LMFDB - Embedded newform 1344.2.bk.d.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(289,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1344.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.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:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			289.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1344.289
Dual form			1344.2.bk.d.865.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.866025 - 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(0.500000 - 0.866025i) q^{9} +(-0.866025 + 0.500000i) q^{11} +2.00000i q^{13} +1.00000 q^{15} +(2.00000 + 3.46410i) q^{17} +(0.866025 + 2.50000i) q^{21} +(-3.00000 + 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} -1.00000i q^{27} -1.00000i q^{29} +(2.50000 + 4.33013i) q^{31} +(-0.500000 + 0.866025i) q^{33} +(-1.73205 + 2.00000i) q^{35} +(6.92820 + 4.00000i) q^{37} +(1.00000 + 1.73205i) q^{39} -2.00000 q^{41} +2.00000i q^{43} +(0.866025 - 0.500000i) q^{45} +(-4.00000 + 6.92820i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(3.46410 + 2.00000i) q^{51} +(7.79423 - 4.50000i) q^{53} -1.00000 q^{55} +(-0.866025 + 0.500000i) q^{59} +(6.92820 + 4.00000i) q^{61} +(2.00000 + 1.73205i) q^{63} +(-1.00000 + 1.73205i) q^{65} +(1.73205 - 1.00000i) q^{67} +6.00000i q^{69} -16.0000 q^{71} +(3.00000 + 5.19615i) q^{73} +(-3.46410 - 2.00000i) q^{75} +(-0.866025 - 2.50000i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -15.0000i q^{83} +4.00000i q^{85} +(-0.500000 - 0.866025i) q^{87} +(4.00000 - 6.92820i) q^{89} +(-5.19615 - 1.00000i) q^{91} +(4.33013 + 2.50000i) q^{93} +11.0000 q^{97} +1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^7 + 2 * q^9	$ 4 q - 2 q^{7} + 2 q^{9} + 4 q^{15} + 8 q^{17} - 12 q^{23} - 8 q^{25} + 10 q^{31} - 2 q^{33} + 4 q^{39} - 8 q^{41} - 16 q^{47} - 26 q^{49} - 4 q^{55} + 8 q^{63} - 4 q^{65} - 64 q^{71} + 12 q^{73} + 2 q^{79} - 2 q^{81} - 2 q^{87} + 16 q^{89} + 44 q^{97}+O(q^{100}) $ 4 * q - 2 * q^7 + 2 * q^9 + 4 * q^15 + 8 * q^17 - 12 * q^23 - 8 * q^25 + 10 * q^31 - 2 * q^33 + 4 * q^39 - 8 * q^41 - 16 * q^47 - 26 * q^49 - 4 * q^55 + 8 * q^63 - 4 * q^65 - 64 * q^71 + 12 * q^73 + 2 * q^79 - 2 * q^81 - 2 * q^87 + 16 * q^89 + 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i
$4$		0			0
$5$	0.866025	+	0.500000i	0.387298	+	0.223607i	0.680989	−	0.732294i	$-0.261550\pi$
$5$	0.866025	+	0.500000i	0.387298	+	0.223607i	−0.293691	+	0.955901i	$0.594884\pi$
$6$		0			0
$7$	−0.500000	+	2.59808i	−0.188982	+	0.981981i
$7$	−0.500000	+	2.59808i	−0.188982	+	0.981981i
$8$		0			0
$9$	0.500000	−	0.866025i	0.166667	−	0.288675i
$10$		0			0
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i	−0.624844	−	0.780750i	$-0.714837\pi$
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i	0.363727	+	0.931505i	$0.381504\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	0.999853	−	0.0171533i	$-0.00546033\pi$
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	−0.514782	+	0.857321i	$0.672127\pi$
$18$		0			0
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	0.866025	+	2.50000i	0.188982	+	0.545545i
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$		−	1.00000i		−	0.185695i	−0.995680	−	0.0928477i	$-0.970403\pi$
$29$		−	1.00000i		−	0.185695i	0.995680	−	0.0928477i	$-0.0295970\pi$
$30$		0			0
$31$	2.50000	+	4.33013i	0.449013	+	0.777714i	0.998322	−	0.0579057i	$-0.0184423\pi$
$31$	2.50000	+	4.33013i	0.449013	+	0.777714i	−0.549309	+	0.835619i	$0.685109\pi$
$32$		0			0
$33$	−0.500000	+	0.866025i	−0.0870388	+	0.150756i
$34$		0			0
$35$	−1.73205	+	2.00000i	−0.292770	+	0.338062i
$36$		0			0
$37$	6.92820	+	4.00000i	1.13899	+	0.657596i	0.946180	−	0.323640i	$-0.104907\pi$
$37$	6.92820	+	4.00000i	1.13899	+	0.657596i	0.192809	+	0.981236i	$0.438240\pi$
$38$		0			0
$39$	1.00000	+	1.73205i	0.160128	+	0.277350i
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$	0.866025	−	0.500000i	0.129099	−	0.0745356i
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$		0			0
$51$	3.46410	+	2.00000i	0.485071	+	0.280056i
$52$		0			0
$53$	7.79423	−	4.50000i	1.07062	−	0.618123i	0.142269	−	0.989828i	$-0.454560\pi$
$53$	7.79423	−	4.50000i	1.07062	−	0.618123i	0.928351	+	0.371706i	$0.121227\pi$
$54$		0			0
$55$	−1.00000			−0.134840
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.866025	+	0.500000i	−0.112747	+	0.0650945i	−0.555313	−	0.831641i	$-0.687402\pi$
$59$	−0.866025	+	0.500000i	−0.112747	+	0.0650945i	0.442566	+	0.896736i	$0.354068\pi$
$60$		0			0
$61$	6.92820	+	4.00000i	0.887066	+	0.512148i	0.872982	−	0.487753i	$-0.162183\pi$
$61$	6.92820	+	4.00000i	0.887066	+	0.512148i	0.0140840	+	0.999901i	$0.495517\pi$
$62$		0			0
$63$	2.00000	+	1.73205i	0.251976	+	0.218218i
$64$		0			0
$65$	−1.00000	+	1.73205i	−0.124035	+	0.214834i
$66$		0			0
$67$	1.73205	−	1.00000i	0.211604	−	0.122169i	−0.390453	−	0.920623i	$-0.627682\pi$
$67$	1.73205	−	1.00000i	0.211604	−	0.122169i	0.602056	+	0.798454i	$0.294348\pi$
$68$		0			0
$69$			6.00000i			0.722315i
$70$		0			0
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$	3.00000	+	5.19615i	0.351123	+	0.608164i	0.986447	−	0.164083i	$-0.0524664\pi$
$73$	3.00000	+	5.19615i	0.351123	+	0.608164i	−0.635323	+	0.772246i	$0.719133\pi$
$74$		0			0
$75$	−3.46410	−	2.00000i	−0.400000	−	0.230940i
$76$		0			0
$77$	−0.866025	−	2.50000i	−0.0986928	−	0.284901i
$78$		0			0
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	−0.836527	−	0.547926i	$-0.815418\pi$
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	0.892781	+	0.450490i	$0.148751\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	15.0000i		−	1.64646i	−0.567705	−	0.823232i	$-0.692169\pi$
$83$		−	15.0000i		−	1.64646i	0.567705	−	0.823232i	$-0.307831\pi$
$84$		0			0
$85$			4.00000i			0.433861i
$86$		0			0
$87$	−0.500000	−	0.866025i	−0.0536056	−	0.0928477i
$88$		0			0
$89$	4.00000	−	6.92820i	0.423999	−	0.734388i	−0.572327	−	0.820025i	$-0.693959\pi$
$89$	4.00000	−	6.92820i	0.423999	−	0.734388i	0.996326	+	0.0856373i	$0.0272926\pi$
$90$		0			0
$91$	−5.19615	−	1.00000i	−0.544705	−	0.104828i
$92$		0			0
$93$	4.33013	+	2.50000i	0.449013	+	0.259238i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	11.0000			1.11688			0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000			1.11688			0.558440	+	0.829545i	$0.311400\pi$
$98$		0			0
$99$			1.00000i			0.100504i
$100$		0			0
$101$	12.1244	−	7.00000i	1.20642	−	0.696526i	0.244443	−	0.969664i	$-0.421395\pi$
$101$	12.1244	−	7.00000i	1.20642	−	0.696526i	0.961975	+	0.273138i	$0.0880614\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	−0.992990	−	0.118199i	$-0.962288\pi$
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	0.598858	+	0.800855i	$0.295621\pi$
$104$		0			0
$105$	−0.500000	+	2.59808i	−0.0487950	+	0.253546i
$106$		0			0
$107$	12.9904	+	7.50000i	1.25583	+	0.725052i	0.972261	−	0.233900i	$-0.0751489\pi$
$107$	12.9904	+	7.50000i	1.25583	+	0.725052i	0.283567	+	0.958952i	$0.408482\pi$
$108$		0			0
$109$	1.73205	−	1.00000i	0.165900	−	0.0957826i	−0.414751	−	0.909935i	$-0.636131\pi$
$109$	1.73205	−	1.00000i	0.165900	−	0.0957826i	0.580651	+	0.814152i	$0.302798\pi$
$110$		0			0
$111$	8.00000			0.759326
$112$		0			0
$113$	2.00000			0.188144			0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000			0.188144			0.0940721	+	0.995565i	$0.470012\pi$
$114$		0			0
$115$	−5.19615	+	3.00000i	−0.484544	+	0.279751i
$116$		0			0
$117$	1.73205	+	1.00000i	0.160128	+	0.0924500i
$118$		0			0
$119$	−10.0000	+	3.46410i	−0.916698	+	0.317554i
$120$		0			0
$121$	−5.00000	+	8.66025i	−0.454545	+	0.787296i
$122$		0			0
$123$	−1.73205	+	1.00000i	−0.156174	+	0.0901670i
$124$		0			0
$125$		−	9.00000i		−	0.804984i
$126$		0			0
$127$	13.0000			1.15356			0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000			1.15356			0.576782	+	0.816898i	$0.304308\pi$
$128$		0			0
$129$	1.00000	+	1.73205i	0.0880451	+	0.152499i
$130$		0			0
$131$	2.59808	+	1.50000i	0.226995	+	0.131056i	0.609185	−	0.793028i	$-0.291497\pi$
$131$	2.59808	+	1.50000i	0.226995	+	0.131056i	−0.382190	+	0.924084i	$0.624830\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	0.500000	−	0.866025i	0.0430331	−	0.0745356i
$136$		0			0
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	0.708942	−	0.705266i	$-0.249173\pi$
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	−0.965250	+	0.261329i	$0.915839\pi$
$138$		0			0
$139$		−	8.00000i		−	0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$		−	8.00000i		−	0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$		0			0
$141$			8.00000i			0.673722i
$142$		0			0
$143$	−1.00000	−	1.73205i	−0.0836242	−	0.144841i
$144$		0			0
$145$	0.500000	−	0.866025i	0.0415227	−	0.0719195i
$146$		0			0
$147$	−6.92820	+	1.00000i	−0.571429	+	0.0824786i
$148$		0			0
$149$	−8.66025	−	5.00000i	−0.709476	−	0.409616i	0.101391	−	0.994847i	$-0.467671\pi$
$149$	−8.66025	−	5.00000i	−0.709476	−	0.409616i	−0.810867	+	0.585231i	$0.801004\pi$
$150$		0			0
$151$	−2.50000	−	4.33013i	−0.203447	−	0.352381i	0.746190	−	0.665733i	$-0.231881\pi$
$151$	−2.50000	−	4.33013i	−0.203447	−	0.352381i	−0.949637	+	0.313353i	$0.898548\pi$
$152$		0			0
$153$	4.00000			0.323381
$154$		0			0
$155$			5.00000i			0.401610i
$156$		0			0
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	−0.631822	−	0.775113i	$-0.717693\pi$
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	0.355357	+	0.934731i	$0.384359\pi$
$158$		0			0
$159$	4.50000	−	7.79423i	0.356873	−	0.618123i
$160$		0			0
$161$	−12.0000	−	10.3923i	−0.945732	−	0.819028i
$162$		0			0
$163$	19.0526	+	11.0000i	1.49231	+	0.861586i	0.999961	−	0.00881059i	$-0.00280453\pi$
$163$	19.0526	+	11.0000i	1.49231	+	0.861586i	0.492350	+	0.870397i	$0.336138\pi$
$164$		0			0
$165$	−0.866025	+	0.500000i	−0.0674200	+	0.0389249i
$166$		0			0
$167$	−18.0000			−1.39288			−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000			−1.39288			−0.696441	+	0.717614i	$0.745234\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−15.5885	−	9.00000i	−1.18517	−	0.684257i	−0.227964	−	0.973670i	$-0.573207\pi$
$173$	−15.5885	−	9.00000i	−1.18517	−	0.684257i	−0.957205	+	0.289412i	$0.906540\pi$
$174$		0			0
$175$	10.0000	−	3.46410i	0.755929	−	0.261861i
$176$		0			0
$177$	−0.500000	+	0.866025i	−0.0375823	+	0.0650945i
$178$		0			0
$179$	−17.3205	+	10.0000i	−1.29460	+	0.747435i	−0.979465	−	0.201613i	$-0.935382\pi$
$179$	−17.3205	+	10.0000i	−1.29460	+	0.747435i	−0.315130	+	0.949048i	$0.602048\pi$
$180$		0			0
$181$			18.0000i			1.33793i	0.743294	+	0.668965i	$0.233262\pi$
$181$			18.0000i			1.33793i	−0.743294	+	0.668965i	$0.766738\pi$
$182$		0			0
$183$	8.00000			0.591377
$184$		0			0
$185$	4.00000	+	6.92820i	0.294086	+	0.509372i
$186$		0			0
$187$	−3.46410	−	2.00000i	−0.253320	−	0.146254i
$188$		0			0
$189$	2.59808	+	0.500000i	0.188982	+	0.0363696i
$190$		0			0
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	−0.684244	−	0.729253i	$-0.739868\pi$
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	0.973674	+	0.227946i	$0.0732010\pi$
$192$		0			0
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	0.531446	−	0.847092i	$-0.321649\pi$
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	−0.999326	+	0.0366998i	$0.988315\pi$
$194$		0			0
$195$			2.00000i			0.143223i
$196$		0			0
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$		0			0
$199$	−4.00000	−	6.92820i	−0.283552	−	0.491127i	0.688705	−	0.725042i	$-0.258180\pi$
$199$	−4.00000	−	6.92820i	−0.283552	−	0.491127i	−0.972257	+	0.233915i	$0.924846\pi$
$200$		0			0
$201$	1.00000	−	1.73205i	0.0705346	−	0.122169i
$202$		0			0
$203$	2.59808	+	0.500000i	0.182349	+	0.0350931i
$204$		0			0
$205$	−1.73205	−	1.00000i	−0.120972	−	0.0698430i
$206$		0			0
$207$	3.00000	+	5.19615i	0.208514	+	0.361158i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	16.0000i		−	1.10149i	−0.834675	−	0.550743i	$-0.814345\pi$
$211$		−	16.0000i		−	1.10149i	0.834675	−	0.550743i	$-0.185655\pi$
$212$		0			0
$213$	−13.8564	+	8.00000i	−0.949425	+	0.548151i
$214$		0			0
$215$	−1.00000	+	1.73205i	−0.0681994	+	0.118125i
$216$		0			0
$217$	−12.5000	+	4.33013i	−0.848555	+	0.293948i
$218$		0			0
$219$	5.19615	+	3.00000i	0.351123	+	0.202721i
$220$		0			0
$221$	−6.92820	+	4.00000i	−0.466041	+	0.269069i
$222$		0			0
$223$	21.0000			1.40626			0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000			1.40626			0.703132	+	0.711059i	$0.251784\pi$
$224$		0			0
$225$	−4.00000			−0.266667
$226$		0			0
$227$	2.59808	−	1.50000i	0.172440	−	0.0995585i	−0.411296	−	0.911502i	$-0.634924\pi$
$227$	2.59808	−	1.50000i	0.172440	−	0.0995585i	0.583736	+	0.811943i	$0.301590\pi$
$228$		0			0
$229$	−8.66025	−	5.00000i	−0.572286	−	0.330409i	0.185776	−	0.982592i	$-0.440520\pi$
$229$	−8.66025	−	5.00000i	−0.572286	−	0.330409i	−0.758062	+	0.652183i	$0.773853\pi$
$230$		0			0
$231$	−2.00000	−	1.73205i	−0.131590	−	0.113961i
$232$		0			0
$233$	−12.0000	+	20.7846i	−0.786146	+	1.36165i	0.142166	+	0.989843i	$0.454593\pi$
$233$	−12.0000	+	20.7846i	−0.786146	+	1.36165i	−0.928312	+	0.371802i	$0.878740\pi$
$234$		0			0
$235$	−6.92820	+	4.00000i	−0.451946	+	0.260931i
$236$		0			0
$237$		−	1.00000i		−	0.0649570i
$238$		0			0
$239$	2.00000			0.129369			0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000			0.129369			0.0646846	+	0.997906i	$0.479396\pi$
$240$		0			0
$241$	6.50000	+	11.2583i	0.418702	+	0.725213i	0.995809	−	0.0914555i	$-0.0291519\pi$
$241$	6.50000	+	11.2583i	0.418702	+	0.725213i	−0.577107	+	0.816668i	$0.695819\pi$
$242$		0			0
$243$	−0.866025	−	0.500000i	−0.0555556	−	0.0320750i
$244$		0			0
$245$	−4.33013	−	5.50000i	−0.276642	−	0.351382i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−7.50000	−	12.9904i	−0.475293	−	0.823232i
$250$		0			0
$251$		−	21.0000i		−	1.32551i	−0.748837	−	0.662754i	$-0.769387\pi$
$251$		−	21.0000i		−	1.32551i	0.748837	−	0.662754i	$-0.230613\pi$
$252$		0			0
$253$		−	6.00000i		−	0.377217i
$254$		0			0
$255$	2.00000	+	3.46410i	0.125245	+	0.216930i
$256$		0			0
$257$	6.00000	−	10.3923i	0.374270	−	0.648254i	−0.615948	−	0.787787i	$-0.711227\pi$
$257$	6.00000	−	10.3923i	0.374270	−	0.648254i	0.990217	+	0.139533i	$0.0445601\pi$
$258$		0			0
$259$	−13.8564	+	16.0000i	−0.860995	+	0.994192i
$260$		0			0
$261$	−0.866025	−	0.500000i	−0.0536056	−	0.0309492i
$262$		0			0
$263$	−5.00000	−	8.66025i	−0.308313	−	0.534014i	0.669680	−	0.742650i	$-0.266431\pi$
$263$	−5.00000	−	8.66025i	−0.308313	−	0.534014i	−0.977993	+	0.208635i	$0.933098\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		0			0
$267$		−	8.00000i		−	0.489592i
$268$		0			0
$269$	−18.1865	+	10.5000i	−1.10885	+	0.640196i	−0.938532	−	0.345192i	$-0.887814\pi$
$269$	−18.1865	+	10.5000i	−1.10885	+	0.640196i	−0.170321	+	0.985389i	$0.554480\pi$
$270$		0			0
$271$	12.5000	−	21.6506i	0.759321	−	1.31518i	−0.183876	−	0.982949i	$-0.558865\pi$
$271$	12.5000	−	21.6506i	0.759321	−	1.31518i	0.943197	−	0.332233i	$-0.107802\pi$
$272$		0			0
$273$	−5.00000	+	1.73205i	−0.302614	+	0.104828i
$274$		0			0
$275$	3.46410	+	2.00000i	0.208893	+	0.120605i
$276$		0			0
$277$	−27.7128	+	16.0000i	−1.66510	+	0.961347i	−0.694881	+	0.719125i	$0.744543\pi$
$277$	−27.7128	+	16.0000i	−1.66510	+	0.961347i	−0.970221	+	0.242222i	$0.922124\pi$
$278$		0			0
$279$	5.00000			0.299342
$280$		0			0
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$	19.0526	−	11.0000i	1.13256	−	0.653882i	0.187980	−	0.982173i	$-0.439806\pi$
$283$	19.0526	−	11.0000i	1.13256	−	0.653882i	0.944577	+	0.328291i	$0.106473\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.00000	−	5.19615i	0.0590281	−	0.306719i
$288$		0			0
$289$	0.500000	−	0.866025i	0.0294118	−	0.0509427i
$290$		0			0
$291$	9.52628	−	5.50000i	0.558440	−	0.322416i
$292$		0			0
$293$		−	31.0000i		−	1.81104i	−0.424304	−	0.905520i	$-0.639481\pi$
$293$		−	31.0000i		−	1.81104i	0.424304	−	0.905520i	$-0.360519\pi$
$294$		0			0
$295$	−1.00000			−0.0582223
$296$		0			0
$297$	0.500000	+	0.866025i	0.0290129	+	0.0502519i
$298$		0			0
$299$	−10.3923	−	6.00000i	−0.601003	−	0.346989i
$300$		0			0
$301$	−5.19615	−	1.00000i	−0.299501	−	0.0576390i
$302$		0			0
$303$	7.00000	−	12.1244i	0.402139	−	0.696526i
$304$		0			0
$305$	4.00000	+	6.92820i	0.229039	+	0.396708i
$306$		0			0
$307$		−	14.0000i		−	0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$		−	14.0000i		−	0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$		0			0
$309$			8.00000i			0.455104i
$310$		0			0
$311$	−7.00000	−	12.1244i	−0.396934	−	0.687509i	0.596412	−	0.802678i	$-0.296592\pi$
$311$	−7.00000	−	12.1244i	−0.396934	−	0.687509i	−0.993346	+	0.115169i	$0.963259\pi$
$312$		0			0
$313$	4.50000	−	7.79423i	0.254355	−	0.440556i	−0.710365	−	0.703833i	$-0.751470\pi$
$313$	4.50000	−	7.79423i	0.254355	−	0.440556i	0.964720	+	0.263278i	$0.0848035\pi$
$314$		0			0
$315$	0.866025	+	2.50000i	0.0487950	+	0.140859i
$316$		0			0
$317$	−9.52628	−	5.50000i	−0.535049	−	0.308911i	0.208021	−	0.978124i	$-0.433298\pi$
$317$	−9.52628	−	5.50000i	−0.535049	−	0.308911i	−0.743070	+	0.669214i	$0.766631\pi$
$318$		0			0
$319$	0.500000	+	0.866025i	0.0279946	+	0.0484881i
$320$		0			0
$321$	15.0000			0.837218
$322$		0			0
$323$		0			0
$324$		0			0
$325$	6.92820	−	4.00000i	0.384308	−	0.221880i
$326$		0			0
$327$	1.00000	−	1.73205i	0.0553001	−	0.0957826i
$328$		0			0
$329$	−16.0000	−	13.8564i	−0.882109	−	0.763928i
$330$		0			0
$331$	20.7846	+	12.0000i	1.14243	+	0.659580i	0.947030	−	0.321145i	$-0.104068\pi$
$331$	20.7846	+	12.0000i	1.14243	+	0.659580i	0.195395	+	0.980725i	$0.437401\pi$
$332$		0			0
$333$	6.92820	−	4.00000i	0.379663	−	0.219199i
$334$		0			0
$335$	2.00000			0.109272
$336$		0			0
$337$	−15.0000			−0.817102			−0.408551	−	0.912735i	$-0.633966\pi$
$337$	−15.0000			−0.817102			−0.408551	+	0.912735i	$0.633966\pi$
$338$		0			0
$339$	1.73205	−	1.00000i	0.0940721	−	0.0543125i
$340$		0			0
$341$	−4.33013	−	2.50000i	−0.234490	−	0.135383i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$	−3.00000	+	5.19615i	−0.161515	+	0.279751i
$346$		0			0
$347$	−20.7846	+	12.0000i	−1.11578	+	0.644194i	−0.940319	−	0.340293i	$-0.889474\pi$
$347$	−20.7846	+	12.0000i	−1.11578	+	0.644194i	−0.175457	+	0.984487i	$0.556140\pi$
$348$		0			0
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$	12.0000	+	20.7846i	0.638696	+	1.10625i	0.985719	+	0.168397i	$0.0538590\pi$
$353$	12.0000	+	20.7846i	0.638696	+	1.10625i	−0.347024	+	0.937856i	$0.612808\pi$
$354$		0			0
$355$	−13.8564	−	8.00000i	−0.735422	−	0.424596i
$356$		0			0
$357$	−6.92820	+	8.00000i	−0.366679	+	0.423405i
$358$		0			0
$359$	−3.00000	+	5.19615i	−0.158334	+	0.274242i	−0.934268	−	0.356572i	$-0.883946\pi$
$359$	−3.00000	+	5.19615i	−0.158334	+	0.274242i	0.775934	+	0.630814i	$0.217279\pi$
$360$		0			0
$361$	−9.50000	−	16.4545i	−0.500000	−	0.866025i
$362$		0			0
$363$			10.0000i			0.524864i
$364$		0			0
$365$			6.00000i			0.314054i
$366$		0			0
$367$	1.50000	+	2.59808i	0.0782994	+	0.135618i	0.902516	−	0.430656i	$-0.141718\pi$
$367$	1.50000	+	2.59808i	0.0782994	+	0.135618i	−0.824217	+	0.566274i	$0.808384\pi$
$368$		0			0
$369$	−1.00000	+	1.73205i	−0.0520579	+	0.0901670i
$370$		0			0
$371$	7.79423	+	22.5000i	0.404656	+	1.16814i
$372$		0			0
$373$	−25.9808	−	15.0000i	−1.34523	−	0.776671i	−0.357663	−	0.933851i	$-0.616426\pi$
$373$	−25.9808	−	15.0000i	−1.34523	−	0.776671i	−0.987570	+	0.157180i	$0.949760\pi$
$374$		0			0
$375$	−4.50000	−	7.79423i	−0.232379	−	0.402492i
$376$		0			0
$377$	2.00000			0.103005
$378$		0			0
$379$			10.0000i			0.513665i	0.966456	+	0.256833i	$0.0826790\pi$
$379$			10.0000i			0.513665i	−0.966456	+	0.256833i	$0.917321\pi$
$380$		0			0
$381$	11.2583	−	6.50000i	0.576782	−	0.333005i
$382$		0			0
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$	0.500000	−	2.59808i	0.0254824	−	0.132410i
$386$		0			0
$387$	1.73205	+	1.00000i	0.0880451	+	0.0508329i
$388$		0			0
$389$	15.5885	−	9.00000i	0.790366	−	0.456318i	−0.0497253	−	0.998763i	$-0.515835\pi$
$389$	15.5885	−	9.00000i	0.790366	−	0.456318i	0.840091	+	0.542445i	$0.182501\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$		0			0
$393$	3.00000			0.151330
$394$		0			0
$395$	0.866025	−	0.500000i	0.0435745	−	0.0251577i
$396$		0			0
$397$	17.3205	+	10.0000i	0.869291	+	0.501886i	0.867113	−	0.498112i	$-0.165973\pi$
$397$	17.3205	+	10.0000i	0.869291	+	0.501886i	0.00217869	+	0.999998i	$0.499307\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.0000	+	24.2487i	−0.699127	+	1.21092i	0.269643	+	0.962960i	$0.413094\pi$
$401$	−14.0000	+	24.2487i	−0.699127	+	1.21092i	−0.968770	+	0.247962i	$0.920239\pi$
$402$		0			0
$403$	−8.66025	+	5.00000i	−0.431398	+	0.249068i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$	−8.00000			−0.396545
$408$		0			0
$409$	−8.50000	−	14.7224i	−0.420298	−	0.727977i	0.575670	−	0.817682i	$-0.304741\pi$
$409$	−8.50000	−	14.7224i	−0.420298	−	0.727977i	−0.995968	+	0.0897044i	$0.971408\pi$
$410$		0			0
$411$	−5.19615	−	3.00000i	−0.256307	−	0.147979i
$412$		0			0
$413$	−0.866025	−	2.50000i	−0.0426143	−	0.123017i
$414$		0			0
$415$	7.50000	−	12.9904i	0.368161	−	0.637673i
$416$		0			0
$417$	−4.00000	−	6.92820i	−0.195881	−	0.339276i
$418$		0			0
$419$		−	28.0000i		−	1.36789i	−0.729534	−	0.683945i	$-0.760263\pi$
$419$		−	28.0000i		−	1.36789i	0.729534	−	0.683945i	$-0.239737\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	4.00000	+	6.92820i	0.194487	+	0.336861i
$424$		0			0
$425$	8.00000	−	13.8564i	0.388057	−	0.672134i
$426$		0			0
$427$	−13.8564	+	16.0000i	−0.670559	+	0.774294i
$428$		0			0
$429$	−1.73205	−	1.00000i	−0.0836242	−	0.0482805i
$430$		0			0
$431$	1.00000	+	1.73205i	0.0481683	+	0.0834300i	0.889104	−	0.457705i	$-0.151328\pi$
$431$	1.00000	+	1.73205i	0.0481683	+	0.0834300i	−0.840936	+	0.541135i	$0.817995\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		0			0
$435$		−	1.00000i		−	0.0479463i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−4.50000	+	7.79423i	−0.214773	+	0.371998i	−0.953202	−	0.302333i	$-0.902235\pi$
$439$	−4.50000	+	7.79423i	−0.214773	+	0.371998i	0.738429	+	0.674331i	$0.235568\pi$
$440$		0			0
$441$	−5.50000	+	4.33013i	−0.261905	+	0.206197i
$442$		0			0
$443$	7.79423	+	4.50000i	0.370315	+	0.213801i	0.673596	−	0.739100i	$-0.264749\pi$
$443$	7.79423	+	4.50000i	0.370315	+	0.213801i	−0.303281	+	0.952901i	$0.598082\pi$
$444$		0			0
$445$	6.92820	−	4.00000i	0.328428	−	0.189618i
$446$		0			0
$447$	−10.0000			−0.472984
$448$		0			0
$449$	−14.0000			−0.660701			−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000			−0.660701			−0.330350	+	0.943858i	$0.607167\pi$
$450$		0			0
$451$	1.73205	−	1.00000i	0.0815591	−	0.0470882i
$452$		0			0
$453$	−4.33013	−	2.50000i	−0.203447	−	0.117460i
$454$		0			0
$455$	−4.00000	−	3.46410i	−0.187523	−	0.162400i
$456$		0			0
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	0.410184	+	0.912003i	$0.365464\pi$
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	−0.994910	+	0.100771i	$0.967869\pi$
$458$		0			0
$459$	3.46410	−	2.00000i	0.161690	−	0.0933520i
$460$		0			0
$461$			22.0000i			1.02464i	0.858794	+	0.512321i	$0.171214\pi$
$461$			22.0000i			1.02464i	−0.858794	+	0.512321i	$0.828786\pi$
$462$		0			0
$463$	4.00000			0.185896			0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000			0.185896			0.0929479	+	0.995671i	$0.470371\pi$
$464$		0			0
$465$	2.50000	+	4.33013i	0.115935	+	0.200805i
$466$		0			0
$467$	27.7128	+	16.0000i	1.28240	+	0.740392i	0.977286	−	0.211925i	$-0.0679734\pi$
$467$	27.7128	+	16.0000i	1.28240	+	0.740392i	0.305110	+	0.952317i	$0.401307\pi$
$468$		0			0
$469$	1.73205	+	5.00000i	0.0799787	+	0.230879i
$470$		0			0
$471$	−2.00000	+	3.46410i	−0.0921551	+	0.159617i
$472$		0			0
$473$	−1.00000	−	1.73205i	−0.0459800	−	0.0796398i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	9.00000i		−	0.412082i
$478$		0			0
$479$	1.00000	+	1.73205i	0.0456912	+	0.0791394i	0.887967	−	0.459908i	$-0.152118\pi$
$479$	1.00000	+	1.73205i	0.0456912	+	0.0791394i	−0.842275	+	0.539048i	$0.818784\pi$
$480$		0			0
$481$	−8.00000	+	13.8564i	−0.364769	+	0.631798i
$482$		0			0
$483$	−15.5885	−	3.00000i	−0.709299	−	0.136505i
$484$		0			0
$485$	9.52628	+	5.50000i	0.432566	+	0.249742i
$486$		0			0
$487$	−3.50000	−	6.06218i	−0.158600	−	0.274703i	0.775764	−	0.631023i	$-0.217365\pi$
$487$	−3.50000	−	6.06218i	−0.158600	−	0.274703i	−0.934364	+	0.356320i	$0.884031\pi$
$488$		0			0
$489$	22.0000			0.994874
$490$		0			0
$491$			3.00000i			0.135388i	0.997706	+	0.0676941i	$0.0215642\pi$
$491$			3.00000i			0.135388i	−0.997706	+	0.0676941i	$0.978436\pi$
$492$		0			0
$493$	3.46410	−	2.00000i	0.156015	−	0.0900755i
$494$		0			0
$495$	−0.500000	+	0.866025i	−0.0224733	+	0.0389249i
$496$		0			0
$497$	8.00000	−	41.5692i	0.358849	−	1.86463i
$498$		0			0
$499$	17.3205	+	10.0000i	0.775372	+	0.447661i	0.834788	−	0.550572i	$-0.185590\pi$
$499$	17.3205	+	10.0000i	0.775372	+	0.447661i	−0.0594153	+	0.998233i	$0.518924\pi$
$500$		0			0
$501$	−15.5885	+	9.00000i	−0.696441	+	0.402090i
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$	14.0000			0.622992
$506$		0			0
$507$	7.79423	−	4.50000i	0.346154	−	0.199852i
$508$		0			0
$509$	7.79423	+	4.50000i	0.345473	+	0.199459i	0.662690	−	0.748894i	$-0.269415\pi$
$509$	7.79423	+	4.50000i	0.345473	+	0.199459i	−0.317217	+	0.948353i	$0.602748\pi$
$510$		0			0
$511$	−15.0000	+	5.19615i	−0.663561	+	0.229864i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.92820	+	4.00000i	−0.305293	+	0.176261i
$516$		0			0
$517$		−	8.00000i		−	0.351840i
$518$		0			0
$519$	−18.0000			−0.790112
$520$		0			0
$521$	−9.00000	−	15.5885i	−0.394297	−	0.682943i	0.598714	−	0.800963i	$-0.295679\pi$
$521$	−9.00000	−	15.5885i	−0.394297	−	0.682943i	−0.993011	+	0.118020i	$0.962345\pi$
$522$		0			0
$523$	−29.4449	−	17.0000i	−1.28753	−	0.743358i	−0.309320	−	0.950958i	$-0.600101\pi$
$523$	−29.4449	−	17.0000i	−1.28753	−	0.743358i	−0.978214	+	0.207600i	$0.933435\pi$
$524$		0			0
$525$	6.92820	−	8.00000i	0.302372	−	0.349149i
$526$		0			0
$527$	−10.0000	+	17.3205i	−0.435607	+	0.754493i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$			1.00000i			0.0433963i
$532$		0			0
$533$		−	4.00000i		−	0.173259i
$534$		0			0
$535$	7.50000	+	12.9904i	0.324253	+	0.561623i
$536$		0			0
$537$	−10.0000	+	17.3205i	−0.431532	+	0.747435i
$538$		0			0
$539$	6.92820	−	1.00000i	0.298419	−	0.0430730i
$540$		0			0
$541$	32.9090	+	19.0000i	1.41487	+	0.816874i	0.995842	−	0.0911008i	$-0.0290385\pi$
$541$	32.9090	+	19.0000i	1.41487	+	0.816874i	0.419025	+	0.907975i	$0.362372\pi$
$542$		0			0
$543$	9.00000	+	15.5885i	0.386227	+	0.668965i
$544$		0			0
$545$	2.00000			0.0856706
$546$		0			0
$547$		−	16.0000i		−	0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$		−	16.0000i		−	0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$		0			0
$549$	6.92820	−	4.00000i	0.295689	−	0.170716i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.00000	+	1.73205i	0.0850487	+	0.0736543i
$554$		0			0
$555$	6.92820	+	4.00000i	0.294086	+	0.169791i
$556$		0			0
$557$	−19.9186	+	11.5000i	−0.843978	+	0.487271i	−0.858614	−	0.512622i	$-0.828674\pi$
$557$	−19.9186	+	11.5000i	−0.843978	+	0.487271i	0.0146368	+	0.999893i	$0.495341\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$	−4.00000			−0.168880
$562$		0			0
$563$	18.1865	−	10.5000i	0.766471	−	0.442522i	−0.0651433	−	0.997876i	$-0.520750\pi$
$563$	18.1865	−	10.5000i	0.766471	−	0.442522i	0.831614	+	0.555354i	$0.187417\pi$
$564$		0			0
$565$	1.73205	+	1.00000i	0.0728679	+	0.0420703i
$566$		0			0
$567$	2.50000	−	0.866025i	0.104990	−	0.0363696i
$568$		0			0
$569$	−20.0000	+	34.6410i	−0.838444	+	1.45223i	0.0527519	+	0.998608i	$0.483201\pi$
$569$	−20.0000	+	34.6410i	−0.838444	+	1.45223i	−0.891196	+	0.453619i	$0.850133\pi$
$570$		0			0
$571$	−10.3923	+	6.00000i	−0.434904	+	0.251092i	−0.701434	−	0.712735i	$-0.747456\pi$
$571$	−10.3923	+	6.00000i	−0.434904	+	0.251092i	0.266529	+	0.963827i	$0.414123\pi$
$572$		0			0
$573$		−	8.00000i		−	0.334205i
$574$		0			0
$575$	24.0000			1.00087
$576$		0			0
$577$	8.50000	+	14.7224i	0.353860	+	0.612903i	0.986922	−	0.161198i	$-0.0515357\pi$
$577$	8.50000	+	14.7224i	0.353860	+	0.612903i	−0.633062	+	0.774101i	$0.718202\pi$
$578$		0			0
$579$	−11.2583	−	6.50000i	−0.467880	−	0.270131i
$580$		0			0
$581$	38.9711	+	7.50000i	1.61680	+	0.311152i
$582$		0			0
$583$	−4.50000	+	7.79423i	−0.186371	+	0.322804i
$584$		0			0
$585$	1.00000	+	1.73205i	0.0413449	+	0.0716115i
$586$		0			0
$587$		−	47.0000i		−	1.93990i	−0.243309	−	0.969949i	$-0.578233\pi$
$587$		−	47.0000i		−	1.93990i	0.243309	−	0.969949i	$-0.421767\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−9.00000	−	15.5885i	−0.370211	−	0.641223i
$592$		0			0
$593$	11.0000	−	19.0526i	0.451716	−	0.782395i	−0.546777	−	0.837278i	$-0.684145\pi$
$593$	11.0000	−	19.0526i	0.451716	−	0.782395i	0.998493	+	0.0548835i	$0.0174787\pi$
$594$		0			0
$595$	−10.3923	−	2.00000i	−0.426043	−	0.0819920i
$596$		0			0
$597$	−6.92820	−	4.00000i	−0.283552	−	0.163709i
$598$		0			0
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	0.990752	+	0.135686i	$0.0433238\pi$
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	−0.377869	+	0.925859i	$0.623343\pi$
$600$		0			0
$601$	25.0000			1.01977			0.509886	−	0.860242i	$-0.329688\pi$
$601$	25.0000			1.01977			0.509886	+	0.860242i	$0.329688\pi$
$602$		0			0
$603$		−	2.00000i		−	0.0814463i
$604$		0			0
$605$	−8.66025	+	5.00000i	−0.352089	+	0.203279i
$606$		0			0
$607$	−22.5000	+	38.9711i	−0.913247	+	1.58179i	−0.103799	+	0.994598i	$0.533100\pi$
$607$	−22.5000	+	38.9711i	−0.913247	+	1.58179i	−0.809448	+	0.587192i	$0.800234\pi$
$608$		0			0
$609$	2.50000	−	0.866025i	0.101305	−	0.0350931i
$610$		0			0
$611$	−13.8564	−	8.00000i	−0.560570	−	0.323645i
$612$		0			0
$613$	22.5167	−	13.0000i	0.909439	−	0.525065i	0.0291886	−	0.999574i	$-0.490708\pi$
$613$	22.5167	−	13.0000i	0.909439	−	0.525065i	0.880251	+	0.474509i	$0.157374\pi$
$614$		0			0
$615$	−2.00000			−0.0806478
$616$		0			0
$617$	16.0000			0.644136			0.322068	−	0.946717i	$-0.395622\pi$
$617$	16.0000			0.644136			0.322068	+	0.946717i	$0.395622\pi$
$618$		0			0
$619$	27.7128	−	16.0000i	1.11387	−	0.643094i	0.174042	−	0.984738i	$-0.444317\pi$
$619$	27.7128	−	16.0000i	1.11387	−	0.643094i	0.939829	+	0.341644i	$0.110984\pi$
$620$		0			0
$621$	5.19615	+	3.00000i	0.208514	+	0.120386i
$622$		0			0
$623$	16.0000	+	13.8564i	0.641026	+	0.555145i
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			32.0000i			1.27592i
$630$		0			0
$631$	−5.00000			−0.199047			−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000			−0.199047			−0.0995234	+	0.995035i	$0.531732\pi$
$632$		0			0
$633$	−8.00000	−	13.8564i	−0.317971	−	0.550743i
$634$		0			0
$635$	11.2583	+	6.50000i	0.446773	+	0.257945i
$636$		0			0
$637$	5.19615	−	13.0000i	0.205879	−	0.515079i
$638$		0			0
$639$	−8.00000	+	13.8564i	−0.316475	+	0.548151i
$640$		0			0
$641$	2.00000	+	3.46410i	0.0789953	+	0.136824i	0.902817	−	0.430026i	$-0.141495\pi$
$641$	2.00000	+	3.46410i	0.0789953	+	0.136824i	−0.823821	+	0.566849i	$0.808162\pi$
$642$		0			0
$643$			14.0000i			0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$			14.0000i			0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$		0			0
$645$			2.00000i			0.0787499i
$646$		0			0
$647$	6.00000	+	10.3923i	0.235884	+	0.408564i	0.959529	−	0.281609i	$-0.0908680\pi$
$647$	6.00000	+	10.3923i	0.235884	+	0.408564i	−0.723645	+	0.690172i	$0.757535\pi$
$648$		0			0
$649$	0.500000	−	0.866025i	0.0196267	−	0.0339945i
$650$		0			0
$651$	−8.66025	+	10.0000i	−0.339422	+	0.391931i
$652$		0			0
$653$	12.9904	+	7.50000i	0.508353	+	0.293498i	0.732156	−	0.681137i	$-0.238514\pi$
$653$	12.9904	+	7.50000i	0.508353	+	0.293498i	−0.223803	+	0.974634i	$0.571847\pi$
$654$		0			0
$655$	1.50000	+	2.59808i	0.0586098	+	0.101515i
$656$		0			0
$657$	6.00000			0.234082
$658$		0			0
$659$		−	8.00000i		−	0.311636i	−0.987786	−	0.155818i	$-0.950199\pi$
$659$		−	8.00000i		−	0.311636i	0.987786	−	0.155818i	$-0.0498013\pi$
$660$		0			0
$661$	−25.9808	+	15.0000i	−1.01053	+	0.583432i	−0.911348	−	0.411636i	$-0.864957\pi$
$661$	−25.9808	+	15.0000i	−1.01053	+	0.583432i	−0.0991864	+	0.995069i	$0.531624\pi$
$662$		0			0
$663$	−4.00000	+	6.92820i	−0.155347	+	0.269069i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	5.19615	+	3.00000i	0.201196	+	0.116160i
$668$		0			0
$669$	18.1865	−	10.5000i	0.703132	−	0.405953i
$670$		0			0
$671$	−8.00000			−0.308837
$672$		0			0
$673$	−39.0000			−1.50334			−0.751670	−	0.659540i	$-0.770751\pi$
$673$	−39.0000			−1.50334			−0.751670	+	0.659540i	$0.770751\pi$
$674$		0			0
$675$	−3.46410	+	2.00000i	−0.133333	+	0.0769800i
$676$		0			0
$677$	28.5788	+	16.5000i	1.09837	+	0.634147i	0.935793	−	0.352549i	$-0.114685\pi$
$677$	28.5788	+	16.5000i	1.09837	+	0.634147i	0.162581	+	0.986695i	$0.448018\pi$
$678$		0			0
$679$	−5.50000	+	28.5788i	−0.211071	+	1.09676i
$680$		0			0
$681$	1.50000	−	2.59808i	0.0574801	−	0.0995585i
$682$		0			0
$683$	−37.2391	+	21.5000i	−1.42491	+	0.822675i	−0.996713	−	0.0810089i	$-0.974186\pi$
$683$	−37.2391	+	21.5000i	−1.42491	+	0.822675i	−0.428201	+	0.903684i	$0.640852\pi$
$684$		0			0
$685$		−	6.00000i		−	0.229248i
$686$		0			0
$687$	−10.0000			−0.381524
$688$		0			0
$689$	9.00000	+	15.5885i	0.342873	+	0.593873i
$690$		0			0
$691$	8.66025	+	5.00000i	0.329452	+	0.190209i	0.655598	−	0.755110i	$-0.272417\pi$
$691$	8.66025	+	5.00000i	0.329452	+	0.190209i	−0.326146	+	0.945319i	$0.605750\pi$
$692$		0			0
$693$	−2.59808	−	0.500000i	−0.0986928	−	0.0189934i
$694$		0			0
$695$	4.00000	−	6.92820i	0.151729	−	0.262802i
$696$		0			0
$697$	−4.00000	−	6.92820i	−0.151511	−	0.262424i
$698$		0			0
$699$			24.0000i			0.907763i
$700$		0			0
$701$			39.0000i			1.47301i	0.676432	+	0.736505i	$0.263525\pi$
$701$			39.0000i			1.47301i	−0.676432	+	0.736505i	$0.736475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−4.00000	+	6.92820i	−0.150649	+	0.260931i
$706$		0			0
$707$	12.1244	+	35.0000i	0.455983	+	1.31631i
$708$		0			0
$709$	13.8564	+	8.00000i	0.520388	+	0.300446i	0.737093	−	0.675791i	$-0.236198\pi$
$709$	13.8564	+	8.00000i	0.520388	+	0.300446i	−0.216705	+	0.976237i	$0.569531\pi$
$710$		0			0
$711$	−0.500000	−	0.866025i	−0.0187515	−	0.0324785i
$712$		0			0
$713$	−30.0000			−1.12351
$714$		0			0
$715$		−	2.00000i		−	0.0747958i
$716$		0			0
$717$	1.73205	−	1.00000i	0.0646846	−	0.0373457i
$718$		0			0
$719$	−14.0000	+	24.2487i	−0.522112	+	0.904324i	0.477557	+	0.878601i	$0.341522\pi$
$719$	−14.0000	+	24.2487i	−0.522112	+	0.904324i	−0.999669	+	0.0257237i	$0.991811\pi$
$720$		0			0
$721$	−16.0000	−	13.8564i	−0.595871	−	0.516040i
$722$		0			0
$723$	11.2583	+	6.50000i	0.418702	+	0.241738i
$724$		0			0
$725$	−3.46410	+	2.00000i	−0.128654	+	0.0742781i
$726$		0			0
$727$	−11.0000			−0.407967			−0.203984	−	0.978974i	$-0.565389\pi$
$727$	−11.0000			−0.407967			−0.203984	+	0.978974i	$0.565389\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−6.92820	+	4.00000i	−0.256249	+	0.147945i
$732$		0			0
$733$	−22.5167	−	13.0000i	−0.831672	−	0.480166i	0.0227529	−	0.999741i	$-0.492757\pi$
$733$	−22.5167	−	13.0000i	−0.831672	−	0.480166i	−0.854425	+	0.519575i	$0.826090\pi$
$734$		0			0
$735$	−6.50000	−	2.59808i	−0.239756	−	0.0958315i
$736$		0			0
$737$	−1.00000	+	1.73205i	−0.0368355	+	0.0638009i
$738$		0			0
$739$	17.3205	−	10.0000i	0.637145	−	0.367856i	−0.146369	−	0.989230i	$-0.546759\pi$
$739$	17.3205	−	10.0000i	0.637145	−	0.367856i	0.783514	+	0.621374i	$0.213425\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	38.0000			1.39408			0.697042	−	0.717030i	$-0.254499\pi$
$743$	38.0000			1.39408			0.697042	+	0.717030i	$0.254499\pi$
$744$		0			0
$745$	−5.00000	−	8.66025i	−0.183186	−	0.317287i
$746$		0			0
$747$	−12.9904	−	7.50000i	−0.475293	−	0.274411i
$748$		0			0
$749$	−25.9808	+	30.0000i	−0.949316	+	1.09618i
$750$		0			0
$751$	−22.5000	+	38.9711i	−0.821037	+	1.42208i	0.0838743	+	0.996476i	$0.473271\pi$
$751$	−22.5000	+	38.9711i	−0.821037	+	1.42208i	−0.904911	+	0.425601i	$0.860063\pi$
$752$		0			0
$753$	−10.5000	−	18.1865i	−0.382641	−	0.662754i
$754$		0			0
$755$		−	5.00000i		−	0.181969i
$756$		0			0
$757$			28.0000i			1.01768i	0.860862	+	0.508839i	$0.169925\pi$
$757$			28.0000i			1.01768i	−0.860862	+	0.508839i	$0.830075\pi$
$758$		0			0
$759$	−3.00000	−	5.19615i	−0.108893	−	0.188608i
$760$		0			0
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	−0.454935	−	0.890525i	$-0.650337\pi$
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	0.998684	−	0.0512772i	$-0.0163292\pi$
$762$		0			0
$763$	1.73205	+	5.00000i	0.0627044	+	0.181012i
$764$		0			0
$765$	3.46410	+	2.00000i	0.125245	+	0.0723102i
$766$		0			0
$767$	−1.00000	−	1.73205i	−0.0361079	−	0.0625407i
$768$		0			0
$769$	49.0000			1.76699			0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000			1.76699			0.883493	+	0.468445i	$0.155186\pi$
$770$		0			0
$771$		−	12.0000i		−	0.432169i
$772$		0			0
$773$	46.7654	−	27.0000i	1.68203	−	0.971123i	0.721726	−	0.692179i	$-0.243349\pi$
$773$	46.7654	−	27.0000i	1.68203	−	0.971123i	0.960307	−	0.278944i	$-0.0899843\pi$
$774$		0			0
$775$	10.0000	−	17.3205i	0.359211	−	0.622171i
$776$		0			0
$777$	−4.00000	+	20.7846i	−0.143499	+	0.745644i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	13.8564	−	8.00000i	0.495821	−	0.286263i
$782$		0			0
$783$	−1.00000			−0.0357371
$784$		0			0
$785$	−4.00000			−0.142766
$786$		0			0
$787$	6.92820	−	4.00000i	0.246964	−	0.142585i	−0.371409	−	0.928469i	$-0.621125\pi$
$787$	6.92820	−	4.00000i	0.246964	−	0.142585i	0.618373	+	0.785885i	$0.287792\pi$
$788$		0			0
$789$	−8.66025	−	5.00000i	−0.308313	−	0.178005i
$790$		0			0
$791$	−1.00000	+	5.19615i	−0.0355559	+	0.184754i
$792$		0			0
$793$	−8.00000	+	13.8564i	−0.284088	+	0.492055i
$794$		0			0
$795$	7.79423	−	4.50000i	0.276433	−	0.159599i
$796$		0			0
$797$			37.0000i			1.31061i	0.755366	+	0.655304i	$0.227459\pi$
$797$			37.0000i			1.31061i	−0.755366	+	0.655304i	$0.772541\pi$
$798$		0			0
$799$	−32.0000			−1.13208
$800$		0			0
$801$	−4.00000	−	6.92820i	−0.141333	−	0.244796i
$802$		0			0
$803$	−5.19615	−	3.00000i	−0.183368	−	0.105868i
$804$		0			0
$805$	−5.19615	−	15.0000i	−0.183140	−	0.528681i
$806$		0			0
$807$	−10.5000	+	18.1865i	−0.369618	+	0.640196i
$808$		0			0
$809$	24.0000	+	41.5692i	0.843795	+	1.46150i	0.886664	+	0.462415i	$0.153017\pi$
$809$	24.0000	+	41.5692i	0.843795	+	1.46150i	−0.0428684	+	0.999081i	$0.513650\pi$
$810$		0			0
$811$			30.0000i			1.05344i	0.850038	+	0.526721i	$0.176579\pi$
$811$			30.0000i			1.05344i	−0.850038	+	0.526721i	$0.823421\pi$
$812$		0			0
$813$		−	25.0000i		−	0.876788i
$814$		0			0
$815$	11.0000	+	19.0526i	0.385313	+	0.667382i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−3.46410	+	4.00000i	−0.121046	+	0.139771i
$820$		0			0
$821$	33.7750	+	19.5000i	1.17876	+	0.680555i	0.955726	−	0.294257i	$-0.0950722\pi$
$821$	33.7750	+	19.5000i	1.17876	+	0.680555i	0.223029	+	0.974812i	$0.428406\pi$
$822$		0			0
$823$	16.0000	+	27.7128i	0.557725	+	0.966008i	0.997686	+	0.0679910i	$0.0216589\pi$
$823$	16.0000	+	27.7128i	0.557725	+	0.966008i	−0.439961	+	0.898017i	$0.645008\pi$
$824$		0			0
$825$	4.00000			0.139262
$826$		0			0
$827$			15.0000i			0.521601i	0.965393	+	0.260801i	$0.0839865\pi$
$827$			15.0000i			0.521601i	−0.965393	+	0.260801i	$0.916014\pi$
$828$		0			0
$829$	41.5692	−	24.0000i	1.44376	−	0.833554i	0.445661	−	0.895202i	$-0.352969\pi$
$829$	41.5692	−	24.0000i	1.44376	−	0.833554i	0.998098	+	0.0616475i	$0.0196355\pi$
$830$		0			0
$831$	−16.0000	+	27.7128i	−0.555034	+	0.961347i
$832$		0			0
$833$	−4.00000	−	27.7128i	−0.138592	−	0.960192i
$834$		0			0
$835$	−15.5885	−	9.00000i	−0.539461	−	0.311458i
$836$		0			0
$837$	4.33013	−	2.50000i	0.149671	−	0.0864126i
$838$		0			0
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	28.0000			0.965517
$842$		0			0
$843$	22.5167	−	13.0000i	0.775515	−	0.447744i
$844$		0			0
$845$	7.79423	+	4.50000i	0.268130	+	0.154805i
$846$		0			0
$847$	−20.0000	−	17.3205i	−0.687208	−	0.595140i
$848$		0			0
$849$	11.0000	−	19.0526i	0.377519	−	0.653882i
$850$		0			0
$851$	−41.5692	+	24.0000i	−1.42497	+	0.822709i
$852$		0			0
$853$		−	6.00000i		−	0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$		−	6.00000i		−	0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.00000	−	6.92820i	−0.136637	−	0.236663i	0.789584	−	0.613642i	$-0.210296\pi$
$857$	−4.00000	−	6.92820i	−0.136637	−	0.236663i	−0.926222	+	0.376979i	$0.876963\pi$
$858$		0			0
$859$	−12.1244	−	7.00000i	−0.413678	−	0.238837i	0.278691	−	0.960381i	$-0.410099\pi$
$859$	−12.1244	−	7.00000i	−0.413678	−	0.238837i	−0.692369	+	0.721544i	$0.743433\pi$
$860$		0			0
$861$	−1.73205	−	5.00000i	−0.0590281	−	0.170400i
$862$		0			0
$863$	18.0000	−	31.1769i	0.612727	−	1.06127i	−0.378052	−	0.925785i	$-0.623406\pi$
$863$	18.0000	−	31.1769i	0.612727	−	1.06127i	0.990779	−	0.135490i	$-0.0432609\pi$
$864$		0			0
$865$	−9.00000	−	15.5885i	−0.306009	−	0.530023i
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$			1.00000i			0.0339227i
$870$		0			0
$871$	2.00000	+	3.46410i	0.0677674	+	0.117377i
$872$		0			0
$873$	5.50000	−	9.52628i	0.186147	−	0.322416i
$874$		0			0
$875$	23.3827	+	4.50000i	0.790479	+	0.152128i
$876$		0			0
$877$	−27.7128	−	16.0000i	−0.935795	−	0.540282i	−0.0471555	−	0.998888i	$-0.515016\pi$
$877$	−27.7128	−	16.0000i	−0.935795	−	0.540282i	−0.888640	+	0.458606i	$0.848349\pi$
$878$		0			0
$879$	−15.5000	−	26.8468i	−0.522802	−	0.905520i
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$		−	6.00000i		−	0.201916i	−0.994891	−	0.100958i	$-0.967809\pi$
$883$		−	6.00000i		−	0.201916i	0.994891	−	0.100958i	$-0.0321908\pi$
$884$		0			0
$885$	−0.866025	+	0.500000i	−0.0291111	+	0.0168073i
$886$		0			0
$887$	3.00000	−	5.19615i	0.100730	−	0.174470i	−0.811256	−	0.584692i	$-0.801215\pi$
$887$	3.00000	−	5.19615i	0.100730	−	0.174470i	0.911986	+	0.410222i	$0.134549\pi$
$888$		0			0
$889$	−6.50000	+	33.7750i	−0.218003	+	1.13278i
$890$		0			0
$891$	0.866025	+	0.500000i	0.0290129	+	0.0167506i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−20.0000			−0.668526
$896$		0			0
$897$	−12.0000			−0.400668
$898$		0			0
$899$	4.33013	−	2.50000i	0.144418	−	0.0833797i
$900$		0			0
$901$	31.1769	+	18.0000i	1.03865	+	0.599667i
$902$		0			0
$903$	−5.00000	+	1.73205i	−0.166390	+	0.0576390i
$904$		0			0
$905$	−9.00000	+	15.5885i	−0.299170	+	0.518178i
$906$		0			0
$907$	22.5167	−	13.0000i	0.747653	−	0.431658i	−0.0771920	−	0.997016i	$-0.524595\pi$
$907$	22.5167	−	13.0000i	0.747653	−	0.431658i	0.824845	+	0.565358i	$0.191262\pi$
$908$		0			0
$909$		−	14.0000i		−	0.464351i
$910$		0			0
$911$	−32.0000			−1.06021			−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000			−1.06021			−0.530104	+	0.847933i	$0.677847\pi$
$912$		0			0
$913$	7.50000	+	12.9904i	0.248214	+	0.429919i
$914$		0			0
$915$	6.92820	+	4.00000i	0.229039	+	0.132236i
$916$		0			0
$917$	−5.19615	+	6.00000i	−0.171592	+	0.198137i
$918$		0			0
$919$	20.0000	−	34.6410i	0.659739	−	1.14270i	−0.320944	−	0.947098i	$-0.604000\pi$
$919$	20.0000	−	34.6410i	0.659739	−	1.14270i	0.980683	−	0.195603i	$-0.0626666\pi$
$920$		0			0
$921$	−7.00000	−	12.1244i	−0.230658	−	0.399511i
$922$		0			0
$923$		−	32.0000i		−	1.05329i
$924$		0			0
$925$		−	32.0000i		−	1.05215i
$926$		0			0
$927$	4.00000	+	6.92820i	0.131377	+	0.227552i
$928$		0			0
$929$	−11.0000	+	19.0526i	−0.360898	+	0.625094i	−0.988109	−	0.153755i	$-0.950863\pi$
$929$	−11.0000	+	19.0526i	−0.360898	+	0.625094i	0.627211	+	0.778850i	$0.284197\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−12.1244	−	7.00000i	−0.396934	−	0.229170i
$934$		0			0
$935$	−2.00000	−	3.46410i	−0.0654070	−	0.113288i
$936$		0			0
$937$	1.00000			0.0326686			0.0163343	−	0.999867i	$-0.494800\pi$
$937$	1.00000			0.0326686			0.0163343	+	0.999867i	$0.494800\pi$
$938$		0			0
$939$		−	9.00000i		−	0.293704i
$940$		0			0
$941$	−30.3109	+	17.5000i	−0.988107	+	0.570484i	−0.904708	−	0.426033i	$-0.859911\pi$
$941$	−30.3109	+	17.5000i	−0.988107	+	0.570484i	−0.0833989	+	0.996516i	$0.526578\pi$
$942$		0			0
$943$	6.00000	−	10.3923i	0.195387	−	0.338420i
$944$		0			0
$945$	2.00000	+	1.73205i	0.0650600	+	0.0563436i
$946$		0			0
$947$	−24.2487	−	14.0000i	−0.787977	−	0.454939i	0.0512727	−	0.998685i	$-0.483672\pi$
$947$	−24.2487	−	14.0000i	−0.787977	−	0.454939i	−0.839250	+	0.543746i	$0.817006\pi$
$948$		0			0
$949$	−10.3923	+	6.00000i	−0.337348	+	0.194768i
$950$		0			0
$951$	−11.0000			−0.356699
$952$		0			0
$953$	30.0000			0.971795			0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000			0.971795			0.485898	+	0.874016i	$0.338493\pi$
$954$		0			0
$955$	6.92820	−	4.00000i	0.224191	−	0.129437i
$956$		0			0
$957$	0.866025	+	0.500000i	0.0279946	+	0.0161627i
$958$		0			0
$959$	15.0000	−	5.19615i	0.484375	−	0.167793i
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$		0			0
$963$	12.9904	−	7.50000i	0.418609	−	0.241684i
$964$		0			0
$965$		−	13.0000i		−	0.418485i
$966$		0			0
$967$	−23.0000			−0.739630			−0.369815	−	0.929105i	$-0.620579\pi$
$967$	−23.0000			−0.739630			−0.369815	+	0.929105i	$0.620579\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.9904	+	7.50000i	0.416881	+	0.240686i	0.693742	−	0.720224i	$-0.255961\pi$
$971$	12.9904	+	7.50000i	0.416881	+	0.240686i	−0.276861	+	0.960910i	$0.589294\pi$
$972$		0			0
$973$	20.7846	+	4.00000i	0.666324	+	0.128234i
$974$		0			0
$975$	4.00000	−	6.92820i	0.128103	−	0.221880i
$976$		0			0
$977$	21.0000	+	36.3731i	0.671850	+	1.16368i	0.977379	+	0.211495i	$0.0678332\pi$
$977$	21.0000	+	36.3731i	0.671850	+	1.16368i	−0.305530	+	0.952183i	$0.598833\pi$
$978$		0			0
$979$			8.00000i			0.255681i
$980$		0			0
$981$		−	2.00000i		−	0.0638551i
$982$		0			0
$983$	7.00000	+	12.1244i	0.223265	+	0.386707i	0.955798	−	0.294025i	$-0.0949950\pi$
$983$	7.00000	+	12.1244i	0.223265	+	0.386707i	−0.732532	+	0.680732i	$0.761662\pi$
$984$		0			0
$985$	9.00000	−	15.5885i	0.286764	−	0.496690i
$986$		0			0
$987$	−20.7846	−	4.00000i	−0.661581	−	0.127321i
$988$		0			0
$989$	−10.3923	−	6.00000i	−0.330456	−	0.190789i
$990$		0			0
$991$	−21.5000	−	37.2391i	−0.682970	−	1.18294i	−0.974070	−	0.226246i	$-0.927355\pi$
$991$	−21.5000	−	37.2391i	−0.682970	−	1.18294i	0.291100	−	0.956693i	$-0.405979\pi$
$992$		0			0
$993$	24.0000			0.761617
$994$		0			0
$995$		−	8.00000i		−	0.253617i
$996$		0			0
$997$	−1.73205	+	1.00000i	−0.0548546	+	0.0316703i	−0.527176	−	0.849756i	$-0.676749\pi$
$997$	−1.73205	+	1.00000i	−0.0548546	+	0.0316703i	0.472322	+	0.881426i	$0.343416\pi$
$998$		0			0
$999$	4.00000	−	6.92820i	0.126554	−	0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.bk.d.289.2	yes	4
4.3	odd	2		1344.2.bk.e.289.1	yes	4
7.4	even	3	inner	1344.2.bk.d.865.1	yes	4
8.3	odd	2		1344.2.bk.e.289.2	yes	4
8.5	even	2	inner	1344.2.bk.d.289.1	✓	4
28.11	odd	6		1344.2.bk.e.865.2	yes	4
56.11	odd	6		1344.2.bk.e.865.1	yes	4
56.53	even	6	inner	1344.2.bk.d.865.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.2.bk.d.289.1	✓	4	8.5	even	2	inner
1344.2.bk.d.289.2	yes	4	1.1	even	1	trivial
1344.2.bk.d.865.1	yes	4	7.4	even	3	inner
1344.2.bk.d.865.2	yes	4	56.53	even	6	inner
1344.2.bk.e.289.1	yes	4	4.3	odd	2
1344.2.bk.e.289.2	yes	4	8.3	odd	2
1344.2.bk.e.865.1	yes	4	56.11	odd	6
1344.2.bk.e.865.2	yes	4	28.11	odd	6

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(0.500000 - 0.866025i) q^{9} +(-0.866025 + 0.500000i) q^{11} +2.00000i q^{13} +1.00000 q^{15} +(2.00000 + 3.46410i) q^{17} +(0.866025 + 2.50000i) q^{21} +(-3.00000 + 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} -1.00000i q^{27} -1.00000i q^{29} +(2.50000 + 4.33013i) q^{31} +(-0.500000 + 0.866025i) q^{33} +(-1.73205 + 2.00000i) q^{35} +(6.92820 + 4.00000i) q^{37} +(1.00000 + 1.73205i) q^{39} -2.00000 q^{41} +2.00000i q^{43} +(0.866025 - 0.500000i) q^{45} +(-4.00000 + 6.92820i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(3.46410 + 2.00000i) q^{51} +(7.79423 - 4.50000i) q^{53} -1.00000 q^{55} +(-0.866025 + 0.500000i) q^{59} +(6.92820 + 4.00000i) q^{61} +(2.00000 + 1.73205i) q^{63} +(-1.00000 + 1.73205i) q^{65} +(1.73205 - 1.00000i) q^{67} +6.00000i q^{69} -16.0000 q^{71} +(3.00000 + 5.19615i) q^{73} +(-3.46410 - 2.00000i) q^{75} +(-0.866025 - 2.50000i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -15.0000i q^{83} +4.00000i q^{85} +(-0.500000 - 0.866025i) q^{87} +(4.00000 - 6.92820i) q^{89} +(-5.19615 - 1.00000i) q^{91} +(4.33013 + 2.50000i) q^{93} +11.0000 q^{97} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^7 + 2 * q^9	\( 4 q - 2 q^{7} + 2 q^{9} + 4 q^{15} + 8 q^{17} - 12 q^{23} - 8 q^{25} + 10 q^{31} - 2 q^{33} + 4 q^{39} - 8 q^{41} - 16 q^{47} - 26 q^{49} - 4 q^{55} + 8 q^{63} - 4 q^{65} - 64 q^{71} + 12 q^{73} + 2 q^{79} - 2 q^{81} - 2 q^{87} + 16 q^{89} + 44 q^{97}+O(q^{100}) \) 4 * q - 2 * q^7 + 2 * q^9 + 4 * q^15 + 8 * q^17 - 12 * q^23 - 8 * q^25 + 10 * q^31 - 2 * q^33 + 4 * q^39 - 8 * q^41 - 16 * q^47 - 26 * q^49 - 4 * q^55 + 8 * q^63 - 4 * q^65 - 64 * q^71 + 12 * q^73 + 2 * q^79 - 2 * q^81 - 2 * q^87 + 16 * q^89 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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