LMFDB - Embedded newform 1008.3.bk.b.143.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(143,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 3, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.143");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.bk (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 44 x^{18} + 1263 x^{16} - 20896 x^{14} + 250941 x^{12} - 2032238 x^{10} + 12100168 x^{8} + \cdots + 5184 $ x^20 - 44x^18 + 1263x^16 - 20896x^14 + 250941x^12 - 2032238x^10 + 12100168x^8 - 44511576x^6 + 104461920x^4 - 736992*x^2 + 5184
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			143.5
Root			$-0.0727418 - 0.0419975i$ of defining polynomial
Character	$\chi$	$=$	1008.143
Dual form			1008.3.bk.b.719.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0727418 - 0.125993i) q^{5} +(-5.11533 + 4.77843i) q^{7} +O(q^{10})$	$q+(-0.0727418 - 0.125993i) q^{5} +(-5.11533 + 4.77843i) q^{7} +(6.19935 - 10.7376i) q^{11} +5.42227i q^{13} +(0.593630 - 1.02820i) q^{17} +(-6.87409 - 11.9063i) q^{19} +(-0.448147 - 0.776213i) q^{23} +(12.4894 - 21.6323i) q^{25} +44.8846i q^{29} +(12.8917 - 22.3291i) q^{31} +(0.974145 + 0.296902i) q^{35} +(4.59810 + 7.96414i) q^{37} +53.1800 q^{41} +13.7583i q^{43} +(13.3530 - 7.70937i) q^{47} +(3.33319 - 48.8865i) q^{49} +(53.4909 + 30.8830i) q^{53} -1.80381 q^{55} +(38.0324 + 21.9580i) q^{59} +(81.6621 - 47.1476i) q^{61} +(0.683166 - 0.394426i) q^{65} +(-52.0398 - 30.0452i) q^{67} +76.2540 q^{71} +(41.4352 + 23.9226i) q^{73} +(19.5971 + 84.5495i) q^{77} +(-6.93118 + 4.00172i) q^{79} -150.217i q^{83} -0.172727 q^{85} +(56.2159 + 97.3688i) q^{89} +(-25.9100 - 27.7367i) q^{91} +(-1.00007 + 1.73217i) q^{95} -9.78971i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 10 q^{7}+O(q^{10}) $ 20 * q + 10 * q^7	$ 20 q + 10 q^{7} + 14 q^{19} - 14 q^{25} + 22 q^{31} - 10 q^{37} + 158 q^{49} - 240 q^{55} - 180 q^{61} + 318 q^{67} - 246 q^{73} + 66 q^{79} + 144 q^{85} - 42 q^{91}+O(q^{100}) $ 20 * q + 10 * q^7 + 14 * q^19 - 14 * q^25 + 22 * q^31 - 10 * q^37 + 158 * q^49 - 240 * q^55 - 180 * q^61 + 318 * q^67 - 246 * q^73 + 66 * q^79 + 144 * q^85 - 42 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$	$-1$

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
$5$	−0.0727418	−	0.125993i	−0.0145484	−	0.0251985i	0.858660	−	0.512546i	$-0.171298\pi$
$5$	−0.0727418	−	0.125993i	−0.0145484	−	0.0251985i	−0.873208	+	0.487348i	$0.837964\pi$
$6$		0			0
$7$	−5.11533	+	4.77843i	−0.730761	+	0.682633i
$7$	−5.11533	+	4.77843i	−0.730761	+	0.682633i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.19935	−	10.7376i	0.563577	−	0.976144i	−0.433603	−	0.901104i	$-0.642758\pi$
$11$	6.19935	−	10.7376i	0.563577	−	0.976144i	0.997180	−	0.0750405i	$-0.0239086\pi$
$12$		0			0
$13$			5.42227i			0.417098i	0.978012	+	0.208549i	$0.0668741\pi$
$13$			5.42227i			0.417098i	−0.978012	+	0.208549i	$0.933126\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.593630	−	1.02820i	0.0349194	−	0.0604822i	−0.848038	−	0.529936i	$-0.822216\pi$
$17$	0.593630	−	1.02820i	0.0349194	−	0.0604822i	0.882957	+	0.469454i	$0.155549\pi$
$18$		0			0
$19$	−6.87409	−	11.9063i	−0.361794	−	0.626646i	0.626462	−	0.779452i	$-0.284502\pi$
$19$	−6.87409	−	11.9063i	−0.361794	−	0.626646i	−0.988256	+	0.152806i	$0.951169\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.448147	−	0.776213i	−0.0194846	−	0.0337484i	0.856119	−	0.516779i	$-0.172869\pi$
$23$	−0.448147	−	0.776213i	−0.0194846	−	0.0337484i	−0.875603	+	0.483031i	$0.839536\pi$
$24$		0			0
$25$	12.4894	−	21.6323i	0.499577	−	0.865292i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			44.8846i			1.54775i	0.633341	+	0.773873i	$0.281683\pi$
$29$			44.8846i			1.54775i	−0.633341	+	0.773873i	$0.718317\pi$
$30$		0			0
$31$	12.8917	−	22.3291i	0.415862	−	0.720295i	−0.579656	−	0.814861i	$-0.696813\pi$
$31$	12.8917	−	22.3291i	0.415862	−	0.720295i	0.995519	+	0.0945664i	$0.0301465\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.974145	+	0.296902i	0.0278327	+	0.00848290i
$36$		0			0
$37$	4.59810	+	7.96414i	0.124273	+	0.215247i	0.921448	−	0.388501i	$-0.127007\pi$
$37$	4.59810	+	7.96414i	0.124273	+	0.215247i	−0.797176	+	0.603747i	$0.793674\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	53.1800			1.29707			0.648537	−	0.761183i	$-0.275381\pi$
$41$	53.1800			1.29707			0.648537	+	0.761183i	$0.275381\pi$
$42$		0			0
$43$			13.7583i			0.319960i	0.987120	+	0.159980i	$0.0511430\pi$
$43$			13.7583i			0.319960i	−0.987120	+	0.159980i	$0.948857\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	13.3530	−	7.70937i	0.284107	−	0.164029i	−0.351174	−	0.936310i	$-0.614218\pi$
$47$	13.3530	−	7.70937i	0.284107	−	0.164029i	0.635281	+	0.772281i	$0.280884\pi$
$48$		0			0
$49$	3.33319	−	48.8865i	0.0680242	−	0.997684i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	53.4909	+	30.8830i	1.00926	+	0.582698i	0.910975	−	0.412461i	$-0.135331\pi$
$53$	53.4909	+	30.8830i	1.00926	+	0.582698i	0.0982863	+	0.995158i	$0.468664\pi$
$54$		0			0
$55$	−1.80381			−0.0327965
$56$		0			0
$57$		0			0
$58$		0			0
$59$	38.0324	+	21.9580i	0.644617	+	0.372170i	0.786391	−	0.617730i	$-0.211947\pi$
$59$	38.0324	+	21.9580i	0.644617	+	0.372170i	−0.141774	+	0.989899i	$0.545281\pi$
$60$		0			0
$61$	81.6621	−	47.1476i	1.33872	−	0.772912i	0.352105	−	0.935961i	$-0.385466\pi$
$61$	81.6621	−	47.1476i	1.33872	−	0.772912i	0.986618	+	0.163049i	$0.0521327\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.683166	−	0.394426i	0.0105102	−	0.00606809i
$66$		0			0
$67$	−52.0398	−	30.0452i	−0.776714	−	0.448436i	0.0585505	−	0.998284i	$-0.481352\pi$
$67$	−52.0398	−	30.0452i	−0.776714	−	0.448436i	−0.835264	+	0.549848i	$0.814685\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	76.2540			1.07400			0.537000	−	0.843582i	$-0.319557\pi$
$71$	76.2540			1.07400			0.537000	+	0.843582i	$0.319557\pi$
$72$		0			0
$73$	41.4352	+	23.9226i	0.567605	+	0.327707i	0.756192	−	0.654349i	$-0.227057\pi$
$73$	41.4352	+	23.9226i	0.567605	+	0.327707i	−0.188587	+	0.982056i	$0.560391\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	19.5971	+	84.5495i	0.254508	+	1.09805i
$78$		0			0
$79$	−6.93118	+	4.00172i	−0.0877365	+	0.0506547i	−0.543226	−	0.839586i	$-0.682797\pi$
$79$	−6.93118	+	4.00172i	−0.0877365	+	0.0506547i	0.455490	+	0.890241i	$0.349464\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	150.217i		−	1.80984i	−0.425578	−	0.904922i	$-0.639929\pi$
$83$		−	150.217i		−	1.80984i	0.425578	−	0.904922i	$-0.360071\pi$
$84$		0			0
$85$	−0.172727			−0.00203208
$86$		0			0
$87$		0			0
$88$		0			0
$89$	56.2159	+	97.3688i	0.631639	+	1.09403i	0.987217	+	0.159385i	$0.0509510\pi$
$89$	56.2159	+	97.3688i	0.631639	+	1.09403i	−0.355577	+	0.934647i	$0.615716\pi$
$90$		0			0
$91$	−25.9100	−	27.7367i	−0.284725	−	0.304799i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00007	+	1.73217i	−0.0105270	+	0.0182333i
$96$		0			0
$97$		−	9.78971i		−	0.100925i	−0.998726	−	0.0504624i	$-0.983930\pi$
$97$		−	9.78971i		−	0.100925i	0.998726	−	0.0504624i	$-0.0160695\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.03158	−	1.78676i	0.0102137	−	0.0176907i	−0.860873	−	0.508819i	$-0.830082\pi$
$101$	1.03158	−	1.78676i	0.0102137	−	0.0176907i	0.871087	+	0.491129i	$0.163415\pi$
$102$		0			0
$103$	−42.7154	−	73.9853i	−0.414713	−	0.718304i	0.580685	−	0.814128i	$-0.302785\pi$
$103$	−42.7154	−	73.9853i	−0.414713	−	0.718304i	−0.995398	+	0.0958242i	$0.969451\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−38.3142	−	66.3621i	−0.358077	−	0.620207i	0.629563	−	0.776950i	$-0.283234\pi$
$107$	−38.3142	−	66.3621i	−0.358077	−	0.620207i	−0.987639	+	0.156743i	$0.949901\pi$
$108$		0			0
$109$	55.5378	−	96.1942i	0.509521	−	0.882516i	−0.490419	−	0.871487i	$-0.663156\pi$
$109$	55.5378	−	96.1942i	0.509521	−	0.882516i	0.999939	−	0.0110286i	$-0.00351060\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			81.4543i			0.720835i	0.932791	+	0.360417i	$0.117366\pi$
$113$			81.4543i			0.720835i	−0.932791	+	0.360417i	$0.882634\pi$
$114$		0			0
$115$	−0.0651980	+	0.112926i	−0.000566939	+	0.000981968i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.87656	+	8.09619i	0.0157694	+	0.0680352i
$120$		0			0
$121$	−16.3639	−	28.3431i	−0.135239	−	0.234240i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.27110			−0.0581688
$126$		0			0
$127$		−	109.943i		−	0.865692i	−0.901468	−	0.432846i	$-0.857509\pi$
$127$		−	109.943i		−	0.865692i	0.901468	−	0.432846i	$-0.142491\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	127.339	−	73.5191i	0.972052	−	0.561214i	0.0721907	−	0.997391i	$-0.477001\pi$
$131$	127.339	−	73.5191i	0.972052	−	0.561214i	0.899861	+	0.436176i	$0.143668\pi$
$132$		0			0
$133$	92.0565	+	28.0571i	0.692154	+	0.210956i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	126.338	+	72.9413i	0.922176	+	0.532419i	0.884329	−	0.466865i	$-0.154617\pi$
$137$	126.338	+	72.9413i	0.922176	+	0.532419i	0.0378474	+	0.999284i	$0.487950\pi$
$138$		0			0
$139$	−119.338			−0.858550			−0.429275	−	0.903174i	$-0.641231\pi$
$139$	−119.338			−0.858550			−0.429275	+	0.903174i	$0.641231\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	58.2221	+	33.6146i	0.407148	+	0.235067i
$144$		0			0
$145$	5.65513	−	3.26499i	0.0390009	−	0.0225172i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	79.6998	−	46.0147i	0.534898	−	0.308823i	−0.208111	−	0.978105i	$-0.566731\pi$
$149$	79.6998	−	46.0147i	0.534898	−	0.308823i	0.743009	+	0.669282i	$0.233398\pi$
$150$		0			0
$151$	190.804	+	110.161i	1.26360	+	0.729542i	0.973770	−	0.227536i	$-0.0730668\pi$
$151$	190.804	+	110.161i	1.26360	+	0.729542i	0.289833	+	0.957077i	$0.406400\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.75107			−0.0242005
$156$		0			0
$157$	−94.9592	−	54.8247i	−0.604836	−	0.349202i	0.166106	−	0.986108i	$-0.446881\pi$
$157$	−94.9592	−	54.8247i	−0.604836	−	0.349202i	−0.770942	+	0.636906i	$0.780214\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00150	+	1.82915i	0.0372764	+	0.0113612i
$162$		0			0
$163$	−74.5037	+	43.0147i	−0.457078	+	0.263894i	−0.710815	−	0.703379i	$-0.751674\pi$
$163$	−74.5037	+	43.0147i	−0.457078	+	0.263894i	0.253737	+	0.967273i	$0.418340\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			170.499i			1.02095i	0.859891	+	0.510477i	$0.170531\pi$
$167$			170.499i			1.02095i	−0.859891	+	0.510477i	$0.829469\pi$
$168$		0			0
$169$	139.599			0.826029
$170$		0			0
$171$		0			0
$172$		0			0
$173$	24.2999	+	42.0886i	0.140462	+	0.243287i	0.927671	−	0.373400i	$-0.121808\pi$
$173$	24.2999	+	42.0886i	0.140462	+	0.243287i	−0.787209	+	0.616687i	$0.788475\pi$
$174$		0			0
$175$	39.4810	+	170.336i	0.225606	+	0.973350i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−59.1463	+	102.444i	−0.330426	+	0.572315i	−0.982595	−	0.185758i	$-0.940526\pi$
$179$	−59.1463	+	102.444i	−0.330426	+	0.572315i	0.652169	+	0.758073i	$0.273859\pi$
$180$		0			0
$181$		−	47.3387i		−	0.261540i	−0.991413	−	0.130770i	$-0.958255\pi$
$181$		−	47.3387i		−	0.261540i	0.991413	−	0.130770i	$-0.0417449\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.668948	−	1.15865i	0.00361593	−	0.00626298i
$186$		0			0
$187$	−7.36025	−	12.7483i	−0.0393596	−	0.0681728i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	66.9201	+	115.909i	0.350367	+	0.606854i	0.986314	−	0.164879i	$-0.0527235\pi$
$191$	66.9201	+	115.909i	0.350367	+	0.606854i	−0.635947	+	0.771733i	$0.719390\pi$
$192$		0			0
$193$	6.38881	−	11.0657i	0.0331026	−	0.0573354i	−0.848999	−	0.528394i	$-0.822795\pi$
$193$	6.38881	−	11.0657i	0.0331026	−	0.0573354i	0.882102	+	0.471058i	$0.156128\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	173.239i		−	0.879387i	−0.898148	−	0.439694i	$-0.855087\pi$
$197$		−	173.239i		−	0.879387i	0.898148	−	0.439694i	$-0.144913\pi$
$198$		0			0
$199$	134.748	−	233.390i	0.677123	−	1.17281i	−0.298720	−	0.954341i	$-0.596560\pi$
$199$	134.748	−	233.390i	0.677123	−	1.17281i	0.975843	−	0.218471i	$-0.0701070\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−214.478	−	229.600i	−1.05654	−	1.13103i
$204$		0			0
$205$	−3.86841	−	6.70029i	−0.0188703	−	0.0326843i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−170.460			−0.815596
$210$		0			0
$211$			13.2604i			0.0628454i	0.999506	+	0.0314227i	$0.0100038\pi$
$211$			13.2604i			0.0628454i	−0.999506	+	0.0314227i	$0.989996\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.73344	−	1.00080i	0.00806253	−	0.00465490i
$216$		0			0
$217$	40.7528	+	175.823i	0.187801	+	0.810245i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.57517	+	3.21883i	0.0252270	+	0.0145648i
$222$		0			0
$223$	−85.5644			−0.383697			−0.191848	−	0.981425i	$-0.561448\pi$
$223$	−85.5644			−0.383697			−0.191848	+	0.981425i	$0.561448\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	129.243	+	74.6186i	0.569353	+	0.328716i	0.756891	−	0.653541i	$-0.226717\pi$
$227$	129.243	+	74.6186i	0.569353	+	0.328716i	−0.187538	+	0.982257i	$0.560051\pi$
$228$		0			0
$229$	−354.138	+	204.462i	−1.54645	+	0.892846i	−0.548046	+	0.836448i	$0.684628\pi$
$229$	−354.138	+	204.462i	−1.54645	+	0.892846i	−0.998408	+	0.0563977i	$0.982039\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−238.297	+	137.581i	−1.02273	+	0.590476i	−0.914895	−	0.403692i	$-0.867727\pi$
$233$	−238.297	+	137.581i	−1.02273	+	0.590476i	−0.107840	+	0.994168i	$0.534393\pi$
$234$		0			0
$235$	−1.94265	−	1.12159i	−0.00826658	−	0.00477271i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−86.3058			−0.361112			−0.180556	−	0.983565i	$-0.557790\pi$
$239$	−86.3058			−0.361112			−0.180556	+	0.983565i	$0.557790\pi$
$240$		0			0
$241$	−174.857	−	100.954i	−0.725546	−	0.418894i	0.0912443	−	0.995829i	$-0.470916\pi$
$241$	−174.857	−	100.954i	−0.725546	−	0.418894i	−0.816791	+	0.576934i	$0.804249\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.40180	+	3.13614i	−0.0261298	+	0.0128006i
$246$		0			0
$247$	64.5591	−	37.2732i	0.261373	−	0.150904i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	408.799i		−	1.62868i	−0.580388	−	0.814340i	$-0.697099\pi$
$251$		−	408.799i		−	1.62868i	0.580388	−	0.814340i	$-0.302901\pi$
$252$		0			0
$253$	−11.1129			−0.0439244
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−226.128	−	391.665i	−0.879876	−	1.52399i	−0.851476	−	0.524393i	$-0.824292\pi$
$257$	−226.128	−	391.665i	−0.879876	−	1.52399i	−0.0283994	−	0.999597i	$-0.509041\pi$
$258$		0			0
$259$	−61.5768	−	18.7675i	−0.237748	−	0.0724614i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−227.716	+	394.415i	−0.865839	+	1.49968i	0.000372217		1.00000i	$0.499882\pi$
$263$	−227.716	+	394.415i	−0.865839	+	1.49968i	−0.866211	+	0.499678i	$0.833452\pi$
$264$		0			0
$265$		−	8.98594i		−	0.0339092i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	123.710	−	214.272i	0.459888	−	0.796549i	−0.539067	−	0.842263i	$-0.681223\pi$
$269$	123.710	−	214.272i	0.459888	−	0.796549i	0.998955	+	0.0457141i	$0.0145563\pi$
$270$		0			0
$271$	87.5261	+	151.600i	0.322975	+	0.559408i	0.981100	−	0.193500i	$-0.0619838\pi$
$271$	87.5261	+	151.600i	0.322975	+	0.559408i	−0.658126	+	0.752908i	$0.728650\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−154.853	−	268.212i	−0.563100	−	0.975318i
$276$		0			0
$277$	−89.8329	+	155.595i	−0.324306	+	0.561715i	−0.981372	−	0.192119i	$-0.938464\pi$
$277$	−89.8329	+	155.595i	−0.324306	+	0.561715i	0.657065	+	0.753834i	$0.271798\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	346.090i		−	1.23164i	−0.787888	−	0.615818i	$-0.788825\pi$
$281$		−	346.090i		−	1.23164i	0.787888	−	0.615818i	$-0.211175\pi$
$282$		0			0
$283$	−157.365	+	272.564i	−0.556061	+	0.963125i	0.441759	+	0.897134i	$0.354354\pi$
$283$	−157.365	+	272.564i	−0.556061	+	0.963125i	−0.997820	+	0.0659919i	$0.978979\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−272.033	+	254.117i	−0.947851	+	0.885425i
$288$		0			0
$289$	143.795	+	249.061i	0.497561	+	0.861801i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−413.196			−1.41023			−0.705113	−	0.709095i	$-0.749104\pi$
$293$	−413.196			−1.41023			−0.705113	+	0.709095i	$0.749104\pi$
$294$		0			0
$295$		−	6.38906i		−	0.0216578i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.20884	−	2.42997i	0.0140764	−	0.00812700i
$300$		0			0
$301$	−65.7431	−	70.3782i	−0.218416	−	0.233815i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−11.8805	−	6.85921i	−0.0389525	−	0.0224892i
$306$		0			0
$307$	−147.200			−0.479479			−0.239740	−	0.970837i	$-0.577062\pi$
$307$	−147.200			−0.479479			−0.239740	+	0.970837i	$0.577062\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	446.413	+	257.736i	1.43541	+	0.828735i	0.997526	−	0.0702968i	$-0.0223946\pi$
$311$	446.413	+	257.736i	1.43541	+	0.828735i	0.437884	+	0.899031i	$0.355728\pi$
$312$		0			0
$313$	−94.3107	+	54.4503i	−0.301312	+	0.173963i	−0.643032	−	0.765839i	$-0.722324\pi$
$313$	−94.3107	+	54.4503i	−0.301312	+	0.173963i	0.341720	+	0.939802i	$0.388991\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−125.385	+	72.3913i	−0.395538	+	0.228364i	−0.684557	−	0.728960i	$-0.740004\pi$
$317$	−125.385	+	72.3913i	−0.395538	+	0.228364i	0.289019	+	0.957323i	$0.406671\pi$
$318$		0			0
$319$	481.953	+	278.255i	1.51082	+	0.872274i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.3227			−0.0505346
$324$		0			0
$325$	117.296	+	67.7210i	0.360912	+	0.208372i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−31.4664	+	103.242i	−0.0956425	+	0.313807i
$330$		0			0
$331$	−96.3115	+	55.6055i	−0.290971	+	0.167992i	−0.638380	−	0.769722i	$-0.720395\pi$
$331$	−96.3115	+	55.6055i	−0.290971	+	0.167992i	0.347408	+	0.937714i	$0.387062\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			8.74218i			0.0260960i
$336$		0			0
$337$	−540.982			−1.60529			−0.802644	−	0.596458i	$-0.796574\pi$
$337$	−540.982			−1.60529			−0.802644	+	0.596458i	$0.796574\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−159.841	−	276.852i	−0.468741	−	0.811884i
$342$		0			0
$343$	216.550	+	265.998i	0.631342	+	0.775504i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−210.670	+	364.891i	−0.607118	+	1.05156i	0.384595	+	0.923086i	$0.374341\pi$
$347$	−210.670	+	364.891i	−0.607118	+	1.05156i	−0.991713	+	0.128474i	$0.958992\pi$
$348$		0			0
$349$			563.602i			1.61491i	0.589932	+	0.807453i	$0.299155\pi$
$349$			563.602i			1.61491i	−0.589932	+	0.807453i	$0.700845\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	269.344	−	466.517i	0.763014	−	1.32158i	−0.178277	−	0.983980i	$-0.557052\pi$
$353$	269.344	−	466.517i	0.763014	−	1.32158i	0.941290	−	0.337598i	$-0.109615\pi$
$354$		0			0
$355$	−5.54686	−	9.60744i	−0.0156250	−	0.0270632i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	107.960	+	186.991i	0.300723	+	0.520867i	0.976300	−	0.216422i	$-0.0694387\pi$
$359$	107.960	+	186.991i	0.300723	+	0.520867i	−0.675577	+	0.737289i	$0.736105\pi$
$360$		0			0
$361$	85.9938	−	148.946i	0.238210	−	0.412592i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	6.96070i		−	0.0190704i
$366$		0			0
$367$	350.937	−	607.841i	0.956232	−	1.65624i	0.224710	−	0.974426i	$-0.427857\pi$
$367$	350.937	−	607.841i	0.956232	−	1.65624i	0.731523	−	0.681817i	$-0.238810\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−421.196	+	97.6259i	−1.13530	+	0.263143i
$372$		0			0
$373$	−138.894	−	240.572i	−0.372371	−	0.644965i	0.617559	−	0.786525i	$-0.288122\pi$
$373$	−138.894	−	240.572i	−0.372371	−	0.644965i	−0.989930	+	0.141559i	$0.954788\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−243.377			−0.645561
$378$		0			0
$379$			50.4237i			0.133044i	0.997785	+	0.0665221i	$0.0211903\pi$
$379$			50.4237i			0.133044i	−0.997785	+	0.0665221i	$0.978810\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−292.039	+	168.609i	−0.762503	+	0.440231i	−0.830194	−	0.557475i	$-0.811770\pi$
$383$	−292.039	+	168.609i	−0.762503	+	0.440231i	0.0676909	+	0.997706i	$0.478437\pi$
$384$		0			0
$385$	9.22707	−	8.61937i	0.0239664	−	0.0223880i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−185.536	−	107.119i	−0.476956	−	0.275371i	0.242191	−	0.970229i	$-0.422134\pi$
$389$	−185.536	−	107.119i	−0.476956	−	0.275371i	−0.719147	+	0.694858i	$0.755467\pi$
$390$		0			0
$391$	−1.06413			−0.00272157
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.00837	+	0.582185i	0.00255284	+	0.00147389i
$396$		0			0
$397$	120.927	−	69.8172i	0.304602	−	0.175862i	−0.339906	−	0.940459i	$-0.610395\pi$
$397$	120.927	−	69.8172i	0.304602	−	0.175862i	0.644508	+	0.764597i	$0.277062\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	212.271	−	122.555i	0.529355	−	0.305623i	−0.211399	−	0.977400i	$-0.567802\pi$
$401$	212.271	−	122.555i	0.529355	−	0.305623i	0.740754	+	0.671777i	$0.234468\pi$
$402$		0			0
$403$	121.075	+	69.9025i	0.300433	+	0.173455i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	114.021			0.280149
$408$		0			0
$409$	−637.451	−	368.033i	−1.55856	−	0.899836i	−0.997395	−	0.0721321i	$-0.977020\pi$
$409$	−637.451	−	368.033i	−1.55856	−	0.899836i	−0.561166	−	0.827703i	$-0.689647\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−299.473	+	69.4127i	−0.725116	+	0.168069i
$414$		0			0
$415$	−18.9262	+	10.9271i	−0.0456054	+	0.0263303i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	509.683i		−	1.21643i	−0.793774	−	0.608213i	$-0.791887\pi$
$419$		−	509.683i		−	1.21643i	0.793774	−	0.608213i	$-0.208113\pi$
$420$		0			0
$421$	−80.5857			−0.191415			−0.0957074	−	0.995410i	$-0.530511\pi$
$421$	−80.5857			−0.191415			−0.0957074	+	0.995410i	$0.530511\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−14.8282	−	25.6832i	−0.0348899	−	0.0604310i
$426$		0			0
$427$	−192.437	+	631.392i	−0.450672	+	1.47867i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−330.390	+	572.252i	−0.766566	+	1.32773i	0.172849	+	0.984948i	$0.444703\pi$
$431$	−330.390	+	572.252i	−0.766566	+	1.32773i	−0.939415	+	0.342782i	$0.888631\pi$
$432$		0			0
$433$			649.927i			1.50099i	0.660879	+	0.750493i	$0.270184\pi$
$433$			649.927i			1.50099i	−0.660879	+	0.750493i	$0.729816\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.16120	+	10.6715i	−0.0140989	+	0.0244199i
$438$		0			0
$439$	−226.552	−	392.399i	−0.516063	−	0.893847i	−0.999826	−	0.0186480i	$-0.994064\pi$
$439$	−226.552	−	392.399i	−0.516063	−	0.893847i	0.483763	−	0.875199i	$-0.339270\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−313.964	−	543.801i	−0.708722	−	1.22754i	−0.965331	−	0.261028i	$-0.915939\pi$
$443$	−313.964	−	543.801i	−0.708722	−	1.22754i	0.256609	−	0.966515i	$-0.417395\pi$
$444$		0			0
$445$	8.17850	−	14.1656i	0.0183786	−	0.0318327i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			91.1829i			0.203080i	0.994831	+	0.101540i	$0.0323770\pi$
$449$			91.1829i			0.203080i	−0.994831	+	0.101540i	$0.967623\pi$
$450$		0			0
$451$	329.682	−	571.025i	0.731001	−	1.26613i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.60988	+	5.28208i	−0.00353820	+	0.0116090i
$456$		0			0
$457$	164.472	+	284.873i	0.359894	+	0.623355i	0.987943	−	0.154819i	$-0.0494794\pi$
$457$	164.472	+	284.873i	0.359894	+	0.623355i	−0.628048	+	0.778174i	$0.716146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	59.7749			0.129664			0.0648318	−	0.997896i	$-0.479349\pi$
$461$	59.7749			0.129664			0.0648318	+	0.997896i	$0.479349\pi$
$462$		0			0
$463$			404.836i			0.874376i	0.899370	+	0.437188i	$0.144026\pi$
$463$			404.836i			0.874376i	−0.899370	+	0.437188i	$0.855974\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	550.325	−	317.730i	1.17843	−	0.680365i	0.222776	−	0.974870i	$-0.428488\pi$
$467$	550.325	−	317.730i	1.17843	−	0.680365i	0.955650	+	0.294505i	$0.0951548\pi$
$468$		0			0
$469$	409.770	−	94.9776i	0.873710	−	0.202511i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	147.731	+	85.2925i	0.312328	+	0.180322i
$474$		0			0
$475$	−343.413			−0.722976
$476$		0			0
$477$		0			0
$478$		0			0
$479$	461.145	+	266.242i	0.962725	+	0.555830i	0.897011	−	0.442009i	$-0.145734\pi$
$479$	461.145	+	266.242i	0.962725	+	0.555830i	0.0657143	+	0.997838i	$0.479067\pi$
$480$		0			0
$481$	−43.1837	+	24.9321i	−0.0897790	+	0.0518340i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.23343	+	0.712122i	−0.00254316	+	0.00146829i
$486$		0			0
$487$	424.265	+	244.949i	0.871180	+	0.502976i	0.867740	−	0.497018i	$-0.165572\pi$
$487$	424.265	+	244.949i	0.871180	+	0.502976i	0.00343968	+	0.999994i	$0.498905\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	127.584			0.259846			0.129923	−	0.991524i	$-0.458527\pi$
$491$	127.584			0.259846			0.129923	+	0.991524i	$0.458527\pi$
$492$		0			0
$493$	46.1503	+	26.6449i	0.0936111	+	0.0540464i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−390.065	+	364.375i	−0.784838	+	0.733148i
$498$		0			0
$499$	−585.142	+	337.832i	−1.17263	+	0.677017i	−0.954298	−	0.298858i	$-0.903394\pi$
$499$	−585.142	+	337.832i	−1.17263	+	0.677017i	−0.218331	+	0.975875i	$0.570061\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	216.020i		−	0.429463i	−0.976673	−	0.214732i	$-0.931112\pi$
$503$		−	216.020i		−	0.429463i	0.976673	−	0.214732i	$-0.0688877\pi$
$504$		0			0
$505$	−0.300157			−0.000594371
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−272.791	−	472.488i	−0.535935	−	0.928267i	−0.999117	−	0.0420041i	$-0.986626\pi$
$509$	−272.791	−	472.488i	−0.535935	−	0.928267i	0.463182	−	0.886263i	$-0.346708\pi$
$510$		0			0
$511$	−326.267	+	75.6231i	−0.638488	+	0.147991i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.21440	+	10.7637i	−0.0120668	+	0.0209003i
$516$		0			0
$517$		−	191.172i		−	0.369772i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	336.829	−	583.404i	0.646504	−	1.11978i	−0.337448	−	0.941344i	$-0.609564\pi$
$521$	336.829	−	583.404i	0.646504	−	1.11978i	0.983952	−	0.178434i	$-0.0571030\pi$
$522$		0			0
$523$	−287.900	−	498.657i	−0.550478	−	0.953455i	−0.998240	−	0.0593028i	$-0.981112\pi$
$523$	−287.900	−	498.657i	−0.550478	−	0.953455i	0.447762	−	0.894153i	$-0.352221\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−15.3059	−	26.5105i	−0.0290434	−	0.0503046i
$528$		0			0
$529$	264.098	−	457.432i	0.499241	−	0.864710i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			288.357i			0.541007i
$534$		0			0
$535$	−5.57409	+	9.65461i	−0.0104189	+	0.0180460i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−504.260	−	338.855i	−0.935546	−	0.628673i
$540$		0			0
$541$	122.638	+	212.416i	0.226688	+	0.392635i	0.956825	−	0.290666i	$-0.0938769\pi$
$541$	122.638	+	212.416i	0.226688	+	0.392635i	−0.730136	+	0.683301i	$0.760544\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−16.1597			−0.0296508
$546$		0			0
$547$		−	242.307i		−	0.442975i	−0.975163	−	0.221487i	$-0.928909\pi$
$547$		−	242.307i		−	0.442975i	0.975163	−	0.221487i	$-0.0710911\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	534.408	−	308.541i	0.969888	−	0.559965i
$552$		0			0
$553$	16.3333	−	53.5903i	0.0295359	−	0.0969083i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	758.383	+	437.853i	1.36155	+	0.786091i	0.989830	−	0.142254i	$-0.0454350\pi$
$557$	758.383	+	437.853i	1.36155	+	0.786091i	0.371720	+	0.928345i	$0.378768\pi$
$558$		0			0
$559$	−74.6013			−0.133455
$560$		0			0
$561$		0			0
$562$		0			0
$563$	683.486	+	394.611i	1.21401	+	0.700907i	0.963630	−	0.267241i	$-0.0861121\pi$
$563$	683.486	+	394.611i	1.21401	+	0.700907i	0.250377	+	0.968148i	$0.419445\pi$
$564$		0			0
$565$	10.2626	−	5.92514i	0.0181640	−	0.0104870i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−540.822	+	312.244i	−0.950478	+	0.548758i	−0.893229	−	0.449602i	$-0.851566\pi$
$569$	−540.822	+	312.244i	−0.950478	+	0.548758i	−0.0572483	+	0.998360i	$0.518233\pi$
$570$		0			0
$571$	160.056	+	92.4084i	0.280308	+	0.161836i	0.633563	−	0.773691i	$-0.281592\pi$
$571$	160.056	+	92.4084i	0.280308	+	0.161836i	−0.353255	+	0.935527i	$0.614925\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−22.3884			−0.0389363
$576$		0			0
$577$	545.184	+	314.762i	0.944859	+	0.545515i	0.891480	−	0.453060i	$-0.149668\pi$
$577$	545.184	+	314.762i	0.944859	+	0.545515i	0.0533789	+	0.998574i	$0.483001\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	717.802	+	768.410i	1.23546	+	1.32256i
$582$		0			0
$583$	663.217	−	382.909i	1.13759	−	0.656790i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	679.276i		−	1.15720i	−0.815612	−	0.578600i	$-0.803599\pi$
$587$		−	679.276i		−	1.15720i	0.815612	−	0.578600i	$-0.196401\pi$
$588$		0			0
$589$	−354.476			−0.601826
$590$		0			0
$591$		0			0
$592$		0			0
$593$	419.050	+	725.815i	0.706660	+	1.22397i	0.966089	+	0.258209i	$0.0831324\pi$
$593$	419.050	+	725.815i	0.706660	+	1.22397i	−0.259429	+	0.965762i	$0.583534\pi$
$594$		0			0
$595$	0.883556	−	0.825364i	0.00148497	−	0.00138717i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−508.898	+	881.437i	−0.849579	+	1.47151i	0.0320050	+	0.999488i	$0.489811\pi$
$599$	−508.898	+	881.437i	−0.849579	+	1.47151i	−0.881584	+	0.472027i	$0.843523\pi$
$600$		0			0
$601$		−	420.661i		−	0.699935i	−0.936762	−	0.349967i	$-0.886193\pi$
$601$		−	420.661i		−	0.699935i	0.936762	−	0.349967i	$-0.113807\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.38068	+	4.12345i	−0.00393500	+	0.00681563i
$606$		0			0
$607$	197.049	+	341.298i	0.324627	+	0.562271i	0.981437	−	0.191786i	$-0.0614279\pi$
$607$	197.049	+	341.298i	0.324627	+	0.562271i	−0.656810	+	0.754056i	$0.728095\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	41.8023	+	72.4037i	0.0684162	+	0.118500i
$612$		0			0
$613$	374.635	−	648.886i	0.611150	−	1.05854i	−0.379897	−	0.925029i	$-0.624041\pi$
$613$	374.635	−	648.886i	0.611150	−	1.05854i	0.991047	−	0.133514i	$-0.0426260\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	55.5379i		−	0.0900129i	−0.998987	−	0.0450064i	$-0.985669\pi$
$617$		−	55.5379i		−	0.0900129i	0.998987	−	0.0450064i	$-0.0143308\pi$
$618$		0			0
$619$	84.5393	−	146.426i	0.136574	−	0.236553i	−0.789624	−	0.613591i	$-0.789724\pi$
$619$	84.5393	−	146.426i	0.136574	−	0.236553i	0.926198	+	0.377038i	$0.123058\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−752.833	−	229.450i	−1.20840	−	0.368298i
$624$		0			0
$625$	−311.707	−	539.892i	−0.498730	−	0.863826i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	10.9183			0.0173582
$630$		0			0
$631$		−	1044.13i		−	1.65472i	−0.561672	−	0.827360i	$-0.689842\pi$
$631$		−	1044.13i		−	1.65472i	0.561672	−	0.827360i	$-0.310158\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.8520	+	7.99745i	−0.0218142	+	0.0125944i
$636$		0			0
$637$	265.076	+	18.0735i	0.416132	+	0.0283728i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1055.19	+	609.213i	1.64616	+	0.950411i	0.978578	+	0.205875i	$0.0660039\pi$
$641$	1055.19	+	609.213i	1.64616	+	0.950411i	0.667582	+	0.744536i	$0.267329\pi$
$642$		0			0
$643$	−671.894			−1.04494			−0.522468	−	0.852659i	$-0.674989\pi$
$643$	−671.894			−1.04494			−0.522468	+	0.852659i	$0.674989\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	879.201	+	507.607i	1.35889	+	0.784555i	0.989474	−	0.144710i	$-0.0462248\pi$
$647$	879.201	+	507.607i	1.35889	+	0.784555i	0.369415	+	0.929265i	$0.379558\pi$
$648$		0			0
$649$	471.552	−	272.251i	0.726582	−	0.419493i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	446.173	−	257.598i	0.683267	−	0.394484i	−0.117818	−	0.993035i	$-0.537590\pi$
$653$	446.173	−	257.598i	0.683267	−	0.394484i	0.801085	+	0.598551i	$0.204257\pi$
$654$		0			0
$655$	−18.5257	−	10.6958i	−0.0282835	−	0.0163295i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	599.600			0.909863			0.454932	−	0.890526i	$-0.349664\pi$
$659$	599.600			0.909863			0.454932	+	0.890526i	$0.349664\pi$
$660$		0			0
$661$	−855.559	−	493.957i	−1.29434	−	0.747288i	−0.314920	−	0.949118i	$-0.601978\pi$
$661$	−855.559	−	493.957i	−1.29434	−	0.747288i	−0.979420	+	0.201831i	$0.935311\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.16137	−	13.6394i	−0.00475394	−	0.0205103i
$666$		0			0
$667$	34.8400	−	20.1149i	0.0522339	−	0.0301573i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1169.14i		−	1.74238i
$672$		0			0
$673$	1147.54			1.70511			0.852555	−	0.522637i	$-0.175052\pi$
$673$	1147.54			1.70511			0.852555	+	0.522637i	$0.175052\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−32.4036	−	56.1246i	−0.0478635	−	0.0829020i	0.841101	−	0.540878i	$-0.181908\pi$
$677$	−32.4036	−	56.1246i	−0.0478635	−	0.0829020i	−0.888965	+	0.457976i	$0.848575\pi$
$678$		0			0
$679$	46.7795	+	50.0776i	0.0688946	+	0.0737520i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−79.7770	+	138.178i	−0.116804	+	0.202310i	−0.918499	−	0.395422i	$-0.870598\pi$
$683$	−79.7770	+	138.178i	−0.116804	+	0.202310i	0.801696	+	0.597733i	$0.203932\pi$
$684$		0			0
$685$		−	21.2235i		−	0.0309833i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−167.456	+	290.042i	−0.243042	+	0.420961i
$690$		0			0
$691$	−128.779	−	223.051i	−0.186365	−	0.322794i	0.757670	−	0.652637i	$-0.226338\pi$
$691$	−128.779	−	223.051i	−0.186365	−	0.322794i	−0.944036	+	0.329843i	$0.893004\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.68090	+	15.0358i	0.0124905	+	0.0216342i
$696$		0			0
$697$	31.5693	−	54.6796i	0.0452931	−	0.0784499i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			90.3621i			0.128905i	0.997921	+	0.0644523i	$0.0205300\pi$
$701$			90.3621i			0.128905i	−0.997921	+	0.0644523i	$0.979470\pi$
$702$		0			0
$703$	63.2154	−	109.492i	0.0899224	−	0.155750i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.26100	+	14.0692i	0.00461245	+	0.0198999i
$708$		0			0
$709$	177.354	+	307.186i	0.250146	+	0.433266i	0.963566	−	0.267471i	$-0.0861880\pi$
$709$	177.354	+	307.186i	0.250146	+	0.433266i	−0.713420	+	0.700737i	$0.752855\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−23.1096			−0.0324117
$714$		0			0
$715$		−	9.78074i		−	0.0136794i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	716.439	−	413.636i	0.996438	−	0.575294i	0.0892457	−	0.996010i	$-0.471554\pi$
$719$	716.439	−	413.636i	0.996438	−	0.575294i	0.907192	+	0.420716i	$0.138221\pi$
$720$		0			0
$721$	572.037	+	174.346i	0.793394	+	0.241812i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	970.958	+	560.583i	1.33925	+	0.773217i
$726$		0			0
$727$	−617.496			−0.849376			−0.424688	−	0.905340i	$-0.639616\pi$
$727$	−617.496			−0.849376			−0.424688	+	0.905340i	$0.639616\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	14.1463	+	8.16734i	0.0193519	+	0.0111728i
$732$		0			0
$733$	−715.188	+	412.914i	−0.975699	+	0.563320i	−0.900969	−	0.433884i	$-0.857143\pi$
$733$	−715.188	+	412.914i	−0.975699	+	0.563320i	−0.0747303	+	0.997204i	$0.523810\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−645.226	+	372.522i	−0.875477	+	0.505457i
$738$		0			0
$739$	603.416	+	348.382i	0.816530	+	0.471424i	0.849218	−	0.528042i	$-0.177074\pi$
$739$	603.416	+	348.382i	0.816530	+	0.471424i	−0.0326885	+	0.999466i	$0.510407\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1271.00			1.71063			0.855315	−	0.518108i	$-0.173363\pi$
$743$	1271.00			1.71063			0.855315	+	0.518108i	$0.173363\pi$
$744$		0			0
$745$	−11.5950	−	6.69439i	−0.0155638	−	0.00898575i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	513.097	+	156.382i	0.685042	+	0.208788i
$750$		0			0
$751$	−1203.19	+	694.662i	−1.60212	+	0.924983i	−0.611055	+	0.791588i	$0.709254\pi$
$751$	−1203.19	+	694.662i	−1.60212	+	0.924983i	−0.991063	+	0.133395i	$0.957412\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	32.0532i		−	0.0424546i
$756$		0			0
$757$	1200.40			1.58573			0.792866	−	0.609396i	$-0.208588\pi$
$757$	1200.40			1.58573			0.792866	+	0.609396i	$0.208588\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	461.357	+	799.094i	0.606251	+	1.05006i	0.991852	+	0.127392i	$0.0406607\pi$
$761$	461.357	+	799.094i	0.606251	+	1.05006i	−0.385601	+	0.922665i	$0.626006\pi$
$762$		0			0
$763$	175.564	+	757.448i	0.230096	+	0.992724i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−119.062	+	206.222i	−0.155231	+	0.268868i
$768$		0			0
$769$		−	471.046i		−	0.612544i	−0.951944	−	0.306272i	$-0.900918\pi$
$769$		−	471.046i		−	0.612544i	0.951944	−	0.306272i	$-0.0990817\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−23.9550	+	41.4913i	−0.0309897	+	0.0536757i	−0.881104	−	0.472922i	$-0.843199\pi$
$773$	−23.9550	+	41.4913i	−0.0309897	+	0.0536757i	0.850115	+	0.526598i	$0.176533\pi$
$774$		0			0
$775$	−322.020	−	557.756i	−0.415510	−	0.719685i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−365.564	−	633.176i	−0.469274	−	0.812806i
$780$		0			0
$781$	472.725	−	818.785i	0.605282	−	1.04838i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			15.9522i			0.0203213i
$786$		0			0
$787$	667.287	−	1155.78i	0.847887	−	1.46858i	−0.0352029	−	0.999380i	$-0.511208\pi$
$787$	667.287	−	1155.78i	0.847887	−	1.46858i	0.883090	−	0.469204i	$-0.155459\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−389.224	−	416.666i	−0.492066	−	0.526758i
$792$		0			0
$793$	255.647	+	442.794i	0.322380	+	0.558379i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1382.72			−1.73490			−0.867452	−	0.497521i	$-0.834244\pi$
$797$	−1382.72			−1.73490			−0.867452	+	0.497521i	$0.834244\pi$
$798$		0			0
$799$		−	18.3061i		−	0.0229112i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	513.743	−	296.609i	0.639779	−	0.369377i
$804$		0			0
$805$	−0.206101	−	0.889200i	−0.000256026	−	0.00110460i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	142.326	+	82.1720i	0.175928	+	0.101572i	0.585378	−	0.810760i	$-0.300946\pi$
$809$	142.326	+	82.1720i	0.175928	+	0.101572i	−0.409450	+	0.912333i	$0.634279\pi$
$810$		0			0
$811$	−1395.32			−1.72049			−0.860247	−	0.509877i	$-0.829691\pi$
$811$	−1395.32			−1.72049			−0.860247	+	0.509877i	$0.829691\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.8391	+	6.25794i	0.0132995	+	0.00767845i
$816$		0			0
$817$	163.810	−	94.5758i	0.200502	−	0.115760i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	435.953	−	251.698i	0.531003	−	0.306575i	−0.210422	−	0.977611i	$-0.567484\pi$
$821$	435.953	−	251.698i	0.531003	−	0.306575i	0.741425	+	0.671036i	$0.234150\pi$
$822$		0			0
$823$	15.4902	+	8.94327i	0.0188216	+	0.0108667i	0.509381	−	0.860541i	$-0.329874\pi$
$823$	15.4902	+	8.94327i	0.0188216	+	0.0108667i	−0.490560	+	0.871408i	$0.663208\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−598.694			−0.723934			−0.361967	−	0.932191i	$-0.617895\pi$
$827$	−598.694			−0.723934			−0.361967	+	0.932191i	$0.617895\pi$
$828$		0			0
$829$	86.9552	+	50.2036i	0.104892	+	0.0605592i	0.551528	−	0.834156i	$-0.314045\pi$
$829$	86.9552	+	50.2036i	0.104892	+	0.0605592i	−0.446636	+	0.894716i	$0.647378\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−48.2863	−	32.4477i	−0.0579668	−	0.0389528i
$834$		0			0
$835$	21.4816	−	12.4024i	0.0257265	−	0.0148532i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	193.555i		−	0.230697i	−0.993325	−	0.115349i	$-0.963202\pi$
$839$		−	193.555i		−	0.230697i	0.993325	−	0.115349i	$-0.0367985\pi$
$840$		0			0
$841$	−1173.63			−1.39552
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10.1547	−	17.5884i	−0.0120174	−	0.0208147i
$846$		0			0
$847$	219.142	+	66.7905i	0.258727	+	0.0788553i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.12124	−	7.13820i	0.00484282	−	0.00838802i
$852$		0			0
$853$			378.522i			0.443754i	0.975075	+	0.221877i	$0.0712183\pi$
$853$			378.522i			0.443754i	−0.975075	+	0.221877i	$0.928782\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	314.236	−	544.272i	0.366670	−	0.635090i	−0.622373	−	0.782721i	$-0.713831\pi$
$857$	314.236	−	544.272i	0.366670	−	0.635090i	0.989043	+	0.147630i	$0.0471646\pi$
$858$		0			0
$859$	−11.6570	−	20.1906i	−0.0135705	−	0.0235047i	0.859160	−	0.511706i	$-0.170986\pi$
$859$	−11.6570	−	20.1906i	−0.0135705	−	0.0235047i	−0.872731	+	0.488202i	$0.837653\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−394.910	−	684.004i	−0.457601	−	0.792588i	0.541233	−	0.840873i	$-0.317958\pi$
$863$	−394.910	−	684.004i	−0.457601	−	0.792588i	−0.998834	+	0.0482847i	$0.984625\pi$
$864$		0			0
$865$	3.53524	−	6.12321i	0.00408698	−	0.00707885i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			99.2322i			0.114191i
$870$		0			0
$871$	162.913	−	282.174i	0.187042	−	0.323966i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	37.1941	−	34.7445i	0.0425075	−	0.0397080i
$876$		0			0
$877$	−429.084	−	743.195i	−0.489263	−	0.847429i	0.510660	−	0.859783i	$-0.329401\pi$
$877$	−429.084	−	743.195i	−0.489263	−	0.847429i	−0.999924	+	0.0123535i	$0.996068\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	664.948			0.754765			0.377383	−	0.926057i	$-0.376824\pi$
$881$	664.948			0.754765			0.377383	+	0.926057i	$0.376824\pi$
$882$		0			0
$883$		−	1109.08i		−	1.25604i	−0.778197	−	0.628020i	$-0.783865\pi$
$883$		−	1109.08i		−	1.25604i	0.778197	−	0.628020i	$-0.216135\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	209.042	−	120.691i	0.235674	−	0.136066i	−0.377513	−	0.926004i	$-0.623221\pi$
$887$	209.042	−	120.691i	0.235674	−	0.136066i	0.613187	+	0.789938i	$0.289887\pi$
$888$		0			0
$889$	525.355	+	562.394i	0.590950	+	0.632614i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−183.580	−	105.990i	−0.205576	−	0.118689i
$894$		0			0
$895$	17.2096			0.0192287
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1002.23	+	578.641i	1.11483	+	0.643649i
$900$		0			0
$901$	63.5076	−	36.6661i	0.0704857	−	0.0406949i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.96433	+	3.44350i	−0.00659041	+	0.00380498i
$906$		0			0
$907$	1177.62	+	679.896i	1.29836	+	0.749610i	0.980121	−	0.198400i	$-0.0635744\pi$
$907$	1177.62	+	679.896i	1.29836	+	0.749610i	0.318242	+	0.948010i	$0.396908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1338.25			−1.46899			−0.734496	−	0.678614i	$-0.762581\pi$
$911$	−1338.25			−1.46899			−0.734496	+	0.678614i	$0.762581\pi$
$912$		0			0
$913$	−1612.97	−	931.248i	−1.76667	−	1.01999i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−300.074	+	984.554i	−0.327234	+	1.07367i
$918$		0			0
$919$	526.729	−	304.107i	0.573154	−	0.330911i	−0.185254	−	0.982691i	$-0.559311\pi$
$919$	526.729	−	304.107i	0.573154	−	0.330911i	0.758408	+	0.651780i	$0.225977\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			413.470i			0.447963i
$924$		0			0
$925$	229.710			0.248335
$926$		0			0
$927$		0			0
$928$		0			0
$929$	84.4858	+	146.334i	0.0909428	+	0.157517i	0.907908	−	0.419169i	$-0.137679\pi$
$929$	84.4858	+	146.334i	0.0909428	+	0.157517i	−0.816965	+	0.576687i	$0.804345\pi$
$930$		0			0
$931$	−604.968	+	296.364i	−0.649805	+	0.318329i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.07080	+	1.85467i	−0.00114524	+	0.00198361i
$936$		0			0
$937$			594.008i			0.633947i	0.948434	+	0.316973i	$0.102667\pi$
$937$			594.008i			0.633947i	−0.948434	+	0.316973i	$0.897333\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	173.814	−	301.055i	0.184712	−	0.319931i	−0.758767	−	0.651362i	$-0.774198\pi$
$941$	173.814	−	301.055i	0.184712	−	0.319931i	0.943480	+	0.331431i	$0.107531\pi$
$942$		0			0
$943$	−23.8325	−	41.2790i	−0.0252730	−	0.0437741i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	785.387	+	1360.33i	0.829342	+	1.43646i	0.898555	+	0.438861i	$0.144618\pi$
$947$	785.387	+	1360.33i	0.829342	+	1.43646i	−0.0692132	+	0.997602i	$0.522049\pi$
$948$		0			0
$949$	−129.715	+	224.673i	−0.136686	+	0.236747i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	404.804i		−	0.424768i	−0.977186	−	0.212384i	$-0.931877\pi$
$953$		−	404.804i		−	0.424768i	0.977186	−	0.212384i	$-0.0681227\pi$
$954$		0			0
$955$	9.73579	−	16.8629i	0.0101945	−	0.0176575i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−994.806	+	230.579i	−1.03734	+	0.240437i
$960$		0			0
$961$	148.106	+	256.528i	0.154117	+	0.266938i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.85893			−0.00192636
$966$		0			0
$967$			1062.40i			1.09866i	0.835607	+	0.549328i	$0.185116\pi$
$967$			1062.40i			1.09866i	−0.835607	+	0.549328i	$0.814884\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−498.275	+	287.679i	−0.513157	+	0.296271i	−0.734130	−	0.679008i	$-0.762410\pi$
$971$	−498.275	+	287.679i	−0.513157	+	0.296271i	0.220973	+	0.975280i	$0.429077\pi$
$972$		0			0
$973$	610.456	−	570.251i	0.627395	−	0.586075i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−617.968	−	356.784i	−0.632515	−	0.365183i	0.149210	−	0.988806i	$-0.452327\pi$
$977$	−617.968	−	356.784i	−0.632515	−	0.365183i	−0.781726	+	0.623623i	$0.785660\pi$
$978$		0			0
$979$	1394.01			1.42391
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1227.18	+	708.513i	1.24840	+	0.720766i	0.970791	−	0.239927i	$-0.0771234\pi$
$983$	1227.18	+	708.513i	1.24840	+	0.720766i	0.277613	+	0.960693i	$0.410457\pi$
$984$		0			0
$985$	−21.8269	+	12.6017i	−0.0221593	+	0.0127937i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10.6794	−	6.16574i	0.0107981	−	0.00623431i
$990$		0			0
$991$	−387.078	−	223.480i	−0.390593	−	0.225509i	0.291824	−	0.956472i	$-0.405738\pi$
$991$	−387.078	−	223.480i	−0.390593	−	0.225509i	−0.682417	+	0.730963i	$0.739071\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−39.2071			−0.0394042
$996$		0			0
$997$	1280.87	+	739.513i	1.28473	+	0.741738i	0.977709	−	0.209965i	$-0.0673351\pi$
$997$	1280.87	+	739.513i	1.28473	+	0.741738i	0.307019	+	0.951703i	$0.400668\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.bk.b.143.5	yes	20
3.2	odd	2	inner	1008.3.bk.b.143.6	yes	20
4.3	odd	2		1008.3.bk.a.143.5	✓	20
7.5	odd	6		1008.3.bk.a.719.6	yes	20
12.11	even	2		1008.3.bk.a.143.6	yes	20
21.5	even	6		1008.3.bk.a.719.5	yes	20
28.19	even	6	inner	1008.3.bk.b.719.6	yes	20
84.47	odd	6	inner	1008.3.bk.b.719.5	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.3.bk.a.143.5	✓	20	4.3	odd	2
1008.3.bk.a.143.6	yes	20	12.11	even	2
1008.3.bk.a.719.5	yes	20	21.5	even	6
1008.3.bk.a.719.6	yes	20	7.5	odd	6
1008.3.bk.b.143.5	yes	20	1.1	even	1	trivial
1008.3.bk.b.143.6	yes	20	3.2	odd	2	inner
1008.3.bk.b.719.5	yes	20	84.47	odd	6	inner
1008.3.bk.b.719.6	yes	20	28.19	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.0727418 - 0.125993i) q^{5} +(-5.11533 + 4.77843i) q^{7} +O(q^{10})\)	\(q+(-0.0727418 - 0.125993i) q^{5} +(-5.11533 + 4.77843i) q^{7} +(6.19935 - 10.7376i) q^{11} +5.42227i q^{13} +(0.593630 - 1.02820i) q^{17} +(-6.87409 - 11.9063i) q^{19} +(-0.448147 - 0.776213i) q^{23} +(12.4894 - 21.6323i) q^{25} +44.8846i q^{29} +(12.8917 - 22.3291i) q^{31} +(0.974145 + 0.296902i) q^{35} +(4.59810 + 7.96414i) q^{37} +53.1800 q^{41} +13.7583i q^{43} +(13.3530 - 7.70937i) q^{47} +(3.33319 - 48.8865i) q^{49} +(53.4909 + 30.8830i) q^{53} -1.80381 q^{55} +(38.0324 + 21.9580i) q^{59} +(81.6621 - 47.1476i) q^{61} +(0.683166 - 0.394426i) q^{65} +(-52.0398 - 30.0452i) q^{67} +76.2540 q^{71} +(41.4352 + 23.9226i) q^{73} +(19.5971 + 84.5495i) q^{77} +(-6.93118 + 4.00172i) q^{79} -150.217i q^{83} -0.172727 q^{85} +(56.2159 + 97.3688i) q^{89} +(-25.9100 - 27.7367i) q^{91} +(-1.00007 + 1.73217i) q^{95} -9.78971i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 10 q^{7}+O(q^{10}) \) 20 * q + 10 * q^7	\( 20 q + 10 q^{7} + 14 q^{19} - 14 q^{25} + 22 q^{31} - 10 q^{37} + 158 q^{49} - 240 q^{55} - 180 q^{61} + 318 q^{67} - 246 q^{73} + 66 q^{79} + 144 q^{85} - 42 q^{91}+O(q^{100}) \) 20 * q + 10 * q^7 + 14 * q^19 - 14 * q^25 + 22 * q^31 - 10 * q^37 + 158 * q^49 - 240 * q^55 - 180 * q^61 + 318 * q^67 - 246 * q^73 + 66 * q^79 + 144 * q^85 - 42 * q^91

Introduction

L-functions

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