LMFDB - Embedded newform 1584.2.b.e.593.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,2,Mod(593,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.593"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1584.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.6483036802$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			593.4
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	1584.593
Dual form			1584.2.b.e.593.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{5} +2.44949i q^{7} +(1.73205 - 2.82843i) q^{11} -4.89898i q^{13} -7.34847i q^{19} +2.82843i q^{23} +3.00000 q^{25} +6.92820 q^{29} +4.00000 q^{31} -3.46410 q^{35} +8.00000 q^{37} -6.92820 q^{41} +2.44949i q^{43} +2.82843i q^{47} +1.00000 q^{49} +9.89949i q^{53} +(4.00000 + 2.44949i) q^{55} +11.3137i q^{59} +4.89898i q^{61} +6.92820 q^{65} +4.00000 q^{67} +2.82843i q^{71} +(6.92820 + 4.24264i) q^{77} -12.2474i q^{79} +13.8564 q^{83} -7.07107i q^{89} +12.0000 q^{91} +10.3923 q^{95} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{25} + 16 q^{31} + 32 q^{37} + 4 q^{49} + 16 q^{55} + 16 q^{67} + 48 q^{91} - 40 q^{97}+O(q^{100}) $ 4 * q + 12 * q^25 + 16 * q^31 + 32 * q^37 + 4 * q^49 + 16 * q^55 + 16 * q^67 + 48 * q^91 - 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			1.41421i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			1.41421i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$			2.44949i			0.925820i	0.886405	+	0.462910i	$0.153195\pi$
$7$			2.44949i			0.925820i	−0.886405	+	0.462910i	$0.846805\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.73205	−	2.82843i	0.522233	−	0.852803i
$11$	1.73205	−	2.82843i	0.522233	−	0.852803i
$12$		0			0
$13$		−	4.89898i		−	1.35873i	−0.733799	−	0.679366i	$-0.762255\pi$
$13$		−	4.89898i		−	1.35873i	0.733799	−	0.679366i	$-0.237745\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		−	7.34847i		−	1.68585i	−0.538028	−	0.842927i	$-0.680830\pi$
$19$		−	7.34847i		−	1.68585i	0.538028	−	0.842927i	$-0.319170\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.82843i			0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$			2.82843i			0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.92820			1.28654			0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820			1.28654			0.643268	+	0.765641i	$0.277578\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.46410			−0.585540
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.92820			−1.08200			−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820			−1.08200			−0.541002	+	0.841021i	$0.681955\pi$
$42$		0			0
$43$			2.44949i			0.373544i	0.982403	+	0.186772i	$0.0598025\pi$
$43$			2.44949i			0.373544i	−0.982403	+	0.186772i	$0.940197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.82843i			0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$			2.82843i			0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$			9.89949i			1.35980i	0.733305	+	0.679900i	$0.237977\pi$
$53$			9.89949i			1.35980i	−0.733305	+	0.679900i	$0.762023\pi$
$54$		0			0
$55$	4.00000	+	2.44949i	0.539360	+	0.330289i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			11.3137i			1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$			11.3137i			1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$		0			0
$61$			4.89898i			0.627250i	0.949547	+	0.313625i	$0.101543\pi$
$61$			4.89898i			0.627250i	−0.949547	+	0.313625i	$0.898457\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.92820			0.859338
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.82843i			0.335673i	0.985815	+	0.167836i	$0.0536780\pi$
$71$			2.82843i			0.335673i	−0.985815	+	0.167836i	$0.946322\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.92820	+	4.24264i	0.789542	+	0.483494i
$78$		0			0
$79$		−	12.2474i		−	1.37795i	−0.724787	−	0.688973i	$-0.758062\pi$
$79$		−	12.2474i		−	1.37795i	0.724787	−	0.688973i	$-0.241938\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.8564			1.52094			0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564			1.52094			0.760469	+	0.649374i	$0.224969\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	7.07107i		−	0.749532i	−0.927119	−	0.374766i	$-0.877723\pi$
$89$		−	7.07107i		−	0.749532i	0.927119	−	0.374766i	$-0.122277\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.3923			1.06623
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.8564			−1.37876			−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564			−1.37876			−0.689382	+	0.724398i	$0.742118\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		−	14.6969i		−	1.40771i	−0.710343	−	0.703856i	$-0.751460\pi$
$109$		−	14.6969i		−	1.40771i	0.710343	−	0.703856i	$-0.248540\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.41421i			0.133038i	0.997785	+	0.0665190i	$0.0211893\pi$
$113$			1.41421i			0.133038i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$		0			0
$115$	−4.00000			−0.373002
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.00000	−	9.79796i	−0.454545	−	0.890724i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$		−	7.34847i		−	0.652071i	−0.945357	−	0.326036i	$-0.894287\pi$
$127$		−	7.34847i		−	0.652071i	0.945357	−	0.326036i	$-0.105713\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.46410			−0.302660			−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410			−0.302660			−0.151330	+	0.988483i	$0.548356\pi$
$132$		0			0
$133$	18.0000			1.56080
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	15.5563i		−	1.32907i	−0.747258	−	0.664534i	$-0.768630\pi$
$137$		−	15.5563i		−	1.32907i	0.747258	−	0.664534i	$-0.231370\pi$
$138$		0			0
$139$		−	2.44949i		−	0.207763i	−0.994590	−	0.103882i	$-0.966874\pi$
$139$		−	2.44949i		−	0.207763i	0.994590	−	0.103882i	$-0.0331263\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−13.8564	−	8.48528i	−1.15873	−	0.709575i
$144$		0			0
$145$			9.79796i			0.813676i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.8564			1.13516			0.567581	−	0.823318i	$-0.307880\pi$
$149$	13.8564			1.13516			0.567581	+	0.823318i	$0.307880\pi$
$150$		0			0
$151$			2.44949i			0.199337i	0.995021	+	0.0996683i	$0.0317782\pi$
$151$			2.44949i			0.199337i	−0.995021	+	0.0996683i	$0.968222\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			5.65685i			0.454369i
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.92820			−0.546019
$162$		0			0
$163$	−8.00000			−0.626608			−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000			−0.626608			−0.313304	+	0.949653i	$0.601436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.46410			−0.268060			−0.134030	−	0.990977i	$-0.542792\pi$
$167$	−3.46410			−0.268060			−0.134030	+	0.990977i	$0.542792\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.92820			0.526742			0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820			0.526742			0.263371	+	0.964695i	$0.415166\pi$
$174$		0			0
$175$			7.34847i			0.555492i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	22.6274i		−	1.69125i	−0.533775	−	0.845626i	$-0.679227\pi$
$179$		−	22.6274i		−	1.69125i	0.533775	−	0.845626i	$-0.320773\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			11.3137i			0.831800i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2.82843i			0.204658i	0.994751	+	0.102329i	$0.0326294\pi$
$191$			2.82843i			0.204658i	−0.994751	+	0.102329i	$0.967371\pi$
$192$		0			0
$193$			9.79796i			0.705273i	0.935760	+	0.352636i	$0.114715\pi$
$193$			9.79796i			0.705273i	−0.935760	+	0.352636i	$0.885285\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			16.9706i			1.19110i
$204$		0			0
$205$		−	9.79796i		−	0.684319i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−20.7846	−	12.7279i	−1.43770	−	0.880409i
$210$		0			0
$211$			12.2474i			0.843149i	0.906794	+	0.421575i	$0.138522\pi$
$211$			12.2474i			0.843149i	−0.906794	+	0.421575i	$0.861478\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.46410			−0.236250
$216$		0			0
$217$			9.79796i			0.665129i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−27.7128			−1.83936			−0.919682	−	0.392664i	$-0.871554\pi$
$227$	−27.7128			−1.83936			−0.919682	+	0.392664i	$0.871554\pi$
$228$		0			0
$229$	−10.0000			−0.660819			−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000			−0.660819			−0.330409	+	0.943838i	$0.607187\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	20.7846			1.36165			0.680823	−	0.732448i	$-0.261622\pi$
$233$	20.7846			1.36165			0.680823	+	0.732448i	$0.261622\pi$
$234$		0			0
$235$	−4.00000			−0.260931
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.3205			1.12037			0.560185	−	0.828367i	$-0.310730\pi$
$239$	17.3205			1.12037			0.560185	+	0.828367i	$0.310730\pi$
$240$		0			0
$241$			19.5959i			1.26228i	0.775667	+	0.631142i	$0.217413\pi$
$241$			19.5959i			1.26228i	−0.775667	+	0.631142i	$0.782587\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			1.41421i			0.0903508i
$246$		0			0
$247$	−36.0000			−2.29063
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	5.65685i		−	0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$		−	5.65685i		−	0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$		0			0
$253$	8.00000	+	4.89898i	0.502956	+	0.307996i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			9.89949i			0.617514i	0.951141	+	0.308757i	$0.0999129\pi$
$257$			9.89949i			0.617514i	−0.951141	+	0.308757i	$0.900087\pi$
$258$		0			0
$259$			19.5959i			1.21763i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.46410			0.213606			0.106803	−	0.994280i	$-0.465939\pi$
$263$	3.46410			0.213606			0.106803	+	0.994280i	$0.465939\pi$
$264$		0			0
$265$	−14.0000			−0.860013
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	15.5563i		−	0.948487i	−0.880394	−	0.474244i	$-0.842722\pi$
$269$		−	15.5563i		−	0.948487i	0.880394	−	0.474244i	$-0.157278\pi$
$270$		0			0
$271$			7.34847i			0.446388i	0.974774	+	0.223194i	$0.0716483\pi$
$271$			7.34847i			0.446388i	−0.974774	+	0.223194i	$0.928352\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.19615	−	8.48528i	0.313340	−	0.511682i
$276$		0			0
$277$		−	24.4949i		−	1.47176i	−0.677114	−	0.735878i	$-0.736770\pi$
$277$		−	24.4949i		−	1.47176i	0.677114	−	0.735878i	$-0.263230\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.8564			−0.826604			−0.413302	−	0.910594i	$-0.635625\pi$
$281$	−13.8564			−0.826604			−0.413302	+	0.910594i	$0.635625\pi$
$282$		0			0
$283$		−	2.44949i		−	0.145607i	−0.997346	−	0.0728035i	$-0.976805\pi$
$283$		−	2.44949i		−	0.145607i	0.997346	−	0.0728035i	$-0.0231946\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	16.9706i		−	1.00174i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.92820			−0.404750			−0.202375	−	0.979308i	$-0.564866\pi$
$293$	−6.92820			−0.404750			−0.202375	+	0.979308i	$0.564866\pi$
$294$		0			0
$295$	−16.0000			−0.931556
$296$		0			0
$297$		0			0
$298$		0			0
$299$	13.8564			0.801337
$300$		0			0
$301$	−6.00000			−0.345834
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.92820			−0.396708
$306$		0			0
$307$		−	7.34847i		−	0.419399i	−0.977766	−	0.209700i	$-0.932751\pi$
$307$		−	7.34847i		−	0.419399i	0.977766	−	0.209700i	$-0.0672486\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	14.1421i		−	0.801927i	−0.916094	−	0.400963i	$-0.868675\pi$
$311$		−	14.1421i		−	0.801927i	0.916094	−	0.400963i	$-0.131325\pi$
$312$		0			0
$313$	−16.0000			−0.904373			−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000			−0.904373			−0.452187	+	0.891923i	$0.649356\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			18.3848i			1.03259i	0.856410	+	0.516296i	$0.172690\pi$
$317$			18.3848i			1.03259i	−0.856410	+	0.516296i	$0.827310\pi$
$318$		0			0
$319$	12.0000	−	19.5959i	0.671871	−	1.09716i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	14.6969i		−	0.815239i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.92820			−0.381964
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			5.65685i			0.309067i
$336$		0			0
$337$			9.79796i			0.533729i	0.963734	+	0.266864i	$0.0859876\pi$
$337$			9.79796i			0.533729i	−0.963734	+	0.266864i	$0.914012\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.92820	−	11.3137i	0.375183	−	0.612672i
$342$		0			0
$343$			19.5959i			1.05808i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.8564			−0.743851			−0.371925	−	0.928263i	$-0.621302\pi$
$347$	−13.8564			−0.743851			−0.371925	+	0.928263i	$0.621302\pi$
$348$		0			0
$349$			4.89898i			0.262236i	0.991367	+	0.131118i	$0.0418567\pi$
$349$			4.89898i			0.262236i	−0.991367	+	0.131118i	$0.958143\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	32.5269i		−	1.73123i	−0.500708	−	0.865616i	$-0.666927\pi$
$353$		−	32.5269i		−	1.73123i	0.500708	−	0.865616i	$-0.333073\pi$
$354$		0			0
$355$	−4.00000			−0.212298
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−35.0000			−1.84211
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−24.2487			−1.25893
$372$		0			0
$373$			24.4949i			1.26830i	0.773211	+	0.634149i	$0.218649\pi$
$373$			24.4949i			1.26830i	−0.773211	+	0.634149i	$0.781351\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	33.9411i		−	1.74806i
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			36.7696i			1.87884i	0.342773	+	0.939418i	$0.388634\pi$
$383$			36.7696i			1.87884i	−0.342773	+	0.939418i	$0.611366\pi$
$384$		0			0
$385$	−6.00000	+	9.79796i	−0.305788	+	0.499350i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			9.89949i			0.501924i	0.967997	+	0.250962i	$0.0807470\pi$
$389$			9.89949i			0.501924i	−0.967997	+	0.250962i	$0.919253\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	17.3205			0.871489
$396$		0			0
$397$	8.00000			0.401508			0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000			0.401508			0.200754	+	0.979642i	$0.435661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			1.41421i			0.0706225i	0.999376	+	0.0353112i	$0.0112422\pi$
$401$			1.41421i			0.0706225i	−0.999376	+	0.0353112i	$0.988758\pi$
$402$		0			0
$403$		−	19.5959i		−	0.976142i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.8564	−	22.6274i	0.686837	−	1.12160i
$408$		0			0
$409$			39.1918i			1.93791i	0.247234	+	0.968956i	$0.420478\pi$
$409$			39.1918i			1.93791i	−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−27.7128			−1.36366
$414$		0			0
$415$			19.5959i			0.961926i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			11.3137i			0.552711i	0.961056	+	0.276355i	$0.0891267\pi$
$419$			11.3137i			0.552711i	−0.961056	+	0.276355i	$0.910873\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−12.0000			−0.580721
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.3923			−0.500580			−0.250290	−	0.968171i	$-0.580526\pi$
$431$	−10.3923			−0.500580			−0.250290	+	0.968171i	$0.580526\pi$
$432$		0			0
$433$	−16.0000			−0.768911			−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000			−0.768911			−0.384455	+	0.923144i	$0.625611\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	20.7846			0.994263
$438$		0			0
$439$			2.44949i			0.116908i	0.998290	+	0.0584539i	$0.0186171\pi$
$439$			2.44949i			0.116908i	−0.998290	+	0.0584539i	$0.981383\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	22.6274i		−	1.07506i	−0.843244	−	0.537531i	$-0.819357\pi$
$443$		−	22.6274i		−	1.07506i	0.843244	−	0.537531i	$-0.180643\pi$
$444$		0			0
$445$	10.0000			0.474045
$446$		0			0
$447$		0			0
$448$		0			0
$449$			9.89949i			0.467186i	0.972334	+	0.233593i	$0.0750483\pi$
$449$			9.89949i			0.467186i	−0.972334	+	0.233593i	$0.924952\pi$
$450$		0			0
$451$	−12.0000	+	19.5959i	−0.565058	+	0.922736i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			16.9706i			0.795592i
$456$		0			0
$457$		−	9.79796i		−	0.458329i	−0.973388	−	0.229165i	$-0.926401\pi$
$457$		−	9.79796i		−	0.458329i	0.973388	−	0.229165i	$-0.0735994\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.92820			0.322679			0.161339	−	0.986899i	$-0.448419\pi$
$461$	6.92820			0.322679			0.161339	+	0.986899i	$0.448419\pi$
$462$		0			0
$463$	28.0000			1.30127			0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000			1.30127			0.650635	+	0.759390i	$0.274503\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	22.6274i		−	1.04707i	−0.852004	−	0.523536i	$-0.824613\pi$
$467$		−	22.6274i		−	1.04707i	0.852004	−	0.523536i	$-0.175387\pi$
$468$		0			0
$469$			9.79796i			0.452428i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.92820	+	4.24264i	0.318559	+	0.195077i
$474$		0			0
$475$		−	22.0454i		−	1.01151i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−38.1051			−1.74107			−0.870534	−	0.492109i	$-0.836226\pi$
$479$	−38.1051			−1.74107			−0.870534	+	0.492109i	$0.836226\pi$
$480$		0			0
$481$		−	39.1918i		−	1.78699i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	14.1421i		−	0.642161i
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	27.7128			1.25066			0.625331	−	0.780360i	$-0.284964\pi$
$491$	27.7128			1.25066			0.625331	+	0.780360i	$0.284964\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.92820			−0.310772
$498$		0			0
$499$	−32.0000			−1.43252			−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000			−1.43252			−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−31.1769			−1.39011			−0.695055	−	0.718957i	$-0.744620\pi$
$503$	−31.1769			−1.39011			−0.695055	+	0.718957i	$0.744620\pi$
$504$		0			0
$505$		−	19.5959i		−	0.872007i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	15.5563i		−	0.689523i	−0.938690	−	0.344762i	$-0.887960\pi$
$509$		−	15.5563i		−	0.689523i	0.938690	−	0.344762i	$-0.112040\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			5.65685i			0.249271i
$516$		0			0
$517$	8.00000	+	4.89898i	0.351840	+	0.215457i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	15.5563i		−	0.681536i	−0.940147	−	0.340768i	$-0.889313\pi$
$521$		−	15.5563i		−	0.681536i	0.940147	−	0.340768i	$-0.110687\pi$
$522$		0			0
$523$		−	22.0454i		−	0.963978i	−0.876177	−	0.481989i	$-0.839914\pi$
$523$		−	22.0454i		−	0.963978i	0.876177	−	0.481989i	$-0.160086\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$		0			0
$532$		0			0
$533$			33.9411i			1.47015i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.73205	−	2.82843i	0.0746047	−	0.121829i
$540$		0			0
$541$			14.6969i			0.631871i	0.948781	+	0.315935i	$0.102318\pi$
$541$			14.6969i			0.631871i	−0.948781	+	0.315935i	$0.897682\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	20.7846			0.890315
$546$		0			0
$547$			31.8434i			1.36152i	0.732505	+	0.680762i	$0.238351\pi$
$547$			31.8434i			1.36152i	−0.732505	+	0.680762i	$0.761649\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	50.9117i		−	2.16891i
$552$		0			0
$553$	30.0000			1.27573
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−20.7846			−0.880672			−0.440336	−	0.897833i	$-0.645141\pi$
$557$	−20.7846			−0.880672			−0.440336	+	0.897833i	$0.645141\pi$
$558$		0			0
$559$	12.0000			0.507546
$560$		0			0
$561$		0			0
$562$		0			0
$563$	17.3205			0.729972			0.364986	−	0.931013i	$-0.381074\pi$
$563$	17.3205			0.729972			0.364986	+	0.931013i	$0.381074\pi$
$564$		0			0
$565$	−2.00000			−0.0841406
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.8564			−0.580891			−0.290445	−	0.956892i	$-0.593803\pi$
$569$	−13.8564			−0.580891			−0.290445	+	0.956892i	$0.593803\pi$
$570$		0			0
$571$		−	2.44949i		−	0.102508i	−0.998686	−	0.0512540i	$-0.983678\pi$
$571$		−	2.44949i		−	0.102508i	0.998686	−	0.0512540i	$-0.0163218\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			8.48528i			0.353861i
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			33.9411i			1.40812i
$582$		0			0
$583$	28.0000	+	17.1464i	1.15964	+	0.710132i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			28.2843i			1.16742i	0.811963	+	0.583708i	$0.198399\pi$
$587$			28.2843i			1.16742i	−0.811963	+	0.583708i	$0.801601\pi$
$588$		0			0
$589$		−	29.3939i		−	1.21115i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−41.5692			−1.70704			−0.853522	−	0.521057i	$-0.825538\pi$
$593$	−41.5692			−1.70704			−0.853522	+	0.521057i	$0.825538\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			2.82843i			0.115566i	0.998329	+	0.0577832i	$0.0184032\pi$
$599$			2.82843i			0.115566i	−0.998329	+	0.0577832i	$0.981597\pi$
$600$		0			0
$601$			19.5959i			0.799334i	0.916660	+	0.399667i	$0.130874\pi$
$601$			19.5959i			0.799334i	−0.916660	+	0.399667i	$0.869126\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.8564	−	7.07107i	0.563343	−	0.287480i
$606$		0			0
$607$			12.2474i			0.497109i	0.968618	+	0.248554i	$0.0799554\pi$
$607$			12.2474i			0.497109i	−0.968618	+	0.248554i	$0.920045\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.8564			0.560570
$612$		0			0
$613$		−	14.6969i		−	0.593604i	−0.954939	−	0.296802i	$-0.904080\pi$
$613$		−	14.6969i		−	0.593604i	0.954939	−	0.296802i	$-0.0959201\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			26.8701i			1.08175i	0.841104	+	0.540874i	$0.181906\pi$
$617$			26.8701i			1.08175i	−0.841104	+	0.540874i	$0.818094\pi$
$618$		0			0
$619$	28.0000			1.12542			0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000			1.12542			0.562708	+	0.826656i	$0.309760\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	17.3205			0.693932
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−44.0000			−1.75161			−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000			−1.75161			−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.3923			0.412406
$636$		0			0
$637$		−	4.89898i		−	0.194105i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	32.5269i		−	1.28474i	−0.766396	−	0.642368i	$-0.777952\pi$
$641$		−	32.5269i		−	1.28474i	0.766396	−	0.642368i	$-0.222048\pi$
$642$		0			0
$643$	28.0000			1.10421			0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000			1.10421			0.552106	+	0.833774i	$0.313824\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			19.7990i			0.778379i	0.921158	+	0.389189i	$0.127245\pi$
$647$			19.7990i			0.778379i	−0.921158	+	0.389189i	$0.872755\pi$
$648$		0			0
$649$	32.0000	+	19.5959i	1.25611	+	0.769207i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			26.8701i			1.05151i	0.850637	+	0.525753i	$0.176216\pi$
$653$			26.8701i			1.05151i	−0.850637	+	0.525753i	$0.823784\pi$
$654$		0			0
$655$		−	4.89898i		−	0.191419i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−38.1051			−1.48436			−0.742182	−	0.670198i	$-0.766209\pi$
$659$	−38.1051			−1.48436			−0.742182	+	0.670198i	$0.766209\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			25.4558i			0.987135i
$666$		0			0
$667$			19.5959i			0.758757i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.8564	+	8.48528i	0.534921	+	0.327571i
$672$		0			0
$673$		−	39.1918i		−	1.51073i	−0.655302	−	0.755367i	$-0.727459\pi$
$673$		−	39.1918i		−	1.51073i	0.655302	−	0.755367i	$-0.272541\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	27.7128			1.06509			0.532545	−	0.846402i	$-0.321236\pi$
$677$	27.7128			1.06509			0.532545	+	0.846402i	$0.321236\pi$
$678$		0			0
$679$		−	24.4949i		−	0.940028i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			28.2843i			1.08227i	0.840937	+	0.541134i	$0.182005\pi$
$683$			28.2843i			1.08227i	−0.840937	+	0.541134i	$0.817995\pi$
$684$		0			0
$685$	22.0000			0.840577
$686$		0			0
$687$		0			0
$688$		0			0
$689$	48.4974			1.84760
$690$		0			0
$691$	−44.0000			−1.67384			−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000			−1.67384			−0.836919	+	0.547326i	$0.815646\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.46410			0.131401
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20.7846			0.785024			0.392512	−	0.919747i	$-0.371606\pi$
$701$	20.7846			0.785024			0.392512	+	0.919747i	$0.371606\pi$
$702$		0			0
$703$		−	58.7878i		−	2.21722i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	33.9411i		−	1.27649i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			11.3137i			0.423702i
$714$		0			0
$715$	12.0000	−	19.5959i	0.448775	−	0.732846i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			2.82843i			0.105483i	0.998608	+	0.0527413i	$0.0167959\pi$
$719$			2.82843i			0.105483i	−0.998608	+	0.0527413i	$0.983204\pi$
$720$		0			0
$721$			9.79796i			0.364895i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	20.7846			0.771921
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	34.2929i		−	1.26664i	−0.773892	−	0.633318i	$-0.781693\pi$
$733$		−	34.2929i		−	1.26664i	0.773892	−	0.633318i	$-0.218307\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.92820	−	11.3137i	0.255204	−	0.416746i
$738$		0			0
$739$			7.34847i			0.270318i	0.990824	+	0.135159i	$0.0431545\pi$
$739$			7.34847i			0.270318i	−0.990824	+	0.135159i	$0.956846\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	27.7128			1.01668			0.508342	−	0.861155i	$-0.330258\pi$
$743$	27.7128			1.01668			0.508342	+	0.861155i	$0.330258\pi$
$744$		0			0
$745$			19.5959i			0.717939i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.00000			0.145962			0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000			0.145962			0.0729810	+	0.997333i	$0.476749\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.46410			−0.126072
$756$		0			0
$757$	8.00000			0.290765			0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000			0.290765			0.145382	+	0.989376i	$0.453559\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	13.8564			0.502294			0.251147	−	0.967949i	$-0.419192\pi$
$761$	13.8564			0.502294			0.251147	+	0.967949i	$0.419192\pi$
$762$		0			0
$763$	36.0000			1.30329
$764$		0			0
$765$		0			0
$766$		0			0
$767$	55.4256			2.00130
$768$		0			0
$769$		−	19.5959i		−	0.706647i	−0.935501	−	0.353323i	$-0.885052\pi$
$769$		−	19.5959i		−	0.706647i	0.935501	−	0.353323i	$-0.114948\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	32.5269i		−	1.16991i	−0.811065	−	0.584956i	$-0.801112\pi$
$773$		−	32.5269i		−	1.16991i	0.811065	−	0.584956i	$-0.198888\pi$
$774$		0			0
$775$	12.0000			0.431053
$776$		0			0
$777$		0			0
$778$		0			0
$779$			50.9117i			1.82410i
$780$		0			0
$781$	8.00000	+	4.89898i	0.286263	+	0.175299i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			19.7990i			0.706656i
$786$		0			0
$787$			12.2474i			0.436574i	0.975885	+	0.218287i	$0.0700470\pi$
$787$			12.2474i			0.436574i	−0.975885	+	0.218287i	$0.929953\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.46410			−0.123169
$792$		0			0
$793$	24.0000			0.852265
$794$		0			0
$795$		0			0
$796$		0			0
$797$			26.8701i			0.951786i	0.879503	+	0.475893i	$0.157875\pi$
$797$			26.8701i			0.951786i	−0.879503	+	0.475893i	$0.842125\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		−	9.79796i		−	0.345333i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−20.7846			−0.730748			−0.365374	−	0.930861i	$-0.619059\pi$
$809$	−20.7846			−0.730748			−0.365374	+	0.930861i	$0.619059\pi$
$810$		0			0
$811$		−	22.0454i		−	0.774119i	−0.922055	−	0.387059i	$-0.873491\pi$
$811$		−	22.0454i		−	0.774119i	0.922055	−	0.387059i	$-0.126509\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	11.3137i		−	0.396302i
$816$		0			0
$817$	18.0000			0.629740
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.8564			−0.483592			−0.241796	−	0.970327i	$-0.577736\pi$
$821$	−13.8564			−0.483592			−0.241796	+	0.970327i	$0.577736\pi$
$822$		0			0
$823$	−8.00000			−0.278862			−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000			−0.278862			−0.139431	+	0.990232i	$0.544527\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−41.5692			−1.44550			−0.722752	−	0.691108i	$-0.757123\pi$
$827$	−41.5692			−1.44550			−0.722752	+	0.691108i	$0.757123\pi$
$828$		0			0
$829$	−46.0000			−1.59765			−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000			−1.59765			−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	4.89898i		−	0.169536i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			19.7990i			0.683537i	0.939784	+	0.341769i	$0.111026\pi$
$839$			19.7990i			0.683537i	−0.939784	+	0.341769i	$0.888974\pi$
$840$		0			0
$841$	19.0000			0.655172
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	15.5563i		−	0.535155i
$846$		0			0
$847$	24.0000	−	12.2474i	0.824650	−	0.420827i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			22.6274i			0.775658i
$852$		0			0
$853$			34.2929i			1.17417i	0.809527	+	0.587083i	$0.199724\pi$
$853$			34.2929i			1.17417i	−0.809527	+	0.587083i	$0.800276\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−34.6410			−1.18331			−0.591657	−	0.806190i	$-0.701526\pi$
$857$	−34.6410			−1.18331			−0.591657	+	0.806190i	$0.701526\pi$
$858$		0			0
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			19.7990i			0.673965i	0.941511	+	0.336983i	$0.109406\pi$
$863$			19.7990i			0.673965i	−0.941511	+	0.336983i	$0.890594\pi$
$864$		0			0
$865$			9.79796i			0.333141i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−34.6410	−	21.2132i	−1.17512	−	0.719609i
$870$		0			0
$871$		−	19.5959i		−	0.663982i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−27.7128			−0.936864
$876$		0			0
$877$			53.8888i			1.81969i	0.414943	+	0.909847i	$0.363801\pi$
$877$			53.8888i			1.81969i	−0.414943	+	0.909847i	$0.636199\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			9.89949i			0.333522i	0.985997	+	0.166761i	$0.0533309\pi$
$881$			9.89949i			0.333522i	−0.985997	+	0.166761i	$0.946669\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−55.4256			−1.86101			−0.930505	−	0.366279i	$-0.880632\pi$
$887$	−55.4256			−1.86101			−0.930505	+	0.366279i	$0.880632\pi$
$888$		0			0
$889$	18.0000			0.603701
$890$		0			0
$891$		0			0
$892$		0			0
$893$	20.7846			0.695530
$894$		0			0
$895$	32.0000			1.06964
$896$		0			0
$897$		0			0
$898$		0			0
$899$	27.7128			0.924274
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			11.3137i			0.376080i
$906$		0			0
$907$	52.0000			1.72663			0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000			1.72663			0.863316	+	0.504664i	$0.168384\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			53.7401i			1.78049i	0.455483	+	0.890245i	$0.349467\pi$
$911$			53.7401i			1.78049i	−0.455483	+	0.890245i	$0.650533\pi$
$912$		0			0
$913$	24.0000	−	39.1918i	0.794284	−	1.29706i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	8.48528i		−	0.280209i
$918$		0			0
$919$			36.7423i			1.21202i	0.795458	+	0.606009i	$0.207230\pi$
$919$			36.7423i			1.21202i	−0.795458	+	0.606009i	$0.792770\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.8564			0.456089
$924$		0			0
$925$	24.0000			0.789115
$926$		0			0
$927$		0			0
$928$		0			0
$929$			9.89949i			0.324792i	0.986726	+	0.162396i	$0.0519222\pi$
$929$			9.89949i			0.324792i	−0.986726	+	0.162396i	$0.948078\pi$
$930$		0			0
$931$		−	7.34847i		−	0.240836i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	29.3939i		−	0.960256i	−0.877198	−	0.480128i	$-0.840590\pi$
$937$		−	29.3939i		−	0.960256i	0.877198	−	0.480128i	$-0.159410\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−27.7128			−0.903412			−0.451706	−	0.892167i	$-0.649184\pi$
$941$	−27.7128			−0.903412			−0.451706	+	0.892167i	$0.649184\pi$
$942$		0			0
$943$		−	19.5959i		−	0.638131i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	22.6274i		−	0.735292i	−0.929966	−	0.367646i	$-0.880164\pi$
$947$		−	22.6274i		−	0.735292i	0.929966	−	0.367646i	$-0.119836\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20.7846			0.673280			0.336640	−	0.941634i	$-0.390710\pi$
$953$	20.7846			0.673280			0.336640	+	0.941634i	$0.390710\pi$
$954$		0			0
$955$	−4.00000			−0.129437
$956$		0			0
$957$		0			0
$958$		0			0
$959$	38.1051			1.23048
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−13.8564			−0.446054
$966$		0			0
$967$			41.6413i			1.33909i	0.742769	+	0.669547i	$0.233512\pi$
$967$			41.6413i			1.33909i	−0.742769	+	0.669547i	$0.766488\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	56.5685i		−	1.81537i	−0.419651	−	0.907685i	$-0.637848\pi$
$971$		−	56.5685i		−	1.81537i	0.419651	−	0.907685i	$-0.362152\pi$
$972$		0			0
$973$	6.00000			0.192351
$974$		0			0
$975$		0			0
$976$		0			0
$977$			35.3553i			1.13112i	0.824708	+	0.565559i	$0.191339\pi$
$977$			35.3553i			1.13112i	−0.824708	+	0.565559i	$0.808661\pi$
$978$		0			0
$979$	−20.0000	−	12.2474i	−0.639203	−	0.391430i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	48.0833i		−	1.53362i	−0.641875	−	0.766809i	$-0.721843\pi$
$983$		−	48.0833i		−	1.53362i	0.641875	−	0.766809i	$-0.278157\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.92820			−0.220304
$990$		0			0
$991$	52.0000			1.65183			0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000			1.65183			0.825917	+	0.563791i	$0.190658\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	11.3137i		−	0.358669i
$996$		0			0
$997$		−	24.4949i		−	0.775761i	−0.921710	−	0.387881i	$-0.873207\pi$
$997$		−	24.4949i		−	0.775761i	0.921710	−	0.387881i	$-0.126793\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.2.b.e.593.4		4
3.2	odd	2	inner	1584.2.b.e.593.2		4
4.3	odd	2		99.2.d.a.98.4	yes	4
8.3	odd	2		6336.2.b.s.2177.1		4
8.5	even	2		6336.2.b.t.2177.2		4
11.10	odd	2	inner	1584.2.b.e.593.3		4
12.11	even	2		99.2.d.a.98.1	✓	4
20.3	even	4		2475.2.d.a.2474.1		8
20.7	even	4		2475.2.d.a.2474.7		8
20.19	odd	2		2475.2.f.e.2276.2		4
24.5	odd	2		6336.2.b.t.2177.4		4
24.11	even	2		6336.2.b.s.2177.3		4
33.32	even	2	inner	1584.2.b.e.593.1		4
36.7	odd	6		891.2.g.c.296.1		8
36.11	even	6		891.2.g.c.296.4		8
36.23	even	6		891.2.g.c.593.3		8
36.31	odd	6		891.2.g.c.593.2		8
44.43	even	2		99.2.d.a.98.2	yes	4
60.23	odd	4		2475.2.d.a.2474.6		8
60.47	odd	4		2475.2.d.a.2474.4		8
60.59	even	2		2475.2.f.e.2276.4		4
88.21	odd	2		6336.2.b.t.2177.1		4
88.43	even	2		6336.2.b.s.2177.2		4
132.131	odd	2		99.2.d.a.98.3	yes	4
220.43	odd	4		2475.2.d.a.2474.8		8
220.87	odd	4		2475.2.d.a.2474.2		8
220.219	even	2		2475.2.f.e.2276.3		4
264.131	odd	2		6336.2.b.s.2177.4		4
264.197	even	2		6336.2.b.t.2177.3		4
396.43	even	6		891.2.g.c.296.3		8
396.131	odd	6		891.2.g.c.593.1		8
396.175	even	6		891.2.g.c.593.4		8
396.263	odd	6		891.2.g.c.296.2		8
660.263	even	4		2475.2.d.a.2474.3		8
660.527	even	4		2475.2.d.a.2474.5		8
660.659	odd	2		2475.2.f.e.2276.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.d.a.98.1	✓	4	12.11	even	2
99.2.d.a.98.2	yes	4	44.43	even	2
99.2.d.a.98.3	yes	4	132.131	odd	2
99.2.d.a.98.4	yes	4	4.3	odd	2
891.2.g.c.296.1		8	36.7	odd	6
891.2.g.c.296.2		8	396.263	odd	6
891.2.g.c.296.3		8	396.43	even	6
891.2.g.c.296.4		8	36.11	even	6
891.2.g.c.593.1		8	396.131	odd	6
891.2.g.c.593.2		8	36.31	odd	6
891.2.g.c.593.3		8	36.23	even	6
891.2.g.c.593.4		8	396.175	even	6
1584.2.b.e.593.1		4	33.32	even	2	inner
1584.2.b.e.593.2		4	3.2	odd	2	inner
1584.2.b.e.593.3		4	11.10	odd	2	inner
1584.2.b.e.593.4		4	1.1	even	1	trivial
2475.2.d.a.2474.1		8	20.3	even	4
2475.2.d.a.2474.2		8	220.87	odd	4
2475.2.d.a.2474.3		8	660.263	even	4
2475.2.d.a.2474.4		8	60.47	odd	4
2475.2.d.a.2474.5		8	660.527	even	4
2475.2.d.a.2474.6		8	60.23	odd	4
2475.2.d.a.2474.7		8	20.7	even	4
2475.2.d.a.2474.8		8	220.43	odd	4
2475.2.f.e.2276.1		4	660.659	odd	2
2475.2.f.e.2276.2		4	20.19	odd	2
2475.2.f.e.2276.3		4	220.219	even	2
2475.2.f.e.2276.4		4	60.59	even	2
6336.2.b.s.2177.1		4	8.3	odd	2
6336.2.b.s.2177.2		4	88.43	even	2
6336.2.b.s.2177.3		4	24.11	even	2
6336.2.b.s.2177.4		4	264.131	odd	2
6336.2.b.t.2177.1		4	88.21	odd	2
6336.2.b.t.2177.2		4	8.5	even	2
6336.2.b.t.2177.3		4	264.197	even	2
6336.2.b.t.2177.4		4	24.5	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{5} +2.44949i q^{7} +(1.73205 - 2.82843i) q^{11} -4.89898i q^{13} -7.34847i q^{19} +2.82843i q^{23} +3.00000 q^{25} +6.92820 q^{29} +4.00000 q^{31} -3.46410 q^{35} +8.00000 q^{37} -6.92820 q^{41} +2.44949i q^{43} +2.82843i q^{47} +1.00000 q^{49} +9.89949i q^{53} +(4.00000 + 2.44949i) q^{55} +11.3137i q^{59} +4.89898i q^{61} +6.92820 q^{65} +4.00000 q^{67} +2.82843i q^{71} +(6.92820 + 4.24264i) q^{77} -12.2474i q^{79} +13.8564 q^{83} -7.07107i q^{89} +12.0000 q^{91} +10.3923 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{25} + 16 q^{31} + 32 q^{37} + 4 q^{49} + 16 q^{55} + 16 q^{67} + 48 q^{91} - 40 q^{97}+O(q^{100}) \) 4 * q + 12 * q^25 + 16 * q^31 + 32 * q^37 + 4 * q^49 + 16 * q^55 + 16 * q^67 + 48 * q^91 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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