LMFDB - Embedded newform 1680.2.bg.a.1201.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(961,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1680.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.bg (of order $3$, degree $2$, not 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:	$13.4148675396$
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:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1201.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1680.1201
Dual form			1680.2.bg.a.961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +1.00000 q^{15} +(2.00000 - 3.46410i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(2.00000 - 1.73205i) q^{21} +(2.00000 + 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} +(-2.50000 + 4.33013i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(0.500000 + 2.59808i) q^{35} +(2.50000 + 4.33013i) q^{37} +(-0.500000 + 0.866025i) q^{39} +2.00000 q^{41} +9.00000 q^{43} +(-0.500000 + 0.866025i) q^{45} +(-1.00000 - 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +(2.00000 + 3.46410i) q^{51} +(-6.00000 + 10.3923i) q^{53} +2.00000 q^{55} +1.00000 q^{57} +(-4.00000 + 6.92820i) q^{59} +(7.00000 + 12.1244i) q^{61} +(0.500000 + 2.59808i) q^{63} +(-0.500000 - 0.866025i) q^{65} +(4.50000 - 7.79423i) q^{67} -4.00000 q^{69} -2.00000 q^{71} +(-0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(4.00000 - 3.46410i) q^{77} +(-1.50000 - 2.59808i) q^{79} +(-0.500000 + 0.866025i) q^{81} +18.0000 q^{83} -4.00000 q^{85} +(2.00000 + 3.46410i) q^{89} +(-2.50000 - 0.866025i) q^{91} +(-2.50000 - 4.33013i) q^{93} +(-0.500000 + 0.866025i) q^{95} +10.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} - 5 q^{7} - q^{9}+O(q^{10}) $ 2 * q - q^3 - q^5 - 5 * q^7 - q^9	$ 2 q - q^{3} - q^{5} - 5 q^{7} - q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - q^{19} + 4 q^{21} + 4 q^{23} - q^{25} + 2 q^{27} - 5 q^{31} - 2 q^{33} + q^{35} + 5 q^{37} - q^{39} + 4 q^{41} + 18 q^{43} - q^{45} - 2 q^{47} + 11 q^{49} + 4 q^{51} - 12 q^{53} + 4 q^{55} + 2 q^{57} - 8 q^{59} + 14 q^{61} + q^{63} - q^{65} + 9 q^{67} - 8 q^{69} - 4 q^{71} - q^{73} - q^{75} + 8 q^{77} - 3 q^{79} - q^{81} + 36 q^{83} - 8 q^{85} + 4 q^{89} - 5 q^{91} - 5 q^{93} - q^{95} + 20 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 - 5 * q^7 - q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 - q^19 + 4 * q^21 + 4 * q^23 - q^25 + 2 * q^27 - 5 * q^31 - 2 * q^33 + q^35 + 5 * q^37 - q^39 + 4 * q^41 + 18 * q^43 - q^45 - 2 * q^47 + 11 * q^49 + 4 * q^51 - 12 * q^53 + 4 * q^55 + 2 * q^57 - 8 * q^59 + 14 * q^61 + q^63 - q^65 + 9 * q^67 - 8 * q^69 - 4 * q^71 - q^73 - q^75 + 8 * q^77 - 3 * q^79 - q^81 + 36 * q^83 - 8 * q^85 + 4 * q^89 - 5 * q^91 - 5 * q^93 - q^95 + 20 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		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
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	−0.514782	−	0.857321i	$-0.672127\pi$
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	0.999853	+	0.0171533i	$0.00546033\pi$
$18$		0			0
$19$	−0.500000	−	0.866025i	−0.114708	−	0.198680i	0.802955	−	0.596040i	$-0.203260\pi$
$19$	−0.500000	−	0.866025i	−0.114708	−	0.198680i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		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
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	0.549309	+	0.835619i	$0.314891\pi$
$32$		0			0
$33$	−1.00000	−	1.73205i	−0.174078	−	0.301511i
$34$		0			0
$35$	0.500000	+	2.59808i	0.0845154	+	0.439155i
$36$		0			0
$37$	2.50000	+	4.33013i	0.410997	+	0.711868i	0.994999	−	0.0998840i	$-0.0318472\pi$
$37$	2.50000	+	4.33013i	0.410997	+	0.711868i	−0.584002	+	0.811752i	$0.698514\pi$
$38$		0			0
$39$	−0.500000	+	0.866025i	−0.0800641	+	0.138675i
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	9.00000			1.37249			0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000			1.37249			0.686244	+	0.727372i	$0.259258\pi$
$44$		0			0
$45$	−0.500000	+	0.866025i	−0.0745356	+	0.129099i
$46$		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
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	0.0783936	+	0.996922i	$0.475021\pi$
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	−0.902557	+	0.430570i	$0.858312\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$		0			0
$57$	1.00000			0.132453
$58$		0			0
$59$	−4.00000	+	6.92820i	−0.520756	+	0.901975i	0.478953	+	0.877841i	$0.341016\pi$
$59$	−4.00000	+	6.92820i	−0.520756	+	0.901975i	−0.999709	+	0.0241347i	$0.992317\pi$
$60$		0			0
$61$	7.00000	+	12.1244i	0.896258	+	1.55236i	0.832240	+	0.554416i	$0.187058\pi$
$61$	7.00000	+	12.1244i	0.896258	+	1.55236i	0.0640184	+	0.997949i	$0.479608\pi$
$62$		0			0
$63$	0.500000	+	2.59808i	0.0629941	+	0.327327i
$64$		0			0
$65$	−0.500000	−	0.866025i	−0.0620174	−	0.107417i
$66$		0			0
$67$	4.50000	−	7.79423i	0.549762	−	0.952217i	−0.448528	−	0.893769i	$-0.648052\pi$
$67$	4.50000	−	7.79423i	0.549762	−	0.952217i	0.998290	−	0.0584478i	$-0.0186151\pi$
$68$		0			0
$69$	−4.00000			−0.481543
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	−0.500000	+	0.866025i	−0.0585206	+	0.101361i	−0.893801	−	0.448463i	$-0.851972\pi$
$73$	−0.500000	+	0.866025i	−0.0585206	+	0.101361i	0.835281	+	0.549823i	$0.185305\pi$
$74$		0			0
$75$	−0.500000	−	0.866025i	−0.0577350	−	0.100000i
$76$		0			0
$77$	4.00000	−	3.46410i	0.455842	−	0.394771i
$78$		0			0
$79$	−1.50000	−	2.59808i	−0.168763	−	0.292306i	0.769222	−	0.638982i	$-0.220644\pi$
$79$	−1.50000	−	2.59808i	−0.168763	−	0.292306i	−0.937985	+	0.346675i	$0.887311\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	18.0000			1.97576			0.987878	−	0.155230i	$-0.0496119\pi$
$83$	18.0000			1.97576			0.987878	+	0.155230i	$0.0496119\pi$
$84$		0			0
$85$	−4.00000			−0.433861
$86$		0			0
$87$		0			0
$88$		0			0
$89$	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
$91$	−2.50000	−	0.866025i	−0.262071	−	0.0907841i
$92$		0			0
$93$	−2.50000	−	4.33013i	−0.259238	−	0.449013i
$94$		0			0
$95$	−0.500000	+	0.866025i	−0.0512989	+	0.0888523i
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		0			0
$103$	−0.500000	−	0.866025i	−0.0492665	−	0.0853320i	0.840341	−	0.542059i	$-0.182355\pi$
$103$	−0.500000	−	0.866025i	−0.0492665	−	0.0853320i	−0.889607	+	0.456727i	$0.849022\pi$
$104$		0			0
$105$	−2.50000	−	0.866025i	−0.243975	−	0.0845154i
$106$		0			0
$107$	2.00000	+	3.46410i	0.193347	+	0.334887i	0.946357	−	0.323122i	$-0.104732\pi$
$107$	2.00000	+	3.46410i	0.193347	+	0.334887i	−0.753010	+	0.658009i	$0.771399\pi$
$108$		0			0
$109$	−6.50000	+	11.2583i	−0.622587	+	1.07835i	0.366415	+	0.930451i	$0.380585\pi$
$109$	−6.50000	+	11.2583i	−0.622587	+	1.07835i	−0.989002	+	0.147901i	$0.952748\pi$
$110$		0			0
$111$	−5.00000			−0.474579
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	2.00000	−	3.46410i	0.186501	−	0.323029i
$116$		0			0
$117$	−0.500000	−	0.866025i	−0.0462250	−	0.0800641i
$118$		0			0
$119$	−8.00000	+	6.92820i	−0.733359	+	0.635107i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$	−1.00000	+	1.73205i	−0.0901670	+	0.156174i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	9.00000			0.798621			0.399310	−	0.916816i	$-0.369250\pi$
$127$	9.00000			0.798621			0.399310	+	0.916816i	$0.369250\pi$
$128$		0			0
$129$	−4.50000	+	7.79423i	−0.396203	+	0.686244i
$130$		0			0
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	−0.928199	−	0.372084i	$-0.878643\pi$
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	0.141865	−	0.989886i	$-0.454690\pi$
$132$		0			0
$133$	0.500000	+	2.59808i	0.0433555	+	0.225282i
$134$		0			0
$135$	−0.500000	−	0.866025i	−0.0430331	−	0.0745356i
$136$		0			0
$137$	−8.00000	+	13.8564i	−0.683486	+	1.18383i	0.290424	+	0.956898i	$0.406204\pi$
$137$	−8.00000	+	13.8564i	−0.683486	+	1.18383i	−0.973910	+	0.226935i	$0.927130\pi$
$138$		0			0
$139$	13.0000			1.10265			0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000			1.10265			0.551323	+	0.834292i	$0.314123\pi$
$140$		0			0
$141$	2.00000			0.168430
$142$		0			0
$143$	−1.00000	+	1.73205i	−0.0836242	+	0.144841i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6.50000	+	2.59808i	−0.536111	+	0.214286i
$148$		0			0
$149$	−6.00000	−	10.3923i	−0.491539	−	0.851371i	0.508413	−	0.861113i	$-0.330232\pi$
$149$	−6.00000	−	10.3923i	−0.491539	−	0.851371i	−0.999953	+	0.00974235i	$0.996899\pi$
$150$		0			0
$151$	−8.00000	+	13.8564i	−0.651031	+	1.12762i	0.331842	+	0.943335i	$0.392330\pi$
$151$	−8.00000	+	13.8564i	−0.651031	+	1.12762i	−0.982873	+	0.184284i	$0.941004\pi$
$152$		0			0
$153$	−4.00000			−0.323381
$154$		0			0
$155$	5.00000			0.401610
$156$		0			0
$157$	11.0000	−	19.0526i	0.877896	−	1.52056i	0.0242497	−	0.999706i	$-0.492280\pi$
$157$	11.0000	−	19.0526i	0.877896	−	1.52056i	0.853646	−	0.520854i	$-0.174386\pi$
$158$		0			0
$159$	−6.00000	−	10.3923i	−0.475831	−	0.824163i
$160$		0			0
$161$	−2.00000	−	10.3923i	−0.157622	−	0.819028i
$162$		0			0
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	0.999410	−	0.0343508i	$-0.0109363\pi$
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	−0.529454	+	0.848339i	$0.677603\pi$
$164$		0			0
$165$	−1.00000	+	1.73205i	−0.0778499	+	0.134840i
$166$		0			0
$167$	−2.00000			−0.154765			−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000			−0.154765			−0.0773823	+	0.997001i	$0.524656\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$	−0.500000	+	0.866025i	−0.0382360	+	0.0662266i
$172$		0			0
$173$	8.00000	+	13.8564i	0.608229	+	1.05348i	0.991532	+	0.129861i	$0.0414530\pi$
$173$	8.00000	+	13.8564i	0.608229	+	1.05348i	−0.383304	+	0.923622i	$0.625214\pi$
$174$		0			0
$175$	2.00000	−	1.73205i	0.151186	−	0.130931i
$176$		0			0
$177$	−4.00000	−	6.92820i	−0.300658	−	0.520756i
$178$		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
$181$	17.0000			1.26360			0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000			1.26360			0.631800	+	0.775131i	$0.282316\pi$
$182$		0			0
$183$	−14.0000			−1.03491
$184$		0			0
$185$	2.50000	−	4.33013i	0.183804	−	0.318357i
$186$		0			0
$187$	4.00000	+	6.92820i	0.292509	+	0.506640i
$188$		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
$193$	8.50000	−	14.7224i	0.611843	−	1.05974i	−0.379086	−	0.925361i	$-0.623762\pi$
$193$	8.50000	−	14.7224i	0.611843	−	1.05974i	0.990930	−	0.134382i	$-0.0429051\pi$
$194$		0			0
$195$	1.00000			0.0716115
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−8.00000	+	13.8564i	−0.567105	+	0.982255i	0.429745	+	0.902950i	$0.358603\pi$
$199$	−8.00000	+	13.8564i	−0.567105	+	0.982255i	−0.996850	+	0.0793045i	$0.974730\pi$
$200$		0			0
$201$	4.50000	+	7.79423i	0.317406	+	0.549762i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.00000	−	1.73205i	−0.0698430	−	0.120972i
$206$		0			0
$207$	2.00000	−	3.46410i	0.139010	−	0.240772i
$208$		0			0
$209$	2.00000			0.138343
$210$		0			0
$211$	28.0000			1.92760			0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000			1.92760			0.963800	+	0.266627i	$0.0859092\pi$
$212$		0			0
$213$	1.00000	−	1.73205i	0.0685189	−	0.118678i
$214$		0			0
$215$	−4.50000	−	7.79423i	−0.306897	−	0.531562i
$216$		0			0
$217$	10.0000	−	8.66025i	0.678844	−	0.587896i
$218$		0			0
$219$	−0.500000	−	0.866025i	−0.0337869	−	0.0585206i
$220$		0			0
$221$	2.00000	−	3.46410i	0.134535	−	0.233021i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	9.00000	−	15.5885i	0.597351	−	1.03464i	−0.395860	−	0.918311i	$-0.629553\pi$
$227$	9.00000	−	15.5885i	0.597351	−	1.03464i	0.993210	−	0.116331i	$-0.0371134\pi$
$228$		0			0
$229$	−0.500000	−	0.866025i	−0.0330409	−	0.0572286i	0.849032	−	0.528341i	$-0.177186\pi$
$229$	−0.500000	−	0.866025i	−0.0330409	−	0.0572286i	−0.882073	+	0.471113i	$0.843853\pi$
$230$		0			0
$231$	1.00000	+	5.19615i	0.0657952	+	0.341882i
$232$		0			0
$233$	1.00000	+	1.73205i	0.0655122	+	0.113470i	0.896921	−	0.442191i	$-0.145799\pi$
$233$	1.00000	+	1.73205i	0.0655122	+	0.113470i	−0.831409	+	0.555661i	$0.812465\pi$
$234$		0			0
$235$	−1.00000	+	1.73205i	−0.0652328	+	0.112987i
$236$		0			0
$237$	3.00000			0.194871
$238$		0			0
$239$	−26.0000			−1.68180			−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000			−1.68180			−0.840900	+	0.541190i	$0.817974\pi$
$240$		0			0
$241$	9.00000	−	15.5885i	0.579741	−	1.00414i	−0.415768	−	0.909471i	$-0.636487\pi$
$241$	9.00000	−	15.5885i	0.579741	−	1.00414i	0.995509	−	0.0946700i	$-0.0301796\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	1.00000	−	6.92820i	0.0638877	−	0.442627i
$246$		0			0
$247$	−0.500000	−	0.866025i	−0.0318142	−	0.0551039i
$248$		0			0
$249$	−9.00000	+	15.5885i	−0.570352	+	0.987878i
$250$		0			0
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		0			0
$253$	−8.00000			−0.502956
$254$		0			0
$255$	2.00000	−	3.46410i	0.125245	−	0.216930i
$256$		0			0
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	0.560781	−	0.827964i	$-0.310501\pi$
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	−0.997429	+	0.0716680i	$0.977168\pi$
$258$		0			0
$259$	−2.50000	−	12.9904i	−0.155342	−	0.807183i
$260$		0			0
$261$		0			0
$262$		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
$265$	12.0000			0.737154
$266$		0			0
$267$	−4.00000			−0.244796
$268$		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
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$	2.00000	−	1.73205i	0.121046	−	0.104828i
$274$		0			0
$275$	−1.00000	−	1.73205i	−0.0603023	−	0.104447i
$276$		0			0
$277$	−2.50000	+	4.33013i	−0.150210	+	0.260172i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−2.50000	+	4.33013i	−0.150210	+	0.260172i	0.781094	+	0.624413i	$0.214662\pi$
$278$		0			0
$279$	5.00000			0.299342
$280$		0			0
$281$	−20.0000			−1.19310			−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000			−1.19310			−0.596550	+	0.802576i	$0.703462\pi$
$282$		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	$0.900046\pi$
$284$		0			0
$285$	−0.500000	−	0.866025i	−0.0296174	−	0.0512989i
$286$		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
$291$	−5.00000	+	8.66025i	−0.293105	+	0.507673i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	8.00000			0.465778
$296$		0			0
$297$	−1.00000	+	1.73205i	−0.0580259	+	0.100504i
$298$		0			0
$299$	2.00000	+	3.46410i	0.115663	+	0.200334i
$300$		0			0
$301$	−22.5000	−	7.79423i	−1.29688	−	0.449252i
$302$		0			0
$303$	−3.00000	−	5.19615i	−0.172345	−	0.298511i
$304$		0			0
$305$	7.00000	−	12.1244i	0.400819	−	0.694239i
$306$		0			0
$307$	19.0000			1.08439			0.542194	−	0.840254i	$-0.317594\pi$
$307$	19.0000			1.08439			0.542194	+	0.840254i	$0.317594\pi$
$308$		0			0
$309$	1.00000			0.0568880
$310$		0			0
$311$	13.0000	−	22.5167i	0.737162	−	1.27680i	−0.216606	−	0.976259i	$-0.569499\pi$
$311$	13.0000	−	22.5167i	0.737162	−	1.27680i	0.953768	−	0.300544i	$-0.0971681\pi$
$312$		0			0
$313$	−9.50000	−	16.4545i	−0.536972	−	0.930062i	−0.999065	−	0.0432311i	$-0.986235\pi$
$313$	−9.50000	−	16.4545i	−0.536972	−	0.930062i	0.462093	−	0.886831i	$-0.347098\pi$
$314$		0			0
$315$	2.00000	−	1.73205i	0.112687	−	0.0975900i
$316$		0			0
$317$	2.00000	+	3.46410i	0.112331	+	0.194563i	0.916710	−	0.399554i	$-0.130835\pi$
$317$	2.00000	+	3.46410i	0.112331	+	0.194563i	−0.804379	+	0.594117i	$0.797502\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$	−4.00000			−0.222566
$324$		0			0
$325$	−0.500000	+	0.866025i	−0.0277350	+	0.0480384i
$326$		0			0
$327$	−6.50000	−	11.2583i	−0.359451	−	0.622587i
$328$		0			0
$329$	1.00000	+	5.19615i	0.0551318	+	0.286473i
$330$		0			0
$331$	0.500000	+	0.866025i	0.0274825	+	0.0476011i	0.879440	−	0.476011i	$-0.157918\pi$
$331$	0.500000	+	0.866025i	0.0274825	+	0.0476011i	−0.851957	+	0.523612i	$0.824584\pi$
$332$		0			0
$333$	2.50000	−	4.33013i	0.136999	−	0.237289i
$334$		0			0
$335$	−9.00000			−0.491723
$336$		0			0
$337$	−7.00000			−0.381314			−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000			−0.381314			−0.190657	+	0.981657i	$0.561062\pi$
$338$		0			0
$339$	7.00000	−	12.1244i	0.380188	−	0.658505i
$340$		0			0
$341$	−5.00000	−	8.66025i	−0.270765	−	0.468979i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$	2.00000	+	3.46410i	0.107676	+	0.186501i
$346$		0			0
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	−0.340293	−	0.940319i	$-0.610526\pi$
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	0.984487	−	0.175457i	$-0.0561403\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$	−15.0000	+	25.9808i	−0.798369	+	1.38282i	0.122308	+	0.992492i	$0.460970\pi$
$353$	−15.0000	+	25.9808i	−0.798369	+	1.38282i	−0.920677	+	0.390324i	$0.872363\pi$
$354$		0			0
$355$	1.00000	+	1.73205i	0.0530745	+	0.0919277i
$356$		0			0
$357$	−2.00000	−	10.3923i	−0.105851	−	0.550019i
$358$		0			0
$359$	8.00000	+	13.8564i	0.422224	+	0.731313i	0.996157	−	0.0875892i	$-0.0279163\pi$
$359$	8.00000	+	13.8564i	0.422224	+	0.731313i	−0.573933	+	0.818902i	$0.694583\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$		0			0
$363$	−7.00000			−0.367405
$364$		0			0
$365$	1.00000			0.0523424
$366$		0			0
$367$	0.500000	−	0.866025i	0.0260998	−	0.0452062i	−0.852680	−	0.522433i	$-0.825025\pi$
$367$	0.500000	−	0.866025i	0.0260998	−	0.0452062i	0.878780	+	0.477227i	$0.158358\pi$
$368$		0			0
$369$	−1.00000	−	1.73205i	−0.0520579	−	0.0901670i
$370$		0			0
$371$	24.0000	−	20.7846i	1.24602	−	1.07908i
$372$		0			0
$373$	12.5000	+	21.6506i	0.647225	+	1.12103i	0.983783	+	0.179364i	$0.0574041\pi$
$373$	12.5000	+	21.6506i	0.647225	+	1.12103i	−0.336557	+	0.941663i	$0.609263\pi$
$374$		0			0
$375$	−0.500000	+	0.866025i	−0.0258199	+	0.0447214i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	11.0000			0.565032			0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000			0.565032			0.282516	+	0.959263i	$0.408831\pi$
$380$		0			0
$381$	−4.50000	+	7.79423i	−0.230542	+	0.399310i
$382$		0			0
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.799783	−	0.600289i	$-0.795052\pi$
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.119974	−	0.992777i	$-0.538281\pi$
$384$		0			0
$385$	−5.00000	−	1.73205i	−0.254824	−	0.0882735i
$386$		0			0
$387$	−4.50000	−	7.79423i	−0.228748	−	0.396203i
$388$		0			0
$389$	−15.0000	+	25.9808i	−0.760530	+	1.31728i	0.182047	+	0.983290i	$0.441728\pi$
$389$	−15.0000	+	25.9808i	−0.760530	+	1.31728i	−0.942578	+	0.333987i	$0.891606\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$	18.0000			0.907980
$394$		0			0
$395$	−1.50000	+	2.59808i	−0.0754732	+	0.130723i
$396$		0			0
$397$	−15.5000	−	26.8468i	−0.777923	−	1.34740i	−0.933137	−	0.359521i	$-0.882940\pi$
$397$	−15.5000	−	26.8468i	−0.777923	−	1.34740i	0.155214	−	0.987881i	$-0.450393\pi$
$398$		0			0
$399$	−2.50000	−	0.866025i	−0.125157	−	0.0433555i
$400$		0			0
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	0.676425	−	0.736512i	$-0.263528\pi$
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	−0.976050	+	0.217545i	$0.930195\pi$
$402$		0			0
$403$	−2.50000	+	4.33013i	−0.124534	+	0.215699i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	−10.0000			−0.495682
$408$		0			0
$409$	−6.50000	+	11.2583i	−0.321404	+	0.556689i	−0.980778	−	0.195127i	$-0.937488\pi$
$409$	−6.50000	+	11.2583i	−0.321404	+	0.556689i	0.659374	+	0.751815i	$0.270822\pi$
$410$		0			0
$411$	−8.00000	−	13.8564i	−0.394611	−	0.683486i
$412$		0			0
$413$	16.0000	−	13.8564i	0.787309	−	0.681829i
$414$		0			0
$415$	−9.00000	−	15.5885i	−0.441793	−	0.765207i
$416$		0			0
$417$	−6.50000	+	11.2583i	−0.318306	+	0.551323i
$418$		0			0
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		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
$427$	−7.00000	−	36.3731i	−0.338754	−	1.76022i
$428$		0			0
$429$	−1.00000	−	1.73205i	−0.0482805	−	0.0836242i
$430$		0			0
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	0.237460	+	0.971397i	$0.423685\pi$
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	−0.959985	+	0.280052i	$0.909648\pi$
$432$		0			0
$433$	27.0000			1.29754			0.648769	−	0.760986i	$-0.275284\pi$
$433$	27.0000			1.29754			0.648769	+	0.760986i	$0.275284\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.00000	−	3.46410i	0.0956730	−	0.165710i
$438$		0			0
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	0.945552	−	0.325471i	$-0.105523\pi$
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	−0.754642	+	0.656136i	$0.772190\pi$
$440$		0			0
$441$	1.00000	−	6.92820i	0.0476190	−	0.329914i
$442$		0			0
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	0.814595	−	0.580030i	$-0.196959\pi$
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	−0.909618	+	0.415445i	$0.863626\pi$
$444$		0			0
$445$	2.00000	−	3.46410i	0.0948091	−	0.164214i
$446$		0			0
$447$	12.0000			0.567581
$448$		0			0
$449$	22.0000			1.03824			0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000			1.03824			0.519122	+	0.854700i	$0.326259\pi$
$450$		0			0
$451$	−2.00000	+	3.46410i	−0.0941763	+	0.163118i
$452$		0			0
$453$	−8.00000	−	13.8564i	−0.375873	−	0.651031i
$454$		0			0
$455$	0.500000	+	2.59808i	0.0234404	+	0.121800i
$456$		0			0
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	0.854094	−	0.520119i	$-0.174112\pi$
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	−0.877483	+	0.479608i	$0.840779\pi$
$458$		0			0
$459$	2.00000	−	3.46410i	0.0933520	−	0.161690i
$460$		0			0
$461$	−16.0000			−0.745194			−0.372597	−	0.927993i	$-0.621533\pi$
$461$	−16.0000			−0.745194			−0.372597	+	0.927993i	$0.621533\pi$
$462$		0			0
$463$	−41.0000			−1.90543			−0.952716	−	0.303863i	$-0.901724\pi$
$463$	−41.0000			−1.90543			−0.952716	+	0.303863i	$0.901724\pi$
$464$		0			0
$465$	−2.50000	+	4.33013i	−0.115935	+	0.200805i
$466$		0			0
$467$	17.0000	+	29.4449i	0.786666	+	1.36255i	0.927999	+	0.372584i	$0.121528\pi$
$467$	17.0000	+	29.4449i	0.786666	+	1.36255i	−0.141332	+	0.989962i	$0.545139\pi$
$468$		0			0
$469$	−18.0000	+	15.5885i	−0.831163	+	0.719808i
$470$		0			0
$471$	11.0000	+	19.0526i	0.506853	+	0.877896i
$472$		0			0
$473$	−9.00000	+	15.5885i	−0.413820	+	0.716758i
$474$		0			0
$475$	1.00000			0.0458831
$476$		0			0
$477$	12.0000			0.549442
$478$		0			0
$479$	6.00000	−	10.3923i	0.274147	−	0.474837i	−0.695773	−	0.718262i	$-0.744938\pi$
$479$	6.00000	−	10.3923i	0.274147	−	0.474837i	0.969920	+	0.243426i	$0.0782712\pi$
$480$		0			0
$481$	2.50000	+	4.33013i	0.113990	+	0.197437i
$482$		0			0
$483$	10.0000	+	3.46410i	0.455016	+	0.157622i
$484$		0			0
$485$	−5.00000	−	8.66025i	−0.227038	−	0.393242i
$486$		0			0
$487$	16.5000	−	28.5788i	0.747686	−	1.29503i	−0.201243	−	0.979541i	$-0.564498\pi$
$487$	16.5000	−	28.5788i	0.747686	−	1.29503i	0.948929	−	0.315489i	$-0.102169\pi$
$488$		0			0
$489$	−12.0000			−0.542659
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−1.00000	−	1.73205i	−0.0449467	−	0.0778499i
$496$		0			0
$497$	5.00000	+	1.73205i	0.224281	+	0.0776931i
$498$		0			0
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	0.804627	−	0.593780i	$-0.202365\pi$
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	−0.916542	+	0.399937i	$0.869032\pi$
$500$		0			0
$501$	1.00000	−	1.73205i	0.0446767	−	0.0773823i
$502$		0			0
$503$	−10.0000			−0.445878			−0.222939	−	0.974832i	$-0.571565\pi$
$503$	−10.0000			−0.445878			−0.222939	+	0.974832i	$0.571565\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$	6.00000	−	10.3923i	0.266469	−	0.461538i
$508$		0			0
$509$	−13.0000	−	22.5167i	−0.576215	−	0.998033i	−0.995908	−	0.0903676i	$-0.971196\pi$
$509$	−13.0000	−	22.5167i	−0.576215	−	0.998033i	0.419694	−	0.907666i	$-0.362138\pi$
$510$		0			0
$511$	2.00000	−	1.73205i	0.0884748	−	0.0766214i
$512$		0			0
$513$	−0.500000	−	0.866025i	−0.0220755	−	0.0382360i
$514$		0			0
$515$	−0.500000	+	0.866025i	−0.0220326	+	0.0381616i
$516$		0			0
$517$	4.00000			0.175920
$518$		0			0
$519$	−16.0000			−0.702322
$520$		0			0
$521$	−18.0000	+	31.1769i	−0.788594	+	1.36589i	0.138234	+	0.990400i	$0.455857\pi$
$521$	−18.0000	+	31.1769i	−0.788594	+	1.36589i	−0.926828	+	0.375486i	$0.877476\pi$
$522$		0			0
$523$	14.5000	+	25.1147i	0.634041	+	1.09819i	0.986718	+	0.162446i	$0.0519382\pi$
$523$	14.5000	+	25.1147i	0.634041	+	1.09819i	−0.352677	+	0.935745i	$0.614728\pi$
$524$		0			0
$525$	0.500000	+	2.59808i	0.0218218	+	0.113389i
$526$		0			0
$527$	10.0000	+	17.3205i	0.435607	+	0.754493i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	8.00000			0.347170
$532$		0			0
$533$	2.00000			0.0866296
$534$		0			0
$535$	2.00000	−	3.46410i	0.0864675	−	0.149766i
$536$		0			0
$537$	−13.0000	−	22.5167i	−0.560991	−	0.971666i
$538$		0			0
$539$	−13.0000	+	5.19615i	−0.559950	+	0.223814i
$540$		0			0
$541$	15.5000	+	26.8468i	0.666397	+	1.15423i	0.978905	+	0.204318i	$0.0654977\pi$
$541$	15.5000	+	26.8468i	0.666397	+	1.15423i	−0.312507	+	0.949915i	$0.601169\pi$
$542$		0			0
$543$	−8.50000	+	14.7224i	−0.364770	+	0.631800i
$544$		0			0
$545$	13.0000			0.556859
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$	7.00000	−	12.1244i	0.298753	−	0.517455i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.50000	+	7.79423i	0.0637865	+	0.331444i
$554$		0			0
$555$	2.50000	+	4.33013i	0.106119	+	0.183804i
$556$		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
$559$	9.00000			0.380659
$560$		0			0
$561$	−8.00000			−0.337760
$562$		0			0
$563$	−15.0000	+	25.9808i	−0.632175	+	1.09496i	0.354932	+	0.934892i	$0.384504\pi$
$563$	−15.0000	+	25.9808i	−0.632175	+	1.09496i	−0.987106	+	0.160066i	$0.948829\pi$
$564$		0			0
$565$	7.00000	+	12.1244i	0.294492	+	0.510075i
$566$		0			0
$567$	2.00000	−	1.73205i	0.0839921	−	0.0727393i
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	−2.50000	+	4.33013i	−0.104622	+	0.181210i	−0.913584	−	0.406651i	$-0.866697\pi$
$571$	−2.50000	+	4.33013i	−0.104622	+	0.181210i	0.808962	+	0.587861i	$0.200030\pi$
$572$		0			0
$573$	10.0000			0.417756
$574$		0			0
$575$	−4.00000			−0.166812
$576$		0			0
$577$	6.50000	−	11.2583i	0.270599	−	0.468690i	−0.698417	−	0.715691i	$-0.746112\pi$
$577$	6.50000	−	11.2583i	0.270599	−	0.468690i	0.969015	+	0.247001i	$0.0794451\pi$
$578$		0			0
$579$	8.50000	+	14.7224i	0.353248	+	0.611843i
$580$		0			0
$581$	−45.0000	−	15.5885i	−1.86691	−	0.646718i
$582$		0			0
$583$	−12.0000	−	20.7846i	−0.496989	−	0.860811i
$584$		0			0
$585$	−0.500000	+	0.866025i	−0.0206725	+	0.0358057i
$586$		0			0
$587$	36.0000			1.48588			0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000			1.48588			0.742940	+	0.669359i	$0.233431\pi$
$588$		0			0
$589$	5.00000			0.206021
$590$		0			0
$591$	−6.00000	+	10.3923i	−0.246807	+	0.427482i
$592$		0			0
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	0.797831	−	0.602881i	$-0.205981\pi$
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	−0.921026	+	0.389501i	$0.872647\pi$
$594$		0			0
$595$	10.0000	+	3.46410i	0.409960	+	0.142014i
$596$		0			0
$597$	−8.00000	−	13.8564i	−0.327418	−	0.567105i
$598$		0			0
$599$	−14.0000	+	24.2487i	−0.572024	+	0.990775i	0.424333	+	0.905506i	$0.360508\pi$
$599$	−14.0000	+	24.2487i	−0.572024	+	0.990775i	−0.996358	+	0.0852695i	$0.972825\pi$
$600$		0			0
$601$	23.0000			0.938190			0.469095	−	0.883148i	$-0.344580\pi$
$601$	23.0000			0.938190			0.469095	+	0.883148i	$0.344580\pi$
$602$		0			0
$603$	−9.00000			−0.366508
$604$		0			0
$605$	3.50000	−	6.06218i	0.142295	−	0.246463i
$606$		0			0
$607$	4.50000	+	7.79423i	0.182649	+	0.316358i	0.942782	−	0.333410i	$-0.108199\pi$
$607$	4.50000	+	7.79423i	0.182649	+	0.316358i	−0.760133	+	0.649768i	$0.774866\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.00000	−	1.73205i	−0.0404557	−	0.0700713i
$612$		0			0
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	−0.885514	−	0.464614i	$-0.846193\pi$
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	0.845124	+	0.534570i	$0.179527\pi$
$614$		0			0
$615$	2.00000			0.0806478
$616$		0			0
$617$	34.0000			1.36879			0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000			1.36879			0.684394	+	0.729112i	$0.260067\pi$
$618$		0			0
$619$	−5.50000	+	9.52628i	−0.221064	+	0.382893i	−0.955131	−	0.296183i	$-0.904286\pi$
$619$	−5.50000	+	9.52628i	−0.221064	+	0.382893i	0.734068	+	0.679076i	$0.237620\pi$
$620$		0			0
$621$	2.00000	+	3.46410i	0.0802572	+	0.139010i
$622$		0			0
$623$	−2.00000	−	10.3923i	−0.0801283	−	0.416359i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	−1.00000	+	1.73205i	−0.0399362	+	0.0691714i
$628$		0			0
$629$	20.0000			0.797452
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	−14.0000	+	24.2487i	−0.556450	+	0.963800i
$634$		0			0
$635$	−4.50000	−	7.79423i	−0.178577	−	0.309305i
$636$		0			0
$637$	5.50000	+	4.33013i	0.217918	+	0.171566i
$638$		0			0
$639$	1.00000	+	1.73205i	0.0395594	+	0.0685189i
$640$		0			0
$641$	18.0000	−	31.1769i	0.710957	−	1.23141i	−0.253541	−	0.967325i	$-0.581595\pi$
$641$	18.0000	−	31.1769i	0.710957	−	1.23141i	0.964498	−	0.264089i	$-0.0850714\pi$
$642$		0			0
$643$	−7.00000			−0.276053			−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000			−0.276053			−0.138027	+	0.990429i	$0.544076\pi$
$644$		0			0
$645$	9.00000			0.354375
$646$		0			0
$647$	−7.00000	+	12.1244i	−0.275198	+	0.476658i	−0.970185	−	0.242365i	$-0.922077\pi$
$647$	−7.00000	+	12.1244i	−0.275198	+	0.476658i	0.694987	+	0.719023i	$0.255410\pi$
$648$		0			0
$649$	−8.00000	−	13.8564i	−0.314027	−	0.543912i
$650$		0			0
$651$	2.50000	+	12.9904i	0.0979827	+	0.509133i
$652$		0			0
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	0.695934	−	0.718106i	$-0.254991\pi$
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	−0.969865	+	0.243643i	$0.921657\pi$
$654$		0			0
$655$	−9.00000	+	15.5885i	−0.351659	+	0.609091i
$656$		0			0
$657$	1.00000			0.0390137
$658$		0			0
$659$	−4.00000			−0.155818			−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000			−0.155818			−0.0779089	+	0.996960i	$0.524824\pi$
$660$		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
$663$	2.00000	+	3.46410i	0.0776736	+	0.134535i
$664$		0			0
$665$	2.00000	−	1.73205i	0.0775567	−	0.0671660i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−28.0000			−1.08093
$672$		0			0
$673$	11.0000			0.424019			0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000			0.424019			0.212009	+	0.977268i	$0.431999\pi$
$674$		0			0
$675$	−0.500000	+	0.866025i	−0.0192450	+	0.0333333i
$676$		0			0
$677$	−12.0000	−	20.7846i	−0.461197	−	0.798817i	0.537823	−	0.843057i	$-0.319247\pi$
$677$	−12.0000	−	20.7846i	−0.461197	−	0.798817i	−0.999021	+	0.0442400i	$0.985913\pi$
$678$		0			0
$679$	−25.0000	−	8.66025i	−0.959412	−	0.332350i
$680$		0			0
$681$	9.00000	+	15.5885i	0.344881	+	0.597351i
$682$		0			0
$683$	−8.00000	+	13.8564i	−0.306111	+	0.530201i	−0.977508	−	0.210898i	$-0.932361\pi$
$683$	−8.00000	+	13.8564i	−0.306111	+	0.530201i	0.671397	+	0.741098i	$0.265695\pi$
$684$		0			0
$685$	16.0000			0.611329
$686$		0			0
$687$	1.00000			0.0381524
$688$		0			0
$689$	−6.00000	+	10.3923i	−0.228582	+	0.395915i
$690$		0			0
$691$	−5.50000	−	9.52628i	−0.209230	−	0.362397i	0.742242	−	0.670132i	$-0.233762\pi$
$691$	−5.50000	−	9.52628i	−0.209230	−	0.362397i	−0.951472	+	0.307735i	$0.900429\pi$
$692$		0			0
$693$	−5.00000	−	1.73205i	−0.189934	−	0.0657952i
$694$		0			0
$695$	−6.50000	−	11.2583i	−0.246559	−	0.427053i
$696$		0			0
$697$	4.00000	−	6.92820i	0.151511	−	0.262424i
$698$		0			0
$699$	−2.00000			−0.0756469
$700$		0			0
$701$	36.0000			1.35970			0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000			1.35970			0.679851	+	0.733351i	$0.262045\pi$
$702$		0			0
$703$	2.50000	−	4.33013i	0.0942893	−	0.163314i
$704$		0			0
$705$	−1.00000	−	1.73205i	−0.0376622	−	0.0652328i
$706$		0			0
$707$	12.0000	−	10.3923i	0.451306	−	0.390843i
$708$		0			0
$709$	13.0000	+	22.5167i	0.488225	+	0.845631i	0.999908	−	0.0135434i	$-0.00431112\pi$
$709$	13.0000	+	22.5167i	0.488225	+	0.845631i	−0.511683	+	0.859174i	$0.670978\pi$
$710$		0			0
$711$	−1.50000	+	2.59808i	−0.0562544	+	0.0974355i
$712$		0			0
$713$	−20.0000			−0.749006
$714$		0			0
$715$	2.00000			0.0747958
$716$		0			0
$717$	13.0000	−	22.5167i	0.485494	−	0.840900i
$718$		0			0
$719$	21.0000	+	36.3731i	0.783168	+	1.35649i	0.930087	+	0.367338i	$0.119731\pi$
$719$	21.0000	+	36.3731i	0.783168	+	1.35649i	−0.146920	+	0.989148i	$0.546936\pi$
$720$		0			0
$721$	0.500000	+	2.59808i	0.0186210	+	0.0967574i
$722$		0			0
$723$	9.00000	+	15.5885i	0.334714	+	0.579741i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	3.00000			0.111264			0.0556319	−	0.998451i	$-0.482283\pi$
$727$	3.00000			0.111264			0.0556319	+	0.998451i	$0.482283\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	18.0000	−	31.1769i	0.665754	−	1.15312i
$732$		0			0
$733$	−4.50000	−	7.79423i	−0.166211	−	0.287886i	0.770873	−	0.636988i	$-0.219820\pi$
$733$	−4.50000	−	7.79423i	−0.166211	−	0.287886i	−0.937085	+	0.349102i	$0.886487\pi$
$734$		0			0
$735$	5.50000	+	4.33013i	0.202871	+	0.159719i
$736$		0			0
$737$	9.00000	+	15.5885i	0.331519	+	0.574208i
$738$		0			0
$739$	7.50000	−	12.9904i	0.275892	−	0.477859i	−0.694468	−	0.719524i	$-0.744360\pi$
$739$	7.50000	−	12.9904i	0.275892	−	0.477859i	0.970360	+	0.241665i	$0.0776935\pi$
$740$		0			0
$741$	1.00000			0.0367359
$742$		0			0
$743$	6.00000			0.220119			0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000			0.220119			0.110059	+	0.993925i	$0.464896\pi$
$744$		0			0
$745$	−6.00000	+	10.3923i	−0.219823	+	0.380745i
$746$		0			0
$747$	−9.00000	−	15.5885i	−0.329293	−	0.570352i
$748$		0			0
$749$	−2.00000	−	10.3923i	−0.0730784	−	0.379727i
$750$		0			0
$751$	15.5000	+	26.8468i	0.565603	+	0.979653i	0.996993	+	0.0774878i	$0.0246899\pi$
$751$	15.5000	+	26.8468i	0.565603	+	0.979653i	−0.431390	+	0.902165i	$0.641977\pi$
$752$		0			0
$753$	2.00000	−	3.46410i	0.0728841	−	0.126239i
$754$		0			0
$755$	16.0000			0.582300
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$	4.00000	−	6.92820i	0.145191	−	0.251478i
$760$		0			0
$761$	−10.0000	−	17.3205i	−0.362500	−	0.627868i	0.625872	−	0.779926i	$-0.284743\pi$
$761$	−10.0000	−	17.3205i	−0.362500	−	0.627868i	−0.988372	+	0.152058i	$0.951410\pi$
$762$		0			0
$763$	26.0000	−	22.5167i	0.941263	−	0.815158i
$764$		0			0
$765$	2.00000	+	3.46410i	0.0723102	+	0.125245i
$766$		0			0
$767$	−4.00000	+	6.92820i	−0.144432	+	0.250163i
$768$		0			0
$769$	31.0000			1.11789			0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000			1.11789			0.558944	+	0.829205i	$0.311207\pi$
$770$		0			0
$771$	14.0000			0.504198
$772$		0			0
$773$	7.00000	−	12.1244i	0.251773	−	0.436083i	−0.712241	−	0.701935i	$-0.752320\pi$
$773$	7.00000	−	12.1244i	0.251773	−	0.436083i	0.964014	+	0.265852i	$0.0856532\pi$
$774$		0			0
$775$	−2.50000	−	4.33013i	−0.0898027	−	0.155543i
$776$		0			0
$777$	12.5000	+	4.33013i	0.448435	+	0.155342i
$778$		0			0
$779$	−1.00000	−	1.73205i	−0.0358287	−	0.0620572i
$780$		0			0
$781$	2.00000	−	3.46410i	0.0715656	−	0.123955i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−22.0000			−0.785214
$786$		0			0
$787$	−8.00000	+	13.8564i	−0.285169	+	0.493928i	−0.972650	−	0.232275i	$-0.925383\pi$
$787$	−8.00000	+	13.8564i	−0.285169	+	0.493928i	0.687481	+	0.726202i	$0.258716\pi$
$788$		0			0
$789$	6.00000	+	10.3923i	0.213606	+	0.369976i
$790$		0			0
$791$	35.0000	+	12.1244i	1.24446	+	0.431092i
$792$		0			0
$793$	7.00000	+	12.1244i	0.248577	+	0.430548i
$794$		0			0
$795$	−6.00000	+	10.3923i	−0.212798	+	0.368577i
$796$		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
$801$	2.00000	−	3.46410i	0.0706665	−	0.122398i
$802$		0			0
$803$	−1.00000	−	1.73205i	−0.0352892	−	0.0611227i
$804$		0			0
$805$	−8.00000	+	6.92820i	−0.281963	+	0.244187i
$806$		0			0
$807$	9.00000	+	15.5885i	0.316815	+	0.548740i
$808$		0			0
$809$	−23.0000	+	39.8372i	−0.808637	+	1.40060i	0.105171	+	0.994454i	$0.466461\pi$
$809$	−23.0000	+	39.8372i	−0.808637	+	1.40060i	−0.913808	+	0.406146i	$0.866872\pi$
$810$		0			0
$811$	−48.0000			−1.68551			−0.842754	−	0.538299i	$-0.819067\pi$
$811$	−48.0000			−1.68551			−0.842754	+	0.538299i	$0.819067\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.00000	−	10.3923i	0.210171	−	0.364027i
$816$		0			0
$817$	−4.50000	−	7.79423i	−0.157435	−	0.272686i
$818$		0			0
$819$	0.500000	+	2.59808i	0.0174714	+	0.0907841i
$820$		0			0
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.859396	+	0.511311i	$0.170840\pi$
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.0131101	+	0.999914i	$0.495827\pi$
$822$		0			0
$823$	8.00000	−	13.8564i	0.278862	−	0.483004i	−0.692240	−	0.721668i	$-0.743376\pi$
$823$	8.00000	−	13.8564i	0.278862	−	0.483004i	0.971102	+	0.238664i	$0.0767093\pi$
$824$		0			0
$825$	2.00000			0.0696311
$826$		0			0
$827$	−6.00000			−0.208640			−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000			−0.208640			−0.104320	+	0.994544i	$0.533267\pi$
$828$		0			0
$829$	−2.50000	+	4.33013i	−0.0868286	+	0.150392i	−0.906169	−	0.422916i	$-0.861007\pi$
$829$	−2.50000	+	4.33013i	−0.0868286	+	0.150392i	0.819340	+	0.573307i	$0.194340\pi$
$830$		0			0
$831$	−2.50000	−	4.33013i	−0.0867240	−	0.150210i
$832$		0			0
$833$	26.0000	−	10.3923i	0.900847	−	0.360072i
$834$		0			0
$835$	1.00000	+	1.73205i	0.0346064	+	0.0599401i
$836$		0			0
$837$	−2.50000	+	4.33013i	−0.0864126	+	0.149671i
$838$		0			0
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	10.0000	−	17.3205i	0.344418	−	0.596550i
$844$		0			0
$845$	6.00000	+	10.3923i	0.206406	+	0.357506i
$846$		0			0
$847$	−3.50000	−	18.1865i	−0.120261	−	0.624897i
$848$		0			0
$849$	−12.5000	−	21.6506i	−0.428999	−	0.743048i
$850$		0			0
$851$	−10.0000	+	17.3205i	−0.342796	+	0.593739i
$852$		0			0
$853$	−29.0000			−0.992941			−0.496471	−	0.868054i	$-0.665371\pi$
$853$	−29.0000			−0.992941			−0.496471	+	0.868054i	$0.665371\pi$
$854$		0			0
$855$	1.00000			0.0341993
$856$		0			0
$857$	−22.0000	+	38.1051i	−0.751506	+	1.30165i	0.195587	+	0.980686i	$0.437339\pi$
$857$	−22.0000	+	38.1051i	−0.751506	+	1.30165i	−0.947093	+	0.320960i	$0.895995\pi$
$858$		0			0
$859$	4.00000	+	6.92820i	0.136478	+	0.236387i	0.926161	−	0.377128i	$-0.123088\pi$
$859$	4.00000	+	6.92820i	0.136478	+	0.236387i	−0.789683	+	0.613515i	$0.789755\pi$
$860$		0			0
$861$	4.00000	−	3.46410i	0.136320	−	0.118056i
$862$		0			0
$863$	15.0000	+	25.9808i	0.510606	+	0.884395i	0.999924	+	0.0122903i	$0.00391222\pi$
$863$	15.0000	+	25.9808i	0.510606	+	0.884395i	−0.489319	+	0.872105i	$0.662754\pi$
$864$		0			0
$865$	8.00000	−	13.8564i	0.272008	−	0.471132i
$866$		0			0
$867$	−1.00000			−0.0339618
$868$		0			0
$869$	6.00000			0.203536
$870$		0			0
$871$	4.50000	−	7.79423i	0.152477	−	0.264097i
$872$		0			0
$873$	−5.00000	−	8.66025i	−0.169224	−	0.293105i
$874$		0			0
$875$	−2.50000	−	0.866025i	−0.0845154	−	0.0292770i
$876$		0			0
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	0.989788	−	0.142548i	$-0.0455296\pi$
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	−0.618344	+	0.785907i	$0.712196\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	24.0000			0.808581			0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000			0.808581			0.404290	+	0.914631i	$0.367519\pi$
$882$		0			0
$883$	−1.00000			−0.0336527			−0.0168263	−	0.999858i	$-0.505356\pi$
$883$	−1.00000			−0.0336527			−0.0168263	+	0.999858i	$0.505356\pi$
$884$		0			0
$885$	−4.00000	+	6.92820i	−0.134459	+	0.232889i
$886$		0			0
$887$	11.0000	+	19.0526i	0.369344	+	0.639722i	0.989463	−	0.144785i	$-0.0462491\pi$
$887$	11.0000	+	19.0526i	0.369344	+	0.639722i	−0.620119	+	0.784508i	$0.712916\pi$
$888$		0			0
$889$	−22.5000	−	7.79423i	−0.754626	−	0.261410i
$890$		0			0
$891$	−1.00000	−	1.73205i	−0.0335013	−	0.0580259i
$892$		0			0
$893$	−1.00000	+	1.73205i	−0.0334637	+	0.0579609i
$894$		0			0
$895$	26.0000			0.869084
$896$		0			0
$897$	−4.00000			−0.133556
$898$		0			0
$899$		0			0
$900$		0			0
$901$	24.0000	+	41.5692i	0.799556	+	1.38487i
$902$		0			0
$903$	18.0000	−	15.5885i	0.599002	−	0.518751i
$904$		0			0
$905$	−8.50000	−	14.7224i	−0.282550	−	0.489390i
$906$		0			0
$907$	4.50000	−	7.79423i	0.149420	−	0.258803i	−0.781593	−	0.623788i	$-0.785593\pi$
$907$	4.50000	−	7.79423i	0.149420	−	0.258803i	0.931013	+	0.364985i	$0.118926\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$	−18.0000	+	31.1769i	−0.595713	+	1.03181i
$914$		0			0
$915$	7.00000	+	12.1244i	0.231413	+	0.400819i
$916$		0			0
$917$	9.00000	+	46.7654i	0.297206	+	1.54433i
$918$		0			0
$919$	−5.50000	−	9.52628i	−0.181428	−	0.314243i	0.760939	−	0.648824i	$-0.224739\pi$
$919$	−5.50000	−	9.52628i	−0.181428	−	0.314243i	−0.942367	+	0.334581i	$0.891405\pi$
$920$		0			0
$921$	−9.50000	+	16.4545i	−0.313036	+	0.542194i
$922$		0			0
$923$	−2.00000			−0.0658308
$924$		0			0
$925$	−5.00000			−0.164399
$926$		0			0
$927$	−0.500000	+	0.866025i	−0.0164222	+	0.0284440i
$928$		0			0
$929$	7.00000	+	12.1244i	0.229663	+	0.397787i	0.957708	−	0.287742i	$-0.0929044\pi$
$929$	7.00000	+	12.1244i	0.229663	+	0.397787i	−0.728046	+	0.685529i	$0.759571\pi$
$930$		0			0
$931$	1.00000	−	6.92820i	0.0327737	−	0.227063i
$932$		0			0
$933$	13.0000	+	22.5167i	0.425601	+	0.737162i
$934$		0			0
$935$	4.00000	−	6.92820i	0.130814	−	0.226576i
$936$		0			0
$937$	−21.0000			−0.686040			−0.343020	−	0.939328i	$-0.611450\pi$
$937$	−21.0000			−0.686040			−0.343020	+	0.939328i	$0.611450\pi$
$938$		0			0
$939$	19.0000			0.620042
$940$		0			0
$941$	24.0000	−	41.5692i	0.782378	−	1.35512i	−0.148176	−	0.988961i	$-0.547340\pi$
$941$	24.0000	−	41.5692i	0.782378	−	1.35512i	0.930553	−	0.366157i	$-0.119327\pi$
$942$		0			0
$943$	4.00000	+	6.92820i	0.130258	+	0.225613i
$944$		0			0
$945$	0.500000	+	2.59808i	0.0162650	+	0.0845154i
$946$		0			0
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	0.681930	−	0.731417i	$-0.261141\pi$
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	−0.974391	+	0.224860i	$0.927807\pi$
$948$		0			0
$949$	−0.500000	+	0.866025i	−0.0162307	+	0.0281124i
$950$		0			0
$951$	−4.00000			−0.129709
$952$		0			0
$953$	−36.0000			−1.16615			−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000			−1.16615			−0.583077	+	0.812417i	$0.698151\pi$
$954$		0			0
$955$	−5.00000	+	8.66025i	−0.161796	+	0.280239i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	32.0000	−	27.7128i	1.03333	−	0.894893i
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$		0			0
$963$	2.00000	−	3.46410i	0.0644491	−	0.111629i
$964$		0			0
$965$	−17.0000			−0.547249
$966$		0			0
$967$	−37.0000			−1.18984			−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000			−1.18984			−0.594920	+	0.803785i	$0.702816\pi$
$968$		0			0
$969$	2.00000	−	3.46410i	0.0642493	−	0.111283i
$970$		0			0
$971$	−18.0000	−	31.1769i	−0.577647	−	1.00051i	−0.995748	−	0.0921142i	$-0.970638\pi$
$971$	−18.0000	−	31.1769i	−0.577647	−	1.00051i	0.418101	−	0.908401i	$-0.362696\pi$
$972$		0			0
$973$	−32.5000	−	11.2583i	−1.04190	−	0.360925i
$974$		0			0
$975$	−0.500000	−	0.866025i	−0.0160128	−	0.0277350i
$976$		0			0
$977$	−9.00000	+	15.5885i	−0.287936	+	0.498719i	−0.973317	−	0.229465i	$-0.926302\pi$
$977$	−9.00000	+	15.5885i	−0.287936	+	0.498719i	0.685381	+	0.728184i	$0.259636\pi$
$978$		0			0
$979$	−8.00000			−0.255681
$980$		0			0
$981$	13.0000			0.415058
$982$		0			0
$983$	−8.00000	+	13.8564i	−0.255160	+	0.441951i	−0.964939	−	0.262474i	$-0.915462\pi$
$983$	−8.00000	+	13.8564i	−0.255160	+	0.441951i	0.709779	+	0.704425i	$0.248795\pi$
$984$		0			0
$985$	−6.00000	−	10.3923i	−0.191176	−	0.331126i
$986$		0			0
$987$	−5.00000	−	1.73205i	−0.159152	−	0.0551318i
$988$		0			0
$989$	18.0000	+	31.1769i	0.572367	+	0.991368i
$990$		0			0
$991$	−26.5000	+	45.8993i	−0.841800	+	1.45804i	0.0465710	+	0.998915i	$0.485171\pi$
$991$	−26.5000	+	45.8993i	−0.841800	+	1.45804i	−0.888371	+	0.459126i	$0.848163\pi$
$992$		0			0
$993$	−1.00000			−0.0317340
$994$		0			0
$995$	16.0000			0.507234
$996$		0			0
$997$	−5.50000	+	9.52628i	−0.174187	+	0.301700i	−0.939880	−	0.341506i	$-0.889063\pi$
$997$	−5.50000	+	9.52628i	−0.174187	+	0.301700i	0.765693	+	0.643206i	$0.222396\pi$
$998$		0			0
$999$	2.50000	+	4.33013i	0.0790965	+	0.136999i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.bg.a.1201.1		2
4.3	odd	2		420.2.q.a.361.1	yes	2
7.2	even	3	inner	1680.2.bg.a.961.1		2
12.11	even	2		1260.2.s.d.361.1		2
20.3	even	4		2100.2.bc.c.949.2		4
20.7	even	4		2100.2.bc.c.949.1		4
20.19	odd	2		2100.2.q.a.1201.1		2
28.3	even	6		2940.2.a.h.1.1		1
28.11	odd	6		2940.2.a.d.1.1		1
28.19	even	6		2940.2.q.h.961.1		2
28.23	odd	6		420.2.q.a.121.1	✓	2
28.27	even	2		2940.2.q.h.361.1		2
84.11	even	6		8820.2.a.j.1.1		1
84.23	even	6		1260.2.s.d.541.1		2
84.59	odd	6		8820.2.a.y.1.1		1
140.23	even	12		2100.2.bc.c.1549.1		4
140.79	odd	6		2100.2.q.a.1801.1		2
140.107	even	12		2100.2.bc.c.1549.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.q.a.121.1	✓	2	28.23	odd	6
420.2.q.a.361.1	yes	2	4.3	odd	2
1260.2.s.d.361.1		2	12.11	even	2
1260.2.s.d.541.1		2	84.23	even	6
1680.2.bg.a.961.1		2	7.2	even	3	inner
1680.2.bg.a.1201.1		2	1.1	even	1	trivial
2100.2.q.a.1201.1		2	20.19	odd	2
2100.2.q.a.1801.1		2	140.79	odd	6
2100.2.bc.c.949.1		4	20.7	even	4
2100.2.bc.c.949.2		4	20.3	even	4
2100.2.bc.c.1549.1		4	140.23	even	12
2100.2.bc.c.1549.2		4	140.107	even	12
2940.2.a.d.1.1		1	28.11	odd	6
2940.2.a.h.1.1		1	28.3	even	6
2940.2.q.h.361.1		2	28.27	even	2
2940.2.q.h.961.1		2	28.19	even	6
8820.2.a.j.1.1		1	84.11	even	6
8820.2.a.y.1.1		1	84.59	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.bg (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +1.00000 q^{15} +(2.00000 - 3.46410i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(2.00000 - 1.73205i) q^{21} +(2.00000 + 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} +(-2.50000 + 4.33013i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(0.500000 + 2.59808i) q^{35} +(2.50000 + 4.33013i) q^{37} +(-0.500000 + 0.866025i) q^{39} +2.00000 q^{41} +9.00000 q^{43} +(-0.500000 + 0.866025i) q^{45} +(-1.00000 - 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +(2.00000 + 3.46410i) q^{51} +(-6.00000 + 10.3923i) q^{53} +2.00000 q^{55} +1.00000 q^{57} +(-4.00000 + 6.92820i) q^{59} +(7.00000 + 12.1244i) q^{61} +(0.500000 + 2.59808i) q^{63} +(-0.500000 - 0.866025i) q^{65} +(4.50000 - 7.79423i) q^{67} -4.00000 q^{69} -2.00000 q^{71} +(-0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(4.00000 - 3.46410i) q^{77} +(-1.50000 - 2.59808i) q^{79} +(-0.500000 + 0.866025i) q^{81} +18.0000 q^{83} -4.00000 q^{85} +(2.00000 + 3.46410i) q^{89} +(-2.50000 - 0.866025i) q^{91} +(-2.50000 - 4.33013i) q^{93} +(-0.500000 + 0.866025i) q^{95} +10.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} - 5 q^{7} - q^{9}+O(q^{10}) \) 2 * q - q^3 - q^5 - 5 * q^7 - q^9	\( 2 q - q^{3} - q^{5} - 5 q^{7} - q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - q^{19} + 4 q^{21} + 4 q^{23} - q^{25} + 2 q^{27} - 5 q^{31} - 2 q^{33} + q^{35} + 5 q^{37} - q^{39} + 4 q^{41} + 18 q^{43} - q^{45} - 2 q^{47} + 11 q^{49} + 4 q^{51} - 12 q^{53} + 4 q^{55} + 2 q^{57} - 8 q^{59} + 14 q^{61} + q^{63} - q^{65} + 9 q^{67} - 8 q^{69} - 4 q^{71} - q^{73} - q^{75} + 8 q^{77} - 3 q^{79} - q^{81} + 36 q^{83} - 8 q^{85} + 4 q^{89} - 5 q^{91} - 5 q^{93} - q^{95} + 20 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 - 5 * q^7 - q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 - q^19 + 4 * q^21 + 4 * q^23 - q^25 + 2 * q^27 - 5 * q^31 - 2 * q^33 + q^35 + 5 * q^37 - q^39 + 4 * q^41 + 18 * q^43 - q^45 - 2 * q^47 + 11 * q^49 + 4 * q^51 - 12 * q^53 + 4 * q^55 + 2 * q^57 - 8 * q^59 + 14 * q^61 + q^63 - q^65 + 9 * q^67 - 8 * q^69 - 4 * q^71 - q^73 - q^75 + 8 * q^77 - 3 * q^79 - q^81 + 36 * q^83 - 8 * q^85 + 4 * q^89 - 5 * q^91 - 5 * q^93 - q^95 + 20 * q^97 + 4 * q^99

Introduction

L-functions

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