159i
$783$		0			0
$784$	−3.00000			−0.107143
$785$		−	14.0000i		−	0.499681i
$786$		0			0
$787$		−	38.0000i		−	1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$		−	38.0000i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$		−	22.0000i		−	0.783718i
$789$		0			0
$790$	8.00000			0.284627
$791$		−	28.0000i		−	0.995565i
$792$		0			0
$793$	−30.0000	−	20.0000i	−1.06533	−	0.710221i
$794$	8.00000			0.283909
$795$		0			0
$796$	−24.0000			−0.850657
$797$	50.0000			1.77109			0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000			1.77109			0.885545	+	0.464553i	$0.153785\pi$
$798$		0			0
$799$		−	8.00000i		−	0.283020i
$800$			5.00000i			0.176777i
$801$		0			0
$802$	10.0000			0.353112
$803$		0			0
$804$		0			0
$805$	−16.0000			−0.563926
$806$	6.00000	+	4.00000i	0.211341	+	0.140894i
$807$		0			0
$808$		−	6.00000i		−	0.211079i
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$		−	14.0000i		−	0.491606i	−0.969320	−	0.245803i	$-0.920948\pi$
$811$		−	14.0000i		−	0.491606i	0.969320	−	0.245803i	$-0.0790517\pi$
$812$			4.00000i			0.140372i
$813$		0			0
$814$		0			0
$815$	−18.0000			−0.630512
$816$		0			0
$817$		−	8.00000i		−	0.279885i
$818$	−12.0000			−0.419570
$819$		0			0
$820$	2.00000			0.0698430
$821$		−	18.0000i		−	0.628204i	−0.949389	−	0.314102i	$-0.898297\pi$
$821$		−	18.0000i		−	0.628204i	0.949389	−	0.314102i	$-0.101703\pi$
$822$		0			0
$823$	24.0000			0.836587			0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000			0.836587			0.418294	+	0.908312i	$0.362628\pi$
$824$		−	48.0000i		−	1.67216i
$825$		0			0
$826$		−	24.0000i		−	0.835067i
$827$		−	8.00000i		−	0.278187i	−0.990279	−	0.139094i	$-0.955581\pi$
$827$		−	8.00000i		−	0.278187i	0.990279	−	0.139094i	$-0.0444189\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$			12.0000i			0.416526i
$831$		0			0
$832$	21.0000	+	14.0000i	0.728044	+	0.485363i
$833$	−6.00000			−0.207888
$834$		0			0
$835$	20.0000			0.692129
$836$		0			0
$837$		0			0
$838$		−	12.0000i		−	0.414533i
$839$		−	24.0000i		−	0.828572i	−0.910147	−	0.414286i	$-0.864031\pi$
$839$		−	24.0000i		−	0.828572i	0.910147	−	0.414286i	$-0.135969\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$	−40.0000			−1.37849
$843$		0			0
$844$	−12.0000			−0.413057
$845$	12.0000	−	5.00000i	0.412813	−	0.172005i
$846$		0			0
$847$		−	22.0000i		−	0.755929i
$848$	−6.00000			−0.206041
$849$		0			0
$850$		−	2.00000i		−	0.0685994i
$851$		−	64.0000i		−	2.19389i
$852$		0			0
$853$			24.0000i			0.821744i	0.911693	+	0.410872i	$0.134776\pi$
$853$			24.0000i			0.821744i	−0.911693	+	0.410872i	$0.865224\pi$
$854$	−20.0000			−0.684386
$855$		0			0
$856$			24.0000i			0.820303i
$857$	−38.0000			−1.29806			−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000			−1.29806			−0.649028	+	0.760765i	$0.724824\pi$
$858$		0			0
$859$	−28.0000			−0.955348			−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000			−0.955348			−0.477674	+	0.878537i	$0.658520\pi$
$860$			4.00000i			0.136399i
$861$		0			0
$862$	−12.0000			−0.408722
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$			10.0000i			0.340010i
$866$		−	14.0000i		−	0.475739i
$867$		0			0
$868$	−4.00000			−0.135769
$869$		0			0
$870$		0			0
$871$	−12.0000	+	18.0000i	−0.406604	+	0.609907i
$872$	48.0000			1.62549
$873$		0			0
$874$	16.0000			0.541208
$875$	2.00000			0.0676123
$876$		0			0
$877$		−	16.0000i		−	0.540282i	−0.962821	−	0.270141i	$-0.912930\pi$
$877$		−	16.0000i		−	0.540282i	0.962821	−	0.270141i	$-0.0870703\pi$
$878$		−	16.0000i		−	0.539974i
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$	6.00000	+	4.00000i	0.201802	+	0.134535i
$885$		0			0
$886$			24.0000i			0.806296i
$887$	16.0000			0.537227			0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000			0.537227			0.268614	+	0.963248i	$0.413434\pi$
$888$		0			0
$889$		0			0
$890$			6.00000i			0.201120i
$891$		0			0
$892$		−	22.0000i		−	0.736614i
$893$	−8.00000			−0.267710
$894$		0			0
$895$			12.0000i			0.401116i
$896$	−6.00000			−0.200446
$897$		0			0
$898$	−6.00000			−0.200223
$899$			4.00000i			0.133407i
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$		−	42.0000i		−	1.39690i
$905$		−	6.00000i		−	0.199447i
$906$		0			0
$907$	−44.0000			−1.46100			−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000			−1.46100			−0.730498	+	0.682915i	$0.760712\pi$
$908$		0			0
$909$		0			0
$910$	4.00000	−	6.00000i	0.132599	−	0.198898i
$911$	−20.0000			−0.662630			−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000			−0.662630			−0.331315	+	0.943520i	$0.607492\pi$
$912$		0			0
$913$		0			0
$914$	8.00000			0.264616
$915$		0			0
$916$			8.00000i			0.264327i
$917$		−	24.0000i		−	0.792550i
$918$		0			0
$919$	−32.0000			−1.05558			−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000			−1.05558			−0.527791	+	0.849374i	$0.676980\pi$
$920$	−24.0000			−0.791257
$921$		0			0
$922$	34.0000			1.11973
$923$	−16.0000	+	24.0000i	−0.526646	+	0.789970i
$924$		0			0
$925$			8.00000i			0.263038i
$926$	14.0000			0.460069
$927$		0			0
$928$			10.0000i			0.328266i
$929$		−	22.0000i		−	0.721797i	−0.932605	−	0.360898i	$-0.882470\pi$
$929$		−	22.0000i		−	0.721797i	0.932605	−	0.360898i	$-0.117530\pi$
$930$		0			0
$931$			6.00000i			0.196642i
$932$	−14.0000			−0.458585
$933$		0			0
$934$			36.0000i			1.17796i
$935$		0			0
$936$		0			0
$937$	38.0000			1.24141			0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000			1.24141			0.620703	+	0.784046i	$0.286847\pi$
$938$			12.0000i			0.391814i
$939$		0			0
$940$	4.00000			0.130466
$941$		−	50.0000i		−	1.62995i	−0.579494	−	0.814977i	$-0.696750\pi$
$941$		−	50.0000i		−	1.62995i	0.579494	−	0.814977i	$-0.303250\pi$
$942$		0			0
$943$			16.0000i			0.521032i
$944$		−	12.0000i		−	0.390567i
$945$		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
$949$	32.0000	−	48.0000i	1.03876	−	1.55815i
$950$	−2.00000			−0.0648886
$951$		0			0
$952$	12.0000			0.388922
$953$	−30.0000			−0.971795			−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000			−0.971795			−0.485898	+	0.874016i	$0.661507\pi$
$954$		0			0
$955$		−	20.0000i		−	0.647185i
$956$		−	8.00000i		−	0.258738i
$957$		0			0
$958$		0			0
$959$	12.0000			0.387500
$960$		0			0
$961$	27.0000			0.870968
$962$	24.0000	+	16.0000i	0.773791	+	0.515861i
$963$		0			0
$964$			12.0000i			0.386494i
$965$	4.00000			0.128765
$966$		0			0
$967$			58.0000i			1.86515i	0.360971	+	0.932577i	$0.382445\pi$
$967$			58.0000i			1.86515i	−0.360971	+	0.932577i	$0.617555\pi$
$968$		−	33.0000i		−	1.06066i
$969$		0			0
$970$		−	16.0000i		−	0.513729i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$	18.0000			0.576757
$975$		0			0
$976$	−10.0000			−0.320092
$977$			38.0000i			1.21573i	0.794041	+	0.607864i	$0.207973\pi$
$977$			38.0000i			1.21573i	−0.794041	+	0.607864i	$0.792027\pi$
$978$		0			0
$979$		0			0
$980$		−	3.00000i		−	0.0958315i
$981$		0			0
$982$		−	40.0000i		−	1.27645i
$983$		−	32.0000i		−	1.02064i	−0.859984	−	0.510321i	$-0.829527\pi$
$983$		−	32.0000i		−	1.02064i	0.859984	−	0.510321i	$-0.170473\pi$
$984$		0			0
$985$	−22.0000			−0.700978
$986$		−	4.00000i		−	0.127386i
$987$		0			0
$988$	4.00000	−	6.00000i	0.127257	−	0.190885i
$989$	−32.0000			−1.01754
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$	−10.0000			−0.317500
$993$		0			0
$994$			16.0000i			0.507489i
$995$			24.0000i			0.760851i
$996$		0			0
$997$	−42.0000			−1.33015			−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000			−1.33015			−0.665077	+	0.746775i	$0.731601\pi$
$998$	−10.0000			−0.316544
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.b.a.181.1		2
3.2	odd	2		195.2.b.b.181.2	yes	2
12.11	even	2		3120.2.g.a.961.2		2
13.5	odd	4		7605.2.a.d.1.1		1
13.8	odd	4		7605.2.a.p.1.1		1
13.12	even	2	inner	585.2.b.a.181.2		2
15.2	even	4		975.2.h.a.649.1		2
15.8	even	4		975.2.h.d.649.2		2
15.14	odd	2		975.2.b.b.376.1		2
39.5	even	4		2535.2.a.l.1.1		1
39.8	even	4		2535.2.a.e.1.1		1
39.38	odd	2		195.2.b.b.181.1	✓	2
156.155	even	2		3120.2.g.a.961.1		2
195.38	even	4		975.2.h.a.649.2		2
195.77	even	4		975.2.h.d.649.1		2
195.194	odd	2		975.2.b.b.376.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.b.b.181.1	✓	2	39.38	odd	2
195.2.b.b.181.2	yes	2	3.2	odd	2
585.2.b.a.181.1		2	1.1	even	1	trivial
585.2.b.a.181.2		2	13.12	even	2	inner
975.2.b.b.376.1		2	15.14	odd	2
975.2.b.b.376.2		2	195.194	odd	2
975.2.h.a.649.1		2	15.2	even	4
975.2.h.a.649.2		2	195.38	even	4
975.2.h.d.649.1		2	195.77	even	4
975.2.h.d.649.2		2	15.8	even	4
2535.2.a.e.1.1		1	39.8	even	4
2535.2.a.l.1.1		1	39.5	even	4
3120.2.g.a.961.1		2	156.155	even	2
3120.2.g.a.961.2		2	12.11	even	2
7605.2.a.d.1.1		1	13.5	odd	4
7605.2.a.p.1.1		1	13.8	odd	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{5} -2.00000i q^{7} -3.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{5} -2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{10} +(-3.00000 - 2.00000i) q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +2.00000i q^{19} -1.00000i q^{20} +8.00000 q^{23} -1.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -2.00000i q^{28} -2.00000 q^{29} -2.00000i q^{31} -5.00000i q^{32} +2.00000i q^{34} -2.00000 q^{35} -8.00000i q^{37} +2.00000 q^{38} -3.00000 q^{40} +2.00000i q^{41} -4.00000 q^{43} -8.00000i q^{46} +4.00000i q^{47} +3.00000 q^{49} +1.00000i q^{50} +(-3.00000 - 2.00000i) q^{52} +6.00000 q^{53} -6.00000 q^{56} +2.00000i q^{58} +12.0000i q^{59} +10.0000 q^{61} -2.00000 q^{62} -7.00000 q^{64} +(-2.00000 + 3.00000i) q^{65} -6.00000i q^{67} -2.00000 q^{68} +2.00000i q^{70} -8.00000i q^{71} +16.0000i q^{73} -8.00000 q^{74} +2.00000i q^{76} -8.00000 q^{79} +1.00000i q^{80} +2.00000 q^{82} -12.0000i q^{83} +2.00000i q^{85} +4.00000i q^{86} -6.00000i q^{89} +(-4.00000 + 6.00000i) q^{91} +8.00000 q^{92} +4.00000 q^{94} +2.00000 q^{95} +16.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} - 2 q^{10} - 6 q^{13} - 4 q^{14} - 2 q^{16} - 4 q^{17} + 16 q^{23} - 2 q^{25} - 4 q^{26} - 4 q^{29} - 4 q^{35} + 4 q^{38} - 6 q^{40} - 8 q^{43} + 6 q^{49} - 6 q^{52} + 12 q^{53} - 12 q^{56} + 20 q^{61} - 4 q^{62} - 14 q^{64} - 4 q^{65} - 4 q^{68} - 16 q^{74} - 16 q^{79} + 4 q^{82} - 8 q^{91} + 16 q^{92} + 8 q^{94} + 4 q^{95}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^10 - 6 * q^13 - 4 * q^14 - 2 * q^16 - 4 * q^17 + 16 * q^23 - 2 * q^25 - 4 * q^26 - 4 * q^29 - 4 * q^35 + 4 * q^38 - 6 * q^40 - 8 * q^43 + 6 * q^49 - 6 * q^52 + 12 * q^53 - 12 * q^56 + 20 * q^61 - 4 * q^62 - 14 * q^64 - 4 * q^65 - 4 * q^68 - 16 * q^74 - 16 * q^79 + 4 * q^82 - 8 * q^91 + 16 * q^92 + 8 * q^94 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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