LMFDB - Embedded newform 840.2.bg.d.361.1

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			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	840.361
Dual form			840.2.bg.d.121.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.50000 + 2.59808i) q^{11} +1.00000 q^{13} +1.00000 q^{15} +(3.50000 + 6.06218i) q^{19} +(0.500000 + 2.59808i) q^{21} +(2.50000 + 4.33013i) q^{23} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +(-3.00000 + 5.19615i) q^{31} +(1.50000 + 2.59808i) q^{33} +(-2.50000 - 0.866025i) q^{35} +(-1.50000 - 2.59808i) q^{37} +(0.500000 - 0.866025i) q^{39} -3.00000 q^{41} +8.00000 q^{43} +(0.500000 - 0.866025i) q^{45} +(0.500000 + 0.866025i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-2.50000 + 4.33013i) q^{53} -3.00000 q^{55} +7.00000 q^{57} +(2.00000 - 3.46410i) q^{59} +(4.00000 + 6.92820i) q^{61} +(2.50000 + 0.866025i) q^{63} +(0.500000 + 0.866025i) q^{65} +5.00000 q^{69} -6.00000 q^{71} +(7.00000 - 12.1244i) q^{73} +(0.500000 + 0.866025i) q^{75} +(-1.50000 - 7.79423i) q^{77} +(8.00000 + 13.8564i) q^{79} +(-0.500000 + 0.866025i) q^{81} -16.0000 q^{83} +(-3.00000 - 5.19615i) q^{89} +(-2.00000 + 1.73205i) q^{91} +(3.00000 + 5.19615i) q^{93} +(-3.50000 + 6.06218i) q^{95} +16.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 3 q^{11} + 2 q^{13} + 2 q^{15} + 7 q^{19} + q^{21} + 5 q^{23} - q^{25} - 2 q^{27} - 6 q^{31} + 3 q^{33} - 5 q^{35} - 3 q^{37} + q^{39} - 6 q^{41} + 16 q^{43} + q^{45} + q^{47} + 2 q^{49} - 5 q^{53} - 6 q^{55} + 14 q^{57} + 4 q^{59} + 8 q^{61} + 5 q^{63} + q^{65} + 10 q^{69} - 12 q^{71} + 14 q^{73} + q^{75} - 3 q^{77} + 16 q^{79} - q^{81} - 32 q^{83} - 6 q^{89} - 4 q^{91} + 6 q^{93} - 7 q^{95} + 32 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^5 - 4 * q^7 - q^9 - 3 * q^11 + 2 * q^13 + 2 * q^15 + 7 * q^19 + q^21 + 5 * q^23 - q^25 - 2 * q^27 - 6 * q^31 + 3 * q^33 - 5 * q^35 - 3 * q^37 + q^39 - 6 * q^41 + 16 * q^43 + q^45 + q^47 + 2 * q^49 - 5 * q^53 - 6 * q^55 + 14 * q^57 + 4 * q^59 + 8 * q^61 + 5 * q^63 + q^65 + 10 * q^69 - 12 * q^71 + 14 * q^73 + q^75 - 3 * q^77 + 16 * q^79 - q^81 - 32 * q^83 - 6 * q^89 - 4 * q^91 + 6 * q^93 - 7 * q^95 + 32 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	−2.00000	+	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
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		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	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$	0.500000	+	2.59808i	0.109109	+	0.566947i
$22$		0			0
$23$	2.50000	+	4.33013i	0.521286	+	0.902894i	0.999694	+	0.0247559i	$0.00788087\pi$
$23$	2.50000	+	4.33013i	0.521286	+	0.902894i	−0.478407	+	0.878138i	$0.658786\pi$
$24$		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
$31$	−3.00000	+	5.19615i	−0.538816	+	0.933257i	0.460152	+	0.887840i	$0.347795\pi$
$31$	−3.00000	+	5.19615i	−0.538816	+	0.933257i	−0.998968	+	0.0454165i	$0.985539\pi$
$32$		0			0
$33$	1.50000	+	2.59808i	0.261116	+	0.452267i
$34$		0			0
$35$	−2.50000	−	0.866025i	−0.422577	−	0.146385i
$36$		0			0
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	0.715981	−	0.698119i	$-0.245980\pi$
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	−0.962580	+	0.270998i	$0.912646\pi$
$38$		0			0
$39$	0.500000	−	0.866025i	0.0800641	−	0.138675i
$40$		0			0
$41$	−3.00000			−0.468521			−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000			−0.468521			−0.234261	+	0.972174i	$0.575267\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$	0.500000	−	0.866025i	0.0745356	−	0.129099i
$46$		0			0
$47$	0.500000	+	0.866025i	0.0729325	+	0.126323i	0.900185	−	0.435507i	$-0.143431\pi$
$47$	0.500000	+	0.866025i	0.0729325	+	0.126323i	−0.827253	+	0.561830i	$0.810098\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.50000	+	4.33013i	−0.343401	+	0.594789i	−0.985062	−	0.172200i	$-0.944912\pi$
$53$	−2.50000	+	4.33013i	−0.343401	+	0.594789i	0.641661	+	0.766989i	$0.278246\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$	7.00000			0.927173
$58$		0			0
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	−0.705965	−	0.708247i	$-0.749486\pi$
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	0.966342	+	0.257260i	$0.0828195\pi$
$60$		0			0
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	0.999901	+	0.0140840i	$0.00448323\pi$
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	−0.487753	+	0.872982i	$0.662183\pi$
$62$		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
$67$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$67$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$68$		0			0
$69$	5.00000			0.601929
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	7.00000	−	12.1244i	0.819288	−	1.41905i	−0.0869195	−	0.996215i	$-0.527702\pi$
$73$	7.00000	−	12.1244i	0.819288	−	1.41905i	0.906208	−	0.422833i	$-0.138964\pi$
$74$		0			0
$75$	0.500000	+	0.866025i	0.0577350	+	0.100000i
$76$		0			0
$77$	−1.50000	−	7.79423i	−0.170941	−	0.888235i
$78$		0			0
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.827401	+	0.561611i	$0.189818\pi$
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.0726692	+	0.997356i	$0.476848\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	0.662071	−	0.749441i	$-0.269678\pi$
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	−0.980071	+	0.198650i	$0.936344\pi$
$90$		0			0
$91$	−2.00000	+	1.73205i	−0.209657	+	0.181568i
$92$		0			0
$93$	3.00000	+	5.19615i	0.311086	+	0.538816i
$94$		0			0
$95$	−3.50000	+	6.06218i	−0.359092	+	0.621966i
$96$		0			0
$97$	16.0000			1.62455			0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000			1.62455			0.812277	+	0.583272i	$0.198228\pi$
$98$		0			0
$99$	3.00000			0.301511
$100$		0			0
$101$	−4.00000	+	6.92820i	−0.398015	+	0.689382i	−0.993481	−	0.113998i	$-0.963634\pi$
$101$	−4.00000	+	6.92820i	−0.398015	+	0.689382i	0.595466	+	0.803380i	$0.296967\pi$
$102$		0			0
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	0.598858	−	0.800855i	$-0.295621\pi$
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	−0.992990	+	0.118199i	$0.962288\pi$
$104$		0			0
$105$	−2.00000	+	1.73205i	−0.195180	+	0.169031i
$106$		0			0
$107$	−7.00000	−	12.1244i	−0.676716	−	1.17211i	−0.975964	−	0.217931i	$-0.930069\pi$
$107$	−7.00000	−	12.1244i	−0.676716	−	1.17211i	0.299249	−	0.954175i	$-0.403264\pi$
$108$		0			0
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	0.909935	+	0.414751i	$0.136131\pi$
$110$		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
$115$	−2.50000	+	4.33013i	−0.233126	+	0.403786i
$116$		0			0
$117$	−0.500000	−	0.866025i	−0.0462250	−	0.0800641i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$	−1.50000	+	2.59808i	−0.135250	+	0.234261i
$124$		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
$129$	4.00000	−	6.92820i	0.352180	−	0.609994i
$130$		0			0
$131$	−3.50000	−	6.06218i	−0.305796	−	0.529655i	0.671642	−	0.740876i	$-0.265589\pi$
$131$	−3.50000	−	6.06218i	−0.305796	−	0.529655i	−0.977438	+	0.211221i	$0.932256\pi$
$132$		0			0
$133$	−17.5000	−	6.06218i	−1.51744	−	0.525657i
$134$		0			0
$135$	−0.500000	−	0.866025i	−0.0430331	−	0.0745356i
$136$		0			0
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	−0.820141	−	0.572161i	$-0.806105\pi$
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	0.905577	+	0.424182i	$0.139438\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	1.00000			0.0842152
$142$		0			0
$143$	−1.50000	+	2.59808i	−0.125436	+	0.217262i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.50000	−	4.33013i	−0.453632	−	0.357143i
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	0.426948	+	0.904276i	$0.359589\pi$
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	−0.996600	+	0.0823900i	$0.973745\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.00000			−0.481932
$156$		0			0
$157$	7.50000	−	12.9904i	0.598565	−	1.03675i	−0.394468	−	0.918910i	$-0.629071\pi$
$157$	7.50000	−	12.9904i	0.598565	−	1.03675i	0.993033	−	0.117836i	$-0.0375956\pi$
$158$		0			0
$159$	2.50000	+	4.33013i	0.198263	+	0.343401i
$160$		0			0
$161$	−12.5000	−	4.33013i	−0.985138	−	0.341262i
$162$		0			0
$163$	−4.00000	−	6.92820i	−0.313304	−	0.542659i	0.665771	−	0.746156i	$-0.268103\pi$
$163$	−4.00000	−	6.92820i	−0.313304	−	0.542659i	−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$	−1.50000	+	2.59808i	−0.116775	+	0.202260i
$166$		0			0
$167$	−23.0000			−1.77979			−0.889897	−	0.456162i	$-0.849224\pi$
$167$	−23.0000			−1.77979			−0.889897	+	0.456162i	$0.849224\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$	3.50000	−	6.06218i	0.267652	−	0.463586i
$172$		0			0
$173$	−4.50000	−	7.79423i	−0.342129	−	0.592584i	0.642699	−	0.766119i	$-0.277815\pi$
$173$	−4.50000	−	7.79423i	−0.342129	−	0.592584i	−0.984828	+	0.173534i	$0.944481\pi$
$174$		0			0
$175$	−0.500000	−	2.59808i	−0.0377964	−	0.196396i
$176$		0			0
$177$	−2.00000	−	3.46410i	−0.150329	−	0.260378i
$178$		0			0
$179$	7.50000	−	12.9904i	0.560576	−	0.970947i	−0.436870	−	0.899525i	$-0.643913\pi$
$179$	7.50000	−	12.9904i	0.560576	−	0.970947i	0.997446	−	0.0714220i	$-0.0227537\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$	8.00000			0.591377
$184$		0			0
$185$	1.50000	−	2.59808i	0.110282	−	0.191014i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	2.00000	−	1.73205i	0.145479	−	0.125988i
$190$		0			0
$191$	11.0000	+	19.0526i	0.795932	+	1.37859i	0.922246	+	0.386604i	$0.126352\pi$
$191$	11.0000	+	19.0526i	0.795932	+	1.37859i	−0.126314	+	0.991990i	$0.540315\pi$
$192$		0			0
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$	1.00000			0.0716115
$196$		0			0
$197$	15.0000			1.06871			0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000			1.06871			0.534353	+	0.845262i	$0.320555\pi$
$198$		0			0
$199$	12.0000	−	20.7846i	0.850657	−	1.47338i	−0.0299585	−	0.999551i	$-0.509538\pi$
$199$	12.0000	−	20.7846i	0.850657	−	1.47338i	0.880616	−	0.473831i	$-0.157129\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.50000	−	2.59808i	−0.104765	−	0.181458i
$206$		0			0
$207$	2.50000	−	4.33013i	0.173762	−	0.300965i
$208$		0			0
$209$	−21.0000			−1.45260
$210$		0			0
$211$	9.00000			0.619586			0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000			0.619586			0.309793	+	0.950804i	$0.399740\pi$
$212$		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
$217$	−3.00000	−	15.5885i	−0.203653	−	1.05821i
$218$		0			0
$219$	−7.00000	−	12.1244i	−0.473016	−	0.819288i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	4.00000			0.267860			0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000			0.267860			0.133930	+	0.990991i	$0.457240\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	12.0000	−	20.7846i	0.796468	−	1.37952i	−0.125435	−	0.992102i	$-0.540033\pi$
$227$	12.0000	−	20.7846i	0.796468	−	1.37952i	0.921903	−	0.387421i	$-0.126634\pi$
$228$		0			0
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	0.231287	−	0.972886i	$-0.425707\pi$
$230$		0			0
$231$	−7.50000	−	2.59808i	−0.493464	−	0.170941i
$232$		0			0
$233$	11.0000	+	19.0526i	0.720634	+	1.24817i	0.960746	+	0.277429i	$0.0894825\pi$
$233$	11.0000	+	19.0526i	0.720634	+	1.24817i	−0.240112	+	0.970745i	$0.577184\pi$
$234$		0			0
$235$	−0.500000	+	0.866025i	−0.0326164	+	0.0564933i
$236$		0			0
$237$	16.0000			1.03931
$238$		0			0
$239$	−2.00000			−0.129369			−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000			−0.129369			−0.0646846	+	0.997906i	$0.520604\pi$
$240$		0			0
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	−0.956456	−	0.291877i	$-0.905720\pi$
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	0.731001	+	0.682376i	$0.239053\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$		0			0
$245$	6.50000	−	2.59808i	0.415270	−	0.165985i
$246$		0			0
$247$	3.50000	+	6.06218i	0.222700	+	0.385727i
$248$		0			0
$249$	−8.00000	+	13.8564i	−0.506979	+	0.878114i
$250$		0			0
$251$	−31.0000			−1.95670			−0.978351	−	0.206951i	$-0.933646\pi$
$251$	−31.0000			−1.95670			−0.978351	+	0.206951i	$0.933646\pi$
$252$		0			0
$253$	−15.0000			−0.943042
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	0.990217	−	0.139533i	$-0.0445601\pi$
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	−0.615948	+	0.787787i	$0.711227\pi$
$258$		0			0
$259$	7.50000	+	2.59808i	0.466027	+	0.161437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	16.0000	−	27.7128i	0.986602	−	1.70885i	0.352014	−	0.935995i	$-0.385497\pi$
$263$	16.0000	−	27.7128i	0.986602	−	1.70885i	0.634588	−	0.772851i	$-0.281170\pi$
$264$		0			0
$265$	−5.00000			−0.307148
$266$		0			0
$267$	−6.00000			−0.367194
$268$		0			0
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	−0.623082	−	0.782157i	$-0.714120\pi$
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	0.988908	+	0.148527i	$0.0474530\pi$
$270$		0			0
$271$	12.0000	+	20.7846i	0.728948	+	1.26258i	0.957328	+	0.289003i	$0.0933238\pi$
$271$	12.0000	+	20.7846i	0.728948	+	1.26258i	−0.228380	+	0.973572i	$0.573343\pi$
$272$		0			0
$273$	0.500000	+	2.59808i	0.0302614	+	0.157243i
$274$		0			0
$275$	−1.50000	−	2.59808i	−0.0904534	−	0.156670i
$276$		0			0
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	0.675810	+	0.737075i	$0.263794\pi$
$278$		0			0
$279$	6.00000			0.359211
$280$		0			0
$281$	−17.0000			−1.01413			−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000			−1.01413			−0.507067	+	0.861906i	$0.669271\pi$
$282$		0			0
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	−0.579437	−	0.815017i	$-0.696728\pi$
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	0.995544	+	0.0942988i	$0.0300609\pi$
$284$		0			0
$285$	3.50000	+	6.06218i	0.207322	+	0.359092i
$286$		0			0
$287$	6.00000	−	5.19615i	0.354169	−	0.306719i
$288$		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
$293$	3.00000			0.175262			0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000			0.175262			0.0876309	+	0.996153i	$0.472070\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$	1.50000	−	2.59808i	0.0870388	−	0.150756i
$298$		0			0
$299$	2.50000	+	4.33013i	0.144579	+	0.250418i
$300$		0			0
$301$	−16.0000	+	13.8564i	−0.922225	+	0.798670i
$302$		0			0
$303$	4.00000	+	6.92820i	0.229794	+	0.398015i
$304$		0			0
$305$	−4.00000	+	6.92820i	−0.229039	+	0.396708i
$306$		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
$309$	−8.00000			−0.455104
$310$		0			0
$311$	2.00000	−	3.46410i	0.113410	−	0.196431i	−0.803733	−	0.594990i	$-0.797156\pi$
$311$	2.00000	−	3.46410i	0.113410	−	0.196431i	0.917143	+	0.398559i	$0.130489\pi$
$312$		0			0
$313$	−6.00000	−	10.3923i	−0.339140	−	0.587408i	0.645131	−	0.764072i	$-0.276803\pi$
$313$	−6.00000	−	10.3923i	−0.339140	−	0.587408i	−0.984271	+	0.176664i	$0.943469\pi$
$314$		0			0
$315$	0.500000	+	2.59808i	0.0281718	+	0.146385i
$316$		0			0
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	0.999980	+	0.00635137i	$0.00202172\pi$
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	−0.494489	+	0.869184i	$0.664645\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−14.0000			−0.781404
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.500000	+	0.866025i	−0.0277350	+	0.0480384i
$326$		0			0
$327$	−1.00000	−	1.73205i	−0.0553001	−	0.0957826i
$328$		0			0
$329$	−2.50000	−	0.866025i	−0.137829	−	0.0477455i
$330$		0			0
$331$	8.50000	+	14.7224i	0.467202	+	0.809218i	0.999298	−	0.0374662i	$-0.0119287\pi$
$331$	8.50000	+	14.7224i	0.467202	+	0.809218i	−0.532096	+	0.846684i	$0.678595\pi$
$332$		0			0
$333$	−1.50000	+	2.59808i	−0.0821995	+	0.142374i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	36.0000			1.96104			0.980522	−	0.196407i	$-0.0629273\pi$
$337$	36.0000			1.96104			0.980522	+	0.196407i	$0.0629273\pi$
$338$		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
$345$	2.50000	+	4.33013i	0.134595	+	0.233126i
$346$		0			0
$347$	17.0000	−	29.4449i	0.912608	−	1.58068i	0.102241	−	0.994760i	$-0.467399\pi$
$347$	17.0000	−	29.4449i	0.912608	−	1.58068i	0.810366	−	0.585923i	$-0.199268\pi$
$348$		0			0
$349$	−4.00000			−0.214115			−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000			−0.214115			−0.107058	+	0.994253i	$0.534143\pi$
$350$		0			0
$351$	−1.00000			−0.0533761
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$	−3.00000	−	5.19615i	−0.159223	−	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	0.999476	+	0.0323801i	$0.0103087\pi$
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	−0.471696	+	0.881761i	$0.656358\pi$
$360$		0			0
$361$	−15.0000	+	25.9808i	−0.789474	+	1.36741i
$362$		0			0
$363$	2.00000			0.104973
$364$		0			0
$365$	14.0000			0.732793
$366$		0			0
$367$	−2.50000	+	4.33013i	−0.130499	+	0.226031i	−0.923869	−	0.382709i	$-0.874991\pi$
$367$	−2.50000	+	4.33013i	−0.130499	+	0.226031i	0.793370	+	0.608740i	$0.208325\pi$
$368$		0			0
$369$	1.50000	+	2.59808i	0.0780869	+	0.135250i
$370$		0			0
$371$	−2.50000	−	12.9904i	−0.129794	−	0.674427i
$372$		0			0
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	−0.996610	−	0.0822766i	$-0.973781\pi$
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	0.427051	−	0.904227i	$-0.359552\pi$
$374$		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
$383$	0.500000	+	0.866025i	0.0255488	+	0.0442518i	0.878517	−	0.477711i	$-0.158533\pi$
$383$	0.500000	+	0.866025i	0.0255488	+	0.0442518i	−0.852968	+	0.521963i	$0.825200\pi$
$384$		0			0
$385$	6.00000	−	5.19615i	0.305788	−	0.264820i
$386$		0			0
$387$	−4.00000	−	6.92820i	−0.203331	−	0.352180i
$388$		0			0
$389$	−13.0000	+	22.5167i	−0.659126	+	1.14164i	0.321716	+	0.946836i	$0.395740\pi$
$389$	−13.0000	+	22.5167i	−0.659126	+	1.14164i	−0.980842	+	0.194804i	$0.937593\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−7.00000			−0.353103
$394$		0			0
$395$	−8.00000	+	13.8564i	−0.402524	+	0.697191i
$396$		0			0
$397$	9.00000	+	15.5885i	0.451697	+	0.782362i	0.998492	−	0.0549046i	$-0.0174855\pi$
$397$	9.00000	+	15.5885i	0.451697	+	0.782362i	−0.546795	+	0.837267i	$0.684152\pi$
$398$		0			0
$399$	−14.0000	+	12.1244i	−0.700877	+	0.606977i
$400$		0			0
$401$	14.5000	+	25.1147i	0.724095	+	1.25417i	0.959345	+	0.282235i	$0.0910758\pi$
$401$	14.5000	+	25.1147i	0.724095	+	1.25417i	−0.235250	+	0.971935i	$0.575591\pi$
$402$		0			0
$403$	−3.00000	+	5.19615i	−0.149441	+	0.258839i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	9.00000			0.446113
$408$		0			0
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	−0.639430	−	0.768849i	$-0.720830\pi$
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	0.985558	+	0.169338i	$0.0541630\pi$
$410$		0			0
$411$	−1.00000	−	1.73205i	−0.0493264	−	0.0854358i
$412$		0			0
$413$	2.00000	+	10.3923i	0.0984136	+	0.511372i
$414$		0			0
$415$	−8.00000	−	13.8564i	−0.392705	−	0.680184i
$416$		0			0
$417$	−2.00000	+	3.46410i	−0.0979404	+	0.169638i
$418$		0			0
$419$	15.0000			0.732798			0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000			0.732798			0.366399	+	0.930458i	$0.380591\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$	0.500000	−	0.866025i	0.0243108	−	0.0421076i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−20.0000	−	6.92820i	−0.967868	−	0.335279i
$428$		0			0
$429$	1.50000	+	2.59808i	0.0724207	+	0.125436i
$430$		0			0
$431$	−8.00000	+	13.8564i	−0.385346	+	0.667440i	−0.991817	−	0.127666i	$-0.959251\pi$
$431$	−8.00000	+	13.8564i	−0.385346	+	0.667440i	0.606471	+	0.795106i	$0.292585\pi$
$432$		0			0
$433$	8.00000			0.384455			0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000			0.384455			0.192228	+	0.981350i	$0.438429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−17.5000	+	30.3109i	−0.837139	+	1.44997i
$438$		0			0
$439$	16.0000	+	27.7128i	0.763638	+	1.32266i	0.940963	+	0.338508i	$0.109922\pi$
$439$	16.0000	+	27.7128i	0.763638	+	1.32266i	−0.177325	+	0.984152i	$0.556744\pi$
$440$		0			0
$441$	−6.50000	+	2.59808i	−0.309524	+	0.123718i
$442$		0			0
$443$	20.0000	+	34.6410i	0.950229	+	1.64584i	0.744927	+	0.667146i	$0.232484\pi$
$443$	20.0000	+	34.6410i	0.950229	+	1.64584i	0.205301	+	0.978699i	$0.434183\pi$
$444$		0			0
$445$	3.00000	−	5.19615i	0.142214	−	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.0000			−0.991051			−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000			−0.991051			−0.495526	+	0.868593i	$0.665025\pi$
$450$		0			0
$451$	4.50000	−	7.79423i	0.211897	−	0.367016i
$452$		0			0
$453$	7.00000	+	12.1244i	0.328889	+	0.569652i
$454$		0			0
$455$	−2.50000	−	0.866025i	−0.117202	−	0.0405999i
$456$		0			0
$457$	9.00000	+	15.5885i	0.421002	+	0.729197i	0.996038	−	0.0889312i	$-0.0283451\pi$
$457$	9.00000	+	15.5885i	0.421002	+	0.729197i	−0.575036	+	0.818128i	$0.695012\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	36.0000			1.67669			0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000			1.67669			0.838344	+	0.545142i	$0.183524\pi$
$462$		0			0
$463$	−9.00000			−0.418265			−0.209133	−	0.977887i	$-0.567064\pi$
$463$	−9.00000			−0.418265			−0.209133	+	0.977887i	$0.567064\pi$
$464$		0			0
$465$	−3.00000	+	5.19615i	−0.139122	+	0.240966i
$466$		0			0
$467$	4.00000	+	6.92820i	0.185098	+	0.320599i	0.943610	−	0.331061i	$-0.107406\pi$
$467$	4.00000	+	6.92820i	0.185098	+	0.320599i	−0.758512	+	0.651660i	$0.774073\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−7.50000	−	12.9904i	−0.345582	−	0.598565i
$472$		0			0
$473$	−12.0000	+	20.7846i	−0.551761	+	0.955677i
$474$		0			0
$475$	−7.00000			−0.321182
$476$		0			0
$477$	5.00000			0.228934
$478$		0			0
$479$	17.0000	−	29.4449i	0.776750	−	1.34537i	−0.157056	−	0.987590i	$-0.550200\pi$
$479$	17.0000	−	29.4449i	0.776750	−	1.34537i	0.933806	−	0.357780i	$-0.116466\pi$
$480$		0			0
$481$	−1.50000	−	2.59808i	−0.0683941	−	0.118462i
$482$		0			0
$483$	−10.0000	+	8.66025i	−0.455016	+	0.394055i
$484$		0			0
$485$	8.00000	+	13.8564i	0.363261	+	0.629187i
$486$		0			0
$487$	−20.0000	+	34.6410i	−0.906287	+	1.56973i	−0.0871056	+	0.996199i	$0.527762\pi$
$487$	−20.0000	+	34.6410i	−0.906287	+	1.56973i	−0.819181	+	0.573535i	$0.805572\pi$
$488$		0			0
$489$	−8.00000			−0.361773
$490$		0			0
$491$	28.0000			1.26362			0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000			1.26362			0.631811	+	0.775122i	$0.282312\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	1.50000	+	2.59808i	0.0674200	+	0.116775i
$496$		0			0
$497$	12.0000	−	10.3923i	0.538274	−	0.466159i
$498$		0			0
$499$	−12.0000	−	20.7846i	−0.537194	−	0.930447i	−0.999054	−	0.0434940i	$-0.986151\pi$
$499$	−12.0000	−	20.7846i	−0.537194	−	0.930447i	0.461860	−	0.886953i	$-0.347182\pi$
$500$		0			0
$501$	−11.5000	+	19.9186i	−0.513782	+	0.889897i
$502$		0			0
$503$	28.0000			1.24846			0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000			1.24846			0.624229	+	0.781241i	$0.285413\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$		0			0
$507$	−6.00000	+	10.3923i	−0.266469	+	0.461538i
$508$		0			0
$509$	5.00000	+	8.66025i	0.221621	+	0.383859i	0.955300	−	0.295637i	$-0.0955319\pi$
$509$	5.00000	+	8.66025i	0.221621	+	0.383859i	−0.733679	+	0.679496i	$0.762199\pi$
$510$		0			0
$511$	7.00000	+	36.3731i	0.309662	+	1.60905i
$512$		0			0
$513$	−3.50000	−	6.06218i	−0.154529	−	0.267652i
$514$		0			0
$515$	4.00000	−	6.92820i	0.176261	−	0.305293i
$516$		0			0
$517$	−3.00000			−0.131940
$518$		0			0
$519$	−9.00000			−0.395056
$520$		0			0
$521$	16.5000	−	28.5788i	0.722878	−	1.25206i	−0.236963	−	0.971519i	$-0.576152\pi$
$521$	16.5000	−	28.5788i	0.722878	−	1.25206i	0.959841	−	0.280543i	$-0.0905145\pi$
$522$		0			0
$523$	−13.0000	−	22.5167i	−0.568450	−	0.984585i	−0.996719	−	0.0809336i	$-0.974210\pi$
$523$	−13.0000	−	22.5167i	−0.568450	−	0.984585i	0.428269	−	0.903651i	$-0.359124\pi$
$524$		0			0
$525$	−2.50000	−	0.866025i	−0.109109	−	0.0377964i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.00000	+	1.73205i	−0.0434783	+	0.0753066i
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$	−3.00000			−0.129944
$534$		0			0
$535$	7.00000	−	12.1244i	0.302636	−	0.524182i
$536$		0			0
$537$	−7.50000	−	12.9904i	−0.323649	−	0.560576i
$538$		0			0
$539$	16.5000	+	12.9904i	0.710705	+	0.559535i
$540$		0			0
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	0.886721	−	0.462304i	$-0.152977\pi$
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	−0.843728	+	0.536771i	$0.819644\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.00000			0.0856706
$546$		0			0
$547$	24.0000			1.02617			0.513083	−	0.858339i	$-0.328503\pi$
$547$	24.0000			1.02617			0.513083	+	0.858339i	$0.328503\pi$
$548$		0			0
$549$	4.00000	−	6.92820i	0.170716	−	0.295689i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−40.0000	−	13.8564i	−1.70097	−	0.589234i
$554$		0			0
$555$	−1.50000	−	2.59808i	−0.0636715	−	0.110282i
$556$		0			0
$557$	4.50000	−	7.79423i	0.190671	−	0.330252i	−0.754802	−	0.655953i	$-0.772267\pi$
$557$	4.50000	−	7.79423i	0.190671	−	0.330252i	0.945473	+	0.325701i	$0.105600\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.00000	+	15.5885i	−0.379305	+	0.656975i	−0.990961	−	0.134148i	$-0.957170\pi$
$563$	−9.00000	+	15.5885i	−0.379305	+	0.656975i	0.611656	+	0.791123i	$0.290503\pi$
$564$		0			0
$565$	3.00000	+	5.19615i	0.126211	+	0.218604i
$566$		0			0
$567$	−0.500000	−	2.59808i	−0.0209980	−	0.109109i
$568$		0			0
$569$	16.5000	+	28.5788i	0.691716	+	1.19809i	0.971275	+	0.237959i	$0.0764783\pi$
$569$	16.5000	+	28.5788i	0.691716	+	1.19809i	−0.279559	+	0.960128i	$0.590188\pi$
$570$		0			0
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	−0.0554391	−	0.998462i	$-0.517656\pi$
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	0.892413	−	0.451219i	$-0.149011\pi$
$572$		0			0
$573$	22.0000			0.919063
$574$		0			0
$575$	−5.00000			−0.208514
$576$		0			0
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	−0.886090	−	0.463513i	$-0.846589\pi$
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	0.844459	+	0.535620i	$0.179922\pi$
$578$		0			0
$579$	−1.00000	−	1.73205i	−0.0415586	−	0.0719816i
$580$		0			0
$581$	32.0000	−	27.7128i	1.32758	−	1.14972i
$582$		0			0
$583$	−7.50000	−	12.9904i	−0.310618	−	0.538007i
$584$		0			0
$585$	0.500000	−	0.866025i	0.0206725	−	0.0358057i
$586$		0			0
$587$	−18.0000			−0.742940			−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000			−0.742940			−0.371470	+	0.928445i	$0.621146\pi$
$588$		0			0
$589$	−42.0000			−1.73058
$590$		0			0
$591$	7.50000	−	12.9904i	0.308509	−	0.534353i
$592$		0			0
$593$	−4.00000	−	6.92820i	−0.164260	−	0.284507i	0.772132	−	0.635462i	$-0.219190\pi$
$593$	−4.00000	−	6.92820i	−0.164260	−	0.284507i	−0.936392	+	0.350955i	$0.885857\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−12.0000	−	20.7846i	−0.491127	−	0.850657i
$598$		0			0
$599$	11.0000	−	19.0526i	0.449448	−	0.778466i	−0.548902	−	0.835887i	$-0.684954\pi$
$599$	11.0000	−	19.0526i	0.449448	−	0.778466i	0.998350	+	0.0574201i	$0.0182874\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.00000	+	1.73205i	−0.0406558	+	0.0704179i
$606$		0			0
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	0.786212	−	0.617957i	$-0.212039\pi$
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	−0.928272	+	0.371901i	$0.878706\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.500000	+	0.866025i	0.0202278	+	0.0350356i
$612$		0			0
$613$	4.50000	−	7.79423i	0.181753	−	0.314806i	−0.760724	−	0.649075i	$-0.775156\pi$
$613$	4.50000	−	7.79423i	0.181753	−	0.314806i	0.942478	+	0.334269i	$0.108489\pi$
$614$		0			0
$615$	−3.00000			−0.120972
$616$		0			0
$617$	4.00000			0.161034			0.0805170	−	0.996753i	$-0.474343\pi$
$617$	4.00000			0.161034			0.0805170	+	0.996753i	$0.474343\pi$
$618$		0			0
$619$	15.5000	−	26.8468i	0.622998	−	1.07906i	−0.365927	−	0.930644i	$-0.619248\pi$
$619$	15.5000	−	26.8468i	0.622998	−	1.07906i	0.988924	−	0.148420i	$-0.0474187\pi$
$620$		0			0
$621$	−2.50000	−	4.33013i	−0.100322	−	0.173762i
$622$		0			0
$623$	15.0000	+	5.19615i	0.600962	+	0.208179i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	−10.5000	+	18.1865i	−0.419330	+	0.726300i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	14.0000			0.557331			0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000			0.557331			0.278666	+	0.960388i	$0.410108\pi$
$632$		0			0
$633$	4.50000	−	7.79423i	0.178859	−	0.309793i
$634$		0			0
$635$	5.50000	+	9.52628i	0.218261	+	0.378039i
$636$		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
$641$	17.5000	−	30.3109i	0.691208	−	1.19721i	−0.280234	−	0.959932i	$-0.590412\pi$
$641$	17.5000	−	30.3109i	0.691208	−	1.19721i	0.971442	−	0.237276i	$-0.0762547\pi$
$642$		0			0
$643$	−34.0000			−1.34083			−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000			−1.34083			−0.670415	+	0.741987i	$0.733884\pi$
$644$		0			0
$645$	8.00000			0.315000
$646$		0			0
$647$	−21.5000	+	37.2391i	−0.845252	+	1.46402i	0.0401498	+	0.999194i	$0.487216\pi$
$647$	−21.5000	+	37.2391i	−0.845252	+	1.46402i	−0.885402	+	0.464826i	$0.846117\pi$
$648$		0			0
$649$	6.00000	+	10.3923i	0.235521	+	0.407934i
$650$		0			0
$651$	−15.0000	−	5.19615i	−0.587896	−	0.203653i
$652$		0			0
$653$	4.50000	+	7.79423i	0.176099	+	0.305012i	0.940541	−	0.339680i	$-0.110319\pi$
$653$	4.50000	+	7.79423i	0.176099	+	0.305012i	−0.764442	+	0.644692i	$0.776986\pi$
$654$		0			0
$655$	3.50000	−	6.06218i	0.136756	−	0.236869i
$656$		0			0
$657$	−14.0000			−0.546192
$658$		0			0
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$	−20.0000	+	34.6410i	−0.777910	+	1.34738i	0.155235	+	0.987878i	$0.450387\pi$
$661$	−20.0000	+	34.6410i	−0.777910	+	1.34738i	−0.933144	+	0.359502i	$0.882947\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.50000	−	18.1865i	−0.135724	−	0.705244i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	2.00000	−	3.46410i	0.0773245	−	0.133930i
$670$		0			0
$671$	−24.0000			−0.926510
$672$		0			0
$673$	12.0000			0.462566			0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000			0.462566			0.231283	+	0.972887i	$0.425708\pi$
$674$		0			0
$675$	0.500000	−	0.866025i	0.0192450	−	0.0333333i
$676$		0			0
$677$	−17.5000	−	30.3109i	−0.672580	−	1.16494i	−0.977170	−	0.212459i	$-0.931853\pi$
$677$	−17.5000	−	30.3109i	−0.672580	−	1.16494i	0.304590	−	0.952483i	$-0.401480\pi$
$678$		0			0
$679$	−32.0000	+	27.7128i	−1.22805	+	1.06352i
$680$		0			0
$681$	−12.0000	−	20.7846i	−0.459841	−	0.796468i
$682$		0			0
$683$	2.00000	−	3.46410i	0.0765279	−	0.132550i	−0.825222	−	0.564809i	$-0.808950\pi$
$683$	2.00000	−	3.46410i	0.0765279	−	0.132550i	0.901750	+	0.432259i	$0.142283\pi$
$684$		0			0
$685$	2.00000			0.0764161
$686$		0			0
$687$	−22.0000			−0.839352
$688$		0			0
$689$	−2.50000	+	4.33013i	−0.0952424	+	0.164965i
$690$		0			0
$691$	−6.00000	−	10.3923i	−0.228251	−	0.395342i	0.729039	−	0.684472i	$-0.239967\pi$
$691$	−6.00000	−	10.3923i	−0.228251	−	0.395342i	−0.957290	+	0.289130i	$0.906634\pi$
$692$		0			0
$693$	−6.00000	+	5.19615i	−0.227921	+	0.197386i
$694$		0			0
$695$	−2.00000	−	3.46410i	−0.0758643	−	0.131401i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	22.0000			0.832116
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		0			0
$703$	10.5000	−	18.1865i	0.396015	−	0.685918i
$704$		0			0
$705$	0.500000	+	0.866025i	0.0188311	+	0.0326164i
$706$		0			0
$707$	−4.00000	−	20.7846i	−0.150435	−	0.781686i
$708$		0			0
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	0.826023	−	0.563636i	$-0.190598\pi$
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	−0.901135	+	0.433539i	$0.857265\pi$
$710$		0			0
$711$	8.00000	−	13.8564i	0.300023	−	0.519656i
$712$		0			0
$713$	−30.0000			−1.12351
$714$		0			0
$715$	−3.00000			−0.112194
$716$		0			0
$717$	−1.00000	+	1.73205i	−0.0373457	+	0.0646846i
$718$		0			0
$719$	−1.00000	−	1.73205i	−0.0372937	−	0.0645946i	0.846776	−	0.531949i	$-0.178540\pi$
$719$	−1.00000	−	1.73205i	−0.0372937	−	0.0645946i	−0.884070	+	0.467355i	$0.845207\pi$
$720$		0			0
$721$	20.0000	+	6.92820i	0.744839	+	0.258020i
$722$		0			0
$723$	3.50000	+	6.06218i	0.130166	+	0.225455i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−33.0000			−1.22390			−0.611951	−	0.790896i	$-0.709615\pi$
$727$	−33.0000			−1.22390			−0.611951	+	0.790896i	$0.709615\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−9.50000	−	16.4545i	−0.350891	−	0.607760i	0.635515	−	0.772088i	$-0.280788\pi$
$733$	−9.50000	−	16.4545i	−0.350891	−	0.607760i	−0.986406	+	0.164328i	$0.947454\pi$
$734$		0			0
$735$	1.00000	−	6.92820i	0.0368856	−	0.255551i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−20.5000	+	35.5070i	−0.754105	+	1.30615i	0.191714	+	0.981451i	$0.438596\pi$
$739$	−20.5000	+	35.5070i	−0.754105	+	1.30615i	−0.945818	+	0.324697i	$0.894738\pi$
$740$		0			0
$741$	7.00000			0.257151
$742$		0			0
$743$	5.00000			0.183432			0.0917161	−	0.995785i	$-0.470765\pi$
$743$	5.00000			0.183432			0.0917161	+	0.995785i	$0.470765\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	8.00000	+	13.8564i	0.292705	+	0.506979i
$748$		0			0
$749$	35.0000	+	12.1244i	1.27887	+	0.443014i
$750$		0			0
$751$	−17.0000	−	29.4449i	−0.620339	−	1.07446i	−0.989423	−	0.145062i	$-0.953662\pi$
$751$	−17.0000	−	29.4449i	−0.620339	−	1.07446i	0.369084	−	0.929396i	$-0.379672\pi$
$752$		0			0
$753$	−15.5000	+	26.8468i	−0.564851	+	0.978351i
$754$		0			0
$755$	−14.0000			−0.509512
$756$		0			0
$757$	−14.0000			−0.508839			−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000			−0.508839			−0.254419	+	0.967094i	$0.581884\pi$
$758$		0			0
$759$	−7.50000	+	12.9904i	−0.272233	+	0.471521i
$760$		0			0
$761$	−21.5000	−	37.2391i	−0.779374	−	1.34992i	−0.932303	−	0.361679i	$-0.882204\pi$
$761$	−21.5000	−	37.2391i	−0.779374	−	1.34992i	0.152928	−	0.988237i	$-0.451130\pi$
$762$		0			0
$763$	1.00000	+	5.19615i	0.0362024	+	0.188113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.00000	−	3.46410i	0.0722158	−	0.125081i
$768$		0			0
$769$	35.0000			1.26213			0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000			1.26213			0.631066	+	0.775729i	$0.282618\pi$
$770$		0			0
$771$	12.0000			0.432169
$772$		0			0
$773$	15.5000	−	26.8468i	0.557496	−	0.965612i	−0.440208	−	0.897896i	$-0.645095\pi$
$773$	15.5000	−	26.8468i	0.557496	−	0.965612i	0.997705	−	0.0677162i	$-0.0215712\pi$
$774$		0			0
$775$	−3.00000	−	5.19615i	−0.107763	−	0.186651i
$776$		0			0
$777$	6.00000	−	5.19615i	0.215249	−	0.186411i
$778$		0			0
$779$	−10.5000	−	18.1865i	−0.376202	−	0.651600i
$780$		0			0
$781$	9.00000	−	15.5885i	0.322045	−	0.557799i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	15.0000			0.535373
$786$		0			0
$787$	−19.0000	+	32.9090i	−0.677277	+	1.17308i	0.298521	+	0.954403i	$0.403507\pi$
$787$	−19.0000	+	32.9090i	−0.677277	+	1.17308i	−0.975798	+	0.218675i	$0.929827\pi$
$788$		0			0
$789$	−16.0000	−	27.7128i	−0.569615	−	0.986602i
$790$		0			0
$791$	−12.0000	+	10.3923i	−0.426671	+	0.369508i
$792$		0			0
$793$	4.00000	+	6.92820i	0.142044	+	0.246028i
$794$		0			0
$795$	−2.50000	+	4.33013i	−0.0886659	+	0.153574i
$796$		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
$801$	−3.00000	+	5.19615i	−0.106000	+	0.183597i
$802$		0			0
$803$	21.0000	+	36.3731i	0.741074	+	1.28358i
$804$		0			0
$805$	−2.50000	−	12.9904i	−0.0881134	−	0.457851i
$806$		0			0
$807$	−6.00000	−	10.3923i	−0.211210	−	0.365826i
$808$		0			0
$809$	27.5000	−	47.6314i	0.966849	−	1.67463i	0.262284	−	0.964991i	$-0.415524\pi$
$809$	27.5000	−	47.6314i	0.966849	−	1.67463i	0.704564	−	0.709640i	$-0.251142\pi$
$810$		0			0
$811$	5.00000			0.175574			0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000			0.175574			0.0877869	+	0.996139i	$0.472021\pi$
$812$		0			0
$813$	24.0000			0.841717
$814$		0			0
$815$	4.00000	−	6.92820i	0.140114	−	0.242684i
$816$		0			0
$817$	28.0000	+	48.4974i	0.979596	+	1.69671i
$818$		0			0
$819$	2.50000	+	0.866025i	0.0873571	+	0.0302614i
$820$		0			0
$821$	−21.0000	−	36.3731i	−0.732905	−	1.26943i	−0.955636	−	0.294549i	$-0.904831\pi$
$821$	−21.0000	−	36.3731i	−0.732905	−	1.26943i	0.222731	−	0.974880i	$-0.428503\pi$
$822$		0			0
$823$	24.0000	−	41.5692i	0.836587	−	1.44901i	−0.0561440	−	0.998423i	$-0.517881\pi$
$823$	24.0000	−	41.5692i	0.836587	−	1.44901i	0.892731	−	0.450589i	$-0.148786\pi$
$824$		0			0
$825$	−3.00000			−0.104447
$826$		0			0
$827$	30.0000			1.04320			0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000			1.04320			0.521601	+	0.853189i	$0.325335\pi$
$828$		0			0
$829$	−12.0000	+	20.7846i	−0.416777	+	0.721879i	−0.995613	−	0.0935647i	$-0.970174\pi$
$829$	−12.0000	+	20.7846i	−0.416777	+	0.721879i	0.578836	+	0.815444i	$0.303507\pi$
$830$		0			0
$831$	5.00000	+	8.66025i	0.173448	+	0.300421i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−11.5000	−	19.9186i	−0.397974	−	0.689311i
$836$		0			0
$837$	3.00000	−	5.19615i	0.103695	−	0.179605i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−8.50000	+	14.7224i	−0.292756	+	0.507067i
$844$		0			0
$845$	−6.00000	−	10.3923i	−0.206406	−	0.357506i
$846$		0			0
$847$	−5.00000	−	1.73205i	−0.171802	−	0.0595140i
$848$		0			0
$849$	−7.00000	−	12.1244i	−0.240239	−	0.416107i
$850$		0			0
$851$	7.50000	−	12.9904i	0.257097	−	0.445305i
$852$		0			0
$853$	23.0000			0.787505			0.393753	−	0.919216i	$-0.371177\pi$
$853$	23.0000			0.787505			0.393753	+	0.919216i	$0.371177\pi$
$854$		0			0
$855$	7.00000			0.239395
$856$		0			0
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	0.244701	+	0.969599i	$0.421310\pi$
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	−0.962048	+	0.272882i	$0.912023\pi$
$858$		0			0
$859$	−2.00000	−	3.46410i	−0.0682391	−	0.118194i	0.829887	−	0.557931i	$-0.188405\pi$
$859$	−2.00000	−	3.46410i	−0.0682391	−	0.118194i	−0.898126	+	0.439738i	$0.855071\pi$
$860$		0			0
$861$	−1.50000	−	7.79423i	−0.0511199	−	0.265627i
$862$		0			0
$863$	4.50000	+	7.79423i	0.153182	+	0.265319i	0.932395	−	0.361440i	$-0.117715\pi$
$863$	4.50000	+	7.79423i	0.153182	+	0.265319i	−0.779214	+	0.626758i	$0.784381\pi$
$864$		0			0
$865$	4.50000	−	7.79423i	0.153005	−	0.265012i
$866$		0			0
$867$	17.0000			0.577350
$868$		0			0
$869$	−48.0000			−1.62829
$870$		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
$877$	−28.5000	−	49.3634i	−0.962377	−	1.66689i	−0.716504	−	0.697583i	$-0.754259\pi$
$877$	−28.5000	−	49.3634i	−0.962377	−	1.66689i	−0.245873	−	0.969302i	$-0.579075\pi$
$878$		0			0
$879$	1.50000	−	2.59808i	0.0505937	−	0.0876309i
$880$		0			0
$881$	3.00000			0.101073			0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000			0.101073			0.0505363	+	0.998722i	$0.483907\pi$
$882$		0			0
$883$	−26.0000			−0.874970			−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000			−0.874970			−0.437485	+	0.899226i	$0.644131\pi$
$884$		0			0
$885$	2.00000	−	3.46410i	0.0672293	−	0.116445i
$886$		0			0
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	0.948999	−	0.315279i	$-0.102098\pi$
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	−0.747539	+	0.664218i	$0.768765\pi$
$888$		0			0
$889$	−22.0000	+	19.0526i	−0.737856	+	0.639002i
$890$		0			0
$891$	−1.50000	−	2.59808i	−0.0502519	−	0.0870388i
$892$		0			0
$893$	−3.50000	+	6.06218i	−0.117123	+	0.202863i
$894$		0			0
$895$	15.0000			0.501395
$896$		0			0
$897$	5.00000			0.166945
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	4.00000	+	20.7846i	0.133112	+	0.691669i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−11.0000	+	19.0526i	−0.365249	+	0.632630i	−0.988816	−	0.149140i	$-0.952349\pi$
$907$	−11.0000	+	19.0526i	−0.365249	+	0.632630i	0.623567	+	0.781770i	$0.285683\pi$
$908$		0			0
$909$	8.00000			0.265343
$910$		0			0
$911$	−42.0000			−1.39152			−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000			−1.39152			−0.695761	+	0.718273i	$0.744933\pi$
$912$		0			0
$913$	24.0000	−	41.5692i	0.794284	−	1.37574i
$914$		0			0
$915$	4.00000	+	6.92820i	0.132236	+	0.229039i
$916$		0			0
$917$	17.5000	+	6.06218i	0.577901	+	0.200191i
$918$		0			0
$919$	−10.0000	−	17.3205i	−0.329870	−	0.571351i	0.652616	−	0.757689i	$-0.273671\pi$
$919$	−10.0000	−	17.3205i	−0.329870	−	0.571351i	−0.982486	+	0.186338i	$0.940338\pi$
$920$		0			0
$921$	−10.0000	+	17.3205i	−0.329511	+	0.570730i
$922$		0			0
$923$	−6.00000			−0.197492
$924$		0			0
$925$	3.00000			0.0986394
$926$		0			0
$927$	−4.00000	+	6.92820i	−0.131377	+	0.227552i
$928$		0			0
$929$	−13.5000	−	23.3827i	−0.442921	−	0.767161i	0.554984	−	0.831861i	$-0.312724\pi$
$929$	−13.5000	−	23.3827i	−0.442921	−	0.767161i	−0.997905	+	0.0646999i	$0.979391\pi$
$930$		0			0
$931$	45.5000	−	18.1865i	1.49120	−	0.596040i
$932$		0			0
$933$	−2.00000	−	3.46410i	−0.0654771	−	0.113410i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−20.0000			−0.653372			−0.326686	−	0.945133i	$-0.605932\pi$
$937$	−20.0000			−0.653372			−0.326686	+	0.945133i	$0.605932\pi$
$938$		0			0
$939$	−12.0000			−0.391605
$940$		0			0
$941$	9.00000	−	15.5885i	0.293392	−	0.508169i	−0.681218	−	0.732081i	$-0.738549\pi$
$941$	9.00000	−	15.5885i	0.293392	−	0.508169i	0.974609	+	0.223912i	$0.0718827\pi$
$942$		0			0
$943$	−7.50000	−	12.9904i	−0.244234	−	0.423025i
$944$		0			0
$945$	2.50000	+	0.866025i	0.0813250	+	0.0281718i
$946$		0			0
$947$	−19.0000	−	32.9090i	−0.617417	−	1.06940i	−0.989955	−	0.141381i	$-0.954846\pi$
$947$	−19.0000	−	32.9090i	−0.617417	−	1.06940i	0.372538	−	0.928017i	$-0.378488\pi$
$948$		0			0
$949$	7.00000	−	12.1244i	0.227230	−	0.393573i
$950$		0			0
$951$	18.0000			0.583690
$952$		0			0
$953$	4.00000			0.129573			0.0647864	−	0.997899i	$-0.479363\pi$
$953$	4.00000			0.129573			0.0647864	+	0.997899i	$0.479363\pi$
$954$		0			0
$955$	−11.0000	+	19.0526i	−0.355952	+	0.616526i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.00000	+	5.19615i	0.0322917	+	0.167793i
$960$		0			0
$961$	−2.50000	−	4.33013i	−0.0806452	−	0.139682i
$962$		0			0
$963$	−7.00000	+	12.1244i	−0.225572	+	0.390702i
$964$		0			0
$965$	2.00000			0.0643823
$966$		0			0
$967$	4.00000			0.128631			0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000			0.128631			0.0643157	+	0.997930i	$0.479514\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.50000	+	7.79423i	0.144412	+	0.250129i	0.929153	−	0.369694i	$-0.120538\pi$
$971$	4.50000	+	7.79423i	0.144412	+	0.250129i	−0.784741	+	0.619823i	$0.787204\pi$
$972$		0			0
$973$	8.00000	−	6.92820i	0.256468	−	0.222108i
$974$		0			0
$975$	0.500000	+	0.866025i	0.0160128	+	0.0277350i
$976$		0			0
$977$	−27.0000	+	46.7654i	−0.863807	+	1.49616i	0.00442082	+	0.999990i	$0.498593\pi$
$977$	−27.0000	+	46.7654i	−0.863807	+	1.49616i	−0.868227	+	0.496167i	$0.834741\pi$
$978$		0			0
$979$	18.0000			0.575282
$980$		0			0
$981$	−2.00000			−0.0638551
$982$		0			0
$983$	−10.5000	+	18.1865i	−0.334898	+	0.580060i	−0.983465	−	0.181097i	$-0.942035\pi$
$983$	−10.5000	+	18.1865i	−0.334898	+	0.580060i	0.648567	+	0.761157i	$0.275369\pi$
$984$		0			0
$985$	7.50000	+	12.9904i	0.238970	+	0.413908i
$986$		0			0
$987$	−2.00000	+	1.73205i	−0.0636607	+	0.0551318i
$988$		0			0
$989$	20.0000	+	34.6410i	0.635963	+	1.10152i
$990$		0			0
$991$	1.00000	−	1.73205i	0.0317660	−	0.0550204i	−0.849705	−	0.527258i	$-0.823220\pi$
$991$	1.00000	−	1.73205i	0.0317660	−	0.0550204i	0.881471	+	0.472237i	$0.156554\pi$
$992$		0			0
$993$	17.0000			0.539479
$994$		0			0
$995$	24.0000			0.760851
$996$		0			0
$997$	−23.0000	+	39.8372i	−0.728417	+	1.26166i	0.229135	+	0.973395i	$0.426410\pi$
$997$	−23.0000	+	39.8372i	−0.728417	+	1.26166i	−0.957552	+	0.288261i	$0.906923\pi$
$998$		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.d.361.1	yes	2
3.2	odd	2		2520.2.bi.b.361.1		2
4.3	odd	2		1680.2.bg.i.1201.1		2
7.2	even	3	inner	840.2.bg.d.121.1	✓	2
7.3	odd	6		5880.2.a.bh.1.1		1
7.4	even	3		5880.2.a.f.1.1		1
21.2	odd	6		2520.2.bi.b.1801.1		2
28.23	odd	6		1680.2.bg.i.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.bg.d.121.1	✓	2	7.2	even	3	inner
840.2.bg.d.361.1	yes	2	1.1	even	1	trivial
1680.2.bg.i.961.1		2	28.23	odd	6
1680.2.bg.i.1201.1		2	4.3	odd	2
2520.2.bi.b.361.1		2	3.2	odd	2
2520.2.bi.b.1801.1		2	21.2	odd	6
5880.2.a.f.1.1		1	7.4	even	3
5880.2.a.bh.1.1		1	7.3	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)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.50000 + 2.59808i) q^{11} +1.00000 q^{13} +1.00000 q^{15} +(3.50000 + 6.06218i) q^{19} +(0.500000 + 2.59808i) q^{21} +(2.50000 + 4.33013i) q^{23} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +(-3.00000 + 5.19615i) q^{31} +(1.50000 + 2.59808i) q^{33} +(-2.50000 - 0.866025i) q^{35} +(-1.50000 - 2.59808i) q^{37} +(0.500000 - 0.866025i) q^{39} -3.00000 q^{41} +8.00000 q^{43} +(0.500000 - 0.866025i) q^{45} +(0.500000 + 0.866025i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-2.50000 + 4.33013i) q^{53} -3.00000 q^{55} +7.00000 q^{57} +(2.00000 - 3.46410i) q^{59} +(4.00000 + 6.92820i) q^{61} +(2.50000 + 0.866025i) q^{63} +(0.500000 + 0.866025i) q^{65} +5.00000 q^{69} -6.00000 q^{71} +(7.00000 - 12.1244i) q^{73} +(0.500000 + 0.866025i) q^{75} +(-1.50000 - 7.79423i) q^{77} +(8.00000 + 13.8564i) q^{79} +(-0.500000 + 0.866025i) q^{81} -16.0000 q^{83} +(-3.00000 - 5.19615i) q^{89} +(-2.00000 + 1.73205i) q^{91} +(3.00000 + 5.19615i) q^{93} +(-3.50000 + 6.06218i) q^{95} +16.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 2 * q + q^3 + q^5 - 4 * q^7 - q^9	\( 2 q + q^{3} + q^{5} - 4 q^{7} - q^{9} - 3 q^{11} + 2 q^{13} + 2 q^{15} + 7 q^{19} + q^{21} + 5 q^{23} - q^{25} - 2 q^{27} - 6 q^{31} + 3 q^{33} - 5 q^{35} - 3 q^{37} + q^{39} - 6 q^{41} + 16 q^{43} + q^{45} + q^{47} + 2 q^{49} - 5 q^{53} - 6 q^{55} + 14 q^{57} + 4 q^{59} + 8 q^{61} + 5 q^{63} + q^{65} + 10 q^{69} - 12 q^{71} + 14 q^{73} + q^{75} - 3 q^{77} + 16 q^{79} - q^{81} - 32 q^{83} - 6 q^{89} - 4 q^{91} + 6 q^{93} - 7 q^{95} + 32 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^5 - 4 * q^7 - q^9 - 3 * q^11 + 2 * q^13 + 2 * q^15 + 7 * q^19 + q^21 + 5 * q^23 - q^25 - 2 * q^27 - 6 * q^31 + 3 * q^33 - 5 * q^35 - 3 * q^37 + q^39 - 6 * q^41 + 16 * q^43 + q^45 + q^47 + 2 * q^49 - 5 * q^53 - 6 * q^55 + 14 * q^57 + 4 * q^59 + 8 * q^61 + 5 * q^63 + q^65 + 10 * q^69 - 12 * q^71 + 14 * q^73 + q^75 - 3 * q^77 + 16 * q^79 - q^81 - 32 * q^83 - 6 * q^89 - 4 * q^91 + 6 * q^93 - 7 * q^95 + 32 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

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