LMFDB - Embedded newform 2394.2.o.a.505.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(505,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("2394.505");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.o (of order $3$, 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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 798)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			505.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2394.505
Dual form			2394.2.o.a.1261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} -1.00000 q^{7} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} -4.00000 q^{11} +(-1.50000 - 2.59808i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.00000 + 1.73205i) q^{17} +(-0.500000 + 4.33013i) q^{19} +3.00000 q^{20} +(2.00000 - 3.46410i) q^{22} +(0.500000 + 0.866025i) q^{23} +(-2.00000 - 3.46410i) q^{25} +3.00000 q^{26} +(0.500000 + 0.866025i) q^{28} -6.00000 q^{31} +(-0.500000 - 0.866025i) q^{32} +(-1.00000 - 1.73205i) q^{34} +(1.50000 - 2.59808i) q^{35} +4.00000 q^{37} +(-3.50000 - 2.59808i) q^{38} +(-1.50000 + 2.59808i) q^{40} +(2.00000 - 3.46410i) q^{41} +(3.00000 - 5.19615i) q^{43} +(2.00000 + 3.46410i) q^{44} -1.00000 q^{46} +(-1.00000 - 1.73205i) q^{47} +1.00000 q^{49} +4.00000 q^{50} +(-1.50000 + 2.59808i) q^{52} +(-2.00000 - 3.46410i) q^{53} +(6.00000 - 10.3923i) q^{55} -1.00000 q^{56} +(6.50000 - 11.2583i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(3.00000 - 5.19615i) q^{62} +1.00000 q^{64} +9.00000 q^{65} +(6.00000 + 10.3923i) q^{67} +2.00000 q^{68} +(1.50000 + 2.59808i) q^{70} +(-3.50000 + 6.06218i) q^{71} +(1.00000 - 1.73205i) q^{73} +(-2.00000 + 3.46410i) q^{74} +(4.00000 - 1.73205i) q^{76} +4.00000 q^{77} +(4.00000 - 6.92820i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(2.00000 + 3.46410i) q^{82} -17.0000 q^{83} +(-3.00000 - 5.19615i) q^{85} +(3.00000 + 5.19615i) q^{86} -4.00000 q^{88} +(2.00000 + 3.46410i) q^{89} +(1.50000 + 2.59808i) q^{91} +(0.500000 - 0.866025i) q^{92} +2.00000 q^{94} +(-10.5000 - 7.79423i) q^{95} +(5.00000 - 8.66025i) q^{97} +(-0.500000 + 0.866025i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 3 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 - 3 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} - 3 q^{5} - 2 q^{7} + 2 q^{8} - 3 q^{10} - 8 q^{11} - 3 q^{13} + q^{14} - q^{16} - 2 q^{17} - q^{19} + 6 q^{20} + 4 q^{22} + q^{23} - 4 q^{25} + 6 q^{26} + q^{28} - 12 q^{31} - q^{32} - 2 q^{34} + 3 q^{35} + 8 q^{37} - 7 q^{38} - 3 q^{40} + 4 q^{41} + 6 q^{43} + 4 q^{44} - 2 q^{46} - 2 q^{47} + 2 q^{49} + 8 q^{50} - 3 q^{52} - 4 q^{53} + 12 q^{55} - 2 q^{56} + 13 q^{59} - 5 q^{61} + 6 q^{62} + 2 q^{64} + 18 q^{65} + 12 q^{67} + 4 q^{68} + 3 q^{70} - 7 q^{71} + 2 q^{73} - 4 q^{74} + 8 q^{76} + 8 q^{77} + 8 q^{79} - 3 q^{80} + 4 q^{82} - 34 q^{83} - 6 q^{85} + 6 q^{86} - 8 q^{88} + 4 q^{89} + 3 q^{91} + q^{92} + 4 q^{94} - 21 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 3 * q^5 - 2 * q^7 + 2 * q^8 - 3 * q^10 - 8 * q^11 - 3 * q^13 + q^14 - q^16 - 2 * q^17 - q^19 + 6 * q^20 + 4 * q^22 + q^23 - 4 * q^25 + 6 * q^26 + q^28 - 12 * q^31 - q^32 - 2 * q^34 + 3 * q^35 + 8 * q^37 - 7 * q^38 - 3 * q^40 + 4 * q^41 + 6 * q^43 + 4 * q^44 - 2 * q^46 - 2 * q^47 + 2 * q^49 + 8 * q^50 - 3 * q^52 - 4 * q^53 + 12 * q^55 - 2 * q^56 + 13 * q^59 - 5 * q^61 + 6 * q^62 + 2 * q^64 + 18 * q^65 + 12 * q^67 + 4 * q^68 + 3 * q^70 - 7 * q^71 + 2 * q^73 - 4 * q^74 + 8 * q^76 + 8 * q^77 + 8 * q^79 - 3 * q^80 + 4 * q^82 - 34 * q^83 - 6 * q^85 + 6 * q^86 - 8 * q^88 + 4 * q^89 + 3 * q^91 + q^92 + 4 * q^94 - 21 * q^95 + 10 * q^97 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	0.306851	+	0.951757i	$0.400725\pi$
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.50000	−	2.59808i	−0.474342	−	0.821584i
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	−1.50000	−	2.59808i	−0.416025	−	0.720577i	0.579510	−	0.814965i	$-0.303244\pi$
$13$	−1.50000	−	2.59808i	−0.416025	−	0.720577i	−0.995535	+	0.0943882i	$0.969911\pi$
$14$	0.500000	−	0.866025i	0.133631	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$		0			0
$19$	−0.500000	+	4.33013i	−0.114708	+	0.993399i
$19$	−0.500000	+	4.33013i	−0.114708	+	0.993399i
$20$	3.00000			0.670820
$21$		0			0
$22$	2.00000	−	3.46410i	0.426401	−	0.738549i
$23$	0.500000	+	0.866025i	0.104257	+	0.180579i	0.913434	−	0.406986i	$-0.133420\pi$
$23$	0.500000	+	0.866025i	0.104257	+	0.180579i	−0.809177	+	0.587565i	$0.800087\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$	3.00000			0.588348
$27$		0			0
$28$	0.500000	+	0.866025i	0.0944911	+	0.163663i
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−1.00000	−	1.73205i	−0.171499	−	0.297044i
$35$	1.50000	−	2.59808i	0.253546	−	0.439155i
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$	−3.50000	−	2.59808i	−0.567775	−	0.421464i
$39$		0			0
$40$	−1.50000	+	2.59808i	−0.237171	+	0.410792i
$41$	2.00000	−	3.46410i	0.312348	−	0.541002i	−0.666523	−	0.745485i	$-0.732218\pi$
$41$	2.00000	−	3.46410i	0.312348	−	0.541002i	0.978870	+	0.204483i	$0.0655513\pi$
$42$		0			0
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	−0.541332	−	0.840809i	$-0.682080\pi$
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	0.998828	+	0.0484030i	$0.0154132\pi$
$44$	2.00000	+	3.46410i	0.301511	+	0.522233i
$45$		0			0
$46$	−1.00000			−0.147442
$47$	−1.00000	−	1.73205i	−0.145865	−	0.252646i	0.783830	−	0.620975i	$-0.213263\pi$
$47$	−1.00000	−	1.73205i	−0.145865	−	0.252646i	−0.929695	+	0.368329i	$0.879930\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$	4.00000			0.565685
$51$		0			0
$52$	−1.50000	+	2.59808i	−0.208013	+	0.360288i
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	0.695344	−	0.718677i	$-0.255252\pi$
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	−0.970065	+	0.242846i	$0.921919\pi$
$54$		0			0
$55$	6.00000	−	10.3923i	0.809040	−	1.40130i
$56$	−1.00000			−0.133631
$57$		0			0
$58$		0			0
$59$	6.50000	−	11.2583i	0.846228	−	1.46571i	−0.0383226	−	0.999265i	$-0.512201\pi$
$59$	6.50000	−	11.2583i	0.846228	−	1.46571i	0.884551	−	0.466444i	$-0.154465\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$	3.00000	−	5.19615i	0.381000	−	0.659912i
$63$		0			0
$64$	1.00000			0.125000
$65$	9.00000			1.11631
$66$		0			0
$67$	6.00000	+	10.3923i	0.733017	+	1.26962i	0.955588	+	0.294706i	$0.0952216\pi$
$67$	6.00000	+	10.3923i	0.733017	+	1.26962i	−0.222571	+	0.974916i	$0.571445\pi$
$68$	2.00000			0.242536
$69$		0			0
$70$	1.50000	+	2.59808i	0.179284	+	0.310530i
$71$	−3.50000	+	6.06218i	−0.415374	+	0.719448i	−0.995468	−	0.0951014i	$-0.969682\pi$
$71$	−3.50000	+	6.06218i	−0.415374	+	0.719448i	0.580094	+	0.814550i	$0.303016\pi$
$72$		0			0
$73$	1.00000	−	1.73205i	0.117041	−	0.202721i	−0.801553	−	0.597924i	$-0.795992\pi$
$73$	1.00000	−	1.73205i	0.117041	−	0.202721i	0.918594	+	0.395203i	$0.129326\pi$
$74$	−2.00000	+	3.46410i	−0.232495	+	0.402694i
$75$		0			0
$76$	4.00000	−	1.73205i	0.458831	−	0.198680i
$77$	4.00000			0.455842
$78$		0			0
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	−0.548352	−	0.836247i	$-0.684745\pi$
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	0.998388	+	0.0567635i	$0.0180781\pi$
$80$	−1.50000	−	2.59808i	−0.167705	−	0.290474i
$81$		0			0
$82$	2.00000	+	3.46410i	0.220863	+	0.382546i
$83$	−17.0000			−1.86599			−0.932996	−	0.359886i	$-0.882816\pi$
$83$	−17.0000			−1.86599			−0.932996	+	0.359886i	$0.882816\pi$
$84$		0			0
$85$	−3.00000	−	5.19615i	−0.325396	−	0.563602i
$86$	3.00000	+	5.19615i	0.323498	+	0.560316i
$87$		0			0
$88$	−4.00000			−0.426401
$89$	2.00000	+	3.46410i	0.212000	+	0.367194i	0.952340	−	0.305038i	$-0.0986691\pi$
$89$	2.00000	+	3.46410i	0.212000	+	0.367194i	−0.740341	+	0.672232i	$0.765336\pi$
$90$		0			0
$91$	1.50000	+	2.59808i	0.157243	+	0.272352i
$92$	0.500000	−	0.866025i	0.0521286	−	0.0902894i
$93$		0			0
$94$	2.00000			0.206284
$95$	−10.5000	−	7.79423i	−1.07728	−	0.799671i
$96$		0			0
$97$	5.00000	−	8.66025i	0.507673	−	0.879316i	−0.492287	−	0.870433i	$-0.663839\pi$
$97$	5.00000	−	8.66025i	0.507673	−	0.879316i	0.999961	−	0.00888289i	$-0.00282755\pi$
$98$	−0.500000	+	0.866025i	−0.0505076	+	0.0874818i
$99$		0			0
$100$	−2.00000	+	3.46410i	−0.200000	+	0.346410i
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	0.969664	+	0.244443i	$0.0786053\pi$
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	−0.273138	+	0.961975i	$0.588061\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	−1.50000	−	2.59808i	−0.147087	−	0.254762i
$105$		0			0
$106$	4.00000			0.388514
$107$	10.0000			0.966736			0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000			0.966736			0.483368	+	0.875417i	$0.339413\pi$
$108$		0			0
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	−0.754204	−	0.656640i	$-0.771977\pi$
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	0.945769	+	0.324840i	$0.105310\pi$
$110$	6.00000	+	10.3923i	0.572078	+	0.990867i
$111$		0			0
$112$	0.500000	−	0.866025i	0.0472456	−	0.0818317i
$113$	5.00000			0.470360			0.235180	−	0.971952i	$-0.424432\pi$
$113$	5.00000			0.470360			0.235180	+	0.971952i	$0.424432\pi$
$114$		0			0
$115$	−3.00000			−0.279751
$116$		0			0
$117$		0			0
$118$	6.50000	+	11.2583i	0.598374	+	1.03641i
$119$	1.00000	−	1.73205i	0.0916698	−	0.158777i
$120$		0			0
$121$	5.00000			0.454545
$122$	5.00000			0.452679
$123$		0			0
$124$	3.00000	+	5.19615i	0.269408	+	0.466628i
$125$	−3.00000			−0.268328
$126$		0			0
$127$	5.50000	+	9.52628i	0.488046	+	0.845321i	0.999905	−	0.0137486i	$-0.00437646\pi$
$127$	5.50000	+	9.52628i	0.488046	+	0.845321i	−0.511859	+	0.859069i	$0.671043\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−4.50000	+	7.79423i	−0.394676	+	0.683599i
$131$	−4.50000	+	7.79423i	−0.393167	+	0.680985i	−0.992865	−	0.119241i	$-0.961954\pi$
$131$	−4.50000	+	7.79423i	−0.393167	+	0.680985i	0.599699	+	0.800226i	$0.295287\pi$
$132$		0			0
$133$	0.500000	−	4.33013i	0.0433555	−	0.375470i
$134$	−12.0000			−1.03664
$135$		0			0
$136$	−1.00000	+	1.73205i	−0.0857493	+	0.148522i
$137$	−10.5000	−	18.1865i	−0.897076	−	1.55378i	−0.831215	−	0.555952i	$-0.812354\pi$
$137$	−10.5000	−	18.1865i	−0.897076	−	1.55378i	−0.0658609	−	0.997829i	$-0.520979\pi$
$138$		0			0
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	−0.296866	+	0.954919i	$0.595942\pi$
$140$	−3.00000			−0.253546
$141$		0			0
$142$	−3.50000	−	6.06218i	−0.293713	−	0.508727i
$143$	6.00000	+	10.3923i	0.501745	+	0.869048i
$144$		0			0
$145$		0			0
$146$	1.00000	+	1.73205i	0.0827606	+	0.143346i
$147$		0			0
$148$	−2.00000	−	3.46410i	−0.164399	−	0.284747i
$149$	10.0000	−	17.3205i	0.819232	−	1.41895i	−0.0870170	−	0.996207i	$-0.527733\pi$
$149$	10.0000	−	17.3205i	0.819232	−	1.41895i	0.906249	−	0.422744i	$-0.138933\pi$
$150$		0			0
$151$	19.0000			1.54620			0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000			1.54620			0.773099	+	0.634285i	$0.218706\pi$
$152$	−0.500000	+	4.33013i	−0.0405554	+	0.351220i
$153$		0			0
$154$	−2.00000	+	3.46410i	−0.161165	+	0.279145i
$155$	9.00000	−	15.5885i	0.722897	−	1.25210i
$156$		0			0
$157$	4.50000	−	7.79423i	0.359139	−	0.622047i	−0.628678	−	0.777666i	$-0.716404\pi$
$157$	4.50000	−	7.79423i	0.359139	−	0.622047i	0.987817	+	0.155618i	$0.0497370\pi$
$158$	4.00000	+	6.92820i	0.318223	+	0.551178i
$159$		0			0
$160$	3.00000			0.237171
$161$	−0.500000	−	0.866025i	−0.0394055	−	0.0682524i
$162$		0			0
$163$	14.0000			1.09656			0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000			1.09656			0.548282	+	0.836293i	$0.315282\pi$
$164$	−4.00000			−0.312348
$165$		0			0
$166$	8.50000	−	14.7224i	0.659728	−	1.14268i
$167$	−9.00000	−	15.5885i	−0.696441	−	1.20627i	−0.969693	−	0.244328i	$-0.921432\pi$
$167$	−9.00000	−	15.5885i	−0.696441	−	1.20627i	0.273252	−	0.961943i	$-0.411901\pi$
$168$		0			0
$169$	2.00000	−	3.46410i	0.153846	−	0.266469i
$170$	6.00000			0.460179
$171$		0			0
$172$	−6.00000			−0.457496
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	−0.426329	−	0.904568i	$-0.640193\pi$
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	0.996544	−	0.0830722i	$-0.0264732\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$	2.00000	−	3.46410i	0.150756	−	0.261116i
$177$		0			0
$178$	−4.00000			−0.299813
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$	−1.50000	−	2.59808i	−0.111494	−	0.193113i	0.804879	−	0.593439i	$-0.202230\pi$
$181$	−1.50000	−	2.59808i	−0.111494	−	0.193113i	−0.916373	+	0.400326i	$0.868897\pi$
$182$	−3.00000			−0.222375
$183$		0			0
$184$	0.500000	+	0.866025i	0.0368605	+	0.0638442i
$185$	−6.00000	+	10.3923i	−0.441129	+	0.764057i
$186$		0			0
$187$	4.00000	−	6.92820i	0.292509	−	0.506640i
$188$	−1.00000	+	1.73205i	−0.0729325	+	0.126323i
$189$		0			0
$190$	12.0000	−	5.19615i	0.870572	−	0.376969i
$191$	11.0000			0.795932			0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000			0.795932			0.397966	+	0.917400i	$0.369716\pi$
$192$		0			0
$193$	−2.50000	+	4.33013i	−0.179954	+	0.311689i	−0.941865	−	0.335993i	$-0.890928\pi$
$193$	−2.50000	+	4.33013i	−0.179954	+	0.311689i	0.761911	+	0.647682i	$0.224262\pi$
$194$	5.00000	+	8.66025i	0.358979	+	0.621770i
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.0357143	−	0.0618590i
$197$	−4.00000			−0.284988			−0.142494	−	0.989796i	$-0.545512\pi$
$197$	−4.00000			−0.284988			−0.142494	+	0.989796i	$0.545512\pi$
$198$		0			0
$199$	−7.00000	−	12.1244i	−0.496217	−	0.859473i	0.503774	−	0.863836i	$-0.331945\pi$
$199$	−7.00000	−	12.1244i	−0.496217	−	0.859473i	−0.999990	+	0.00436292i	$0.998611\pi$
$200$	−2.00000	−	3.46410i	−0.141421	−	0.244949i
$201$		0			0
$202$	−14.0000			−0.985037
$203$		0			0
$204$		0			0
$205$	6.00000	+	10.3923i	0.419058	+	0.725830i
$206$		0			0
$207$		0			0
$208$	3.00000			0.208013
$209$	2.00000	−	17.3205i	0.138343	−	1.19808i
$210$		0			0
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	−0.694857	−	0.719148i	$-0.744533\pi$
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	0.970229	+	0.242190i	$0.0778659\pi$
$212$	−2.00000	+	3.46410i	−0.137361	+	0.237915i
$213$		0			0
$214$	−5.00000	+	8.66025i	−0.341793	+	0.592003i
$215$	9.00000	+	15.5885i	0.613795	+	1.06312i
$216$		0			0
$217$	6.00000			0.407307
$218$	2.00000	+	3.46410i	0.135457	+	0.234619i
$219$		0			0
$220$	−12.0000			−0.809040
$221$	6.00000			0.403604
$222$		0			0
$223$	9.00000	−	15.5885i	0.602685	−	1.04388i	−0.389728	−	0.920930i	$-0.627431\pi$
$223$	9.00000	−	15.5885i	0.602685	−	1.04388i	0.992413	−	0.122950i	$-0.0392356\pi$
$224$	0.500000	+	0.866025i	0.0334077	+	0.0578638i
$225$		0			0
$226$	−2.50000	+	4.33013i	−0.166298	+	0.288036i
$227$	−5.00000			−0.331862			−0.165931	−	0.986137i	$-0.553063\pi$
$227$	−5.00000			−0.331862			−0.165931	+	0.986137i	$0.553063\pi$
$228$		0			0
$229$	−17.0000			−1.12339			−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000			−1.12339			−0.561696	+	0.827344i	$0.689851\pi$
$230$	1.50000	−	2.59808i	0.0989071	−	0.171312i
$231$		0			0
$232$		0			0
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	−0.680135	−	0.733087i	$-0.738079\pi$
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	0.974939	+	0.222470i	$0.0714120\pi$
$234$		0			0
$235$	6.00000			0.391397
$236$	−13.0000			−0.846228
$237$		0			0
$238$	1.00000	+	1.73205i	0.0648204	+	0.112272i
$239$	−27.0000			−1.74648			−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000			−1.74648			−0.873242	+	0.487286i	$0.837987\pi$
$240$		0			0
$241$	−11.0000	−	19.0526i	−0.708572	−	1.22728i	−0.965387	−	0.260822i	$-0.916006\pi$
$241$	−11.0000	−	19.0526i	−0.708572	−	1.22728i	0.256814	−	0.966461i	$-0.417327\pi$
$242$	−2.50000	+	4.33013i	−0.160706	+	0.278351i
$243$		0			0
$244$	−2.50000	+	4.33013i	−0.160046	+	0.277208i
$245$	−1.50000	+	2.59808i	−0.0958315	+	0.165985i
$246$		0			0
$247$	12.0000	−	5.19615i	0.763542	−	0.330623i
$248$	−6.00000			−0.381000
$249$		0			0
$250$	1.50000	−	2.59808i	0.0948683	−	0.164317i
$251$	10.5000	+	18.1865i	0.662754	+	1.14792i	0.979889	+	0.199543i	$0.0639459\pi$
$251$	10.5000	+	18.1865i	0.662754	+	1.14792i	−0.317135	+	0.948380i	$0.602721\pi$
$252$		0			0
$253$	−2.00000	−	3.46410i	−0.125739	−	0.217786i
$254$	−11.0000			−0.690201
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	0.990217	−	0.139533i	$-0.0445601\pi$
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	−0.615948	+	0.787787i	$0.711227\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$	−4.50000	−	7.79423i	−0.279078	−	0.483378i
$261$		0			0
$262$	−4.50000	−	7.79423i	−0.278011	−	0.481529i
$263$	−1.50000	+	2.59808i	−0.0924940	+	0.160204i	−0.908560	−	0.417755i	$-0.862817\pi$
$263$	−1.50000	+	2.59808i	−0.0924940	+	0.160204i	0.816066	+	0.577959i	$0.196151\pi$
$264$		0			0
$265$	12.0000			0.737154
$266$	3.50000	+	2.59808i	0.214599	+	0.159298i
$267$		0			0
$268$	6.00000	−	10.3923i	0.366508	−	0.634811i
$269$	7.00000	−	12.1244i	0.426798	−	0.739235i	−0.569789	−	0.821791i	$-0.692975\pi$
$269$	7.00000	−	12.1244i	0.426798	−	0.739235i	0.996586	+	0.0825561i	$0.0263084\pi$
$270$		0			0
$271$	−13.0000	+	22.5167i	−0.789694	+	1.36779i	0.136461	+	0.990645i	$0.456427\pi$
$271$	−13.0000	+	22.5167i	−0.789694	+	1.36779i	−0.926155	+	0.377144i	$0.876906\pi$
$272$	−1.00000	−	1.73205i	−0.0606339	−	0.105021i
$273$		0			0
$274$	21.0000			1.26866
$275$	8.00000	+	13.8564i	0.482418	+	0.835573i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−16.0000			−0.959616
$279$		0			0
$280$	1.50000	−	2.59808i	0.0896421	−	0.155265i
$281$	−13.0000	−	22.5167i	−0.775515	−	1.34323i	−0.934505	−	0.355951i	$-0.884157\pi$
$281$	−13.0000	−	22.5167i	−0.775515	−	1.34323i	0.158990	−	0.987280i	$-0.449176\pi$
$282$		0			0
$283$	−12.5000	+	21.6506i	−0.743048	+	1.28700i	0.208053	+	0.978117i	$0.433287\pi$
$283$	−12.5000	+	21.6506i	−0.743048	+	1.28700i	−0.951101	+	0.308879i	$0.900046\pi$
$284$	7.00000			0.415374
$285$		0			0
$286$	−12.0000			−0.709575
$287$	−2.00000	+	3.46410i	−0.118056	+	0.204479i
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$		0			0
$292$	−2.00000			−0.117041
$293$	19.0000			1.10999			0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000			1.10999			0.554996	+	0.831853i	$0.312720\pi$
$294$		0			0
$295$	19.5000	+	33.7750i	1.13533	+	1.96646i
$296$	4.00000			0.232495
$297$		0			0
$298$	10.0000	+	17.3205i	0.579284	+	1.00335i
$299$	1.50000	−	2.59808i	0.0867472	−	0.150251i
$300$		0			0
$301$	−3.00000	+	5.19615i	−0.172917	+	0.299501i
$302$	−9.50000	+	16.4545i	−0.546664	+	0.946849i
$303$		0			0
$304$	−3.50000	−	2.59808i	−0.200739	−	0.149010i
$305$	15.0000			0.858898
$306$		0			0
$307$	−7.50000	+	12.9904i	−0.428048	+	0.741400i	−0.996700	−	0.0811780i	$-0.974132\pi$
$307$	−7.50000	+	12.9904i	−0.428048	+	0.741400i	0.568652	+	0.822578i	$0.307465\pi$
$308$	−2.00000	−	3.46410i	−0.113961	−	0.197386i
$309$		0			0
$310$	9.00000	+	15.5885i	0.511166	+	0.885365i
$311$	2.00000			0.113410			0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000			0.113410			0.0567048	+	0.998391i	$0.481941\pi$
$312$		0			0
$313$	−12.0000	−	20.7846i	−0.678280	−	1.17482i	−0.975499	−	0.220006i	$-0.929392\pi$
$313$	−12.0000	−	20.7846i	−0.678280	−	1.17482i	0.297218	−	0.954810i	$-0.403941\pi$
$314$	4.50000	+	7.79423i	0.253950	+	0.439854i
$315$		0			0
$316$	−8.00000			−0.450035
$317$	3.00000	+	5.19615i	0.168497	+	0.291845i	0.937892	−	0.346929i	$-0.112775\pi$
$317$	3.00000	+	5.19615i	0.168497	+	0.291845i	−0.769395	+	0.638774i	$0.779442\pi$
$318$		0			0
$319$		0			0
$320$	−1.50000	+	2.59808i	−0.0838525	+	0.145237i
$321$		0			0
$322$	1.00000			0.0557278
$323$	−7.00000	−	5.19615i	−0.389490	−	0.289122i
$324$		0			0
$325$	−6.00000	+	10.3923i	−0.332820	+	0.576461i
$326$	−7.00000	+	12.1244i	−0.387694	+	0.671506i
$327$		0			0
$328$	2.00000	−	3.46410i	0.110432	−	0.191273i
$329$	1.00000	+	1.73205i	0.0551318	+	0.0954911i
$330$		0			0
$331$	−24.0000			−1.31916			−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000			−1.31916			−0.659580	+	0.751635i	$0.729266\pi$
$332$	8.50000	+	14.7224i	0.466498	+	0.807998i
$333$		0			0
$334$	18.0000			0.984916
$335$	−36.0000			−1.96689
$336$		0			0
$337$	−8.50000	+	14.7224i	−0.463025	+	0.801982i	−0.999110	−	0.0421818i	$-0.986569\pi$
$337$	−8.50000	+	14.7224i	−0.463025	+	0.801982i	0.536085	+	0.844164i	$0.319902\pi$
$338$	2.00000	+	3.46410i	0.108786	+	0.188422i
$339$		0			0
$340$	−3.00000	+	5.19615i	−0.162698	+	0.281801i
$341$	24.0000			1.29967
$342$		0			0
$343$	−1.00000			−0.0539949
$344$	3.00000	−	5.19615i	0.161749	−	0.280158i
$345$		0			0
$346$	7.50000	+	12.9904i	0.403202	+	0.698367i
$347$	11.0000	−	19.0526i	0.590511	−	1.02279i	−0.403653	−	0.914912i	$-0.632260\pi$
$347$	11.0000	−	19.0526i	0.590511	−	1.02279i	0.994164	−	0.107883i	$-0.0344071\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$	−4.00000			−0.213809
$351$		0			0
$352$	2.00000	+	3.46410i	0.106600	+	0.184637i
$353$	−12.0000			−0.638696			−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000			−0.638696			−0.319348	+	0.947638i	$0.603464\pi$
$354$		0			0
$355$	−10.5000	−	18.1865i	−0.557282	−	0.965241i
$356$	2.00000	−	3.46410i	0.106000	−	0.183597i
$357$		0			0
$358$	5.00000	−	8.66025i	0.264258	−	0.457709i
$359$	12.0000	−	20.7846i	0.633336	−	1.09697i	−0.353529	−	0.935423i	$-0.615019\pi$
$359$	12.0000	−	20.7846i	0.633336	−	1.09697i	0.986865	−	0.161546i	$-0.0516481\pi$
$360$		0			0
$361$	−18.5000	−	4.33013i	−0.973684	−	0.227901i
$362$	3.00000			0.157676
$363$		0			0
$364$	1.50000	−	2.59808i	0.0786214	−	0.136176i
$365$	3.00000	+	5.19615i	0.157027	+	0.271979i
$366$		0			0
$367$	−16.0000	−	27.7128i	−0.835193	−	1.44660i	−0.893873	−	0.448320i	$-0.852022\pi$
$367$	−16.0000	−	27.7128i	−0.835193	−	1.44660i	0.0586798	−	0.998277i	$-0.481311\pi$
$368$	−1.00000			−0.0521286
$369$		0			0
$370$	−6.00000	−	10.3923i	−0.311925	−	0.540270i
$371$	2.00000	+	3.46410i	0.103835	+	0.179847i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$	4.00000	+	6.92820i	0.206835	+	0.358249i
$375$		0			0
$376$	−1.00000	−	1.73205i	−0.0515711	−	0.0893237i
$377$		0			0
$378$		0			0
$379$	−34.0000			−1.74646			−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000			−1.74646			−0.873231	+	0.487306i	$0.837980\pi$
$380$	−1.50000	+	12.9904i	−0.0769484	+	0.666392i
$381$		0			0
$382$	−5.50000	+	9.52628i	−0.281404	+	0.487407i
$383$	−2.00000	+	3.46410i	−0.102195	+	0.177007i	−0.912589	−	0.408879i	$-0.865920\pi$
$383$	−2.00000	+	3.46410i	−0.102195	+	0.177007i	0.810394	+	0.585886i	$0.199253\pi$
$384$		0			0
$385$	−6.00000	+	10.3923i	−0.305788	+	0.529641i
$386$	−2.50000	−	4.33013i	−0.127247	−	0.220398i
$387$		0			0
$388$	−10.0000			−0.507673
$389$	4.00000	+	6.92820i	0.202808	+	0.351274i	0.949432	−	0.313972i	$-0.101660\pi$
$389$	4.00000	+	6.92820i	0.202808	+	0.351274i	−0.746624	+	0.665246i	$0.768327\pi$
$390$		0			0
$391$	−2.00000			−0.101144
$392$	1.00000			0.0505076
$393$		0			0
$394$	2.00000	−	3.46410i	0.100759	−	0.174519i
$395$	12.0000	+	20.7846i	0.603786	+	1.04579i
$396$		0			0
$397$	−9.00000	+	15.5885i	−0.451697	+	0.782362i	−0.998492	−	0.0549046i	$-0.982515\pi$
$397$	−9.00000	+	15.5885i	−0.451697	+	0.782362i	0.546795	+	0.837267i	$0.315848\pi$
$398$	14.0000			0.701757
$399$		0			0
$400$	4.00000			0.200000
$401$	1.50000	−	2.59808i	0.0749064	−	0.129742i	−0.826139	−	0.563466i	$-0.809468\pi$
$401$	1.50000	−	2.59808i	0.0749064	−	0.129742i	0.901046	+	0.433724i	$0.142801\pi$
$402$		0			0
$403$	9.00000	+	15.5885i	0.448322	+	0.776516i
$404$	7.00000	−	12.1244i	0.348263	−	0.603209i
$405$		0			0
$406$		0			0
$407$	−16.0000			−0.793091
$408$		0			0
$409$	−12.0000	−	20.7846i	−0.593362	−	1.02773i	−0.993776	−	0.111398i	$-0.964467\pi$
$409$	−12.0000	−	20.7846i	−0.593362	−	1.02773i	0.400414	−	0.916334i	$-0.368866\pi$
$410$	−12.0000			−0.592638
$411$		0			0
$412$		0			0
$413$	−6.50000	+	11.2583i	−0.319844	+	0.553986i
$414$		0			0
$415$	25.5000	−	44.1673i	1.25175	−	2.16809i
$416$	−1.50000	+	2.59808i	−0.0735436	+	0.127381i
$417$		0			0
$418$	14.0000	+	10.3923i	0.684762	+	0.508304i
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	4.00000	−	6.92820i	0.194948	−	0.337660i	−0.751935	−	0.659237i	$-0.770879\pi$
$421$	4.00000	−	6.92820i	0.194948	−	0.337660i	0.946883	+	0.321577i	$0.104213\pi$
$422$	4.00000	+	6.92820i	0.194717	+	0.337260i
$423$		0			0
$424$	−2.00000	−	3.46410i	−0.0971286	−	0.168232i
$425$	8.00000			0.388057
$426$		0			0
$427$	2.50000	+	4.33013i	0.120983	+	0.209550i
$428$	−5.00000	−	8.66025i	−0.241684	−	0.418609i
$429$		0			0
$430$	−18.0000			−0.868037
$431$	−8.00000	−	13.8564i	−0.385346	−	0.667440i	0.606471	−	0.795106i	$-0.292585\pi$
$431$	−8.00000	−	13.8564i	−0.385346	−	0.667440i	−0.991817	+	0.127666i	$0.959251\pi$
$432$		0			0
$433$	12.0000	+	20.7846i	0.576683	+	0.998845i	0.995857	+	0.0909384i	$0.0289866\pi$
$433$	12.0000	+	20.7846i	0.576683	+	0.998845i	−0.419173	+	0.907906i	$0.637680\pi$
$434$	−3.00000	+	5.19615i	−0.144005	+	0.249423i
$435$		0			0
$436$	−4.00000			−0.191565
$437$	−4.00000	+	1.73205i	−0.191346	+	0.0828552i
$438$		0			0
$439$	−5.00000	+	8.66025i	−0.238637	+	0.413331i	−0.960323	−	0.278889i	$-0.910034\pi$
$439$	−5.00000	+	8.66025i	−0.238637	+	0.413331i	0.721686	+	0.692220i	$0.243367\pi$
$440$	6.00000	−	10.3923i	0.286039	−	0.495434i
$441$		0			0
$442$	−3.00000	+	5.19615i	−0.142695	+	0.247156i
$443$	9.00000	+	15.5885i	0.427603	+	0.740630i	0.996660	−	0.0816684i	$-0.0260248\pi$
$443$	9.00000	+	15.5885i	0.427603	+	0.740630i	−0.569057	+	0.822298i	$0.692691\pi$
$444$		0			0
$445$	−12.0000			−0.568855
$446$	9.00000	+	15.5885i	0.426162	+	0.738135i
$447$		0			0
$448$	−1.00000			−0.0472456
$449$	39.0000			1.84052			0.920262	−	0.391303i	$-0.127976\pi$
$449$	39.0000			1.84052			0.920262	+	0.391303i	$0.127976\pi$
$450$		0			0
$451$	−8.00000	+	13.8564i	−0.376705	+	0.652473i
$452$	−2.50000	−	4.33013i	−0.117590	−	0.203672i
$453$		0			0
$454$	2.50000	−	4.33013i	0.117331	−	0.203223i
$455$	−9.00000			−0.421927
$456$		0			0
$457$	−7.00000			−0.327446			−0.163723	−	0.986506i	$-0.552350\pi$
$457$	−7.00000			−0.327446			−0.163723	+	0.986506i	$0.552350\pi$
$458$	8.50000	−	14.7224i	0.397179	−	0.687934i
$459$		0			0
$460$	1.50000	+	2.59808i	0.0699379	+	0.121136i
$461$	−16.5000	+	28.5788i	−0.768482	+	1.33105i	0.169904	+	0.985461i	$0.445654\pi$
$461$	−16.5000	+	28.5788i	−0.768482	+	1.33105i	−0.938386	+	0.345589i	$0.887679\pi$
$462$		0			0
$463$	−9.00000			−0.418265			−0.209133	−	0.977887i	$-0.567064\pi$
$463$	−9.00000			−0.418265			−0.209133	+	0.977887i	$0.567064\pi$
$464$		0			0
$465$		0			0
$466$	4.50000	+	7.79423i	0.208458	+	0.361061i
$467$	4.00000			0.185098			0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000			0.185098			0.0925490	+	0.995708i	$0.470499\pi$
$468$		0			0
$469$	−6.00000	−	10.3923i	−0.277054	−	0.479872i
$470$	−3.00000	+	5.19615i	−0.138380	+	0.239681i
$471$		0			0
$472$	6.50000	−	11.2583i	0.299187	−	0.518207i
$473$	−12.0000	+	20.7846i	−0.551761	+	0.955677i
$474$		0			0
$475$	16.0000	−	6.92820i	0.734130	−	0.317888i
$476$	−2.00000			−0.0916698
$477$		0			0
$478$	13.5000	−	23.3827i	0.617476	−	1.06950i
$479$	−21.0000	−	36.3731i	−0.959514	−	1.66193i	−0.723681	−	0.690134i	$-0.757551\pi$
$479$	−21.0000	−	36.3731i	−0.959514	−	1.66193i	−0.235833	−	0.971794i	$-0.575782\pi$
$480$		0			0
$481$	−6.00000	−	10.3923i	−0.273576	−	0.473848i
$482$	22.0000			1.00207
$483$		0			0
$484$	−2.50000	−	4.33013i	−0.113636	−	0.196824i
$485$	15.0000	+	25.9808i	0.681115	+	1.17973i
$486$		0			0
$487$	−28.0000			−1.26880			−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000			−1.26880			−0.634401	+	0.773004i	$0.718753\pi$
$488$	−2.50000	−	4.33013i	−0.113170	−	0.196016i
$489$		0			0
$490$	−1.50000	−	2.59808i	−0.0677631	−	0.117369i
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	−0.698285	−	0.715820i	$-0.746053\pi$
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	0.969061	+	0.246822i	$0.0793863\pi$
$492$		0			0
$493$		0			0
$494$	−1.50000	+	12.9904i	−0.0674882	+	0.584465i
$495$		0			0
$496$	3.00000	−	5.19615i	0.134704	−	0.233314i
$497$	3.50000	−	6.06218i	0.156996	−	0.271926i
$498$		0			0
$499$	5.00000	−	8.66025i	0.223831	−	0.387686i	−0.732137	−	0.681157i	$-0.761477\pi$
$499$	5.00000	−	8.66025i	0.223831	−	0.387686i	0.955968	+	0.293471i	$0.0948104\pi$
$500$	1.50000	+	2.59808i	0.0670820	+	0.116190i
$501$		0			0
$502$	−21.0000			−0.937276
$503$	11.0000	+	19.0526i	0.490466	+	0.849512i	0.999940	−	0.0109744i	$-0.00349334\pi$
$503$	11.0000	+	19.0526i	0.490466	+	0.849512i	−0.509474	+	0.860486i	$0.670160\pi$
$504$		0			0
$505$	−42.0000			−1.86898
$506$	4.00000			0.177822
$507$		0			0
$508$	5.50000	−	9.52628i	0.244023	−	0.422660i
$509$	−6.50000	−	11.2583i	−0.288107	−	0.499017i	0.685251	−	0.728307i	$-0.259693\pi$
$509$	−6.50000	−	11.2583i	−0.288107	−	0.499017i	−0.973358	+	0.229291i	$0.926359\pi$
$510$		0			0
$511$	−1.00000	+	1.73205i	−0.0442374	+	0.0766214i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−12.0000			−0.529297
$515$		0			0
$516$		0			0
$517$	4.00000	+	6.92820i	0.175920	+	0.304702i
$518$	2.00000	−	3.46410i	0.0878750	−	0.152204i
$519$		0			0
$520$	9.00000			0.394676
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$	16.0000	+	27.7128i	0.699631	+	1.21180i	0.968594	+	0.248646i	$0.0799857\pi$
$523$	16.0000	+	27.7128i	0.699631	+	1.21180i	−0.268963	+	0.963150i	$0.586681\pi$
$524$	9.00000			0.393167
$525$		0			0
$526$	−1.50000	−	2.59808i	−0.0654031	−	0.113282i
$527$	6.00000	−	10.3923i	0.261364	−	0.452696i
$528$		0			0
$529$	11.0000	−	19.0526i	0.478261	−	0.828372i
$530$	−6.00000	+	10.3923i	−0.260623	+	0.451413i
$531$		0			0
$532$	−4.00000	+	1.73205i	−0.173422	+	0.0750939i
$533$	−12.0000			−0.519778
$534$		0			0
$535$	−15.0000	+	25.9808i	−0.648507	+	1.12325i
$536$	6.00000	+	10.3923i	0.259161	+	0.448879i
$537$		0			0
$538$	7.00000	+	12.1244i	0.301791	+	0.522718i
$539$	−4.00000			−0.172292
$540$		0			0
$541$	−16.0000	−	27.7128i	−0.687894	−	1.19147i	−0.972518	−	0.232828i	$-0.925202\pi$
$541$	−16.0000	−	27.7128i	−0.687894	−	1.19147i	0.284624	−	0.958639i	$-0.408131\pi$
$542$	−13.0000	−	22.5167i	−0.558398	−	0.967173i
$543$		0			0
$544$	2.00000			0.0857493
$545$	6.00000	+	10.3923i	0.257012	+	0.445157i
$546$		0			0
$547$	23.0000	+	39.8372i	0.983409	+	1.70331i	0.648803	+	0.760956i	$0.275270\pi$
$547$	23.0000	+	39.8372i	0.983409	+	1.70331i	0.334606	+	0.942358i	$0.391397\pi$
$548$	−10.5000	+	18.1865i	−0.448538	+	0.776890i
$549$		0			0
$550$	−16.0000			−0.682242
$551$		0			0
$552$		0			0
$553$	−4.00000	+	6.92820i	−0.170097	+	0.294617i
$554$	1.00000	−	1.73205i	0.0424859	−	0.0735878i
$555$		0			0
$556$	8.00000	−	13.8564i	0.339276	−	0.587643i
$557$	11.0000	+	19.0526i	0.466085	+	0.807283i	0.999250	−	0.0387286i	$-0.0123308\pi$
$557$	11.0000	+	19.0526i	0.466085	+	0.807283i	−0.533165	+	0.846011i	$0.678997\pi$
$558$		0			0
$559$	−18.0000			−0.761319
$560$	1.50000	+	2.59808i	0.0633866	+	0.109789i
$561$		0			0
$562$	26.0000			1.09674
$563$	−33.0000			−1.39078			−0.695392	−	0.718631i	$-0.744769\pi$
$563$	−33.0000			−1.39078			−0.695392	+	0.718631i	$0.744769\pi$
$564$		0			0
$565$	−7.50000	+	12.9904i	−0.315527	+	0.546509i
$566$	−12.5000	−	21.6506i	−0.525414	−	0.910044i
$567$		0			0
$568$	−3.50000	+	6.06218i	−0.146857	+	0.254363i
$569$	9.00000			0.377300			0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000			0.377300			0.188650	+	0.982044i	$0.439589\pi$
$570$		0			0
$571$	2.00000			0.0836974			0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000			0.0836974			0.0418487	+	0.999124i	$0.486675\pi$
$572$	6.00000	−	10.3923i	0.250873	−	0.434524i
$573$		0			0
$574$	−2.00000	−	3.46410i	−0.0834784	−	0.144589i
$575$	2.00000	−	3.46410i	0.0834058	−	0.144463i
$576$		0			0
$577$	−8.00000			−0.333044			−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000			−0.333044			−0.166522	+	0.986038i	$0.553254\pi$
$578$	−13.0000			−0.540729
$579$		0			0
$580$		0			0
$581$	17.0000			0.705279
$582$		0			0
$583$	8.00000	+	13.8564i	0.331326	+	0.573874i
$584$	1.00000	−	1.73205i	0.0413803	−	0.0716728i
$585$		0			0
$586$	−9.50000	+	16.4545i	−0.392441	+	0.679728i
$587$	−10.0000	+	17.3205i	−0.412744	+	0.714894i	−0.995189	−	0.0979766i	$-0.968763\pi$
$587$	−10.0000	+	17.3205i	−0.412744	+	0.714894i	0.582445	+	0.812870i	$0.302096\pi$
$588$		0			0
$589$	3.00000	−	25.9808i	0.123613	−	1.07052i
$590$	−39.0000			−1.60560
$591$		0			0
$592$	−2.00000	+	3.46410i	−0.0821995	+	0.142374i
$593$	−16.0000	−	27.7128i	−0.657041	−	1.13803i	−0.981378	−	0.192087i	$-0.938474\pi$
$593$	−16.0000	−	27.7128i	−0.657041	−	1.13803i	0.324337	−	0.945942i	$-0.394859\pi$
$594$		0			0
$595$	3.00000	+	5.19615i	0.122988	+	0.213021i
$596$	−20.0000			−0.819232
$597$		0			0
$598$	1.50000	+	2.59808i	0.0613396	+	0.106243i
$599$	−21.5000	−	37.2391i	−0.878466	−	1.52155i	−0.853024	−	0.521872i	$-0.825234\pi$
$599$	−21.5000	−	37.2391i	−0.878466	−	1.52155i	−0.0254422	−	0.999676i	$-0.508099\pi$
$600$		0			0
$601$	−6.00000			−0.244745			−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000			−0.244745			−0.122373	+	0.992484i	$0.539050\pi$
$602$	−3.00000	−	5.19615i	−0.122271	−	0.211779i
$603$		0			0
$604$	−9.50000	−	16.4545i	−0.386550	−	0.669523i
$605$	−7.50000	+	12.9904i	−0.304918	+	0.528134i
$606$		0			0
$607$	−16.0000			−0.649420			−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000			−0.649420			−0.324710	+	0.945814i	$0.605267\pi$
$608$	4.00000	−	1.73205i	0.162221	−	0.0702439i
$609$		0			0
$610$	−7.50000	+	12.9904i	−0.303666	+	0.525965i
$611$	−3.00000	+	5.19615i	−0.121367	+	0.210214i
$612$		0			0
$613$	5.00000	−	8.66025i	0.201948	−	0.349784i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	5.00000	−	8.66025i	0.201948	−	0.349784i	0.949156	+	0.314806i	$0.101939\pi$
$614$	−7.50000	−	12.9904i	−0.302675	−	0.524249i
$615$		0			0
$616$	4.00000			0.161165
$617$	−13.5000	−	23.3827i	−0.543490	−	0.941351i	−0.998700	−	0.0509678i	$-0.983769\pi$
$617$	−13.5000	−	23.3827i	−0.543490	−	0.941351i	0.455211	−	0.890384i	$-0.349564\pi$
$618$		0			0
$619$	−5.00000			−0.200967			−0.100483	−	0.994939i	$-0.532039\pi$
$619$	−5.00000			−0.200967			−0.100483	+	0.994939i	$0.532039\pi$
$620$	−18.0000			−0.722897
$621$		0			0
$622$	−1.00000	+	1.73205i	−0.0400963	+	0.0694489i
$623$	−2.00000	−	3.46410i	−0.0801283	−	0.138786i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$	24.0000			0.959233
$627$		0			0
$628$	−9.00000			−0.359139
$629$	−4.00000	+	6.92820i	−0.159490	+	0.276246i
$630$		0			0
$631$	−4.00000	−	6.92820i	−0.159237	−	0.275807i	0.775356	−	0.631524i	$-0.217570\pi$
$631$	−4.00000	−	6.92820i	−0.159237	−	0.275807i	−0.934594	+	0.355716i	$0.884237\pi$
$632$	4.00000	−	6.92820i	0.159111	−	0.275589i
$633$		0			0
$634$	−6.00000			−0.238290
$635$	−33.0000			−1.30957
$636$		0			0
$637$	−1.50000	−	2.59808i	−0.0594322	−	0.102940i
$638$		0			0
$639$		0			0
$640$	−1.50000	−	2.59808i	−0.0592927	−	0.102698i
$641$	−8.50000	+	14.7224i	−0.335730	+	0.581501i	−0.983625	−	0.180229i	$-0.942316\pi$
$641$	−8.50000	+	14.7224i	−0.335730	+	0.581501i	0.647895	+	0.761730i	$0.275650\pi$
$642$		0			0
$643$	20.5000	−	35.5070i	0.808441	−	1.40026i	−0.105502	−	0.994419i	$-0.533645\pi$
$643$	20.5000	−	35.5070i	0.808441	−	1.40026i	0.913943	−	0.405842i	$-0.133022\pi$
$644$	−0.500000	+	0.866025i	−0.0197028	+	0.0341262i
$645$		0			0
$646$	8.00000	−	3.46410i	0.314756	−	0.136293i
$647$	−30.0000			−1.17942			−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000			−1.17942			−0.589711	+	0.807614i	$0.700758\pi$
$648$		0			0
$649$	−26.0000	+	45.0333i	−1.02059	+	1.76771i
$650$	−6.00000	−	10.3923i	−0.235339	−	0.407620i
$651$		0			0
$652$	−7.00000	−	12.1244i	−0.274141	−	0.474826i
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$	−13.5000	−	23.3827i	−0.527489	−	0.913637i
$656$	2.00000	+	3.46410i	0.0780869	+	0.135250i
$657$		0			0
$658$	−2.00000			−0.0779681
$659$	−13.0000	−	22.5167i	−0.506408	−	0.877125i	−0.999973	−	0.00741531i	$-0.997640\pi$
$659$	−13.0000	−	22.5167i	−0.506408	−	0.877125i	0.493564	−	0.869709i	$-0.335694\pi$
$660$		0			0
$661$	5.50000	+	9.52628i	0.213925	+	0.370529i	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	5.50000	+	9.52628i	0.213925	+	0.370529i	−0.739014	+	0.673690i	$0.764708\pi$
$662$	12.0000	−	20.7846i	0.466393	−	0.807817i
$663$		0			0
$664$	−17.0000			−0.659728
$665$	10.5000	+	7.79423i	0.407173	+	0.302247i
$666$		0			0
$667$		0			0
$668$	−9.00000	+	15.5885i	−0.348220	+	0.603136i
$669$		0			0
$670$	18.0000	−	31.1769i	0.695401	−	1.20447i
$671$	10.0000	+	17.3205i	0.386046	+	0.668651i
$672$		0			0
$673$	11.0000			0.424019			0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000			0.424019			0.212009	+	0.977268i	$0.431999\pi$
$674$	−8.50000	−	14.7224i	−0.327408	−	0.567087i
$675$		0			0
$676$	−4.00000			−0.153846
$677$	−42.0000			−1.61419			−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000			−1.61419			−0.807096	+	0.590421i	$0.798962\pi$
$678$		0			0
$679$	−5.00000	+	8.66025i	−0.191882	+	0.332350i
$680$	−3.00000	−	5.19615i	−0.115045	−	0.199263i
$681$		0			0
$682$	−12.0000	+	20.7846i	−0.459504	+	0.795884i
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$	63.0000			2.40711
$686$	0.500000	−	0.866025i	0.0190901	−	0.0330650i
$687$		0			0
$688$	3.00000	+	5.19615i	0.114374	+	0.198101i
$689$	−6.00000	+	10.3923i	−0.228582	+	0.395915i
$690$		0			0
$691$	−41.0000			−1.55971			−0.779857	−	0.625958i	$-0.784708\pi$
$691$	−41.0000			−1.55971			−0.779857	+	0.625958i	$0.784708\pi$
$692$	−15.0000			−0.570214
$693$		0			0
$694$	11.0000	+	19.0526i	0.417554	+	0.723225i
$695$	−48.0000			−1.82074
$696$		0			0
$697$	4.00000	+	6.92820i	0.151511	+	0.262424i
$698$	−5.00000	+	8.66025i	−0.189253	+	0.327795i
$699$		0			0
$700$	2.00000	−	3.46410i	0.0755929	−	0.130931i
$701$	−1.00000	+	1.73205i	−0.0377695	+	0.0654187i	−0.884292	−	0.466934i	$-0.845359\pi$
$701$	−1.00000	+	1.73205i	−0.0377695	+	0.0654187i	0.846523	+	0.532353i	$0.178692\pi$
$702$		0			0
$703$	−2.00000	+	17.3205i	−0.0754314	+	0.653255i
$704$	−4.00000			−0.150756
$705$		0			0
$706$	6.00000	−	10.3923i	0.225813	−	0.391120i
$707$	−7.00000	−	12.1244i	−0.263262	−	0.455983i
$708$		0			0
$709$	2.00000	+	3.46410i	0.0751116	+	0.130097i	0.901135	−	0.433539i	$-0.142735\pi$
$709$	2.00000	+	3.46410i	0.0751116	+	0.130097i	−0.826023	+	0.563636i	$0.809402\pi$
$710$	21.0000			0.788116
$711$		0			0
$712$	2.00000	+	3.46410i	0.0749532	+	0.129823i
$713$	−3.00000	−	5.19615i	−0.112351	−	0.194597i
$714$		0			0
$715$	−36.0000			−1.34632
$716$	5.00000	+	8.66025i	0.186859	+	0.323649i
$717$		0			0
$718$	12.0000	+	20.7846i	0.447836	+	0.775675i
$719$	11.0000	−	19.0526i	0.410231	−	0.710541i	−0.584684	−	0.811261i	$-0.698781\pi$
$719$	11.0000	−	19.0526i	0.410231	−	0.710541i	0.994915	+	0.100721i	$0.0321148\pi$
$720$		0			0
$721$		0			0
$722$	13.0000	−	13.8564i	0.483810	−	0.515682i
$723$		0			0
$724$	−1.50000	+	2.59808i	−0.0557471	+	0.0965567i
$725$		0			0
$726$		0			0
$727$	−22.0000	+	38.1051i	−0.815935	+	1.41324i	0.0927199	+	0.995692i	$0.470444\pi$
$727$	−22.0000	+	38.1051i	−0.815935	+	1.41324i	−0.908655	+	0.417548i	$0.862889\pi$
$728$	1.50000	+	2.59808i	0.0555937	+	0.0962911i
$729$		0			0
$730$	−6.00000			−0.222070
$731$	6.00000	+	10.3923i	0.221918	+	0.384373i
$732$		0			0
$733$	−11.0000			−0.406294			−0.203147	−	0.979148i	$-0.565117\pi$
$733$	−11.0000			−0.406294			−0.203147	+	0.979148i	$0.565117\pi$
$734$	32.0000			1.18114
$735$		0			0
$736$	0.500000	−	0.866025i	0.0184302	−	0.0319221i
$737$	−24.0000	−	41.5692i	−0.884051	−	1.53122i
$738$		0			0
$739$	22.0000	−	38.1051i	0.809283	−	1.40172i	−0.104078	−	0.994569i	$-0.533189\pi$
$739$	22.0000	−	38.1051i	0.809283	−	1.40172i	0.913361	−	0.407150i	$-0.133477\pi$
$740$	12.0000			0.441129
$741$		0			0
$742$	−4.00000			−0.146845
$743$	−1.50000	+	2.59808i	−0.0550297	+	0.0953142i	−0.892228	−	0.451585i	$-0.850859\pi$
$743$	−1.50000	+	2.59808i	−0.0550297	+	0.0953142i	0.837198	+	0.546899i	$0.184192\pi$
$744$		0			0
$745$	30.0000	+	51.9615i	1.09911	+	1.90372i
$746$	5.00000	−	8.66025i	0.183063	−	0.317074i
$747$		0			0
$748$	−8.00000			−0.292509
$749$	−10.0000			−0.365392
$750$		0			0
$751$	−20.0000	−	34.6410i	−0.729810	−	1.26407i	−0.956963	−	0.290209i	$-0.906275\pi$
$751$	−20.0000	−	34.6410i	−0.729810	−	1.26407i	0.227153	−	0.973859i	$-0.427058\pi$
$752$	2.00000			0.0729325
$753$		0			0
$754$		0			0
$755$	−28.5000	+	49.3634i	−1.03722	+	1.79652i
$756$		0			0
$757$	23.0000	−	39.8372i	0.835949	−	1.44791i	−0.0573060	−	0.998357i	$-0.518251\pi$
$757$	23.0000	−	39.8372i	0.835949	−	1.44791i	0.893255	−	0.449550i	$-0.148416\pi$
$758$	17.0000	−	29.4449i	0.617468	−	1.06949i
$759$		0			0
$760$	−10.5000	−	7.79423i	−0.380875	−	0.282726i
$761$	−8.00000			−0.290000			−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000			−0.290000			−0.145000	+	0.989432i	$0.546318\pi$
$762$		0			0
$763$	−2.00000	+	3.46410i	−0.0724049	+	0.125409i
$764$	−5.50000	−	9.52628i	−0.198983	−	0.344649i
$765$		0			0
$766$	−2.00000	−	3.46410i	−0.0722629	−	0.125163i
$767$	−39.0000			−1.40821
$768$		0			0
$769$	15.0000	+	25.9808i	0.540914	+	0.936890i	0.998852	+	0.0479061i	$0.0152548\pi$
$769$	15.0000	+	25.9808i	0.540914	+	0.936890i	−0.457938	+	0.888984i	$0.651412\pi$
$770$	−6.00000	−	10.3923i	−0.216225	−	0.374513i
$771$		0			0
$772$	5.00000			0.179954
$773$	4.50000	+	7.79423i	0.161854	+	0.280339i	0.935534	−	0.353238i	$-0.114919\pi$
$773$	4.50000	+	7.79423i	0.161854	+	0.280339i	−0.773680	+	0.633577i	$0.781586\pi$
$774$		0			0
$775$	12.0000	+	20.7846i	0.431053	+	0.746605i
$776$	5.00000	−	8.66025i	0.179490	−	0.310885i
$777$		0			0
$778$	−8.00000			−0.286814
$779$	14.0000	+	10.3923i	0.501602	+	0.372343i
$780$		0			0
$781$	14.0000	−	24.2487i	0.500959	−	0.867687i
$782$	1.00000	−	1.73205i	0.0357599	−	0.0619380i
$783$		0			0
$784$	−0.500000	+	0.866025i	−0.0178571	+	0.0309295i
$785$	13.5000	+	23.3827i	0.481836	+	0.834564i
$786$		0			0
$787$	−47.0000			−1.67537			−0.837685	−	0.546154i	$-0.816091\pi$
$787$	−47.0000			−1.67537			−0.837685	+	0.546154i	$0.816091\pi$
$788$	2.00000	+	3.46410i	0.0712470	+	0.123404i
$789$		0			0
$790$	−24.0000			−0.853882
$791$	−5.00000			−0.177780
$792$		0			0
$793$	−7.50000	+	12.9904i	−0.266333	+	0.461302i
$794$	−9.00000	−	15.5885i	−0.319398	−	0.553214i
$795$		0			0
$796$	−7.00000	+	12.1244i	−0.248108	+	0.429736i
$797$	49.0000			1.73567			0.867835	−	0.496853i	$-0.165511\pi$
$797$	49.0000			1.73567			0.867835	+	0.496853i	$0.165511\pi$
$798$		0			0
$799$	4.00000			0.141510
$800$	−2.00000	+	3.46410i	−0.0707107	+	0.122474i
$801$		0			0
$802$	1.50000	+	2.59808i	0.0529668	+	0.0917413i
$803$	−4.00000	+	6.92820i	−0.141157	+	0.244491i
$804$		0			0
$805$	3.00000			0.105736
$806$	−18.0000			−0.634023
$807$		0			0
$808$	7.00000	+	12.1244i	0.246259	+	0.426533i
$809$	−9.00000			−0.316423			−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000			−0.316423			−0.158212	+	0.987405i	$0.550573\pi$
$810$		0			0
$811$	10.0000	+	17.3205i	0.351147	+	0.608205i	0.986451	−	0.164057i	$-0.0524582\pi$
$811$	10.0000	+	17.3205i	0.351147	+	0.608205i	−0.635303	+	0.772263i	$0.719125\pi$
$812$		0			0
$813$		0			0
$814$	8.00000	−	13.8564i	0.280400	−	0.485667i
$815$	−21.0000	+	36.3731i	−0.735598	+	1.27409i
$816$		0			0
$817$	21.0000	+	15.5885i	0.734697	+	0.545371i
$818$	24.0000			0.839140
$819$		0			0
$820$	6.00000	−	10.3923i	0.209529	−	0.362915i
$821$	−24.0000	−	41.5692i	−0.837606	−	1.45078i	−0.891891	−	0.452250i	$-0.850621\pi$
$821$	−24.0000	−	41.5692i	−0.837606	−	1.45078i	0.0542853	−	0.998525i	$-0.482712\pi$
$822$		0			0
$823$	−18.5000	−	32.0429i	−0.644869	−	1.11695i	−0.984332	−	0.176327i	$-0.943578\pi$
$823$	−18.5000	−	32.0429i	−0.644869	−	1.11695i	0.339462	−	0.940620i	$-0.389755\pi$
$824$		0			0
$825$		0			0
$826$	−6.50000	−	11.2583i	−0.226164	−	0.391727i
$827$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$827$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$828$		0			0
$829$	41.0000			1.42399			0.711994	−	0.702185i	$-0.247792\pi$
$829$	41.0000			1.42399			0.711994	+	0.702185i	$0.247792\pi$
$830$	25.5000	+	44.1673i	0.885118	+	1.53307i
$831$		0			0
$832$	−1.50000	−	2.59808i	−0.0520031	−	0.0900721i
$833$	−1.00000	+	1.73205i	−0.0346479	+	0.0600120i
$834$		0			0
$835$	54.0000			1.86875
$836$	−16.0000	+	6.92820i	−0.553372	+	0.239617i
$837$		0			0
$838$	−6.00000	+	10.3923i	−0.207267	+	0.358996i
$839$	15.0000	−	25.9808i	0.517858	−	0.896956i	−0.481927	−	0.876211i	$-0.660063\pi$
$839$	15.0000	−	25.9808i	0.517858	−	0.896956i	0.999785	−	0.0207443i	$-0.00660359\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$	4.00000	+	6.92820i	0.137849	+	0.238762i
$843$		0			0
$844$	−8.00000			−0.275371
$845$	6.00000	+	10.3923i	0.206406	+	0.357506i
$846$		0			0
$847$	−5.00000			−0.171802
$848$	4.00000			0.137361
$849$		0			0
$850$	−4.00000	+	6.92820i	−0.137199	+	0.237635i
$851$	2.00000	+	3.46410i	0.0685591	+	0.118748i
$852$		0			0
$853$	15.0000	−	25.9808i	0.513590	−	0.889564i	−0.486286	−	0.873800i	$-0.661649\pi$
$853$	15.0000	−	25.9808i	0.513590	−	0.889564i	0.999876	−	0.0157644i	$-0.00501816\pi$
$854$	−5.00000			−0.171096
$855$		0			0
$856$	10.0000			0.341793
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	−0.244701	−	0.969599i	$-0.578690\pi$
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	0.962048	−	0.272882i	$-0.0879768\pi$
$858$		0			0
$859$	10.0000	+	17.3205i	0.341196	+	0.590968i	0.984655	−	0.174512i	$-0.0558348\pi$
$859$	10.0000	+	17.3205i	0.341196	+	0.590968i	−0.643459	+	0.765480i	$0.722501\pi$
$860$	9.00000	−	15.5885i	0.306897	−	0.531562i
$861$		0			0
$862$	16.0000			0.544962
$863$	52.0000			1.77010			0.885050	−	0.465495i	$-0.154124\pi$
$863$	52.0000			1.77010			0.885050	+	0.465495i	$0.154124\pi$
$864$		0			0
$865$	22.5000	+	38.9711i	0.765023	+	1.32506i
$866$	−24.0000			−0.815553
$867$		0			0
$868$	−3.00000	−	5.19615i	−0.101827	−	0.176369i
$869$	−16.0000	+	27.7128i	−0.542763	+	0.940093i
$870$		0			0
$871$	18.0000	−	31.1769i	0.609907	−	1.05639i
$872$	2.00000	−	3.46410i	0.0677285	−	0.117309i
$873$		0			0
$874$	0.500000	−	4.33013i	0.0169128	−	0.146469i
$875$	3.00000			0.101419
$876$		0			0
$877$	−10.0000	+	17.3205i	−0.337676	+	0.584872i	−0.983995	−	0.178195i	$-0.942974\pi$
$877$	−10.0000	+	17.3205i	−0.337676	+	0.584872i	0.646319	+	0.763067i	$0.276307\pi$
$878$	−5.00000	−	8.66025i	−0.168742	−	0.292269i
$879$		0			0
$880$	6.00000	+	10.3923i	0.202260	+	0.350325i
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	15.0000	+	25.9808i	0.504790	+	0.874322i	0.999985	+	0.00554010i	$0.00176348\pi$
$883$	15.0000	+	25.9808i	0.504790	+	0.874322i	−0.495194	+	0.868782i	$0.664903\pi$
$884$	−3.00000	−	5.19615i	−0.100901	−	0.174766i
$885$		0			0
$886$	−18.0000			−0.604722
$887$	−21.0000	−	36.3731i	−0.705111	−	1.22129i	−0.966651	−	0.256096i	$-0.917564\pi$
$887$	−21.0000	−	36.3731i	−0.705111	−	1.22129i	0.261540	−	0.965193i	$-0.415770\pi$
$888$		0			0
$889$	−5.50000	−	9.52628i	−0.184464	−	0.319501i
$890$	6.00000	−	10.3923i	0.201120	−	0.348351i
$891$		0			0
$892$	−18.0000			−0.602685
$893$	8.00000	−	3.46410i	0.267710	−	0.115922i
$894$		0			0
$895$	15.0000	−	25.9808i	0.501395	−	0.868441i
$896$	0.500000	−	0.866025i	0.0167038	−	0.0289319i
$897$		0			0
$898$	−19.5000	+	33.7750i	−0.650723	+	1.12709i
$899$		0			0
$900$		0			0
$901$	8.00000			0.266519
$902$	−8.00000	−	13.8564i	−0.266371	−	0.461368i
$903$		0			0
$904$	5.00000			0.166298
$905$	9.00000			0.299170
$906$		0			0
$907$	−29.0000	+	50.2295i	−0.962929	+	1.66784i	−0.247851	+	0.968798i	$0.579724\pi$
$907$	−29.0000	+	50.2295i	−0.962929	+	1.66784i	−0.715079	+	0.699044i	$0.753609\pi$
$908$	2.50000	+	4.33013i	0.0829654	+	0.143700i
$909$		0			0
$910$	4.50000	−	7.79423i	0.149174	−	0.258376i
$911$	−33.0000			−1.09334			−0.546669	−	0.837349i	$-0.684105\pi$
$911$	−33.0000			−1.09334			−0.546669	+	0.837349i	$0.684105\pi$
$912$		0			0
$913$	68.0000			2.25047
$914$	3.50000	−	6.06218i	0.115770	−	0.200519i
$915$		0			0
$916$	8.50000	+	14.7224i	0.280848	+	0.486443i
$917$	4.50000	−	7.79423i	0.148603	−	0.257388i
$918$		0			0
$919$	−17.0000			−0.560778			−0.280389	−	0.959886i	$-0.590464\pi$
$919$	−17.0000			−0.560778			−0.280389	+	0.959886i	$0.590464\pi$
$920$	−3.00000			−0.0989071
$921$		0			0
$922$	−16.5000	−	28.5788i	−0.543399	−	0.941194i
$923$	21.0000			0.691223
$924$		0			0
$925$	−8.00000	−	13.8564i	−0.263038	−	0.455596i
$926$	4.50000	−	7.79423i	0.147879	−	0.256134i
$927$		0			0
$928$		0			0
$929$	24.0000	−	41.5692i	0.787414	−	1.36384i	−0.140132	−	0.990133i	$-0.544753\pi$
$929$	24.0000	−	41.5692i	0.787414	−	1.36384i	0.927546	−	0.373709i	$-0.121914\pi$
$930$		0			0
$931$	−0.500000	+	4.33013i	−0.0163868	+	0.141914i
$932$	−9.00000			−0.294805
$933$		0			0
$934$	−2.00000	+	3.46410i	−0.0654420	+	0.113349i
$935$	12.0000	+	20.7846i	0.392442	+	0.679729i
$936$		0			0
$937$	23.0000	+	39.8372i	0.751377	+	1.30142i	0.947155	+	0.320775i	$0.103943\pi$
$937$	23.0000	+	39.8372i	0.751377	+	1.30142i	−0.195778	+	0.980648i	$0.562723\pi$
$938$	12.0000			0.391814
$939$		0			0
$940$	−3.00000	−	5.19615i	−0.0978492	−	0.169480i
$941$	6.50000	+	11.2583i	0.211894	+	0.367011i	0.952307	−	0.305141i	$-0.0987035\pi$
$941$	6.50000	+	11.2583i	0.211894	+	0.367011i	−0.740413	+	0.672152i	$0.765370\pi$
$942$		0			0
$943$	4.00000			0.130258
$944$	6.50000	+	11.2583i	0.211557	+	0.366427i
$945$		0			0
$946$	−12.0000	−	20.7846i	−0.390154	−	0.675766i
$947$	11.0000	−	19.0526i	0.357452	−	0.619125i	−0.630082	−	0.776528i	$-0.716979\pi$
$947$	11.0000	−	19.0526i	0.357452	−	0.619125i	0.987534	+	0.157403i	$0.0503122\pi$
$948$		0			0
$949$	−6.00000			−0.194768
$950$	−2.00000	+	17.3205i	−0.0648886	+	0.561951i
$951$		0			0
$952$	1.00000	−	1.73205i	0.0324102	−	0.0561361i
$953$	−3.00000	+	5.19615i	−0.0971795	+	0.168320i	−0.910516	−	0.413473i	$-0.864315\pi$
$953$	−3.00000	+	5.19615i	−0.0971795	+	0.168320i	0.813337	+	0.581793i	$0.197649\pi$
$954$		0			0
$955$	−16.5000	+	28.5788i	−0.533927	+	0.924789i
$956$	13.5000	+	23.3827i	0.436621	+	0.756250i
$957$		0			0
$958$	42.0000			1.35696
$959$	10.5000	+	18.1865i	0.339063	+	0.587274i
$960$		0			0
$961$	5.00000			0.161290
$962$	12.0000			0.386896
$963$		0			0
$964$	−11.0000	+	19.0526i	−0.354286	+	0.613642i
$965$	−7.50000	−	12.9904i	−0.241434	−	0.418175i
$966$		0			0
$967$	22.5000	−	38.9711i	0.723551	−	1.25323i	−0.236016	−	0.971749i	$-0.575842\pi$
$967$	22.5000	−	38.9711i	0.723551	−	1.25323i	0.959568	−	0.281478i	$-0.0908248\pi$
$968$	5.00000			0.160706
$969$		0			0
$970$	−30.0000			−0.963242
$971$	−16.5000	+	28.5788i	−0.529510	+	0.917139i	0.469897	+	0.882721i	$0.344291\pi$
$971$	−16.5000	+	28.5788i	−0.529510	+	0.917139i	−0.999408	+	0.0344175i	$0.989042\pi$
$972$		0			0
$973$	−8.00000	−	13.8564i	−0.256468	−	0.444216i
$974$	14.0000	−	24.2487i	0.448589	−	0.776979i
$975$		0			0
$976$	5.00000			0.160046
$977$	9.00000			0.287936			0.143968	−	0.989582i	$-0.454014\pi$
$977$	9.00000			0.287936			0.143968	+	0.989582i	$0.454014\pi$
$978$		0			0
$979$	−8.00000	−	13.8564i	−0.255681	−	0.442853i
$980$	3.00000			0.0958315
$981$		0			0
$982$	6.00000	+	10.3923i	0.191468	+	0.331632i
$983$	−21.0000	+	36.3731i	−0.669796	+	1.16012i	0.308165	+	0.951333i	$0.400285\pi$
$983$	−21.0000	+	36.3731i	−0.669796	+	1.16012i	−0.977961	+	0.208788i	$0.933048\pi$
$984$		0			0
$985$	6.00000	−	10.3923i	0.191176	−	0.331126i
$986$		0			0
$987$		0			0
$988$	−10.5000	−	7.79423i	−0.334050	−	0.247967i
$989$	6.00000			0.190789
$990$		0			0
$991$	17.5000	−	30.3109i	0.555906	−	0.962857i	−0.441927	−	0.897051i	$-0.645705\pi$
$991$	17.5000	−	30.3109i	0.555906	−	0.962857i	0.997832	−	0.0658059i	$-0.0209618\pi$
$992$	3.00000	+	5.19615i	0.0952501	+	0.164978i
$993$		0			0
$994$	3.50000	+	6.06218i	0.111013	+	0.192281i
$995$	42.0000			1.33149
$996$		0			0
$997$	21.5000	+	37.2391i	0.680912	+	1.17937i	0.974703	+	0.223504i	$0.0717497\pi$
$997$	21.5000	+	37.2391i	0.680912	+	1.17937i	−0.293791	+	0.955870i	$0.594917\pi$
$998$	5.00000	+	8.66025i	0.158272	+	0.274136i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.o.a.505.1		2
3.2	odd	2		798.2.k.i.505.1	yes	2
19.7	even	3	inner	2394.2.o.a.1261.1		2
57.26	odd	6		798.2.k.i.463.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.k.i.463.1	✓	2	57.26	odd	6
798.2.k.i.505.1	yes	2	3.2	odd	2
2394.2.o.a.505.1		2	1.1	even	1	trivial
2394.2.o.a.1261.1		2	19.7	even	3	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.o (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} -1.00000 q^{7} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} -4.00000 q^{11} +(-1.50000 - 2.59808i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.00000 + 1.73205i) q^{17} +(-0.500000 + 4.33013i) q^{19} +3.00000 q^{20} +(2.00000 - 3.46410i) q^{22} +(0.500000 + 0.866025i) q^{23} +(-2.00000 - 3.46410i) q^{25} +3.00000 q^{26} +(0.500000 + 0.866025i) q^{28} -6.00000 q^{31} +(-0.500000 - 0.866025i) q^{32} +(-1.00000 - 1.73205i) q^{34} +(1.50000 - 2.59808i) q^{35} +4.00000 q^{37} +(-3.50000 - 2.59808i) q^{38} +(-1.50000 + 2.59808i) q^{40} +(2.00000 - 3.46410i) q^{41} +(3.00000 - 5.19615i) q^{43} +(2.00000 + 3.46410i) q^{44} -1.00000 q^{46} +(-1.00000 - 1.73205i) q^{47} +1.00000 q^{49} +4.00000 q^{50} +(-1.50000 + 2.59808i) q^{52} +(-2.00000 - 3.46410i) q^{53} +(6.00000 - 10.3923i) q^{55} -1.00000 q^{56} +(6.50000 - 11.2583i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(3.00000 - 5.19615i) q^{62} +1.00000 q^{64} +9.00000 q^{65} +(6.00000 + 10.3923i) q^{67} +2.00000 q^{68} +(1.50000 + 2.59808i) q^{70} +(-3.50000 + 6.06218i) q^{71} +(1.00000 - 1.73205i) q^{73} +(-2.00000 + 3.46410i) q^{74} +(4.00000 - 1.73205i) q^{76} +4.00000 q^{77} +(4.00000 - 6.92820i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(2.00000 + 3.46410i) q^{82} -17.0000 q^{83} +(-3.00000 - 5.19615i) q^{85} +(3.00000 + 5.19615i) q^{86} -4.00000 q^{88} +(2.00000 + 3.46410i) q^{89} +(1.50000 + 2.59808i) q^{91} +(0.500000 - 0.866025i) q^{92} +2.00000 q^{94} +(-10.5000 - 7.79423i) q^{95} +(5.00000 - 8.66025i) q^{97} +(-0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 3 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 - 3 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} - 3 q^{5} - 2 q^{7} + 2 q^{8} - 3 q^{10} - 8 q^{11} - 3 q^{13} + q^{14} - q^{16} - 2 q^{17} - q^{19} + 6 q^{20} + 4 q^{22} + q^{23} - 4 q^{25} + 6 q^{26} + q^{28} - 12 q^{31} - q^{32} - 2 q^{34} + 3 q^{35} + 8 q^{37} - 7 q^{38} - 3 q^{40} + 4 q^{41} + 6 q^{43} + 4 q^{44} - 2 q^{46} - 2 q^{47} + 2 q^{49} + 8 q^{50} - 3 q^{52} - 4 q^{53} + 12 q^{55} - 2 q^{56} + 13 q^{59} - 5 q^{61} + 6 q^{62} + 2 q^{64} + 18 q^{65} + 12 q^{67} + 4 q^{68} + 3 q^{70} - 7 q^{71} + 2 q^{73} - 4 q^{74} + 8 q^{76} + 8 q^{77} + 8 q^{79} - 3 q^{80} + 4 q^{82} - 34 q^{83} - 6 q^{85} + 6 q^{86} - 8 q^{88} + 4 q^{89} + 3 q^{91} + q^{92} + 4 q^{94} - 21 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 3 * q^5 - 2 * q^7 + 2 * q^8 - 3 * q^10 - 8 * q^11 - 3 * q^13 + q^14 - q^16 - 2 * q^17 - q^19 + 6 * q^20 + 4 * q^22 + q^23 - 4 * q^25 + 6 * q^26 + q^28 - 12 * q^31 - q^32 - 2 * q^34 + 3 * q^35 + 8 * q^37 - 7 * q^38 - 3 * q^40 + 4 * q^41 + 6 * q^43 + 4 * q^44 - 2 * q^46 - 2 * q^47 + 2 * q^49 + 8 * q^50 - 3 * q^52 - 4 * q^53 + 12 * q^55 - 2 * q^56 + 13 * q^59 - 5 * q^61 + 6 * q^62 + 2 * q^64 + 18 * q^65 + 12 * q^67 + 4 * q^68 + 3 * q^70 - 7 * q^71 + 2 * q^73 - 4 * q^74 + 8 * q^76 + 8 * q^77 + 8 * q^79 - 3 * q^80 + 4 * q^82 - 34 * q^83 - 6 * q^85 + 6 * q^86 - 8 * q^88 + 4 * q^89 + 3 * q^91 + q^92 + 4 * q^94 - 21 * q^95 + 10 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more