LMFDB - Embedded newform 2700.2.s.d.1549.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,2,Mod(1549,2700)] 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("2700.1549"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.1333317747165888577536.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 $ x^16 + 3x^14 + 5x^12 + 15x^10 + 45x^8 + 60x^6 + 80x^4 + 192*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{10} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1549.8
Root			$1.15347 - 0.818235i$ of defining polynomial
Character	$\chi$	$=$	2700.1549
Dual form			2700.2.s.d.2449.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.32643 - 2.49787i) q^{7} +(1.99787 + 3.46041i) q^{11} +(-1.33642 - 0.771582i) q^{13} -6.99574i q^{17} +2.25667 q^{19} +(6.75116 + 3.89778i) q^{23} +(-3.08304 - 5.33998i) q^{29} +(0.271582 - 0.470394i) q^{31} +6.25240i q^{37} +(0.0979532 - 0.169660i) q^{41} +(-0.0747616 + 0.0431636i) q^{43} +(-3.31658 + 1.91483i) q^{47} +(8.97869 - 15.5515i) q^{49} -4.19164i q^{53} +(-3.51278 + 6.08432i) q^{59} +(1.45470 + 2.51962i) q^{61} +(7.76470 + 4.48295i) q^{67} -8.79130 q^{71} -2.28650i q^{73} +(17.2873 + 9.98082i) q^{77} +(-6.32211 - 10.9502i) q^{79} +(12.0911 - 6.98082i) q^{83} -10.3577 q^{89} -7.70924 q^{91} +(-8.08758 + 4.66936i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{11} + 16 q^{19} - 18 q^{29} - 4 q^{31} - 18 q^{41} + 18 q^{49} - 30 q^{59} + 2 q^{61} + 48 q^{71} - 14 q^{79} + 12 q^{89} - 44 q^{91}+O(q^{100}) $ 16 * q - 6 * q^11 + 16 * q^19 - 18 * q^29 - 4 * q^31 - 18 * q^41 + 18 * q^49 - 30 * q^59 + 2 * q^61 + 48 * q^71 - 14 * q^79 + 12 * q^89 - 44 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$-1$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	4.32643	−	2.49787i	1.63524	−	0.944105i	0.652797	−	0.757533i	$-0.273595\pi$
$7$	4.32643	−	2.49787i	1.63524	−	0.944105i	0.982441	−	0.186573i	$-0.0597379\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.99787	+	3.46041i	0.602380	+	1.04335i	0.992460	+	0.122571i	$0.0391141\pi$
$11$	1.99787	+	3.46041i	0.602380	+	1.04335i	−0.390080	+	0.920781i	$0.627553\pi$
$12$		0			0
$13$	−1.33642	−	0.771582i	−0.370656	−	0.213998i	0.303089	−	0.952962i	$-0.401982\pi$
$13$	−1.33642	−	0.771582i	−0.370656	−	0.213998i	−0.673745	+	0.738964i	$0.735315\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.99574i		−	1.69672i	−0.529424	−	0.848358i	$-0.677592\pi$
$17$		−	6.99574i		−	1.69672i	0.529424	−	0.848358i	$-0.322408\pi$
$18$		0			0
$19$	2.25667			0.517715			0.258857	−	0.965916i	$-0.416654\pi$
$19$	2.25667			0.517715			0.258857	+	0.965916i	$0.416654\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.75116	+	3.89778i	1.40771	+	0.812744i	0.995167	−	0.0981929i	$-0.0313062\pi$
$23$	6.75116	+	3.89778i	1.40771	+	0.812744i	0.412546	+	0.910937i	$0.364640\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.08304	−	5.33998i	−0.572506	−	0.991609i	−0.996308	−	0.0858540i	$-0.972638\pi$
$29$	−3.08304	−	5.33998i	−0.572506	−	0.991609i	0.423802	−	0.905755i	$-0.360695\pi$
$30$		0			0
$31$	0.271582	−	0.470394i	0.0487775	−	0.0844852i	−0.840606	−	0.541647i	$-0.817801\pi$
$31$	0.271582	−	0.470394i	0.0487775	−	0.0844852i	0.889383	+	0.457162i	$0.151134\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.25240i			1.02789i	0.857824	+	0.513944i	$0.171816\pi$
$37$			6.25240i			1.02789i	−0.857824	+	0.513944i	$0.828184\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.0979532	−	0.169660i	0.0152977	−	0.0264964i	−0.858275	−	0.513190i	$-0.828464\pi$
$41$	0.0979532	−	0.169660i	0.0152977	−	0.0264964i	0.873573	+	0.486693i	$0.161797\pi$
$42$		0			0
$43$	−0.0747616	+	0.0431636i	−0.0114010	+	0.00658239i	−0.505690	−	0.862715i	$-0.668762\pi$
$43$	−0.0747616	+	0.0431636i	−0.0114010	+	0.00658239i	0.494289	+	0.869298i	$0.335429\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.31658	+	1.91483i	−0.483774	+	0.279307i	−0.721988	−	0.691906i	$-0.756771\pi$
$47$	−3.31658	+	1.91483i	−0.483774	+	0.279307i	0.238214	+	0.971213i	$0.423438\pi$
$48$		0			0
$49$	8.97869	−	15.5515i	1.28267	−	2.22165i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4.19164i		−	0.575766i	−0.957666	−	0.287883i	$-0.907049\pi$
$53$		−	4.19164i		−	0.575766i	0.957666	−	0.287883i	$-0.0929515\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.51278	+	6.08432i	−0.457326	+	0.792111i	−0.998819	−	0.0485942i	$-0.984526\pi$
$59$	−3.51278	+	6.08432i	−0.457326	+	0.792111i	0.541493	+	0.840705i	$0.317859\pi$
$60$		0			0
$61$	1.45470	+	2.51962i	0.186256	+	0.322604i	0.943999	−	0.329948i	$-0.107031\pi$
$61$	1.45470	+	2.51962i	0.186256	+	0.322604i	−0.757743	+	0.652553i	$0.773698\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	7.76470	+	4.48295i	0.948609	+	0.547680i	0.892649	−	0.450753i	$-0.148845\pi$
$67$	7.76470	+	4.48295i	0.948609	+	0.547680i	0.0559605	+	0.998433i	$0.482178\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.79130			−1.04334			−0.521668	−	0.853149i	$-0.674690\pi$
$71$	−8.79130			−1.04334			−0.521668	+	0.853149i	$0.674690\pi$
$72$		0			0
$73$		−	2.28650i		−	0.267614i	−0.991007	−	0.133807i	$-0.957280\pi$
$73$		−	2.28650i		−	0.267614i	0.991007	−	0.133807i	$-0.0427203\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	17.2873	+	9.98082i	1.97007	+	1.13742i
$78$		0			0
$79$	−6.32211	−	10.9502i	−0.711293	−	1.23199i	−0.964372	−	0.264549i	$-0.914777\pi$
$79$	−6.32211	−	10.9502i	−0.711293	−	1.23199i	0.253080	−	0.967445i	$-0.418557\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.0911	−	6.98082i	1.32717	−	0.766244i	0.342313	−	0.939586i	$-0.388790\pi$
$83$	12.0911	−	6.98082i	1.32717	−	0.766244i	0.984862	+	0.173342i	$0.0554565\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.3577			−1.09792			−0.548958	−	0.835850i	$-0.684975\pi$
$89$	−10.3577			−1.09792			−0.548958	+	0.835850i	$0.684975\pi$
$90$		0			0
$91$	−7.70924			−0.808148
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.08758	+	4.66936i	−0.821169	+	0.474102i	−0.850819	−	0.525458i	$-0.823894\pi$
$97$	−8.08758	+	4.66936i	−0.821169	+	0.474102i	0.0296505	+	0.999560i	$0.490561\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.98082	+	6.89498i	0.396106	+	0.686076i	0.993242	−	0.116064i	$-0.0370278\pi$
$101$	3.98082	+	6.89498i	0.396106	+	0.686076i	−0.597135	+	0.802140i	$0.703694\pi$
$102$		0			0
$103$	14.2151	+	8.20711i	1.40066	+	0.808670i	0.994460	−	0.105115i	$-0.0335211\pi$
$103$	14.2151	+	8.20711i	1.40066	+	0.808670i	0.406198	+	0.913785i	$0.366854\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	11.3662i		−	1.09882i	−0.835554	−	0.549408i	$-0.814853\pi$
$107$		−	11.3662i		−	1.09882i	0.835554	−	0.549408i	$-0.185147\pi$
$108$		0			0
$109$	2.22683			0.213292			0.106646	−	0.994297i	$-0.465989\pi$
$109$	2.22683			0.213292			0.106646	+	0.994297i	$0.465989\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.45425	−	1.41696i	−0.230877	−	0.133297i	0.380100	−	0.924945i	$-0.375890\pi$
$113$	−2.45425	−	1.41696i	−0.230877	−	0.133297i	−0.610976	+	0.791649i	$0.709223\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17.4744	−	30.2666i	−1.60188	−	2.77453i
$120$		0			0
$121$	−2.48295	+	4.30060i	−0.225723	+	0.390963i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	7.02130i		−	0.623040i	−0.950240	−	0.311520i	$-0.899162\pi$
$127$		−	7.02130i		−	0.623040i	0.950240	−	0.311520i	$-0.100838\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.18312	+	7.24538i	−0.365481	+	0.633032i	−0.988853	−	0.148893i	$-0.952429\pi$
$131$	−4.18312	+	7.24538i	−0.365481	+	0.633032i	0.623372	+	0.781925i	$0.285762\pi$
$132$		0			0
$133$	9.76331	−	5.63685i	0.846587	−	0.488777i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.41734	−	1.39565i	0.206527	−	0.119238i	−0.393169	−	0.919466i	$-0.628621\pi$
$137$	2.41734	−	1.39565i	0.206527	−	0.119238i	0.599696	+	0.800228i	$0.295288\pi$
$138$		0			0
$139$	−1.95799	+	3.39135i	−0.166075	+	0.287650i	−0.937036	−	0.349231i	$-0.886443\pi$
$139$	−1.95799	+	3.39135i	−0.166075	+	0.287650i	0.770962	+	0.636882i	$0.219776\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	6.16607i		−	0.515633i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.37860	−	11.0481i	0.522555	−	0.905093i	−0.477100	−	0.878849i	$-0.658312\pi$
$149$	6.37860	−	11.0481i	0.522555	−	0.905093i	0.999656	−	0.0262436i	$-0.00835456\pi$
$150$		0			0
$151$	0.188545	+	0.326569i	0.0153436	+	0.0265758i	0.873595	−	0.486653i	$-0.161782\pi$
$151$	0.188545	+	0.326569i	0.0153436	+	0.0265758i	−0.858252	+	0.513229i	$0.828449\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	10.4761	+	6.04839i	0.836086	+	0.482714i	0.855932	−	0.517089i	$-0.172984\pi$
$157$	10.4761	+	6.04839i	0.836086	+	0.482714i	−0.0198461	+	0.999803i	$0.506318\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	38.9446			3.06926
$162$		0			0
$163$			10.8563i			0.850333i	0.905115	+	0.425166i	$0.139784\pi$
$163$			10.8563i			0.850333i	−0.905115	+	0.425166i	$0.860216\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.49240	−	2.59369i	−0.347632	−	0.200706i	0.316010	−	0.948756i	$-0.397657\pi$
$167$	−4.49240	−	2.59369i	−0.347632	−	0.200706i	−0.663642	+	0.748050i	$0.730990\pi$
$168$		0			0
$169$	−5.30932	−	9.19602i	−0.408409	−	0.707386i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.92264	−	1.68739i	0.222204	−	0.128290i	−0.384766	−	0.923014i	$-0.625718\pi$
$173$	2.92264	−	1.68739i	0.222204	−	0.128290i	0.606970	+	0.794724i	$0.292385\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	19.5997			1.46495			0.732474	−	0.680795i	$-0.238366\pi$
$179$	19.5997			1.46495			0.732474	+	0.680795i	$0.238366\pi$
$180$		0			0
$181$	5.84865			0.434727			0.217363	−	0.976091i	$-0.430254\pi$
$181$	5.84865			0.434727			0.217363	+	0.976091i	$0.430254\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	24.2081	−	13.9766i	1.77027	−	1.02207i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.2956	−	21.2965i	−0.889676	−	1.54096i	−0.840259	−	0.542186i	$-0.817597\pi$
$191$	−12.2956	−	21.2965i	−0.889676	−	1.54096i	−0.0494171	−	0.998778i	$-0.515736\pi$
$192$		0			0
$193$	8.95729	+	5.17150i	0.644760	+	0.372252i	0.786446	−	0.617659i	$-0.211919\pi$
$193$	8.95729	+	5.17150i	0.644760	+	0.372252i	−0.141686	+	0.989912i	$0.545252\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	16.0256i		−	1.14177i	−0.821028	−	0.570887i	$-0.806599\pi$
$197$		−	16.0256i		−	1.14177i	0.821028	−	0.570887i	$-0.193401\pi$
$198$		0			0
$199$	22.2353			1.57622			0.788111	−	0.615533i	$-0.211059\pi$
$199$	22.2353			1.57622			0.788111	+	0.615533i	$0.211059\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−26.6771	−	15.4020i	−1.87237	−	1.08101i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.50852	+	7.80899i	0.311861	+	0.540159i
$210$		0			0
$211$	−12.1779	+	21.0927i	−0.838360	+	1.45208i	0.0529050	+	0.998600i	$0.483152\pi$
$211$	−12.1779	+	21.0927i	−0.838360	+	1.45208i	−0.891265	+	0.453483i	$0.850181\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	2.71350i		−	0.184205i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.39778	+	9.34923i	−0.363094	+	0.628897i
$222$		0			0
$223$	9.74117	−	5.62407i	0.652317	−	0.376615i	−0.137026	−	0.990567i	$-0.543754\pi$
$223$	9.74117	−	5.62407i	0.652317	−	0.376615i	0.789343	+	0.613952i	$0.210421\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.78931	−	5.07451i	0.583367	−	0.336807i	−0.179103	−	0.983830i	$-0.557320\pi$
$227$	8.78931	−	5.07451i	0.583367	−	0.336807i	0.762470	+	0.647023i	$0.223986\pi$
$228$		0			0
$229$	−1.18428	+	2.05123i	−0.0782595	+	0.135549i	−0.902499	−	0.430692i	$-0.858270\pi$
$229$	−1.18428	+	2.05123i	−0.0782595	+	0.135549i	0.824240	+	0.566241i	$0.191603\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			24.1831i			1.58429i	0.610334	+	0.792144i	$0.291035\pi$
$233$			24.1831i			1.58429i	−0.610334	+	0.792144i	$0.708965\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.897782	−	1.55500i	0.0580727	−	0.100585i	−0.835528	−	0.549449i	$-0.814838\pi$
$239$	0.897782	−	1.55500i	0.0580727	−	0.100585i	0.893600	+	0.448864i	$0.148171\pi$
$240$		0			0
$241$	7.43339	+	12.8750i	0.478827	+	0.829352i	0.999705	−	0.0242785i	$-0.00772884\pi$
$241$	7.43339	+	12.8750i	0.478827	+	0.829352i	−0.520878	+	0.853631i	$0.674396\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.01585	−	1.74120i	−0.191894	−	0.110790i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.8297			0.999159			0.499580	−	0.866268i	$-0.333488\pi$
$251$	15.8297			0.999159			0.499580	+	0.866268i	$0.333488\pi$
$252$		0			0
$253$			31.1490i			1.95832i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.0489	+	10.9979i	1.18824	+	0.686028i	0.957905	−	0.287084i	$-0.0926860\pi$
$257$	19.0489	+	10.9979i	1.18824	+	0.686028i	0.230330	+	0.973113i	$0.426019\pi$
$258$		0			0
$259$	15.6177	+	27.0506i	0.970435	+	1.68084i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.1231	−	7.57664i	0.809207	−	0.467196i	−0.0374737	−	0.999298i	$-0.511931\pi$
$263$	13.1231	−	7.57664i	0.809207	−	0.467196i	0.846680	+	0.532102i	$0.178598\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.99147			−0.487249			−0.243624	−	0.969870i	$-0.578336\pi$
$269$	−7.99147			−0.487249			−0.243624	+	0.969870i	$0.578336\pi$
$270$		0			0
$271$	3.97248			0.241311			0.120656	−	0.992694i	$-0.461500\pi$
$271$	3.97248			0.241311			0.120656	+	0.992694i	$0.461500\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.7738	+	9.10702i	−0.947757	+	0.547188i	−0.892383	−	0.451278i	$-0.850968\pi$
$277$	−15.7738	+	9.10702i	−0.947757	+	0.547188i	−0.0553736	+	0.998466i	$0.517635\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−11.1043	−	19.2333i	−0.662429	−	1.14736i	−0.979975	−	0.199119i	$-0.936192\pi$
$281$	−11.1043	−	19.2333i	−0.662429	−	1.14736i	0.317546	−	0.948243i	$-0.397141\pi$
$282$		0			0
$283$	−10.0916	−	5.82637i	−0.599882	−	0.346342i	0.169113	−	0.985597i	$-0.445910\pi$
$283$	−10.0916	−	5.82637i	−0.599882	−	0.346342i	−0.768995	+	0.639255i	$0.779243\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	0.978696i		−	0.0577706i
$288$		0			0
$289$	−31.9403			−1.87884
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.35473	+	3.09156i	0.312827	+	0.180611i	0.648191	−	0.761478i	$-0.275526\pi$
$293$	5.35473	+	3.09156i	0.312827	+	0.180611i	−0.335364	+	0.942089i	$0.608859\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.01492	−	10.4181i	−0.347852	−	0.602497i
$300$		0			0
$301$	−0.215634	+	0.373489i	−0.0124289	+	0.0215276i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			12.9872i			0.741219i	0.928789	+	0.370610i	$0.120851\pi$
$307$			12.9872i			0.741219i	−0.928789	+	0.370610i	$0.879149\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.0958	+	19.2185i	−0.629186	+	1.08978i	0.358529	+	0.933519i	$0.383278\pi$
$311$	−11.0958	+	19.2185i	−0.629186	+	1.08978i	−0.987715	+	0.156264i	$0.950055\pi$
$312$		0			0
$313$	−14.4559	+	8.34609i	−0.817093	+	0.471749i	−0.849413	−	0.527729i	$-0.823044\pi$
$313$	−14.4559	+	8.34609i	−0.817093	+	0.471749i	0.0323199	+	0.999478i	$0.489710\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−28.5419	+	16.4787i	−1.60307	+	0.925535i	−0.612206	+	0.790698i	$0.709718\pi$
$317$	−28.5419	+	16.4787i	−1.60307	+	0.925535i	−0.990868	+	0.134837i	$0.956949\pi$
$318$		0			0
$319$	12.3190	−	21.3371i	0.689732	−	1.19465i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	15.7870i		−	0.878414i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.56599	+	16.5688i	−0.527390	+	0.913466i
$330$		0			0
$331$	−14.3333	−	24.8260i	−0.787830	−	1.36456i	−0.927294	−	0.374334i	$-0.877871\pi$
$331$	−14.3333	−	24.8260i	−0.787830	−	1.36456i	0.139464	−	0.990227i	$-0.455462\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−25.8543	−	14.9270i	−1.40837	−	0.813125i	−0.413143	−	0.910666i	$-0.635569\pi$
$337$	−25.8543	−	14.9270i	−1.40837	−	0.813125i	−0.995231	+	0.0975411i	$0.968902\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.17034			0.117530
$342$		0			0
$343$		−	54.7401i		−	2.95569i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.19000	+	2.41910i	0.224931	+	0.129864i	0.608231	−	0.793760i	$-0.291879\pi$
$347$	4.19000	+	2.41910i	0.224931	+	0.129864i	−0.383301	+	0.923624i	$0.625213\pi$
$348$		0			0
$349$	−16.0325	−	27.7691i	−0.858200	−	1.48645i	−0.873644	−	0.486566i	$-0.838249\pi$
$349$	−16.0325	−	27.7691i	−0.858200	−	1.48645i	0.0154438	−	0.999881i	$-0.495084\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.53857	−	4.92975i	0.454462	−	0.262384i	−0.255251	−	0.966875i	$-0.582158\pi$
$353$	8.53857	−	4.92975i	0.454462	−	0.262384i	0.709713	+	0.704491i	$0.248825\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.93266			0.524226			0.262113	−	0.965037i	$-0.415581\pi$
$359$	9.93266			0.524226			0.262113	+	0.965037i	$0.415581\pi$
$360$		0			0
$361$	−13.9075			−0.731972
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−16.4729	+	9.51065i	−0.859880	+	0.496452i	−0.863972	−	0.503540i	$-0.832031\pi$
$367$	−16.4729	+	9.51065i	−0.859880	+	0.496452i	0.00409206	+	0.999992i	$0.498697\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10.4702	−	18.1349i	−0.543584	−	0.941515i
$372$		0			0
$373$	1.62681	+	0.939241i	0.0842332	+	0.0486320i	0.541525	−	0.840685i	$-0.317847\pi$
$373$	1.62681	+	0.939241i	0.0842332	+	0.0486320i	−0.457292	+	0.889317i	$0.651180\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			9.51526i			0.490061i
$378$		0			0
$379$	−22.8435			−1.17339			−0.586697	−	0.809807i	$-0.699572\pi$
$379$	−22.8435			−1.17339			−0.586697	+	0.809807i	$0.699572\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.07262	−	2.35133i	−0.208101	−	0.120147i	0.392327	−	0.919826i	$-0.371670\pi$
$383$	−4.07262	−	2.35133i	−0.208101	−	0.120147i	−0.600429	+	0.799678i	$0.705003\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.38287	+	9.32340i	0.272922	+	0.472715i	0.969609	−	0.244661i	$-0.0786765\pi$
$389$	5.38287	+	9.32340i	0.272922	+	0.472715i	−0.696687	+	0.717376i	$0.745343\pi$
$390$		0			0
$391$	27.2679	−	47.2293i	1.37899	−	2.38849i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			8.10958i			0.407008i	0.979074	+	0.203504i	$0.0652330\pi$
$397$			8.10958i			0.407008i	−0.979074	+	0.203504i	$0.934767\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.22744	+	10.7862i	−0.310984	+	0.538640i	−0.978576	−	0.205888i	$-0.933992\pi$
$401$	−6.22744	+	10.7862i	−0.310984	+	0.538640i	0.667592	+	0.744527i	$0.267325\pi$
$402$		0			0
$403$	−0.725894	+	0.419095i	−0.0361594	+	0.0208766i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.6359	+	12.4915i	−1.07245	+	0.619179i
$408$		0			0
$409$	−15.7166	+	27.2219i	−0.777136	+	1.34604i	0.156450	+	0.987686i	$0.449995\pi$
$409$	−15.7166	+	27.2219i	−0.777136	+	1.34604i	−0.933586	+	0.358353i	$0.883338\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			35.0979i			1.72705i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.1848	+	22.8368i	−0.644121	+	1.11565i	0.340383	+	0.940287i	$0.389443\pi$
$419$	−13.1848	+	22.8368i	−0.644121	+	1.11565i	−0.984504	+	0.175363i	$0.943890\pi$
$420$		0			0
$421$	6.61555	+	11.4585i	0.322422	+	0.558451i	0.980987	−	0.194072i	$-0.0621696\pi$
$421$	6.61555	+	11.4585i	0.322422	+	0.558451i	−0.658565	+	0.752524i	$0.728836\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	12.5874	+	7.26732i	0.609145	+	0.351690i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.2895			−1.16999			−0.584993	−	0.811039i	$-0.698903\pi$
$431$	−24.2895			−1.16999			−0.584993	+	0.811039i	$0.698903\pi$
$432$		0			0
$433$		−	17.9840i		−	0.864258i	−0.901812	−	0.432129i	$-0.857763\pi$
$433$		−	17.9840i		−	0.864258i	0.901812	−	0.432129i	$-0.142237\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.2351	+	8.79599i	0.728794	+	0.420769i
$438$		0			0
$439$	11.3577	+	19.6721i	0.542074	+	0.938900i	0.998785	+	0.0492847i	$0.0156942\pi$
$439$	11.3577	+	19.6721i	0.542074	+	0.938900i	−0.456711	+	0.889615i	$0.650972\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.8455	+	11.4578i	−0.942888	+	0.544377i	−0.890864	−	0.454269i	$-0.849900\pi$
$443$	−19.8455	+	11.4578i	−0.942888	+	0.544377i	−0.0520235	+	0.998646i	$0.516567\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.36966			−0.253410			−0.126705	−	0.991940i	$-0.540440\pi$
$449$	−5.36966			−0.253410			−0.126705	+	0.991940i	$0.540440\pi$
$450$		0			0
$451$	0.782790			0.0368601
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	16.6677	−	9.62309i	0.779682	−	0.450149i	−0.0566358	−	0.998395i	$-0.518037\pi$
$457$	16.6677	−	9.62309i	0.779682	−	0.450149i	0.836317	+	0.548246i	$0.184704\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.6870	+	30.6347i	0.823763	+	1.42680i	0.902861	+	0.429933i	$0.141463\pi$
$461$	17.6870	+	30.6347i	0.823763	+	1.42680i	−0.0790973	+	0.996867i	$0.525204\pi$
$462$		0			0
$463$	−14.8277	−	8.56075i	−0.689100	−	0.397852i	0.114175	−	0.993461i	$-0.463578\pi$
$463$	−14.8277	−	8.56075i	−0.689100	−	0.397852i	−0.803275	+	0.595609i	$0.796911\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	25.9318i		−	1.19998i	−0.800007	−	0.599990i	$-0.795171\pi$
$467$		−	25.9318i		−	1.19998i	0.800007	−	0.599990i	$-0.204829\pi$
$468$		0			0
$469$	44.7913			2.06827
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.298728	−	0.172471i	−0.0137355	−	0.00793020i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.4574	+	18.1127i	0.477810	+	0.827591i	0.999676	−	0.0254364i	$-0.00809754\pi$
$479$	10.4574	+	18.1127i	0.477810	+	0.827591i	−0.521867	+	0.853027i	$0.674764\pi$
$480$		0			0
$481$	4.82424	−	8.35583i	0.219966	−	0.380993i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	20.2373i		−	0.917039i	−0.888684	−	0.458520i	$-0.848380\pi$
$487$		−	20.2373i		−	0.917039i	0.888684	−	0.458520i	$-0.151620\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7.10009	+	12.2977i	−0.320422	+	0.554988i	−0.980575	−	0.196143i	$-0.937158\pi$
$491$	−7.10009	+	12.2977i	−0.320422	+	0.554988i	0.660153	+	0.751131i	$0.270491\pi$
$492$		0			0
$493$	−37.3571	+	21.5681i	−1.68248	+	0.971379i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−38.0350	+	21.9595i	−1.70610	+	0.985018i
$498$		0			0
$499$	−14.1384	+	24.4884i	−0.632920	+	1.09625i	0.354032	+	0.935233i	$0.384811\pi$
$499$	−14.1384	+	24.4884i	−0.632920	+	1.09625i	−0.986952	+	0.161016i	$0.948523\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			17.4165i			0.776565i	0.921540	+	0.388282i	$0.126931\pi$
$503$			17.4165i			0.776565i	−0.921540	+	0.388282i	$0.873069\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.1060	+	19.2362i	−0.492267	+	0.852631i	−0.999960	−	0.00890659i	$-0.997165\pi$
$509$	−11.1060	+	19.2362i	−0.492267	+	0.852631i	0.507694	+	0.861538i	$0.330498\pi$
$510$		0			0
$511$	−5.71137	−	9.89238i	−0.252656	−	0.437613i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−13.2522	−	7.65116i	−0.582831	−	0.336498i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.48898			−0.240477			−0.120238	−	0.992745i	$-0.538366\pi$
$521$	−5.48898			−0.240477			−0.120238	+	0.992745i	$0.538366\pi$
$522$		0			0
$523$			26.0991i			1.14123i	0.821216	+	0.570617i	$0.193296\pi$
$523$			26.0991i			1.14123i	−0.821216	+	0.570617i	$0.806704\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.29075	−	1.89991i	−0.143347	−	0.0827616i
$528$		0			0
$529$	18.8854	+	32.7105i	0.821105	+	1.42220i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.261813	+	0.151158i	−0.0113404	+	0.00654737i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	71.7529			3.09062
$540$		0			0
$541$	18.7828			0.807534			0.403767	−	0.914862i	$-0.367701\pi$
$541$	18.7828			0.807534			0.403767	+	0.914862i	$0.367701\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	7.84315	−	4.52825i	0.335349	−	0.193614i	−0.322864	−	0.946445i	$-0.604646\pi$
$547$	7.84315	−	4.52825i	0.335349	−	0.193614i	0.658213	+	0.752831i	$0.271312\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.95738	−	12.0505i	−0.296395	−	0.513370i
$552$		0			0
$553$	−54.7044	−	31.5836i	−2.32627	−	1.34307i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	24.8467i		−	1.05279i	−0.850241	−	0.526394i	$-0.823544\pi$
$557$		−	24.8467i		−	1.05279i	0.850241	−	0.526394i	$-0.176456\pi$
$558$		0			0
$559$	0.133217			0.00563448
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.2247	+	11.6767i	0.852369	+	0.492115i	0.861449	−	0.507843i	$-0.169557\pi$
$563$	20.2247	+	11.6767i	0.852369	+	0.492115i	−0.00908064	+	0.999959i	$0.502890\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	7.54510	+	13.0685i	0.316307	+	0.547860i	0.979714	−	0.200398i	$-0.0642236\pi$
$569$	7.54510	+	13.0685i	0.316307	+	0.547860i	−0.663407	+	0.748258i	$0.730890\pi$
$570$		0			0
$571$	3.26786	−	5.66011i	0.136756	−	0.236868i	−0.789511	−	0.613736i	$-0.789666\pi$
$571$	3.26786	−	5.66011i	0.136756	−	0.236868i	0.926267	+	0.376868i	$0.122999\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			8.12662i			0.338316i	0.985589	+	0.169158i	$0.0541047\pi$
$577$			8.12662i			0.338316i	−0.985589	+	0.169158i	$0.945895\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	34.8743	−	60.4041i	1.44683	−	2.50598i
$582$		0			0
$583$	14.5048	−	8.37435i	0.600727	−	0.346830i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.3438	+	6.54937i	−0.468210	+	0.270321i	−0.715490	−	0.698623i	$-0.753797\pi$
$587$	−11.3438	+	6.54937i	−0.468210	+	0.270321i	0.247280	+	0.968944i	$0.420463\pi$
$588$		0			0
$589$	0.612869	−	1.06152i	0.0252528	−	0.0437392i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			8.60478i			0.353356i	0.984269	+	0.176678i	$0.0565351\pi$
$593$			8.60478i			0.353356i	−0.984269	+	0.176678i	$0.943465\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.1827	+	21.1010i	−0.497771	+	0.862165i	−0.999997	−	0.00257140i	$-0.999181\pi$
$599$	−12.1827	+	21.1010i	−0.497771	+	0.862165i	0.502225	+	0.864737i	$0.332515\pi$
$600$		0			0
$601$	12.0900	+	20.9404i	0.493160	+	0.854179i	0.999969	−	0.00787996i	$-0.00250829\pi$
$601$	12.0900	+	20.9404i	0.493160	+	0.854179i	−0.506809	+	0.862059i	$0.669175\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−1.23582	−	0.713502i	−0.0501605	−	0.0289602i	0.474710	−	0.880142i	$-0.342553\pi$
$607$	−1.23582	−	0.713502i	−0.0501605	−	0.0289602i	−0.524871	+	0.851182i	$0.675886\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.90979			0.239085
$612$		0			0
$613$		−	6.92878i		−	0.279851i	−0.990162	−	0.139925i	$-0.955314\pi$
$613$		−	6.92878i		−	0.279851i	0.990162	−	0.139925i	$-0.0446863\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.9437	−	13.2466i	−0.923679	−	0.533286i	−0.0388718	−	0.999244i	$-0.512376\pi$
$617$	−22.9437	−	13.2466i	−0.923679	−	0.533286i	−0.884807	+	0.465958i	$0.845710\pi$
$618$		0			0
$619$	−13.4798	−	23.3478i	−0.541801	−	0.938426i	−0.998801	−	0.0489597i	$-0.984409\pi$
$619$	−13.4798	−	23.3478i	−0.541801	−	0.938426i	0.457000	−	0.889467i	$-0.348924\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−44.8120	+	25.8722i	−1.79535	+	1.03655i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	43.7401			1.74403
$630$		0			0
$631$	−15.9350			−0.634361			−0.317181	−	0.948365i	$-0.602736\pi$
$631$	−15.9350			−0.634361			−0.317181	+	0.948365i	$0.602736\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−23.9986	+	13.8556i	−0.950858	+	0.548978i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.58090	−	6.20231i	−0.141437	−	0.244976i	0.786601	−	0.617462i	$-0.211839\pi$
$641$	−3.58090	−	6.20231i	−0.141437	−	0.244976i	−0.928038	+	0.372485i	$0.878506\pi$
$642$		0			0
$643$	6.11680	+	3.53154i	0.241223	+	0.139270i	0.615739	−	0.787950i	$-0.288858\pi$
$643$	6.11680	+	3.53154i	0.241223	+	0.139270i	−0.374516	+	0.927221i	$0.622191\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	26.4037i		−	1.03804i	−0.854763	−	0.519019i	$-0.826297\pi$
$647$		−	26.4037i		−	1.03804i	0.854763	−	0.519019i	$-0.173703\pi$
$648$		0			0
$649$	−28.0723			−1.10193
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.68545	+	3.85985i	0.261622	+	0.151048i	0.625074	−	0.780565i	$-0.285069\pi$
$653$	6.68545	+	3.85985i	0.261622	+	0.151048i	−0.363452	+	0.931613i	$0.618402\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11.1426	+	19.2996i	0.434055	+	0.751806i	0.997218	−	0.0745403i	$-0.0237489\pi$
$659$	11.1426	+	19.2996i	0.434055	+	0.751806i	−0.563163	+	0.826346i	$0.690416\pi$
$660$		0			0
$661$	−16.0544	+	27.8070i	−0.624442	+	1.08157i	0.364207	+	0.931318i	$0.381340\pi$
$661$	−16.0544	+	27.8070i	−0.624442	+	1.08157i	−0.988648	+	0.150247i	$0.951993\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	48.0680i		−	1.86120i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.81261	+	10.0677i	−0.224393	+	0.388661i
$672$		0			0
$673$	−6.78237	+	3.91580i	−0.261441	+	0.150943i	−0.624992	−	0.780631i	$-0.714898\pi$
$673$	−6.78237	+	3.91580i	−0.261441	+	0.150943i	0.363551	+	0.931574i	$0.381564\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.5914	+	9.00171i	−0.599227	+	0.345964i	−0.768737	−	0.639565i	$-0.779115\pi$
$677$	−15.5914	+	9.00171i	−0.599227	+	0.345964i	0.169511	+	0.985528i	$0.445781\pi$
$678$		0			0
$679$	−23.3269	+	40.4034i	−0.895205	+	1.55054i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	30.3356i		−	1.16076i	−0.814347	−	0.580379i	$-0.802904\pi$
$683$		−	30.3356i		−	1.16076i	0.814347	−	0.580379i	$-0.197096\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.23419	+	5.60179i	−0.123213	+	0.213411i
$690$		0			0
$691$	−14.0575	−	24.3482i	−0.534771	−	0.926251i	−0.999174	−	0.0406267i	$-0.987065\pi$
$691$	−14.0575	−	24.3482i	−0.534771	−	0.926251i	0.464403	−	0.885624i	$-0.346269\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.18690	−	0.685255i	−0.0449569	−	0.0259559i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−28.4378			−1.07408			−0.537041	−	0.843556i	$-0.680458\pi$
$701$	−28.4378			−1.07408			−0.537041	+	0.843556i	$0.680458\pi$
$702$		0			0
$703$			14.1096i			0.532153i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	34.4455	+	19.8871i	1.29546	+	0.747932i
$708$		0			0
$709$	15.4500	+	26.7602i	0.580237	+	1.00500i	0.995451	+	0.0952767i	$0.0303736\pi$
$709$	15.4500	+	26.7602i	0.580237	+	1.00500i	−0.415213	+	0.909724i	$0.636293\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.66698	−	2.11713i	0.137330	−	0.0792873i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.2462			−0.792349			−0.396175	−	0.918175i	$-0.629662\pi$
$719$	−21.2462			−0.792349			−0.396175	+	0.918175i	$0.629662\pi$
$720$		0			0
$721$	82.0011			3.05388
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−21.6351	+	12.4910i	−0.802402	+	0.463267i	−0.844310	−	0.535854i	$-0.819990\pi$
$727$	−21.6351	+	12.4910i	−0.802402	+	0.463267i	0.0419082	+	0.999121i	$0.486656\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.301961	+	0.523013i	0.0111684	+	0.0193443i
$732$		0			0
$733$	9.06528	+	5.23384i	0.334834	+	0.193316i	0.657985	−	0.753031i	$-0.271409\pi$
$733$	9.06528	+	5.23384i	0.334834	+	0.193316i	−0.323151	+	0.946347i	$0.604742\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			35.8254i			1.31964i
$738$		0			0
$739$	−35.6244			−1.31046			−0.655232	−	0.755428i	$-0.727429\pi$
$739$	−35.6244			−1.31046			−0.655232	+	0.755428i	$0.727429\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.70823	+	5.02770i	0.319474	+	0.184448i	0.651158	−	0.758942i	$-0.274283\pi$
$743$	8.70823	+	5.02770i	0.319474	+	0.184448i	−0.331684	+	0.943391i	$0.607617\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−28.3914	−	49.1753i	−1.03740	−	1.79683i
$750$		0			0
$751$	−8.86740	+	15.3588i	−0.323576	+	0.560450i	−0.981223	−	0.192876i	$-0.938219\pi$
$751$	−8.86740	+	15.3588i	−0.323576	+	0.560450i	0.657647	+	0.753326i	$0.271552\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			6.42042i			0.233354i	0.993170	+	0.116677i	$0.0372243\pi$
$757$			6.42042i			0.233354i	−0.993170	+	0.116677i	$0.962776\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.6955	−	21.9892i	0.460211	−	0.797108i	−0.538760	−	0.842459i	$-0.681107\pi$
$761$	12.6955	−	21.9892i	0.460211	−	0.797108i	0.998971	+	0.0453506i	$0.0144405\pi$
$762$		0			0
$763$	9.63425	−	5.56234i	0.348783	−	0.201370i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9.38910	−	5.42080i	0.339021	−	0.195734i
$768$		0			0
$769$	−7.40088	+	12.8187i	−0.266883	+	0.462254i	−0.968055	−	0.250737i	$-0.919327\pi$
$769$	−7.40088	+	12.8187i	−0.266883	+	0.462254i	0.701173	+	0.712992i	$0.252660\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	7.48046i		−	0.269053i	−0.990910	−	0.134527i	$-0.957049\pi$
$773$		−	7.48046i		−	0.269053i	0.990910	−	0.134527i	$-0.0429514\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.221048	−	0.382866i	0.00791985	−	0.0137176i
$780$		0			0
$781$	−17.5639	−	30.4215i	−0.628484	−	1.08857i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−45.3082	−	26.1587i	−1.61506	−	0.932457i	−0.988172	−	0.153350i	$-0.950994\pi$
$787$	−45.3082	−	26.1587i	−1.61506	−	0.932457i	−0.626891	−	0.779107i	$-0.715673\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14.1575			−0.503384
$792$		0			0
$793$		−	4.48969i		−	0.159434i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	46.5366	+	26.8679i	1.64841	+	0.951711i	0.977703	+	0.209991i	$0.0673434\pi$
$797$	46.5366	+	26.8679i	1.64841	+	0.951711i	0.670709	+	0.741721i	$0.265990\pi$
$798$		0			0
$799$	13.3957	+	23.2019i	0.473904	+	0.820826i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.91222	−	4.56812i	0.279216	−	0.161205i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−26.1908			−0.920819			−0.460410	−	0.887707i	$-0.652297\pi$
$809$	−26.1908			−0.920819			−0.460410	+	0.887707i	$0.652297\pi$
$810$		0			0
$811$	−52.3506			−1.83828			−0.919140	−	0.393931i	$-0.871115\pi$
$811$	−52.3506			−1.83828			−0.919140	+	0.393931i	$0.871115\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.168712	+	0.0974059i	−0.00590249	+	0.00340780i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.42932	−	4.20771i	−0.0847839	−	0.146850i	0.820515	−	0.571625i	$-0.193687\pi$
$821$	−2.42932	−	4.20771i	−0.0847839	−	0.146850i	−0.905299	+	0.424775i	$0.860353\pi$
$822$		0			0
$823$	−1.01998	−	0.588886i	−0.0355543	−	0.0205273i	0.482117	−	0.876107i	$-0.339868\pi$
$823$	−1.01998	−	0.588886i	−0.0355543	−	0.0205273i	−0.517672	+	0.855579i	$0.673201\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			32.7188i			1.13775i	0.822426	+	0.568873i	$0.192620\pi$
$827$			32.7188i			1.13775i	−0.822426	+	0.568873i	$0.807380\pi$
$828$		0			0
$829$	5.99342			0.208160			0.104080	−	0.994569i	$-0.466810\pi$
$829$	5.99342			0.208160			0.104080	+	0.994569i	$0.466810\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−108.794	−	62.8125i	−3.76951	−	2.17632i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19.1933	+	33.2439i	0.662628	+	1.14771i	0.979923	+	0.199378i	$0.0638922\pi$
$839$	19.1933	+	33.2439i	0.662628	+	1.14771i	−0.317295	+	0.948327i	$0.602774\pi$
$840$		0			0
$841$	−4.51023	+	7.81195i	−0.155525	+	0.269378i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$			24.8083i			0.852425i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.3705	+	42.2109i	−0.835410	+	1.44697i
$852$		0			0
$853$	18.8092	−	10.8595i	0.644015	−	0.371822i	−0.142145	−	0.989846i	$-0.545400\pi$
$853$	18.8092	−	10.8595i	0.644015	−	0.371822i	0.786159	+	0.618024i	$0.212067\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.98342	+	2.29983i	−0.136071	+	0.0785607i	−0.566490	−	0.824068i	$-0.691699\pi$
$857$	−3.98342	+	2.29983i	−0.136071	+	0.0785607i	0.430419	+	0.902629i	$0.358366\pi$
$858$		0			0
$859$	7.45896	−	12.9193i	0.254496	−	0.440801i	−0.710262	−	0.703937i	$-0.751424\pi$
$859$	7.45896	−	12.9193i	0.254496	−	0.440801i	0.964759	+	0.263137i	$0.0847570\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	11.4089i		−	0.388362i	−0.980966	−	0.194181i	$-0.937795\pi$
$863$		−	11.4089i		−	0.388362i	0.980966	−	0.194181i	$-0.0622050\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	25.2615	−	43.7541i	0.856937	−	1.48426i
$870$		0			0
$871$	−6.91793	−	11.9822i	−0.234405	−	0.406001i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	24.1867	+	13.9642i	0.816727	+	0.471537i	0.849286	−	0.527932i	$-0.177033\pi$
$877$	24.1867	+	13.9642i	0.816727	+	0.471537i	−0.0325596	+	0.999470i	$0.510366\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	42.3270			1.42603			0.713017	−	0.701147i	$-0.247328\pi$
$881$	42.3270			1.42603			0.713017	+	0.701147i	$0.247328\pi$
$882$		0			0
$883$		−	20.3895i		−	0.686161i	−0.939306	−	0.343081i	$-0.888530\pi$
$883$		−	20.3895i		−	0.686161i	0.939306	−	0.343081i	$-0.111470\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−34.2471	−	19.7725i	−1.14990	−	0.663897i	−0.201040	−	0.979583i	$-0.564432\pi$
$887$	−34.2471	−	19.7725i	−1.14990	−	0.663897i	−0.948864	+	0.315686i	$0.897765\pi$
$888$		0			0
$889$	−17.5383	−	30.3772i	−0.588215	−	1.01882i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.48442	+	4.32113i	−0.250457	+	0.144601i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.34919			−0.111702
$900$		0			0
$901$	−29.3236			−0.976911
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−32.0080	+	18.4798i	−1.06281	+	0.613613i	−0.926208	−	0.377013i	$-0.876951\pi$
$907$	−32.0080	+	18.4798i	−1.06281	+	0.613613i	−0.136601	+	0.990626i	$0.543618\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.92335	−	8.52749i	−0.163118	−	0.282528i	0.772867	−	0.634568i	$-0.218822\pi$
$911$	−4.92335	−	8.52749i	−0.163118	−	0.282528i	−0.935985	+	0.352039i	$0.885488\pi$
$912$		0			0
$913$	48.3130	+	27.8935i	1.59893	+	0.923140i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			41.7955i			1.38021i
$918$		0			0
$919$	8.42079			0.277776			0.138888	−	0.990308i	$-0.455647\pi$
$919$	8.42079			0.277776			0.138888	+	0.990308i	$0.455647\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11.7489	+	6.78321i	0.386718	+	0.223272i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.1938	+	24.5843i	0.465683	+	0.806586i	0.999232	−	0.0391829i	$-0.0124755\pi$
$929$	14.1938	+	24.5843i	0.465683	+	0.806586i	−0.533549	+	0.845769i	$0.679142\pi$
$930$		0			0
$931$	20.2619	−	35.0946i	0.664057	−	1.15018i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			4.69342i			0.153327i	0.997057	+	0.0766637i	$0.0244268\pi$
$937$			4.69342i			0.153327i	−0.997057	+	0.0766637i	$0.975573\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.51278	−	16.4766i	0.310108	−	0.537123i	−0.668278	−	0.743912i	$-0.732968\pi$
$941$	9.51278	−	16.4766i	0.310108	−	0.537123i	0.978385	+	0.206789i	$0.0663015\pi$
$942$		0			0
$943$	1.32259	−	0.763600i	0.0430696	−	0.0248662i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.0997	+	6.40843i	−0.360693	+	0.208246i	−0.669385	−	0.742916i	$-0.733442\pi$
$947$	−11.0997	+	6.40843i	−0.360693	+	0.208246i	0.308692	+	0.951162i	$0.400109\pi$
$948$		0			0
$949$	−1.76422	+	3.05572i	−0.0572690	+	0.0991928i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			45.9701i			1.48912i	0.667556	+	0.744560i	$0.267340\pi$
$953$			45.9701i			1.48912i	−0.667556	+	0.744560i	$0.732660\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.97230	−	12.0764i	0.225147	−	0.389967i
$960$		0			0
$961$	15.3525	+	26.5913i	0.495242	+	0.857783i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	4.45751	+	2.57354i	0.143344	+	0.0827596i	0.569957	−	0.821675i	$-0.306960\pi$
$967$	4.45751	+	2.57354i	0.143344	+	0.0827596i	−0.426613	+	0.904434i	$0.640293\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10.3875			0.333352			0.166676	−	0.986012i	$-0.446697\pi$
$971$	10.3875			0.333352			0.166676	+	0.986012i	$0.446697\pi$
$972$		0			0
$973$			19.5632i			0.627169i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−34.2618	−	19.7811i	−1.09613	−	0.632852i	−0.160930	−	0.986966i	$-0.551449\pi$
$977$	−34.2618	−	19.7811i	−1.09613	−	0.632852i	−0.935202	+	0.354113i	$0.884783\pi$
$978$		0			0
$979$	−20.6933	−	35.8419i	−0.661362	−	1.14551i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.7292	+	10.2360i	−0.565474	+	0.326477i	−0.755340	−	0.655333i	$-0.772528\pi$
$983$	−17.7292	+	10.2360i	−0.565474	+	0.326477i	0.189865	+	0.981810i	$0.439195\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.672970			−0.0213992
$990$		0			0
$991$	−19.2415			−0.611228			−0.305614	−	0.952156i	$-0.598862\pi$
$991$	−19.2415			−0.611228			−0.305614	+	0.952156i	$0.598862\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	20.1251	−	11.6193i	0.637370	−	0.367986i	−0.146231	−	0.989250i	$-0.546714\pi$
$997$	20.1251	−	11.6193i	0.637370	−	0.367986i	0.783601	+	0.621265i	$0.213381\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.s.d.1549.8		16
3.2	odd	2		900.2.s.d.49.7		16
5.2	odd	4		2700.2.i.e.901.4		8
5.3	odd	4		2700.2.i.d.901.1		8
5.4	even	2	inner	2700.2.s.d.1549.1		16
9.2	odd	6		900.2.s.d.349.2		16
9.4	even	3		8100.2.d.s.649.8		8
9.5	odd	6		8100.2.d.q.649.8		8
9.7	even	3	inner	2700.2.s.d.2449.1		16
15.2	even	4		900.2.i.d.301.2	✓	8
15.8	even	4		900.2.i.e.301.3	yes	8
15.14	odd	2		900.2.s.d.49.2		16
45.2	even	12		900.2.i.d.601.2	yes	8
45.4	even	6		8100.2.d.s.649.1		8
45.7	odd	12		2700.2.i.e.1801.4		8
45.13	odd	12		8100.2.a.ba.1.4		4
45.14	odd	6		8100.2.d.q.649.1		8
45.22	odd	12		8100.2.a.y.1.1		4
45.23	even	12		8100.2.a.z.1.4		4
45.29	odd	6		900.2.s.d.349.7		16
45.32	even	12		8100.2.a.x.1.1		4
45.34	even	6	inner	2700.2.s.d.2449.8		16
45.38	even	12		900.2.i.e.601.3	yes	8
45.43	odd	12		2700.2.i.d.1801.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.2	✓	8	15.2	even	4
900.2.i.d.601.2	yes	8	45.2	even	12
900.2.i.e.301.3	yes	8	15.8	even	4
900.2.i.e.601.3	yes	8	45.38	even	12
900.2.s.d.49.2		16	15.14	odd	2
900.2.s.d.49.7		16	3.2	odd	2
900.2.s.d.349.2		16	9.2	odd	6
900.2.s.d.349.7		16	45.29	odd	6
2700.2.i.d.901.1		8	5.3	odd	4
2700.2.i.d.1801.1		8	45.43	odd	12
2700.2.i.e.901.4		8	5.2	odd	4
2700.2.i.e.1801.4		8	45.7	odd	12
2700.2.s.d.1549.1		16	5.4	even	2	inner
2700.2.s.d.1549.8		16	1.1	even	1	trivial
2700.2.s.d.2449.1		16	9.7	even	3	inner
2700.2.s.d.2449.8		16	45.34	even	6	inner
8100.2.a.x.1.1		4	45.32	even	12
8100.2.a.y.1.1		4	45.22	odd	12
8100.2.a.z.1.4		4	45.23	even	12
8100.2.a.ba.1.4		4	45.13	odd	12
8100.2.d.q.649.1		8	45.14	odd	6
8100.2.d.q.649.8		8	9.5	odd	6
8100.2.d.s.649.1		8	45.4	even	6
8100.2.d.s.649.8		8	9.4	even	3

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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.32643 - 2.49787i) q^{7} +(1.99787 + 3.46041i) q^{11} +(-1.33642 - 0.771582i) q^{13} -6.99574i q^{17} +2.25667 q^{19} +(6.75116 + 3.89778i) q^{23} +(-3.08304 - 5.33998i) q^{29} +(0.271582 - 0.470394i) q^{31} +6.25240i q^{37} +(0.0979532 - 0.169660i) q^{41} +(-0.0747616 + 0.0431636i) q^{43} +(-3.31658 + 1.91483i) q^{47} +(8.97869 - 15.5515i) q^{49} -4.19164i q^{53} +(-3.51278 + 6.08432i) q^{59} +(1.45470 + 2.51962i) q^{61} +(7.76470 + 4.48295i) q^{67} -8.79130 q^{71} -2.28650i q^{73} +(17.2873 + 9.98082i) q^{77} +(-6.32211 - 10.9502i) q^{79} +(12.0911 - 6.98082i) q^{83} -10.3577 q^{89} -7.70924 q^{91} +(-8.08758 + 4.66936i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{11} + 16 q^{19} - 18 q^{29} - 4 q^{31} - 18 q^{41} + 18 q^{49} - 30 q^{59} + 2 q^{61} + 48 q^{71} - 14 q^{79} + 12 q^{89} - 44 q^{91}+O(q^{100}) \) 16 * q - 6 * q^11 + 16 * q^19 - 18 * q^29 - 4 * q^31 - 18 * q^41 + 18 * q^49 - 30 * q^59 + 2 * q^61 + 48 * q^71 - 14 * q^79 + 12 * q^89 - 44 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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