LMFDB - Embedded newform 840.2.bg.e.121.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [840,2,Mod(121,840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(840, 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("840.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.bg (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.70743376979$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	840.121
Dual form			840.2.bg.e.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +1.00000 q^{15} +(-1.50000 + 2.59808i) q^{19} +(-2.00000 + 1.73205i) q^{21} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +(4.50000 + 7.79423i) q^{31} +(-1.00000 + 1.73205i) q^{33} +(2.50000 + 0.866025i) q^{35} +(-1.50000 + 2.59808i) q^{37} +(0.500000 + 0.866025i) q^{39} +2.00000 q^{41} +3.00000 q^{43} +(0.500000 + 0.866025i) q^{45} +(3.00000 - 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +2.00000 q^{55} -3.00000 q^{57} +(2.00000 + 3.46410i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-2.50000 - 0.866025i) q^{63} +(0.500000 - 0.866025i) q^{65} +(-2.50000 - 4.33013i) q^{67} +14.0000 q^{71} +(-0.500000 - 0.866025i) q^{73} +(0.500000 - 0.866025i) q^{75} +(-4.00000 + 3.46410i) q^{77} +(-4.50000 + 7.79423i) q^{79} +(-0.500000 - 0.866025i) q^{81} -6.00000 q^{83} +(2.00000 - 3.46410i) q^{89} +(0.500000 + 2.59808i) q^{91} +(-4.50000 + 7.79423i) q^{93} +(1.50000 + 2.59808i) q^{95} -14.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{5} + q^{7} - q^{9}+O(q^{10}) $ 2 * q + q^3 + q^5 + q^7 - q^9	$ 2 q + q^{3} + q^{5} + q^{7} - q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} - 3 q^{19} - 4 q^{21} - q^{25} - 2 q^{27} + 9 q^{31} - 2 q^{33} + 5 q^{35} - 3 q^{37} + q^{39} + 4 q^{41} + 6 q^{43} + q^{45} + 6 q^{47} - 13 q^{49} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 2 q^{61} - 5 q^{63} + q^{65} - 5 q^{67} + 28 q^{71} - q^{73} + q^{75} - 8 q^{77} - 9 q^{79} - q^{81} - 12 q^{83} + 4 q^{89} + q^{91} - 9 q^{93} + 3 q^{95} - 28 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^5 + q^7 - q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 - 3 * q^19 - 4 * q^21 - q^25 - 2 * q^27 + 9 * q^31 - 2 * q^33 + 5 * q^35 - 3 * q^37 + q^39 + 4 * q^41 + 6 * q^43 + q^45 + 6 * q^47 - 13 * q^49 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 2 * q^61 - 5 * q^63 + q^65 - 5 * q^67 + 28 * q^71 - q^73 + q^75 - 8 * q^77 - 9 * q^79 - q^81 - 12 * q^83 + 4 * q^89 + q^91 - 9 * q^93 + 3 * q^95 - 28 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$	$1$

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
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		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
$21$	−2.00000	+	1.73205i	−0.436436	+	0.377964i
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		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
$31$	4.50000	+	7.79423i	0.808224	+	1.39988i	0.914093	+	0.405505i	$0.132904\pi$
$31$	4.50000	+	7.79423i	0.808224	+	1.39988i	−0.105869	+	0.994380i	$0.533762\pi$
$32$		0			0
$33$	−1.00000	+	1.73205i	−0.174078	+	0.301511i
$34$		0			0
$35$	2.50000	+	0.866025i	0.422577	+	0.146385i
$36$		0			0
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	−0.962580	−	0.270998i	$-0.912646\pi$
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	0.715981	+	0.698119i	$0.245980\pi$
$38$		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
$43$	3.00000			0.457496			0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000			0.457496			0.228748	+	0.973486i	$0.426537\pi$
$44$		0			0
$45$	0.500000	+	0.866025i	0.0745356	+	0.129099i
$46$		0			0
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	$-0.689164\pi$
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	0.997503	+	0.0706177i	$0.0224970\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$		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
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$		0			0
$63$	−2.50000	−	0.866025i	−0.314970	−	0.109109i
$64$		0			0
$65$	0.500000	−	0.866025i	0.0620174	−	0.107417i
$66$		0			0
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$	−0.500000	−	0.866025i	−0.0585206	−	0.101361i	0.835281	−	0.549823i	$-0.185305\pi$
$73$	−0.500000	−	0.866025i	−0.0585206	−	0.101361i	−0.893801	+	0.448463i	$0.851972\pi$
$74$		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
$79$	−4.50000	+	7.79423i	−0.506290	+	0.876919i	0.493684	+	0.869641i	$0.335650\pi$
$79$	−4.50000	+	7.79423i	−0.506290	+	0.876919i	−0.999974	+	0.00727784i	$0.997683\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.00000	−	3.46410i	0.212000	−	0.367194i	−0.740341	−	0.672232i	$-0.765336\pi$
$89$	2.00000	−	3.46410i	0.212000	−	0.367194i	0.952340	+	0.305038i	$0.0986691\pi$
$90$		0			0
$91$	0.500000	+	2.59808i	0.0524142	+	0.272352i
$92$		0			0
$93$	−4.50000	+	7.79423i	−0.466628	+	0.808224i
$94$		0			0
$95$	1.50000	+	2.59808i	0.153897	+	0.266557i
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	−2.00000			−0.201008
$100$		0			0
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	0.911479	−	0.411346i	$-0.134941\pi$
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	−0.811976	+	0.583691i	$0.801608\pi$
$102$		0			0
$103$	−1.50000	+	2.59808i	−0.147799	+	0.255996i	−0.930414	−	0.366511i	$-0.880552\pi$
$103$	−1.50000	+	2.59808i	−0.147799	+	0.255996i	0.782614	+	0.622507i	$0.213886\pi$
$104$		0			0
$105$	0.500000	+	2.59808i	0.0487950	+	0.253546i
$106$		0			0
$107$	8.00000	−	13.8564i	0.773389	−	1.33955i	−0.162306	−	0.986740i	$-0.551893\pi$
$107$	8.00000	−	13.8564i	0.773389	−	1.33955i	0.935695	−	0.352809i	$-0.114773\pi$
$108$		0			0
$109$	−6.50000	−	11.2583i	−0.622587	−	1.07835i	−0.989002	−	0.147901i	$-0.952748\pi$
$109$	−6.50000	−	11.2583i	−0.622587	−	1.07835i	0.366415	−	0.930451i	$-0.380585\pi$
$110$		0			0
$111$	−3.00000			−0.284747
$112$		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
$115$		0			0
$116$		0			0
$117$	−0.500000	+	0.866025i	−0.0462250	+	0.0800641i
$118$		0			0
$119$		0			0
$120$		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
$127$	11.0000			0.976092			0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000			0.976092			0.488046	+	0.872818i	$0.337710\pi$
$128$		0			0
$129$	1.50000	+	2.59808i	0.132068	+	0.228748i
$130$		0			0
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	−0.141865	−	0.989886i	$-0.545310\pi$
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	0.928199	−	0.372084i	$-0.121357\pi$
$132$		0			0
$133$	−7.50000	−	2.59808i	−0.650332	−	0.225282i
$134$		0			0
$135$	−0.500000	+	0.866025i	−0.0430331	+	0.0745356i
$136$		0			0
$137$	−4.00000	−	6.92820i	−0.341743	−	0.591916i	0.643013	−	0.765855i	$-0.277684\pi$
$137$	−4.00000	−	6.92820i	−0.341743	−	0.591916i	−0.984757	+	0.173939i	$0.944351\pi$
$138$		0			0
$139$	−9.00000			−0.763370			−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000			−0.763370			−0.381685	+	0.924292i	$0.624656\pi$
$140$		0			0
$141$	6.00000			0.505291
$142$		0			0
$143$	1.00000	+	1.73205i	0.0836242	+	0.144841i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.50000	−	4.33013i	−0.453632	−	0.357143i
$148$		0			0
$149$	10.0000	−	17.3205i	0.819232	−	1.41895i	−0.0870170	−	0.996207i	$-0.527733\pi$
$149$	10.0000	−	17.3205i	0.819232	−	1.41895i	0.906249	−	0.422744i	$-0.138933\pi$
$150$		0			0
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	0.982873	+	0.184284i	$0.0589965\pi$
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	−0.331842	+	0.943335i	$0.607670\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.00000			0.722897
$156$		0			0
$157$	−5.00000	−	8.66025i	−0.399043	−	0.691164i	0.594565	−	0.804048i	$-0.297324\pi$
$157$	−5.00000	−	8.66025i	−0.399043	−	0.691164i	−0.993608	+	0.112884i	$0.963991\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	−	10.3923i	0.469956	−	0.813988i	−0.529454	−	0.848339i	$-0.677603\pi$
$163$	6.00000	−	10.3923i	0.469956	−	0.813988i	0.999410	+	0.0343508i	$0.0109363\pi$
$164$		0			0
$165$	1.00000	+	1.73205i	0.0778499	+	0.134840i
$166$		0			0
$167$	−18.0000			−1.39288			−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000			−1.39288			−0.696441	+	0.717614i	$0.745234\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$	−1.50000	−	2.59808i	−0.114708	−	0.198680i
$172$		0			0
$173$	−2.00000	+	3.46410i	−0.152057	+	0.263371i	−0.931984	−	0.362500i	$-0.881923\pi$
$173$	−2.00000	+	3.46410i	−0.152057	+	0.263371i	0.779926	+	0.625871i	$0.215256\pi$
$174$		0			0
$175$	2.00000	−	1.73205i	0.151186	−	0.130931i
$176$		0			0
$177$	−2.00000	+	3.46410i	−0.150329	+	0.260378i
$178$		0			0
$179$	−5.00000	−	8.66025i	−0.373718	−	0.647298i	0.616417	−	0.787420i	$-0.288584\pi$
$179$	−5.00000	−	8.66025i	−0.373718	−	0.647298i	−0.990134	+	0.140122i	$0.955250\pi$
$180$		0			0
$181$	25.0000			1.85824			0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000			1.85824			0.929118	+	0.369784i	$0.120568\pi$
$182$		0			0
$183$	−2.00000			−0.147844
$184$		0			0
$185$	1.50000	+	2.59808i	0.110282	+	0.191014i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.500000	−	2.59808i	−0.0363696	−	0.188982i
$190$		0			0
$191$	11.0000	−	19.0526i	0.795932	−	1.37859i	−0.126314	−	0.991990i	$-0.540315\pi$
$191$	11.0000	−	19.0526i	0.795932	−	1.37859i	0.922246	−	0.386604i	$-0.126352\pi$
$192$		0			0
$193$	−11.5000	−	19.9186i	−0.827788	−	1.43377i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−11.5000	−	19.9186i	−0.827788	−	1.43377i	0.0719816	−	0.997406i	$-0.477068\pi$
$194$		0			0
$195$	1.00000			0.0716115
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	12.0000	+	20.7846i	0.850657	+	1.47338i	0.880616	+	0.473831i	$0.157129\pi$
$199$	12.0000	+	20.7846i	0.850657	+	1.47338i	−0.0299585	+	0.999551i	$0.509538\pi$
$200$		0			0
$201$	2.50000	−	4.33013i	0.176336	−	0.305424i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1.00000	−	1.73205i	0.0698430	−	0.120972i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000			−0.415029
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	7.00000	+	12.1244i	0.479632	+	0.830747i
$214$		0			0
$215$	1.50000	−	2.59808i	0.102299	−	0.177187i
$216$		0			0
$217$	−18.0000	+	15.5885i	−1.22192	+	1.05821i
$218$		0			0
$219$	0.500000	−	0.866025i	0.0337869	−	0.0585206i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−16.0000			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−13.0000	−	22.5167i	−0.862840	−	1.49448i	−0.869176	−	0.494503i	$-0.835350\pi$
$227$	−13.0000	−	22.5167i	−0.862840	−	1.49448i	0.00633544	−	0.999980i	$-0.497983\pi$
$228$		0			0
$229$	11.5000	−	19.9186i	0.759941	−	1.31626i	−0.182939	−	0.983124i	$-0.558561\pi$
$229$	11.5000	−	19.9186i	0.759941	−	1.31626i	0.942880	−	0.333133i	$-0.108106\pi$
$230$		0			0
$231$	−5.00000	−	1.73205i	−0.328976	−	0.113961i
$232$		0			0
$233$	11.0000	−	19.0526i	0.720634	−	1.24817i	−0.240112	−	0.970745i	$-0.577184\pi$
$233$	11.0000	−	19.0526i	0.720634	−	1.24817i	0.960746	−	0.277429i	$-0.0894825\pi$
$234$		0			0
$235$	−3.00000	−	5.19615i	−0.195698	−	0.338960i
$236$		0			0
$237$	−9.00000			−0.584613
$238$		0			0
$239$	−22.0000			−1.42306			−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000			−1.42306			−0.711531	+	0.702655i	$0.751998\pi$
$240$		0			0
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	0.995509	+	0.0946700i	$0.0301796\pi$
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	−0.415768	+	0.909471i	$0.636487\pi$
$242$		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
$247$	−1.50000	+	2.59808i	−0.0954427	+	0.165312i
$248$		0			0
$249$	−3.00000	−	5.19615i	−0.190117	−	0.329293i
$250$		0			0
$251$	−16.0000			−1.00991			−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000			−1.00991			−0.504956	+	0.863145i	$0.668491\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	0.435970	+	0.899961i	$0.356405\pi$
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	−0.997374	+	0.0724199i	$0.976928\pi$
$258$		0			0
$259$	−7.50000	−	2.59808i	−0.466027	−	0.161437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	0.989561	−	0.144112i	$-0.0460326\pi$
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	−0.619586	+	0.784929i	$0.712699\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	4.00000			0.244796
$268$		0			0
$269$	1.00000	+	1.73205i	0.0609711	+	0.105605i	0.894900	−	0.446267i	$-0.147247\pi$
$269$	1.00000	+	1.73205i	0.0609711	+	0.105605i	−0.833929	+	0.551872i	$0.813914\pi$
$270$		0			0
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	−0.999870	−	0.0161307i	$-0.994865\pi$
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	0.513905	+	0.857847i	$0.328199\pi$
$272$		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	0.781094	−	0.624413i	$-0.214662\pi$
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$	−9.00000			−0.538816
$280$		0			0
$281$	8.00000			0.477240			0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000			0.477240			0.238620	+	0.971113i	$0.423305\pi$
$282$		0			0
$283$	−5.50000	−	9.52628i	−0.326941	−	0.566279i	0.654962	−	0.755662i	$-0.272685\pi$
$283$	−5.50000	−	9.52628i	−0.326941	−	0.566279i	−0.981903	+	0.189383i	$0.939351\pi$
$284$		0			0
$285$	−1.50000	+	2.59808i	−0.0888523	+	0.153897i
$286$		0			0
$287$	1.00000	+	5.19615i	0.0590281	+	0.306719i
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$	−7.00000	−	12.1244i	−0.410347	−	0.710742i
$292$		0			0
$293$	8.00000			0.467365			0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000			0.467365			0.233682	+	0.972313i	$0.424922\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$	−1.00000	−	1.73205i	−0.0580259	−	0.100504i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.50000	+	7.79423i	0.0864586	+	0.449252i
$302$		0			0
$303$	−1.00000	+	1.73205i	−0.0574485	+	0.0995037i
$304$		0			0
$305$	1.00000	+	1.73205i	0.0572598	+	0.0991769i
$306$		0			0
$307$	−15.0000			−0.856095			−0.428048	−	0.903756i	$-0.640798\pi$
$307$	−15.0000			−0.856095			−0.428048	+	0.903756i	$0.640798\pi$
$308$		0			0
$309$	−3.00000			−0.170664
$310$		0			0
$311$	7.00000	+	12.1244i	0.396934	+	0.687509i	0.993346	−	0.115169i	$-0.0367410\pi$
$311$	7.00000	+	12.1244i	0.396934	+	0.687509i	−0.596412	+	0.802678i	$0.703408\pi$
$312$		0			0
$313$	6.50000	−	11.2583i	0.367402	−	0.636358i	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	6.50000	−	11.2583i	0.367402	−	0.636358i	0.989158	+	0.146852i	$0.0469141\pi$
$314$		0			0
$315$	−2.00000	+	1.73205i	−0.112687	+	0.0975900i
$316$		0			0
$317$	−6.00000	+	10.3923i	−0.336994	+	0.583690i	−0.983866	−	0.178908i	$-0.942743\pi$
$317$	−6.00000	+	10.3923i	−0.336994	+	0.583690i	0.646872	+	0.762598i	$0.276077\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	16.0000			0.893033
$322$		0			0
$323$		0			0
$324$		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
$329$	15.0000	+	5.19615i	0.826977	+	0.286473i
$330$		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
$333$	−1.50000	−	2.59808i	−0.0821995	−	0.142374i
$334$		0			0
$335$	−5.00000			−0.273179
$336$		0			0
$337$	1.00000			0.0544735			0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000			0.0544735			0.0272367	+	0.999629i	$0.491329\pi$
$338$		0			0
$339$	3.00000	+	5.19615i	0.162938	+	0.282216i
$340$		0			0
$341$	−9.00000	+	15.5885i	−0.487377	+	0.844162i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	0.914702	−	0.404128i	$-0.132425\pi$
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	−0.807337	+	0.590091i	$0.799092\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$	−1.00000			−0.0533761
$352$		0			0
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	0.920677	+	0.390324i	$0.127637\pi$
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	−0.122308	+	0.992492i	$0.539030\pi$
$354$		0			0
$355$	7.00000	−	12.1244i	0.371521	−	0.643494i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	5.00000	+	8.66025i	0.263158	+	0.455803i
$362$		0			0
$363$	7.00000			0.367405
$364$		0			0
$365$	−1.00000			−0.0523424
$366$		0			0
$367$	17.5000	+	30.3109i	0.913493	+	1.58222i	0.809093	+	0.587680i	$0.199959\pi$
$367$	17.5000	+	30.3109i	0.913493	+	1.58222i	0.104399	+	0.994535i	$0.466708\pi$
$368$		0			0
$369$	−1.00000	+	1.73205i	−0.0520579	+	0.0901670i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	16.5000	−	28.5788i	0.854338	−	1.47976i	−0.0229205	−	0.999737i	$-0.507296\pi$
$373$	16.5000	−	28.5788i	0.854338	−	1.47976i	0.877258	−	0.480019i	$-0.159370\pi$
$374$		0			0
$375$	−0.500000	−	0.866025i	−0.0258199	−	0.0447214i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.00000			0.0513665			0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000			0.0513665			0.0256833	+	0.999670i	$0.491824\pi$
$380$		0			0
$381$	5.50000	+	9.52628i	0.281774	+	0.488046i
$382$		0			0
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	0.377531	+	0.925997i	$0.376773\pi$
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	−0.990702	+	0.136047i	$0.956560\pi$
$384$		0			0
$385$	1.00000	+	5.19615i	0.0509647	+	0.264820i
$386$		0			0
$387$	−1.50000	+	2.59808i	−0.0762493	+	0.132068i
$388$		0			0
$389$	7.00000	+	12.1244i	0.354914	+	0.614729i	0.987103	−	0.160085i	$-0.0511768\pi$
$389$	7.00000	+	12.1244i	0.354914	+	0.614729i	−0.632189	+	0.774814i	$0.717843\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	18.0000			0.907980
$394$		0			0
$395$	4.50000	+	7.79423i	0.226420	+	0.392170i
$396$		0			0
$397$	−3.50000	+	6.06218i	−0.175660	+	0.304252i	−0.940389	−	0.340099i	$-0.889539\pi$
$397$	−3.50000	+	6.06218i	−0.175660	+	0.304252i	0.764730	+	0.644351i	$0.222873\pi$
$398$		0			0
$399$	−1.50000	−	7.79423i	−0.0750939	−	0.390199i
$400$		0			0
$401$	12.0000	−	20.7846i	0.599251	−	1.03793i	−0.393680	−	0.919247i	$-0.628798\pi$
$401$	12.0000	−	20.7846i	0.599251	−	1.03793i	0.992932	−	0.118686i	$-0.0378683\pi$
$402$		0			0
$403$	4.50000	+	7.79423i	0.224161	+	0.388258i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	−6.00000			−0.297409
$408$		0			0
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	0.999402	−	0.0345902i	$-0.0110126\pi$
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	−0.529657	+	0.848212i	$0.677679\pi$
$410$		0			0
$411$	4.00000	−	6.92820i	0.197305	−	0.341743i
$412$		0			0
$413$	−8.00000	+	6.92820i	−0.393654	+	0.340915i
$414$		0			0
$415$	−3.00000	+	5.19615i	−0.147264	+	0.255069i
$416$		0			0
$417$	−4.50000	−	7.79423i	−0.220366	−	0.381685i
$418$		0			0
$419$	−30.0000			−1.46560			−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000			−1.46560			−0.732798	+	0.680446i	$0.761786\pi$
$420$		0			0
$421$	5.00000			0.243685			0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000			0.243685			0.121843	+	0.992549i	$0.461120\pi$
$422$		0			0
$423$	3.00000	+	5.19615i	0.145865	+	0.252646i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−5.00000	−	1.73205i	−0.241967	−	0.0838198i
$428$		0			0
$429$	−1.00000	+	1.73205i	−0.0482805	+	0.0836242i
$430$		0			0
$431$	17.0000	+	29.4449i	0.818861	+	1.41831i	0.906522	+	0.422159i	$0.138728\pi$
$431$	17.0000	+	29.4449i	0.818861	+	1.41831i	−0.0876607	+	0.996150i	$0.527939\pi$
$432$		0			0
$433$	3.00000			0.144171			0.0720854	−	0.997398i	$-0.477035\pi$
$433$	3.00000			0.144171			0.0720854	+	0.997398i	$0.477035\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$		0			0
$441$	1.00000	−	6.92820i	0.0476190	−	0.329914i
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$	−2.00000	−	3.46410i	−0.0948091	−	0.164214i
$446$		0			0
$447$	20.0000			0.945968
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		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
$455$	2.50000	+	0.866025i	0.117202	+	0.0405999i
$456$		0			0
$457$	−8.50000	+	14.7224i	−0.397613	+	0.688686i	−0.993431	−	0.114433i	$-0.963495\pi$
$457$	−8.50000	+	14.7224i	−0.397613	+	0.688686i	0.595818	+	0.803120i	$0.296828\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−24.0000			−1.11779			−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000			−1.11779			−0.558896	+	0.829238i	$0.688775\pi$
$462$		0			0
$463$	−19.0000			−0.883005			−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000			−0.883005			−0.441502	+	0.897260i	$0.645554\pi$
$464$		0			0
$465$	4.50000	+	7.79423i	0.208683	+	0.361449i
$466$		0			0
$467$	9.00000	−	15.5885i	0.416470	−	0.721348i	−0.579111	−	0.815249i	$-0.696600\pi$
$467$	9.00000	−	15.5885i	0.416470	−	0.721348i	0.995582	+	0.0939008i	$0.0299336\pi$
$468$		0			0
$469$	10.0000	−	8.66025i	0.461757	−	0.399893i
$470$		0			0
$471$	5.00000	−	8.66025i	0.230388	−	0.399043i
$472$		0			0
$473$	3.00000	+	5.19615i	0.137940	+	0.238919i
$474$		0			0
$475$	3.00000			0.137649
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	0.998392	+	0.0566937i	$0.0180558\pi$
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	−0.450098	+	0.892979i	$0.648611\pi$
$480$		0			0
$481$	−1.50000	+	2.59808i	−0.0683941	+	0.118462i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.00000	+	12.1244i	−0.317854	+	0.550539i
$486$		0			0
$487$	−2.50000	−	4.33013i	−0.113286	−	0.196217i	0.803807	−	0.594890i	$-0.202804\pi$
$487$	−2.50000	−	4.33013i	−0.113286	−	0.196217i	−0.917093	+	0.398673i	$0.869471\pi$
$488$		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
$497$	7.00000	+	36.3731i	0.313993	+	1.63156i
$498$		0			0
$499$	0.500000	−	0.866025i	0.0223831	−	0.0387686i	−0.854617	−	0.519259i	$-0.826208\pi$
$499$	0.500000	−	0.866025i	0.0223831	−	0.0387686i	0.877000	+	0.480490i	$0.159541\pi$
$500$		0			0
$501$	−9.00000	−	15.5885i	−0.402090	−	0.696441i
$502$		0			0
$503$	−22.0000			−0.980932			−0.490466	−	0.871460i	$-0.663173\pi$
$503$	−22.0000			−0.980932			−0.490466	+	0.871460i	$0.663173\pi$
$504$		0			0
$505$	2.00000			0.0889988
$506$		0			0
$507$	−6.00000	−	10.3923i	−0.266469	−	0.461538i
$508$		0			0
$509$	−15.0000	+	25.9808i	−0.664863	+	1.15158i	0.314459	+	0.949271i	$0.398177\pi$
$509$	−15.0000	+	25.9808i	−0.664863	+	1.15158i	−0.979322	+	0.202306i	$0.935156\pi$
$510$		0			0
$511$	2.00000	−	1.73205i	0.0884748	−	0.0766214i
$512$		0			0
$513$	1.50000	−	2.59808i	0.0662266	−	0.114708i
$514$		0			0
$515$	1.50000	+	2.59808i	0.0660979	+	0.114485i
$516$		0			0
$517$	12.0000			0.527759
$518$		0			0
$519$	−4.00000			−0.175581
$520$		0			0
$521$	−6.00000	−	10.3923i	−0.262865	−	0.455295i	0.704137	−	0.710064i	$-0.251334\pi$
$521$	−6.00000	−	10.3923i	−0.262865	−	0.455295i	−0.967002	+	0.254769i	$0.918001\pi$
$522$		0			0
$523$	−0.500000	+	0.866025i	−0.0218635	+	0.0378686i	−0.876750	−	0.480946i	$-0.840293\pi$
$523$	−0.500000	+	0.866025i	−0.0218635	+	0.0378686i	0.854887	+	0.518815i	$0.173627\pi$
$524$		0			0
$525$	2.50000	+	0.866025i	0.109109	+	0.0377964i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$	2.00000			0.0866296
$534$		0			0
$535$	−8.00000	−	13.8564i	−0.345870	−	0.599065i
$536$		0			0
$537$	5.00000	−	8.66025i	0.215766	−	0.373718i
$538$		0			0
$539$	−11.0000	−	8.66025i	−0.473804	−	0.373024i
$540$		0			0
$541$	−16.5000	+	28.5788i	−0.709390	+	1.22870i	0.255693	+	0.966758i	$0.417696\pi$
$541$	−16.5000	+	28.5788i	−0.709390	+	1.22870i	−0.965084	+	0.261942i	$0.915637\pi$
$542$		0			0
$543$	12.5000	+	21.6506i	0.536426	+	0.929118i
$544$		0			0
$545$	−13.0000			−0.556859
$546$		0			0
$547$	44.0000			1.88130			0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000			1.88130			0.940652	+	0.339372i	$0.110215\pi$
$548$		0			0
$549$	−1.00000	−	1.73205i	−0.0426790	−	0.0739221i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−22.5000	−	7.79423i	−0.956797	−	0.331444i
$554$		0			0
$555$	−1.50000	+	2.59808i	−0.0636715	+	0.110282i
$556$		0			0
$557$	−3.00000	−	5.19615i	−0.127114	−	0.220168i	0.795443	−	0.606028i	$-0.207238\pi$
$557$	−3.00000	−	5.19615i	−0.127114	−	0.220168i	−0.922557	+	0.385860i	$0.873905\pi$
$558$		0			0
$559$	3.00000			0.126886
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.00000	+	1.73205i	0.0421450	+	0.0729972i	0.886328	−	0.463057i	$-0.153248\pi$
$563$	1.00000	+	1.73205i	0.0421450	+	0.0729972i	−0.844183	+	0.536054i	$0.819914\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$		0			0
$567$	2.00000	−	1.73205i	0.0839921	−	0.0727393i
$568$		0			0
$569$	19.0000	−	32.9090i	0.796521	−	1.37962i	−0.125347	−	0.992113i	$-0.540004\pi$
$569$	19.0000	−	32.9090i	0.796521	−	1.37962i	0.921869	−	0.387503i	$-0.126662\pi$
$570$		0			0
$571$	−7.50000	−	12.9904i	−0.313865	−	0.543631i	0.665330	−	0.746549i	$-0.268291\pi$
$571$	−7.50000	−	12.9904i	−0.313865	−	0.543631i	−0.979196	+	0.202919i	$0.934957\pi$
$572$		0			0
$573$	22.0000			0.919063
$574$		0			0
$575$		0			0
$576$		0			0
$577$	6.50000	+	11.2583i	0.270599	+	0.468690i	0.969015	−	0.247001i	$-0.0794451\pi$
$577$	6.50000	+	11.2583i	0.270599	+	0.468690i	−0.698417	+	0.715691i	$0.746112\pi$
$578$		0			0
$579$	11.5000	−	19.9186i	0.477924	−	0.827788i
$580$		0			0
$581$	−3.00000	−	15.5885i	−0.124461	−	0.646718i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	0.500000	+	0.866025i	0.0206725	+	0.0358057i
$586$		0			0
$587$	−28.0000			−1.15568			−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000			−1.15568			−0.577842	+	0.816149i	$0.696105\pi$
$588$		0			0
$589$	−27.0000			−1.11252
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−19.0000	+	32.9090i	−0.780236	+	1.35141i	0.151567	+	0.988447i	$0.451568\pi$
$593$	−19.0000	+	32.9090i	−0.780236	+	1.35141i	−0.931804	+	0.362962i	$0.881765\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−12.0000	+	20.7846i	−0.491127	+	0.850657i
$598$		0			0
$599$	6.00000	+	10.3923i	0.245153	+	0.424618i	0.962175	−	0.272433i	$-0.0878284\pi$
$599$	6.00000	+	10.3923i	0.245153	+	0.424618i	−0.717021	+	0.697051i	$0.754495\pi$
$600$		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
$603$	5.00000			0.203616
$604$		0			0
$605$	−3.50000	−	6.06218i	−0.142295	−	0.246463i
$606$		0			0
$607$	21.5000	−	37.2391i	0.872658	−	1.51149i	0.0134214	−	0.999910i	$-0.495728\pi$
$607$	21.5000	−	37.2391i	0.872658	−	1.51149i	0.859237	−	0.511578i	$-0.170939\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.00000	−	5.19615i	0.121367	−	0.210214i
$612$		0			0
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	−0.999574	−	0.0291886i	$-0.990708\pi$
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	0.474509	−	0.880251i	$-0.342626\pi$
$614$		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
$619$	15.5000	+	26.8468i	0.622998	+	1.07906i	0.988924	+	0.148420i	$0.0474187\pi$
$619$	15.5000	+	26.8468i	0.622998	+	1.07906i	−0.365927	+	0.930644i	$0.619248\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.0000	+	3.46410i	0.400642	+	0.138786i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−3.00000	−	5.19615i	−0.119808	−	0.207514i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	24.0000			0.955425			0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000			0.955425			0.477712	+	0.878516i	$0.341466\pi$
$632$		0			0
$633$	2.00000	+	3.46410i	0.0794929	+	0.137686i
$634$		0			0
$635$	5.50000	−	9.52628i	0.218261	−	0.378039i
$636$		0			0
$637$	−6.50000	+	2.59808i	−0.257539	+	0.102940i
$638$		0			0
$639$	−7.00000	+	12.1244i	−0.276916	+	0.479632i
$640$		0			0
$641$	−20.0000	−	34.6410i	−0.789953	−	1.36824i	−0.925995	−	0.377535i	$-0.876772\pi$
$641$	−20.0000	−	34.6410i	−0.789953	−	1.36824i	0.136043	−	0.990703i	$-0.456562\pi$
$642$		0			0
$643$	11.0000			0.433798			0.216899	−	0.976194i	$-0.430406\pi$
$643$	11.0000			0.433798			0.216899	+	0.976194i	$0.430406\pi$
$644$		0			0
$645$	3.00000			0.118125
$646$		0			0
$647$	−9.00000	−	15.5885i	−0.353827	−	0.612845i	0.633090	−	0.774078i	$-0.281786\pi$
$647$	−9.00000	−	15.5885i	−0.353827	−	0.612845i	−0.986916	+	0.161233i	$0.948453\pi$
$648$		0			0
$649$	−4.00000	+	6.92820i	−0.157014	+	0.271956i
$650$		0			0
$651$	−22.5000	−	7.79423i	−0.881845	−	0.305480i
$652$		0			0
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	−0.918736	−	0.394872i	$-0.870789\pi$
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	0.801337	+	0.598213i	$0.204122\pi$
$654$		0			0
$655$	−9.00000	−	15.5885i	−0.351659	−	0.609091i
$656$		0			0
$657$	1.00000			0.0390137
$658$		0			0
$659$	−40.0000			−1.55818			−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000			−1.55818			−0.779089	+	0.626913i	$0.784318\pi$
$660$		0			0
$661$	−17.5000	−	30.3109i	−0.680671	−	1.17896i	−0.974776	−	0.223184i	$-0.928355\pi$
$661$	−17.5000	−	30.3109i	−0.680671	−	1.17896i	0.294105	−	0.955773i	$-0.404978\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000	+	5.19615i	−0.232670	+	0.201498i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−8.00000	−	13.8564i	−0.309298	−	0.535720i
$670$		0			0
$671$	−4.00000			−0.154418
$672$		0			0
$673$	−13.0000			−0.501113			−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000			−0.501113			−0.250557	+	0.968102i	$0.580614\pi$
$674$		0			0
$675$	0.500000	+	0.866025i	0.0192450	+	0.0333333i
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−7.00000	−	36.3731i	−0.268635	−	1.39587i
$680$		0			0
$681$	13.0000	−	22.5167i	0.498161	−	0.862840i
$682$		0			0
$683$	22.0000	+	38.1051i	0.841807	+	1.45805i	0.888366	+	0.459136i	$0.151841\pi$
$683$	22.0000	+	38.1051i	0.841807	+	1.45805i	−0.0465592	+	0.998916i	$0.514826\pi$
$684$		0			0
$685$	−8.00000			−0.305664
$686$		0			0
$687$	23.0000			0.877505
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−8.50000	+	14.7224i	−0.323355	+	0.560068i	−0.981178	−	0.193105i	$-0.938144\pi$
$691$	−8.50000	+	14.7224i	−0.323355	+	0.560068i	0.657823	+	0.753173i	$0.271478\pi$
$692$		0			0
$693$	−1.00000	−	5.19615i	−0.0379869	−	0.197386i
$694$		0			0
$695$	−4.50000	+	7.79423i	−0.170695	+	0.295652i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	22.0000			0.832116
$700$		0			0
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$		0			0
$703$	−4.50000	−	7.79423i	−0.169721	−	0.293965i
$704$		0			0
$705$	3.00000	−	5.19615i	0.112987	−	0.195698i
$706$		0			0
$707$	−4.00000	+	3.46410i	−0.150435	+	0.130281i
$708$		0			0
$709$	13.0000	−	22.5167i	0.488225	−	0.845631i	−0.511683	−	0.859174i	$-0.670978\pi$
$709$	13.0000	−	22.5167i	0.488225	−	0.845631i	0.999908	+	0.0135434i	$0.00431112\pi$
$710$		0			0
$711$	−4.50000	−	7.79423i	−0.168763	−	0.292306i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	2.00000			0.0747958
$716$		0			0
$717$	−11.0000	−	19.0526i	−0.410803	−	0.711531i
$718$		0			0
$719$	−1.00000	+	1.73205i	−0.0372937	+	0.0645946i	−0.884070	−	0.467355i	$-0.845207\pi$
$719$	−1.00000	+	1.73205i	−0.0372937	+	0.0645946i	0.846776	+	0.531949i	$0.178540\pi$
$720$		0			0
$721$	−7.50000	−	2.59808i	−0.279315	−	0.0967574i
$722$		0			0
$723$	−9.00000	+	15.5885i	−0.334714	+	0.579741i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	17.0000			0.630495			0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000			0.630495			0.315248	+	0.949009i	$0.397912\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$	15.5000	−	26.8468i	0.572506	−	0.991609i	−0.423802	−	0.905755i	$-0.639305\pi$
$733$	15.5000	−	26.8468i	0.572506	−	0.991609i	0.996308	−	0.0858539i	$-0.0273618\pi$
$734$		0			0
$735$	−6.50000	+	2.59808i	−0.239756	+	0.0958315i
$736$		0			0
$737$	5.00000	−	8.66025i	0.184177	−	0.319005i
$738$		0			0
$739$	−5.50000	−	9.52628i	−0.202321	−	0.350430i	0.746955	−	0.664875i	$-0.231515\pi$
$739$	−5.50000	−	9.52628i	−0.202321	−	0.350430i	−0.949276	+	0.314445i	$0.898182\pi$
$740$		0			0
$741$	−3.00000			−0.110208
$742$		0			0
$743$	−30.0000			−1.10059			−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000			−1.10059			−0.550297	+	0.834969i	$0.685485\pi$
$744$		0			0
$745$	−10.0000	−	17.3205i	−0.366372	−	0.634574i
$746$		0			0
$747$	3.00000	−	5.19615i	0.109764	−	0.190117i
$748$		0			0
$749$	40.0000	+	13.8564i	1.46157	+	0.506302i
$750$		0			0
$751$	−9.50000	+	16.4545i	−0.346660	+	0.600433i	−0.985654	−	0.168779i	$-0.946018\pi$
$751$	−9.50000	+	16.4545i	−0.346660	+	0.600433i	0.638994	+	0.769212i	$0.279351\pi$
$752$		0			0
$753$	−8.00000	−	13.8564i	−0.291536	−	0.504956i
$754$		0			0
$755$	16.0000			0.582300
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.00000	+	6.92820i	−0.145000	+	0.251147i	−0.929373	−	0.369142i	$-0.879652\pi$
$761$	−4.00000	+	6.92820i	−0.145000	+	0.251147i	0.784373	+	0.620289i	$0.212985\pi$
$762$		0			0
$763$	26.0000	−	22.5167i	0.941263	−	0.815158i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.00000	+	3.46410i	0.0722158	+	0.125081i
$768$		0			0
$769$	−25.0000			−0.901523			−0.450762	−	0.892644i	$-0.648848\pi$
$769$	−25.0000			−0.901523			−0.450762	+	0.892644i	$0.648848\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$	−17.0000	−	29.4449i	−0.611448	−	1.05906i	−0.990997	−	0.133887i	$-0.957254\pi$
$773$	−17.0000	−	29.4449i	−0.611448	−	1.05906i	0.379549	−	0.925172i	$-0.376079\pi$
$774$		0			0
$775$	4.50000	−	7.79423i	0.161645	−	0.279977i
$776$		0			0
$777$	−1.50000	−	7.79423i	−0.0538122	−	0.279616i
$778$		0			0
$779$	−3.00000	+	5.19615i	−0.107486	+	0.186171i
$780$		0			0
$781$	14.0000	+	24.2487i	0.500959	+	0.867687i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.0000			−0.356915
$786$		0			0
$787$	−4.00000	−	6.92820i	−0.142585	−	0.246964i	0.785885	−	0.618373i	$-0.212208\pi$
$787$	−4.00000	−	6.92820i	−0.142585	−	0.246964i	−0.928469	+	0.371409i	$0.878875\pi$
$788$		0			0
$789$	−6.00000	+	10.3923i	−0.213606	+	0.369976i
$790$		0			0
$791$	3.00000	+	15.5885i	0.106668	+	0.554262i
$792$		0			0
$793$	−1.00000	+	1.73205i	−0.0355110	+	0.0615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	32.0000			1.13350			0.566749	−	0.823890i	$-0.308201\pi$
$797$	32.0000			1.13350			0.566749	+	0.823890i	$0.308201\pi$
$798$		0			0
$799$		0			0
$800$		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
$805$		0			0
$806$		0			0
$807$	−1.00000	+	1.73205i	−0.0352017	+	0.0609711i
$808$		0			0
$809$	−5.00000	−	8.66025i	−0.175791	−	0.304478i	0.764644	−	0.644453i	$-0.222915\pi$
$809$	−5.00000	−	8.66025i	−0.175791	−	0.304478i	−0.940435	+	0.339975i	$0.889582\pi$
$810$		0			0
$811$	−40.0000			−1.40459			−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000			−1.40459			−0.702295	+	0.711886i	$0.747841\pi$
$812$		0			0
$813$	−16.0000			−0.561144
$814$		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
$819$	−2.50000	−	0.866025i	−0.0873571	−	0.0302614i
$820$		0			0
$821$	−1.00000	+	1.73205i	−0.0349002	+	0.0604490i	−0.882948	−	0.469471i	$-0.844445\pi$
$821$	−1.00000	+	1.73205i	−0.0349002	+	0.0604490i	0.848048	+	0.529920i	$0.177778\pi$
$822$		0			0
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	−0.997686	−	0.0679910i	$-0.978341\pi$
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	0.439961	−	0.898017i	$-0.354992\pi$
$824$		0			0
$825$	2.00000			0.0696311
$826$		0			0
$827$	50.0000			1.73867			0.869335	−	0.494223i	$-0.164547\pi$
$827$	50.0000			1.73867			0.869335	+	0.494223i	$0.164547\pi$
$828$		0			0
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	0.945589	−	0.325362i	$-0.105486\pi$
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	−0.754567	+	0.656223i	$0.772153\pi$
$830$		0			0
$831$	2.50000	−	4.33013i	0.0867240	−	0.150210i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−9.00000	+	15.5885i	−0.311458	+	0.539461i
$836$		0			0
$837$	−4.50000	−	7.79423i	−0.155543	−	0.269408i
$838$		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
$841$	−29.0000			−1.00000
$842$		0			0
$843$	4.00000	+	6.92820i	0.137767	+	0.238620i
$844$		0			0
$845$	−6.00000	+	10.3923i	−0.206406	+	0.357506i
$846$		0			0
$847$	17.5000	+	6.06218i	0.601307	+	0.208299i
$848$		0			0
$849$	5.50000	−	9.52628i	0.188760	−	0.326941i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	43.0000			1.47229			0.736146	−	0.676823i	$-0.236644\pi$
$853$	43.0000			1.47229			0.736146	+	0.676823i	$0.236644\pi$
$854$		0			0
$855$	−3.00000			−0.102598
$856$		0			0
$857$	−26.0000	−	45.0333i	−0.888143	−	1.53831i	−0.842068	−	0.539371i	$-0.818662\pi$
$857$	−26.0000	−	45.0333i	−0.888143	−	1.53831i	−0.0460748	−	0.998938i	$-0.514671\pi$
$858$		0			0
$859$	8.00000	−	13.8564i	0.272956	−	0.472774i	−0.696661	−	0.717400i	$-0.745332\pi$
$859$	8.00000	−	13.8564i	0.272956	−	0.472774i	0.969618	+	0.244626i	$0.0786652\pi$
$860$		0			0
$861$	−4.00000	+	3.46410i	−0.136320	+	0.118056i
$862$		0			0
$863$	17.0000	−	29.4449i	0.578687	−	1.00231i	−0.416944	−	0.908932i	$-0.636899\pi$
$863$	17.0000	−	29.4449i	0.578687	−	1.00231i	0.995630	−	0.0933825i	$-0.0297679\pi$
$864$		0			0
$865$	2.00000	+	3.46410i	0.0680020	+	0.117783i
$866$		0			0
$867$	17.0000			0.577350
$868$		0			0
$869$	−18.0000			−0.610608
$870$		0			0
$871$	−2.50000	−	4.33013i	−0.0847093	−	0.146721i
$872$		0			0
$873$	7.00000	−	12.1244i	0.236914	−	0.410347i
$874$		0			0
$875$	−0.500000	−	2.59808i	−0.0169031	−	0.0878310i
$876$		0			0
$877$	−1.00000	+	1.73205i	−0.0337676	+	0.0584872i	−0.882415	−	0.470471i	$-0.844084\pi$
$877$	−1.00000	+	1.73205i	−0.0337676	+	0.0584872i	0.848648	+	0.528958i	$0.177417\pi$
$878$		0			0
$879$	4.00000	+	6.92820i	0.134917	+	0.233682i
$880$		0			0
$881$	−52.0000			−1.75192			−0.875962	−	0.482380i	$-0.839773\pi$
$881$	−52.0000			−1.75192			−0.875962	+	0.482380i	$0.839773\pi$
$882$		0			0
$883$	29.0000			0.975928			0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000			0.975928			0.487964	+	0.872864i	$0.337740\pi$
$884$		0			0
$885$	2.00000	+	3.46410i	0.0672293	+	0.116445i
$886$		0			0
$887$	1.00000	−	1.73205i	0.0335767	−	0.0581566i	−0.848749	−	0.528796i	$-0.822644\pi$
$887$	1.00000	−	1.73205i	0.0335767	−	0.0581566i	0.882325	+	0.470640i	$0.155977\pi$
$888$		0			0
$889$	5.50000	+	28.5788i	0.184464	+	0.958503i
$890$		0			0
$891$	1.00000	−	1.73205i	0.0335013	−	0.0580259i
$892$		0			0
$893$	9.00000	+	15.5885i	0.301174	+	0.521648i
$894$		0			0
$895$	−10.0000			−0.334263
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−6.00000	+	5.19615i	−0.199667	+	0.172917i
$904$		0			0
$905$	12.5000	−	21.6506i	0.415514	−	0.719691i
$906$		0			0
$907$	1.50000	+	2.59808i	0.0498067	+	0.0862677i	0.889854	−	0.456246i	$-0.150806\pi$
$907$	1.50000	+	2.59808i	0.0498067	+	0.0862677i	−0.840047	+	0.542513i	$0.817473\pi$
$908$		0			0
$909$	−2.00000			−0.0663358
$910$		0			0
$911$	8.00000			0.265052			0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000			0.265052			0.132526	+	0.991180i	$0.457691\pi$
$912$		0			0
$913$	−6.00000	−	10.3923i	−0.198571	−	0.343935i
$914$		0			0
$915$	−1.00000	+	1.73205i	−0.0330590	+	0.0572598i
$916$		0			0
$917$	45.0000	+	15.5885i	1.48603	+	0.514776i
$918$		0			0
$919$	−12.5000	+	21.6506i	−0.412337	+	0.714189i	−0.995145	−	0.0984214i	$-0.968621\pi$
$919$	−12.5000	+	21.6506i	−0.412337	+	0.714189i	0.582808	+	0.812610i	$0.301954\pi$
$920$		0			0
$921$	−7.50000	−	12.9904i	−0.247133	−	0.428048i
$922$		0			0
$923$	14.0000			0.460816
$924$		0			0
$925$	3.00000			0.0986394
$926$		0			0
$927$	−1.50000	−	2.59808i	−0.0492665	−	0.0853320i
$928$		0			0
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	−0.679770	−	0.733426i	$-0.737920\pi$
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	0.975050	+	0.221985i	$0.0712536\pi$
$930$		0			0
$931$	3.00000	−	20.7846i	0.0983210	−	0.681188i
$932$		0			0
$933$	−7.00000	+	12.1244i	−0.229170	+	0.396934i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−5.00000			−0.163343			−0.0816714	−	0.996659i	$-0.526026\pi$
$937$	−5.00000			−0.163343			−0.0816714	+	0.996659i	$0.526026\pi$
$938$		0			0
$939$	13.0000			0.424239
$940$		0			0
$941$	−26.0000	−	45.0333i	−0.847576	−	1.46804i	−0.883365	−	0.468685i	$-0.844728\pi$
$941$	−26.0000	−	45.0333i	−0.847576	−	1.46804i	0.0357896	−	0.999359i	$-0.488605\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−2.50000	−	0.866025i	−0.0813250	−	0.0281718i
$946$		0			0
$947$	−19.0000	+	32.9090i	−0.617417	+	1.06940i	0.372538	+	0.928017i	$0.378488\pi$
$947$	−19.0000	+	32.9090i	−0.617417	+	1.06940i	−0.989955	+	0.141381i	$0.954846\pi$
$948$		0			0
$949$	−0.500000	−	0.866025i	−0.0162307	−	0.0281124i
$950$		0			0
$951$	−12.0000			−0.389127
$952$		0			0
$953$	−16.0000			−0.518291			−0.259145	−	0.965838i	$-0.583441\pi$
$953$	−16.0000			−0.518291			−0.259145	+	0.965838i	$0.583441\pi$
$954$		0			0
$955$	−11.0000	−	19.0526i	−0.355952	−	0.616526i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	16.0000	−	13.8564i	0.516667	−	0.447447i
$960$		0			0
$961$	−25.0000	+	43.3013i	−0.806452	+	1.39682i
$962$		0			0
$963$	8.00000	+	13.8564i	0.257796	+	0.446516i
$964$		0			0
$965$	−23.0000			−0.740396
$966$		0			0
$967$	49.0000			1.57573			0.787867	−	0.615846i	$-0.211185\pi$
$967$	49.0000			1.57573			0.787867	+	0.615846i	$0.211185\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8.00000	+	13.8564i	−0.256732	+	0.444673i	−0.965365	−	0.260905i	$-0.915979\pi$
$971$	−8.00000	+	13.8564i	−0.256732	+	0.444673i	0.708632	+	0.705578i	$0.249313\pi$
$972$		0			0
$973$	−4.50000	−	23.3827i	−0.144263	−	0.749614i
$974$		0			0
$975$	0.500000	−	0.866025i	0.0160128	−	0.0277350i
$976$		0			0
$977$	3.00000	+	5.19615i	0.0959785	+	0.166240i	0.910017	−	0.414572i	$-0.136069\pi$
$977$	3.00000	+	5.19615i	0.0959785	+	0.166240i	−0.814038	+	0.580812i	$0.802735\pi$
$978$		0			0
$979$	8.00000			0.255681
$980$		0			0
$981$	13.0000			0.415058
$982$		0			0
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	−0.996138	−	0.0878058i	$-0.972015\pi$
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	0.422027	−	0.906583i	$-0.361319\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	3.00000	+	15.5885i	0.0954911	+	0.496186i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−11.5000	−	19.9186i	−0.365310	−	0.632735i	0.623516	−	0.781810i	$-0.285704\pi$
$991$	−11.5000	−	19.9186i	−0.365310	−	0.632735i	−0.988826	+	0.149076i	$0.952370\pi$
$992$		0			0
$993$	−13.0000			−0.412543
$994$		0			0
$995$	24.0000			0.760851
$996$		0			0
$997$	−5.50000	−	9.52628i	−0.174187	−	0.301700i	0.765693	−	0.643206i	$-0.222396\pi$
$997$	−5.50000	−	9.52628i	−0.174187	−	0.301700i	−0.939880	+	0.341506i	$0.889063\pi$
$998$		0			0
$999$	1.50000	−	2.59808i	0.0474579	−	0.0821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.bg.e.121.1	✓	2
3.2	odd	2		2520.2.bi.c.1801.1		2
4.3	odd	2		1680.2.bg.h.961.1		2
7.2	even	3		5880.2.a.c.1.1		1
7.4	even	3	inner	840.2.bg.e.361.1	yes	2
7.5	odd	6		5880.2.a.bc.1.1		1
21.11	odd	6		2520.2.bi.c.361.1		2
28.11	odd	6		1680.2.bg.h.1201.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.bg.e.121.1	✓	2	1.1	even	1	trivial
840.2.bg.e.361.1	yes	2	7.4	even	3	inner
1680.2.bg.h.961.1		2	4.3	odd	2
1680.2.bg.h.1201.1		2	28.11	odd	6
2520.2.bi.c.361.1		2	21.11	odd	6
2520.2.bi.c.1801.1		2	3.2	odd	2
5880.2.a.c.1.1		1	7.2	even	3
5880.2.a.bc.1.1		1	7.5	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:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.bg (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +1.00000 q^{15} +(-1.50000 + 2.59808i) q^{19} +(-2.00000 + 1.73205i) q^{21} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +(4.50000 + 7.79423i) q^{31} +(-1.00000 + 1.73205i) q^{33} +(2.50000 + 0.866025i) q^{35} +(-1.50000 + 2.59808i) q^{37} +(0.500000 + 0.866025i) q^{39} +2.00000 q^{41} +3.00000 q^{43} +(0.500000 + 0.866025i) q^{45} +(3.00000 - 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +2.00000 q^{55} -3.00000 q^{57} +(2.00000 + 3.46410i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-2.50000 - 0.866025i) q^{63} +(0.500000 - 0.866025i) q^{65} +(-2.50000 - 4.33013i) q^{67} +14.0000 q^{71} +(-0.500000 - 0.866025i) q^{73} +(0.500000 - 0.866025i) q^{75} +(-4.00000 + 3.46410i) q^{77} +(-4.50000 + 7.79423i) q^{79} +(-0.500000 - 0.866025i) q^{81} -6.00000 q^{83} +(2.00000 - 3.46410i) q^{89} +(0.500000 + 2.59808i) q^{91} +(-4.50000 + 7.79423i) q^{93} +(1.50000 + 2.59808i) q^{95} -14.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} + q^{7} - q^{9}+O(q^{10}) \) 2 * q + q^3 + q^5 + q^7 - q^9	\( 2 q + q^{3} + q^{5} + q^{7} - q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} - 3 q^{19} - 4 q^{21} - q^{25} - 2 q^{27} + 9 q^{31} - 2 q^{33} + 5 q^{35} - 3 q^{37} + q^{39} + 4 q^{41} + 6 q^{43} + q^{45} + 6 q^{47} - 13 q^{49} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 2 q^{61} - 5 q^{63} + q^{65} - 5 q^{67} + 28 q^{71} - q^{73} + q^{75} - 8 q^{77} - 9 q^{79} - q^{81} - 12 q^{83} + 4 q^{89} + q^{91} - 9 q^{93} + 3 q^{95} - 28 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^5 + q^7 - q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 - 3 * q^19 - 4 * q^21 - q^25 - 2 * q^27 + 9 * q^31 - 2 * q^33 + 5 * q^35 - 3 * q^37 + q^39 + 4 * q^41 + 6 * q^43 + q^45 + 6 * q^47 - 13 * q^49 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 2 * q^61 - 5 * q^63 + q^65 - 5 * q^67 + 28 * q^71 - q^73 + q^75 - 8 * q^77 - 9 * q^79 - q^81 - 12 * q^83 + 4 * q^89 + q^91 - 9 * q^93 + 3 * q^95 - 28 * q^97 - 4 * q^99

Introduction

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