LMFDB - Embedded newform 168.2.q.b.25.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(25,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("168.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	168.q (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:	$1.34148675396$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	168.25
Dual form			168.2.q.b.121.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{5} +(2.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{5} +(2.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(3.00000 - 5.19615i) q^{11} -3.00000 q^{13} -2.00000 q^{15} +(-2.00000 + 3.46410i) q^{17} +(2.50000 + 4.33013i) q^{19} +(2.00000 - 1.73205i) q^{21} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} -1.00000 q^{27} -4.00000 q^{29} +(-3.50000 + 6.06218i) q^{31} +(-3.00000 - 5.19615i) q^{33} +(-1.00000 - 5.19615i) q^{35} +(4.50000 + 7.79423i) q^{37} +(-1.50000 + 2.59808i) q^{39} -2.00000 q^{41} -1.00000 q^{43} +(-1.00000 + 1.73205i) q^{45} +(-1.00000 - 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +(2.00000 + 3.46410i) q^{51} +(-4.00000 + 6.92820i) q^{53} -12.0000 q^{55} +5.00000 q^{57} +(-5.00000 - 8.66025i) q^{61} +(-0.500000 - 2.59808i) q^{63} +(3.00000 + 5.19615i) q^{65} +(7.50000 - 12.9904i) q^{67} +4.00000 q^{69} -6.00000 q^{71} +(5.50000 - 9.52628i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(12.0000 - 10.3923i) q^{77} +(-0.500000 - 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} +6.00000 q^{83} +8.00000 q^{85} +(-2.00000 + 3.46410i) q^{87} +(4.00000 + 6.92820i) q^{89} +(-7.50000 - 2.59808i) q^{91} +(3.50000 + 6.06218i) q^{93} +(5.00000 - 8.66025i) q^{95} -14.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 2 q^{5} + 5 q^{7} - q^{9}+O(q^{10}) $ 2 * q + q^3 - 2 * q^5 + 5 * q^7 - q^9	$ 2 q + q^{3} - 2 q^{5} + 5 q^{7} - q^{9} + 6 q^{11} - 6 q^{13} - 4 q^{15} - 4 q^{17} + 5 q^{19} + 4 q^{21} + 4 q^{23} + q^{25} - 2 q^{27} - 8 q^{29} - 7 q^{31} - 6 q^{33} - 2 q^{35} + 9 q^{37} - 3 q^{39} - 4 q^{41} - 2 q^{43} - 2 q^{45} - 2 q^{47} + 11 q^{49} + 4 q^{51} - 8 q^{53} - 24 q^{55} + 10 q^{57} - 10 q^{61} - q^{63} + 6 q^{65} + 15 q^{67} + 8 q^{69} - 12 q^{71} + 11 q^{73} - q^{75} + 24 q^{77} - q^{79} - q^{81} + 12 q^{83} + 16 q^{85} - 4 q^{87} + 8 q^{89} - 15 q^{91} + 7 q^{93} + 10 q^{95} - 28 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + q^3 - 2 * q^5 + 5 * q^7 - q^9 + 6 * q^11 - 6 * q^13 - 4 * q^15 - 4 * q^17 + 5 * q^19 + 4 * q^21 + 4 * q^23 + q^25 - 2 * q^27 - 8 * q^29 - 7 * q^31 - 6 * q^33 - 2 * q^35 + 9 * q^37 - 3 * q^39 - 4 * q^41 - 2 * q^43 - 2 * q^45 - 2 * q^47 + 11 * q^49 + 4 * q^51 - 8 * q^53 - 24 * q^55 + 10 * q^57 - 10 * q^61 - q^63 + 6 * q^65 + 15 * q^67 + 8 * q^69 - 12 * q^71 + 11 * q^73 - q^75 + 24 * q^77 - q^79 - q^81 + 12 * q^83 + 16 * q^85 - 4 * q^87 + 8 * q^89 - 15 * q^91 + 7 * q^93 + 10 * q^95 - 28 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$4$		0			0
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.0829925	−	0.996550i	$-0.473552\pi$
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.821541	−	0.570149i	$-0.193114\pi$
$12$		0			0
$13$	−3.00000			−0.832050			−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000			−0.832050			−0.416025	+	0.909353i	$0.636577\pi$
$14$		0			0
$15$	−2.00000			−0.516398
$16$		0			0
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	−0.999853	−	0.0171533i	$-0.994540\pi$
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	0.514782	+	0.857321i	$0.327873\pi$
$18$		0			0
$19$	2.50000	+	4.33013i	0.573539	+	0.993399i	0.996199	+	0.0871106i	$0.0277634\pi$
$19$	2.50000	+	4.33013i	0.573539	+	0.993399i	−0.422659	+	0.906289i	$0.638903\pi$
$20$		0			0
$21$	2.00000	−	1.73205i	0.436436	−	0.377964i
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$	−3.50000	+	6.06218i	−0.628619	+	1.08880i	0.359211	+	0.933257i	$0.383046\pi$
$31$	−3.50000	+	6.06218i	−0.628619	+	1.08880i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$	−3.00000	−	5.19615i	−0.522233	−	0.904534i
$34$		0			0
$35$	−1.00000	−	5.19615i	−0.169031	−	0.878310i
$36$		0			0
$37$	4.50000	+	7.79423i	0.739795	+	1.28136i	0.952587	+	0.304266i	$0.0984111\pi$
$37$	4.50000	+	7.79423i	0.739795	+	1.28136i	−0.212792	+	0.977098i	$0.568256\pi$
$38$		0			0
$39$	−1.50000	+	2.59808i	−0.240192	+	0.416025i
$40$		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
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		0			0
$45$	−1.00000	+	1.73205i	−0.149071	+	0.258199i
$46$		0			0
$47$	−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
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$	2.00000	+	3.46410i	0.280056	+	0.485071i
$52$		0			0
$53$	−4.00000	+	6.92820i	−0.549442	+	0.951662i	0.448871	+	0.893597i	$0.351826\pi$
$53$	−4.00000	+	6.92820i	−0.549442	+	0.951662i	−0.998313	+	0.0580651i	$0.981507\pi$
$54$		0			0
$55$	−12.0000			−1.61808
$56$		0			0
$57$	5.00000			0.662266
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	−0.985391	−	0.170305i	$-0.945525\pi$
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	0.345207	−	0.938527i	$-0.387809\pi$
$62$		0			0
$63$	−0.500000	−	2.59808i	−0.0629941	−	0.327327i
$64$		0			0
$65$	3.00000	+	5.19615i	0.372104	+	0.644503i
$66$		0			0
$67$	7.50000	−	12.9904i	0.916271	−	1.58703i	0.111241	−	0.993793i	$-0.464517\pi$
$67$	7.50000	−	12.9904i	0.916271	−	1.58703i	0.805030	−	0.593234i	$-0.202149\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	5.50000	−	9.52628i	0.643726	−	1.11497i	−0.340868	−	0.940111i	$-0.610721\pi$
$73$	5.50000	−	9.52628i	0.643726	−	1.11497i	0.984594	−	0.174855i	$-0.0559458\pi$
$74$		0			0
$75$	−0.500000	−	0.866025i	−0.0577350	−	0.100000i
$76$		0			0
$77$	12.0000	−	10.3923i	1.36753	−	1.18431i
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	$-0.184582\pi$
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$	−2.00000	+	3.46410i	−0.214423	+	0.371391i
$88$		0			0
$89$	4.00000	+	6.92820i	0.423999	+	0.734388i	0.996326	−	0.0856373i	$-0.0272926\pi$
$89$	4.00000	+	6.92820i	0.423999	+	0.734388i	−0.572327	+	0.820025i	$0.693959\pi$
$90$		0			0
$91$	−7.50000	−	2.59808i	−0.786214	−	0.272352i
$92$		0			0
$93$	3.50000	+	6.06218i	0.362933	+	0.628619i
$94$		0			0
$95$	5.00000	−	8.66025i	0.512989	−	0.888523i
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	−6.00000			−0.603023
$100$		0			0
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	−0.677284	−	0.735721i	$-0.736843\pi$
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	0.975796	+	0.218685i	$0.0701767\pi$
$102$		0			0
$103$	4.50000	+	7.79423i	0.443398	+	0.767988i	0.997939	−	0.0641683i	$-0.0204394\pi$
$103$	4.50000	+	7.79423i	0.443398	+	0.767988i	−0.554541	+	0.832156i	$0.687106\pi$
$104$		0			0
$105$	−5.00000	−	1.73205i	−0.487950	−	0.169031i
$106$		0			0
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	0.995474	+	0.0950377i	$0.0302972\pi$
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	−0.415432	+	0.909624i	$0.636370\pi$
$108$		0			0
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	−0.472708	−	0.881219i	$-0.656723\pi$
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	0.999512	−	0.0312328i	$-0.00994332\pi$
$110$		0			0
$111$	9.00000			0.854242
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	4.00000	−	6.92820i	0.373002	−	0.646058i
$116$		0			0
$117$	1.50000	+	2.59808i	0.138675	+	0.240192i
$118$		0			0
$119$	−8.00000	+	6.92820i	−0.733359	+	0.635107i
$120$		0			0
$121$	−12.5000	−	21.6506i	−1.13636	−	1.96824i
$122$		0			0
$123$	−1.00000	+	1.73205i	−0.0901670	+	0.156174i
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$	−1.00000			−0.0887357			−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000			−0.0887357			−0.0443678	+	0.999015i	$0.514127\pi$
$128$		0			0
$129$	−0.500000	+	0.866025i	−0.0440225	+	0.0762493i
$130$		0			0
$131$	−7.00000	−	12.1244i	−0.611593	−	1.05931i	−0.990972	−	0.134069i	$-0.957196\pi$
$131$	−7.00000	−	12.1244i	−0.611593	−	1.05931i	0.379379	−	0.925241i	$-0.376138\pi$
$132$		0			0
$133$	2.50000	+	12.9904i	0.216777	+	1.12641i
$134$		0			0
$135$	1.00000	+	1.73205i	0.0860663	+	0.149071i
$136$		0			0
$137$	−10.0000	+	17.3205i	−0.854358	+	1.47979i	0.0228820	+	0.999738i	$0.492716\pi$
$137$	−10.0000	+	17.3205i	−0.854358	+	1.47979i	−0.877240	+	0.480053i	$0.840618\pi$
$138$		0			0
$139$	−9.00000			−0.763370			−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000			−0.763370			−0.381685	+	0.924292i	$0.624656\pi$
$140$		0			0
$141$	−2.00000			−0.168430
$142$		0			0
$143$	−9.00000	+	15.5885i	−0.752618	+	1.30357i
$144$		0			0
$145$	4.00000	+	6.92820i	0.332182	+	0.575356i
$146$		0			0
$147$	6.50000	−	2.59808i	0.536111	−	0.214286i
$148$		0			0
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	0.772399	−	0.635138i	$-0.219057\pi$
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	−0.936245	+	0.351348i	$0.885723\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$		0			0
$153$	4.00000			0.323381
$154$		0			0
$155$	14.0000			1.12451
$156$		0			0
$157$	−9.00000	+	15.5885i	−0.718278	+	1.24409i	0.243403	+	0.969925i	$0.421736\pi$
$157$	−9.00000	+	15.5885i	−0.718278	+	1.24409i	−0.961681	+	0.274169i	$0.911597\pi$
$158$		0			0
$159$	4.00000	+	6.92820i	0.317221	+	0.549442i
$160$		0			0
$161$	2.00000	+	10.3923i	0.157622	+	0.819028i
$162$		0			0
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	0.777007	−	0.629492i	$-0.216737\pi$
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	−0.933659	+	0.358162i	$0.883403\pi$
$164$		0			0
$165$	−6.00000	+	10.3923i	−0.467099	+	0.809040i
$166$		0			0
$167$	18.0000			1.39288			0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000			1.39288			0.696441	+	0.717614i	$0.254766\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$	2.50000	−	4.33013i	0.191180	−	0.331133i
$172$		0			0
$173$	−10.0000	−	17.3205i	−0.760286	−	1.31685i	−0.942703	−	0.333633i	$-0.891725\pi$
$173$	−10.0000	−	17.3205i	−0.760286	−	1.31685i	0.182417	−	0.983221i	$-0.441608\pi$
$174$		0			0
$175$	2.00000	−	1.73205i	0.151186	−	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.0000	+	22.5167i	−0.971666	+	1.68297i	−0.281139	+	0.959667i	$0.590712\pi$
$179$	−13.0000	+	22.5167i	−0.971666	+	1.68297i	−0.690526	+	0.723307i	$0.742621\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$	9.00000	−	15.5885i	0.661693	−	1.14609i
$186$		0			0
$187$	12.0000	+	20.7846i	0.877527	+	1.51992i
$188$		0			0
$189$	−2.50000	−	0.866025i	−0.181848	−	0.0629941i
$190$		0			0
$191$	−5.00000	−	8.66025i	−0.361787	−	0.626634i	0.626468	−	0.779447i	$-0.284500\pi$
$191$	−5.00000	−	8.66025i	−0.361787	−	0.626634i	−0.988255	+	0.152813i	$0.951167\pi$
$192$		0			0
$193$	−1.50000	+	2.59808i	−0.107972	+	0.187014i	−0.914949	−	0.403570i	$-0.867769\pi$
$193$	−1.50000	+	2.59808i	−0.107972	+	0.187014i	0.806976	+	0.590584i	$0.201102\pi$
$194$		0			0
$195$	6.00000			0.429669
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	8.00000	−	13.8564i	0.567105	−	0.982255i	−0.429745	−	0.902950i	$-0.641397\pi$
$199$	8.00000	−	13.8564i	0.567105	−	0.982255i	0.996850	−	0.0793045i	$-0.0252700\pi$
$200$		0			0
$201$	−7.50000	−	12.9904i	−0.529009	−	0.916271i
$202$		0			0
$203$	−10.0000	−	3.46410i	−0.701862	−	0.243132i
$204$		0			0
$205$	2.00000	+	3.46410i	0.139686	+	0.241943i
$206$		0			0
$207$	2.00000	−	3.46410i	0.139010	−	0.240772i
$208$		0			0
$209$	30.0000			2.07514
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$	−3.00000	+	5.19615i	−0.205557	+	0.356034i
$214$		0			0
$215$	1.00000	+	1.73205i	0.0681994	+	0.118125i
$216$		0			0
$217$	−14.0000	+	12.1244i	−0.950382	+	0.823055i
$218$		0			0
$219$	−5.50000	−	9.52628i	−0.371656	−	0.643726i
$220$		0			0
$221$	6.00000	−	10.3923i	0.403604	−	0.699062i
$222$		0			0
$223$	24.0000			1.60716			0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000			1.60716			0.803579	+	0.595198i	$0.202926\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−7.00000	+	12.1244i	−0.464606	+	0.804722i	−0.999184	−	0.0403978i	$-0.987137\pi$
$227$	−7.00000	+	12.1244i	−0.464606	+	0.804722i	0.534577	+	0.845120i	$0.320471\pi$
$228$		0			0
$229$	3.50000	+	6.06218i	0.231287	+	0.400600i	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	3.50000	+	6.06218i	0.231287	+	0.400600i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$	−3.00000	−	15.5885i	−0.197386	−	1.02565i
$232$		0			0
$233$	−13.0000	−	22.5167i	−0.851658	−	1.47512i	−0.879711	−	0.475509i	$-0.842264\pi$
$233$	−13.0000	−	22.5167i	−0.851658	−	1.47512i	0.0280525	−	0.999606i	$-0.491069\pi$
$234$		0			0
$235$	−2.00000	+	3.46410i	−0.130466	+	0.225973i
$236$		0			0
$237$	−1.00000			−0.0649570
$238$		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
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	−0.832019	−	0.554747i	$-0.812815\pi$
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	0.896435	+	0.443176i	$0.146148\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$		0			0
$245$	2.00000	−	13.8564i	0.127775	−	0.885253i
$246$		0			0
$247$	−7.50000	−	12.9904i	−0.477214	−	0.826558i
$248$		0			0
$249$	3.00000	−	5.19615i	0.190117	−	0.329293i
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$	24.0000			1.50887
$254$		0			0
$255$	4.00000	−	6.92820i	0.250490	−	0.433861i
$256$		0			0
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	0.997374	+	0.0724199i	$0.0230722\pi$
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		0			0
$259$	4.50000	+	23.3827i	0.279616	+	1.45293i
$260$		0			0
$261$	2.00000	+	3.46410i	0.123797	+	0.214423i
$262$		0			0
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$	16.0000			0.982872
$266$		0			0
$267$	8.00000			0.489592
$268$		0			0
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	−0.449622	−	0.893219i	$-0.648441\pi$
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	0.998361	−	0.0572259i	$-0.0182255\pi$
$270$		0			0
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	0.718580	−	0.695444i	$-0.244792\pi$
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	−0.961563	+	0.274586i	$0.911459\pi$
$272$		0			0
$273$	−6.00000	+	5.19615i	−0.363137	+	0.314485i
$274$		0			0
$275$	−3.00000	−	5.19615i	−0.180907	−	0.313340i
$276$		0			0
$277$	−0.500000	+	0.866025i	−0.0300421	+	0.0520344i	−0.880656	−	0.473757i	$-0.842897\pi$
$277$	−0.500000	+	0.866025i	−0.0300421	+	0.0520344i	0.850613	+	0.525792i	$0.176231\pi$
$278$		0			0
$279$	7.00000			0.419079
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	0.500000	−	0.866025i	0.0297219	−	0.0514799i	−0.850782	−	0.525519i	$-0.823871\pi$
$283$	0.500000	−	0.866025i	0.0297219	−	0.0514799i	0.880504	+	0.474039i	$0.157204\pi$
$284$		0			0
$285$	−5.00000	−	8.66025i	−0.296174	−	0.512989i
$286$		0			0
$287$	−5.00000	−	1.73205i	−0.295141	−	0.102240i
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$	−7.00000	+	12.1244i	−0.410347	+	0.710742i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.00000	+	5.19615i	−0.174078	+	0.301511i
$298$		0			0
$299$	−6.00000	−	10.3923i	−0.346989	−	0.601003i
$300$		0			0
$301$	−2.50000	−	0.866025i	−0.144098	−	0.0499169i
$302$		0			0
$303$	−3.00000	−	5.19615i	−0.172345	−	0.298511i
$304$		0			0
$305$	−10.0000	+	17.3205i	−0.572598	+	0.991769i
$306$		0			0
$307$	−11.0000			−0.627803			−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000			−0.627803			−0.313902	+	0.949456i	$0.601636\pi$
$308$		0			0
$309$	9.00000			0.511992
$310$		0			0
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	−0.489585	−	0.871956i	$-0.662852\pi$
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	0.999928	−	0.0119847i	$-0.00381495\pi$
$312$		0			0
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	0.879810	−	0.475325i	$-0.157669\pi$
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	−0.851549	+	0.524276i	$0.824336\pi$
$314$		0			0
$315$	−4.00000	+	3.46410i	−0.225374	+	0.195180i
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$	−12.0000	+	20.7846i	−0.671871	+	1.16371i
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$	−20.0000			−1.11283
$324$		0			0
$325$	−1.50000	+	2.59808i	−0.0832050	+	0.144115i
$326$		0			0
$327$	−5.50000	−	9.52628i	−0.304151	−	0.526804i
$328$		0			0
$329$	−1.00000	−	5.19615i	−0.0551318	−	0.286473i
$330$		0			0
$331$	−2.50000	−	4.33013i	−0.137412	−	0.238005i	0.789104	−	0.614260i	$-0.210545\pi$
$331$	−2.50000	−	4.33013i	−0.137412	−	0.238005i	−0.926516	+	0.376254i	$0.877212\pi$
$332$		0			0
$333$	4.50000	−	7.79423i	0.246598	−	0.427121i
$334$		0			0
$335$	−30.0000			−1.63908
$336$		0			0
$337$	29.0000			1.57973			0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000			1.57973			0.789865	+	0.613280i	$0.210150\pi$
$338$		0			0
$339$	3.00000	−	5.19615i	0.162938	−	0.282216i
$340$		0			0
$341$	21.0000	+	36.3731i	1.13721	+	1.96971i
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$	−4.00000	−	6.92820i	−0.215353	−	0.373002i
$346$		0			0
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	−22.0000			−1.17763			−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000			−1.17763			−0.588817	+	0.808267i	$0.700406\pi$
$350$		0			0
$351$	3.00000			0.160128
$352$		0			0
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	−0.934751	−	0.355303i	$-0.884378\pi$
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	0.775077	+	0.631867i	$0.217711\pi$
$354$		0			0
$355$	6.00000	+	10.3923i	0.318447	+	0.551566i
$356$		0			0
$357$	2.00000	+	10.3923i	0.105851	+	0.550019i
$358$		0			0
$359$	6.00000	+	10.3923i	0.316668	+	0.548485i	0.979791	−	0.200026i	$-0.0641026\pi$
$359$	6.00000	+	10.3923i	0.316668	+	0.548485i	−0.663123	+	0.748511i	$0.730769\pi$
$360$		0			0
$361$	−3.00000	+	5.19615i	−0.157895	+	0.273482i
$362$		0			0
$363$	−25.0000			−1.31216
$364$		0			0
$365$	−22.0000			−1.15153
$366$		0			0
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	−0.760100	−	0.649806i	$-0.774850\pi$
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	0.942799	+	0.333363i	$0.108183\pi$
$368$		0			0
$369$	1.00000	+	1.73205i	0.0520579	+	0.0901670i
$370$		0			0
$371$	−16.0000	+	13.8564i	−0.830679	+	0.719389i
$372$		0			0
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$	−6.00000	+	10.3923i	−0.309839	+	0.536656i
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	−15.0000			−0.770498			−0.385249	−	0.922813i	$-0.625884\pi$
$379$	−15.0000			−0.770498			−0.385249	+	0.922813i	$0.625884\pi$
$380$		0			0
$381$	−0.500000	+	0.866025i	−0.0256158	+	0.0443678i
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$	−30.0000	−	10.3923i	−1.52894	−	0.529641i
$386$		0			0
$387$	0.500000	+	0.866025i	0.0254164	+	0.0440225i
$388$		0			0
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	−0.321716	−	0.946836i	$-0.604260\pi$
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	0.980842	−	0.194804i	$-0.0624070\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$	−14.0000			−0.706207
$394$		0			0
$395$	−1.00000	+	1.73205i	−0.0503155	+	0.0871489i
$396$		0			0
$397$	2.50000	+	4.33013i	0.125471	+	0.217323i	0.921917	−	0.387387i	$-0.126622\pi$
$397$	2.50000	+	4.33013i	0.125471	+	0.217323i	−0.796446	+	0.604710i	$0.793289\pi$
$398$		0			0
$399$	12.5000	+	4.33013i	0.625783	+	0.216777i
$400$		0			0
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	10.5000	−	18.1865i	0.523042	−	0.905936i
$404$		0			0
$405$	2.00000			0.0993808
$406$		0			0
$407$	54.0000			2.67668
$408$		0			0
$409$	1.50000	−	2.59808i	0.0741702	−	0.128467i	−0.826555	−	0.562856i	$-0.809703\pi$
$409$	1.50000	−	2.59808i	0.0741702	−	0.128467i	0.900725	+	0.434389i	$0.143036\pi$
$410$		0			0
$411$	10.0000	+	17.3205i	0.493264	+	0.854358i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−6.00000	−	10.3923i	−0.294528	−	0.510138i
$416$		0			0
$417$	−4.50000	+	7.79423i	−0.220366	+	0.381685i
$418$		0			0
$419$	26.0000			1.27018			0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000			1.27018			0.635092	+	0.772437i	$0.280962\pi$
$420$		0			0
$421$	−35.0000			−1.70580			−0.852898	−	0.522078i	$-0.825157\pi$
$421$	−35.0000			−1.70580			−0.852898	+	0.522078i	$0.825157\pi$
$422$		0			0
$423$	−1.00000	+	1.73205i	−0.0486217	+	0.0842152i
$424$		0			0
$425$	2.00000	+	3.46410i	0.0970143	+	0.168034i
$426$		0			0
$427$	−5.00000	−	25.9808i	−0.241967	−	1.25730i
$428$		0			0
$429$	9.00000	+	15.5885i	0.434524	+	0.752618i
$430$		0			0
$431$	9.00000	−	15.5885i	0.433515	−	0.750870i	−0.563658	−	0.826008i	$-0.690607\pi$
$431$	9.00000	−	15.5885i	0.433515	−	0.750870i	0.997173	+	0.0751385i	$0.0239399\pi$
$432$		0			0
$433$	31.0000			1.48976			0.744882	−	0.667196i	$-0.232506\pi$
$433$	31.0000			1.48976			0.744882	+	0.667196i	$0.232506\pi$
$434$		0			0
$435$	8.00000			0.383571
$436$		0			0
$437$	−10.0000	+	17.3205i	−0.478365	+	0.828552i
$438$		0			0
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	−0.996284	−	0.0861252i	$-0.972552\pi$
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	0.423556	−	0.905870i	$-0.360782\pi$
$440$		0			0
$441$	1.00000	−	6.92820i	0.0476190	−	0.329914i
$442$		0			0
$443$	8.00000	+	13.8564i	0.380091	+	0.658338i	0.991075	−	0.133306i	$-0.0425592\pi$
$443$	8.00000	+	13.8564i	0.380091	+	0.658338i	−0.610984	+	0.791643i	$0.709226\pi$
$444$		0			0
$445$	8.00000	−	13.8564i	0.379236	−	0.656857i
$446$		0			0
$447$	−4.00000			−0.189194
$448$		0			0
$449$	−38.0000			−1.79333			−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000			−1.79333			−0.896665	+	0.442709i	$0.854018\pi$
$450$		0			0
$451$	−6.00000	+	10.3923i	−0.282529	+	0.489355i
$452$		0			0
$453$	−4.00000	−	6.92820i	−0.187936	−	0.325515i
$454$		0			0
$455$	3.00000	+	15.5885i	0.140642	+	0.730798i
$456$		0			0
$457$	−6.50000	−	11.2583i	−0.304057	−	0.526642i	0.672994	−	0.739648i	$-0.265008\pi$
$457$	−6.50000	−	11.2583i	−0.304057	−	0.526642i	−0.977051	+	0.213006i	$0.931675\pi$
$458$		0			0
$459$	2.00000	−	3.46410i	0.0933520	−	0.161690i
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$	17.0000			0.790057			0.395029	−	0.918669i	$-0.370735\pi$
$463$	17.0000			0.790057			0.395029	+	0.918669i	$0.370735\pi$
$464$		0			0
$465$	7.00000	−	12.1244i	0.324617	−	0.562254i
$466$		0			0
$467$	−15.0000	−	25.9808i	−0.694117	−	1.20225i	−0.970477	−	0.241192i	$-0.922462\pi$
$467$	−15.0000	−	25.9808i	−0.694117	−	1.20225i	0.276360	−	0.961054i	$-0.410872\pi$
$468$		0			0
$469$	30.0000	−	25.9808i	1.38527	−	1.19968i
$470$		0			0
$471$	9.00000	+	15.5885i	0.414698	+	0.718278i
$472$		0			0
$473$	−3.00000	+	5.19615i	−0.137940	+	0.238919i
$474$		0			0
$475$	5.00000			0.229416
$476$		0			0
$477$	8.00000			0.366295
$478$		0			0
$479$	−8.00000	+	13.8564i	−0.365529	+	0.633115i	−0.988861	−	0.148842i	$-0.952445\pi$
$479$	−8.00000	+	13.8564i	−0.365529	+	0.633115i	0.623332	+	0.781958i	$0.285779\pi$
$480$		0			0
$481$	−13.5000	−	23.3827i	−0.615547	−	1.06616i
$482$		0			0
$483$	10.0000	+	3.46410i	0.455016	+	0.157622i
$484$		0			0
$485$	14.0000	+	24.2487i	0.635707	+	1.10108i
$486$		0			0
$487$	−12.5000	+	21.6506i	−0.566429	+	0.981084i	0.430486	+	0.902597i	$0.358342\pi$
$487$	−12.5000	+	21.6506i	−0.566429	+	0.981084i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$		0			0
$493$	8.00000	−	13.8564i	0.360302	−	0.624061i
$494$		0			0
$495$	6.00000	+	10.3923i	0.269680	+	0.467099i
$496$		0			0
$497$	−15.0000	−	5.19615i	−0.672842	−	0.233079i
$498$		0			0
$499$	8.50000	+	14.7224i	0.380512	+	0.659067i	0.991136	−	0.132855i	$-0.0424144\pi$
$499$	8.50000	+	14.7224i	0.380512	+	0.659067i	−0.610623	+	0.791921i	$0.709081\pi$
$500$		0			0
$501$	9.00000	−	15.5885i	0.402090	−	0.696441i
$502$		0			0
$503$	−14.0000			−0.624229			−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000			−0.624229			−0.312115	+	0.950044i	$0.601037\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$	−2.00000	+	3.46410i	−0.0888231	+	0.153846i
$508$		0			0
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	0.791849	−	0.610718i	$-0.209119\pi$
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	−0.924821	+	0.380402i	$0.875786\pi$
$510$		0			0
$511$	22.0000	−	19.0526i	0.973223	−	0.842836i
$512$		0			0
$513$	−2.50000	−	4.33013i	−0.110378	−	0.191180i
$514$		0			0
$515$	9.00000	−	15.5885i	0.396587	−	0.686909i
$516$		0			0
$517$	−12.0000			−0.527759
$518$		0			0
$519$	−20.0000			−0.877903
$520$		0			0
$521$	6.00000	−	10.3923i	0.262865	−	0.455295i	−0.704137	−	0.710064i	$-0.748666\pi$
$521$	6.00000	−	10.3923i	0.262865	−	0.455295i	0.967002	+	0.254769i	$0.0819994\pi$
$522$		0			0
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	−0.986718	−	0.162446i	$-0.948062\pi$
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	0.352677	−	0.935745i	$-0.385272\pi$
$524$		0			0
$525$	−0.500000	−	2.59808i	−0.0218218	−	0.113389i
$526$		0			0
$527$	−14.0000	−	24.2487i	−0.609850	−	1.05629i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.00000			0.259889
$534$		0			0
$535$	12.0000	−	20.7846i	0.518805	−	0.898597i
$536$		0			0
$537$	13.0000	+	22.5167i	0.560991	+	0.971666i
$538$		0			0
$539$	39.0000	−	15.5885i	1.67985	−	0.671442i
$540$		0			0
$541$	−0.500000	−	0.866025i	−0.0214967	−	0.0372333i	0.855077	−	0.518501i	$-0.173510\pi$
$541$	−0.500000	−	0.866025i	−0.0214967	−	0.0372333i	−0.876574	+	0.481268i	$0.840176\pi$
$542$		0			0
$543$	−3.50000	+	6.06218i	−0.150199	+	0.260153i
$544$		0			0
$545$	−22.0000			−0.942376
$546$		0			0
$547$	4.00000			0.171028			0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000			0.171028			0.0855138	+	0.996337i	$0.472747\pi$
$548$		0			0
$549$	−5.00000	+	8.66025i	−0.213395	+	0.369611i
$550$		0			0
$551$	−10.0000	−	17.3205i	−0.426014	−	0.737878i
$552$		0			0
$553$	−0.500000	−	2.59808i	−0.0212622	−	0.110481i
$554$		0			0
$555$	−9.00000	−	15.5885i	−0.382029	−	0.661693i
$556$		0			0
$557$	1.00000	−	1.73205i	0.0423714	−	0.0733893i	−0.844062	−	0.536246i	$-0.819842\pi$
$557$	1.00000	−	1.73205i	0.0423714	−	0.0733893i	0.886433	+	0.462856i	$0.153175\pi$
$558$		0			0
$559$	3.00000			0.126886
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$	1.00000	−	1.73205i	0.0421450	−	0.0729972i	−0.844183	−	0.536054i	$-0.819914\pi$
$563$	1.00000	−	1.73205i	0.0421450	−	0.0729972i	0.886328	+	0.463057i	$0.153248\pi$
$564$		0			0
$565$	−6.00000	−	10.3923i	−0.252422	−	0.437208i
$566$		0			0
$567$	−2.00000	+	1.73205i	−0.0839921	+	0.0727393i
$568$		0			0
$569$	9.00000	+	15.5885i	0.377300	+	0.653502i	0.990668	−	0.136295i	$-0.0435194\pi$
$569$	9.00000	+	15.5885i	0.377300	+	0.653502i	−0.613369	+	0.789797i	$0.710186\pi$
$570$		0			0
$571$	−11.5000	+	19.9186i	−0.481260	+	0.833567i	−0.999769	−	0.0215055i	$-0.993154\pi$
$571$	−11.5000	+	19.9186i	−0.481260	+	0.833567i	0.518509	+	0.855072i	$0.326487\pi$
$572$		0			0
$573$	−10.0000			−0.417756
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$	−19.5000	+	33.7750i	−0.811796	+	1.40607i	0.0998105	+	0.995006i	$0.468176\pi$
$577$	−19.5000	+	33.7750i	−0.811796	+	1.40607i	−0.911606	+	0.411065i	$0.865157\pi$
$578$		0			0
$579$	1.50000	+	2.59808i	0.0623379	+	0.107972i
$580$		0			0
$581$	15.0000	+	5.19615i	0.622305	+	0.215573i
$582$		0			0
$583$	24.0000	+	41.5692i	0.993978	+	1.72162i
$584$		0			0
$585$	3.00000	−	5.19615i	0.124035	−	0.214834i
$586$		0			0
$587$	16.0000			0.660391			0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000			0.660391			0.330195	+	0.943913i	$0.392885\pi$
$588$		0			0
$589$	−35.0000			−1.44215
$590$		0			0
$591$	−6.00000	+	10.3923i	−0.246807	+	0.427482i
$592$		0			0
$593$	15.0000	+	25.9808i	0.615976	+	1.06690i	0.990212	+	0.139569i	$0.0445716\pi$
$593$	15.0000	+	25.9808i	0.615976	+	1.06690i	−0.374236	+	0.927333i	$0.622095\pi$
$594$		0			0
$595$	20.0000	+	6.92820i	0.819920	+	0.284029i
$596$		0			0
$597$	−8.00000	−	13.8564i	−0.327418	−	0.567105i
$598$		0			0
$599$	−2.00000	+	3.46410i	−0.0817178	+	0.141539i	−0.903988	−	0.427558i	$-0.859374\pi$
$599$	−2.00000	+	3.46410i	−0.0817178	+	0.141539i	0.822270	+	0.569097i	$0.192707\pi$
$600$		0			0
$601$	31.0000			1.26452			0.632258	−	0.774758i	$-0.282128\pi$
$601$	31.0000			1.26452			0.632258	+	0.774758i	$0.282128\pi$
$602$		0			0
$603$	−15.0000			−0.610847
$604$		0			0
$605$	−25.0000	+	43.3013i	−1.01639	+	1.76045i
$606$		0			0
$607$	−0.500000	−	0.866025i	−0.0202944	−	0.0351509i	0.855700	−	0.517472i	$-0.173127\pi$
$607$	−0.500000	−	0.866025i	−0.0202944	−	0.0351509i	−0.875994	+	0.482322i	$0.839794\pi$
$608$		0			0
$609$	−8.00000	+	6.92820i	−0.324176	+	0.280745i
$610$		0			0
$611$	3.00000	+	5.19615i	0.121367	+	0.210214i
$612$		0			0
$613$	19.0000	−	32.9090i	0.767403	−	1.32918i	−0.171564	−	0.985173i	$-0.554882\pi$
$613$	19.0000	−	32.9090i	0.767403	−	1.32918i	0.938967	−	0.344008i	$-0.111785\pi$
$614$		0			0
$615$	4.00000			0.161296
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$		0			0
$619$	−4.50000	+	7.79423i	−0.180870	+	0.313276i	−0.942177	−	0.335115i	$-0.891225\pi$
$619$	−4.50000	+	7.79423i	−0.180870	+	0.313276i	0.761307	+	0.648392i	$0.224558\pi$
$620$		0			0
$621$	−2.00000	−	3.46410i	−0.0802572	−	0.139010i
$622$		0			0
$623$	4.00000	+	20.7846i	0.160257	+	0.832718i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$	15.0000	−	25.9808i	0.599042	−	1.03757i
$628$		0			0
$629$	−36.0000			−1.43541
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$	−2.00000	+	3.46410i	−0.0794929	+	0.137686i
$634$		0			0
$635$	1.00000	+	1.73205i	0.0396838	+	0.0687343i
$636$		0			0
$637$	−16.5000	−	12.9904i	−0.653754	−	0.514698i
$638$		0			0
$639$	3.00000	+	5.19615i	0.118678	+	0.205557i
$640$		0			0
$641$	10.0000	−	17.3205i	0.394976	−	0.684119i	−0.598122	−	0.801405i	$-0.704086\pi$
$641$	10.0000	−	17.3205i	0.394976	−	0.684119i	0.993098	+	0.117286i	$0.0374195\pi$
$642$		0			0
$643$	−17.0000			−0.670415			−0.335207	−	0.942144i	$-0.608806\pi$
$643$	−17.0000			−0.670415			−0.335207	+	0.942144i	$0.608806\pi$
$644$		0			0
$645$	2.00000			0.0787499
$646$		0			0
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	−0.633090	−	0.774078i	$-0.718214\pi$
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	0.986916	+	0.161233i	$0.0515470\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	3.50000	+	18.1865i	0.137176	+	0.712786i
$652$		0			0
$653$	11.0000	+	19.0526i	0.430463	+	0.745584i	0.996913	−	0.0785119i	$-0.0250169\pi$
$653$	11.0000	+	19.0526i	0.430463	+	0.745584i	−0.566450	+	0.824096i	$0.691684\pi$
$654$		0			0
$655$	−14.0000	+	24.2487i	−0.547025	+	0.947476i
$656$		0			0
$657$	−11.0000			−0.429151
$658$		0			0
$659$	−40.0000			−1.55818			−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000			−1.55818			−0.779089	+	0.626913i	$0.784318\pi$
$660$		0			0
$661$	−17.5000	+	30.3109i	−0.680671	+	1.17896i	0.294105	+	0.955773i	$0.404978\pi$
$661$	−17.5000	+	30.3109i	−0.680671	+	1.17896i	−0.974776	+	0.223184i	$0.928355\pi$
$662$		0			0
$663$	−6.00000	−	10.3923i	−0.233021	−	0.403604i
$664$		0			0
$665$	20.0000	−	17.3205i	0.775567	−	0.671660i
$666$		0			0
$667$	−8.00000	−	13.8564i	−0.309761	−	0.536522i
$668$		0			0
$669$	12.0000	−	20.7846i	0.463947	−	0.803579i
$670$		0			0
$671$	−60.0000			−2.31627
$672$		0			0
$673$	7.00000			0.269830			0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000			0.269830			0.134915	+	0.990857i	$0.456924\pi$
$674$		0			0
$675$	−0.500000	+	0.866025i	−0.0192450	+	0.0333333i
$676$		0			0
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	0.727386	−	0.686229i	$-0.240735\pi$
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	−0.957984	+	0.286820i	$0.907402\pi$
$678$		0			0
$679$	−35.0000	−	12.1244i	−1.34318	−	0.465290i
$680$		0			0
$681$	7.00000	+	12.1244i	0.268241	+	0.464606i
$682$		0			0
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$	40.0000			1.52832
$686$		0			0
$687$	7.00000			0.267067
$688$		0			0
$689$	12.0000	−	20.7846i	0.457164	−	0.791831i
$690$		0			0
$691$	3.50000	+	6.06218i	0.133146	+	0.230616i	0.924888	−	0.380240i	$-0.124159\pi$
$691$	3.50000	+	6.06218i	0.133146	+	0.230616i	−0.791742	+	0.610856i	$0.790825\pi$
$692$		0			0
$693$	−15.0000	−	5.19615i	−0.569803	−	0.197386i
$694$		0			0
$695$	9.00000	+	15.5885i	0.341389	+	0.591304i
$696$		0			0
$697$	4.00000	−	6.92820i	0.151511	−	0.262424i
$698$		0			0
$699$	−26.0000			−0.983410
$700$		0			0
$701$	28.0000			1.05755			0.528773	−	0.848763i	$-0.322652\pi$
$701$	28.0000			1.05755			0.528773	+	0.848763i	$0.322652\pi$
$702$		0			0
$703$	−22.5000	+	38.9711i	−0.848604	+	1.46982i
$704$		0			0
$705$	2.00000	+	3.46410i	0.0753244	+	0.130466i
$706$		0			0
$707$	12.0000	−	10.3923i	0.451306	−	0.390843i
$708$		0			0
$709$	25.0000	+	43.3013i	0.938895	+	1.62621i	0.767537	+	0.641004i	$0.221482\pi$
$709$	25.0000	+	43.3013i	0.938895	+	1.62621i	0.171358	+	0.985209i	$0.445185\pi$
$710$		0			0
$711$	−0.500000	+	0.866025i	−0.0187515	+	0.0324785i
$712$		0			0
$713$	−28.0000			−1.04861
$714$		0			0
$715$	36.0000			1.34632
$716$		0			0
$717$	1.00000	−	1.73205i	0.0373457	−	0.0646846i
$718$		0			0
$719$	15.0000	+	25.9808i	0.559406	+	0.968919i	0.997546	+	0.0700124i	$0.0223039\pi$
$719$	15.0000	+	25.9808i	0.559406	+	0.968919i	−0.438141	+	0.898906i	$0.644363\pi$
$720$		0			0
$721$	4.50000	+	23.3827i	0.167589	+	0.870817i
$722$		0			0
$723$	−1.00000	−	1.73205i	−0.0371904	−	0.0644157i
$724$		0			0
$725$	−2.00000	+	3.46410i	−0.0742781	+	0.128654i
$726$		0			0
$727$	5.00000			0.185440			0.0927199	−	0.995692i	$-0.470444\pi$
$727$	5.00000			0.185440			0.0927199	+	0.995692i	$0.470444\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	2.00000	−	3.46410i	0.0739727	−	0.128124i
$732$		0			0
$733$	5.50000	+	9.52628i	0.203147	+	0.351861i	0.949541	−	0.313644i	$-0.101550\pi$
$733$	5.50000	+	9.52628i	0.203147	+	0.351861i	−0.746394	+	0.665505i	$0.768216\pi$
$734$		0			0
$735$	−11.0000	−	8.66025i	−0.405741	−	0.319438i
$736$		0			0
$737$	−45.0000	−	77.9423i	−1.65760	−	2.87104i
$738$		0			0
$739$	2.50000	−	4.33013i	0.0919640	−	0.159286i	−0.816373	−	0.577524i	$-0.804019\pi$
$739$	2.50000	−	4.33013i	0.0919640	−	0.159286i	0.908337	+	0.418238i	$0.137352\pi$
$740$		0			0
$741$	−15.0000			−0.551039
$742$		0			0
$743$	−34.0000			−1.24734			−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000			−1.24734			−0.623670	+	0.781688i	$0.714359\pi$
$744$		0			0
$745$	−4.00000	+	6.92820i	−0.146549	+	0.253830i
$746$		0			0
$747$	−3.00000	−	5.19615i	−0.109764	−	0.190117i
$748$		0			0
$749$	6.00000	+	31.1769i	0.219235	+	1.13918i
$750$		0			0
$751$	18.5000	+	32.0429i	0.675075	+	1.16926i	0.976447	+	0.215757i	$0.0692219\pi$
$751$	18.5000	+	32.0429i	0.675075	+	1.16926i	−0.301373	+	0.953506i	$0.597445\pi$
$752$		0			0
$753$	2.00000	−	3.46410i	0.0728841	−	0.126239i
$754$		0			0
$755$	−16.0000			−0.582300
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$	12.0000	−	20.7846i	0.435572	−	0.754434i
$760$		0			0
$761$	6.00000	+	10.3923i	0.217500	+	0.376721i	0.954043	−	0.299670i	$-0.0968765\pi$
$761$	6.00000	+	10.3923i	0.217500	+	0.376721i	−0.736543	+	0.676391i	$0.763543\pi$
$762$		0			0
$763$	22.0000	−	19.0526i	0.796453	−	0.689749i
$764$		0			0
$765$	−4.00000	−	6.92820i	−0.144620	−	0.250490i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	7.00000			0.252426			0.126213	−	0.992003i	$-0.459718\pi$
$769$	7.00000			0.252426			0.126213	+	0.992003i	$0.459718\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		0			0
$773$	25.0000	−	43.3013i	0.899188	−	1.55744i	0.0706526	−	0.997501i	$-0.477492\pi$
$773$	25.0000	−	43.3013i	0.899188	−	1.55744i	0.828535	−	0.559937i	$-0.189175\pi$
$774$		0			0
$775$	3.50000	+	6.06218i	0.125724	+	0.217760i
$776$		0			0
$777$	22.5000	+	7.79423i	0.807183	+	0.279616i
$778$		0			0
$779$	−5.00000	−	8.66025i	−0.179144	−	0.310286i
$780$		0			0
$781$	−18.0000	+	31.1769i	−0.644091	+	1.11560i
$782$		0			0
$783$	4.00000			0.142948
$784$		0			0
$785$	36.0000			1.28490
$786$		0			0
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	−0.426193	−	0.904632i	$-0.640145\pi$
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	0.996531	−	0.0832226i	$-0.0265213\pi$
$788$		0			0
$789$	−6.00000	−	10.3923i	−0.213606	−	0.369976i
$790$		0			0
$791$	15.0000	+	5.19615i	0.533339	+	0.184754i
$792$		0			0
$793$	15.0000	+	25.9808i	0.532666	+	0.922604i
$794$		0			0
$795$	8.00000	−	13.8564i	0.283731	−	0.491436i
$796$		0			0
$797$	12.0000			0.425062			0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000			0.425062			0.212531	+	0.977154i	$0.431829\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$	4.00000	−	6.92820i	0.141333	−	0.244796i
$802$		0			0
$803$	−33.0000	−	57.1577i	−1.16454	−	2.01705i
$804$		0			0
$805$	16.0000	−	13.8564i	0.563926	−	0.488374i
$806$		0			0
$807$	−9.00000	−	15.5885i	−0.316815	−	0.548740i
$808$		0			0
$809$	15.0000	−	25.9808i	0.527372	−	0.913435i	−0.472119	−	0.881535i	$-0.656511\pi$
$809$	15.0000	−	25.9808i	0.527372	−	0.913435i	0.999491	−	0.0319002i	$-0.0101559\pi$
$810$		0			0
$811$	24.0000			0.842754			0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000			0.842754			0.421377	+	0.906886i	$0.361547\pi$
$812$		0			0
$813$	−8.00000			−0.280572
$814$		0			0
$815$	−4.00000	+	6.92820i	−0.140114	+	0.242684i
$816$		0			0
$817$	−2.50000	−	4.33013i	−0.0874639	−	0.151492i
$818$		0			0
$819$	1.50000	+	7.79423i	0.0524142	+	0.272352i
$820$		0			0
$821$	9.00000	+	15.5885i	0.314102	+	0.544041i	0.979246	−	0.202674i	$-0.0649632\pi$
$821$	9.00000	+	15.5885i	0.314102	+	0.544041i	−0.665144	+	0.746715i	$0.731630\pi$
$822$		0			0
$823$	−4.00000	+	6.92820i	−0.139431	+	0.241502i	−0.927281	−	0.374365i	$-0.877861\pi$
$823$	−4.00000	+	6.92820i	−0.139431	+	0.241502i	0.787850	+	0.615867i	$0.211194\pi$
$824$		0			0
$825$	−6.00000			−0.208893
$826$		0			0
$827$	18.0000			0.625921			0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000			0.625921			0.312961	+	0.949766i	$0.398679\pi$
$828$		0			0
$829$	21.5000	−	37.2391i	0.746726	−	1.29337i	−0.202658	−	0.979250i	$-0.564958\pi$
$829$	21.5000	−	37.2391i	0.746726	−	1.29337i	0.949384	−	0.314118i	$-0.101709\pi$
$830$		0			0
$831$	0.500000	+	0.866025i	0.0173448	+	0.0300421i
$832$		0			0
$833$	−26.0000	+	10.3923i	−0.900847	+	0.360072i
$834$		0			0
$835$	−18.0000	−	31.1769i	−0.622916	−	1.07892i
$836$		0			0
$837$	3.50000	−	6.06218i	0.120978	−	0.209540i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.00000	+	6.92820i	0.137604	+	0.238337i
$846$		0			0
$847$	−12.5000	−	64.9519i	−0.429505	−	2.23177i
$848$		0			0
$849$	−0.500000	−	0.866025i	−0.0171600	−	0.0297219i
$850$		0			0
$851$	−18.0000	+	31.1769i	−0.617032	+	1.06873i
$852$		0			0
$853$	−17.0000			−0.582069			−0.291034	−	0.956713i	$-0.593999\pi$
$853$	−17.0000			−0.582069			−0.291034	+	0.956713i	$0.593999\pi$
$854$		0			0
$855$	−10.0000			−0.341993
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−8.00000	−	13.8564i	−0.272956	−	0.472774i	0.696661	−	0.717400i	$-0.254668\pi$
$859$	−8.00000	−	13.8564i	−0.272956	−	0.472774i	−0.969618	+	0.244626i	$0.921335\pi$
$860$		0			0
$861$	−4.00000	+	3.46410i	−0.136320	+	0.118056i
$862$		0			0
$863$	−7.00000	−	12.1244i	−0.238283	−	0.412718i	0.721939	−	0.691957i	$-0.243251\pi$
$863$	−7.00000	−	12.1244i	−0.238283	−	0.412718i	−0.960222	+	0.279239i	$0.909918\pi$
$864$		0			0
$865$	−20.0000	+	34.6410i	−0.680020	+	1.17783i
$866$		0			0
$867$	1.00000			0.0339618
$868$		0			0
$869$	−6.00000			−0.203536
$870$		0			0
$871$	−22.5000	+	38.9711i	−0.762383	+	1.32049i
$872$		0			0
$873$	7.00000	+	12.1244i	0.236914	+	0.410347i
$874$		0			0
$875$	−30.0000	−	10.3923i	−1.01419	−	0.351324i
$876$		0			0
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	0.848648	−	0.528958i	$-0.177417\pi$
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	−0.882415	+	0.470471i	$0.844084\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	52.0000			1.75192			0.875962	−	0.482380i	$-0.160227\pi$
$881$	52.0000			1.75192			0.875962	+	0.482380i	$0.160227\pi$
$882$		0			0
$883$	1.00000			0.0336527			0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000			0.0336527			0.0168263	+	0.999858i	$0.494644\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.0000	+	46.7654i	0.906571	+	1.57023i	0.818794	+	0.574087i	$0.194643\pi$
$887$	27.0000	+	46.7654i	0.906571	+	1.57023i	0.0877772	+	0.996140i	$0.472024\pi$
$888$		0			0
$889$	−2.50000	−	0.866025i	−0.0838473	−	0.0290456i
$890$		0			0
$891$	3.00000	+	5.19615i	0.100504	+	0.174078i
$892$		0			0
$893$	5.00000	−	8.66025i	0.167319	−	0.289804i
$894$		0			0
$895$	52.0000			1.73817
$896$		0			0
$897$	−12.0000			−0.400668
$898$		0			0
$899$	14.0000	−	24.2487i	0.466926	−	0.808740i
$900$		0			0
$901$	−16.0000	−	27.7128i	−0.533037	−	0.923248i
$902$		0			0
$903$	−2.00000	+	1.73205i	−0.0665558	+	0.0576390i
$904$		0			0
$905$	7.00000	+	12.1244i	0.232688	+	0.403027i
$906$		0			0
$907$	−8.50000	+	14.7224i	−0.282238	+	0.488850i	−0.971936	−	0.235247i	$-0.924410\pi$
$907$	−8.50000	+	14.7224i	−0.282238	+	0.488850i	0.689698	+	0.724097i	$0.257743\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	28.0000			0.927681			0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000			0.927681			0.463841	+	0.885919i	$0.346471\pi$
$912$		0			0
$913$	18.0000	−	31.1769i	0.595713	−	1.03181i
$914$		0			0
$915$	10.0000	+	17.3205i	0.330590	+	0.572598i
$916$		0			0
$917$	−7.00000	−	36.3731i	−0.231160	−	1.20114i
$918$		0			0
$919$	−0.500000	−	0.866025i	−0.0164935	−	0.0285675i	0.857661	−	0.514216i	$-0.171917\pi$
$919$	−0.500000	−	0.866025i	−0.0164935	−	0.0285675i	−0.874154	+	0.485648i	$0.838584\pi$
$920$		0			0
$921$	−5.50000	+	9.52628i	−0.181231	+	0.313902i
$922$		0			0
$923$	18.0000			0.592477
$924$		0			0
$925$	9.00000			0.295918
$926$		0			0
$927$	4.50000	−	7.79423i	0.147799	−	0.255996i
$928$		0			0
$929$	−29.0000	−	50.2295i	−0.951459	−	1.64798i	−0.742271	−	0.670100i	$-0.766251\pi$
$929$	−29.0000	−	50.2295i	−0.951459	−	1.64798i	−0.209189	−	0.977875i	$-0.567082\pi$
$930$		0			0
$931$	−5.00000	+	34.6410i	−0.163868	+	1.13531i
$932$		0			0
$933$	−9.00000	−	15.5885i	−0.294647	−	0.510343i
$934$		0			0
$935$	24.0000	−	41.5692i	0.784884	−	1.35946i
$936$		0			0
$937$	−33.0000			−1.07806			−0.539032	−	0.842286i	$-0.681210\pi$
$937$	−33.0000			−1.07806			−0.539032	+	0.842286i	$0.681210\pi$
$938$		0			0
$939$	1.00000			0.0326338
$940$		0			0
$941$	−24.0000	+	41.5692i	−0.782378	+	1.35512i	0.148176	+	0.988961i	$0.452660\pi$
$941$	−24.0000	+	41.5692i	−0.782378	+	1.35512i	−0.930553	+	0.366157i	$0.880673\pi$
$942$		0			0
$943$	−4.00000	−	6.92820i	−0.130258	−	0.225613i
$944$		0			0
$945$	1.00000	+	5.19615i	0.0325300	+	0.169031i
$946$		0			0
$947$	19.0000	+	32.9090i	0.617417	+	1.06940i	0.989955	+	0.141381i	$0.0451542\pi$
$947$	19.0000	+	32.9090i	0.617417	+	1.06940i	−0.372538	+	0.928017i	$0.621512\pi$
$948$		0			0
$949$	−16.5000	+	28.5788i	−0.535613	+	0.927708i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−8.00000			−0.259145			−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000			−0.259145			−0.129573	+	0.991570i	$0.541361\pi$
$954$		0			0
$955$	−10.0000	+	17.3205i	−0.323592	+	0.560478i
$956$		0			0
$957$	12.0000	+	20.7846i	0.387905	+	0.671871i
$958$		0			0
$959$	−40.0000	+	34.6410i	−1.29167	+	1.11862i
$960$		0			0
$961$	−9.00000	−	15.5885i	−0.290323	−	0.502853i
$962$		0			0
$963$	6.00000	−	10.3923i	0.193347	−	0.334887i
$964$		0			0
$965$	6.00000			0.193147
$966$		0			0
$967$	−27.0000			−0.868261			−0.434131	−	0.900850i	$-0.642944\pi$
$967$	−27.0000			−0.868261			−0.434131	+	0.900850i	$0.642944\pi$
$968$		0			0
$969$	−10.0000	+	17.3205i	−0.321246	+	0.556415i
$970$		0			0
$971$	28.0000	+	48.4974i	0.898563	+	1.55636i	0.829332	+	0.558756i	$0.188721\pi$
$971$	28.0000	+	48.4974i	0.898563	+	1.55636i	0.0692304	+	0.997601i	$0.477946\pi$
$972$		0			0
$973$	−22.5000	−	7.79423i	−0.721317	−	0.249871i
$974$		0			0
$975$	1.50000	+	2.59808i	0.0480384	+	0.0832050i
$976$		0			0
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	−0.685381	−	0.728184i	$-0.740364\pi$
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	0.973317	+	0.229465i	$0.0736978\pi$
$978$		0			0
$979$	48.0000			1.53409
$980$		0			0
$981$	−11.0000			−0.351203
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	12.0000	+	20.7846i	0.382352	+	0.662253i
$986$		0			0
$987$	−5.00000	−	1.73205i	−0.159152	−	0.0551318i
$988$		0			0
$989$	−2.00000	−	3.46410i	−0.0635963	−	0.110152i
$990$		0			0
$991$	16.5000	−	28.5788i	0.524140	−	0.907837i	−0.475465	−	0.879734i	$-0.657720\pi$
$991$	16.5000	−	28.5788i	0.524140	−	0.907837i	0.999605	−	0.0281022i	$-0.00894640\pi$
$992$		0			0
$993$	−5.00000			−0.158670
$994$		0			0
$995$	−32.0000			−1.01447
$996$		0			0
$997$	8.50000	−	14.7224i	0.269198	−	0.466264i	−0.699457	−	0.714675i	$-0.746575\pi$
$997$	8.50000	−	14.7224i	0.269198	−	0.466264i	0.968655	+	0.248410i	$0.0799082\pi$
$998$		0			0
$999$	−4.50000	−	7.79423i	−0.142374	−	0.246598i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.q.b.25.1	✓	2
3.2	odd	2		504.2.s.g.361.1		2
4.3	odd	2		336.2.q.a.193.1		2
7.2	even	3	inner	168.2.q.b.121.1	yes	2
7.3	odd	6		1176.2.a.e.1.1		1
7.4	even	3		1176.2.a.d.1.1		1
7.5	odd	6		1176.2.q.e.961.1		2
7.6	odd	2		1176.2.q.e.361.1		2
8.3	odd	2		1344.2.q.t.193.1		2
8.5	even	2		1344.2.q.i.193.1		2
12.11	even	2		1008.2.s.m.865.1		2
21.2	odd	6		504.2.s.g.289.1		2
21.5	even	6		3528.2.s.d.3313.1		2
21.11	odd	6		3528.2.a.f.1.1		1
21.17	even	6		3528.2.a.y.1.1		1
21.20	even	2		3528.2.s.d.361.1		2
28.3	even	6		2352.2.a.e.1.1		1
28.11	odd	6		2352.2.a.x.1.1		1
28.19	even	6		2352.2.q.v.961.1		2
28.23	odd	6		336.2.q.a.289.1		2
28.27	even	2		2352.2.q.v.1537.1		2
56.3	even	6		9408.2.a.cs.1.1		1
56.11	odd	6		9408.2.a.f.1.1		1
56.37	even	6		1344.2.q.i.961.1		2
56.45	odd	6		9408.2.a.bk.1.1		1
56.51	odd	6		1344.2.q.t.961.1		2
56.53	even	6		9408.2.a.cd.1.1		1
84.11	even	6		7056.2.a.i.1.1		1
84.23	even	6		1008.2.s.m.289.1		2
84.59	odd	6		7056.2.a.bn.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.q.b.25.1	✓	2	1.1	even	1	trivial
168.2.q.b.121.1	yes	2	7.2	even	3	inner
336.2.q.a.193.1		2	4.3	odd	2
336.2.q.a.289.1		2	28.23	odd	6
504.2.s.g.289.1		2	21.2	odd	6
504.2.s.g.361.1		2	3.2	odd	2
1008.2.s.m.289.1		2	84.23	even	6
1008.2.s.m.865.1		2	12.11	even	2
1176.2.a.d.1.1		1	7.4	even	3
1176.2.a.e.1.1		1	7.3	odd	6
1176.2.q.e.361.1		2	7.6	odd	2
1176.2.q.e.961.1		2	7.5	odd	6
1344.2.q.i.193.1		2	8.5	even	2
1344.2.q.i.961.1		2	56.37	even	6
1344.2.q.t.193.1		2	8.3	odd	2
1344.2.q.t.961.1		2	56.51	odd	6
2352.2.a.e.1.1		1	28.3	even	6
2352.2.a.x.1.1		1	28.11	odd	6
2352.2.q.v.961.1		2	28.19	even	6
2352.2.q.v.1537.1		2	28.27	even	2
3528.2.a.f.1.1		1	21.11	odd	6
3528.2.a.y.1.1		1	21.17	even	6
3528.2.s.d.361.1		2	21.20	even	2
3528.2.s.d.3313.1		2	21.5	even	6
7056.2.a.i.1.1		1	84.11	even	6
7056.2.a.bn.1.1		1	84.59	odd	6
9408.2.a.f.1.1		1	56.11	odd	6
9408.2.a.bk.1.1		1	56.45	odd	6
9408.2.a.cd.1.1		1	56.53	even	6
9408.2.a.cs.1.1		1	56.3	even	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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{5} +(2.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{5} +(2.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(3.00000 - 5.19615i) q^{11} -3.00000 q^{13} -2.00000 q^{15} +(-2.00000 + 3.46410i) q^{17} +(2.50000 + 4.33013i) q^{19} +(2.00000 - 1.73205i) q^{21} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} -1.00000 q^{27} -4.00000 q^{29} +(-3.50000 + 6.06218i) q^{31} +(-3.00000 - 5.19615i) q^{33} +(-1.00000 - 5.19615i) q^{35} +(4.50000 + 7.79423i) q^{37} +(-1.50000 + 2.59808i) q^{39} -2.00000 q^{41} -1.00000 q^{43} +(-1.00000 + 1.73205i) q^{45} +(-1.00000 - 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +(2.00000 + 3.46410i) q^{51} +(-4.00000 + 6.92820i) q^{53} -12.0000 q^{55} +5.00000 q^{57} +(-5.00000 - 8.66025i) q^{61} +(-0.500000 - 2.59808i) q^{63} +(3.00000 + 5.19615i) q^{65} +(7.50000 - 12.9904i) q^{67} +4.00000 q^{69} -6.00000 q^{71} +(5.50000 - 9.52628i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(12.0000 - 10.3923i) q^{77} +(-0.500000 - 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} +6.00000 q^{83} +8.00000 q^{85} +(-2.00000 + 3.46410i) q^{87} +(4.00000 + 6.92820i) q^{89} +(-7.50000 - 2.59808i) q^{91} +(3.50000 + 6.06218i) q^{93} +(5.00000 - 8.66025i) q^{95} -14.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 2 q^{5} + 5 q^{7} - q^{9}+O(q^{10}) \) 2 * q + q^3 - 2 * q^5 + 5 * q^7 - q^9	\( 2 q + q^{3} - 2 q^{5} + 5 q^{7} - q^{9} + 6 q^{11} - 6 q^{13} - 4 q^{15} - 4 q^{17} + 5 q^{19} + 4 q^{21} + 4 q^{23} + q^{25} - 2 q^{27} - 8 q^{29} - 7 q^{31} - 6 q^{33} - 2 q^{35} + 9 q^{37} - 3 q^{39} - 4 q^{41} - 2 q^{43} - 2 q^{45} - 2 q^{47} + 11 q^{49} + 4 q^{51} - 8 q^{53} - 24 q^{55} + 10 q^{57} - 10 q^{61} - q^{63} + 6 q^{65} + 15 q^{67} + 8 q^{69} - 12 q^{71} + 11 q^{73} - q^{75} + 24 q^{77} - q^{79} - q^{81} + 12 q^{83} + 16 q^{85} - 4 q^{87} + 8 q^{89} - 15 q^{91} + 7 q^{93} + 10 q^{95} - 28 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + q^3 - 2 * q^5 + 5 * q^7 - q^9 + 6 * q^11 - 6 * q^13 - 4 * q^15 - 4 * q^17 + 5 * q^19 + 4 * q^21 + 4 * q^23 + q^25 - 2 * q^27 - 8 * q^29 - 7 * q^31 - 6 * q^33 - 2 * q^35 + 9 * q^37 - 3 * q^39 - 4 * q^41 - 2 * q^43 - 2 * q^45 - 2 * q^47 + 11 * q^49 + 4 * q^51 - 8 * q^53 - 24 * q^55 + 10 * q^57 - 10 * q^61 - q^63 + 6 * q^65 + 15 * q^67 + 8 * q^69 - 12 * q^71 + 11 * q^73 - q^75 + 24 * q^77 - q^79 - q^81 + 12 * q^83 + 16 * q^85 - 4 * q^87 + 8 * q^89 - 15 * q^91 + 7 * q^93 + 10 * q^95 - 28 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more