LMFDB - Embedded newform 5220.2.b.d.289.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5220,2,Mod(289,5220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5220.289"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,-6,0,0,0,-8, 0,0,0,0,0,18,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	no
Analytic conductor:	$41.6819098551$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} + 11 x^{10} - 30 x^{9} - 49 x^{8} + 176 x^{7} - 245 x^{6} + \cdots + 78125 $ x^14 - 2x^13 + 5x^12 - 8x^11 + 11x^10 - 30x^9 - 49x^8 + 176x^7 - 245x^6 - 750x^5 + 1375x^4 - 5000x^3 + 15625x^2 - 31250*x + 78125
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1740)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.7
Root			$0.591682 + 2.15637i$ of defining polynomial
Character	$\chi$	$=$	5220.289
Dual form			5220.2.b.d.289.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.591682 - 2.15637i) q^{5} -2.69732i q^{7} +0.968368i q^{11} +2.99709i q^{13} -1.02889 q^{17} -7.89515i q^{19} +0.0832686i q^{23} +(-4.29982 + 2.55177i) q^{25} +(-4.28803 + 3.25773i) q^{29} -2.02337i q^{31} +(-5.81641 + 1.59596i) q^{35} +3.06250 q^{37} -3.92969i q^{41} -7.15897 q^{43} +0.554230 q^{47} -0.275549 q^{49} +6.25482i q^{53} +(2.08815 - 0.572966i) q^{55} -11.5363 q^{59} -8.77259i q^{61} +(6.46283 - 1.77333i) q^{65} +2.40621i q^{67} -11.1708 q^{71} +7.48700 q^{73} +2.61200 q^{77} -6.55123i q^{79} +7.10098i q^{83} +(0.608773 + 2.21865i) q^{85} +6.83268i q^{89} +8.08413 q^{91} +(-17.0248 + 4.67142i) q^{95} +4.18357 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{5} + 4 q^{17} - 6 q^{25} - 8 q^{29} + 18 q^{35} - 14 q^{37} + 6 q^{43} - 16 q^{47} - 14 q^{49} - 8 q^{55} - 12 q^{59} - 10 q^{65} - 4 q^{71} - 14 q^{73} - 24 q^{77} - 14 q^{85} + 16 q^{91} - 8 q^{95}+ \cdots + 6 q^{97}+O(q^{100}) $ 14 * q - 2 * q^5 + 4 * q^17 - 6 * q^25 - 8 * q^29 + 18 * q^35 - 14 * q^37 + 6 * q^43 - 16 * q^47 - 14 * q^49 - 8 * q^55 - 12 * q^59 - 10 * q^65 - 4 * q^71 - 14 * q^73 - 24 * q^77 - 14 * q^85 + 16 * q^91 - 8 * q^95 + 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5220\mathbb{Z}\right)^\times$.

$n$	$901$	$2611$	$4061$	$4177$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.591682	−	2.15637i	−0.264608	−	0.964356i
$5$	−0.591682	−	2.15637i	−0.264608	−	0.964356i
$6$		0			0
$7$		−	2.69732i		−	1.01949i	−0.860325	−	0.509746i	$-0.829739\pi$
$7$		−	2.69732i		−	1.01949i	0.860325	−	0.509746i	$-0.170261\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.968368i			0.291974i	0.989287	+	0.145987i	$0.0466357\pi$
$11$			0.968368i			0.291974i	−0.989287	+	0.145987i	$0.953364\pi$
$12$		0			0
$13$			2.99709i			0.831244i	0.909537	+	0.415622i	$0.136436\pi$
$13$			2.99709i			0.831244i	−0.909537	+	0.415622i	$0.863564\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.02889			−0.249541			−0.124771	−	0.992186i	$-0.539820\pi$
$17$	−1.02889			−0.249541			−0.124771	+	0.992186i	$0.539820\pi$
$18$		0			0
$19$		−	7.89515i		−	1.81127i	−0.424056	−	0.905636i	$-0.639394\pi$
$19$		−	7.89515i		−	1.81127i	0.424056	−	0.905636i	$-0.360606\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.0832686i			0.0173627i	0.999962	+	0.00868135i	$0.00276339\pi$
$23$			0.0832686i			0.0173627i	−0.999962	+	0.00868135i	$0.997237\pi$
$24$		0			0
$25$	−4.29982	+	2.55177i	−0.859965	+	0.510353i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.28803	+	3.25773i	−0.796267	+	0.604945i
$29$	−4.28803	+	3.25773i	−0.796267	+	0.604945i
$30$		0			0
$31$		−	2.02337i		−	0.363408i	−0.983353	−	0.181704i	$-0.941839\pi$
$31$		−	2.02337i		−	0.363408i	0.983353	−	0.181704i	$-0.0581612\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.81641	+	1.59596i	−0.983153	+	0.269766i
$36$		0			0
$37$	3.06250			0.503471			0.251736	−	0.967796i	$-0.418999\pi$
$37$	3.06250			0.503471			0.251736	+	0.967796i	$0.418999\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3.92969i		−	0.613715i	−0.951755	−	0.306857i	$-0.900723\pi$
$41$		−	3.92969i		−	0.613715i	0.951755	−	0.306857i	$-0.0992775\pi$
$42$		0			0
$43$	−7.15897			−1.09173			−0.545867	−	0.837872i	$-0.683799\pi$
$43$	−7.15897			−1.09173			−0.545867	+	0.837872i	$0.683799\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.554230			0.0808427			0.0404213	−	0.999183i	$-0.487130\pi$
$47$	0.554230			0.0808427			0.0404213	+	0.999183i	$0.487130\pi$
$48$		0			0
$49$	−0.275549			−0.0393641
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.25482i			0.859166i	0.903027	+	0.429583i	$0.141339\pi$
$53$			6.25482i			0.859166i	−0.903027	+	0.429583i	$0.858661\pi$
$54$		0			0
$55$	2.08815	−	0.572966i	0.281567	−	0.0772587i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.5363			−1.50190			−0.750952	−	0.660356i	$-0.770405\pi$
$59$	−11.5363			−1.50190			−0.750952	+	0.660356i	$0.770405\pi$
$60$		0			0
$61$		−	8.77259i		−	1.12321i	−0.827404	−	0.561607i	$-0.810183\pi$
$61$		−	8.77259i		−	1.12321i	0.827404	−	0.561607i	$-0.189817\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.46283	−	1.77333i	0.801615	−	0.219954i
$66$		0			0
$67$			2.40621i			0.293966i	0.989139	+	0.146983i	$0.0469562\pi$
$67$			2.40621i			0.293966i	−0.989139	+	0.146983i	$0.953044\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−11.1708			−1.32573			−0.662864	−	0.748740i	$-0.730659\pi$
$71$	−11.1708			−1.32573			−0.662864	+	0.748740i	$0.730659\pi$
$72$		0			0
$73$	7.48700			0.876287			0.438143	−	0.898905i	$-0.355636\pi$
$73$	7.48700			0.876287			0.438143	+	0.898905i	$0.355636\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.61200			0.297665
$78$		0			0
$79$		−	6.55123i		−	0.737071i	−0.929614	−	0.368535i	$-0.879859\pi$
$79$		−	6.55123i		−	0.737071i	0.929614	−	0.368535i	$-0.120141\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			7.10098i			0.779434i	0.920935	+	0.389717i	$0.127427\pi$
$83$			7.10098i			0.779434i	−0.920935	+	0.389717i	$0.872573\pi$
$84$		0			0
$85$	0.608773	+	2.21865i	0.0660307	+	0.240647i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			6.83268i			0.724263i	0.932127	+	0.362132i	$0.117951\pi$
$89$			6.83268i			0.724263i	−0.932127	+	0.362132i	$0.882049\pi$
$90$		0			0
$91$	8.08413			0.847447
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−17.0248	+	4.67142i	−1.74671	+	0.479278i
$96$		0			0
$97$	4.18357			0.424777			0.212388	−	0.977185i	$-0.431876\pi$
$97$	4.18357			0.424777			0.212388	+	0.977185i	$0.431876\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	2.96662i		−	0.295190i	−0.989048	−	0.147595i	$-0.952847\pi$
$101$		−	2.96662i		−	0.295190i	0.989048	−	0.147595i	$-0.0471531\pi$
$102$		0			0
$103$			1.64370i			0.161958i	0.996716	+	0.0809791i	$0.0258047\pi$
$103$			1.64370i			0.161958i	−0.996716	+	0.0809791i	$0.974195\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	7.01322i		−	0.677994i	−0.940787	−	0.338997i	$-0.889912\pi$
$107$		−	7.01322i		−	0.677994i	0.940787	−	0.338997i	$-0.110088\pi$
$108$		0			0
$109$	−15.9709			−1.52973			−0.764866	−	0.644189i	$-0.777195\pi$
$109$	−15.9709			−1.52973			−0.764866	+	0.644189i	$0.777195\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.86105			−0.927650			−0.463825	−	0.885927i	$-0.653523\pi$
$113$	−9.86105			−0.927650			−0.463825	+	0.885927i	$0.653523\pi$
$114$		0			0
$115$	0.179557	−	0.0492685i	0.0167438	−	0.00459432i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.77524i			0.254405i
$120$		0			0
$121$	10.0623			0.914751
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.04667	+	7.76216i	0.719716	+	0.694268i
$126$		0			0
$127$	12.8032			1.13610			0.568052	−	0.822992i	$-0.307697\pi$
$127$	12.8032			1.13610			0.568052	+	0.822992i	$0.307697\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			0.560603i			0.0489801i	0.999700	+	0.0244901i	$0.00779621\pi$
$131$			0.560603i			0.0489801i	−0.999700	+	0.0244901i	$0.992204\pi$
$132$		0			0
$133$	−21.2958			−1.84658
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−21.3923			−1.82766			−0.913832	−	0.406093i	$-0.866891\pi$
$137$	−21.3923			−1.82766			−0.913832	+	0.406093i	$0.866891\pi$
$138$		0			0
$139$	11.3948			0.966495			0.483248	−	0.875484i	$-0.339457\pi$
$139$	11.3948			0.966495			0.483248	+	0.875484i	$0.339457\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.90229			−0.242701
$144$		0			0
$145$	9.56201	+	7.31902i	0.794081	+	0.607811i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.08745			−0.662549			−0.331275	−	0.943534i	$-0.607479\pi$
$149$	−8.08745			−0.662549			−0.331275	+	0.943534i	$0.607479\pi$
$150$		0			0
$151$	6.69693			0.544988			0.272494	−	0.962157i	$-0.412151\pi$
$151$	6.69693			0.544988			0.272494	+	0.962157i	$0.412151\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.36312	+	1.19719i	−0.350454	+	0.0961608i
$156$		0			0
$157$	−8.31996			−0.664005			−0.332003	−	0.943278i	$-0.607724\pi$
$157$	−8.31996			−0.664005			−0.332003	+	0.943278i	$0.607724\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.224602			0.0177011
$162$		0			0
$163$	−7.34334			−0.575175			−0.287587	−	0.957754i	$-0.592853\pi$
$163$	−7.34334			−0.575175			−0.287587	+	0.957754i	$0.592853\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			12.7347i			0.985439i	0.870188	+	0.492720i	$0.163997\pi$
$167$			12.7347i			0.985439i	−0.870188	+	0.492720i	$0.836003\pi$
$168$		0			0
$169$	4.01744			0.309034
$170$		0			0
$171$		0			0
$172$		0			0
$173$			23.4796i			1.78512i	0.450927	+	0.892561i	$0.351093\pi$
$173$			23.4796i			1.78512i	−0.450927	+	0.892561i	$0.648907\pi$
$174$		0			0
$175$	6.88294	+	11.5980i	0.520301	+	0.876727i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.6460			1.09469			0.547346	−	0.836907i	$-0.315638\pi$
$179$	14.6460			1.09469			0.547346	+	0.836907i	$0.315638\pi$
$180$		0			0
$181$	0.680025			0.0505458			0.0252729	−	0.999681i	$-0.491955\pi$
$181$	0.680025			0.0505458			0.0252729	+	0.999681i	$0.491955\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.81203	−	6.60386i	−0.133223	−	0.485525i
$186$		0			0
$187$		−	0.996339i		−	0.0728595i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			7.03498i			0.509033i	0.967068	+	0.254516i	$0.0819163\pi$
$191$			7.03498i			0.509033i	−0.967068	+	0.254516i	$0.918084\pi$
$192$		0			0
$193$	−27.2083			−1.95850			−0.979248	−	0.202668i	$-0.935039\pi$
$193$	−27.2083			−1.95850			−0.979248	+	0.202668i	$0.935039\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			10.8772i			0.774971i	0.921876	+	0.387485i	$0.126656\pi$
$197$			10.8772i			0.774971i	−0.921876	+	0.387485i	$0.873344\pi$
$198$		0			0
$199$	−9.52278			−0.675052			−0.337526	−	0.941316i	$-0.609590\pi$
$199$	−9.52278			−0.675052			−0.337526	+	0.941316i	$0.609590\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.78715	+	11.5662i	0.616737	+	0.811788i
$204$		0			0
$205$	−8.47385	+	2.32513i	−0.591840	+	0.162394i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.64541			0.528844
$210$		0			0
$211$			5.67254i			0.390514i	0.980752	+	0.195257i	$0.0625541\pi$
$211$			5.67254i			0.390514i	−0.980752	+	0.195257i	$0.937446\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.23584	+	15.4374i	0.288882	+	1.05282i
$216$		0			0
$217$	−5.45768			−0.370491
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	3.08366i		−	0.207430i
$222$		0			0
$223$		−	3.05206i		−	0.204381i	−0.994765	−	0.102191i	$-0.967415\pi$
$223$		−	3.05206i		−	0.204381i	0.994765	−	0.102191i	$-0.0325851\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			3.05026i			0.202453i	0.994863	+	0.101227i	$0.0322767\pi$
$227$			3.05026i			0.202453i	−0.994863	+	0.101227i	$0.967723\pi$
$228$		0			0
$229$			16.3929i			1.08327i	0.840613	+	0.541637i	$0.182195\pi$
$229$			16.3929i			1.08327i	−0.840613	+	0.541637i	$0.817805\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			7.87789i			0.516098i	0.966132	+	0.258049i	$0.0830796\pi$
$233$			7.87789i			0.516098i	−0.966132	+	0.258049i	$0.916920\pi$
$234$		0			0
$235$	−0.327928	−	1.19512i	−0.0213917	−	0.0779611i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.0566089			0.00366173			0.00183086	−	0.999998i	$-0.499417\pi$
$239$	0.0566089			0.00366173			0.00183086	+	0.999998i	$0.499417\pi$
$240$		0			0
$241$	−6.32053			−0.407141			−0.203571	−	0.979060i	$-0.565255\pi$
$241$	−6.32053			−0.407141			−0.203571	+	0.979060i	$0.565255\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.163037	+	0.594184i	0.0104161	+	0.0379610i
$246$		0			0
$247$	23.6625			1.50561
$248$		0			0
$249$		0			0
$250$		0			0
$251$			12.2141i			0.770948i	0.922719	+	0.385474i	$0.125962\pi$
$251$			12.2141i			0.770948i	−0.922719	+	0.385474i	$0.874038\pi$
$252$		0			0
$253$	−0.0806346			−0.00506945
$254$		0			0
$255$		0			0
$256$		0			0
$257$			4.65163i			0.290161i	0.989420	+	0.145080i	$0.0463441\pi$
$257$			4.65163i			0.290161i	−0.989420	+	0.145080i	$0.953656\pi$
$258$		0			0
$259$		−	8.26054i		−	0.513285i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.47603			0.276004			0.138002	−	0.990432i	$-0.455932\pi$
$263$	4.47603			0.276004			0.138002	+	0.990432i	$0.455932\pi$
$264$		0			0
$265$	13.4877	−	3.70087i	0.828542	−	0.227343i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	26.0495i		−	1.58827i	−0.607743	−	0.794134i	$-0.707925\pi$
$269$		−	26.0495i		−	1.58827i	0.607743	−	0.794134i	$-0.292075\pi$
$270$		0			0
$271$			21.0732i			1.28011i	0.768331	+	0.640053i	$0.221088\pi$
$271$			21.0732i			1.28011i	−0.768331	+	0.640053i	$0.778912\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.47105	−	4.16381i	−0.149010	−	0.251087i
$276$		0			0
$277$			13.8437i			0.831786i	0.909414	+	0.415893i	$0.136531\pi$
$277$			13.8437i			0.831786i	−0.909414	+	0.415893i	$0.863469\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.777928			−0.0464073			−0.0232037	−	0.999731i	$-0.507387\pi$
$281$	−0.777928			−0.0464073			−0.0232037	+	0.999731i	$0.507387\pi$
$282$		0			0
$283$			19.5373i			1.16137i	0.814128	+	0.580685i	$0.197215\pi$
$283$			19.5373i			1.16137i	−0.814128	+	0.580685i	$0.802785\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.5996			−0.625677
$288$		0			0
$289$	−15.9414			−0.937729
$290$		0			0
$291$		0			0
$292$		0			0
$293$	29.7470			1.73784			0.868920	−	0.494953i	$-0.164815\pi$
$293$	29.7470			1.73784			0.868920	+	0.494953i	$0.164815\pi$
$294$		0			0
$295$	6.82585	+	24.8766i	0.397417	+	1.44837i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.249564			−0.0144326
$300$		0			0
$301$			19.3101i			1.11301i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−18.9169	+	5.19059i	−1.08318	+	0.297212i
$306$		0			0
$307$	−13.5448			−0.773041			−0.386520	−	0.922281i	$-0.626323\pi$
$307$	−13.5448			−0.773041			−0.386520	+	0.922281i	$0.626323\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	24.5544i		−	1.39235i	−0.717870	−	0.696177i	$-0.754883\pi$
$311$		−	24.5544i		−	1.39235i	0.717870	−	0.696177i	$-0.245117\pi$
$312$		0			0
$313$			13.0879i			0.739770i	0.929078	+	0.369885i	$0.120603\pi$
$313$			13.0879i			0.739770i	−0.929078	+	0.369885i	$0.879397\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.7472			−0.715953			−0.357977	−	0.933731i	$-0.616533\pi$
$317$	−12.7472			−0.715953			−0.357977	+	0.933731i	$0.616533\pi$
$318$		0			0
$319$	−3.15468	−	4.15239i	−0.176628	−	0.232489i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8.12321i			0.451987i
$324$		0			0
$325$	−7.64788	−	12.8870i	−0.424228	−	0.714840i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	1.49494i		−	0.0824185i
$330$		0			0
$331$			17.7465i			0.975437i	0.873001	+	0.487718i	$0.162171\pi$
$331$			17.7465i			0.975437i	−0.873001	+	0.487718i	$0.837829\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.18867	−	1.42371i	0.283487	−	0.0777858i
$336$		0			0
$337$	15.3425			0.835761			0.417881	−	0.908502i	$-0.362773\pi$
$337$	15.3425			0.835761			0.417881	+	0.908502i	$0.362773\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.95936			0.106106
$342$		0			0
$343$		−	18.1380i		−	0.979361i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	24.5489i		−	1.31785i	−0.752207	−	0.658926i	$-0.771011\pi$
$347$		−	24.5489i		−	1.31785i	0.752207	−	0.658926i	$-0.228989\pi$
$348$		0			0
$349$	−32.5782			−1.74387			−0.871936	−	0.489621i	$-0.837135\pi$
$349$	−32.5782			−1.74387			−0.871936	+	0.489621i	$0.837135\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			14.5351i			0.773627i	0.922158	+	0.386813i	$0.126424\pi$
$353$			14.5351i			0.773627i	−0.922158	+	0.386813i	$0.873576\pi$
$354$		0			0
$355$	6.60955	+	24.0883i	0.350799	+	1.27847i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	30.7334i		−	1.62204i	−0.585015	−	0.811022i	$-0.698911\pi$
$359$		−	30.7334i		−	1.62204i	0.585015	−	0.811022i	$-0.301089\pi$
$360$		0			0
$361$	−43.3334			−2.28071
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.42992	−	16.1447i	−0.231873	−	0.845052i
$366$		0			0
$367$	−23.7620			−1.24037			−0.620183	−	0.784457i	$-0.712942\pi$
$367$	−23.7620			−1.24037			−0.620183	+	0.784457i	$0.712942\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.8713			0.875913
$372$		0			0
$373$		−	27.9373i		−	1.44654i	−0.690566	−	0.723269i	$-0.742639\pi$
$373$		−	27.9373i		−	1.44654i	0.690566	−	0.723269i	$-0.257361\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9.76372	−	12.8516i	−0.502857	−	0.661892i
$378$		0			0
$379$		−	30.8404i		−	1.58416i	−0.610415	−	0.792082i	$-0.708997\pi$
$379$		−	30.8404i		−	1.58416i	0.610415	−	0.792082i	$-0.291003\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	16.1205i		−	0.823722i	−0.911247	−	0.411861i	$-0.864879\pi$
$383$		−	16.1205i		−	0.823722i	0.911247	−	0.411861i	$-0.135121\pi$
$384$		0			0
$385$	−1.54547	−	5.63243i	−0.0787647	−	0.287055i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	27.2842i		−	1.38337i	−0.722201	−	0.691683i	$-0.756870\pi$
$389$		−	27.2842i		−	1.38337i	0.722201	−	0.691683i	$-0.243130\pi$
$390$		0			0
$391$		−	0.0856738i		−	0.00433271i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−14.1268	+	3.87625i	−0.710798	+	0.195035i
$396$		0			0
$397$			6.33045i			0.317716i	0.987301	+	0.158858i	$0.0507813\pi$
$397$			6.33045i			0.317716i	−0.987301	+	0.158858i	$0.949219\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.40233			−0.469530			−0.234765	−	0.972052i	$-0.575432\pi$
$401$	−9.40233			−0.469530			−0.234765	+	0.972052i	$0.575432\pi$
$402$		0			0
$403$	6.06422			0.302080
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.96562i			0.147000i
$408$		0			0
$409$			11.6998i			0.578519i	0.957251	+	0.289260i	$0.0934091\pi$
$409$			11.6998i			0.578519i	−0.957251	+	0.289260i	$0.906591\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			31.1173i			1.53118i
$414$		0			0
$415$	15.3123	−	4.20153i	0.751652	−	0.206245i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−19.8844			−0.971417			−0.485709	−	0.874121i	$-0.661438\pi$
$419$	−19.8844			−0.971417			−0.485709	+	0.874121i	$0.661438\pi$
$420$		0			0
$421$		−	2.46576i		−	0.120174i	−0.998193	−	0.0600869i	$-0.980862\pi$
$421$		−	2.46576i		−	0.120174i	0.998193	−	0.0600869i	$-0.0191378\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.42402	−	2.62548i	0.214597	−	0.127354i
$426$		0			0
$427$	−23.6625			−1.14511
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.4340			−0.502586			−0.251293	−	0.967911i	$-0.580856\pi$
$431$	−10.4340			−0.502586			−0.251293	+	0.967911i	$0.580856\pi$
$432$		0			0
$433$	−18.9489			−0.910626			−0.455313	−	0.890331i	$-0.650473\pi$
$433$	−18.9489			−0.910626			−0.455313	+	0.890331i	$0.650473\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.657418			0.0314486
$438$		0			0
$439$	−26.0344			−1.24255			−0.621276	−	0.783592i	$-0.713386\pi$
$439$	−26.0344			−1.24255			−0.621276	+	0.783592i	$0.713386\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−26.3026			−1.24968			−0.624838	−	0.780755i	$-0.714835\pi$
$443$	−26.3026			−1.24968			−0.624838	+	0.780755i	$0.714835\pi$
$444$		0			0
$445$	14.7338	−	4.04278i	0.698447	−	0.191646i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	20.7531i		−	0.979398i	−0.871892	−	0.489699i	$-0.837107\pi$
$449$		−	20.7531i		−	0.979398i	0.871892	−	0.489699i	$-0.162893\pi$
$450$		0			0
$451$	3.80539			0.179189
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.78323	−	17.4323i	−0.224241	−	0.817240i
$456$		0			0
$457$			14.7395i			0.689485i	0.938697	+	0.344743i	$0.112034\pi$
$457$			14.7395i			0.689485i	−0.938697	+	0.344743i	$0.887966\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	20.7265i		−	0.965329i	−0.875805	−	0.482665i	$-0.839669\pi$
$461$		−	20.7265i		−	0.965329i	0.875805	−	0.482665i	$-0.160331\pi$
$462$		0			0
$463$		−	39.3103i		−	1.82691i	−0.406945	−	0.913453i	$-0.633406\pi$
$463$		−	39.3103i		−	1.82691i	0.406945	−	0.913453i	$-0.366594\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.8091			0.916657			0.458329	−	0.888783i	$-0.348448\pi$
$467$	19.8091			0.916657			0.458329	+	0.888783i	$0.348448\pi$
$468$		0			0
$469$	6.49033			0.299696
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	6.93252i		−	0.318758i
$474$		0			0
$475$	20.1466	+	33.9478i	0.924389	+	1.55763i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			19.6122i			0.896104i	0.894008	+	0.448052i	$0.147882\pi$
$479$			19.6122i			0.896104i	−0.894008	+	0.448052i	$0.852118\pi$
$480$		0			0
$481$			9.17858i			0.418507i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.47534	−	9.02130i	−0.112399	−	0.409636i
$486$		0			0
$487$		−	24.6810i		−	1.11840i	−0.829032	−	0.559202i	$-0.811108\pi$
$487$		−	24.6810i		−	1.11840i	0.829032	−	0.559202i	$-0.188892\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			8.69557i			0.392426i	0.980561	+	0.196213i	$0.0628644\pi$
$491$			8.69557i			0.392426i	−0.980561	+	0.196213i	$0.937136\pi$
$492$		0			0
$493$	4.41189	−	3.35183i	0.198702	−	0.150959i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			30.1312i			1.35157i
$498$		0			0
$499$	−12.1525			−0.544020			−0.272010	−	0.962294i	$-0.587688\pi$
$499$	−12.1525			−0.544020			−0.272010	+	0.962294i	$0.587688\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21.9465			0.978545			0.489273	−	0.872131i	$-0.337262\pi$
$503$	21.9465			0.978545			0.489273	+	0.872131i	$0.337262\pi$
$504$		0			0
$505$	−6.39711	+	1.75530i	−0.284668	+	0.0781096i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−22.3545			−0.990848			−0.495424	−	0.868651i	$-0.664987\pi$
$509$	−22.3545			−0.990848			−0.495424	+	0.868651i	$0.664987\pi$
$510$		0			0
$511$		−	20.1948i		−	0.893367i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.54441	−	0.972546i	0.156185	−	0.0428555i
$516$		0			0
$517$			0.536698i			0.0236040i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.33761			−0.146223			−0.0731117	−	0.997324i	$-0.523293\pi$
$521$	−3.33761			−0.146223			−0.0731117	+	0.997324i	$0.523293\pi$
$522$		0			0
$523$		−	35.3198i		−	1.54443i	−0.635363	−	0.772213i	$-0.719150\pi$
$523$		−	35.3198i		−	1.54443i	0.635363	−	0.772213i	$-0.280850\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			2.08181i			0.0906853i
$528$		0			0
$529$	22.9931			0.999699
$530$		0			0
$531$		0			0
$532$		0			0
$533$	11.7777			0.510147
$534$		0			0
$535$	−15.1231	+	4.14960i	−0.653827	+	0.179403i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	0.266833i		−	0.0114933i
$540$		0			0
$541$		−	36.1020i		−	1.55214i	−0.630644	−	0.776072i	$-0.717209\pi$
$541$		−	36.1020i		−	1.55214i	0.630644	−	0.776072i	$-0.282791\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.44968	+	34.4390i	0.404780	+	1.47521i
$546$		0			0
$547$			24.7415i			1.05787i	0.848662	+	0.528936i	$0.177409\pi$
$547$			24.7415i			1.05787i	−0.848662	+	0.528936i	$0.822591\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	25.7203	+	33.8546i	1.09572	+	1.44226i
$552$		0			0
$553$	−17.6708			−0.751438
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	2.37690i		−	0.100712i	−0.998731	−	0.0503562i	$-0.983964\pi$
$557$		−	2.37690i		−	0.100712i	0.998731	−	0.0503562i	$-0.0160357\pi$
$558$		0			0
$559$		−	21.4561i		−	0.907497i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	37.0866			1.56302			0.781508	−	0.623896i	$-0.214451\pi$
$563$	37.0866			1.56302			0.781508	+	0.623896i	$0.214451\pi$
$564$		0			0
$565$	5.83461	+	21.2640i	0.245464	+	0.894584i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			5.73356i			0.240363i	0.992752	+	0.120182i	$0.0383477\pi$
$569$			5.73356i			0.240363i	−0.992752	+	0.120182i	$0.961652\pi$
$570$		0			0
$571$	−12.4614			−0.521495			−0.260747	−	0.965407i	$-0.583969\pi$
$571$	−12.4614			−0.521495			−0.260747	+	0.965407i	$0.583969\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.212482	−	0.358040i	−0.00886111	−	0.0149313i
$576$		0			0
$577$	32.0421			1.33393			0.666964	−	0.745090i	$-0.267593\pi$
$577$	32.0421			1.33393			0.666964	+	0.745090i	$0.267593\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	19.1536			0.794627
$582$		0			0
$583$	−6.05697			−0.250854
$584$		0			0
$585$		0			0
$586$		0			0
$587$			23.8050i			0.982538i	0.871008	+	0.491269i	$0.163467\pi$
$587$			23.8050i			0.982538i	−0.871008	+	0.491269i	$0.836533\pi$
$588$		0			0
$589$	−15.9748			−0.658230
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9.48683i			0.389577i	0.980845	+	0.194789i	$0.0624021\pi$
$593$			9.48683i			0.389577i	−0.980845	+	0.194789i	$0.937598\pi$
$594$		0			0
$595$	5.98442	−	1.64206i	0.245337	−	0.0673178i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			3.68070i			0.150389i	0.997169	+	0.0751947i	$0.0239578\pi$
$599$			3.68070i			0.150389i	−0.997169	+	0.0751947i	$0.976042\pi$
$600$		0			0
$601$			20.0281i			0.816962i	0.912767	+	0.408481i	$0.133941\pi$
$601$			20.0281i			0.816962i	−0.912767	+	0.408481i	$0.866059\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.95366	−	21.6979i	−0.242051	−	0.882146i
$606$		0			0
$607$	6.19497			0.251446			0.125723	−	0.992065i	$-0.459875\pi$
$607$	6.19497			0.251446			0.125723	+	0.992065i	$0.459875\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1.66108i			0.0672000i
$612$		0			0
$613$			15.8284i			0.639302i	0.947535	+	0.319651i	$0.103566\pi$
$613$			15.8284i			0.639302i	−0.947535	+	0.319651i	$0.896434\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.09084			−0.325725			−0.162862	−	0.986649i	$-0.552073\pi$
$617$	−8.09084			−0.325725			−0.162862	+	0.986649i	$0.552073\pi$
$618$		0			0
$619$		−	25.5105i		−	1.02535i	−0.858582	−	0.512677i	$-0.828654\pi$
$619$		−	25.5105i		−	1.02535i	0.858582	−	0.512677i	$-0.171346\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.4300			0.738381
$624$		0			0
$625$	11.9770	−	21.9443i	0.479079	−	0.877772i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.15096			−0.125637
$630$		0			0
$631$	26.7233			1.06384			0.531918	−	0.846796i	$-0.321471\pi$
$631$	26.7233			1.06384			0.531918	+	0.846796i	$0.321471\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.57546	−	27.6085i	−0.300623	−	1.09561i
$636$		0			0
$637$		−	0.825845i		−	0.0327212i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			7.84397i			0.309818i	0.987929	+	0.154909i	$0.0495085\pi$
$641$			7.84397i			0.309818i	−0.987929	+	0.154909i	$0.950492\pi$
$642$		0			0
$643$		−	0.673789i		−	0.0265716i	−0.999912	−	0.0132858i	$-0.995771\pi$
$643$		−	0.673789i		−	0.0265716i	0.999912	−	0.0132858i	$-0.00422913\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	5.32492i		−	0.209344i	−0.994507	−	0.104672i	$-0.966621\pi$
$647$		−	5.32492i		−	0.209344i	0.994507	−	0.104672i	$-0.0333793\pi$
$648$		0			0
$649$		−	11.1714i		−	0.438517i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.7861			−0.422092			−0.211046	−	0.977476i	$-0.567687\pi$
$653$	−10.7861			−0.422092			−0.211046	+	0.977476i	$0.567687\pi$
$654$		0			0
$655$	1.20887	−	0.331699i	0.0472343	−	0.0129606i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			35.9464i			1.40027i	0.714009	+	0.700137i	$0.246878\pi$
$659$			35.9464i			1.40027i	−0.714009	+	0.700137i	$0.753122\pi$
$660$		0			0
$661$	20.6119			0.801711			0.400855	−	0.916141i	$-0.368713\pi$
$661$	20.6119			0.801711			0.400855	+	0.916141i	$0.368713\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.6003	+	45.9215i	0.488620	+	1.78076i
$666$		0			0
$667$	−0.271266	−	0.357058i	−0.0105035	−	0.0138253i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8.49509			0.327949
$672$		0			0
$673$			6.76765i			0.260874i	0.991457	+	0.130437i	$0.0416380\pi$
$673$			6.76765i			0.260874i	−0.991457	+	0.130437i	$0.958362\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.43219			−0.170343			−0.0851715	−	0.996366i	$-0.527144\pi$
$677$	−4.43219			−0.170343			−0.0851715	+	0.996366i	$0.527144\pi$
$678$		0			0
$679$		−	11.2844i		−	0.433056i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	37.8482i		−	1.44822i	−0.689684	−	0.724110i	$-0.742251\pi$
$683$		−	37.8482i		−	1.44822i	0.689684	−	0.724110i	$-0.257749\pi$
$684$		0			0
$685$	12.6574	+	46.1295i	0.483615	+	1.76252i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.7463			−0.714176
$690$		0			0
$691$	29.2078			1.11112			0.555559	−	0.831477i	$-0.312504\pi$
$691$	29.2078			1.11112			0.555559	+	0.831477i	$0.312504\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.74211	−	24.5714i	−0.255743	−	0.932046i
$696$		0			0
$697$			4.04320i			0.153147i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.7710			0.708969			0.354485	−	0.935062i	$-0.384656\pi$
$701$	18.7710			0.708969			0.354485	+	0.935062i	$0.384656\pi$
$702$		0			0
$703$		−	24.1789i		−	0.911923i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.00193			−0.300943
$708$		0			0
$709$	−36.7978			−1.38197			−0.690986	−	0.722868i	$-0.742823\pi$
$709$	−36.7978			−1.38197			−0.690986	+	0.722868i	$0.742823\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.168483			0.00630974
$714$		0			0
$715$	1.71723	+	6.25839i	0.0642209	+	0.234051i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.7934			0.850051			0.425025	−	0.905181i	$-0.360265\pi$
$719$	22.7934			0.850051			0.425025	+	0.905181i	$0.360265\pi$
$720$		0			0
$721$	4.43358			0.165115
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10.1248	−	24.9497i	0.376026	−	0.926609i
$726$		0			0
$727$	12.4337			0.461142			0.230571	−	0.973056i	$-0.425941\pi$
$727$	12.4337			0.461142			0.230571	+	0.973056i	$0.425941\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.36576			0.272433
$732$		0			0
$733$	1.40998			0.0520789			0.0260395	−	0.999661i	$-0.491710\pi$
$733$	1.40998			0.0520789			0.0260395	+	0.999661i	$0.491710\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.33010			−0.0858302
$738$		0			0
$739$			44.1174i			1.62289i	0.584432	+	0.811443i	$0.301317\pi$
$739$			44.1174i			1.62289i	−0.584432	+	0.811443i	$0.698683\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.86033			0.214995			0.107497	−	0.994205i	$-0.465716\pi$
$743$	5.86033			0.214995			0.107497	+	0.994205i	$0.465716\pi$
$744$		0			0
$745$	4.78520	+	17.4395i	0.175316	+	0.638933i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.9169			−0.691209
$750$		0			0
$751$		−	11.7992i		−	0.430559i	−0.976552	−	0.215280i	$-0.930934\pi$
$751$		−	11.7992i		−	0.430559i	0.976552	−	0.215280i	$-0.0690663\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.96245	−	14.4410i	−0.144208	−	0.525563i
$756$		0			0
$757$	0.997808			0.0362660			0.0181330	−	0.999836i	$-0.494228\pi$
$757$	0.997808			0.0362660			0.0181330	+	0.999836i	$0.494228\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.47998			0.234899			0.117450	−	0.993079i	$-0.462528\pi$
$761$	6.47998			0.234899			0.117450	+	0.993079i	$0.462528\pi$
$762$		0			0
$763$			43.0786i			1.55955i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	34.5755i		−	1.24845i
$768$		0			0
$769$		−	31.5502i		−	1.13773i	−0.822431	−	0.568864i	$-0.807383\pi$
$769$		−	31.5502i		−	1.13773i	0.822431	−	0.568864i	$-0.192617\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−31.7747			−1.14286			−0.571428	−	0.820652i	$-0.693610\pi$
$773$	−31.7747			−1.14286			−0.571428	+	0.820652i	$0.693610\pi$
$774$		0			0
$775$	5.16317	+	8.70013i	0.185466	+	0.312518i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−31.0255			−1.11160
$780$		0			0
$781$		−	10.8174i		−	0.387078i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.92278	+	17.9409i	0.175701	+	0.640338i
$786$		0			0
$787$		−	15.6567i		−	0.558100i	−0.960277	−	0.279050i	$-0.909980\pi$
$787$		−	15.6567i		−	0.558100i	0.960277	−	0.279050i	$-0.0900196\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			26.5984i			0.945731i
$792$		0			0
$793$	26.2923			0.933666
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−55.4594			−1.96447			−0.982237	−	0.187647i	$-0.939914\pi$
$797$	−55.4594			−1.96447			−0.982237	+	0.187647i	$0.939914\pi$
$798$		0			0
$799$	−0.570239			−0.0201736
$800$		0			0
$801$		0			0
$802$		0			0
$803$			7.25016i			0.255853i
$804$		0			0
$805$	−0.132893	−	0.484324i	−0.00468387	−	0.0170702i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			5.45022i			0.191619i	0.995400	+	0.0958097i	$0.0305440\pi$
$809$			5.45022i			0.191619i	−0.995400	+	0.0958097i	$0.969456\pi$
$810$		0			0
$811$	2.51950			0.0884717			0.0442359	−	0.999021i	$-0.485915\pi$
$811$	2.51950			0.0884717			0.0442359	+	0.999021i	$0.485915\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.34493	+	15.8349i	0.152196	+	0.554673i
$816$		0			0
$817$			56.5212i			1.97743i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.04758			0.280862			0.140431	−	0.990090i	$-0.455151\pi$
$821$	8.04758			0.280862			0.140431	+	0.990090i	$0.455151\pi$
$822$		0			0
$823$	41.5365			1.44787			0.723936	−	0.689867i	$-0.242331\pi$
$823$	41.5365			1.44787			0.723936	+	0.689867i	$0.242331\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.3771			0.569487			0.284744	−	0.958604i	$-0.408092\pi$
$827$	16.3771			0.569487			0.284744	+	0.958604i	$0.408092\pi$
$828$		0			0
$829$		−	7.41324i		−	0.257472i	−0.991679	−	0.128736i	$-0.958908\pi$
$829$		−	7.41324i		−	0.257472i	0.991679	−	0.128736i	$-0.0410920\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.283508			0.00982297
$834$		0			0
$835$	27.4606	−	7.53489i	0.950314	−	0.260756i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	34.0179i		−	1.17443i	−0.809431	−	0.587214i	$-0.800225\pi$
$839$		−	34.0179i		−	1.17443i	0.809431	−	0.587214i	$-0.199775\pi$
$840$		0			0
$841$	7.77439	−	27.9385i	0.268083	−	0.963396i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.37705	−	8.66306i	−0.0817729	−	0.298018i
$846$		0			0
$847$		−	27.1412i		−	0.932582i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			0.255010i			0.00874162i
$852$		0			0
$853$	−52.1341			−1.78504			−0.892519	−	0.451011i	$-0.851064\pi$
$853$	−52.1341			−1.78504			−0.892519	+	0.451011i	$0.851064\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	37.8222i		−	1.29198i	−0.763346	−	0.645990i	$-0.776445\pi$
$857$		−	37.8222i		−	1.29198i	0.763346	−	0.645990i	$-0.223555\pi$
$858$		0			0
$859$		−	26.6791i		−	0.910281i	−0.890420	−	0.455140i	$-0.849589\pi$
$859$		−	26.6791i		−	0.910281i	0.890420	−	0.455140i	$-0.150411\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	34.0283i		−	1.15834i	−0.815208	−	0.579168i	$-0.803378\pi$
$863$		−	34.0283i		−	1.15834i	0.815208	−	0.579168i	$-0.196622\pi$
$864$		0			0
$865$	50.6306	−	13.8925i	1.72149	−	0.472358i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6.34400			0.215205
$870$		0			0
$871$	−7.21164			−0.244357
$872$		0			0
$873$		0			0
$874$		0			0
$875$	20.9370	−	21.7045i	0.707801	−	0.733745i
$876$		0			0
$877$			40.7864i			1.37726i	0.725114	+	0.688629i	$0.241787\pi$
$877$			40.7864i			1.37726i	−0.725114	+	0.688629i	$0.758213\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			4.93942i			0.166413i	0.996532	+	0.0832066i	$0.0265161\pi$
$881$			4.93942i			0.166413i	−0.996532	+	0.0832066i	$0.973484\pi$
$882$		0			0
$883$			1.88143i			0.0633151i	0.999499	+	0.0316575i	$0.0100786\pi$
$883$			1.88143i			0.0633151i	−0.999499	+	0.0316575i	$0.989921\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	54.3735			1.82568			0.912842	−	0.408312i	$-0.133883\pi$
$887$	54.3735			1.82568			0.912842	+	0.408312i	$0.133883\pi$
$888$		0			0
$889$		−	34.5345i		−	1.15825i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	4.37573i		−	0.146428i
$894$		0			0
$895$	−8.66576	−	31.5821i	−0.289664	−	1.05567i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.59159	+	8.67627i	0.219842	+	0.289370i
$900$		0			0
$901$		−	6.43549i		−	0.214397i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.402359	−	1.46638i	−0.0133749	−	0.0487442i
$906$		0			0
$907$	−0.482567			−0.0160234			−0.00801169	−	0.999968i	$-0.502550\pi$
$907$	−0.482567			−0.0160234			−0.00801169	+	0.999968i	$0.502550\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	26.5859i		−	0.880831i	−0.897794	−	0.440416i	$-0.854831\pi$
$911$		−	26.5859i		−	0.880831i	0.897794	−	0.440416i	$-0.145169\pi$
$912$		0			0
$913$	−6.87636			−0.227574
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.51213			0.0499348
$918$		0			0
$919$	−7.53009			−0.248395			−0.124197	−	0.992258i	$-0.539636\pi$
$919$	−7.53009			−0.248395			−0.124197	+	0.992258i	$0.539636\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	33.4799i		−	1.10200i
$924$		0			0
$925$	−13.1682	+	7.81478i	−0.432968	+	0.256948i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−40.7004			−1.33534			−0.667669	−	0.744458i	$-0.732708\pi$
$929$	−40.7004			−1.33534			−0.667669	+	0.744458i	$0.732708\pi$
$930$		0			0
$931$			2.17550i			0.0712991i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.14847	+	0.589516i	−0.0702625	+	0.0192792i
$936$		0			0
$937$		−	16.7717i		−	0.547908i	−0.961743	−	0.273954i	$-0.911668\pi$
$937$		−	16.7717i		−	0.547908i	0.961743	−	0.273954i	$-0.0883316\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−44.2670			−1.44306			−0.721531	−	0.692382i	$-0.756561\pi$
$941$	−44.2670			−1.44306			−0.721531	+	0.692382i	$0.756561\pi$
$942$		0			0
$943$	0.327220			0.0106557
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.91810			0.257304			0.128652	−	0.991690i	$-0.458935\pi$
$947$	7.91810			0.257304			0.128652	+	0.991690i	$0.458935\pi$
$948$		0			0
$949$			22.4392i			0.728408i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			32.1276i			1.04072i	0.853948	+	0.520358i	$0.174201\pi$
$953$			32.1276i			1.04072i	−0.853948	+	0.520358i	$0.825799\pi$
$954$		0			0
$955$	15.1700	−	4.16247i	0.490889	−	0.134694i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			57.7018i			1.86329i
$960$		0			0
$961$	26.9060			0.867935
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.0987	+	58.6710i	0.518234	+	1.88869i
$966$		0			0
$967$	−32.7437			−1.05296			−0.526482	−	0.850186i	$-0.676489\pi$
$967$	−32.7437			−1.05296			−0.526482	+	0.850186i	$0.676489\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	8.19697i		−	0.263053i	−0.991313	−	0.131527i	$-0.958012\pi$
$971$		−	8.19697i		−	0.263053i	0.991313	−	0.131527i	$-0.0419879\pi$
$972$		0			0
$973$		−	30.7355i		−	0.985334i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	7.55744i		−	0.241784i	−0.992666	−	0.120892i	$-0.961425\pi$
$977$		−	7.55744i		−	0.241784i	0.992666	−	0.120892i	$-0.0385755\pi$
$978$		0			0
$979$	−6.61655			−0.211466
$980$		0			0
$981$		0			0
$982$		0			0
$983$	32.0276			1.02152			0.510761	−	0.859723i	$-0.329364\pi$
$983$	32.0276			1.02152			0.510761	+	0.859723i	$0.329364\pi$
$984$		0			0
$985$	23.4553	−	6.43587i	0.747348	−	0.205064i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	0.596117i		−	0.0189554i
$990$		0			0
$991$	52.0196			1.65246			0.826229	−	0.563335i	$-0.190482\pi$
$991$	52.0196			1.65246			0.826229	+	0.563335i	$0.190482\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.63446	+	20.5346i	0.178624	+	0.650991i
$996$		0			0
$997$	−54.1997			−1.71652			−0.858261	−	0.513213i	$-0.828455\pi$
$997$	−54.1997			−1.71652			−0.858261	+	0.513213i	$0.828455\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.b.d.289.7		14
3.2	odd	2		1740.2.b.a.289.8	yes	14
5.4	even	2		5220.2.b.c.289.8		14
15.2	even	4		8700.2.l.j.6901.20		28
15.8	even	4		8700.2.l.j.6901.17		28
15.14	odd	2		1740.2.b.b.289.7	yes	14
29.28	even	2		5220.2.b.c.289.7		14
87.86	odd	2		1740.2.b.b.289.8	yes	14
145.144	even	2	inner	5220.2.b.d.289.8		14
435.173	even	4		8700.2.l.j.6901.19		28
435.347	even	4		8700.2.l.j.6901.18		28
435.434	odd	2		1740.2.b.a.289.7	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1740.2.b.a.289.7	✓	14	435.434	odd	2
1740.2.b.a.289.8	yes	14	3.2	odd	2
1740.2.b.b.289.7	yes	14	15.14	odd	2
1740.2.b.b.289.8	yes	14	87.86	odd	2
5220.2.b.c.289.7		14	29.28	even	2
5220.2.b.c.289.8		14	5.4	even	2
5220.2.b.d.289.7		14	1.1	even	1	trivial
5220.2.b.d.289.8		14	145.144	even	2	inner
8700.2.l.j.6901.17		28	15.8	even	4
8700.2.l.j.6901.18		28	435.347	even	4
8700.2.l.j.6901.19		28	435.173	even	4
8700.2.l.j.6901.20		28	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.591682 - 2.15637i) q^{5} -2.69732i q^{7} +0.968368i q^{11} +2.99709i q^{13} -1.02889 q^{17} -7.89515i q^{19} +0.0832686i q^{23} +(-4.29982 + 2.55177i) q^{25} +(-4.28803 + 3.25773i) q^{29} -2.02337i q^{31} +(-5.81641 + 1.59596i) q^{35} +3.06250 q^{37} -3.92969i q^{41} -7.15897 q^{43} +0.554230 q^{47} -0.275549 q^{49} +6.25482i q^{53} +(2.08815 - 0.572966i) q^{55} -11.5363 q^{59} -8.77259i q^{61} +(6.46283 - 1.77333i) q^{65} +2.40621i q^{67} -11.1708 q^{71} +7.48700 q^{73} +2.61200 q^{77} -6.55123i q^{79} +7.10098i q^{83} +(0.608773 + 2.21865i) q^{85} +6.83268i q^{89} +8.08413 q^{91} +(-17.0248 + 4.67142i) q^{95} +4.18357 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{5} + 4 q^{17} - 6 q^{25} - 8 q^{29} + 18 q^{35} - 14 q^{37} + 6 q^{43} - 16 q^{47} - 14 q^{49} - 8 q^{55} - 12 q^{59} - 10 q^{65} - 4 q^{71} - 14 q^{73} - 24 q^{77} - 14 q^{85} + 16 q^{91} - 8 q^{95}+ \cdots + 6 q^{97}+O(q^{100}) \) 14 * q - 2 * q^5 + 4 * q^17 - 6 * q^25 - 8 * q^29 + 18 * q^35 - 14 * q^37 + 6 * q^43 - 16 * q^47 - 14 * q^49 - 8 * q^55 - 12 * q^59 - 10 * q^65 - 4 * q^71 - 14 * q^73 - 24 * q^77 - 14 * q^85 + 16 * q^91 - 8 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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