LMFDB - Embedded newform 1680.2.bg.g.961.1

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 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1680.961
Dual form			1680.2.bg.g.1201.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{11} +1.00000 q^{13} -1.00000 q^{15} +(-1.50000 + 2.59808i) q^{19} +(2.50000 + 0.866025i) q^{21} +(3.50000 - 6.06218i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -8.00000 q^{29} +(-1.00000 - 1.73205i) q^{31} +(-0.500000 + 0.866025i) q^{33} +(0.500000 + 2.59808i) q^{35} +(-5.50000 + 9.52628i) q^{37} +(-0.500000 - 0.866025i) q^{39} -11.0000 q^{41} -8.00000 q^{43} +(0.500000 + 0.866025i) q^{45} +(-2.50000 + 4.33013i) q^{47} +(1.00000 - 6.92820i) q^{49} +(5.50000 + 9.52628i) q^{53} -1.00000 q^{55} +3.00000 q^{57} +(2.00000 + 3.46410i) q^{59} +(-0.500000 - 2.59808i) q^{63} +(0.500000 - 0.866025i) q^{65} -7.00000 q^{69} +6.00000 q^{71} +(3.00000 + 5.19615i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(2.50000 + 0.866025i) q^{77} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 - 0.866025i) q^{81} -8.00000 q^{83} +(4.00000 + 6.92820i) q^{87} +(5.00000 - 8.66025i) q^{89} +(-2.00000 + 1.73205i) q^{91} +(-1.00000 + 1.73205i) q^{93} +(1.50000 + 2.59808i) q^{95} -16.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 2 * q - q^3 + q^5 - 4 * q^7 - q^9	$ 2 q - q^{3} + q^{5} - 4 q^{7} - q^{9} - q^{11} + 2 q^{13} - 2 q^{15} - 3 q^{19} + 5 q^{21} + 7 q^{23} - q^{25} + 2 q^{27} - 16 q^{29} - 2 q^{31} - q^{33} + q^{35} - 11 q^{37} - q^{39} - 22 q^{41} - 16 q^{43} + q^{45} - 5 q^{47} + 2 q^{49} + 11 q^{53} - 2 q^{55} + 6 q^{57} + 4 q^{59} - q^{63} + q^{65} - 14 q^{69} + 12 q^{71} + 6 q^{73} - q^{75} + 5 q^{77} - 8 q^{79} - q^{81} - 16 q^{83} + 8 q^{87} + 10 q^{89} - 4 q^{91} - 2 q^{93} + 3 q^{95} - 32 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - q^3 + q^5 - 4 * q^7 - q^9 - q^11 + 2 * q^13 - 2 * q^15 - 3 * q^19 + 5 * q^21 + 7 * q^23 - q^25 + 2 * q^27 - 16 * q^29 - 2 * q^31 - q^33 + q^35 - 11 * q^37 - q^39 - 22 * q^41 - 16 * q^43 + q^45 - 5 * q^47 + 2 * q^49 + 11 * q^53 - 2 * q^55 + 6 * q^57 + 4 * q^59 - q^63 + q^65 - 14 * q^69 + 12 * q^71 + 6 * q^73 - q^75 + 5 * q^77 - 8 * q^79 - q^81 - 16 * q^83 + 8 * q^87 + 10 * q^89 - 4 * q^91 - 2 * q^93 + 3 * q^95 - 32 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	−0.500000	−	0.866025i	−0.150756	−	0.261116i	0.780750	−	0.624844i	$-0.214837\pi$
$11$	−0.500000	−	0.866025i	−0.150756	−	0.261116i	−0.931505	+	0.363727i	$0.881504\pi$
$12$		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
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	−0.985194	−	0.171442i	$-0.945157\pi$
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	0.641071	+	0.767482i	$0.278491\pi$
$20$		0			0
$21$	2.50000	+	0.866025i	0.545545	+	0.188982i
$22$		0			0
$23$	3.50000	−	6.06218i	0.729800	−	1.26405i	−0.227167	−	0.973856i	$-0.572946\pi$
$23$	3.50000	−	6.06218i	0.729800	−	1.26405i	0.956967	−	0.290196i	$-0.0937204\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$		0			0
$33$	−0.500000	+	0.866025i	−0.0870388	+	0.150756i
$34$		0			0
$35$	0.500000	+	2.59808i	0.0845154	+	0.439155i
$36$		0			0
$37$	−5.50000	+	9.52628i	−0.904194	+	1.56611i	−0.0821995	+	0.996616i	$0.526194\pi$
$37$	−5.50000	+	9.52628i	−0.904194	+	1.56611i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$	−0.500000	−	0.866025i	−0.0800641	−	0.138675i
$40$		0			0
$41$	−11.0000			−1.71791			−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000			−1.71791			−0.858956	+	0.512050i	$0.828886\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$	0.500000	+	0.866025i	0.0745356	+	0.129099i
$46$		0			0
$47$	−2.50000	+	4.33013i	−0.364662	+	0.631614i	−0.988722	−	0.149763i	$-0.952149\pi$
$47$	−2.50000	+	4.33013i	−0.364662	+	0.631614i	0.624059	+	0.781377i	$0.285482\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.50000	+	9.52628i	0.755483	+	1.30854i	0.945134	+	0.326683i	$0.105931\pi$
$53$	5.50000	+	9.52628i	0.755483	+	1.30854i	−0.189651	+	0.981852i	$0.560736\pi$
$54$		0			0
$55$	−1.00000			−0.134840
$56$		0			0
$57$	3.00000			0.397360
$58$		0			0
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	$-0.0828195\pi$
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	−0.705965	+	0.708247i	$0.749486\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		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
$67$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$67$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$68$		0			0
$69$	−7.00000			−0.842701
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	3.00000	+	5.19615i	0.351123	+	0.608164i	0.986447	−	0.164083i	$-0.0524664\pi$
$73$	3.00000	+	5.19615i	0.351123	+	0.608164i	−0.635323	+	0.772246i	$0.719133\pi$
$74$		0			0
$75$	−0.500000	+	0.866025i	−0.0577350	+	0.100000i
$76$		0			0
$77$	2.50000	+	0.866025i	0.284901	+	0.0986928i
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−8.00000			−0.878114			−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000			−0.878114			−0.439057	+	0.898459i	$0.644687\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	4.00000	+	6.92820i	0.428845	+	0.742781i
$88$		0			0
$89$	5.00000	−	8.66025i	0.529999	−	0.917985i	−0.469389	−	0.882992i	$-0.655526\pi$
$89$	5.00000	−	8.66025i	0.529999	−	0.917985i	0.999388	−	0.0349934i	$-0.0111410\pi$
$90$		0			0
$91$	−2.00000	+	1.73205i	−0.209657	+	0.181568i
$92$		0			0
$93$	−1.00000	+	1.73205i	−0.103695	+	0.179605i
$94$		0			0
$95$	1.50000	+	2.59808i	0.153897	+	0.266557i
$96$		0			0
$97$	−16.0000			−1.62455			−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000			−1.62455			−0.812277	+	0.583272i	$0.801772\pi$
$98$		0			0
$99$	1.00000			0.100504
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	−8.00000	+	13.8564i	−0.788263	+	1.36531i	0.138767	+	0.990325i	$0.455686\pi$
$103$	−8.00000	+	13.8564i	−0.788263	+	1.36531i	−0.927030	+	0.374987i	$0.877647\pi$
$104$		0			0
$105$	2.00000	−	1.73205i	0.195180	−	0.169031i
$106$		0			0
$107$	−5.00000	+	8.66025i	−0.483368	+	0.837218i	−0.999818	−	0.0190994i	$-0.993920\pi$
$107$	−5.00000	+	8.66025i	−0.483368	+	0.837218i	0.516449	+	0.856318i	$0.327253\pi$
$108$		0			0
$109$	−3.00000	−	5.19615i	−0.287348	−	0.497701i	0.685828	−	0.727764i	$-0.259440\pi$
$109$	−3.00000	−	5.19615i	−0.287348	−	0.497701i	−0.973176	+	0.230063i	$0.926107\pi$
$110$		0			0
$111$	11.0000			1.04407
$112$		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
$115$	−3.50000	−	6.06218i	−0.326377	−	0.565301i
$116$		0			0
$117$	−0.500000	+	0.866025i	−0.0462250	+	0.0800641i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$		0			0
$123$	5.50000	+	9.52628i	0.495918	+	0.858956i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	17.0000			1.50851			0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000			1.50851			0.754253	+	0.656584i	$0.227999\pi$
$128$		0			0
$129$	4.00000	+	6.92820i	0.352180	+	0.609994i
$130$		0			0
$131$	−2.50000	+	4.33013i	−0.218426	+	0.378325i	−0.954327	−	0.298764i	$-0.903426\pi$
$131$	−2.50000	+	4.33013i	−0.218426	+	0.378325i	0.735901	+	0.677089i	$0.236759\pi$
$132$		0			0
$133$	−1.50000	−	7.79423i	−0.130066	−	0.675845i
$134$		0			0
$135$	0.500000	−	0.866025i	0.0430331	−	0.0745356i
$136$		0			0
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	0.938148	+	0.346235i	$0.112540\pi$
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	−0.169226	+	0.985577i	$0.554127\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$		0			0
$141$	5.00000			0.421076
$142$		0			0
$143$	−0.500000	−	0.866025i	−0.0418121	−	0.0724207i
$144$		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
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	3.00000	+	5.19615i	0.244137	+	0.422857i	0.961888	−	0.273442i	$-0.0881622\pi$
$151$	3.00000	+	5.19615i	0.244137	+	0.422857i	−0.717752	+	0.696299i	$0.754829\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	3.50000	+	6.06218i	0.279330	+	0.483814i	0.971219	−	0.238190i	$-0.0765542\pi$
$157$	3.50000	+	6.06218i	0.279330	+	0.483814i	−0.691888	+	0.722005i	$0.743221\pi$
$158$		0			0
$159$	5.50000	−	9.52628i	0.436178	−	0.755483i
$160$		0			0
$161$	3.50000	+	18.1865i	0.275839	+	1.43330i
$162$		0			0
$163$	8.00000	−	13.8564i	0.626608	−	1.08532i	−0.361619	−	0.932326i	$-0.617776\pi$
$163$	8.00000	−	13.8564i	0.626608	−	1.08532i	0.988227	−	0.152992i	$-0.0488907\pi$
$164$		0			0
$165$	0.500000	+	0.866025i	0.0389249	+	0.0674200i
$166$		0			0
$167$	3.00000			0.232147			0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000			0.232147			0.116073	+	0.993241i	$0.462969\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$	−1.50000	−	2.59808i	−0.114708	−	0.198680i
$172$		0			0
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	−0.426329	−	0.904568i	$-0.640193\pi$
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	0.996544	−	0.0830722i	$-0.0264732\pi$
$174$		0			0
$175$	2.50000	+	0.866025i	0.188982	+	0.0654654i
$176$		0			0
$177$	2.00000	−	3.46410i	0.150329	−	0.260378i
$178$		0			0
$179$	−9.50000	−	16.4545i	−0.710063	−	1.22987i	−0.964833	−	0.262864i	$-0.915333\pi$
$179$	−9.50000	−	16.4545i	−0.710063	−	1.22987i	0.254770	−	0.967002i	$-0.418000\pi$
$180$		0			0
$181$	−24.0000			−1.78391			−0.891953	−	0.452128i	$-0.850665\pi$
$181$	−24.0000			−1.78391			−0.891953	+	0.452128i	$0.850665\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.50000	+	9.52628i	0.404368	+	0.700386i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.00000	+	1.73205i	−0.145479	+	0.125988i
$190$		0			0
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	−0.953912	−	0.300088i	$-0.902984\pi$
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	0.736839	+	0.676068i	$0.236317\pi$
$192$		0			0
$193$	−11.0000	−	19.0526i	−0.791797	−	1.37143i	−0.924853	−	0.380325i	$-0.875812\pi$
$193$	−11.0000	−	19.0526i	−0.791797	−	1.37143i	0.133056	−	0.991109i	$-0.457521\pi$
$194$		0			0
$195$	−1.00000			−0.0716115
$196$		0			0
$197$	−1.00000			−0.0712470			−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000			−0.0712470			−0.0356235	+	0.999365i	$0.511342\pi$
$198$		0			0
$199$	−12.0000	−	20.7846i	−0.850657	−	1.47338i	−0.880616	−	0.473831i	$-0.842871\pi$
$199$	−12.0000	−	20.7846i	−0.850657	−	1.47338i	0.0299585	−	0.999551i	$-0.490462\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	16.0000	−	13.8564i	1.12298	−	0.972529i
$204$		0			0
$205$	−5.50000	+	9.52628i	−0.384137	+	0.665344i
$206$		0			0
$207$	3.50000	+	6.06218i	0.243267	+	0.421350i
$208$		0			0
$209$	3.00000			0.207514
$210$		0			0
$211$	−5.00000			−0.344214			−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000			−0.344214			−0.172107	+	0.985078i	$0.555058\pi$
$212$		0			0
$213$	−3.00000	−	5.19615i	−0.205557	−	0.356034i
$214$		0			0
$215$	−4.00000	+	6.92820i	−0.272798	+	0.472500i
$216$		0			0
$217$	5.00000	+	1.73205i	0.339422	+	0.117579i
$218$		0			0
$219$	3.00000	−	5.19615i	0.202721	−	0.351123i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−12.0000			−0.803579			−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000			−0.803579			−0.401790	+	0.915732i	$0.631612\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	4.00000	+	6.92820i	0.265489	+	0.459841i	0.967692	−	0.252136i	$-0.0811332\pi$
$227$	4.00000	+	6.92820i	0.265489	+	0.459841i	−0.702202	+	0.711977i	$0.747800\pi$
$228$		0			0
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	−0.999088	−	0.0426906i	$-0.986407\pi$
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	0.536515	+	0.843891i	$0.319740\pi$
$230$		0			0
$231$	−0.500000	−	2.59808i	−0.0328976	−	0.170941i
$232$		0			0
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	0.404674	+	0.914461i	$0.367385\pi$
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	−0.994283	+	0.106773i	$0.965948\pi$
$234$		0			0
$235$	2.50000	+	4.33013i	0.163082	+	0.282466i
$236$		0			0
$237$	8.00000			0.519656
$238$		0			0
$239$	18.0000			1.16432			0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000			1.16432			0.582162	+	0.813073i	$0.302207\pi$
$240$		0			0
$241$	−3.50000	−	6.06218i	−0.225455	−	0.390499i	0.731001	−	0.682376i	$-0.239053\pi$
$241$	−3.50000	−	6.06218i	−0.225455	−	0.390499i	−0.956456	+	0.291877i	$0.905720\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	−5.50000	−	4.33013i	−0.351382	−	0.276642i
$246$		0			0
$247$	−1.50000	+	2.59808i	−0.0954427	+	0.165312i
$248$		0			0
$249$	4.00000	+	6.92820i	0.253490	+	0.439057i
$250$		0			0
$251$	−13.0000			−0.820553			−0.410276	−	0.911961i	$-0.634568\pi$
$251$	−13.0000			−0.820553			−0.410276	+	0.911961i	$0.634568\pi$
$252$		0			0
$253$	−7.00000			−0.440086
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.0000	−	17.3205i	0.623783	−	1.08042i	−0.364992	−	0.931011i	$-0.618928\pi$
$257$	10.0000	−	17.3205i	0.623783	−	1.08042i	0.988775	−	0.149413i	$-0.0477384\pi$
$258$		0			0
$259$	−5.50000	−	28.5788i	−0.341753	−	1.77580i
$260$		0			0
$261$	4.00000	−	6.92820i	0.247594	−	0.428845i
$262$		0			0
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$	11.0000			0.675725
$266$		0			0
$267$	−10.0000			−0.611990
$268$		0			0
$269$	−10.0000	−	17.3205i	−0.609711	−	1.05605i	−0.991288	−	0.131713i	$-0.957952\pi$
$269$	−10.0000	−	17.3205i	−0.609711	−	1.05605i	0.381577	−	0.924337i	$-0.375381\pi$
$270$		0			0
$271$	16.0000	−	27.7128i	0.971931	−	1.68343i	0.282218	−	0.959350i	$-0.408930\pi$
$271$	16.0000	−	27.7128i	0.971931	−	1.68343i	0.689713	−	0.724083i	$-0.257737\pi$
$272$		0			0
$273$	2.50000	+	0.866025i	0.151307	+	0.0524142i
$274$		0			0
$275$	−0.500000	+	0.866025i	−0.0301511	+	0.0522233i
$276$		0			0
$277$	11.0000	+	19.0526i	0.660926	+	1.14476i	0.980373	+	0.197153i	$0.0631696\pi$
$277$	11.0000	+	19.0526i	0.660926	+	1.14476i	−0.319447	+	0.947604i	$0.603497\pi$
$278$		0			0
$279$	2.00000			0.119737
$280$		0			0
$281$	−1.00000			−0.0596550			−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000			−0.0596550			−0.0298275	+	0.999555i	$0.509496\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$		0			0
$285$	1.50000	−	2.59808i	0.0888523	−	0.153897i
$286$		0			0
$287$	22.0000	−	19.0526i	1.29862	−	1.12464i
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$	8.00000	+	13.8564i	0.468968	+	0.812277i
$292$		0			0
$293$	27.0000			1.57736			0.788678	−	0.614806i	$-0.210766\pi$
$293$	27.0000			1.57736			0.788678	+	0.614806i	$0.210766\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$	−0.500000	−	0.866025i	−0.0290129	−	0.0502519i
$298$		0			0
$299$	3.50000	−	6.06218i	0.202410	−	0.350585i
$300$		0			0
$301$	16.0000	−	13.8564i	0.922225	−	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$	16.0000			0.910208
$310$		0			0
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	0.984475	−	0.175525i	$-0.0561621\pi$
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	−0.644246	+	0.764818i	$0.722829\pi$
$312$		0			0
$313$	6.00000	−	10.3923i	0.339140	−	0.587408i	−0.645131	−	0.764072i	$-0.723197\pi$
$313$	6.00000	−	10.3923i	0.339140	−	0.587408i	0.984271	+	0.176664i	$0.0565306\pi$
$314$		0			0
$315$	−2.50000	−	0.866025i	−0.140859	−	0.0487950i
$316$		0			0
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	−0.494489	−	0.869184i	$-0.664645\pi$
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	0.999980	−	0.00635137i	$-0.00202172\pi$
$318$		0			0
$319$	4.00000	+	6.92820i	0.223957	+	0.387905i
$320$		0			0
$321$	10.0000			0.558146
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.500000	−	0.866025i	−0.0277350	−	0.0480384i
$326$		0			0
$327$	−3.00000	+	5.19615i	−0.165900	+	0.287348i
$328$		0			0
$329$	−2.50000	−	12.9904i	−0.137829	−	0.716183i
$330$		0			0
$331$	−6.50000	+	11.2583i	−0.357272	+	0.618814i	−0.987504	−	0.157593i	$-0.949627\pi$
$331$	−6.50000	+	11.2583i	−0.357272	+	0.618814i	0.630232	+	0.776407i	$0.282960\pi$
$332$		0			0
$333$	−5.50000	−	9.52628i	−0.301398	−	0.522037i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−12.0000			−0.653682			−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000			−0.653682			−0.326841	+	0.945079i	$0.605984\pi$
$338$		0			0
$339$	−3.00000	−	5.19615i	−0.162938	−	0.282216i
$340$		0			0
$341$	−1.00000	+	1.73205i	−0.0541530	+	0.0937958i
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$	−3.50000	+	6.06218i	−0.188434	+	0.326377i
$346$		0			0
$347$	7.00000	+	12.1244i	0.375780	+	0.650870i	0.990443	−	0.137920i	$-0.0440416\pi$
$347$	7.00000	+	12.1244i	0.375780	+	0.650870i	−0.614664	+	0.788789i	$0.710708\pi$
$348$		0			0
$349$	12.0000			0.642345			0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000			0.642345			0.321173	+	0.947021i	$0.395923\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$	−12.0000	−	20.7846i	−0.638696	−	1.10625i	−0.985719	−	0.168397i	$-0.946141\pi$
$353$	−12.0000	−	20.7846i	−0.638696	−	1.10625i	0.347024	−	0.937856i	$-0.387192\pi$
$354$		0			0
$355$	3.00000	−	5.19615i	0.159223	−	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.00000	−	3.46410i	0.105556	−	0.182828i	−0.808409	−	0.588621i	$-0.799671\pi$
$359$	2.00000	−	3.46410i	0.105556	−	0.182828i	0.913965	+	0.405793i	$0.133004\pi$
$360$		0			0
$361$	5.00000	+	8.66025i	0.263158	+	0.455803i
$362$		0			0
$363$	−10.0000			−0.524864
$364$		0			0
$365$	6.00000			0.314054
$366$		0			0
$367$	12.5000	+	21.6506i	0.652495	+	1.13015i	0.982516	+	0.186180i	$0.0596109\pi$
$367$	12.5000	+	21.6506i	0.652495	+	1.13015i	−0.330021	+	0.943974i	$0.607056\pi$
$368$		0			0
$369$	5.50000	−	9.52628i	0.286319	−	0.495918i
$370$		0			0
$371$	−27.5000	−	9.52628i	−1.42773	−	0.494580i
$372$		0			0
$373$	−3.00000	+	5.19615i	−0.155334	+	0.269047i	−0.933181	−	0.359408i	$-0.882979\pi$
$373$	−3.00000	+	5.19615i	−0.155334	+	0.269047i	0.777847	+	0.628454i	$0.216312\pi$
$374$		0			0
$375$	0.500000	+	0.866025i	0.0258199	+	0.0447214i
$376$		0			0
$377$	−8.00000			−0.412021
$378$		0			0
$379$	19.0000			0.975964			0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000			0.975964			0.487982	+	0.872854i	$0.337733\pi$
$380$		0			0
$381$	−8.50000	−	14.7224i	−0.435468	−	0.754253i
$382$		0			0
$383$	17.5000	−	30.3109i	0.894208	−	1.54881i	0.0594268	−	0.998233i	$-0.481073\pi$
$383$	17.5000	−	30.3109i	0.894208	−	1.54881i	0.834781	−	0.550581i	$-0.185594\pi$
$384$		0			0
$385$	2.00000	−	1.73205i	0.101929	−	0.0882735i
$386$		0			0
$387$	4.00000	−	6.92820i	0.203331	−	0.352180i
$388$		0			0
$389$	−5.00000	−	8.66025i	−0.253510	−	0.439092i	0.710980	−	0.703213i	$-0.248252\pi$
$389$	−5.00000	−	8.66025i	−0.253510	−	0.439092i	−0.964490	+	0.264120i	$0.914918\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	5.00000			0.252217
$394$		0			0
$395$	4.00000	+	6.92820i	0.201262	+	0.348596i
$396$		0			0
$397$	−7.00000	+	12.1244i	−0.351320	+	0.608504i	−0.986481	−	0.163876i	$-0.947600\pi$
$397$	−7.00000	+	12.1244i	−0.351320	+	0.608504i	0.635161	+	0.772380i	$0.280934\pi$
$398$		0			0
$399$	−6.00000	+	5.19615i	−0.300376	+	0.260133i
$400$		0			0
$401$	2.50000	−	4.33013i	0.124844	−	0.216236i	−0.796828	−	0.604206i	$-0.793490\pi$
$401$	2.50000	−	4.33013i	0.124844	−	0.216236i	0.921672	+	0.387970i	$0.126824\pi$
$402$		0			0
$403$	−1.00000	−	1.73205i	−0.0498135	−	0.0862796i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	11.0000			0.545250
$408$		0			0
$409$	−1.00000	−	1.73205i	−0.0494468	−	0.0856444i	0.840243	−	0.542211i	$-0.182412\pi$
$409$	−1.00000	−	1.73205i	−0.0494468	−	0.0856444i	−0.889689	+	0.456566i	$0.849079\pi$
$410$		0			0
$411$	9.00000	−	15.5885i	0.443937	−	0.768922i
$412$		0			0
$413$	−10.0000	−	3.46410i	−0.492068	−	0.170457i
$414$		0			0
$415$	−4.00000	+	6.92820i	−0.196352	+	0.340092i
$416$		0			0
$417$	10.0000	+	17.3205i	0.489702	+	0.848189i
$418$		0			0
$419$	5.00000			0.244266			0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000			0.244266			0.122133	+	0.992514i	$0.461027\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$	−2.50000	−	4.33013i	−0.121554	−	0.210538i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−0.500000	+	0.866025i	−0.0241402	+	0.0418121i
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	−32.0000			−1.53782			−0.768911	−	0.639356i	$-0.779201\pi$
$433$	−32.0000			−1.53782			−0.768911	+	0.639356i	$0.779201\pi$
$434$		0			0
$435$	8.00000			0.383571
$436$		0			0
$437$	10.5000	+	18.1865i	0.502283	+	0.869980i
$438$		0			0
$439$	16.0000	−	27.7128i	0.763638	−	1.32266i	−0.177325	−	0.984152i	$-0.556744\pi$
$439$	16.0000	−	27.7128i	0.763638	−	1.32266i	0.940963	−	0.338508i	$-0.109922\pi$
$440$		0			0
$441$	5.50000	+	4.33013i	0.261905	+	0.206197i
$442$		0			0
$443$	8.00000	−	13.8564i	0.380091	−	0.658338i	−0.610984	−	0.791643i	$-0.709226\pi$
$443$	8.00000	−	13.8564i	0.380091	−	0.658338i	0.991075	+	0.133306i	$0.0425592\pi$
$444$		0			0
$445$	−5.00000	−	8.66025i	−0.237023	−	0.410535i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	11.0000			0.519122			0.259561	−	0.965727i	$-0.416422\pi$
$449$	11.0000			0.519122			0.259561	+	0.965727i	$0.416422\pi$
$450$		0			0
$451$	5.50000	+	9.52628i	0.258985	+	0.448575i
$452$		0			0
$453$	3.00000	−	5.19615i	0.140952	−	0.244137i
$454$		0			0
$455$	0.500000	+	2.59808i	0.0234404	+	0.121800i
$456$		0			0
$457$	9.00000	−	15.5885i	0.421002	−	0.729197i	−0.575036	−	0.818128i	$-0.695012\pi$
$457$	9.00000	−	15.5885i	0.421002	−	0.729197i	0.996038	+	0.0889312i	$0.0283451\pi$
$458$		0			0
$459$		0			0
$460$		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
$463$	13.0000			0.604161			0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000			0.604161			0.302081	+	0.953282i	$0.402319\pi$
$464$		0			0
$465$	1.00000	+	1.73205i	0.0463739	+	0.0803219i
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	3.50000	−	6.06218i	0.161271	−	0.279330i
$472$		0			0
$473$	4.00000	+	6.92820i	0.183920	+	0.318559i
$474$		0			0
$475$	3.00000			0.137649
$476$		0			0
$477$	−11.0000			−0.503655
$478$		0			0
$479$	11.0000	+	19.0526i	0.502603	+	0.870534i	0.999995	+	0.00300810i	$0.000957509\pi$
$479$	11.0000	+	19.0526i	0.502603	+	0.870534i	−0.497393	+	0.867526i	$0.665709\pi$
$480$		0			0
$481$	−5.50000	+	9.52628i	−0.250778	+	0.434361i
$482$		0			0
$483$	14.0000	−	12.1244i	0.637022	−	0.551677i
$484$		0			0
$485$	−8.00000	+	13.8564i	−0.363261	+	0.629187i
$486$		0			0
$487$	−8.00000	−	13.8564i	−0.362515	−	0.627894i	0.625859	−	0.779936i	$-0.284748\pi$
$487$	−8.00000	−	13.8564i	−0.362515	−	0.627894i	−0.988374	+	0.152042i	$0.951415\pi$
$488$		0			0
$489$	−16.0000			−0.723545
$490$		0			0
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	0.500000	−	0.866025i	0.0224733	−	0.0389249i
$496$		0			0
$497$	−12.0000	+	10.3923i	−0.538274	+	0.466159i
$498$		0			0
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	0.246214	+	0.969216i	$0.420813\pi$
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	−0.962472	+	0.271380i	$0.912520\pi$
$500$		0			0
$501$	−1.50000	−	2.59808i	−0.0670151	−	0.116073i
$502$		0			0
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	6.00000	+	10.3923i	0.266469	+	0.461538i
$508$		0			0
$509$	17.0000	−	29.4449i	0.753512	−	1.30512i	−0.192599	−	0.981278i	$-0.561692\pi$
$509$	17.0000	−	29.4449i	0.753512	−	1.30512i	0.946111	−	0.323843i	$-0.104975\pi$
$510$		0			0
$511$	−15.0000	−	5.19615i	−0.663561	−	0.229864i
$512$		0			0
$513$	−1.50000	+	2.59808i	−0.0662266	+	0.114708i
$514$		0			0
$515$	8.00000	+	13.8564i	0.352522	+	0.610586i
$516$		0			0
$517$	5.00000			0.219900
$518$		0			0
$519$	−15.0000			−0.658427
$520$		0			0
$521$	16.5000	+	28.5788i	0.722878	+	1.25206i	0.959841	+	0.280543i	$0.0905145\pi$
$521$	16.5000	+	28.5788i	0.722878	+	1.25206i	−0.236963	+	0.971519i	$0.576152\pi$
$522$		0			0
$523$	1.00000	−	1.73205i	0.0437269	−	0.0757373i	−0.843334	−	0.537390i	$-0.819410\pi$
$523$	1.00000	−	1.73205i	0.0437269	−	0.0757373i	0.887061	+	0.461653i	$0.152744\pi$
$524$		0			0
$525$	−0.500000	−	2.59808i	−0.0218218	−	0.113389i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−13.0000	−	22.5167i	−0.565217	−	0.978985i
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$	−11.0000			−0.476463
$534$		0			0
$535$	5.00000	+	8.66025i	0.216169	+	0.374415i
$536$		0			0
$537$	−9.50000	+	16.4545i	−0.409955	+	0.710063i
$538$		0			0
$539$	−6.50000	+	2.59808i	−0.279975	+	0.111907i
$540$		0			0
$541$	5.00000	−	8.66025i	0.214967	−	0.372333i	−0.738296	−	0.674477i	$-0.764369\pi$
$541$	5.00000	−	8.66025i	0.214967	−	0.372333i	0.953262	+	0.302144i	$0.0977023\pi$
$542$		0			0
$543$	12.0000	+	20.7846i	0.514969	+	0.891953i
$544$		0			0
$545$	−6.00000			−0.257012
$546$		0			0
$547$	−16.0000			−0.684111			−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000			−0.684111			−0.342055	+	0.939680i	$0.611123\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.0000	−	20.7846i	0.511217	−	0.885454i
$552$		0			0
$553$	−4.00000	−	20.7846i	−0.170097	−	0.883852i
$554$		0			0
$555$	5.50000	−	9.52628i	0.233462	−	0.404368i
$556$		0			0
$557$	16.5000	+	28.5788i	0.699127	+	1.21092i	0.968769	+	0.247964i	$0.0797613\pi$
$557$	16.5000	+	28.5788i	0.699127	+	1.21092i	−0.269642	+	0.962961i	$0.586905\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.0000	−	32.9090i	−0.800755	−	1.38695i	−0.919120	−	0.393977i	$-0.871099\pi$
$563$	−19.0000	−	32.9090i	−0.800755	−	1.38695i	0.118366	−	0.992970i	$-0.462235\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$		0			0
$567$	2.50000	+	0.866025i	0.104990	+	0.0363696i
$568$		0			0
$569$	4.50000	−	7.79423i	0.188650	−	0.326751i	−0.756151	−	0.654398i	$-0.772922\pi$
$569$	4.50000	−	7.79423i	0.188650	−	0.326751i	0.944800	+	0.327647i	$0.106256\pi$
$570$		0			0
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	−0.978022	−	0.208502i	$-0.933141\pi$
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	0.308443	−	0.951243i	$-0.400192\pi$
$572$		0			0
$573$	6.00000			0.250654
$574$		0			0
$575$	−7.00000			−0.291920
$576$		0			0
$577$	7.00000	+	12.1244i	0.291414	+	0.504744i	0.974144	−	0.225927i	$-0.0725410\pi$
$577$	7.00000	+	12.1244i	0.291414	+	0.504744i	−0.682730	+	0.730670i	$0.739208\pi$
$578$		0			0
$579$	−11.0000	+	19.0526i	−0.457144	+	0.791797i
$580$		0			0
$581$	16.0000	−	13.8564i	0.663792	−	0.574861i
$582$		0			0
$583$	5.50000	−	9.52628i	0.227787	−	0.394538i
$584$		0			0
$585$	0.500000	+	0.866025i	0.0206725	+	0.0358057i
$586$		0			0
$587$	18.0000			0.742940			0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000			0.742940			0.371470	+	0.928445i	$0.378854\pi$
$588$		0			0
$589$	6.00000			0.247226
$590$		0			0
$591$	0.500000	+	0.866025i	0.0205673	+	0.0356235i
$592$		0			0
$593$	−8.00000	+	13.8564i	−0.328521	+	0.569014i	−0.982219	−	0.187741i	$-0.939883\pi$
$593$	−8.00000	+	13.8564i	−0.328521	+	0.569014i	0.653698	+	0.756756i	$0.273217\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−12.0000	+	20.7846i	−0.491127	+	0.850657i
$598$		0			0
$599$	1.00000	+	1.73205i	0.0408589	+	0.0707697i	0.885732	−	0.464198i	$-0.153657\pi$
$599$	1.00000	+	1.73205i	0.0408589	+	0.0707697i	−0.844873	+	0.534967i	$0.820324\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.00000	−	8.66025i	−0.203279	−	0.352089i
$606$		0			0
$607$	−18.5000	+	32.0429i	−0.750892	+	1.30058i	0.196499	+	0.980504i	$0.437043\pi$
$607$	−18.5000	+	32.0429i	−0.750892	+	1.30058i	−0.947391	+	0.320079i	$0.896291\pi$
$608$		0			0
$609$	−20.0000	−	6.92820i	−0.810441	−	0.280745i
$610$		0			0
$611$	−2.50000	+	4.33013i	−0.101139	+	0.175178i
$612$		0			0
$613$	20.5000	+	35.5070i	0.827987	+	1.43412i	0.899615	+	0.436684i	$0.143847\pi$
$613$	20.5000	+	35.5070i	0.827987	+	1.43412i	−0.0716275	+	0.997431i	$0.522819\pi$
$614$		0			0
$615$	11.0000			0.443563
$616$		0			0
$617$	28.0000			1.12724			0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000			1.12724			0.563619	+	0.826035i	$0.309409\pi$
$618$		0			0
$619$	14.5000	+	25.1147i	0.582804	+	1.00945i	0.995145	+	0.0984169i	$0.0313779\pi$
$619$	14.5000	+	25.1147i	0.582804	+	1.00945i	−0.412341	+	0.911030i	$0.635289\pi$
$620$		0			0
$621$	3.50000	−	6.06218i	0.140450	−	0.243267i
$622$		0			0
$623$	5.00000	+	25.9808i	0.200321	+	1.04090i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−1.50000	−	2.59808i	−0.0599042	−	0.103757i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	26.0000			1.03504			0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000			1.03504			0.517522	+	0.855670i	$0.326855\pi$
$632$		0			0
$633$	2.50000	+	4.33013i	0.0993661	+	0.172107i
$634$		0			0
$635$	8.50000	−	14.7224i	0.337312	−	0.584242i
$636$		0			0
$637$	1.00000	−	6.92820i	0.0396214	−	0.274505i
$638$		0			0
$639$	−3.00000	+	5.19615i	−0.118678	+	0.205557i
$640$		0			0
$641$	1.50000	+	2.59808i	0.0592464	+	0.102618i	0.894127	−	0.447813i	$-0.147797\pi$
$641$	1.50000	+	2.59808i	0.0592464	+	0.102618i	−0.834881	+	0.550431i	$0.814464\pi$
$642$		0			0
$643$	10.0000			0.394362			0.197181	−	0.980367i	$-0.436821\pi$
$643$	10.0000			0.394362			0.197181	+	0.980367i	$0.436821\pi$
$644$		0			0
$645$	8.00000			0.315000
$646$		0			0
$647$	−8.50000	−	14.7224i	−0.334169	−	0.578799i	0.649155	−	0.760656i	$-0.275122\pi$
$647$	−8.50000	−	14.7224i	−0.334169	−	0.578799i	−0.983325	+	0.181857i	$0.941789\pi$
$648$		0			0
$649$	2.00000	−	3.46410i	0.0785069	−	0.135978i
$650$		0			0
$651$	−1.00000	−	5.19615i	−0.0391931	−	0.203653i
$652$		0			0
$653$	−3.50000	+	6.06218i	−0.136966	+	0.237231i	−0.926347	−	0.376672i	$-0.877068\pi$
$653$	−3.50000	+	6.06218i	−0.136966	+	0.237231i	0.789381	+	0.613904i	$0.210402\pi$
$654$		0			0
$655$	2.50000	+	4.33013i	0.0976831	+	0.169192i
$656$		0			0
$657$	−6.00000			−0.234082
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$	12.0000	+	20.7846i	0.466746	+	0.808428i	0.999278	−	0.0379819i	$-0.0120929\pi$
$661$	12.0000	+	20.7846i	0.466746	+	0.808428i	−0.532533	+	0.846410i	$0.678760\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.50000	−	2.59808i	−0.290838	−	0.100749i
$666$		0			0
$667$	−28.0000	+	48.4974i	−1.08416	+	1.87783i
$668$		0			0
$669$	6.00000	+	10.3923i	0.231973	+	0.401790i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	28.0000			1.07932			0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000			1.07932			0.539660	+	0.841883i	$0.318553\pi$
$674$		0			0
$675$	−0.500000	−	0.866025i	−0.0192450	−	0.0333333i
$676$		0			0
$677$	6.50000	−	11.2583i	0.249815	−	0.432693i	−0.713659	−	0.700493i	$-0.752963\pi$
$677$	6.50000	−	11.2583i	0.249815	−	0.432693i	0.963474	+	0.267800i	$0.0862968\pi$
$678$		0			0
$679$	32.0000	−	27.7128i	1.22805	−	1.06352i
$680$		0			0
$681$	4.00000	−	6.92820i	0.153280	−	0.265489i
$682$		0			0
$683$	2.00000	+	3.46410i	0.0765279	+	0.132550i	0.901750	−	0.432259i	$-0.142283\pi$
$683$	2.00000	+	3.46410i	0.0765279	+	0.132550i	−0.825222	+	0.564809i	$0.808950\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$		0			0
$687$	14.0000			0.534133
$688$		0			0
$689$	5.50000	+	9.52628i	0.209533	+	0.362922i
$690$		0			0
$691$	−6.00000	+	10.3923i	−0.228251	+	0.395342i	−0.957290	−	0.289130i	$-0.906634\pi$
$691$	−6.00000	+	10.3923i	−0.228251	+	0.395342i	0.729039	+	0.684472i	$0.239967\pi$
$692$		0			0
$693$	−2.00000	+	1.73205i	−0.0759737	+	0.0657952i
$694$		0			0
$695$	−10.0000	+	17.3205i	−0.379322	+	0.657004i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	18.0000			0.680823
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	−16.5000	−	28.5788i	−0.622309	−	1.07787i
$704$		0			0
$705$	2.50000	−	4.33013i	0.0941554	−	0.163082i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−22.0000	+	38.1051i	−0.826227	+	1.43107i	0.0747503	+	0.997202i	$0.476184\pi$
$709$	−22.0000	+	38.1051i	−0.826227	+	1.43107i	−0.900978	+	0.433865i	$0.857149\pi$
$710$		0			0
$711$	−4.00000	−	6.92820i	−0.150012	−	0.259828i
$712$		0			0
$713$	−14.0000			−0.524304
$714$		0			0
$715$	−1.00000			−0.0373979
$716$		0			0
$717$	−9.00000	−	15.5885i	−0.336111	−	0.582162i
$718$		0			0
$719$	1.00000	−	1.73205i	0.0372937	−	0.0645946i	−0.846776	−	0.531949i	$-0.821460\pi$
$719$	1.00000	−	1.73205i	0.0372937	−	0.0645946i	0.884070	+	0.467355i	$0.154793\pi$
$720$		0			0
$721$	−8.00000	−	41.5692i	−0.297936	−	1.54812i
$722$		0			0
$723$	−3.50000	+	6.06218i	−0.130166	+	0.225455i
$724$		0			0
$725$	4.00000	+	6.92820i	0.148556	+	0.257307i
$726$		0			0
$727$	−11.0000			−0.407967			−0.203984	−	0.978974i	$-0.565389\pi$
$727$	−11.0000			−0.407967			−0.203984	+	0.978974i	$0.565389\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−5.50000	+	9.52628i	−0.203147	+	0.351861i	−0.949541	−	0.313644i	$-0.898450\pi$
$733$	−5.50000	+	9.52628i	−0.203147	+	0.351861i	0.746394	+	0.665505i	$0.231784\pi$
$734$		0			0
$735$	−1.00000	+	6.92820i	−0.0368856	+	0.255551i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−9.50000	−	16.4545i	−0.349463	−	0.605288i	0.636691	−	0.771119i	$-0.280303\pi$
$739$	−9.50000	−	16.4545i	−0.349463	−	0.605288i	−0.986154	+	0.165831i	$0.946969\pi$
$740$		0			0
$741$	3.00000			0.110208
$742$		0			0
$743$	−49.0000			−1.79764			−0.898818	−	0.438322i	$-0.855573\pi$
$743$	−49.0000			−1.79764			−0.898818	+	0.438322i	$0.855573\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.00000	−	6.92820i	0.146352	−	0.253490i
$748$		0			0
$749$	−5.00000	−	25.9808i	−0.182696	−	0.949316i
$750$		0			0
$751$	13.0000	−	22.5167i	0.474377	−	0.821645i	−0.525193	−	0.850983i	$-0.676007\pi$
$751$	13.0000	−	22.5167i	0.474377	−	0.821645i	0.999570	+	0.0293387i	$0.00934013\pi$
$752$		0			0
$753$	6.50000	+	11.2583i	0.236873	+	0.410276i
$754$		0			0
$755$	6.00000			0.218362
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$	3.50000	+	6.06218i	0.127042	+	0.220043i
$760$		0			0
$761$	−13.5000	+	23.3827i	−0.489375	+	0.847622i	−0.999925	−	0.0122260i	$-0.996108\pi$
$761$	−13.5000	+	23.3827i	−0.489375	+	0.847622i	0.510551	+	0.859848i	$0.329442\pi$
$762$		0			0
$763$	15.0000	+	5.19615i	0.543036	+	0.188113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.00000	+	3.46410i	0.0722158	+	0.125081i
$768$		0			0
$769$	−5.00000			−0.180305			−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000			−0.180305			−0.0901523	+	0.995928i	$0.528735\pi$
$770$		0			0
$771$	−20.0000			−0.720282
$772$		0			0
$773$	−16.5000	−	28.5788i	−0.593464	−	1.02791i	−0.993762	−	0.111524i	$-0.964427\pi$
$773$	−16.5000	−	28.5788i	−0.593464	−	1.02791i	0.400298	−	0.916385i	$-0.368907\pi$
$774$		0			0
$775$	−1.00000	+	1.73205i	−0.0359211	+	0.0622171i
$776$		0			0
$777$	−22.0000	+	19.0526i	−0.789246	+	0.683507i
$778$		0			0
$779$	16.5000	−	28.5788i	0.591174	−	1.02394i
$780$		0			0
$781$	−3.00000	−	5.19615i	−0.107348	−	0.185933i
$782$		0			0
$783$	−8.00000			−0.285897
$784$		0			0
$785$	7.00000			0.249841
$786$		0			0
$787$	11.0000	+	19.0526i	0.392108	+	0.679150i	0.992727	−	0.120384i	$-0.0384127\pi$
$787$	11.0000	+	19.0526i	0.392108	+	0.679150i	−0.600620	+	0.799535i	$0.705079\pi$
$788$		0			0
$789$	12.0000	−	20.7846i	0.427211	−	0.739952i
$790$		0			0
$791$	−12.0000	+	10.3923i	−0.426671	+	0.369508i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−5.50000	−	9.52628i	−0.195065	−	0.337862i
$796$		0			0
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	5.00000	+	8.66025i	0.176666	+	0.305995i
$802$		0			0
$803$	3.00000	−	5.19615i	0.105868	−	0.183368i
$804$		0			0
$805$	17.5000	+	6.06218i	0.616794	+	0.213664i
$806$		0			0
$807$	−10.0000	+	17.3205i	−0.352017	+	0.609711i
$808$		0			0
$809$	19.5000	+	33.7750i	0.685583	+	1.18747i	0.973253	+	0.229736i	$0.0737862\pi$
$809$	19.5000	+	33.7750i	0.685583	+	1.18747i	−0.287670	+	0.957730i	$0.592880\pi$
$810$		0			0
$811$	−9.00000			−0.316033			−0.158016	−	0.987436i	$-0.550510\pi$
$811$	−9.00000			−0.316033			−0.158016	+	0.987436i	$0.550510\pi$
$812$		0			0
$813$	−32.0000			−1.12229
$814$		0			0
$815$	−8.00000	−	13.8564i	−0.280228	−	0.485369i
$816$		0			0
$817$	12.0000	−	20.7846i	0.419827	−	0.727161i
$818$		0			0
$819$	−0.500000	−	2.59808i	−0.0174714	−	0.0907841i
$820$		0			0
$821$	−21.0000	+	36.3731i	−0.732905	+	1.26943i	0.222731	+	0.974880i	$0.428503\pi$
$821$	−21.0000	+	36.3731i	−0.732905	+	1.26943i	−0.955636	+	0.294549i	$0.904831\pi$
$822$		0			0
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	0.577474	−	0.816409i	$-0.304038\pi$
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	−0.995768	+	0.0919029i	$0.970705\pi$
$824$		0			0
$825$	1.00000			0.0348155
$826$		0			0
$827$	10.0000			0.347734			0.173867	−	0.984769i	$-0.444374\pi$
$827$	10.0000			0.347734			0.173867	+	0.984769i	$0.444374\pi$
$828$		0			0
$829$	4.00000	+	6.92820i	0.138926	+	0.240626i	0.927090	−	0.374838i	$-0.122302\pi$
$829$	4.00000	+	6.92820i	0.138926	+	0.240626i	−0.788165	+	0.615465i	$0.788968\pi$
$830$		0			0
$831$	11.0000	−	19.0526i	0.381586	−	0.660926i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	1.50000	−	2.59808i	0.0519096	−	0.0899101i
$836$		0			0
$837$	−1.00000	−	1.73205i	−0.0345651	−	0.0598684i
$838$		0			0
$839$	40.0000			1.38095			0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000			1.38095			0.690477	+	0.723355i	$0.257401\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$	0.500000	+	0.866025i	0.0172209	+	0.0298275i
$844$		0			0
$845$	−6.00000	+	10.3923i	−0.206406	+	0.357506i
$846$		0			0
$847$	5.00000	+	25.9808i	0.171802	+	0.892710i
$848$		0			0
$849$	−7.00000	+	12.1244i	−0.240239	+	0.416107i
$850$		0			0
$851$	38.5000	+	66.6840i	1.31976	+	2.28590i
$852$		0			0
$853$	−41.0000			−1.40381			−0.701907	−	0.712269i	$-0.747668\pi$
$853$	−41.0000			−1.40381			−0.701907	+	0.712269i	$0.747668\pi$
$854$		0			0
$855$	−3.00000			−0.102598
$856$		0			0
$857$	7.00000	+	12.1244i	0.239115	+	0.414160i	0.960461	−	0.278416i	$-0.0898092\pi$
$857$	7.00000	+	12.1244i	0.239115	+	0.414160i	−0.721345	+	0.692576i	$0.756476\pi$
$858$		0			0
$859$	6.00000	−	10.3923i	0.204717	−	0.354581i	−0.745325	−	0.666701i	$-0.767706\pi$
$859$	6.00000	−	10.3923i	0.204717	−	0.354581i	0.950043	+	0.312120i	$0.101039\pi$
$860$		0			0
$861$	−27.5000	−	9.52628i	−0.937197	−	0.324655i
$862$		0			0
$863$	−26.5000	+	45.8993i	−0.902070	+	1.56243i	−0.0772684	+	0.997010i	$0.524620\pi$
$863$	−26.5000	+	45.8993i	−0.902070	+	1.56243i	−0.824802	+	0.565422i	$0.808713\pi$
$864$		0			0
$865$	−7.50000	−	12.9904i	−0.255008	−	0.441686i
$866$		0			0
$867$	−17.0000			−0.577350
$868$		0			0
$869$	8.00000			0.271381
$870$		0			0
$871$		0			0
$872$		0			0
$873$	8.00000	−	13.8564i	0.270759	−	0.468968i
$874$		0			0
$875$	2.00000	−	1.73205i	0.0676123	−	0.0585540i
$876$		0			0
$877$	−8.50000	+	14.7224i	−0.287025	+	0.497141i	−0.973098	−	0.230391i	$-0.925999\pi$
$877$	−8.50000	+	14.7224i	−0.287025	+	0.497141i	0.686074	+	0.727532i	$0.259333\pi$
$878$		0			0
$879$	−13.5000	−	23.3827i	−0.455344	−	0.788678i
$880$		0			0
$881$	−53.0000			−1.78562			−0.892808	−	0.450438i	$-0.851268\pi$
$881$	−53.0000			−1.78562			−0.892808	+	0.450438i	$0.851268\pi$
$882$		0			0
$883$	42.0000			1.41341			0.706706	−	0.707507i	$-0.250180\pi$
$883$	42.0000			1.41341			0.706706	+	0.707507i	$0.250180\pi$
$884$		0			0
$885$	−2.00000	−	3.46410i	−0.0672293	−	0.116445i
$886$		0			0
$887$	22.0000	−	38.1051i	0.738688	−	1.27944i	−0.214399	−	0.976746i	$-0.568779\pi$
$887$	22.0000	−	38.1051i	0.738688	−	1.27944i	0.953086	−	0.302698i	$-0.0978875\pi$
$888$		0			0
$889$	−34.0000	+	29.4449i	−1.14032	+	0.987549i
$890$		0			0
$891$	−0.500000	+	0.866025i	−0.0167506	+	0.0290129i
$892$		0			0
$893$	−7.50000	−	12.9904i	−0.250978	−	0.434707i
$894$		0			0
$895$	−19.0000			−0.635100
$896$		0			0
$897$	−7.00000			−0.233723
$898$		0			0
$899$	8.00000	+	13.8564i	0.266815	+	0.462137i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−20.0000	−	6.92820i	−0.665558	−	0.230556i
$904$		0			0
$905$	−12.0000	+	20.7846i	−0.398893	+	0.690904i
$906$		0			0
$907$	−5.00000	−	8.66025i	−0.166022	−	0.287559i	0.770996	−	0.636841i	$-0.219759\pi$
$907$	−5.00000	−	8.66025i	−0.166022	−	0.287559i	−0.937018	+	0.349281i	$0.886426\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.00000			0.0662630			0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000			0.0662630			0.0331315	+	0.999451i	$0.489452\pi$
$912$		0			0
$913$	4.00000	+	6.92820i	0.132381	+	0.229290i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.50000	−	12.9904i	−0.0825573	−	0.428980i
$918$		0			0
$919$	−18.0000	+	31.1769i	−0.593765	+	1.02843i	0.399955	+	0.916535i	$0.369026\pi$
$919$	−18.0000	+	31.1769i	−0.593765	+	1.02843i	−0.993720	+	0.111897i	$0.964307\pi$
$920$		0			0
$921$	10.0000	+	17.3205i	0.329511	+	0.570730i
$922$		0			0
$923$	6.00000			0.197492
$924$		0			0
$925$	11.0000			0.361678
$926$		0			0
$927$	−8.00000	−	13.8564i	−0.262754	−	0.455104i
$928$		0			0
$929$	10.5000	−	18.1865i	0.344494	−	0.596681i	−0.640768	−	0.767735i	$-0.721384\pi$
$929$	10.5000	−	18.1865i	0.344494	−	0.596681i	0.985262	+	0.171054i	$0.0547172\pi$
$930$		0			0
$931$	16.5000	+	12.9904i	0.540766	+	0.425743i
$932$		0			0
$933$	6.00000	−	10.3923i	0.196431	−	0.340229i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	12.0000			0.392023			0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000			0.392023			0.196011	+	0.980602i	$0.437201\pi$
$938$		0			0
$939$	−12.0000			−0.391605
$940$		0			0
$941$	−27.0000	−	46.7654i	−0.880175	−	1.52451i	−0.851146	−	0.524929i	$-0.824092\pi$
$941$	−27.0000	−	46.7654i	−0.880175	−	1.52451i	−0.0290288	−	0.999579i	$-0.509241\pi$
$942$		0			0
$943$	−38.5000	+	66.6840i	−1.25373	+	2.17153i
$944$		0			0
$945$	0.500000	+	2.59808i	0.0162650	+	0.0845154i
$946$		0			0
$947$	11.0000	−	19.0526i	0.357452	−	0.619125i	−0.630082	−	0.776528i	$-0.716979\pi$
$947$	11.0000	−	19.0526i	0.357452	−	0.619125i	0.987534	+	0.157403i	$0.0503122\pi$
$948$		0			0
$949$	3.00000	+	5.19615i	0.0973841	+	0.168674i
$950$		0			0
$951$	−18.0000			−0.583690
$952$		0			0
$953$	−20.0000			−0.647864			−0.323932	−	0.946080i	$-0.605005\pi$
$953$	−20.0000			−0.647864			−0.323932	+	0.946080i	$0.605005\pi$
$954$		0			0
$955$	3.00000	+	5.19615i	0.0970777	+	0.168144i
$956$		0			0
$957$	4.00000	−	6.92820i	0.129302	−	0.223957i
$958$		0			0
$959$	−45.0000	−	15.5885i	−1.45313	−	0.503378i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$		0			0
$963$	−5.00000	−	8.66025i	−0.161123	−	0.279073i
$964$		0			0
$965$	−22.0000			−0.708205
$966$		0			0
$967$	20.0000			0.643157			0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000			0.643157			0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.5000	+	38.9711i	−0.722059	+	1.25064i	0.238114	+	0.971237i	$0.423471\pi$
$971$	−22.5000	+	38.9711i	−0.722059	+	1.25064i	−0.960173	+	0.279406i	$0.909862\pi$
$972$		0			0
$973$	40.0000	−	34.6410i	1.28234	−	1.11054i
$974$		0			0
$975$	−0.500000	+	0.866025i	−0.0160128	+	0.0277350i
$976$		0			0
$977$	−27.0000	−	46.7654i	−0.863807	−	1.49616i	−0.868227	−	0.496167i	$-0.834741\pi$
$977$	−27.0000	−	46.7654i	−0.863807	−	1.49616i	0.00442082	−	0.999990i	$-0.498593\pi$
$978$		0			0
$979$	−10.0000			−0.319601
$980$		0			0
$981$	6.00000			0.191565
$982$		0			0
$983$	24.5000	+	42.4352i	0.781429	+	1.35347i	0.931110	+	0.364740i	$0.118842\pi$
$983$	24.5000	+	42.4352i	0.781429	+	1.35347i	−0.149681	+	0.988734i	$0.547825\pi$
$984$		0			0
$985$	−0.500000	+	0.866025i	−0.0159313	+	0.0275939i
$986$		0			0
$987$	−10.0000	+	8.66025i	−0.318304	+	0.275659i
$988$		0			0
$989$	−28.0000	+	48.4974i	−0.890348	+	1.54213i
$990$		0			0
$991$	11.0000	+	19.0526i	0.349427	+	0.605224i	0.986148	−	0.165870i	$-0.0530431\pi$
$991$	11.0000	+	19.0526i	0.349427	+	0.605224i	−0.636721	+	0.771094i	$0.719710\pi$
$992$		0			0
$993$	13.0000			0.412543
$994$		0			0
$995$	−24.0000			−0.760851
$996$		0			0
$997$	−7.00000	−	12.1244i	−0.221692	−	0.383982i	0.733630	−	0.679549i	$-0.237825\pi$
$997$	−7.00000	−	12.1244i	−0.221692	−	0.383982i	−0.955322	+	0.295567i	$0.904491\pi$
$998$		0			0
$999$	−5.50000	+	9.52628i	−0.174012	+	0.301398i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.bg.g.961.1		2
4.3	odd	2		210.2.i.d.121.1	✓	2
7.4	even	3	inner	1680.2.bg.g.1201.1		2
12.11	even	2		630.2.k.c.541.1		2
20.3	even	4		1050.2.o.i.499.1		4
20.7	even	4		1050.2.o.i.499.2		4
20.19	odd	2		1050.2.i.b.751.1		2
28.3	even	6		1470.2.i.m.361.1		2
28.11	odd	6		210.2.i.d.151.1	yes	2
28.19	even	6		1470.2.a.h.1.1		1
28.23	odd	6		1470.2.a.a.1.1		1
28.27	even	2		1470.2.i.m.961.1		2
84.11	even	6		630.2.k.c.361.1		2
84.23	even	6		4410.2.a.bj.1.1		1
84.47	odd	6		4410.2.a.ba.1.1		1
140.19	even	6		7350.2.a.bu.1.1		1
140.39	odd	6		1050.2.i.b.151.1		2
140.67	even	12		1050.2.o.i.949.1		4
140.79	odd	6		7350.2.a.cp.1.1		1
140.123	even	12		1050.2.o.i.949.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.i.d.121.1	✓	2	4.3	odd	2
210.2.i.d.151.1	yes	2	28.11	odd	6
630.2.k.c.361.1		2	84.11	even	6
630.2.k.c.541.1		2	12.11	even	2
1050.2.i.b.151.1		2	140.39	odd	6
1050.2.i.b.751.1		2	20.19	odd	2
1050.2.o.i.499.1		4	20.3	even	4
1050.2.o.i.499.2		4	20.7	even	4
1050.2.o.i.949.1		4	140.67	even	12
1050.2.o.i.949.2		4	140.123	even	12
1470.2.a.a.1.1		1	28.23	odd	6
1470.2.a.h.1.1		1	28.19	even	6
1470.2.i.m.361.1		2	28.3	even	6
1470.2.i.m.961.1		2	28.27	even	2
1680.2.bg.g.961.1		2	1.1	even	1	trivial
1680.2.bg.g.1201.1		2	7.4	even	3	inner
4410.2.a.ba.1.1		1	84.47	odd	6
4410.2.a.bj.1.1		1	84.23	even	6
7350.2.a.bu.1.1		1	140.19	even	6
7350.2.a.cp.1.1		1	140.79	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)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{11} +1.00000 q^{13} -1.00000 q^{15} +(-1.50000 + 2.59808i) q^{19} +(2.50000 + 0.866025i) q^{21} +(3.50000 - 6.06218i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -8.00000 q^{29} +(-1.00000 - 1.73205i) q^{31} +(-0.500000 + 0.866025i) q^{33} +(0.500000 + 2.59808i) q^{35} +(-5.50000 + 9.52628i) q^{37} +(-0.500000 - 0.866025i) q^{39} -11.0000 q^{41} -8.00000 q^{43} +(0.500000 + 0.866025i) q^{45} +(-2.50000 + 4.33013i) q^{47} +(1.00000 - 6.92820i) q^{49} +(5.50000 + 9.52628i) q^{53} -1.00000 q^{55} +3.00000 q^{57} +(2.00000 + 3.46410i) q^{59} +(-0.500000 - 2.59808i) q^{63} +(0.500000 - 0.866025i) q^{65} -7.00000 q^{69} +6.00000 q^{71} +(3.00000 + 5.19615i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(2.50000 + 0.866025i) q^{77} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 - 0.866025i) q^{81} -8.00000 q^{83} +(4.00000 + 6.92820i) q^{87} +(5.00000 - 8.66025i) q^{89} +(-2.00000 + 1.73205i) q^{91} +(-1.00000 + 1.73205i) q^{93} +(1.50000 + 2.59808i) q^{95} -16.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 2 * q - q^3 + q^5 - 4 * q^7 - q^9	\( 2 q - q^{3} + q^{5} - 4 q^{7} - q^{9} - q^{11} + 2 q^{13} - 2 q^{15} - 3 q^{19} + 5 q^{21} + 7 q^{23} - q^{25} + 2 q^{27} - 16 q^{29} - 2 q^{31} - q^{33} + q^{35} - 11 q^{37} - q^{39} - 22 q^{41} - 16 q^{43} + q^{45} - 5 q^{47} + 2 q^{49} + 11 q^{53} - 2 q^{55} + 6 q^{57} + 4 q^{59} - q^{63} + q^{65} - 14 q^{69} + 12 q^{71} + 6 q^{73} - q^{75} + 5 q^{77} - 8 q^{79} - q^{81} - 16 q^{83} + 8 q^{87} + 10 q^{89} - 4 q^{91} - 2 q^{93} + 3 q^{95} - 32 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - q^3 + q^5 - 4 * q^7 - q^9 - q^11 + 2 * q^13 - 2 * q^15 - 3 * q^19 + 5 * q^21 + 7 * q^23 - q^25 + 2 * q^27 - 16 * q^29 - 2 * q^31 - q^33 + q^35 - 11 * q^37 - q^39 - 22 * q^41 - 16 * q^43 + q^45 - 5 * q^47 + 2 * q^49 + 11 * q^53 - 2 * q^55 + 6 * q^57 + 4 * q^59 - q^63 + q^65 - 14 * q^69 + 12 * q^71 + 6 * q^73 - q^75 + 5 * q^77 - 8 * q^79 - q^81 - 16 * q^83 + 8 * q^87 + 10 * q^89 - 4 * q^91 - 2 * q^93 + 3 * q^95 - 32 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more