LMFDB - Embedded newform 2106.2.e.z.1405.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2106,2,Mod(703,2106)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2106, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("2106.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2106 = 2 \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2106.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$16.8164946657$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1405.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2106.1405
Dual form			2106.2.e.z.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +(1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +2.00000 q^{10} +(2.00000 + 3.46410i) q^{11} +(0.500000 - 0.866025i) q^{13} +(-1.00000 + 1.73205i) q^{14} +(-0.500000 - 0.866025i) q^{16} -6.00000 q^{19} +(1.00000 + 1.73205i) q^{20} +(-2.00000 + 3.46410i) q^{22} +(-2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +1.00000 q^{26} -2.00000 q^{28} +(4.00000 + 6.92820i) q^{29} +(1.00000 - 1.73205i) q^{31} +(0.500000 - 0.866025i) q^{32} +4.00000 q^{35} +6.00000 q^{37} +(-3.00000 - 5.19615i) q^{38} +(-1.00000 + 1.73205i) q^{40} +(-3.00000 + 5.19615i) q^{41} +(4.00000 + 6.92820i) q^{43} -4.00000 q^{44} -4.00000 q^{46} +(-4.00000 - 6.92820i) q^{47} +(1.50000 - 2.59808i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(0.500000 + 0.866025i) q^{52} +12.0000 q^{53} +8.00000 q^{55} +(-1.00000 - 1.73205i) q^{56} +(-4.00000 + 6.92820i) q^{58} +(-2.00000 + 3.46410i) q^{59} +(-5.00000 - 8.66025i) q^{61} +2.00000 q^{62} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{65} +(1.00000 - 1.73205i) q^{67} +(2.00000 + 3.46410i) q^{70} -16.0000 q^{71} +14.0000 q^{73} +(3.00000 + 5.19615i) q^{74} +(3.00000 - 5.19615i) q^{76} +(-4.00000 + 6.92820i) q^{77} +(2.00000 + 3.46410i) q^{79} -2.00000 q^{80} -6.00000 q^{82} +(6.00000 + 10.3923i) q^{83} +(-4.00000 + 6.92820i) q^{86} +(-2.00000 - 3.46410i) q^{88} -6.00000 q^{89} +2.00000 q^{91} +(-2.00000 - 3.46410i) q^{92} +(4.00000 - 6.92820i) q^{94} +(-6.00000 + 10.3923i) q^{95} +(5.00000 + 8.66025i) q^{97} +3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{10} + 4 q^{11} + q^{13} - 2 q^{14} - q^{16} - 12 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} + q^{25} + 2 q^{26} - 4 q^{28} + 8 q^{29} + 2 q^{31}+ \cdots + 6 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 2 * q^5 + 2 * q^7 - 2 * q^8 + 4 * q^10 + 4 * q^11 + q^13 - 2 * q^14 - q^16 - 12 * q^19 + 2 * q^20 - 4 * q^22 - 4 * q^23 + q^25 + 2 * q^26 - 4 * q^28 + 8 * q^29 + 2 * q^31 + q^32 + 8 * q^35 + 12 * q^37 - 6 * q^38 - 2 * q^40 - 6 * q^41 + 8 * q^43 - 8 * q^44 - 8 * q^46 - 8 * q^47 + 3 * q^49 - q^50 + q^52 + 24 * q^53 + 16 * q^55 - 2 * q^56 - 8 * q^58 - 4 * q^59 - 10 * q^61 + 4 * q^62 + 2 * q^64 - 2 * q^65 + 2 * q^67 + 4 * q^70 - 32 * q^71 + 28 * q^73 + 6 * q^74 + 6 * q^76 - 8 * q^77 + 4 * q^79 - 4 * q^80 - 12 * q^82 + 12 * q^83 - 8 * q^86 - 4 * q^88 - 12 * q^89 + 4 * q^91 - 4 * q^92 + 8 * q^94 - 12 * q^95 + 10 * q^97 + 6 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times$.

$n$	$1379$	$1783$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	0.990766	−	0.135583i	$-0.0432908\pi$
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	−0.612801	+	0.790237i	$0.709957\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$	2.00000			0.632456
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i
$14$	−1.00000	+	1.73205i	−0.267261	+	0.462910i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	+	1.73205i	0.223607	+	0.387298i
$21$		0			0
$22$	−2.00000	+	3.46410i	−0.426401	+	0.738549i
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	−0.995639	−	0.0932891i	$-0.970262\pi$
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	0.578610	+	0.815604i	$0.303595\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$	1.00000			0.196116
$27$		0			0
$28$	−2.00000			−0.377964
$29$	4.00000	+	6.92820i	0.742781	+	1.28654i	0.951224	+	0.308500i	$0.0998271\pi$
$29$	4.00000	+	6.92820i	0.742781	+	1.28654i	−0.208443	+	0.978035i	$0.566840\pi$
$30$		0			0
$31$	1.00000	−	1.73205i	0.179605	−	0.311086i	−0.762140	−	0.647412i	$-0.775851\pi$
$31$	1.00000	−	1.73205i	0.179605	−	0.311086i	0.941745	+	0.336327i	$0.109185\pi$
$32$	0.500000	−	0.866025i	0.0883883	−	0.153093i
$33$		0			0
$34$		0			0
$35$	4.00000			0.676123
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$	−3.00000	−	5.19615i	−0.486664	−	0.842927i
$39$		0			0
$40$	−1.00000	+	1.73205i	−0.158114	+	0.273861i
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	0.530831	+	0.847477i	$0.321880\pi$
$42$		0			0
$43$	4.00000	+	6.92820i	0.609994	+	1.05654i	0.991241	+	0.132068i	$0.0421616\pi$
$43$	4.00000	+	6.92820i	0.609994	+	1.05654i	−0.381246	+	0.924473i	$0.624505\pi$
$44$	−4.00000			−0.603023
$45$		0			0
$46$	−4.00000			−0.589768
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	−0.995066	−	0.0992202i	$-0.968365\pi$
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	0.411606	−	0.911362i	$-0.364968\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$	−0.500000	+	0.866025i	−0.0707107	+	0.122474i
$51$		0			0
$52$	0.500000	+	0.866025i	0.0693375	+	0.120096i
$53$	12.0000			1.64833			0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000			1.64833			0.824163	+	0.566352i	$0.191646\pi$
$54$		0			0
$55$	8.00000			1.07872
$56$	−1.00000	−	1.73205i	−0.133631	−	0.231455i
$57$		0			0
$58$	−4.00000	+	6.92820i	−0.525226	+	0.909718i
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	−0.966342	−	0.257260i	$-0.917180\pi$
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	0.705965	+	0.708247i	$0.250514\pi$
$60$		0			0
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	−0.985391	−	0.170305i	$-0.945525\pi$
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	0.345207	−	0.938527i	$-0.387809\pi$
$62$	2.00000			0.254000
$63$		0			0
$64$	1.00000			0.125000
$65$	−1.00000	−	1.73205i	−0.124035	−	0.214834i
$66$		0			0
$67$	1.00000	−	1.73205i	0.122169	−	0.211604i	−0.798454	−	0.602056i	$-0.794348\pi$
$67$	1.00000	−	1.73205i	0.122169	−	0.211604i	0.920623	+	0.390453i	$0.127682\pi$
$68$		0			0
$69$		0			0
$70$	2.00000	+	3.46410i	0.239046	+	0.414039i
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$	14.0000			1.63858			0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000			1.63858			0.819288	+	0.573382i	$0.194369\pi$
$74$	3.00000	+	5.19615i	0.348743	+	0.604040i
$75$		0			0
$76$	3.00000	−	5.19615i	0.344124	−	0.596040i
$77$	−4.00000	+	6.92820i	−0.455842	+	0.789542i
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$	−2.00000			−0.223607
$81$		0			0
$82$	−6.00000			−0.662589
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	0.980982	+	0.194099i	$0.0621783\pi$
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	−0.322396	+	0.946605i	$0.604488\pi$
$84$		0			0
$85$		0			0
$86$	−4.00000	+	6.92820i	−0.431331	+	0.747087i
$87$		0			0
$88$	−2.00000	−	3.46410i	−0.213201	−	0.369274i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$	−2.00000	−	3.46410i	−0.208514	−	0.361158i
$93$		0			0
$94$	4.00000	−	6.92820i	0.412568	−	0.714590i
$95$	−6.00000	+	10.3923i	−0.615587	+	1.06623i
$96$		0			0
$97$	5.00000	+	8.66025i	0.507673	+	0.879316i	0.999961	+	0.00888289i	$0.00282755\pi$
$97$	5.00000	+	8.66025i	0.507673	+	0.879316i	−0.492287	+	0.870433i	$0.663839\pi$
$98$	3.00000			0.303046
$99$		0			0
$100$	−1.00000			−0.100000
$101$	8.00000	+	13.8564i	0.796030	+	1.37876i	0.922183	+	0.386753i	$0.126403\pi$
$101$	8.00000	+	13.8564i	0.796030	+	1.37876i	−0.126153	+	0.992011i	$0.540263\pi$
$102$		0			0
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	−0.947576	−	0.319531i	$-0.896475\pi$
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	0.750510	+	0.660859i	$0.229808\pi$
$104$	−0.500000	+	0.866025i	−0.0490290	+	0.0849208i
$105$		0			0
$106$	6.00000	+	10.3923i	0.582772	+	1.00939i
$107$	−20.0000			−1.93347			−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000			−1.93347			−0.966736	+	0.255774i	$0.917670\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$	4.00000	+	6.92820i	0.381385	+	0.660578i
$111$		0			0
$112$	1.00000	−	1.73205i	0.0944911	−	0.163663i
$113$	−2.00000	+	3.46410i	−0.188144	+	0.325875i	−0.944632	−	0.328133i	$-0.893581\pi$
$113$	−2.00000	+	3.46410i	−0.188144	+	0.325875i	0.756487	+	0.654008i	$0.226914\pi$
$114$		0			0
$115$	4.00000	+	6.92820i	0.373002	+	0.646058i
$116$	−8.00000			−0.742781
$117$		0			0
$118$	−4.00000			−0.368230
$119$		0			0
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$	5.00000	−	8.66025i	0.452679	−	0.784063i
$123$		0			0
$124$	1.00000	+	1.73205i	0.0898027	+	0.155543i
$125$	12.0000			1.07331
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$	0.500000	+	0.866025i	0.0441942	+	0.0765466i
$129$		0			0
$130$	1.00000	−	1.73205i	0.0877058	−	0.151911i
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	−0.940072	−	0.340977i	$-0.889242\pi$
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	0.765331	+	0.643637i	$0.222575\pi$
$132$		0			0
$133$	−6.00000	−	10.3923i	−0.520266	−	0.901127i
$134$	2.00000			0.172774
$135$		0			0
$136$		0			0
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	0.938148	+	0.346235i	$0.112540\pi$
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	−0.169226	+	0.985577i	$0.554127\pi$
$138$		0			0
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	−0.0346338	−	0.999400i	$-0.511026\pi$
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	0.882823	−	0.469706i	$-0.155640\pi$
$140$	−2.00000	+	3.46410i	−0.169031	+	0.292770i
$141$		0			0
$142$	−8.00000	−	13.8564i	−0.671345	−	1.16280i
$143$	4.00000			0.334497
$144$		0			0
$145$	16.0000			1.32873
$146$	7.00000	+	12.1244i	0.579324	+	1.00342i
$147$		0			0
$148$	−3.00000	+	5.19615i	−0.246598	+	0.427121i
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	−9.00000	−	15.5885i	−0.732410	−	1.26857i	−0.955851	−	0.293853i	$-0.905062\pi$
$151$	−9.00000	−	15.5885i	−0.732410	−	1.26857i	0.223441	−	0.974717i	$-0.428271\pi$
$152$	6.00000			0.486664
$153$		0			0
$154$	−8.00000			−0.644658
$155$	−2.00000	−	3.46410i	−0.160644	−	0.278243i
$156$		0			0
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	−0.903167	−	0.429289i	$-0.858764\pi$
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	0.823359	+	0.567521i	$0.192098\pi$
$158$	−2.00000	+	3.46410i	−0.159111	+	0.275589i
$159$		0			0
$160$	−1.00000	−	1.73205i	−0.0790569	−	0.136931i
$161$	−8.00000			−0.630488
$162$		0			0
$163$	−10.0000			−0.783260			−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000			−0.783260			−0.391630	+	0.920123i	$0.628089\pi$
$164$	−3.00000	−	5.19615i	−0.234261	−	0.405751i
$165$		0			0
$166$	−6.00000	+	10.3923i	−0.465690	+	0.806599i
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.0384615	−	0.0666173i
$170$		0			0
$171$		0			0
$172$	−8.00000			−0.609994
$173$	−8.00000	−	13.8564i	−0.608229	−	1.05348i	−0.991532	−	0.129861i	$-0.958547\pi$
$173$	−8.00000	−	13.8564i	−0.608229	−	1.05348i	0.383304	−	0.923622i	$-0.374786\pi$
$174$		0			0
$175$	−1.00000	+	1.73205i	−0.0755929	+	0.130931i
$176$	2.00000	−	3.46410i	0.150756	−	0.261116i
$177$		0			0
$178$	−3.00000	−	5.19615i	−0.224860	−	0.389468i
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$	1.00000	+	1.73205i	0.0741249	+	0.128388i
$183$		0			0
$184$	2.00000	−	3.46410i	0.147442	−	0.255377i
$185$	6.00000	−	10.3923i	0.441129	−	0.764057i
$186$		0			0
$187$		0			0
$188$	8.00000			0.583460
$189$		0			0
$190$	−12.0000			−0.870572
$191$	2.00000	+	3.46410i	0.144715	+	0.250654i	0.929267	−	0.369410i	$-0.120440\pi$
$191$	2.00000	+	3.46410i	0.144715	+	0.250654i	−0.784552	+	0.620063i	$0.787107\pi$
$192$		0			0
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	0.827788	+	0.561041i	$0.189599\pi$
$194$	−5.00000	+	8.66025i	−0.358979	+	0.621770i
$195$		0			0
$196$	1.50000	+	2.59808i	0.107143	+	0.185577i
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$	−0.500000	−	0.866025i	−0.0353553	−	0.0612372i
$201$		0			0
$202$	−8.00000	+	13.8564i	−0.562878	+	0.974933i
$203$	−8.00000	+	13.8564i	−0.561490	+	0.972529i
$204$		0			0
$205$	6.00000	+	10.3923i	0.419058	+	0.725830i
$206$	−4.00000			−0.278693
$207$		0			0
$208$	−1.00000			−0.0693375
$209$	−12.0000	−	20.7846i	−0.830057	−	1.43770i
$210$		0			0
$211$	−4.00000	+	6.92820i	−0.275371	+	0.476957i	−0.970229	−	0.242190i	$-0.922134\pi$
$211$	−4.00000	+	6.92820i	−0.275371	+	0.476957i	0.694857	+	0.719148i	$0.255467\pi$
$212$	−6.00000	+	10.3923i	−0.412082	+	0.713746i
$213$		0			0
$214$	−10.0000	−	17.3205i	−0.683586	−	1.18401i
$215$	16.0000			1.09119
$216$		0			0
$217$	4.00000			0.271538
$218$	5.00000	+	8.66025i	0.338643	+	0.586546i
$219$		0			0
$220$	−4.00000	+	6.92820i	−0.269680	+	0.467099i
$221$		0			0
$222$		0			0
$223$	−3.00000	−	5.19615i	−0.200895	−	0.347960i	0.747922	−	0.663786i	$-0.231052\pi$
$223$	−3.00000	−	5.19615i	−0.200895	−	0.347960i	−0.948817	+	0.315826i	$0.897718\pi$
$224$	2.00000			0.133631
$225$		0			0
$226$	−4.00000			−0.266076
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	0.595274	−	0.803523i	$-0.297043\pi$
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	−0.993508	+	0.113761i	$0.963710\pi$
$228$		0			0
$229$	15.0000	−	25.9808i	0.991228	−	1.71686i	0.381157	−	0.924510i	$-0.375526\pi$
$229$	15.0000	−	25.9808i	0.991228	−	1.71686i	0.610071	−	0.792347i	$-0.291141\pi$
$230$	−4.00000	+	6.92820i	−0.263752	+	0.456832i
$231$		0			0
$232$	−4.00000	−	6.92820i	−0.262613	−	0.454859i
$233$	20.0000			1.31024			0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000			1.31024			0.655122	+	0.755523i	$0.272617\pi$
$234$		0			0
$235$	−16.0000			−1.04372
$236$	−2.00000	−	3.46410i	−0.130189	−	0.225494i
$237$		0			0
$238$		0			0
$239$	12.0000	−	20.7846i	0.776215	−	1.34444i	−0.157893	−	0.987456i	$-0.550470\pi$
$239$	12.0000	−	20.7846i	0.776215	−	1.34444i	0.934109	−	0.356988i	$-0.116196\pi$
$240$		0			0
$241$	−3.00000	−	5.19615i	−0.193247	−	0.334714i	0.753077	−	0.657932i	$-0.228569\pi$
$241$	−3.00000	−	5.19615i	−0.193247	−	0.334714i	−0.946324	+	0.323218i	$0.895235\pi$
$242$	−5.00000			−0.321412
$243$		0			0
$244$	10.0000			0.640184
$245$	−3.00000	−	5.19615i	−0.191663	−	0.331970i
$246$		0			0
$247$	−3.00000	+	5.19615i	−0.190885	+	0.330623i
$248$	−1.00000	+	1.73205i	−0.0635001	+	0.109985i
$249$		0			0
$250$	6.00000	+	10.3923i	0.379473	+	0.657267i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−16.0000			−1.00591
$254$	−4.00000	−	6.92820i	−0.250982	−	0.434714i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−6.00000	+	10.3923i	−0.374270	+	0.648254i	−0.990217	−	0.139533i	$-0.955440\pi$
$257$	−6.00000	+	10.3923i	−0.374270	+	0.648254i	0.615948	+	0.787787i	$0.288773\pi$
$258$		0			0
$259$	6.00000	+	10.3923i	0.372822	+	0.645746i
$260$	2.00000			0.124035
$261$		0			0
$262$	−4.00000			−0.247121
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	12.0000	−	20.7846i	0.737154	−	1.27679i
$266$	6.00000	−	10.3923i	0.367884	−	0.637193i
$267$		0			0
$268$	1.00000	+	1.73205i	0.0610847	+	0.105802i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	22.0000			1.33640			0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000			1.33640			0.668202	+	0.743980i	$0.267064\pi$
$272$		0			0
$273$		0			0
$274$	−9.00000	+	15.5885i	−0.543710	+	0.941733i
$275$	−2.00000	+	3.46410i	−0.120605	+	0.208893i
$276$		0			0
$277$	3.00000	+	5.19615i	0.180253	+	0.312207i	0.941966	−	0.335707i	$-0.108975\pi$
$277$	3.00000	+	5.19615i	0.180253	+	0.312207i	−0.761714	+	0.647913i	$0.775642\pi$
$278$	20.0000			1.19952
$279$		0			0
$280$	−4.00000			−0.239046
$281$	5.00000	+	8.66025i	0.298275	+	0.516627i	0.975741	−	0.218926i	$-0.0702554\pi$
$281$	5.00000	+	8.66025i	0.298275	+	0.516627i	−0.677466	+	0.735554i	$0.736922\pi$
$282$		0			0
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	−0.399188	−	0.916869i	$-0.630708\pi$
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	0.993626	−	0.112728i	$-0.0359589\pi$
$284$	8.00000	−	13.8564i	0.474713	−	0.822226i
$285$		0			0
$286$	2.00000	+	3.46410i	0.118262	+	0.204837i
$287$	−12.0000			−0.708338
$288$		0			0
$289$	−17.0000			−1.00000
$290$	8.00000	+	13.8564i	0.469776	+	0.813676i
$291$		0			0
$292$	−7.00000	+	12.1244i	−0.409644	+	0.709524i
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	−0.893757	−	0.448552i	$-0.851940\pi$
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	0.835336	+	0.549740i	$0.185273\pi$
$294$		0			0
$295$	4.00000	+	6.92820i	0.232889	+	0.403376i
$296$	−6.00000			−0.348743
$297$		0			0
$298$	6.00000			0.347571
$299$	2.00000	+	3.46410i	0.115663	+	0.200334i
$300$		0			0
$301$	−8.00000	+	13.8564i	−0.461112	+	0.798670i
$302$	9.00000	−	15.5885i	0.517892	−	0.897015i
$303$		0			0
$304$	3.00000	+	5.19615i	0.172062	+	0.298020i
$305$	−20.0000			−1.14520
$306$		0			0
$307$	2.00000			0.114146			0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$	−4.00000	−	6.92820i	−0.227921	−	0.394771i
$309$		0			0
$310$	2.00000	−	3.46410i	0.113592	−	0.196748i
$311$	6.00000	−	10.3923i	0.340229	−	0.589294i	−0.644246	−	0.764818i	$-0.722829\pi$
$311$	6.00000	−	10.3923i	0.340229	−	0.589294i	0.984475	+	0.175525i	$0.0561621\pi$
$312$		0			0
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	−0.954810	−	0.297218i	$-0.903941\pi$
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	0.220006	−	0.975499i	$-0.429392\pi$
$314$	−2.00000			−0.112867
$315$		0			0
$316$	−4.00000			−0.225018
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	0.769395	−	0.638774i	$-0.220558\pi$
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	−0.937892	+	0.346929i	$0.887225\pi$
$318$		0			0
$319$	−16.0000	+	27.7128i	−0.895828	+	1.55162i
$320$	1.00000	−	1.73205i	0.0559017	−	0.0968246i
$321$		0			0
$322$	−4.00000	−	6.92820i	−0.222911	−	0.386094i
$323$		0			0
$324$		0			0
$325$	1.00000			0.0554700
$326$	−5.00000	−	8.66025i	−0.276924	−	0.479647i
$327$		0			0
$328$	3.00000	−	5.19615i	0.165647	−	0.286910i
$329$	8.00000	−	13.8564i	0.441054	−	0.763928i
$330$		0			0
$331$	5.00000	+	8.66025i	0.274825	+	0.476011i	0.970091	−	0.242742i	$-0.0780468\pi$
$331$	5.00000	+	8.66025i	0.274825	+	0.476011i	−0.695266	+	0.718752i	$0.744713\pi$
$332$	−12.0000			−0.658586
$333$		0			0
$334$		0			0
$335$	−2.00000	−	3.46410i	−0.109272	−	0.189264i
$336$		0			0
$337$	17.0000	−	29.4449i	0.926049	−	1.60396i	0.136184	−	0.990684i	$-0.456516\pi$
$337$	17.0000	−	29.4449i	0.926049	−	1.60396i	0.789865	−	0.613280i	$-0.210150\pi$
$338$	0.500000	−	0.866025i	0.0271964	−	0.0471056i
$339$		0			0
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$	20.0000			1.07990
$344$	−4.00000	−	6.92820i	−0.215666	−	0.373544i
$345$		0			0
$346$	8.00000	−	13.8564i	0.430083	−	0.744925i
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	−0.340293	−	0.940319i	$-0.610526\pi$
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	0.984487	−	0.175457i	$-0.0561403\pi$
$348$		0			0
$349$	−11.0000	−	19.0526i	−0.588817	−	1.01986i	−0.994388	−	0.105797i	$-0.966261\pi$
$349$	−11.0000	−	19.0526i	−0.588817	−	1.01986i	0.405571	−	0.914063i	$-0.367073\pi$
$350$	−2.00000			−0.106904
$351$		0			0
$352$	4.00000			0.213201
$353$	−7.00000	−	12.1244i	−0.372572	−	0.645314i	0.617388	−	0.786659i	$-0.288191\pi$
$353$	−7.00000	−	12.1244i	−0.372572	−	0.645314i	−0.989960	+	0.141344i	$0.954858\pi$
$354$		0			0
$355$	−16.0000	+	27.7128i	−0.849192	+	1.47084i
$356$	3.00000	−	5.19615i	0.159000	−	0.275396i
$357$		0			0
$358$	6.00000	+	10.3923i	0.317110	+	0.549250i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$	−9.00000	−	15.5885i	−0.473029	−	0.819311i
$363$		0			0
$364$	−1.00000	+	1.73205i	−0.0524142	+	0.0907841i
$365$	14.0000	−	24.2487i	0.732793	−	1.26924i
$366$		0			0
$367$	−4.00000	−	6.92820i	−0.208798	−	0.361649i	0.742538	−	0.669804i	$-0.233622\pi$
$367$	−4.00000	−	6.92820i	−0.208798	−	0.361649i	−0.951336	+	0.308155i	$0.900289\pi$
$368$	4.00000			0.208514
$369$		0			0
$370$	12.0000			0.623850
$371$	12.0000	+	20.7846i	0.623009	+	1.07908i
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$	4.00000	+	6.92820i	0.206284	+	0.357295i
$377$	8.00000			0.412021
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$	−6.00000	−	10.3923i	−0.307794	−	0.533114i
$381$		0			0
$382$	−2.00000	+	3.46410i	−0.102329	+	0.177239i
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$	8.00000	+	13.8564i	0.407718	+	0.706188i
$386$	−2.00000			−0.101797
$387$		0			0
$388$	−10.0000			−0.507673
$389$	6.00000	+	10.3923i	0.304212	+	0.526911i	0.977086	−	0.212847i	$-0.0682735\pi$
$389$	6.00000	+	10.3923i	0.304212	+	0.526911i	−0.672874	+	0.739758i	$0.734940\pi$
$390$		0			0
$391$		0			0
$392$	−1.50000	+	2.59808i	−0.0757614	+	0.131223i
$393$		0			0
$394$	−9.00000	−	15.5885i	−0.453413	−	0.785335i
$395$	8.00000			0.402524
$396$		0			0
$397$	−10.0000			−0.501886			−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000			−0.501886			−0.250943	+	0.968002i	$0.580741\pi$
$398$	4.00000	+	6.92820i	0.200502	+	0.347279i
$399$		0			0
$400$	0.500000	−	0.866025i	0.0250000	−	0.0433013i
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	−0.781345	−	0.624099i	$-0.785466\pi$
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	0.931158	+	0.364615i	$0.118800\pi$
$402$		0			0
$403$	−1.00000	−	1.73205i	−0.0498135	−	0.0862796i
$404$	−16.0000			−0.796030
$405$		0			0
$406$	−16.0000			−0.794067
$407$	12.0000	+	20.7846i	0.594818	+	1.03025i
$408$		0			0
$409$	13.0000	−	22.5167i	0.642809	−	1.11338i	−0.341994	−	0.939702i	$-0.611102\pi$
$409$	13.0000	−	22.5167i	0.642809	−	1.11338i	0.984803	−	0.173675i	$-0.0555643\pi$
$410$	−6.00000	+	10.3923i	−0.296319	+	0.513239i
$411$		0			0
$412$	−2.00000	−	3.46410i	−0.0985329	−	0.170664i
$413$	−8.00000			−0.393654
$414$		0			0
$415$	24.0000			1.17811
$416$	−0.500000	−	0.866025i	−0.0245145	−	0.0424604i
$417$		0			0
$418$	12.0000	−	20.7846i	0.586939	−	1.01661i
$419$	−18.0000	+	31.1769i	−0.879358	+	1.52309i	−0.0273103	+	0.999627i	$0.508694\pi$
$419$	−18.0000	+	31.1769i	−0.879358	+	1.52309i	−0.852047	+	0.523465i	$0.824639\pi$
$420$		0			0
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	0.718076	−	0.695965i	$-0.245023\pi$
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	−0.961761	+	0.273890i	$0.911690\pi$
$422$	−8.00000			−0.389434
$423$		0			0
$424$	−12.0000			−0.582772
$425$		0			0
$426$		0			0
$427$	10.0000	−	17.3205i	0.483934	−	0.838198i
$428$	10.0000	−	17.3205i	0.483368	−	0.837218i
$429$		0			0
$430$	8.00000	+	13.8564i	0.385794	+	0.668215i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$	2.00000	+	3.46410i	0.0960031	+	0.166282i
$435$		0			0
$436$	−5.00000	+	8.66025i	−0.239457	+	0.414751i
$437$	12.0000	−	20.7846i	0.574038	−	0.994263i
$438$		0			0
$439$	2.00000	+	3.46410i	0.0954548	+	0.165333i	0.909798	−	0.415051i	$-0.136236\pi$
$439$	2.00000	+	3.46410i	0.0954548	+	0.165333i	−0.814344	+	0.580383i	$0.802903\pi$
$440$	−8.00000			−0.381385
$441$		0			0
$442$		0			0
$443$	−18.0000	−	31.1769i	−0.855206	−	1.48126i	−0.876454	−	0.481486i	$-0.840097\pi$
$443$	−18.0000	−	31.1769i	−0.855206	−	1.48126i	0.0212481	−	0.999774i	$-0.493236\pi$
$444$		0			0
$445$	−6.00000	+	10.3923i	−0.284427	+	0.492642i
$446$	3.00000	−	5.19615i	0.142054	−	0.246045i
$447$		0			0
$448$	1.00000	+	1.73205i	0.0472456	+	0.0818317i
$449$	26.0000			1.22702			0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000			1.22702			0.613508	+	0.789689i	$0.289758\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$	−2.00000	−	3.46410i	−0.0940721	−	0.162938i
$453$		0			0
$454$	6.00000	−	10.3923i	0.281594	−	0.487735i
$455$	2.00000	−	3.46410i	0.0937614	−	0.162400i
$456$		0			0
$457$	1.00000	+	1.73205i	0.0467780	+	0.0810219i	0.888466	−	0.458942i	$-0.151771\pi$
$457$	1.00000	+	1.73205i	0.0467780	+	0.0810219i	−0.841688	+	0.539964i	$0.818438\pi$
$458$	30.0000			1.40181
$459$		0			0
$460$	−8.00000			−0.373002
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	−0.968945	−	0.247276i	$-0.920465\pi$
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	0.270326	−	0.962769i	$-0.412869\pi$
$462$		0			0
$463$	−11.0000	+	19.0526i	−0.511213	+	0.885448i	0.488702	+	0.872451i	$0.337470\pi$
$463$	−11.0000	+	19.0526i	−0.511213	+	0.885448i	−0.999916	+	0.0129968i	$0.995863\pi$
$464$	4.00000	−	6.92820i	0.185695	−	0.321634i
$465$		0			0
$466$	10.0000	+	17.3205i	0.463241	+	0.802357i
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$	−8.00000	−	13.8564i	−0.369012	−	0.639148i
$471$		0			0
$472$	2.00000	−	3.46410i	0.0920575	−	0.159448i
$473$	−16.0000	+	27.7128i	−0.735681	+	1.27424i
$474$		0			0
$475$	−3.00000	−	5.19615i	−0.137649	−	0.238416i
$476$		0			0
$477$		0			0
$478$	24.0000			1.09773
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	3.00000	−	5.19615i	0.136788	−	0.236924i
$482$	3.00000	−	5.19615i	0.136646	−	0.236678i
$483$		0			0
$484$	−2.50000	−	4.33013i	−0.113636	−	0.196824i
$485$	20.0000			0.908153
$486$		0			0
$487$	38.0000			1.72194			0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000			1.72194			0.860972	+	0.508652i	$0.169856\pi$
$488$	5.00000	+	8.66025i	0.226339	+	0.392031i
$489$		0			0
$490$	3.00000	−	5.19615i	0.135526	−	0.234738i
$491$	8.00000	−	13.8564i	0.361035	−	0.625331i	−0.627096	−	0.778942i	$-0.715757\pi$
$491$	8.00000	−	13.8564i	0.361035	−	0.625331i	0.988131	+	0.153611i	$0.0490902\pi$
$492$		0			0
$493$		0			0
$494$	−6.00000			−0.269953
$495$		0			0
$496$	−2.00000			−0.0898027
$497$	−16.0000	−	27.7128i	−0.717698	−	1.24309i
$498$		0			0
$499$	−11.0000	+	19.0526i	−0.492428	+	0.852910i	−0.999962	−	0.00872186i	$-0.997224\pi$
$499$	−11.0000	+	19.0526i	−0.492428	+	0.852910i	0.507534	+	0.861632i	$0.330557\pi$
$500$	−6.00000	+	10.3923i	−0.268328	+	0.464758i
$501$		0			0
$502$		0			0
$503$	32.0000			1.42681			0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000			1.42681			0.713405	+	0.700752i	$0.247152\pi$
$504$		0			0
$505$	32.0000			1.42398
$506$	−8.00000	−	13.8564i	−0.355643	−	0.615992i
$507$		0			0
$508$	4.00000	−	6.92820i	0.177471	−	0.307389i
$509$	5.00000	−	8.66025i	0.221621	−	0.383859i	−0.733679	−	0.679496i	$-0.762199\pi$
$509$	5.00000	−	8.66025i	0.221621	−	0.383859i	0.955300	+	0.295637i	$0.0955319\pi$
$510$		0			0
$511$	14.0000	+	24.2487i	0.619324	+	1.07270i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−12.0000			−0.529297
$515$	4.00000	+	6.92820i	0.176261	+	0.305293i
$516$		0			0
$517$	16.0000	−	27.7128i	0.703679	−	1.21881i
$518$	−6.00000	+	10.3923i	−0.263625	+	0.456612i
$519$		0			0
$520$	1.00000	+	1.73205i	0.0438529	+	0.0759555i
$521$	−36.0000			−1.57719			−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000			−1.57719			−0.788594	+	0.614914i	$0.789191\pi$
$522$		0			0
$523$	−12.0000			−0.524723			−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000			−0.524723			−0.262362	+	0.964970i	$0.584501\pi$
$524$	−2.00000	−	3.46410i	−0.0873704	−	0.151330i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$	24.0000			1.04249
$531$		0			0
$532$	12.0000			0.520266
$533$	3.00000	+	5.19615i	0.129944	+	0.225070i
$534$		0			0
$535$	−20.0000	+	34.6410i	−0.864675	+	1.49766i
$536$	−1.00000	+	1.73205i	−0.0431934	+	0.0748132i
$537$		0			0
$538$		0			0
$539$	12.0000			0.516877
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$	11.0000	+	19.0526i	0.472490	+	0.818377i
$543$		0			0
$544$		0			0
$545$	10.0000	−	17.3205i	0.428353	−	0.741929i
$546$		0			0
$547$	10.0000	+	17.3205i	0.427569	+	0.740571i	0.996657	−	0.0817056i	$-0.0260367\pi$
$547$	10.0000	+	17.3205i	0.427569	+	0.740571i	−0.569087	+	0.822277i	$0.692703\pi$
$548$	−18.0000			−0.768922
$549$		0			0
$550$	−4.00000			−0.170561
$551$	−24.0000	−	41.5692i	−1.02243	−	1.77091i
$552$		0			0
$553$	−4.00000	+	6.92820i	−0.170097	+	0.294617i
$554$	−3.00000	+	5.19615i	−0.127458	+	0.220763i
$555$		0			0
$556$	10.0000	+	17.3205i	0.424094	+	0.734553i
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$	−2.00000	−	3.46410i	−0.0845154	−	0.146385i
$561$		0			0
$562$	−5.00000	+	8.66025i	−0.210912	+	0.365311i
$563$	10.0000	−	17.3205i	0.421450	−	0.729972i	−0.574632	−	0.818412i	$-0.694855\pi$
$563$	10.0000	−	17.3205i	0.421450	−	0.729972i	0.996082	+	0.0884397i	$0.0281881\pi$
$564$		0			0
$565$	4.00000	+	6.92820i	0.168281	+	0.291472i
$566$	20.0000			0.840663
$567$		0			0
$568$	16.0000			0.671345
$569$	2.00000	+	3.46410i	0.0838444	+	0.145223i	0.904898	−	0.425628i	$-0.139947\pi$
$569$	2.00000	+	3.46410i	0.0838444	+	0.145223i	−0.821054	+	0.570851i	$0.806613\pi$
$570$		0			0
$571$	−18.0000	+	31.1769i	−0.753277	+	1.30471i	0.192950	+	0.981209i	$0.438194\pi$
$571$	−18.0000	+	31.1769i	−0.753277	+	1.30471i	−0.946227	+	0.323505i	$0.895139\pi$
$572$	−2.00000	+	3.46410i	−0.0836242	+	0.144841i
$573$		0			0
$574$	−6.00000	−	10.3923i	−0.250435	−	0.433766i
$575$	−4.00000			−0.166812
$576$		0			0
$577$	−30.0000			−1.24892			−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000			−1.24892			−0.624458	+	0.781058i	$0.714680\pi$
$578$	−8.50000	−	14.7224i	−0.353553	−	0.612372i
$579$		0			0
$580$	−8.00000	+	13.8564i	−0.332182	+	0.575356i
$581$	−12.0000	+	20.7846i	−0.497844	+	0.862291i
$582$		0			0
$583$	24.0000	+	41.5692i	0.993978	+	1.72162i
$584$	−14.0000			−0.579324
$585$		0			0
$586$	−2.00000			−0.0826192
$587$	−14.0000	−	24.2487i	−0.577842	−	1.00085i	−0.995726	−	0.0923513i	$-0.970562\pi$
$587$	−14.0000	−	24.2487i	−0.577842	−	1.00085i	0.417885	−	0.908500i	$-0.362772\pi$
$588$		0			0
$589$	−6.00000	+	10.3923i	−0.247226	+	0.428207i
$590$	−4.00000	+	6.92820i	−0.164677	+	0.285230i
$591$		0			0
$592$	−3.00000	−	5.19615i	−0.123299	−	0.213561i
$593$	−38.0000			−1.56047			−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000			−1.56047			−0.780236	+	0.625485i	$0.784901\pi$
$594$		0			0
$595$		0			0
$596$	3.00000	+	5.19615i	0.122885	+	0.212843i
$597$		0			0
$598$	−2.00000	+	3.46410i	−0.0817861	+	0.141658i
$599$	20.0000	−	34.6410i	0.817178	−	1.41539i	−0.0905757	−	0.995890i	$-0.528871\pi$
$599$	20.0000	−	34.6410i	0.817178	−	1.41539i	0.907754	−	0.419504i	$-0.137796\pi$
$600$		0			0
$601$	−5.00000	−	8.66025i	−0.203954	−	0.353259i	0.745845	−	0.666120i	$-0.232046\pi$
$601$	−5.00000	−	8.66025i	−0.203954	−	0.353259i	−0.949799	+	0.312861i	$0.898713\pi$
$602$	−16.0000			−0.652111
$603$		0			0
$604$	18.0000			0.732410
$605$	5.00000	+	8.66025i	0.203279	+	0.352089i
$606$		0			0
$607$	−12.0000	+	20.7846i	−0.487065	+	0.843621i	−0.999889	−	0.0148722i	$-0.995266\pi$
$607$	−12.0000	+	20.7846i	−0.487065	+	0.843621i	0.512824	+	0.858494i	$0.328599\pi$
$608$	−3.00000	+	5.19615i	−0.121666	+	0.210732i
$609$		0			0
$610$	−10.0000	−	17.3205i	−0.404888	−	0.701287i
$611$	−8.00000			−0.323645
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$	1.00000	+	1.73205i	0.0403567	+	0.0698999i
$615$		0			0
$616$	4.00000	−	6.92820i	0.161165	−	0.279145i
$617$	−7.00000	+	12.1244i	−0.281809	+	0.488108i	−0.971830	−	0.235681i	$-0.924268\pi$
$617$	−7.00000	+	12.1244i	−0.281809	+	0.488108i	0.690021	+	0.723789i	$0.257601\pi$
$618$		0			0
$619$	−17.0000	−	29.4449i	−0.683288	−	1.18349i	−0.973972	−	0.226670i	$-0.927216\pi$
$619$	−17.0000	−	29.4449i	−0.683288	−	1.18349i	0.290684	−	0.956819i	$-0.406117\pi$
$620$	4.00000			0.160644
$621$		0			0
$622$	12.0000			0.481156
$623$	−6.00000	−	10.3923i	−0.240385	−	0.416359i
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$	13.0000	−	22.5167i	0.519584	−	0.899947i
$627$		0			0
$628$	−1.00000	−	1.73205i	−0.0399043	−	0.0691164i
$629$		0			0
$630$		0			0
$631$	−14.0000			−0.557331			−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000			−0.557331			−0.278666	+	0.960388i	$0.589892\pi$
$632$	−2.00000	−	3.46410i	−0.0795557	−	0.137795i
$633$		0			0
$634$	3.00000	−	5.19615i	0.119145	−	0.206366i
$635$	−8.00000	+	13.8564i	−0.317470	+	0.549875i
$636$		0			0
$637$	−1.50000	−	2.59808i	−0.0594322	−	0.102940i
$638$	−32.0000			−1.26689
$639$		0			0
$640$	2.00000			0.0790569
$641$	20.0000	+	34.6410i	0.789953	+	1.36824i	0.925995	+	0.377535i	$0.123228\pi$
$641$	20.0000	+	34.6410i	0.789953	+	1.36824i	−0.136043	+	0.990703i	$0.543438\pi$
$642$		0			0
$643$	5.00000	−	8.66025i	0.197181	−	0.341527i	−0.750432	−	0.660947i	$-0.770155\pi$
$643$	5.00000	−	8.66025i	0.197181	−	0.341527i	0.947613	+	0.319420i	$0.103488\pi$
$644$	4.00000	−	6.92820i	0.157622	−	0.273009i
$645$		0			0
$646$		0			0
$647$	8.00000			0.314512			0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000			0.314512			0.157256	+	0.987558i	$0.449735\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$	0.500000	+	0.866025i	0.0196116	+	0.0339683i
$651$		0			0
$652$	5.00000	−	8.66025i	0.195815	−	0.339162i
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$	4.00000	+	6.92820i	0.156293	+	0.270707i
$656$	6.00000			0.234261
$657$		0			0
$658$	16.0000			0.623745
$659$	24.0000	+	41.5692i	0.934907	+	1.61931i	0.774799	+	0.632207i	$0.217851\pi$
$659$	24.0000	+	41.5692i	0.934907	+	1.61931i	0.160108	+	0.987099i	$0.448816\pi$
$660$		0			0
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	−0.969442	−	0.245319i	$-0.921107\pi$
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	0.697174	+	0.716902i	$0.254441\pi$
$662$	−5.00000	+	8.66025i	−0.194331	+	0.336590i
$663$		0			0
$664$	−6.00000	−	10.3923i	−0.232845	−	0.403300i
$665$	−24.0000			−0.930680
$666$		0			0
$667$	−32.0000			−1.23904
$668$		0			0
$669$		0			0
$670$	2.00000	−	3.46410i	0.0772667	−	0.133830i
$671$	20.0000	−	34.6410i	0.772091	−	1.33730i
$672$		0			0
$673$	−11.0000	−	19.0526i	−0.424019	−	0.734422i	0.572309	−	0.820038i	$-0.306048\pi$
$673$	−11.0000	−	19.0526i	−0.424019	−	0.734422i	−0.996328	+	0.0856156i	$0.972714\pi$
$674$	34.0000			1.30963
$675$		0			0
$676$	1.00000			0.0384615
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	0.957984	−	0.286820i	$-0.0925982\pi$
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	−0.727386	+	0.686229i	$0.759265\pi$
$678$		0			0
$679$	−10.0000	+	17.3205i	−0.383765	+	0.664700i
$680$		0			0
$681$		0			0
$682$	4.00000	+	6.92820i	0.153168	+	0.265295i
$683$	−4.00000			−0.153056			−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000			−0.153056			−0.0765279	+	0.997067i	$0.524383\pi$
$684$		0			0
$685$	36.0000			1.37549
$686$	10.0000	+	17.3205i	0.381802	+	0.661300i
$687$		0			0
$688$	4.00000	−	6.92820i	0.152499	−	0.264135i
$689$	6.00000	−	10.3923i	0.228582	−	0.395915i
$690$		0			0
$691$	13.0000	+	22.5167i	0.494543	+	0.856574i	0.999980	−	0.00628943i	$-0.00200200\pi$
$691$	13.0000	+	22.5167i	0.494543	+	0.856574i	−0.505437	+	0.862864i	$0.668669\pi$
$692$	16.0000			0.608229
$693$		0			0
$694$	24.0000			0.911028
$695$	−20.0000	−	34.6410i	−0.758643	−	1.31401i
$696$		0			0
$697$		0			0
$698$	11.0000	−	19.0526i	0.416356	−	0.721150i
$699$		0			0
$700$	−1.00000	−	1.73205i	−0.0377964	−	0.0654654i
$701$	−12.0000			−0.453234			−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000			−0.453234			−0.226617	+	0.973984i	$0.572767\pi$
$702$		0			0
$703$	−36.0000			−1.35777
$704$	2.00000	+	3.46410i	0.0753778	+	0.130558i
$705$		0			0
$706$	7.00000	−	12.1244i	0.263448	−	0.456306i
$707$	−16.0000	+	27.7128i	−0.601742	+	1.04225i
$708$		0			0
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	0.995228	−	0.0975728i	$-0.0311079\pi$
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	−0.582115	+	0.813107i	$0.697775\pi$
$710$	−32.0000			−1.20094
$711$		0			0
$712$	6.00000			0.224860
$713$	4.00000	+	6.92820i	0.149801	+	0.259463i
$714$		0			0
$715$	4.00000	−	6.92820i	0.149592	−	0.259100i
$716$	−6.00000	+	10.3923i	−0.224231	+	0.388379i
$717$		0			0
$718$		0			0
$719$	28.0000			1.04422			0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000			1.04422			0.522112	+	0.852877i	$0.325144\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$	8.50000	+	14.7224i	0.316337	+	0.547912i
$723$		0			0
$724$	9.00000	−	15.5885i	0.334482	−	0.579340i
$725$	−4.00000	+	6.92820i	−0.148556	+	0.257307i
$726$		0			0
$727$	4.00000	+	6.92820i	0.148352	+	0.256953i	0.930618	−	0.365991i	$-0.119270\pi$
$727$	4.00000	+	6.92820i	0.148352	+	0.256953i	−0.782267	+	0.622944i	$0.785937\pi$
$728$	−2.00000			−0.0741249
$729$		0			0
$730$	28.0000			1.03633
$731$		0			0
$732$		0			0
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	−0.588177	−	0.808732i	$-0.700154\pi$
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	0.994471	+	0.105010i	$0.0334875\pi$
$734$	4.00000	−	6.92820i	0.147643	−	0.255725i
$735$		0			0
$736$	2.00000	+	3.46410i	0.0737210	+	0.127688i
$737$	8.00000			0.294684
$738$		0			0
$739$	−54.0000			−1.98642			−0.993211	−	0.116326i	$-0.962888\pi$
$739$	−54.0000			−1.98642			−0.993211	+	0.116326i	$0.962888\pi$
$740$	6.00000	+	10.3923i	0.220564	+	0.382029i
$741$		0			0
$742$	−12.0000	+	20.7846i	−0.440534	+	0.763027i
$743$	20.0000	−	34.6410i	0.733729	−	1.27086i	−0.221550	−	0.975149i	$-0.571112\pi$
$743$	20.0000	−	34.6410i	0.733729	−	1.27086i	0.955279	−	0.295707i	$-0.0955551\pi$
$744$		0			0
$745$	−6.00000	−	10.3923i	−0.219823	−	0.380745i
$746$	10.0000			0.366126
$747$		0			0
$748$		0			0
$749$	−20.0000	−	34.6410i	−0.730784	−	1.26576i
$750$		0			0
$751$	6.00000	−	10.3923i	0.218943	−	0.379221i	−0.735542	−	0.677479i	$-0.763072\pi$
$751$	6.00000	−	10.3923i	0.218943	−	0.379221i	0.954485	+	0.298259i	$0.0964058\pi$
$752$	−4.00000	+	6.92820i	−0.145865	+	0.252646i
$753$		0			0
$754$	4.00000	+	6.92820i	0.145671	+	0.252310i
$755$	−36.0000			−1.31017
$756$		0			0
$757$	−18.0000			−0.654221			−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000			−0.654221			−0.327111	+	0.944986i	$0.606075\pi$
$758$	5.00000	+	8.66025i	0.181608	+	0.314555i
$759$		0			0
$760$	6.00000	−	10.3923i	0.217643	−	0.376969i
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	0.454935	+	0.890525i	$0.349663\pi$
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	−0.998684	+	0.0512772i	$0.983671\pi$
$762$		0			0
$763$	10.0000	+	17.3205i	0.362024	+	0.627044i
$764$	−4.00000			−0.144715
$765$		0			0
$766$		0			0
$767$	2.00000	+	3.46410i	0.0722158	+	0.125081i
$768$		0			0
$769$	15.0000	−	25.9808i	0.540914	−	0.936890i	−0.457938	−	0.888984i	$-0.651412\pi$
$769$	15.0000	−	25.9808i	0.540914	−	0.936890i	0.998852	−	0.0479061i	$-0.0152548\pi$
$770$	−8.00000	+	13.8564i	−0.288300	+	0.499350i
$771$		0			0
$772$	−1.00000	−	1.73205i	−0.0359908	−	0.0623379i
$773$	−14.0000			−0.503545			−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000			−0.503545			−0.251773	+	0.967786i	$0.581013\pi$
$774$		0			0
$775$	2.00000			0.0718421
$776$	−5.00000	−	8.66025i	−0.179490	−	0.310885i
$777$		0			0
$778$	−6.00000	+	10.3923i	−0.215110	+	0.372582i
$779$	18.0000	−	31.1769i	0.644917	−	1.11703i
$780$		0			0
$781$	−32.0000	−	55.4256i	−1.14505	−	1.98328i
$782$		0			0
$783$		0			0
$784$	−3.00000			−0.107143
$785$	2.00000	+	3.46410i	0.0713831	+	0.123639i
$786$		0			0
$787$	11.0000	−	19.0526i	0.392108	−	0.679150i	−0.600620	−	0.799535i	$-0.705079\pi$
$787$	11.0000	−	19.0526i	0.392108	−	0.679150i	0.992727	+	0.120384i	$0.0384127\pi$
$788$	9.00000	−	15.5885i	0.320612	−	0.555316i
$789$		0			0
$790$	4.00000	+	6.92820i	0.142314	+	0.246494i
$791$	−8.00000			−0.284447
$792$		0			0
$793$	−10.0000			−0.355110
$794$	−5.00000	−	8.66025i	−0.177443	−	0.307341i
$795$		0			0
$796$	−4.00000	+	6.92820i	−0.141776	+	0.245564i
$797$	6.00000	−	10.3923i	0.212531	−	0.368114i	−0.739975	−	0.672634i	$-0.765163\pi$
$797$	6.00000	−	10.3923i	0.212531	−	0.368114i	0.952506	+	0.304520i	$0.0984960\pi$
$798$		0			0
$799$		0			0
$800$	1.00000			0.0353553
$801$		0			0
$802$	6.00000			0.211867
$803$	28.0000	+	48.4974i	0.988099	+	1.71144i
$804$		0			0
$805$	−8.00000	+	13.8564i	−0.281963	+	0.488374i
$806$	1.00000	−	1.73205i	0.0352235	−	0.0610089i
$807$		0			0
$808$	−8.00000	−	13.8564i	−0.281439	−	0.487467i
$809$	36.0000			1.26569			0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000			1.26569			0.632846	+	0.774277i	$0.281886\pi$
$810$		0			0
$811$	50.0000			1.75574			0.877869	−	0.478901i	$-0.158965\pi$
$811$	50.0000			1.75574			0.877869	+	0.478901i	$0.158965\pi$
$812$	−8.00000	−	13.8564i	−0.280745	−	0.486265i
$813$		0			0
$814$	−12.0000	+	20.7846i	−0.420600	+	0.728500i
$815$	−10.0000	+	17.3205i	−0.350285	+	0.606711i
$816$		0			0
$817$	−24.0000	−	41.5692i	−0.839654	−	1.45432i
$818$	26.0000			0.909069
$819$		0			0
$820$	−12.0000			−0.419058
$821$	−17.0000	−	29.4449i	−0.593304	−	1.02763i	−0.993784	−	0.111327i	$-0.964490\pi$
$821$	−17.0000	−	29.4449i	−0.593304	−	1.02763i	0.400480	−	0.916306i	$-0.368843\pi$
$822$		0			0
$823$	6.00000	−	10.3923i	0.209147	−	0.362253i	−0.742299	−	0.670069i	$-0.766265\pi$
$823$	6.00000	−	10.3923i	0.209147	−	0.362253i	0.951446	+	0.307816i	$0.0995980\pi$
$824$	2.00000	−	3.46410i	0.0696733	−	0.120678i
$825$		0			0
$826$	−4.00000	−	6.92820i	−0.139178	−	0.241063i
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	46.0000			1.59765			0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000			1.59765			0.798823	+	0.601566i	$0.205456\pi$
$830$	12.0000	+	20.7846i	0.416526	+	0.721444i
$831$		0			0
$832$	0.500000	−	0.866025i	0.0173344	−	0.0300240i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	24.0000			0.830057
$837$		0			0
$838$	−36.0000			−1.24360
$839$	−8.00000	−	13.8564i	−0.276191	−	0.478376i	0.694244	−	0.719740i	$-0.255739\pi$
$839$	−8.00000	−	13.8564i	−0.276191	−	0.478376i	−0.970435	+	0.241363i	$0.922405\pi$
$840$		0			0
$841$	−17.5000	+	30.3109i	−0.603448	+	1.04520i
$842$	5.00000	−	8.66025i	0.172311	−	0.298452i
$843$		0			0
$844$	−4.00000	−	6.92820i	−0.137686	−	0.238479i
$845$	−2.00000			−0.0688021
$846$		0			0
$847$	−10.0000			−0.343604
$848$	−6.00000	−	10.3923i	−0.206041	−	0.356873i
$849$		0			0
$850$		0			0
$851$	−12.0000	+	20.7846i	−0.411355	+	0.712487i
$852$		0			0
$853$	13.0000	+	22.5167i	0.445112	+	0.770956i	0.998060	−	0.0622597i	$-0.0198307\pi$
$853$	13.0000	+	22.5167i	0.445112	+	0.770956i	−0.552948	+	0.833215i	$0.686497\pi$
$854$	20.0000			0.684386
$855$		0			0
$856$	20.0000			0.683586
$857$	14.0000	+	24.2487i	0.478231	+	0.828320i	0.999689	−	0.0249570i	$-0.00794488\pi$
$857$	14.0000	+	24.2487i	0.478231	+	0.828320i	−0.521458	+	0.853277i	$0.674612\pi$
$858$		0			0
$859$	−4.00000	+	6.92820i	−0.136478	+	0.236387i	−0.926161	−	0.377128i	$-0.876912\pi$
$859$	−4.00000	+	6.92820i	−0.136478	+	0.236387i	0.789683	+	0.613515i	$0.210245\pi$
$860$	−8.00000	+	13.8564i	−0.272798	+	0.472500i
$861$		0			0
$862$		0			0
$863$	8.00000			0.272323			0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000			0.272323			0.136162	+	0.990687i	$0.456523\pi$
$864$		0			0
$865$	−32.0000			−1.08803
$866$	−1.00000	−	1.73205i	−0.0339814	−	0.0588575i
$867$		0			0
$868$	−2.00000	+	3.46410i	−0.0678844	+	0.117579i
$869$	−8.00000	+	13.8564i	−0.271381	+	0.470046i
$870$		0			0
$871$	−1.00000	−	1.73205i	−0.0338837	−	0.0586883i
$872$	−10.0000			−0.338643
$873$		0			0
$874$	24.0000			0.811812
$875$	12.0000	+	20.7846i	0.405674	+	0.702648i
$876$		0			0
$877$	−3.00000	+	5.19615i	−0.101303	+	0.175462i	−0.912222	−	0.409697i	$-0.865634\pi$
$877$	−3.00000	+	5.19615i	−0.101303	+	0.175462i	0.810919	+	0.585159i	$0.198968\pi$
$878$	−2.00000	+	3.46410i	−0.0674967	+	0.116908i
$879$		0			0
$880$	−4.00000	−	6.92820i	−0.134840	−	0.233550i
$881$	36.0000			1.21287			0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000			1.21287			0.606435	+	0.795133i	$0.292599\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$		0			0
$886$	18.0000	−	31.1769i	0.604722	−	1.04741i
$887$	−22.0000	+	38.1051i	−0.738688	+	1.27944i	0.214399	+	0.976746i	$0.431221\pi$
$887$	−22.0000	+	38.1051i	−0.738688	+	1.27944i	−0.953086	+	0.302698i	$0.902113\pi$
$888$		0			0
$889$	−8.00000	−	13.8564i	−0.268311	−	0.464729i
$890$	−12.0000			−0.402241
$891$		0			0
$892$	6.00000			0.200895
$893$	24.0000	+	41.5692i	0.803129	+	1.39106i
$894$		0			0
$895$	12.0000	−	20.7846i	0.401116	−	0.694753i
$896$	−1.00000	+	1.73205i	−0.0334077	+	0.0578638i
$897$		0			0
$898$	13.0000	+	22.5167i	0.433816	+	0.751391i
$899$	16.0000			0.533630
$900$		0			0
$901$		0			0
$902$	−12.0000	−	20.7846i	−0.399556	−	0.692052i
$903$		0			0
$904$	2.00000	−	3.46410i	0.0665190	−	0.115214i
$905$	−18.0000	+	31.1769i	−0.598340	+	1.03636i
$906$		0			0
$907$	−6.00000	−	10.3923i	−0.199227	−	0.345071i	0.749051	−	0.662512i	$-0.230510\pi$
$907$	−6.00000	−	10.3923i	−0.199227	−	0.345071i	−0.948278	+	0.317441i	$0.897176\pi$
$908$	12.0000			0.398234
$909$		0			0
$910$	4.00000			0.132599
$911$	−6.00000	−	10.3923i	−0.198789	−	0.344312i	0.749347	−	0.662177i	$-0.230367\pi$
$911$	−6.00000	−	10.3923i	−0.198789	−	0.344312i	−0.948136	+	0.317865i	$0.897034\pi$
$912$		0			0
$913$	−24.0000	+	41.5692i	−0.794284	+	1.37574i
$914$	−1.00000	+	1.73205i	−0.0330771	+	0.0572911i
$915$		0			0
$916$	15.0000	+	25.9808i	0.495614	+	0.858429i
$917$	−8.00000			−0.264183
$918$		0			0
$919$	12.0000			0.395843			0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000			0.395843			0.197922	+	0.980218i	$0.436581\pi$
$920$	−4.00000	−	6.92820i	−0.131876	−	0.228416i
$921$		0			0
$922$	15.0000	−	25.9808i	0.493999	−	0.855631i
$923$	−8.00000	+	13.8564i	−0.263323	+	0.456089i
$924$		0			0
$925$	3.00000	+	5.19615i	0.0986394	+	0.170848i
$926$	−22.0000			−0.722965
$927$		0			0
$928$	8.00000			0.262613
$929$	−15.0000	−	25.9808i	−0.492134	−	0.852401i	0.507825	−	0.861460i	$-0.330450\pi$
$929$	−15.0000	−	25.9808i	−0.492134	−	0.852401i	−0.999959	+	0.00905914i	$0.997116\pi$
$930$		0			0
$931$	−9.00000	+	15.5885i	−0.294963	+	0.510891i
$932$	−10.0000	+	17.3205i	−0.327561	+	0.567352i
$933$		0			0
$934$	−4.00000	−	6.92820i	−0.130884	−	0.226698i
$935$		0			0
$936$		0			0
$937$	30.0000			0.980057			0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000			0.980057			0.490029	+	0.871706i	$0.336986\pi$
$938$	2.00000	+	3.46410i	0.0653023	+	0.113107i
$939$		0			0
$940$	8.00000	−	13.8564i	0.260931	−	0.451946i
$941$	−13.0000	+	22.5167i	−0.423788	+	0.734022i	−0.996306	−	0.0858697i	$-0.972633\pi$
$941$	−13.0000	+	22.5167i	−0.423788	+	0.734022i	0.572518	+	0.819892i	$0.305966\pi$
$942$		0			0
$943$	−12.0000	−	20.7846i	−0.390774	−	0.676840i
$944$	4.00000			0.130189
$945$		0			0
$946$	−32.0000			−1.04041
$947$	−18.0000	−	31.1769i	−0.584921	−	1.01311i	−0.994885	−	0.101012i	$-0.967792\pi$
$947$	−18.0000	−	31.1769i	−0.584921	−	1.01311i	0.409964	−	0.912102i	$-0.365541\pi$
$948$		0			0
$949$	7.00000	−	12.1244i	0.227230	−	0.393573i
$950$	3.00000	−	5.19615i	0.0973329	−	0.168585i
$951$		0			0
$952$		0			0
$953$	28.0000			0.907009			0.453504	−	0.891254i	$-0.350174\pi$
$953$	28.0000			0.907009			0.453504	+	0.891254i	$0.350174\pi$
$954$		0			0
$955$	8.00000			0.258874
$956$	12.0000	+	20.7846i	0.388108	+	0.672222i
$957$		0			0
$958$		0			0
$959$	−18.0000	+	31.1769i	−0.581250	+	1.00676i
$960$		0			0
$961$	13.5000	+	23.3827i	0.435484	+	0.754280i
$962$	6.00000			0.193448
$963$		0			0
$964$	6.00000			0.193247
$965$	2.00000	+	3.46410i	0.0643823	+	0.111513i
$966$		0			0
$967$	29.0000	−	50.2295i	0.932577	−	1.61527i	0.153679	−	0.988121i	$-0.450888\pi$
$967$	29.0000	−	50.2295i	0.932577	−	1.61527i	0.778898	−	0.627150i	$-0.215779\pi$
$968$	2.50000	−	4.33013i	0.0803530	−	0.139176i
$969$		0			0
$970$	10.0000	+	17.3205i	0.321081	+	0.556128i
$971$	48.0000			1.54039			0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000			1.54039			0.770197	+	0.637806i	$0.220158\pi$
$972$		0			0
$973$	40.0000			1.28234
$974$	19.0000	+	32.9090i	0.608799	+	1.05447i
$975$		0			0
$976$	−5.00000	+	8.66025i	−0.160046	+	0.277208i
$977$	−9.00000	+	15.5885i	−0.287936	+	0.498719i	−0.973317	−	0.229465i	$-0.926302\pi$
$977$	−9.00000	+	15.5885i	−0.287936	+	0.498719i	0.685381	+	0.728184i	$0.259636\pi$
$978$		0			0
$979$	−12.0000	−	20.7846i	−0.383522	−	0.664279i
$980$	6.00000			0.191663
$981$		0			0
$982$	16.0000			0.510581
$983$	16.0000	+	27.7128i	0.510321	+	0.883901i	0.999928	+	0.0119587i	$0.00380665\pi$
$983$	16.0000	+	27.7128i	0.510321	+	0.883901i	−0.489608	+	0.871943i	$0.662860\pi$
$984$		0			0
$985$	−18.0000	+	31.1769i	−0.573528	+	0.993379i
$986$		0			0
$987$		0			0
$988$	−3.00000	−	5.19615i	−0.0954427	−	0.165312i
$989$	−32.0000			−1.01754
$990$		0			0
$991$	−48.0000			−1.52477			−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000			−1.52477			−0.762385	+	0.647124i	$0.775972\pi$
$992$	−1.00000	−	1.73205i	−0.0317500	−	0.0549927i
$993$		0			0
$994$	16.0000	−	27.7128i	0.507489	−	0.878997i
$995$	8.00000	−	13.8564i	0.253617	−	0.439278i
$996$		0			0
$997$	−19.0000	−	32.9090i	−0.601736	−	1.04224i	−0.992558	−	0.121771i	$-0.961143\pi$
$997$	−19.0000	−	32.9090i	−0.601736	−	1.04224i	0.390822	−	0.920466i	$-0.372191\pi$
$998$	−22.0000			−0.696398
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.e.z.1405.1		2
3.2	odd	2		2106.2.e.e.1405.1		2
9.2	odd	6		2106.2.e.e.703.1		2
9.4	even	3		234.2.a.a.1.1	✓	1
9.5	odd	6		234.2.a.d.1.1	yes	1
9.7	even	3	inner	2106.2.e.z.703.1		2
36.23	even	6		1872.2.a.p.1.1		1
36.31	odd	6		1872.2.a.g.1.1		1
45.4	even	6		5850.2.a.bv.1.1		1
45.13	odd	12		5850.2.e.d.5149.2		2
45.14	odd	6		5850.2.a.v.1.1		1
45.22	odd	12		5850.2.e.d.5149.1		2
45.23	even	12		5850.2.e.bd.5149.1		2
45.32	even	12		5850.2.e.bd.5149.2		2
72.5	odd	6		7488.2.a.j.1.1		1
72.13	even	6		7488.2.a.bp.1.1		1
72.59	even	6		7488.2.a.s.1.1		1
72.67	odd	6		7488.2.a.bu.1.1		1
117.5	even	12		3042.2.b.b.1351.1		2
117.31	odd	12		3042.2.b.c.1351.2		2
117.77	odd	6		3042.2.a.b.1.1		1
117.86	even	12		3042.2.b.b.1351.2		2
117.103	even	6		3042.2.a.o.1.1		1
117.112	odd	12		3042.2.b.c.1351.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.a.a.1.1	✓	1	9.4	even	3
234.2.a.d.1.1	yes	1	9.5	odd	6
1872.2.a.g.1.1		1	36.31	odd	6
1872.2.a.p.1.1		1	36.23	even	6
2106.2.e.e.703.1		2	9.2	odd	6
2106.2.e.e.1405.1		2	3.2	odd	2
2106.2.e.z.703.1		2	9.7	even	3	inner
2106.2.e.z.1405.1		2	1.1	even	1	trivial
3042.2.a.b.1.1		1	117.77	odd	6
3042.2.a.o.1.1		1	117.103	even	6
3042.2.b.b.1351.1		2	117.5	even	12
3042.2.b.b.1351.2		2	117.86	even	12
3042.2.b.c.1351.1		2	117.112	odd	12
3042.2.b.c.1351.2		2	117.31	odd	12
5850.2.a.v.1.1		1	45.14	odd	6
5850.2.a.bv.1.1		1	45.4	even	6
5850.2.e.d.5149.1		2	45.22	odd	12
5850.2.e.d.5149.2		2	45.13	odd	12
5850.2.e.bd.5149.1		2	45.23	even	12
5850.2.e.bd.5149.2		2	45.32	even	12
7488.2.a.j.1.1		1	72.5	odd	6
7488.2.a.s.1.1		1	72.59	even	6
7488.2.a.bp.1.1		1	72.13	even	6
7488.2.a.bu.1.1		1	72.67	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:	\( N \)	\(=\)	\( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2106.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +(1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +2.00000 q^{10} +(2.00000 + 3.46410i) q^{11} +(0.500000 - 0.866025i) q^{13} +(-1.00000 + 1.73205i) q^{14} +(-0.500000 - 0.866025i) q^{16} -6.00000 q^{19} +(1.00000 + 1.73205i) q^{20} +(-2.00000 + 3.46410i) q^{22} +(-2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +1.00000 q^{26} -2.00000 q^{28} +(4.00000 + 6.92820i) q^{29} +(1.00000 - 1.73205i) q^{31} +(0.500000 - 0.866025i) q^{32} +4.00000 q^{35} +6.00000 q^{37} +(-3.00000 - 5.19615i) q^{38} +(-1.00000 + 1.73205i) q^{40} +(-3.00000 + 5.19615i) q^{41} +(4.00000 + 6.92820i) q^{43} -4.00000 q^{44} -4.00000 q^{46} +(-4.00000 - 6.92820i) q^{47} +(1.50000 - 2.59808i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(0.500000 + 0.866025i) q^{52} +12.0000 q^{53} +8.00000 q^{55} +(-1.00000 - 1.73205i) q^{56} +(-4.00000 + 6.92820i) q^{58} +(-2.00000 + 3.46410i) q^{59} +(-5.00000 - 8.66025i) q^{61} +2.00000 q^{62} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{65} +(1.00000 - 1.73205i) q^{67} +(2.00000 + 3.46410i) q^{70} -16.0000 q^{71} +14.0000 q^{73} +(3.00000 + 5.19615i) q^{74} +(3.00000 - 5.19615i) q^{76} +(-4.00000 + 6.92820i) q^{77} +(2.00000 + 3.46410i) q^{79} -2.00000 q^{80} -6.00000 q^{82} +(6.00000 + 10.3923i) q^{83} +(-4.00000 + 6.92820i) q^{86} +(-2.00000 - 3.46410i) q^{88} -6.00000 q^{89} +2.00000 q^{91} +(-2.00000 - 3.46410i) q^{92} +(4.00000 - 6.92820i) q^{94} +(-6.00000 + 10.3923i) q^{95} +(5.00000 + 8.66025i) q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{10} + 4 q^{11} + q^{13} - 2 q^{14} - q^{16} - 12 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} + q^{25} + 2 q^{26} - 4 q^{28} + 8 q^{29} + 2 q^{31}+ \cdots + 6 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 2 * q^5 + 2 * q^7 - 2 * q^8 + 4 * q^10 + 4 * q^11 + q^13 - 2 * q^14 - q^16 - 12 * q^19 + 2 * q^20 - 4 * q^22 - 4 * q^23 + q^25 + 2 * q^26 - 4 * q^28 + 8 * q^29 + 2 * q^31 + q^32 + 8 * q^35 + 12 * q^37 - 6 * q^38 - 2 * q^40 - 6 * q^41 + 8 * q^43 - 8 * q^44 - 8 * q^46 - 8 * q^47 + 3 * q^49 - q^50 + q^52 + 24 * q^53 + 16 * q^55 - 2 * q^56 - 8 * q^58 - 4 * q^59 - 10 * q^61 + 4 * q^62 + 2 * q^64 - 2 * q^65 + 2 * q^67 + 4 * q^70 - 32 * q^71 + 28 * q^73 + 6 * q^74 + 6 * q^76 - 8 * q^77 + 4 * q^79 - 4 * q^80 - 12 * q^82 + 12 * q^83 - 8 * q^86 - 4 * q^88 - 12 * q^89 + 4 * q^91 - 4 * q^92 + 8 * q^94 - 12 * q^95 + 10 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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