LMFDB - Embedded newform 2880.2.t.c.2161.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2880,2,Mod(721,2880)] 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("2880.721"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2880, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.t (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 $ x^16 - 4x^15 + 4x^14 + 7x^12 - 8x^11 - 28x^10 + 28x^9 + 17x^8 + 56x^7 - 112x^6 - 64x^5 + 112x^4 + 256x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2161.4
Root			$-0.966675 + 1.03225i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.c.721.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{5} +1.73696i q^{7} +(0.505430 + 0.505430i) q^{11} +(-1.88750 + 1.88750i) q^{13} -4.53524 q^{17} +(3.22022 - 3.22022i) q^{19} -8.85045i q^{23} +1.00000i q^{25} +(2.44059 - 2.44059i) q^{29} +5.70401 q^{31} +(1.22822 - 1.22822i) q^{35} +(-5.35670 - 5.35670i) q^{37} +10.0343i q^{41} +(2.10564 + 2.10564i) q^{43} +4.32303 q^{47} +3.98295 q^{49} +(1.37458 + 1.37458i) q^{53} -0.714786i q^{55} +(6.64140 + 6.64140i) q^{59} +(5.26208 - 5.26208i) q^{61} +2.66933 q^{65} +(10.5578 - 10.5578i) q^{67} +14.0437i q^{71} +6.63830i q^{73} +(-0.877914 + 0.877914i) q^{77} -4.27297 q^{79} +(9.15483 - 9.15483i) q^{83} +(3.20690 + 3.20690i) q^{85} +3.23826i q^{89} +(-3.27852 - 3.27852i) q^{91} -4.55407 q^{95} +1.94129 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) $ 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$6$		0			0
$7$			1.73696i			0.656511i	0.944589	+	0.328255i	$0.106461\pi$
$7$			1.73696i			0.656511i	−0.944589	+	0.328255i	$0.893539\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.505430	+	0.505430i	0.152393	+	0.152393i	0.779186	−	0.626793i	$-0.215633\pi$
$11$	0.505430	+	0.505430i	0.152393	+	0.152393i	−0.626793	+	0.779186i	$0.715633\pi$
$12$		0			0
$13$	−1.88750	+	1.88750i	−0.523498	+	0.523498i	−0.918626	−	0.395128i	$-0.870700\pi$
$13$	−1.88750	+	1.88750i	−0.523498	+	0.523498i	0.395128	+	0.918626i	$0.370700\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.53524			−1.09996			−0.549979	−	0.835178i	$-0.685364\pi$
$17$	−4.53524			−1.09996			−0.549979	+	0.835178i	$0.685364\pi$
$18$		0			0
$19$	3.22022	−	3.22022i	0.738768	−	0.738768i	−0.233571	−	0.972340i	$-0.575041\pi$
$19$	3.22022	−	3.22022i	0.738768	−	0.738768i	0.972340	+	0.233571i	$0.0750413\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.85045i		−	1.84545i	−0.385463	−	0.922723i	$-0.625958\pi$
$23$		−	8.85045i		−	1.84545i	0.385463	−	0.922723i	$-0.374042\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.44059	−	2.44059i	0.453205	−	0.453205i	−0.443212	−	0.896417i	$-0.646161\pi$
$29$	2.44059	−	2.44059i	0.453205	−	0.453205i	0.896417	+	0.443212i	$0.146161\pi$
$30$		0			0
$31$	5.70401			1.02447			0.512235	−	0.858845i	$-0.328818\pi$
$31$	5.70401			1.02447			0.512235	+	0.858845i	$0.328818\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.22822	−	1.22822i	0.207607	−	0.207607i
$36$		0			0
$37$	−5.35670	−	5.35670i	−0.880636	−	0.880636i	0.112963	−	0.993599i	$-0.463966\pi$
$37$	−5.35670	−	5.35670i	−0.880636	−	0.880636i	−0.993599	+	0.112963i	$0.963966\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.0343i			1.56709i	0.621335	+	0.783545i	$0.286591\pi$
$41$			10.0343i			1.56709i	−0.621335	+	0.783545i	$0.713409\pi$
$42$		0			0
$43$	2.10564	+	2.10564i	0.321107	+	0.321107i	0.849192	−	0.528085i	$-0.177090\pi$
$43$	2.10564	+	2.10564i	0.321107	+	0.321107i	−0.528085	+	0.849192i	$0.677090\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.32303			0.630578			0.315289	−	0.948996i	$-0.397899\pi$
$47$	4.32303			0.630578			0.315289	+	0.948996i	$0.397899\pi$
$48$		0			0
$49$	3.98295			0.568993
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.37458	+	1.37458i	0.188814	+	0.188814i	0.795183	−	0.606369i	$-0.207375\pi$
$53$	1.37458	+	1.37458i	0.188814	+	0.188814i	−0.606369	+	0.795183i	$0.707375\pi$
$54$		0			0
$55$		−	0.714786i		−	0.0963817i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.64140	+	6.64140i	0.864637	+	0.864637i	0.991872	−	0.127236i	$-0.0406105\pi$
$59$	6.64140	+	6.64140i	0.864637	+	0.864637i	−0.127236	+	0.991872i	$0.540611\pi$
$60$		0			0
$61$	5.26208	−	5.26208i	0.673741	−	0.673741i	−0.284836	−	0.958576i	$-0.591939\pi$
$61$	5.26208	−	5.26208i	0.673741	−	0.673741i	0.958576	+	0.284836i	$0.0919391\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.66933			0.331089
$66$		0			0
$67$	10.5578	−	10.5578i	1.28984	−	1.28984i	0.354954	−	0.934884i	$-0.384497\pi$
$67$	10.5578	−	10.5578i	1.28984	−	1.28984i	0.934884	−	0.354954i	$-0.115503\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			14.0437i			1.66668i	0.552764	+	0.833338i	$0.313573\pi$
$71$			14.0437i			1.66668i	−0.552764	+	0.833338i	$0.686427\pi$
$72$		0			0
$73$			6.63830i			0.776954i	0.921458	+	0.388477i	$0.126999\pi$
$73$			6.63830i			0.776954i	−0.921458	+	0.388477i	$0.873001\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.877914	+	0.877914i	−0.100048	+	0.100048i
$78$		0			0
$79$	−4.27297			−0.480746			−0.240373	−	0.970681i	$-0.577270\pi$
$79$	−4.27297			−0.480746			−0.240373	+	0.970681i	$0.577270\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.15483	−	9.15483i	1.00487	−	1.00487i	0.00488547	−	0.999988i	$-0.498445\pi$
$83$	9.15483	−	9.15483i	1.00487	−	1.00487i	0.999988	−	0.00488547i	$-0.00155510\pi$
$84$		0			0
$85$	3.20690	+	3.20690i	0.347837	+	0.347837i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3.23826i			0.343255i	0.985162	+	0.171627i	$0.0549025\pi$
$89$			3.23826i			0.343255i	−0.985162	+	0.171627i	$0.945097\pi$
$90$		0			0
$91$	−3.27852	−	3.27852i	−0.343682	−	0.343682i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.55407			−0.467238
$96$		0			0
$97$	1.94129			0.197108			0.0985541	−	0.995132i	$-0.468578\pi$
$97$	1.94129			0.197108			0.0985541	+	0.995132i	$0.468578\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.3395	−	10.3395i	−1.02882	−	1.02882i	−0.999572	−	0.0292464i	$-0.990689\pi$
$101$	−10.3395	−	10.3395i	−1.02882	−	1.02882i	−0.0292464	−	0.999572i	$-0.509311\pi$
$102$		0			0
$103$		−	4.96401i		−	0.489118i	−0.969634	−	0.244559i	$-0.921357\pi$
$103$		−	4.96401i		−	0.489118i	0.969634	−	0.244559i	$-0.0786433\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.74631	−	2.74631i	−0.265496	−	0.265496i	0.561787	−	0.827282i	$-0.310114\pi$
$107$	−2.74631	−	2.74631i	−0.265496	−	0.265496i	−0.827282	+	0.561787i	$0.810114\pi$
$108$		0			0
$109$	6.99959	−	6.99959i	0.670439	−	0.670439i	−0.287378	−	0.957817i	$-0.592784\pi$
$109$	6.99959	−	6.99959i	0.670439	−	0.670439i	0.957817	+	0.287378i	$0.0927837\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.53194			0.614474			0.307237	−	0.951633i	$-0.400596\pi$
$113$	6.53194			0.614474			0.307237	+	0.951633i	$0.400596\pi$
$114$		0			0
$115$	−6.25821	+	6.25821i	−0.583582	+	0.583582i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	7.87756i		−	0.722134i
$120$		0			0
$121$		−	10.4891i		−	0.953553i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$	2.50861			0.222603			0.111302	−	0.993787i	$-0.464498\pi$
$127$	2.50861			0.222603			0.111302	+	0.993787i	$0.464498\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.55783	−	8.55783i	0.747701	−	0.747701i	−0.226346	−	0.974047i	$-0.572678\pi$
$131$	8.55783	−	8.55783i	0.747701	−	0.747701i	0.974047	+	0.226346i	$0.0726780\pi$
$132$		0			0
$133$	5.59340	+	5.59340i	0.485009	+	0.485009i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	6.47131i		−	0.552881i	−0.961031	−	0.276440i	$-0.910845\pi$
$137$		−	6.47131i		−	0.552881i	0.961031	−	0.276440i	$-0.0891549\pi$
$138$		0			0
$139$	16.4430	+	16.4430i	1.39468	+	1.39468i	0.814458	+	0.580223i	$0.197035\pi$
$139$	16.4430	+	16.4430i	1.39468	+	1.39468i	0.580223	+	0.814458i	$0.302965\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.90800			−0.159555
$144$		0			0
$145$	−3.45151			−0.286632
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.72803	+	2.72803i	0.223489	+	0.223489i	0.809966	−	0.586477i	$-0.199486\pi$
$149$	2.72803	+	2.72803i	0.223489	+	0.223489i	−0.586477	+	0.809966i	$0.699486\pi$
$150$		0			0
$151$		−	11.5196i		−	0.937453i	−0.883343	−	0.468726i	$-0.844713\pi$
$151$		−	11.5196i		−	0.937453i	0.883343	−	0.468726i	$-0.155287\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.03334	−	4.03334i	−0.323966	−	0.323966i
$156$		0			0
$157$	3.28013	−	3.28013i	0.261783	−	0.261783i	−0.563995	−	0.825778i	$-0.690736\pi$
$157$	3.28013	−	3.28013i	0.261783	−	0.261783i	0.825778	+	0.563995i	$0.190736\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	15.3729			1.21156
$162$		0			0
$163$	9.27367	−	9.27367i	0.726370	−	0.726370i	−0.243525	−	0.969895i	$-0.578304\pi$
$163$	9.27367	−	9.27367i	0.726370	−	0.726370i	0.969895	+	0.243525i	$0.0783037\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.08065i		−	0.547917i	−0.961742	−	0.273958i	$-0.911667\pi$
$167$		−	7.08065i		−	0.547917i	0.961742	−	0.273958i	$-0.0883331\pi$
$168$		0			0
$169$			5.87470i			0.451900i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.21471	−	5.21471i	0.396467	−	0.396467i	−0.480518	−	0.876985i	$-0.659551\pi$
$173$	5.21471	−	5.21471i	0.396467	−	0.396467i	0.876985	+	0.480518i	$0.159551\pi$
$174$		0			0
$175$	−1.73696			−0.131302
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.32196	−	6.32196i	0.472525	−	0.472525i	−0.430206	−	0.902731i	$-0.641559\pi$
$179$	6.32196	−	6.32196i	0.472525	−	0.472525i	0.902731	+	0.430206i	$0.141559\pi$
$180$		0			0
$181$	13.0695	+	13.0695i	0.971448	+	0.971448i	0.999604	−	0.0281553i	$-0.00896329\pi$
$181$	13.0695	+	13.0695i	0.971448	+	0.971448i	−0.0281553	+	0.999604i	$0.508963\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			7.57552i			0.556963i
$186$		0			0
$187$	−2.29225	−	2.29225i	−0.167626	−	0.167626i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−22.1722			−1.60433			−0.802164	−	0.597104i	$-0.796318\pi$
$191$	−22.1722			−1.60433			−0.802164	+	0.597104i	$0.796318\pi$
$192$		0			0
$193$	7.97695			0.574193			0.287097	−	0.957902i	$-0.407310\pi$
$193$	7.97695			0.574193			0.287097	+	0.957902i	$0.407310\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.76327	−	5.76327i	−0.410616	−	0.410616i	0.471337	−	0.881953i	$-0.343772\pi$
$197$	−5.76327	−	5.76327i	−0.410616	−	0.410616i	−0.881953	+	0.471337i	$0.843772\pi$
$198$		0			0
$199$		−	5.38869i		−	0.381994i	−0.981591	−	0.190997i	$-0.938828\pi$
$199$		−	5.38869i		−	0.381994i	0.981591	−	0.190997i	$-0.0611721\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.23921	+	4.23921i	0.297534	+	0.297534i
$204$		0			0
$205$	7.09530	−	7.09530i	0.495557	−	0.495557i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.25519			0.225166
$210$		0			0
$211$	−10.7547	+	10.7547i	−0.740384	+	0.740384i	−0.972652	−	0.232268i	$-0.925385\pi$
$211$	−10.7547	+	10.7547i	−0.740384	+	0.740384i	0.232268	+	0.972652i	$0.425385\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2.97782i		−	0.203086i
$216$		0			0
$217$			9.90766i			0.672576i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.56026	−	8.56026i	0.575826	−	0.575826i
$222$		0			0
$223$	3.98714			0.266998			0.133499	−	0.991049i	$-0.457379\pi$
$223$	3.98714			0.266998			0.133499	+	0.991049i	$0.457379\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.82103	+	3.82103i	−0.253611	+	0.253611i	−0.822449	−	0.568839i	$-0.807393\pi$
$227$	−3.82103	+	3.82103i	−0.253611	+	0.253611i	0.568839	+	0.822449i	$0.307393\pi$
$228$		0			0
$229$	−8.80687	−	8.80687i	−0.581974	−	0.581974i	0.353471	−	0.935445i	$-0.385001\pi$
$229$	−8.80687	−	8.80687i	−0.581974	−	0.581974i	−0.935445	+	0.353471i	$0.885001\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.6042i			1.08778i	0.839157	+	0.543889i	$0.183049\pi$
$233$			16.6042i			1.08778i	−0.839157	+	0.543889i	$0.816951\pi$
$234$		0			0
$235$	−3.05684	−	3.05684i	−0.199406	−	0.199406i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.81234			0.246600			0.123300	−	0.992369i	$-0.460652\pi$
$239$	3.81234			0.246600			0.123300	+	0.992369i	$0.460652\pi$
$240$		0			0
$241$	9.54985			0.615160			0.307580	−	0.951522i	$-0.400481\pi$
$241$	9.54985			0.615160			0.307580	+	0.951522i	$0.400481\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.81637	−	2.81637i	−0.179932	−	0.179932i
$246$		0			0
$247$			12.1563i			0.773487i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	11.9933	+	11.9933i	0.757010	+	0.757010i	0.975777	−	0.218767i	$-0.0702034\pi$
$251$	11.9933	+	11.9933i	0.757010	+	0.757010i	−0.218767	+	0.975777i	$0.570203\pi$
$252$		0			0
$253$	4.47328	−	4.47328i	0.281233	−	0.281233i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.8752			−1.17740			−0.588702	−	0.808350i	$-0.700361\pi$
$257$	−18.8752			−1.17740			−0.588702	+	0.808350i	$0.700361\pi$
$258$		0			0
$259$	9.30440	−	9.30440i	0.578147	−	0.578147i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			23.1398i			1.42686i	0.700727	+	0.713429i	$0.252859\pi$
$263$			23.1398i			1.42686i	−0.700727	+	0.713429i	$0.747141\pi$
$264$		0			0
$265$		−	1.94396i		−	0.119416i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.6368	−	10.6368i	0.648539	−	0.648539i	−0.304101	−	0.952640i	$-0.598356\pi$
$269$	10.6368	−	10.6368i	0.648539	−	0.648539i	0.952640	+	0.304101i	$0.0983560\pi$
$270$		0			0
$271$	−19.9763			−1.21348			−0.606738	−	0.794902i	$-0.707522\pi$
$271$	−19.9763			−1.21348			−0.606738	+	0.794902i	$0.707522\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.505430	+	0.505430i	−0.0304786	+	0.0304786i
$276$		0			0
$277$	−16.1534	−	16.1534i	−0.970563	−	0.970563i	0.0290160	−	0.999579i	$-0.490763\pi$
$277$	−16.1534	−	16.1534i	−0.970563	−	0.970563i	−0.999579	+	0.0290160i	$0.990763\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	9.43520i		−	0.562857i	−0.959582	−	0.281429i	$-0.909192\pi$
$281$		−	9.43520i		−	0.562857i	0.959582	−	0.281429i	$-0.0908082\pi$
$282$		0			0
$283$	−8.71287	−	8.71287i	−0.517926	−	0.517926i	0.399017	−	0.916943i	$-0.369351\pi$
$283$	−8.71287	−	8.71287i	−0.517926	−	0.517926i	−0.916943	+	0.399017i	$0.869351\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−17.4292			−1.02881
$288$		0			0
$289$	3.56843			0.209908
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.1045	+	11.1045i	0.648729	+	0.648729i	0.952686	−	0.303957i	$-0.0983079\pi$
$293$	11.1045	+	11.1045i	0.648729	+	0.648729i	−0.303957	+	0.952686i	$0.598308\pi$
$294$		0			0
$295$		−	9.39236i		−	0.546844i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.7052	+	16.7052i	0.966087	+	0.966087i
$300$		0			0
$301$	−3.65742	+	3.65742i	−0.210810	+	0.210810i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.44171			−0.426111
$306$		0			0
$307$	2.99854	−	2.99854i	0.171136	−	0.171136i	−0.616343	−	0.787478i	$-0.711386\pi$
$307$	2.99854	−	2.99854i	0.171136	−	0.171136i	0.787478	+	0.616343i	$0.211386\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	9.06099i		−	0.513802i	−0.966438	−	0.256901i	$-0.917299\pi$
$311$		−	9.06099i		−	0.513802i	0.966438	−	0.256901i	$-0.0827014\pi$
$312$		0			0
$313$		−	19.5699i		−	1.10616i	−0.833129	−	0.553078i	$-0.813453\pi$
$313$		−	19.5699i		−	1.10616i	0.833129	−	0.553078i	$-0.186547\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.1019	−	11.1019i	0.623546	−	0.623546i	−0.322890	−	0.946436i	$-0.604654\pi$
$317$	11.1019	−	11.1019i	0.623546	−	0.623546i	0.946436	+	0.322890i	$0.104654\pi$
$318$		0			0
$319$	2.46709			0.138131
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−14.6045	+	14.6045i	−0.812614	+	0.812614i
$324$		0			0
$325$	−1.88750	−	1.88750i	−0.104700	−	0.104700i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			7.50894i			0.413981i
$330$		0			0
$331$	8.14718	+	8.14718i	0.447810	+	0.447810i	0.894626	−	0.446816i	$-0.147442\pi$
$331$	8.14718	+	8.14718i	0.447810	+	0.447810i	−0.446816	+	0.894626i	$0.647442\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.9309			−0.815765
$336$		0			0
$337$	−25.1380			−1.36935			−0.684677	−	0.728847i	$-0.740057\pi$
$337$	−25.1380			−1.36935			−0.684677	+	0.728847i	$0.740057\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.88298	+	2.88298i	0.156122	+	0.156122i
$342$		0			0
$343$			19.0770i			1.03006i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.36719	+	7.36719i	0.395491	+	0.395491i	0.876639	−	0.481148i	$-0.159780\pi$
$347$	7.36719	+	7.36719i	0.395491	+	0.395491i	−0.481148	+	0.876639i	$0.659780\pi$
$348$		0			0
$349$	−3.25982	+	3.25982i	−0.174494	+	0.174494i	−0.788951	−	0.614457i	$-0.789375\pi$
$349$	−3.25982	+	3.25982i	−0.174494	+	0.174494i	0.614457	+	0.788951i	$0.289375\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.502832			−0.0267630			−0.0133815	−	0.999910i	$-0.504260\pi$
$353$	−0.502832			−0.0267630			−0.0133815	+	0.999910i	$0.504260\pi$
$354$		0			0
$355$	9.93037	−	9.93037i	0.527049	−	0.527049i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5.95161i			0.314114i	0.987590	+	0.157057i	$0.0502007\pi$
$359$			5.95161i			0.314114i	−0.987590	+	0.157057i	$0.949799\pi$
$360$		0			0
$361$		−	1.73958i		−	0.0915571i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.69399	−	4.69399i	0.245695	−	0.245695i
$366$		0			0
$367$	−1.95365			−0.101980			−0.0509898	−	0.998699i	$-0.516238\pi$
$367$	−1.95365			−0.101980			−0.0509898	+	0.998699i	$0.516238\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.38760	+	2.38760i	−0.123958	+	0.123958i
$372$		0			0
$373$	−18.6509	−	18.6509i	−0.965708	−	0.965708i	0.0337233	−	0.999431i	$-0.489264\pi$
$373$	−18.6509	−	18.6509i	−0.965708	−	0.965708i	−0.999431	+	0.0337233i	$0.989264\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			9.21320i			0.474504i
$378$		0			0
$379$	3.85143	+	3.85143i	0.197835	+	0.197835i	0.799071	−	0.601236i	$-0.205325\pi$
$379$	3.85143	+	3.85143i	0.197835	+	0.197835i	−0.601236	+	0.799071i	$0.705325\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.29258			0.117145			0.0585726	−	0.998283i	$-0.481345\pi$
$383$	2.29258			0.117145			0.0585726	+	0.998283i	$0.481345\pi$
$384$		0			0
$385$	1.24156			0.0632757
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.90500	−	4.90500i	−0.248693	−	0.248693i	0.571741	−	0.820434i	$-0.306268\pi$
$389$	−4.90500	−	4.90500i	−0.248693	−	0.248693i	−0.820434	+	0.571741i	$0.806268\pi$
$390$		0			0
$391$			40.1389i			2.02991i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.02144	+	3.02144i	0.152025	+	0.152025i
$396$		0			0
$397$	−10.8616	+	10.8616i	−0.545126	+	0.545126i	−0.925027	−	0.379901i	$-0.875958\pi$
$397$	−10.8616	+	10.8616i	−0.545126	+	0.545126i	0.379901	+	0.925027i	$0.375958\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.10783			0.354948			0.177474	−	0.984125i	$-0.443207\pi$
$401$	7.10783			0.354948			0.177474	+	0.984125i	$0.443207\pi$
$402$		0			0
$403$	−10.7663	+	10.7663i	−0.536308	+	0.536308i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.41487i		−	0.268405i
$408$		0			0
$409$			29.1697i			1.44235i	0.692752	+	0.721176i	$0.256398\pi$
$409$			29.1697i			1.44235i	−0.692752	+	0.721176i	$0.743602\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−11.5359	+	11.5359i	−0.567643	+	0.567643i
$414$		0			0
$415$	−12.9469			−0.635538
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.06616	−	3.06616i	0.149792	−	0.149792i	−0.628233	−	0.778025i	$-0.716222\pi$
$419$	3.06616	−	3.06616i	0.149792	−	0.149792i	0.778025	+	0.628233i	$0.216222\pi$
$420$		0			0
$421$	−0.532242	−	0.532242i	−0.0259399	−	0.0259399i	0.694018	−	0.719958i	$-0.255839\pi$
$421$	−0.532242	−	0.532242i	−0.0259399	−	0.0259399i	−0.719958	+	0.694018i	$0.755839\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	4.53524i		−	0.219992i
$426$		0			0
$427$	9.14005	+	9.14005i	0.442318	+	0.442318i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.7237			0.805555			0.402777	−	0.915298i	$-0.368045\pi$
$431$	16.7237			0.805555			0.402777	+	0.915298i	$0.368045\pi$
$432$		0			0
$433$	28.3675			1.36326			0.681628	−	0.731699i	$-0.261272\pi$
$433$	28.3675			1.36326			0.681628	+	0.731699i	$0.261272\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−28.5004	−	28.5004i	−1.36336	−	1.36336i
$438$		0			0
$439$		−	13.5018i		−	0.644405i	−0.946671	−	0.322203i	$-0.895577\pi$
$439$		−	13.5018i		−	0.644405i	0.946671	−	0.322203i	$-0.104423\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.55246	−	9.55246i	−0.453851	−	0.453851i	0.442780	−	0.896630i	$-0.353992\pi$
$443$	−9.55246	−	9.55246i	−0.453851	−	0.453851i	−0.896630	+	0.442780i	$0.853992\pi$
$444$		0			0
$445$	2.28980	−	2.28980i	0.108547	−	0.108547i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.35573			0.441524			0.220762	−	0.975328i	$-0.429146\pi$
$449$	9.35573			0.441524			0.220762	+	0.975328i	$0.429146\pi$
$450$		0			0
$451$	−5.07162	+	5.07162i	−0.238813	+	0.238813i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			4.63652i			0.217364i
$456$		0			0
$457$			6.84779i			0.320326i	0.987091	+	0.160163i	$0.0512020\pi$
$457$			6.84779i			0.320326i	−0.987091	+	0.160163i	$0.948798\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.7403	−	11.7403i	0.546801	−	0.546801i	−0.378713	−	0.925514i	$-0.623633\pi$
$461$	11.7403	−	11.7403i	0.546801	−	0.546801i	0.925514	+	0.378713i	$0.123633\pi$
$462$		0			0
$463$	26.6096			1.23665			0.618326	−	0.785922i	$-0.287811\pi$
$463$	26.6096			1.23665			0.618326	+	0.785922i	$0.287811\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.47583	−	1.47583i	0.0682933	−	0.0682933i	−0.672135	−	0.740428i	$-0.734623\pi$
$467$	1.47583	−	1.47583i	0.0682933	−	0.0682933i	0.740428	+	0.672135i	$0.234623\pi$
$468$		0			0
$469$	18.3385	+	18.3385i	0.846792	+	0.846792i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2.12851i			0.0978688i
$474$		0			0
$475$	3.22022	+	3.22022i	0.147754	+	0.147754i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.78600			−0.127296			−0.0636479	−	0.997972i	$-0.520273\pi$
$479$	−2.78600			−0.127296			−0.0636479	+	0.997972i	$0.520273\pi$
$480$		0			0
$481$	20.2215			0.922022
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.37270	−	1.37270i	−0.0623311	−	0.0623311i
$486$		0			0
$487$		−	16.9499i		−	0.768073i	−0.923318	−	0.384036i	$-0.874534\pi$
$487$		−	16.9499i		−	0.768073i	0.923318	−	0.384036i	$-0.125466\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−22.8390	−	22.8390i	−1.03071	−	1.03071i	−0.999513	−	0.0311972i	$-0.990068\pi$
$491$	−22.8390	−	22.8390i	−1.03071	−	1.03071i	−0.0311972	−	0.999513i	$-0.509932\pi$
$492$		0			0
$493$	−11.0687	+	11.0687i	−0.498507	+	0.498507i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−24.3933			−1.09419
$498$		0			0
$499$	−2.33906	+	2.33906i	−0.104711	+	0.104711i	−0.757521	−	0.652811i	$-0.773590\pi$
$499$	−2.33906	+	2.33906i	−0.104711	+	0.104711i	0.652811	+	0.757521i	$0.273590\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	1.58801i		−	0.0708057i	−0.999373	−	0.0354029i	$-0.988729\pi$
$503$		−	1.58801i		−	0.0708057i	0.999373	−	0.0354029i	$-0.0112714\pi$
$504$		0			0
$505$			14.6223i			0.650682i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.61613	−	3.61613i	0.160282	−	0.160282i	−0.622410	−	0.782692i	$-0.713846\pi$
$509$	3.61613	−	3.61613i	0.160282	−	0.160282i	0.782692	+	0.622410i	$0.213846\pi$
$510$		0			0
$511$	−11.5305			−0.510079
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.51009	+	3.51009i	−0.154673	+	0.154673i
$516$		0			0
$517$	2.18499	+	2.18499i	0.0960956	+	0.0960956i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	8.93031i		−	0.391244i	−0.980679	−	0.195622i	$-0.937327\pi$
$521$		−	8.93031i		−	0.391244i	0.980679	−	0.195622i	$-0.0626725\pi$
$522$		0			0
$523$	−15.0355	−	15.0355i	−0.657455	−	0.657455i	0.297323	−	0.954777i	$-0.403906\pi$
$523$	−15.0355	−	15.0355i	−0.657455	−	0.657455i	−0.954777	+	0.297323i	$0.903906\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−25.8691			−1.12687
$528$		0			0
$529$	−55.3305			−2.40567
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−18.9397	−	18.9397i	−0.820368	−	0.820368i
$534$		0			0
$535$			3.88387i			0.167914i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.01310	+	2.01310i	0.0867106	+	0.0867106i
$540$		0			0
$541$	5.57591	−	5.57591i	0.239727	−	0.239727i	−0.577010	−	0.816737i	$-0.695781\pi$
$541$	5.57591	−	5.57591i	0.239727	−	0.239727i	0.816737	+	0.577010i	$0.195781\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.89891			−0.424023
$546$		0			0
$547$	−32.8366	+	32.8366i	−1.40399	+	1.40399i	−0.617136	+	0.786856i	$0.711707\pi$
$547$	−32.8366	+	32.8366i	−1.40399	+	1.40399i	−0.786856	+	0.617136i	$0.788293\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	15.7184i		−	0.669628i
$552$		0			0
$553$		−	7.42199i		−	0.315615i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−24.2077	+	24.2077i	−1.02571	+	1.02571i	−0.0260537	+	0.999661i	$0.508294\pi$
$557$	−24.2077	+	24.2077i	−1.02571	+	1.02571i	−0.999661	+	0.0260537i	$0.991706\pi$
$558$		0			0
$559$	−7.94877			−0.336197
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.3407	−	22.3407i	0.941547	−	0.941547i	−0.0568365	−	0.998384i	$-0.518101\pi$
$563$	22.3407	−	22.3407i	0.941547	−	0.941547i	0.998384	+	0.0568365i	$0.0181014\pi$
$564$		0			0
$565$	−4.61878	−	4.61878i	−0.194314	−	0.194314i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			29.3339i			1.22974i	0.788629	+	0.614870i	$0.210791\pi$
$569$			29.3339i			1.22974i	−0.788629	+	0.614870i	$0.789209\pi$
$570$		0			0
$571$	−23.9934	−	23.9934i	−1.00409	−	1.00409i	−0.999992	−	0.00410070i	$-0.998695\pi$
$571$	−23.9934	−	23.9934i	−1.00409	−	1.00409i	−0.00410070	−	0.999992i	$-0.501305\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.85045			0.369089
$576$		0			0
$577$	−31.9232			−1.32898			−0.664490	−	0.747297i	$-0.731351\pi$
$577$	−31.9232			−1.32898			−0.664490	+	0.747297i	$0.731351\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.9016	+	15.9016i	0.659710	+	0.659710i
$582$		0			0
$583$			1.38951i			0.0575477i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−26.2847	−	26.2847i	−1.08488	−	1.08488i	−0.996046	−	0.0888379i	$-0.971685\pi$
$587$	−26.2847	−	26.2847i	−1.08488	−	1.08488i	−0.0888379	−	0.996046i	$-0.528315\pi$
$588$		0			0
$589$	18.3681	−	18.3681i	0.756846	−	0.756846i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	38.2085			1.56904			0.784518	−	0.620106i	$-0.212910\pi$
$593$	38.2085			1.56904			0.784518	+	0.620106i	$0.212910\pi$
$594$		0			0
$595$	−5.57027	+	5.57027i	−0.228359	+	0.228359i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			25.1150i			1.02617i	0.858337	+	0.513086i	$0.171498\pi$
$599$			25.1150i			1.02617i	−0.858337	+	0.513086i	$0.828502\pi$
$600$		0			0
$601$			22.2022i			0.905647i	0.891600	+	0.452823i	$0.149583\pi$
$601$			22.2022i			0.905647i	−0.891600	+	0.452823i	$0.850417\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.41690	+	7.41690i	−0.301540	+	0.301540i
$606$		0			0
$607$	−12.9648			−0.526226			−0.263113	−	0.964765i	$-0.584749\pi$
$607$	−12.9648			−0.526226			−0.263113	+	0.964765i	$0.584749\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.15970	+	8.15970i	−0.330106	+	0.330106i
$612$		0			0
$613$	7.42804	+	7.42804i	0.300016	+	0.300016i	0.841020	−	0.541004i	$-0.181956\pi$
$613$	7.42804	+	7.42804i	0.300016	+	0.300016i	−0.541004	+	0.841020i	$0.681956\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			23.2743i			0.936989i	0.883467	+	0.468494i	$0.155203\pi$
$617$			23.2743i			0.936989i	−0.883467	+	0.468494i	$0.844797\pi$
$618$		0			0
$619$	31.6213	+	31.6213i	1.27097	+	1.27097i	0.945581	+	0.325386i	$0.105494\pi$
$619$	31.6213	+	31.6213i	1.27097	+	1.27097i	0.325386	+	0.945581i	$0.394506\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.62474			−0.225351
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	24.2939	+	24.2939i	0.968663	+	0.968663i
$630$		0			0
$631$		−	29.9258i		−	1.19133i	−0.803234	−	0.595663i	$-0.796889\pi$
$631$		−	29.9258i		−	1.19133i	0.803234	−	0.595663i	$-0.203111\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.77386	−	1.77386i	−0.0703933	−	0.0703933i
$636$		0			0
$637$	−7.51782	+	7.51782i	−0.297867	+	0.297867i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.2240			−0.403825			−0.201912	−	0.979404i	$-0.564716\pi$
$641$	−10.2240			−0.403825			−0.201912	+	0.979404i	$0.564716\pi$
$642$		0			0
$643$	13.7202	−	13.7202i	0.541074	−	0.541074i	−0.382770	−	0.923844i	$-0.625030\pi$
$643$	13.7202	−	13.7202i	0.541074	−	0.541074i	0.923844	+	0.382770i	$0.125030\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			18.6767i			0.734255i	0.930171	+	0.367128i	$0.119659\pi$
$647$			18.6767i			0.734255i	−0.930171	+	0.367128i	$0.880341\pi$
$648$		0			0
$649$			6.71353i			0.263529i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.7935	+	12.7935i	−0.500647	+	0.500647i	−0.911639	−	0.410992i	$-0.865183\pi$
$653$	−12.7935	+	12.7935i	−0.500647	+	0.500647i	0.410992	+	0.911639i	$0.365183\pi$
$654$		0			0
$655$	−12.1026			−0.472888
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.3193	−	12.3193i	0.479893	−	0.479893i	−0.425204	−	0.905097i	$-0.639798\pi$
$659$	12.3193	−	12.3193i	0.479893	−	0.479893i	0.905097	+	0.425204i	$0.139798\pi$
$660$		0			0
$661$	−24.0352	−	24.0352i	−0.934862	−	0.934862i	0.0631421	−	0.998005i	$-0.479888\pi$
$661$	−24.0352	−	24.0352i	−0.934862	−	0.934862i	−0.998005	+	0.0631421i	$0.979888\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	7.91026i		−	0.306747i
$666$		0			0
$667$	−21.6003	−	21.6003i	−0.836367	−	0.836367i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.31923			0.205347
$672$		0			0
$673$	−21.5360			−0.830150			−0.415075	−	0.909787i	$-0.636245\pi$
$673$	−21.5360			−0.830150			−0.415075	+	0.909787i	$0.636245\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.1852	−	13.1852i	−0.506750	−	0.506750i	0.406778	−	0.913527i	$-0.366652\pi$
$677$	−13.1852	−	13.1852i	−0.506750	−	0.506750i	−0.913527	+	0.406778i	$0.866652\pi$
$678$		0			0
$679$			3.37195i			0.129404i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	30.6011	+	30.6011i	1.17092	+	1.17092i	0.981991	+	0.188926i	$0.0605008\pi$
$683$	30.6011	+	30.6011i	1.17092	+	1.17092i	0.188926	+	0.981991i	$0.439499\pi$
$684$		0			0
$685$	−4.57590	+	4.57590i	−0.174836	+	0.174836i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.18905			−0.197687
$690$		0			0
$691$	25.2675	−	25.2675i	0.961220	−	0.961220i	−0.0380558	−	0.999276i	$-0.512116\pi$
$691$	25.2675	−	25.2675i	0.961220	−	0.961220i	0.999276	+	0.0380558i	$0.0121165\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	23.2540i		−	0.882074i
$696$		0			0
$697$		−	45.5079i		−	1.72373i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.5583	+	18.5583i	−0.700937	+	0.700937i	−0.964612	−	0.263675i	$-0.915065\pi$
$701$	−18.5583	+	18.5583i	−0.700937	+	0.700937i	0.263675	+	0.964612i	$0.415065\pi$
$702$		0			0
$703$	−34.4995			−1.30117
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.9593	−	17.9593i	0.675431	−	0.675431i
$708$		0			0
$709$	4.38093	+	4.38093i	0.164529	+	0.164529i	0.784570	−	0.620040i	$-0.212884\pi$
$709$	4.38093	+	4.38093i	0.164529	+	0.164529i	−0.620040	+	0.784570i	$0.712884\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	50.4831i		−	1.89061i
$714$		0			0
$715$	1.34916	+	1.34916i	0.0504556	+	0.0504556i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.61691			0.0603007			0.0301503	−	0.999545i	$-0.490401\pi$
$719$	1.61691			0.0603007			0.0301503	+	0.999545i	$0.490401\pi$
$720$		0			0
$721$	8.62231			0.321112
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.44059	+	2.44059i	0.0906411	+	0.0906411i
$726$		0			0
$727$		−	39.3600i		−	1.45978i	−0.683563	−	0.729891i	$-0.739571\pi$
$727$		−	39.3600i		−	1.45978i	0.683563	−	0.729891i	$-0.260429\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.54958	−	9.54958i	−0.353204	−	0.353204i
$732$		0			0
$733$	−34.0787	+	34.0787i	−1.25873	+	1.25873i	−0.307026	+	0.951701i	$0.599334\pi$
$733$	−34.0787	+	34.0787i	−1.25873	+	1.25873i	−0.951701	+	0.307026i	$0.900666\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.6724			0.393124
$738$		0			0
$739$	−15.4278	+	15.4278i	−0.567520	+	0.567520i	−0.931433	−	0.363913i	$-0.881441\pi$
$739$	−15.4278	+	15.4278i	−0.567520	+	0.567520i	0.363913	+	0.931433i	$0.381441\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			23.5004i			0.862147i	0.902317	+	0.431074i	$0.141865\pi$
$743$			23.5004i			0.862147i	−0.902317	+	0.431074i	$0.858135\pi$
$744$		0			0
$745$		−	3.85801i		−	0.141347i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.77024	−	4.77024i	0.174301	−	0.174301i
$750$		0			0
$751$	−10.8586			−0.396236			−0.198118	−	0.980178i	$-0.563483\pi$
$751$	−10.8586			−0.396236			−0.198118	+	0.980178i	$0.563483\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.14560	+	8.14560i	−0.296449	+	0.296449i
$756$		0			0
$757$	18.8434	+	18.8434i	0.684874	+	0.684874i	0.961094	−	0.276220i	$-0.0890819\pi$
$757$	18.8434	+	18.8434i	0.684874	+	0.684874i	−0.276220	+	0.961094i	$0.589082\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			22.2837i			0.807783i	0.914807	+	0.403891i	$0.132343\pi$
$761$			22.2837i			0.807783i	−0.914807	+	0.403891i	$0.867657\pi$
$762$		0			0
$763$	12.1580	+	12.1580i	0.440151	+	0.440151i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−25.0713			−0.905271
$768$		0			0
$769$	10.5399			0.380077			0.190039	−	0.981777i	$-0.439139\pi$
$769$	10.5399			0.380077			0.190039	+	0.981777i	$0.439139\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.07768	−	4.07768i	−0.146664	−	0.146664i	0.629962	−	0.776626i	$-0.283070\pi$
$773$	−4.07768	−	4.07768i	−0.146664	−	0.146664i	−0.776626	+	0.629962i	$0.783070\pi$
$774$		0			0
$775$			5.70401i			0.204894i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	32.3125	+	32.3125i	1.15772	+	1.15772i
$780$		0			0
$781$	−7.09809	+	7.09809i	−0.253990	+	0.253990i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.63881			−0.165566
$786$		0			0
$787$	8.16669	−	8.16669i	0.291111	−	0.291111i	−0.546408	−	0.837519i	$-0.684005\pi$
$787$	8.16669	−	8.16669i	0.291111	−	0.291111i	0.837519	+	0.546408i	$0.184005\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			11.3458i			0.403409i
$792$		0			0
$793$			19.8643i			0.705403i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−17.9971	+	17.9971i	−0.637491	+	0.637491i	−0.949936	−	0.312445i	$-0.898852\pi$
$797$	−17.9971	+	17.9971i	−0.637491	+	0.637491i	0.312445	+	0.949936i	$0.398852\pi$
$798$		0			0
$799$	−19.6060			−0.693609
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.35520	+	3.35520i	−0.118402	+	0.118402i
$804$		0			0
$805$	−10.8703	−	10.8703i	−0.383128	−	0.383128i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			42.0296i			1.47768i	0.673879	+	0.738841i	$0.264627\pi$
$809$			42.0296i			1.47768i	−0.673879	+	0.738841i	$0.735373\pi$
$810$		0			0
$811$	−18.7601	−	18.7601i	−0.658757	−	0.658757i	0.296329	−	0.955086i	$-0.404238\pi$
$811$	−18.7601	−	18.7601i	−0.658757	−	0.658757i	−0.955086	+	0.296329i	$0.904238\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.1149			−0.459397
$816$		0			0
$817$	13.5612			0.474447
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.4050	+	21.4050i	0.747038	+	0.747038i	0.973922	−	0.226884i	$-0.0728538\pi$
$821$	21.4050	+	21.4050i	0.747038	+	0.747038i	−0.226884	+	0.973922i	$0.572854\pi$
$822$		0			0
$823$			43.7323i			1.52441i	0.647334	+	0.762206i	$0.275884\pi$
$823$			43.7323i			1.52441i	−0.647334	+	0.762206i	$0.724116\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−19.9621	−	19.9621i	−0.694149	−	0.694149i	0.268993	−	0.963142i	$-0.413309\pi$
$827$	−19.9621	−	19.9621i	−0.694149	−	0.694149i	−0.963142	+	0.268993i	$0.913309\pi$
$828$		0			0
$829$	31.3869	−	31.3869i	1.09011	−	1.09011i	0.0945964	−	0.995516i	$-0.469844\pi$
$829$	31.3869	−	31.3869i	1.09011	−	1.09011i	0.995516	−	0.0945964i	$-0.0301561\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.0637			−0.625869
$834$		0			0
$835$	−5.00677	+	5.00677i	−0.173267	+	0.173267i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	54.5335i		−	1.88271i	−0.337423	−	0.941353i	$-0.609555\pi$
$839$		−	54.5335i		−	1.88271i	0.337423	−	0.941353i	$-0.390445\pi$
$840$		0			0
$841$			17.0871i			0.589210i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.15404	−	4.15404i	0.142903	−	0.142903i
$846$		0			0
$847$	18.2192			0.626018
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−47.4092	+	47.4092i	−1.62517	+	1.62517i
$852$		0			0
$853$	21.5932	+	21.5932i	0.739336	+	0.739336i	0.972449	−	0.233114i	$-0.0748914\pi$
$853$	21.5932	+	21.5932i	0.739336	+	0.739336i	−0.233114	+	0.972449i	$0.574891\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	41.3609i		−	1.41286i	−0.707782	−	0.706431i	$-0.750304\pi$
$857$		−	41.3609i		−	1.41286i	0.707782	−	0.706431i	$-0.249696\pi$
$858$		0			0
$859$	0.700596	+	0.700596i	0.0239040	+	0.0239040i	0.718958	−	0.695054i	$-0.244619\pi$
$859$	0.700596	+	0.700596i	0.0239040	+	0.0239040i	−0.695054	+	0.718958i	$0.744619\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	55.0780			1.87488			0.937439	−	0.348150i	$-0.113190\pi$
$863$	55.0780			1.87488			0.937439	+	0.348150i	$0.113190\pi$
$864$		0			0
$865$	−7.37471			−0.250748
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.15969	−	2.15969i	−0.0732623	−	0.0732623i
$870$		0			0
$871$			39.8556i			1.35045i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.22822	+	1.22822i	0.0415214	+	0.0415214i
$876$		0			0
$877$	−36.5100	+	36.5100i	−1.23285	+	1.23285i	−0.269992	+	0.962863i	$0.587021\pi$
$877$	−36.5100	+	36.5100i	−1.23285	+	1.23285i	−0.962863	+	0.269992i	$0.912979\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−54.3503			−1.83111			−0.915554	−	0.402196i	$-0.868247\pi$
$881$	−54.3503			−1.83111			−0.915554	+	0.402196i	$0.868247\pi$
$882$		0			0
$883$	−35.5476	+	35.5476i	−1.19627	+	1.19627i	−0.220999	+	0.975274i	$0.570932\pi$
$883$	−35.5476	+	35.5476i	−1.19627	+	1.19627i	−0.975274	+	0.220999i	$0.929068\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			0.817003i			0.0274323i	0.999906	+	0.0137161i	$0.00436612\pi$
$887$			0.817003i			0.0274323i	−0.999906	+	0.0137161i	$0.995634\pi$
$888$		0			0
$889$			4.35737i			0.146141i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.9211	−	13.9211i	0.465851	−	0.465851i
$894$		0			0
$895$	−8.94060			−0.298851
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13.9211	−	13.9211i	0.464296	−	0.464296i
$900$		0			0
$901$	−6.23407	−	6.23407i	−0.207687	−	0.207687i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	18.4831i		−	0.614398i
$906$		0			0
$907$	3.36159	+	3.36159i	0.111620	+	0.111620i	0.760711	−	0.649091i	$-0.224851\pi$
$907$	3.36159	+	3.36159i	0.111620	+	0.111620i	−0.649091	+	0.760711i	$0.724851\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	34.6568			1.14823			0.574116	−	0.818774i	$-0.305346\pi$
$911$	34.6568			1.14823			0.574116	+	0.818774i	$0.305346\pi$
$912$		0			0
$913$	9.25426			0.306271
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.8646	+	14.8646i	0.490874	+	0.490874i
$918$		0			0
$919$		−	24.3452i		−	0.803074i	−0.915843	−	0.401537i	$-0.868476\pi$
$919$		−	24.3452i		−	0.803074i	0.915843	−	0.401537i	$-0.131524\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−26.5074	−	26.5074i	−0.872501	−	0.872501i
$924$		0			0
$925$	5.35670	−	5.35670i	0.176127	−	0.176127i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.16600			−0.103873			−0.0519366	−	0.998650i	$-0.516539\pi$
$929$	−3.16600			−0.103873			−0.0519366	+	0.998650i	$0.516539\pi$
$930$		0			0
$931$	12.8260	−	12.8260i	0.420354	−	0.420354i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			3.24173i			0.106016i
$936$		0			0
$937$		−	23.4847i		−	0.767211i	−0.923497	−	0.383606i	$-0.874682\pi$
$937$		−	23.4847i		−	0.767211i	0.923497	−	0.383606i	$-0.125318\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−27.7583	+	27.7583i	−0.904896	+	0.904896i	−0.995855	−	0.0909585i	$-0.971007\pi$
$941$	−27.7583	+	27.7583i	−0.904896	+	0.904896i	0.0909585	+	0.995855i	$0.471007\pi$
$942$		0			0
$943$	88.8078			2.89198
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.2916	+	27.2916i	−0.886857	+	0.886857i	−0.994220	−	0.107363i	$-0.965759\pi$
$947$	−27.2916	+	27.2916i	−0.886857	+	0.886857i	0.107363	+	0.994220i	$0.465759\pi$
$948$		0			0
$949$	−12.5298	−	12.5298i	−0.406734	−	0.406734i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	12.1516i		−	0.393630i	−0.980441	−	0.196815i	$-0.936940\pi$
$953$		−	12.1516i		−	0.393630i	0.980441	−	0.196815i	$-0.0630598\pi$
$954$		0			0
$955$	15.6781	+	15.6781i	0.507333	+	0.507333i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11.2404			0.362972
$960$		0			0
$961$	1.53571			0.0495392
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.64056	−	5.64056i	−0.181576	−	0.181576i
$966$		0			0
$967$			48.2694i			1.55224i	0.630585	+	0.776120i	$0.282815\pi$
$967$			48.2694i			1.55224i	−0.630585	+	0.776120i	$0.717185\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.92047	+	5.92047i	0.189997	+	0.189997i	0.795695	−	0.605698i	$-0.207106\pi$
$971$	5.92047	+	5.92047i	0.189997	+	0.189997i	−0.605698	+	0.795695i	$0.707106\pi$
$972$		0			0
$973$	−28.5610	+	28.5610i	−0.915623	+	0.915623i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.7522			0.887872			0.443936	−	0.896059i	$-0.353582\pi$
$977$	27.7522			0.887872			0.443936	+	0.896059i	$0.353582\pi$
$978$		0			0
$979$	−1.63671	+	1.63671i	−0.0523096	+	0.0523096i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	28.3604i		−	0.904556i	−0.891877	−	0.452278i	$-0.850611\pi$
$983$		−	28.3604i		−	0.904556i	0.891877	−	0.452278i	$-0.149389\pi$
$984$		0			0
$985$			8.15050i			0.259697i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	18.6358	−	18.6358i	0.592585	−	0.592585i
$990$		0			0
$991$	43.7506			1.38979			0.694893	−	0.719114i	$-0.255452\pi$
$991$	43.7506			1.38979			0.694893	+	0.719114i	$0.255452\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.81038	+	3.81038i	−0.120797	+	0.120797i
$996$		0			0
$997$	10.5572	+	10.5572i	0.334349	+	0.334349i	0.854235	−	0.519887i	$-0.174026\pi$
$997$	10.5572	+	10.5572i	0.334349	+	0.334349i	−0.519887	+	0.854235i	$0.674026\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.c.2161.4		16
3.2	odd	2		320.2.l.a.241.4		16
4.3	odd	2		720.2.t.c.181.3		16
12.11	even	2		80.2.l.a.21.6	✓	16
15.2	even	4		1600.2.q.h.49.5		16
15.8	even	4		1600.2.q.g.49.4		16
15.14	odd	2		1600.2.l.i.1201.5		16
16.3	odd	4		720.2.t.c.541.3		16
16.13	even	4	inner	2880.2.t.c.721.1		16
24.5	odd	2		640.2.l.a.481.5		16
24.11	even	2		640.2.l.b.481.4		16
48.5	odd	4		640.2.l.a.161.5		16
48.11	even	4		640.2.l.b.161.4		16
48.29	odd	4		320.2.l.a.81.4		16
48.35	even	4		80.2.l.a.61.6	yes	16
60.23	odd	4		400.2.q.h.149.2		16
60.47	odd	4		400.2.q.g.149.7		16
60.59	even	2		400.2.l.h.101.3		16
96.29	odd	8		5120.2.a.u.1.3		8
96.35	even	8		5120.2.a.s.1.6		8
96.77	odd	8		5120.2.a.t.1.6		8
96.83	even	8		5120.2.a.v.1.3		8
240.29	odd	4		1600.2.l.i.401.5		16
240.77	even	4		1600.2.q.g.849.4		16
240.83	odd	4		400.2.q.g.349.7		16
240.173	even	4		1600.2.q.h.849.5		16
240.179	even	4		400.2.l.h.301.3		16
240.227	odd	4		400.2.q.h.349.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.6	✓	16	12.11	even	2
80.2.l.a.61.6	yes	16	48.35	even	4
320.2.l.a.81.4		16	48.29	odd	4
320.2.l.a.241.4		16	3.2	odd	2
400.2.l.h.101.3		16	60.59	even	2
400.2.l.h.301.3		16	240.179	even	4
400.2.q.g.149.7		16	60.47	odd	4
400.2.q.g.349.7		16	240.83	odd	4
400.2.q.h.149.2		16	60.23	odd	4
400.2.q.h.349.2		16	240.227	odd	4
640.2.l.a.161.5		16	48.5	odd	4
640.2.l.a.481.5		16	24.5	odd	2
640.2.l.b.161.4		16	48.11	even	4
640.2.l.b.481.4		16	24.11	even	2
720.2.t.c.181.3		16	4.3	odd	2
720.2.t.c.541.3		16	16.3	odd	4
1600.2.l.i.401.5		16	240.29	odd	4
1600.2.l.i.1201.5		16	15.14	odd	2
1600.2.q.g.49.4		16	15.8	even	4
1600.2.q.g.849.4		16	240.77	even	4
1600.2.q.h.49.5		16	15.2	even	4
1600.2.q.h.849.5		16	240.173	even	4
2880.2.t.c.721.1		16	16.13	even	4	inner
2880.2.t.c.2161.4		16	1.1	even	1	trivial
5120.2.a.s.1.6		8	96.35	even	8
5120.2.a.t.1.6		8	96.77	odd	8
5120.2.a.u.1.3		8	96.29	odd	8
5120.2.a.v.1.3		8	96.83	even	8

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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{5} +1.73696i q^{7} +(0.505430 + 0.505430i) q^{11} +(-1.88750 + 1.88750i) q^{13} -4.53524 q^{17} +(3.22022 - 3.22022i) q^{19} -8.85045i q^{23} +1.00000i q^{25} +(2.44059 - 2.44059i) q^{29} +5.70401 q^{31} +(1.22822 - 1.22822i) q^{35} +(-5.35670 - 5.35670i) q^{37} +10.0343i q^{41} +(2.10564 + 2.10564i) q^{43} +4.32303 q^{47} +3.98295 q^{49} +(1.37458 + 1.37458i) q^{53} -0.714786i q^{55} +(6.64140 + 6.64140i) q^{59} +(5.26208 - 5.26208i) q^{61} +2.66933 q^{65} +(10.5578 - 10.5578i) q^{67} +14.0437i q^{71} +6.63830i q^{73} +(-0.877914 + 0.877914i) q^{77} -4.27297 q^{79} +(9.15483 - 9.15483i) q^{83} +(3.20690 + 3.20690i) q^{85} +3.23826i q^{89} +(-3.27852 - 3.27852i) q^{91} -4.55407 q^{95} +1.94129 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) \) 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Introduction

L-functions

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