LMFDB - Embedded newform 2106.2.e.ba.703.1

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 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			703.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2106.703
Dual form			2106.2.e.ba.1405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.50000 + 2.59808i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.50000 + 2.59808i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +3.00000 q^{10} +(-3.00000 + 5.19615i) q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -3.00000 q^{17} +2.00000 q^{19} +(1.50000 - 2.59808i) q^{20} +(3.00000 + 5.19615i) q^{22} +(-2.00000 + 3.46410i) q^{25} -1.00000 q^{26} -1.00000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(2.00000 + 3.46410i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-1.50000 + 2.59808i) q^{34} +3.00000 q^{35} -7.00000 q^{37} +(1.00000 - 1.73205i) q^{38} +(-1.50000 - 2.59808i) q^{40} +(0.500000 - 0.866025i) q^{43} +6.00000 q^{44} +(-1.50000 + 2.59808i) q^{47} +(3.00000 + 5.19615i) q^{49} +(2.00000 + 3.46410i) q^{50} +(-0.500000 + 0.866025i) q^{52} -18.0000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(3.00000 + 5.19615i) q^{58} +(3.00000 + 5.19615i) q^{59} +(-4.00000 + 6.92820i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(1.50000 - 2.59808i) q^{65} +(-7.00000 - 12.1244i) q^{67} +(1.50000 + 2.59808i) q^{68} +(1.50000 - 2.59808i) q^{70} -3.00000 q^{71} +2.00000 q^{73} +(-3.50000 + 6.06218i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(3.00000 + 5.19615i) q^{77} +(-4.00000 + 6.92820i) q^{79} -3.00000 q^{80} +(-6.00000 + 10.3923i) q^{83} +(-4.50000 - 7.79423i) q^{85} +(-0.500000 - 0.866025i) q^{86} +(3.00000 - 5.19615i) q^{88} -6.00000 q^{89} -1.00000 q^{91} +(1.50000 + 2.59808i) q^{94} +(3.00000 + 5.19615i) q^{95} +(5.00000 - 8.66025i) q^{97} +6.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 3 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 + 3 * q^5 + q^7 - 2 * q^8	$ 2 q + q^{2} - q^{4} + 3 q^{5} + q^{7} - 2 q^{8} + 6 q^{10} - 6 q^{11} - q^{13} - q^{14} - q^{16} - 6 q^{17} + 4 q^{19} + 3 q^{20} + 6 q^{22} - 4 q^{25} - 2 q^{26} - 2 q^{28} - 6 q^{29} + 4 q^{31} + q^{32} - 3 q^{34} + 6 q^{35} - 14 q^{37} + 2 q^{38} - 3 q^{40} + q^{43} + 12 q^{44} - 3 q^{47} + 6 q^{49} + 4 q^{50} - q^{52} - 36 q^{55} - q^{56} + 6 q^{58} + 6 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} + 3 q^{65} - 14 q^{67} + 3 q^{68} + 3 q^{70} - 6 q^{71} + 4 q^{73} - 7 q^{74} - 2 q^{76} + 6 q^{77} - 8 q^{79} - 6 q^{80} - 12 q^{83} - 9 q^{85} - q^{86} + 6 q^{88} - 12 q^{89} - 2 q^{91} + 3 q^{94} + 6 q^{95} + 10 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 3 * q^5 + q^7 - 2 * q^8 + 6 * q^10 - 6 * q^11 - q^13 - q^14 - q^16 - 6 * q^17 + 4 * q^19 + 3 * q^20 + 6 * q^22 - 4 * q^25 - 2 * q^26 - 2 * q^28 - 6 * q^29 + 4 * q^31 + q^32 - 3 * q^34 + 6 * q^35 - 14 * q^37 + 2 * q^38 - 3 * q^40 + q^43 + 12 * q^44 - 3 * q^47 + 6 * q^49 + 4 * q^50 - q^52 - 36 * q^55 - q^56 + 6 * q^58 + 6 * q^59 - 8 * q^61 + 8 * q^62 + 2 * q^64 + 3 * q^65 - 14 * q^67 + 3 * q^68 + 3 * q^70 - 6 * q^71 + 4 * q^73 - 7 * q^74 - 2 * q^76 + 6 * q^77 - 8 * q^79 - 6 * q^80 - 12 * q^83 - 9 * q^85 - q^86 + 6 * q^88 - 12 * q^89 - 2 * q^91 + 3 * q^94 + 6 * q^95 + 10 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1379$	$1783$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	0.977672	+	0.210138i	$0.0673912\pi$
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	−0.306851	+	0.951757i	$0.599275\pi$
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	0.944911	+	0.327327i	$0.106148\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$	3.00000			0.948683
$11$	−3.00000	+	5.19615i	−0.904534	+	1.56670i	−0.0829925	+	0.996550i	$0.526448\pi$
$11$	−3.00000	+	5.19615i	−0.904534	+	1.56670i	−0.821541	+	0.570149i	$0.806886\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$14$	−0.500000	−	0.866025i	−0.133631	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$	1.50000	−	2.59808i	0.335410	−	0.580948i
$21$		0			0
$22$	3.00000	+	5.19615i	0.639602	+	1.10782i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$	−1.00000			−0.196116
$27$		0			0
$28$	−1.00000			−0.188982
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$		0			0
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$	−1.50000	+	2.59808i	−0.257248	+	0.445566i
$35$	3.00000			0.507093
$36$		0			0
$37$	−7.00000			−1.15079			−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000			−1.15079			−0.575396	+	0.817875i	$0.695152\pi$
$38$	1.00000	−	1.73205i	0.162221	−	0.280976i
$39$		0			0
$40$	−1.50000	−	2.59808i	−0.237171	−	0.410792i
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	0.901629	+	0.432511i	$0.142372\pi$
$44$	6.00000			0.904534
$45$		0			0
$46$		0			0
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	−0.954441	−	0.298401i	$-0.903547\pi$
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	0.735643	+	0.677369i	$0.236880\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$	2.00000	+	3.46410i	0.282843	+	0.489898i
$51$		0			0
$52$	−0.500000	+	0.866025i	−0.0693375	+	0.120096i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−18.0000			−2.42712
$56$	−0.500000	+	0.866025i	−0.0668153	+	0.115728i
$57$		0			0
$58$	3.00000	+	5.19615i	0.393919	+	0.682288i
$59$	3.00000	+	5.19615i	0.390567	+	0.676481i	0.992524	−	0.122047i	$-0.0389457\pi$
$59$	3.00000	+	5.19615i	0.390567	+	0.676481i	−0.601958	+	0.798528i	$0.705612\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	1.00000			0.125000
$65$	1.50000	−	2.59808i	0.186052	−	0.322252i
$66$		0			0
$67$	−7.00000	−	12.1244i	−0.855186	−	1.48123i	−0.876472	−	0.481452i	$-0.840109\pi$
$67$	−7.00000	−	12.1244i	−0.855186	−	1.48123i	0.0212861	−	0.999773i	$-0.493224\pi$
$68$	1.50000	+	2.59808i	0.181902	+	0.315063i
$69$		0			0
$70$	1.50000	−	2.59808i	0.179284	−	0.310530i
$71$	−3.00000			−0.356034			−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000			−0.356034			−0.178017	+	0.984027i	$0.556968\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$	−3.50000	+	6.06218i	−0.406867	+	0.704714i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$	3.00000	+	5.19615i	0.341882	+	0.592157i
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$	−3.00000			−0.335410
$81$		0			0
$82$		0			0
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	0.322396	+	0.946605i	$0.395512\pi$
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	−0.980982	+	0.194099i	$0.937822\pi$
$84$		0			0
$85$	−4.50000	−	7.79423i	−0.488094	−	0.845403i
$86$	−0.500000	−	0.866025i	−0.0539164	−	0.0933859i
$87$		0			0
$88$	3.00000	−	5.19615i	0.319801	−	0.553912i
$89$	−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
$91$	−1.00000			−0.104828
$92$		0			0
$93$		0			0
$94$	1.50000	+	2.59808i	0.154713	+	0.267971i
$95$	3.00000	+	5.19615i	0.307794	+	0.533114i
$96$		0			0
$97$	5.00000	−	8.66025i	0.507673	−	0.879316i	−0.492287	−	0.870433i	$-0.663839\pi$
$97$	5.00000	−	8.66025i	0.507673	−	0.879316i	0.999961	−	0.00888289i	$-0.00282755\pi$
$98$	6.00000			0.606092
$99$		0			0
$100$	4.00000			0.400000
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	−0.396236	−	0.918149i	$-0.629684\pi$
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	0.993258	−	0.115924i	$-0.0369830\pi$
$102$		0			0
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	0.947576	−	0.319531i	$-0.103525\pi$
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	−0.750510	+	0.660859i	$0.770192\pi$
$104$	0.500000	+	0.866025i	0.0490290	+	0.0849208i
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$	−9.00000	+	15.5885i	−0.858116	+	1.48630i
$111$		0			0
$112$	0.500000	+	0.866025i	0.0472456	+	0.0818317i
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$		0			0
$116$	6.00000			0.557086
$117$		0			0
$118$	6.00000			0.552345
$119$	−1.50000	+	2.59808i	−0.137505	+	0.238165i
$120$		0			0
$121$	−12.5000	−	21.6506i	−1.13636	−	1.96824i
$122$	4.00000	+	6.92820i	0.362143	+	0.627250i
$123$		0			0
$124$	2.00000	−	3.46410i	0.179605	−	0.311086i
$125$	3.00000			0.268328
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$	−1.50000	−	2.59808i	−0.131559	−	0.227866i
$131$	10.5000	+	18.1865i	0.917389	+	1.58896i	0.803365	+	0.595487i	$0.203041\pi$
$131$	10.5000	+	18.1865i	0.917389	+	1.58896i	0.114024	+	0.993478i	$0.463626\pi$
$132$		0			0
$133$	1.00000	−	1.73205i	0.0867110	−	0.150188i
$134$	−14.0000			−1.20942
$135$		0			0
$136$	3.00000			0.257248
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	6.50000	+	11.2583i	0.551323	+	0.954919i	0.998179	+	0.0603135i	$0.0192101\pi$
$139$	6.50000	+	11.2583i	0.551323	+	0.954919i	−0.446857	+	0.894606i	$0.647457\pi$
$140$	−1.50000	−	2.59808i	−0.126773	−	0.219578i
$141$		0			0
$142$	−1.50000	+	2.59808i	−0.125877	+	0.218026i
$143$	6.00000			0.501745
$144$		0			0
$145$	−18.0000			−1.49482
$146$	1.00000	−	1.73205i	0.0827606	−	0.143346i
$147$		0			0
$148$	3.50000	+	6.06218i	0.287698	+	0.498308i
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	−8.50000	+	14.7224i	−0.691720	+	1.19809i	0.279554	+	0.960130i	$0.409814\pi$
$151$	−8.50000	+	14.7224i	−0.691720	+	1.19809i	−0.971274	+	0.237964i	$0.923520\pi$
$152$	−2.00000			−0.162221
$153$		0			0
$154$	6.00000			0.483494
$155$	−6.00000	+	10.3923i	−0.481932	+	0.834730i
$156$		0			0
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	0.438948	−	0.898513i	$-0.355351\pi$
$158$	4.00000	+	6.92820i	0.318223	+	0.551178i
$159$		0			0
$160$	−1.50000	+	2.59808i	−0.118585	+	0.205396i
$161$		0			0
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$		0			0
$166$	6.00000	+	10.3923i	0.465690	+	0.806599i
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.0384615	+	0.0666173i
$170$	−9.00000			−0.690268
$171$		0			0
$172$	−1.00000			−0.0762493
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$	−3.00000	−	5.19615i	−0.226134	−	0.391675i
$177$		0			0
$178$	−3.00000	+	5.19615i	−0.224860	+	0.389468i
$179$	3.00000			0.224231			0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000			0.224231			0.112115	+	0.993695i	$0.464237\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$	−0.500000	+	0.866025i	−0.0370625	+	0.0641941i
$183$		0			0
$184$		0			0
$185$	−10.5000	−	18.1865i	−0.771975	−	1.33710i
$186$		0			0
$187$	9.00000	−	15.5885i	0.658145	−	1.13994i
$188$	3.00000			0.218797
$189$		0			0
$190$	6.00000			0.435286
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	−0.331611	−	0.943416i	$-0.607592\pi$
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	0.982828	−	0.184525i	$-0.0590746\pi$
$192$		0			0
$193$	2.00000	+	3.46410i	0.143963	+	0.249351i	0.928986	−	0.370116i	$-0.120682\pi$
$193$	2.00000	+	3.46410i	0.143963	+	0.249351i	−0.785022	+	0.619467i	$0.787349\pi$
$194$	−5.00000	−	8.66025i	−0.358979	−	0.621770i
$195$		0			0
$196$	3.00000	−	5.19615i	0.214286	−	0.371154i
$197$	3.00000			0.213741			0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000			0.213741			0.106871	+	0.994273i	$0.465917\pi$
$198$		0			0
$199$	2.00000			0.141776			0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000			0.141776			0.0708881	+	0.997484i	$0.477417\pi$
$200$	2.00000	−	3.46410i	0.141421	−	0.244949i
$201$		0			0
$202$	−6.00000	−	10.3923i	−0.422159	−	0.731200i
$203$	3.00000	+	5.19615i	0.210559	+	0.364698i
$204$		0			0
$205$		0			0
$206$	4.00000			0.278693
$207$		0			0
$208$	1.00000			0.0693375
$209$	−6.00000	+	10.3923i	−0.415029	+	0.718851i
$210$		0			0
$211$	6.50000	+	11.2583i	0.447478	+	0.775055i	0.998221	−	0.0596196i	$-0.0189888\pi$
$211$	6.50000	+	11.2583i	0.447478	+	0.775055i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$	6.00000	−	10.3923i	0.410152	−	0.710403i
$215$	3.00000			0.204598
$216$		0			0
$217$	4.00000			0.271538
$218$	−3.50000	+	6.06218i	−0.237050	+	0.410582i
$219$		0			0
$220$	9.00000	+	15.5885i	0.606780	+	1.05097i
$221$	1.50000	+	2.59808i	0.100901	+	0.174766i
$222$		0			0
$223$	9.50000	−	16.4545i	0.636167	−	1.10187i	−0.350100	−	0.936713i	$-0.613852\pi$
$223$	9.50000	−	16.4545i	0.636167	−	1.10187i	0.986267	−	0.165161i	$-0.0528144\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	6.00000			0.399114
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	0.996832	−	0.0795401i	$-0.0253452\pi$
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	−0.567300	+	0.823511i	$0.692012\pi$
$230$		0			0
$231$		0			0
$232$	3.00000	−	5.19615i	0.196960	−	0.341144i
$233$	−27.0000			−1.76883			−0.884414	−	0.466702i	$-0.845442\pi$
$233$	−27.0000			−1.76883			−0.884414	+	0.466702i	$0.845442\pi$
$234$		0			0
$235$	−9.00000			−0.587095
$236$	3.00000	−	5.19615i	0.195283	−	0.338241i
$237$		0			0
$238$	1.50000	+	2.59808i	0.0972306	+	0.168408i
$239$	−7.50000	−	12.9904i	−0.485135	−	0.840278i	0.514719	−	0.857359i	$-0.327896\pi$
$239$	−7.50000	−	12.9904i	−0.485135	−	0.840278i	−0.999854	+	0.0170808i	$0.994563\pi$
$240$		0			0
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	0.980917	+	0.194429i	$0.0622852\pi$
$242$	−25.0000			−1.60706
$243$		0			0
$244$	8.00000			0.512148
$245$	−9.00000	+	15.5885i	−0.574989	+	0.995910i
$246$		0			0
$247$	−1.00000	−	1.73205i	−0.0636285	−	0.110208i
$248$	−2.00000	−	3.46410i	−0.127000	−	0.219971i
$249$		0			0
$250$	1.50000	−	2.59808i	0.0948683	−	0.164317i
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$		0			0
$254$	10.0000	−	17.3205i	0.627456	−	1.08679i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−4.50000	−	7.79423i	−0.280702	−	0.486191i	0.690856	−	0.722993i	$-0.257234\pi$
$257$	−4.50000	−	7.79423i	−0.280702	−	0.486191i	−0.971558	+	0.236802i	$0.923901\pi$
$258$		0			0
$259$	−3.50000	+	6.06218i	−0.217479	+	0.376685i
$260$	−3.00000			−0.186052
$261$		0			0
$262$	21.0000			1.29738
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$		0			0
$266$	−1.00000	−	1.73205i	−0.0613139	−	0.106199i
$267$		0			0
$268$	−7.00000	+	12.1244i	−0.427593	+	0.740613i
$269$	24.0000			1.46331			0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000			1.46331			0.731653	+	0.681677i	$0.238749\pi$
$270$		0			0
$271$	11.0000			0.668202			0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000			0.668202			0.334101	+	0.942537i	$0.391567\pi$
$272$	1.50000	−	2.59808i	0.0909509	−	0.157532i
$273$		0			0
$274$		0			0
$275$	−12.0000	−	20.7846i	−0.723627	−	1.25336i
$276$		0			0
$277$	14.0000	−	24.2487i	0.841178	−	1.45696i	−0.0477206	−	0.998861i	$-0.515196\pi$
$277$	14.0000	−	24.2487i	0.841178	−	1.45696i	0.888899	−	0.458103i	$-0.151471\pi$
$278$	13.0000			0.779688
$279$		0			0
$280$	−3.00000			−0.179284
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	−0.762561	−	0.646916i	$-0.776058\pi$
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	0.941526	+	0.336939i	$0.109392\pi$
$282$		0			0
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$	1.50000	+	2.59808i	0.0890086	+	0.154167i
$285$		0			0
$286$	3.00000	−	5.19615i	0.177394	−	0.307255i
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$	−9.00000	+	15.5885i	−0.528498	+	0.915386i
$291$		0			0
$292$	−1.00000	−	1.73205i	−0.0585206	−	0.101361i
$293$	−10.5000	−	18.1865i	−0.613417	−	1.06247i	−0.990660	−	0.136355i	$-0.956461\pi$
$293$	−10.5000	−	18.1865i	−0.613417	−	1.06247i	0.377244	−	0.926114i	$-0.376872\pi$
$294$		0			0
$295$	−9.00000	+	15.5885i	−0.524000	+	0.907595i
$296$	7.00000			0.406867
$297$		0			0
$298$	6.00000			0.347571
$299$		0			0
$300$		0			0
$301$	−0.500000	−	0.866025i	−0.0288195	−	0.0499169i
$302$	8.50000	+	14.7224i	0.489120	+	0.847181i
$303$		0			0
$304$	−1.00000	+	1.73205i	−0.0573539	+	0.0993399i
$305$	−24.0000			−1.37424
$306$		0			0
$307$	2.00000			0.114146			0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$	3.00000	−	5.19615i	0.170941	−	0.296078i
$309$		0			0
$310$	6.00000	+	10.3923i	0.340777	+	0.590243i
$311$	15.0000	+	25.9808i	0.850572	+	1.47323i	0.880693	+	0.473688i	$0.157077\pi$
$311$	15.0000	+	25.9808i	0.850572	+	1.47323i	−0.0301210	+	0.999546i	$0.509589\pi$
$312$		0			0
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	−0.851549	−	0.524276i	$-0.824336\pi$
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	0.879810	+	0.475325i	$0.157669\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$	8.00000			0.450035
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	−0.769395	−	0.638774i	$-0.779442\pi$
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	0.937892	+	0.346929i	$0.112775\pi$
$318$		0			0
$319$	−18.0000	−	31.1769i	−1.00781	−	1.74557i
$320$	1.50000	+	2.59808i	0.0838525	+	0.145237i
$321$		0			0
$322$		0			0
$323$	−6.00000			−0.333849
$324$		0			0
$325$	4.00000			0.221880
$326$	−8.00000	+	13.8564i	−0.443079	+	0.767435i
$327$		0			0
$328$		0			0
$329$	1.50000	+	2.59808i	0.0826977	+	0.143237i
$330$		0			0
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	−0.954765	−	0.297361i	$-0.903893\pi$
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	0.734905	+	0.678170i	$0.237227\pi$
$332$	12.0000			0.658586
$333$		0			0
$334$		0			0
$335$	21.0000	−	36.3731i	1.14735	−	1.98727i
$336$		0			0
$337$	−11.5000	−	19.9186i	−0.626445	−	1.08503i	−0.988260	−	0.152784i	$-0.951176\pi$
$337$	−11.5000	−	19.9186i	−0.626445	−	1.08503i	0.361815	−	0.932250i	$-0.382157\pi$
$338$	0.500000	+	0.866025i	0.0271964	+	0.0471056i
$339$		0			0
$340$	−4.50000	+	7.79423i	−0.244047	+	0.422701i
$341$	−24.0000			−1.29967
$342$		0			0
$343$	13.0000			0.701934
$344$	−0.500000	+	0.866025i	−0.0269582	+	0.0466930i
$345$		0			0
$346$		0			0
$347$	−1.50000	−	2.59808i	−0.0805242	−	0.139472i	0.822951	−	0.568112i	$-0.192326\pi$
$347$	−1.50000	−	2.59808i	−0.0805242	−	0.139472i	−0.903475	+	0.428640i	$0.858993\pi$
$348$		0			0
$349$	9.50000	−	16.4545i	0.508523	−	0.880788i	−0.491428	−	0.870918i	$-0.663525\pi$
$349$	9.50000	−	16.4545i	0.508523	−	0.880788i	0.999951	−	0.00987003i	$-0.00314178\pi$
$350$	4.00000			0.213809
$351$		0			0
$352$	−6.00000			−0.319801
$353$	−12.0000	+	20.7846i	−0.638696	+	1.10625i	0.347024	+	0.937856i	$0.387192\pi$
$353$	−12.0000	+	20.7846i	−0.638696	+	1.10625i	−0.985719	+	0.168397i	$0.946141\pi$
$354$		0			0
$355$	−4.50000	−	7.79423i	−0.238835	−	0.413675i
$356$	3.00000	+	5.19615i	0.159000	+	0.275396i
$357$		0			0
$358$	1.50000	−	2.59808i	0.0792775	−	0.137313i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$	10.0000	−	17.3205i	0.525588	−	0.910346i
$363$		0			0
$364$	0.500000	+	0.866025i	0.0262071	+	0.0453921i
$365$	3.00000	+	5.19615i	0.157027	+	0.271979i
$366$		0			0
$367$	−13.0000	+	22.5167i	−0.678594	+	1.17536i	0.296810	+	0.954937i	$0.404077\pi$
$367$	−13.0000	+	22.5167i	−0.678594	+	1.17536i	−0.975404	+	0.220423i	$0.929256\pi$
$368$		0			0
$369$		0			0
$370$	−21.0000			−1.09174
$371$		0			0
$372$		0			0
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	0.913147	−	0.407630i	$-0.133645\pi$
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	−0.809591	+	0.586994i	$0.800311\pi$
$374$	−9.00000	−	15.5885i	−0.465379	−	0.806060i
$375$		0			0
$376$	1.50000	−	2.59808i	0.0773566	−	0.133986i
$377$	6.00000			0.309016
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$	3.00000	−	5.19615i	0.153897	−	0.266557i
$381$		0			0
$382$	−9.00000	−	15.5885i	−0.460480	−	0.797575i
$383$	−10.5000	−	18.1865i	−0.536525	−	0.929288i	−0.999088	−	0.0427020i	$-0.986403\pi$
$383$	−10.5000	−	18.1865i	−0.536525	−	0.929288i	0.462563	−	0.886586i	$-0.346930\pi$
$384$		0			0
$385$	−9.00000	+	15.5885i	−0.458682	+	0.794461i
$386$	4.00000			0.203595
$387$		0			0
$388$	−10.0000			−0.507673
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	−0.779895	−	0.625910i	$-0.784728\pi$
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	0.932002	+	0.362454i	$0.118061\pi$
$390$		0			0
$391$		0			0
$392$	−3.00000	−	5.19615i	−0.151523	−	0.262445i
$393$		0			0
$394$	1.50000	−	2.59808i	0.0755689	−	0.130889i
$395$	−24.0000			−1.20757
$396$		0			0
$397$	−34.0000			−1.70641			−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000			−1.70641			−0.853206	+	0.521575i	$0.825345\pi$
$398$	1.00000	−	1.73205i	0.0501255	−	0.0868199i
$399$		0			0
$400$	−2.00000	−	3.46410i	−0.100000	−	0.173205i
$401$	−18.0000	−	31.1769i	−0.898877	−	1.55690i	−0.828932	−	0.559350i	$-0.811051\pi$
$401$	−18.0000	−	31.1769i	−0.898877	−	1.55690i	−0.0699455	−	0.997551i	$-0.522283\pi$
$402$		0			0
$403$	2.00000	−	3.46410i	0.0996271	−	0.172559i
$404$	−12.0000			−0.597022
$405$		0			0
$406$	6.00000			0.297775
$407$	21.0000	−	36.3731i	1.04093	−	1.80295i
$408$		0			0
$409$	−16.0000	−	27.7128i	−0.791149	−	1.37031i	−0.925256	−	0.379344i	$-0.876150\pi$
$409$	−16.0000	−	27.7128i	−0.791149	−	1.37031i	0.134107	−	0.990967i	$-0.457183\pi$
$410$		0			0
$411$		0			0
$412$	2.00000	−	3.46410i	0.0985329	−	0.170664i
$413$	6.00000			0.295241
$414$		0			0
$415$	−36.0000			−1.76717
$416$	0.500000	−	0.866025i	0.0245145	−	0.0424604i
$417$		0			0
$418$	6.00000	+	10.3923i	0.293470	+	0.508304i
$419$	−4.50000	−	7.79423i	−0.219839	−	0.380773i	0.734919	−	0.678155i	$-0.237220\pi$
$419$	−4.50000	−	7.79423i	−0.219839	−	0.380773i	−0.954759	+	0.297382i	$0.903887\pi$
$420$		0			0
$421$	−8.50000	+	14.7224i	−0.414265	+	0.717527i	−0.995351	−	0.0963145i	$-0.969295\pi$
$421$	−8.50000	+	14.7224i	−0.414265	+	0.717527i	0.581086	+	0.813842i	$0.302628\pi$
$422$	13.0000			0.632830
$423$		0			0
$424$		0			0
$425$	6.00000	−	10.3923i	0.291043	−	0.504101i
$426$		0			0
$427$	4.00000	+	6.92820i	0.193574	+	0.335279i
$428$	−6.00000	−	10.3923i	−0.290021	−	0.502331i
$429$		0			0
$430$	1.50000	−	2.59808i	0.0723364	−	0.125290i
$431$	−33.0000			−1.58955			−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000			−1.58955			−0.794777	+	0.606902i	$0.792412\pi$
$432$		0			0
$433$	−25.0000			−1.20142			−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000			−1.20142			−0.600712	+	0.799466i	$0.705116\pi$
$434$	2.00000	−	3.46410i	0.0960031	−	0.166282i
$435$		0			0
$436$	3.50000	+	6.06218i	0.167620	+	0.290326i
$437$		0			0
$438$		0			0
$439$	−13.0000	+	22.5167i	−0.620456	+	1.07466i	0.368945	+	0.929451i	$0.379719\pi$
$439$	−13.0000	+	22.5167i	−0.620456	+	1.07466i	−0.989401	+	0.145210i	$0.953614\pi$
$440$	18.0000			0.858116
$441$		0			0
$442$	3.00000			0.142695
$443$	−10.5000	+	18.1865i	−0.498870	+	0.864068i	−0.999999	−	0.00130426i	$-0.999585\pi$
$443$	−10.5000	+	18.1865i	−0.498870	+	0.864068i	0.501129	+	0.865373i	$0.332918\pi$
$444$		0			0
$445$	−9.00000	−	15.5885i	−0.426641	−	0.738964i
$446$	−9.50000	−	16.4545i	−0.449838	−	0.779142i
$447$		0			0
$448$	0.500000	−	0.866025i	0.0236228	−	0.0409159i
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$		0			0
$452$	3.00000	−	5.19615i	0.141108	−	0.244406i
$453$		0			0
$454$		0			0
$455$	−1.50000	−	2.59808i	−0.0703211	−	0.121800i
$456$		0			0
$457$	5.00000	−	8.66025i	0.233890	−	0.405110i	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	5.00000	−	8.66025i	0.233890	−	0.405110i	0.958950	+	0.283577i	$0.0915211\pi$
$458$	13.0000			0.607450
$459$		0			0
$460$		0			0
$461$	−4.50000	+	7.79423i	−0.209586	+	0.363013i	−0.951584	−	0.307388i	$-0.900545\pi$
$461$	−4.50000	+	7.79423i	−0.209586	+	0.363013i	0.741998	+	0.670402i	$0.233878\pi$
$462$		0			0
$463$	20.0000	+	34.6410i	0.929479	+	1.60990i	0.784195	+	0.620515i	$0.213076\pi$
$463$	20.0000	+	34.6410i	0.929479	+	1.60990i	0.145284	+	0.989390i	$0.453590\pi$
$464$	−3.00000	−	5.19615i	−0.139272	−	0.241225i
$465$		0			0
$466$	−13.5000	+	23.3827i	−0.625375	+	1.08318i
$467$	36.0000			1.66588			0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000			1.66588			0.832941	+	0.553362i	$0.186655\pi$
$468$		0			0
$469$	−14.0000			−0.646460
$470$	−4.50000	+	7.79423i	−0.207570	+	0.359521i
$471$		0			0
$472$	−3.00000	−	5.19615i	−0.138086	−	0.239172i
$473$	3.00000	+	5.19615i	0.137940	+	0.238919i
$474$		0			0
$475$	−4.00000	+	6.92820i	−0.183533	+	0.317888i
$476$	3.00000			0.137505
$477$		0			0
$478$	−15.0000			−0.686084
$479$	10.5000	−	18.1865i	0.479757	−	0.830964i	−0.519973	−	0.854183i	$-0.674058\pi$
$479$	10.5000	−	18.1865i	0.479757	−	0.830964i	0.999730	+	0.0232187i	$0.00739140\pi$
$480$		0			0
$481$	3.50000	+	6.06218i	0.159586	+	0.276412i
$482$	−5.00000	−	8.66025i	−0.227744	−	0.394464i
$483$		0			0
$484$	−12.5000	+	21.6506i	−0.568182	+	0.984120i
$485$	30.0000			1.36223
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$	4.00000	−	6.92820i	0.181071	−	0.313625i
$489$		0			0
$490$	9.00000	+	15.5885i	0.406579	+	0.704215i
$491$	4.50000	+	7.79423i	0.203082	+	0.351749i	0.949520	−	0.313707i	$-0.101571\pi$
$491$	4.50000	+	7.79423i	0.203082	+	0.351749i	−0.746438	+	0.665455i	$0.768237\pi$
$492$		0			0
$493$	9.00000	−	15.5885i	0.405340	−	0.702069i
$494$	−2.00000			−0.0899843
$495$		0			0
$496$	−4.00000			−0.179605
$497$	−1.50000	+	2.59808i	−0.0672842	+	0.116540i
$498$		0			0
$499$	20.0000	+	34.6410i	0.895323	+	1.55074i	0.833404	+	0.552664i	$0.186389\pi$
$499$	20.0000	+	34.6410i	0.895323	+	1.55074i	0.0619186	+	0.998081i	$0.480278\pi$
$500$	−1.50000	−	2.59808i	−0.0670820	−	0.116190i
$501$		0			0
$502$	12.0000	−	20.7846i	0.535586	−	0.927663i
$503$	−30.0000			−1.33763			−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000			−1.33763			−0.668817	+	0.743427i	$0.733199\pi$
$504$		0			0
$505$	36.0000			1.60198
$506$		0			0
$507$		0			0
$508$	−10.0000	−	17.3205i	−0.443678	−	0.768473i
$509$	9.00000	+	15.5885i	0.398918	+	0.690946i	0.993593	−	0.113020i	$-0.0360525\pi$
$509$	9.00000	+	15.5885i	0.398918	+	0.690946i	−0.594675	+	0.803966i	$0.702719\pi$
$510$		0			0
$511$	1.00000	−	1.73205i	0.0442374	−	0.0766214i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−9.00000			−0.396973
$515$	−6.00000	+	10.3923i	−0.264392	+	0.457940i
$516$		0			0
$517$	−9.00000	−	15.5885i	−0.395820	−	0.685580i
$518$	3.50000	+	6.06218i	0.153781	+	0.266357i
$519$		0			0
$520$	−1.50000	+	2.59808i	−0.0657794	+	0.113933i
$521$	−9.00000			−0.394297			−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000			−0.394297			−0.197149	+	0.980374i	$0.563168\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$	10.5000	−	18.1865i	0.458695	−	0.794482i
$525$		0			0
$526$	−6.00000	−	10.3923i	−0.261612	−	0.453126i
$527$	−6.00000	−	10.3923i	−0.261364	−	0.452696i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−2.00000			−0.0867110
$533$		0			0
$534$		0			0
$535$	18.0000	+	31.1769i	0.778208	+	1.34790i
$536$	7.00000	+	12.1244i	0.302354	+	0.523692i
$537$		0			0
$538$	12.0000	−	20.7846i	0.517357	−	0.896088i
$539$	−36.0000			−1.55063
$540$		0			0
$541$	11.0000			0.472927			0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000			0.472927			0.236463	+	0.971640i	$0.424012\pi$
$542$	5.50000	−	9.52628i	0.236245	−	0.409189i
$543$		0			0
$544$	−1.50000	−	2.59808i	−0.0643120	−	0.111392i
$545$	−10.5000	−	18.1865i	−0.449771	−	0.779026i
$546$		0			0
$547$	−8.50000	+	14.7224i	−0.363434	+	0.629486i	−0.988524	−	0.151067i	$-0.951729\pi$
$547$	−8.50000	+	14.7224i	−0.363434	+	0.629486i	0.625090	+	0.780553i	$0.285062\pi$
$548$		0			0
$549$		0			0
$550$	−24.0000			−1.02336
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$	4.00000	+	6.92820i	0.170097	+	0.294617i
$554$	−14.0000	−	24.2487i	−0.594803	−	1.03023i
$555$		0			0
$556$	6.50000	−	11.2583i	0.275661	−	0.477460i
$557$	3.00000			0.127114			0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000			0.127114			0.0635570	+	0.997978i	$0.479756\pi$
$558$		0			0
$559$	−1.00000			−0.0422955
$560$	−1.50000	+	2.59808i	−0.0633866	+	0.109789i
$561$		0			0
$562$	−3.00000	−	5.19615i	−0.126547	−	0.219186i
$563$	−19.5000	−	33.7750i	−0.821827	−	1.42345i	−0.904320	−	0.426855i	$-0.859622\pi$
$563$	−19.5000	−	33.7750i	−0.821827	−	1.42345i	0.0824933	−	0.996592i	$-0.473712\pi$
$564$		0			0
$565$	−9.00000	+	15.5885i	−0.378633	+	0.655811i
$566$	4.00000			0.168133
$567$		0			0
$568$	3.00000			0.125877
$569$	−7.50000	+	12.9904i	−0.314416	+	0.544585i	−0.979313	−	0.202350i	$-0.935142\pi$
$569$	−7.50000	+	12.9904i	−0.314416	+	0.544585i	0.664897	+	0.746935i	$0.268475\pi$
$570$		0			0
$571$	−2.50000	−	4.33013i	−0.104622	−	0.181210i	0.808962	−	0.587861i	$-0.200030\pi$
$571$	−2.50000	−	4.33013i	−0.104622	−	0.181210i	−0.913584	+	0.406651i	$0.866697\pi$
$572$	−3.00000	−	5.19615i	−0.125436	−	0.217262i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$	−4.00000	+	6.92820i	−0.166378	+	0.288175i
$579$		0			0
$580$	9.00000	+	15.5885i	0.373705	+	0.647275i
$581$	6.00000	+	10.3923i	0.248922	+	0.431145i
$582$		0			0
$583$		0			0
$584$	−2.00000			−0.0827606
$585$		0			0
$586$	−21.0000			−0.867502
$587$	−12.0000	+	20.7846i	−0.495293	+	0.857873i	−0.999985	−	0.00542667i	$-0.998273\pi$
$587$	−12.0000	+	20.7846i	−0.495293	+	0.857873i	0.504692	+	0.863299i	$0.331606\pi$
$588$		0			0
$589$	4.00000	+	6.92820i	0.164817	+	0.285472i
$590$	9.00000	+	15.5885i	0.370524	+	0.641767i
$591$		0			0
$592$	3.50000	−	6.06218i	0.143849	−	0.249154i
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$	−9.00000			−0.368964
$596$	3.00000	−	5.19615i	0.122885	−	0.212843i
$597$		0			0
$598$		0			0
$599$	−3.00000	−	5.19615i	−0.122577	−	0.212309i	0.798206	−	0.602384i	$-0.205782\pi$
$599$	−3.00000	−	5.19615i	−0.122577	−	0.212309i	−0.920783	+	0.390075i	$0.872449\pi$
$600$		0			0
$601$	9.50000	−	16.4545i	0.387513	−	0.671192i	−0.604601	−	0.796528i	$-0.706668\pi$
$601$	9.50000	−	16.4545i	0.387513	−	0.671192i	0.992114	+	0.125336i	$0.0400009\pi$
$602$	−1.00000			−0.0407570
$603$		0			0
$604$	17.0000			0.691720
$605$	37.5000	−	64.9519i	1.52459	−	2.64067i
$606$		0			0
$607$	−7.00000	−	12.1244i	−0.284121	−	0.492112i	0.688274	−	0.725450i	$-0.258368\pi$
$607$	−7.00000	−	12.1244i	−0.284121	−	0.492112i	−0.972396	+	0.233338i	$0.925035\pi$
$608$	1.00000	+	1.73205i	0.0405554	+	0.0702439i
$609$		0			0
$610$	−12.0000	+	20.7846i	−0.485866	+	0.841544i
$611$	3.00000			0.121367
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$	1.00000	−	1.73205i	0.0403567	−	0.0698999i
$615$		0			0
$616$	−3.00000	−	5.19615i	−0.120873	−	0.209359i
$617$	12.0000	+	20.7846i	0.483102	+	0.836757i	0.999812	−	0.0194037i	$-0.00617676\pi$
$617$	12.0000	+	20.7846i	0.483102	+	0.836757i	−0.516710	+	0.856161i	$0.672843\pi$
$618$		0			0
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	−0.434551	−	0.900647i	$-0.643093\pi$
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	0.997259	−	0.0739910i	$-0.0235736\pi$
$620$	12.0000			0.481932
$621$		0			0
$622$	30.0000			1.20289
$623$	−3.00000	+	5.19615i	−0.120192	+	0.208179i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$	−0.500000	−	0.866025i	−0.0199840	−	0.0346133i
$627$		0			0
$628$	−7.00000	+	12.1244i	−0.279330	+	0.483814i
$629$	21.0000			0.837325
$630$		0			0
$631$	29.0000			1.15447			0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000			1.15447			0.577236	+	0.816577i	$0.304131\pi$
$632$	4.00000	−	6.92820i	0.159111	−	0.275589i
$633$		0			0
$634$	−3.00000	−	5.19615i	−0.119145	−	0.206366i
$635$	30.0000	+	51.9615i	1.19051	+	2.06203i
$636$		0			0
$637$	3.00000	−	5.19615i	0.118864	−	0.205879i
$638$	−36.0000			−1.42525
$639$		0			0
$640$	3.00000			0.118585
$641$	9.00000	−	15.5885i	0.355479	−	0.615707i	−0.631721	−	0.775196i	$-0.717651\pi$
$641$	9.00000	−	15.5885i	0.355479	−	0.615707i	0.987200	+	0.159489i	$0.0509845\pi$
$642$		0			0
$643$	−7.00000	−	12.1244i	−0.276053	−	0.478138i	0.694347	−	0.719640i	$-0.255693\pi$
$643$	−7.00000	−	12.1244i	−0.276053	−	0.478138i	−0.970400	+	0.241502i	$0.922360\pi$
$644$		0			0
$645$		0			0
$646$	−3.00000	+	5.19615i	−0.118033	+	0.204440i
$647$	−6.00000			−0.235884			−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000			−0.235884			−0.117942	+	0.993020i	$0.537630\pi$
$648$		0			0
$649$	−36.0000			−1.41312
$650$	2.00000	−	3.46410i	0.0784465	−	0.135873i
$651$		0			0
$652$	8.00000	+	13.8564i	0.313304	+	0.542659i
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$	−31.5000	+	54.5596i	−1.23081	+	2.13182i
$656$		0			0
$657$		0			0
$658$	3.00000			0.116952
$659$	18.0000	−	31.1769i	0.701180	−	1.21448i	−0.266872	−	0.963732i	$-0.585990\pi$
$659$	18.0000	−	31.1769i	0.701180	−	1.21448i	0.968052	−	0.250748i	$-0.0806766\pi$
$660$		0			0
$661$	11.0000	+	19.0526i	0.427850	+	0.741059i	0.996682	−	0.0813955i	$-0.0259377\pi$
$661$	11.0000	+	19.0526i	0.427850	+	0.741059i	−0.568831	+	0.822454i	$0.692604\pi$
$662$	4.00000	+	6.92820i	0.155464	+	0.269272i
$663$		0			0
$664$	6.00000	−	10.3923i	0.232845	−	0.403300i
$665$	6.00000			0.232670
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−21.0000	−	36.3731i	−0.811301	−	1.40521i
$671$	−24.0000	−	41.5692i	−0.926510	−	1.60476i
$672$		0			0
$673$	9.50000	−	16.4545i	0.366198	−	0.634274i	−0.622770	−	0.782405i	$-0.713993\pi$
$673$	9.50000	−	16.4545i	0.366198	−	0.634274i	0.988968	+	0.148132i	$0.0473259\pi$
$674$	−23.0000			−0.885927
$675$		0			0
$676$	1.00000			0.0384615
$677$	−24.0000	+	41.5692i	−0.922395	+	1.59763i	−0.126697	+	0.991941i	$0.540438\pi$
$677$	−24.0000	+	41.5692i	−0.922395	+	1.59763i	−0.795698	+	0.605693i	$0.792896\pi$
$678$		0			0
$679$	−5.00000	−	8.66025i	−0.191882	−	0.332350i
$680$	4.50000	+	7.79423i	0.172567	+	0.298895i
$681$		0			0
$682$	−12.0000	+	20.7846i	−0.459504	+	0.795884i
$683$	24.0000			0.918334			0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000			0.918334			0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$		0			0
$686$	6.50000	−	11.2583i	0.248171	−	0.429845i
$687$		0			0
$688$	0.500000	+	0.866025i	0.0190623	+	0.0330169i
$689$		0			0
$690$		0			0
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	0.779857	+	0.625958i	$0.215292\pi$
$692$		0			0
$693$		0			0
$694$	−3.00000			−0.113878
$695$	−19.5000	+	33.7750i	−0.739677	+	1.28116i
$696$		0			0
$697$		0			0
$698$	−9.50000	−	16.4545i	−0.359580	−	0.622811i
$699$		0			0
$700$	2.00000	−	3.46410i	0.0755929	−	0.130931i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−14.0000			−0.528020
$704$	−3.00000	+	5.19615i	−0.113067	+	0.195837i
$705$		0			0
$706$	12.0000	+	20.7846i	0.451626	+	0.782239i
$707$	−6.00000	−	10.3923i	−0.225653	−	0.390843i
$708$		0			0
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	−0.999908	−	0.0135434i	$-0.995689\pi$
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	0.511683	+	0.859174i	$0.329022\pi$
$710$	−9.00000			−0.337764
$711$		0			0
$712$	6.00000			0.224860
$713$		0			0
$714$		0			0
$715$	9.00000	+	15.5885i	0.336581	+	0.582975i
$716$	−1.50000	−	2.59808i	−0.0560576	−	0.0970947i
$717$		0			0
$718$		0			0
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$		0			0
$721$	4.00000			0.148968
$722$	−7.50000	+	12.9904i	−0.279121	+	0.483452i
$723$		0			0
$724$	−10.0000	−	17.3205i	−0.371647	−	0.643712i
$725$	−12.0000	−	20.7846i	−0.445669	−	0.771921i
$726$		0			0
$727$	5.00000	−	8.66025i	0.185440	−	0.321191i	−0.758285	−	0.651923i	$-0.773962\pi$
$727$	5.00000	−	8.66025i	0.185440	−	0.321191i	0.943725	+	0.330732i	$0.107296\pi$
$728$	1.00000			0.0370625
$729$		0			0
$730$	6.00000			0.222070
$731$	−1.50000	+	2.59808i	−0.0554795	+	0.0960933i
$732$		0			0
$733$	−11.5000	−	19.9186i	−0.424762	−	0.735710i	0.571636	−	0.820507i	$-0.306309\pi$
$733$	−11.5000	−	19.9186i	−0.424762	−	0.735710i	−0.996398	+	0.0847976i	$0.972976\pi$
$734$	13.0000	+	22.5167i	0.479839	+	0.831105i
$735$		0			0
$736$		0			0
$737$	84.0000			3.09418
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	−10.5000	+	18.1865i	−0.385988	+	0.668550i
$741$		0			0
$742$		0			0
$743$	4.50000	+	7.79423i	0.165089	+	0.285943i	0.936687	−	0.350168i	$-0.113876\pi$
$743$	4.50000	+	7.79423i	0.165089	+	0.285943i	−0.771598	+	0.636111i	$0.780542\pi$
$744$		0			0
$745$	−9.00000	+	15.5885i	−0.329734	+	0.571117i
$746$	4.00000			0.146450
$747$		0			0
$748$	−18.0000			−0.658145
$749$	6.00000	−	10.3923i	0.219235	−	0.379727i
$750$		0			0
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	0.956963	+	0.290209i	$0.0937250\pi$
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	−0.227153	+	0.973859i	$0.572942\pi$
$752$	−1.50000	−	2.59808i	−0.0546994	−	0.0947421i
$753$		0			0
$754$	3.00000	−	5.19615i	0.109254	−	0.189233i
$755$	−51.0000			−1.85608
$756$		0			0
$757$	−16.0000			−0.581530			−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000			−0.581530			−0.290765	+	0.956795i	$0.593910\pi$
$758$	10.0000	−	17.3205i	0.363216	−	0.629109i
$759$		0			0
$760$	−3.00000	−	5.19615i	−0.108821	−	0.188484i
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	0.915264	−	0.402854i	$-0.131982\pi$
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	−0.806514	+	0.591215i	$0.798649\pi$
$762$		0			0
$763$	−3.50000	+	6.06218i	−0.126709	+	0.219466i
$764$	−18.0000			−0.651217
$765$		0			0
$766$	−21.0000			−0.758761
$767$	3.00000	−	5.19615i	0.108324	−	0.187622i
$768$		0			0
$769$	−16.0000	−	27.7128i	−0.576975	−	0.999350i	−0.995824	−	0.0912938i	$-0.970900\pi$
$769$	−16.0000	−	27.7128i	−0.576975	−	0.999350i	0.418849	−	0.908056i	$-0.362434\pi$
$770$	9.00000	+	15.5885i	0.324337	+	0.561769i
$771$		0			0
$772$	2.00000	−	3.46410i	0.0719816	−	0.124676i
$773$	−39.0000			−1.40273			−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000			−1.40273			−0.701366	+	0.712801i	$0.747426\pi$
$774$		0			0
$775$	−16.0000			−0.574737
$776$	−5.00000	+	8.66025i	−0.179490	+	0.310885i
$777$		0			0
$778$	−3.00000	−	5.19615i	−0.107555	−	0.186291i
$779$		0			0
$780$		0			0
$781$	9.00000	−	15.5885i	0.322045	−	0.557799i
$782$		0			0
$783$		0			0
$784$	−6.00000			−0.214286
$785$	21.0000	−	36.3731i	0.749522	−	1.29821i
$786$		0			0
$787$	20.0000	+	34.6410i	0.712923	+	1.23482i	0.963755	+	0.266788i	$0.0859624\pi$
$787$	20.0000	+	34.6410i	0.712923	+	1.23482i	−0.250832	+	0.968031i	$0.580704\pi$
$788$	−1.50000	−	2.59808i	−0.0534353	−	0.0925526i
$789$		0			0
$790$	−12.0000	+	20.7846i	−0.426941	+	0.739483i
$791$	6.00000			0.213335
$792$		0			0
$793$	8.00000			0.284088
$794$	−17.0000	+	29.4449i	−0.603307	+	1.04496i
$795$		0			0
$796$	−1.00000	−	1.73205i	−0.0354441	−	0.0613909i
$797$	21.0000	+	36.3731i	0.743858	+	1.28840i	0.950726	+	0.310031i	$0.100340\pi$
$797$	21.0000	+	36.3731i	0.743858	+	1.28840i	−0.206868	+	0.978369i	$0.566327\pi$
$798$		0			0
$799$	4.50000	−	7.79423i	0.159199	−	0.275740i
$800$	−4.00000			−0.141421
$801$		0			0
$802$	−36.0000			−1.27120
$803$	−6.00000	+	10.3923i	−0.211735	+	0.366736i
$804$		0			0
$805$		0			0
$806$	−2.00000	−	3.46410i	−0.0704470	−	0.122018i
$807$		0			0
$808$	−6.00000	+	10.3923i	−0.211079	+	0.365600i
$809$	−33.0000			−1.16022			−0.580109	−	0.814539i	$-0.696990\pi$
$809$	−33.0000			−1.16022			−0.580109	+	0.814539i	$0.696990\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$	3.00000	−	5.19615i	0.105279	−	0.182349i
$813$		0			0
$814$	−21.0000	−	36.3731i	−0.736050	−	1.27488i
$815$	−24.0000	−	41.5692i	−0.840683	−	1.45611i
$816$		0			0
$817$	1.00000	−	1.73205i	0.0349856	−	0.0605968i
$818$	−32.0000			−1.11885
$819$		0			0
$820$		0			0
$821$	1.50000	−	2.59808i	0.0523504	−	0.0906735i	−0.838663	−	0.544651i	$-0.816662\pi$
$821$	1.50000	−	2.59808i	0.0523504	−	0.0906735i	0.891013	+	0.453978i	$0.149995\pi$
$822$		0			0
$823$	−7.00000	−	12.1244i	−0.244005	−	0.422628i	0.717847	−	0.696201i	$-0.245128\pi$
$823$	−7.00000	−	12.1244i	−0.244005	−	0.422628i	−0.961851	+	0.273573i	$0.911795\pi$
$824$	−2.00000	−	3.46410i	−0.0696733	−	0.120678i
$825$		0			0
$826$	3.00000	−	5.19615i	0.104383	−	0.180797i
$827$	18.0000			0.625921			0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000			0.625921			0.312961	+	0.949766i	$0.398679\pi$
$828$		0			0
$829$	38.0000			1.31979			0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000			1.31979			0.659897	+	0.751356i	$0.270600\pi$
$830$	−18.0000	+	31.1769i	−0.624789	+	1.08217i
$831$		0			0
$832$	−0.500000	−	0.866025i	−0.0173344	−	0.0300240i
$833$	−9.00000	−	15.5885i	−0.311832	−	0.540108i
$834$		0			0
$835$		0			0
$836$	12.0000			0.415029
$837$		0			0
$838$	−9.00000			−0.310900
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$	8.50000	+	14.7224i	0.292929	+	0.507369i
$843$		0			0
$844$	6.50000	−	11.2583i	0.223739	−	0.387528i
$845$	−3.00000			−0.103203
$846$		0			0
$847$	−25.0000			−0.859010
$848$		0			0
$849$		0			0
$850$	−6.00000	−	10.3923i	−0.205798	−	0.356453i
$851$		0			0
$852$		0			0
$853$	18.5000	−	32.0429i	0.633428	−	1.09713i	−0.353418	−	0.935466i	$-0.614981\pi$
$853$	18.5000	−	32.0429i	0.633428	−	1.09713i	0.986846	−	0.161664i	$-0.0516860\pi$
$854$	8.00000			0.273754
$855$		0			0
$856$	−12.0000			−0.410152
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	−0.244701	−	0.969599i	$-0.578690\pi$
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	0.962048	−	0.272882i	$-0.0879768\pi$
$858$		0			0
$859$	2.00000	+	3.46410i	0.0682391	+	0.118194i	0.898126	−	0.439738i	$-0.144929\pi$
$859$	2.00000	+	3.46410i	0.0682391	+	0.118194i	−0.829887	+	0.557931i	$0.811595\pi$
$860$	−1.50000	−	2.59808i	−0.0511496	−	0.0885937i
$861$		0			0
$862$	−16.5000	+	28.5788i	−0.561992	+	0.973399i
$863$	−45.0000			−1.53182			−0.765909	−	0.642949i	$-0.777711\pi$
$863$	−45.0000			−1.53182			−0.765909	+	0.642949i	$0.777711\pi$
$864$		0			0
$865$		0			0
$866$	−12.5000	+	21.6506i	−0.424767	+	0.735719i
$867$		0			0
$868$	−2.00000	−	3.46410i	−0.0678844	−	0.117579i
$869$	−24.0000	−	41.5692i	−0.814144	−	1.41014i
$870$		0			0
$871$	−7.00000	+	12.1244i	−0.237186	+	0.410818i
$872$	7.00000			0.237050
$873$		0			0
$874$		0			0
$875$	1.50000	−	2.59808i	0.0507093	−	0.0878310i
$876$		0			0
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	0.954652	−	0.297724i	$-0.0962275\pi$
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	−0.735163	+	0.677891i	$0.762894\pi$
$878$	13.0000	+	22.5167i	0.438729	+	0.759900i
$879$		0			0
$880$	9.00000	−	15.5885i	0.303390	−	0.525487i
$881$	21.0000			0.707508			0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000			0.707508			0.353754	+	0.935339i	$0.384905\pi$
$882$		0			0
$883$	29.0000			0.975928			0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000			0.975928			0.487964	+	0.872864i	$0.337740\pi$
$884$	1.50000	−	2.59808i	0.0504505	−	0.0873828i
$885$		0			0
$886$	10.5000	+	18.1865i	0.352754	+	0.610989i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	10.0000	−	17.3205i	0.335389	−	0.580911i
$890$	−18.0000			−0.603361
$891$		0			0
$892$	−19.0000			−0.636167
$893$	−3.00000	+	5.19615i	−0.100391	+	0.173883i
$894$		0			0
$895$	4.50000	+	7.79423i	0.150418	+	0.260532i
$896$	−0.500000	−	0.866025i	−0.0167038	−	0.0289319i
$897$		0			0
$898$	3.00000	−	5.19615i	0.100111	−	0.173398i
$899$	−24.0000			−0.800445
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−3.00000	−	5.19615i	−0.0997785	−	0.172821i
$905$	30.0000	+	51.9615i	0.997234	+	1.72726i
$906$		0			0
$907$	18.5000	−	32.0429i	0.614282	−	1.06397i	−0.376228	−	0.926527i	$-0.622779\pi$
$907$	18.5000	−	32.0429i	0.614282	−	1.06397i	0.990510	−	0.137441i	$-0.0438878\pi$
$908$		0			0
$909$		0			0
$910$	−3.00000			−0.0994490
$911$	−15.0000	+	25.9808i	−0.496972	+	0.860781i	−0.999994	−	0.00349271i	$-0.998888\pi$
$911$	−15.0000	+	25.9808i	−0.496972	+	0.860781i	0.503022	+	0.864274i	$0.332222\pi$
$912$		0			0
$913$	−36.0000	−	62.3538i	−1.19143	−	2.06361i
$914$	−5.00000	−	8.66025i	−0.165385	−	0.286456i
$915$		0			0
$916$	6.50000	−	11.2583i	0.214766	−	0.371986i
$917$	21.0000			0.693481
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$		0			0
$922$	4.50000	+	7.79423i	0.148200	+	0.256689i
$923$	1.50000	+	2.59808i	0.0493731	+	0.0855167i
$924$		0			0
$925$	14.0000	−	24.2487i	0.460317	−	0.797293i
$926$	40.0000			1.31448
$927$		0			0
$928$	−6.00000			−0.196960
$929$	−18.0000	+	31.1769i	−0.590561	+	1.02288i	0.403596	+	0.914937i	$0.367760\pi$
$929$	−18.0000	+	31.1769i	−0.590561	+	1.02288i	−0.994157	+	0.107944i	$0.965573\pi$
$930$		0			0
$931$	6.00000	+	10.3923i	0.196642	+	0.340594i
$932$	13.5000	+	23.3827i	0.442207	+	0.765925i
$933$		0			0
$934$	18.0000	−	31.1769i	0.588978	−	1.02014i
$935$	54.0000			1.76599
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$	−7.00000	+	12.1244i	−0.228558	+	0.395874i
$939$		0			0
$940$	4.50000	+	7.79423i	0.146774	+	0.254220i
$941$	10.5000	+	18.1865i	0.342290	+	0.592864i	0.984858	−	0.173365i	$-0.0554641\pi$
$941$	10.5000	+	18.1865i	0.342290	+	0.592864i	−0.642567	+	0.766229i	$0.722131\pi$
$942$		0			0
$943$		0			0
$944$	−6.00000			−0.195283
$945$		0			0
$946$	6.00000			0.195077
$947$	−3.00000	+	5.19615i	−0.0974869	+	0.168852i	−0.910644	−	0.413192i	$-0.864414\pi$
$947$	−3.00000	+	5.19615i	−0.0974869	+	0.168852i	0.813157	+	0.582045i	$0.197747\pi$
$948$		0			0
$949$	−1.00000	−	1.73205i	−0.0324614	−	0.0562247i
$950$	4.00000	+	6.92820i	0.129777	+	0.224781i
$951$		0			0
$952$	1.50000	−	2.59808i	0.0486153	−	0.0842041i
$953$	15.0000			0.485898			0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000			0.485898			0.242949	+	0.970039i	$0.421885\pi$
$954$		0			0
$955$	54.0000			1.74740
$956$	−7.50000	+	12.9904i	−0.242567	+	0.420139i
$957$		0			0
$958$	−10.5000	−	18.1865i	−0.339240	−	0.587580i
$959$		0			0
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$	7.00000			0.225689
$963$		0			0
$964$	−10.0000			−0.322078
$965$	−6.00000	+	10.3923i	−0.193147	+	0.334540i
$966$		0			0
$967$	15.5000	+	26.8468i	0.498446	+	0.863334i	0.999998	−	0.00179302i	$-0.000570736\pi$
$967$	15.5000	+	26.8468i	0.498446	+	0.863334i	−0.501552	+	0.865128i	$0.667237\pi$
$968$	12.5000	+	21.6506i	0.401765	+	0.695878i
$969$		0			0
$970$	15.0000	−	25.9808i	0.481621	−	0.834192i
$971$	−3.00000			−0.0962746			−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000			−0.0962746			−0.0481373	+	0.998841i	$0.515328\pi$
$972$		0			0
$973$	13.0000			0.416761
$974$	−8.00000	+	13.8564i	−0.256337	+	0.443988i
$975$		0			0
$976$	−4.00000	−	6.92820i	−0.128037	−	0.221766i
$977$	27.0000	+	46.7654i	0.863807	+	1.49616i	0.868227	+	0.496167i	$0.165259\pi$
$977$	27.0000	+	46.7654i	0.863807	+	1.49616i	−0.00442082	+	0.999990i	$0.501407\pi$
$978$		0			0
$979$	18.0000	−	31.1769i	0.575282	−	0.996419i
$980$	18.0000			0.574989
$981$		0			0
$982$	9.00000			0.287202
$983$	−19.5000	+	33.7750i	−0.621953	+	1.07725i	0.367168	+	0.930155i	$0.380327\pi$
$983$	−19.5000	+	33.7750i	−0.621953	+	1.07725i	−0.989122	+	0.147100i	$0.953006\pi$
$984$		0			0
$985$	4.50000	+	7.79423i	0.143382	+	0.248345i
$986$	−9.00000	−	15.5885i	−0.286618	−	0.496438i
$987$		0			0
$988$	−1.00000	+	1.73205i	−0.0318142	+	0.0551039i
$989$		0			0
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$	−2.00000	+	3.46410i	−0.0635001	+	0.109985i
$993$		0			0
$994$	1.50000	+	2.59808i	0.0475771	+	0.0824060i
$995$	3.00000	+	5.19615i	0.0951064	+	0.164729i
$996$		0			0
$997$	23.0000	−	39.8372i	0.728417	−	1.26166i	−0.229135	−	0.973395i	$-0.573590\pi$
$997$	23.0000	−	39.8372i	0.728417	−	1.26166i	0.957552	−	0.288261i	$-0.0930771\pi$
$998$	40.0000			1.26618
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.e.ba.703.1		2
3.2	odd	2		2106.2.e.b.703.1		2
9.2	odd	6		234.2.a.e.1.1		1
9.4	even	3	inner	2106.2.e.ba.1405.1		2
9.5	odd	6		2106.2.e.b.1405.1		2
9.7	even	3		26.2.a.a.1.1	✓	1
36.7	odd	6		208.2.a.a.1.1		1
36.11	even	6		1872.2.a.q.1.1		1
45.2	even	12		5850.2.e.a.5149.2		2
45.7	odd	12		650.2.b.d.599.1		2
45.29	odd	6		5850.2.a.p.1.1		1
45.34	even	6		650.2.a.j.1.1		1
45.38	even	12		5850.2.e.a.5149.1		2
45.43	odd	12		650.2.b.d.599.2		2
63.16	even	3		1274.2.f.p.1145.1		2
63.25	even	3		1274.2.f.p.79.1		2
63.34	odd	6		1274.2.a.d.1.1		1
63.52	odd	6		1274.2.f.r.79.1		2
63.61	odd	6		1274.2.f.r.1145.1		2
72.11	even	6		7488.2.a.h.1.1		1
72.29	odd	6		7488.2.a.g.1.1		1
72.43	odd	6		832.2.a.i.1.1		1
72.61	even	6		832.2.a.d.1.1		1
99.43	odd	6		3146.2.a.n.1.1		1
117.7	odd	12		338.2.e.a.23.1		4
117.16	even	3		338.2.c.d.191.1		2
117.25	even	6		338.2.a.f.1.1		1
117.34	odd	12		338.2.b.c.337.1		2
117.38	odd	6		3042.2.a.a.1.1		1
117.43	even	6		338.2.c.a.315.1		2
117.47	even	12		3042.2.b.a.1351.2		2
117.61	even	3		338.2.c.d.315.1		2
117.70	odd	12		338.2.b.c.337.2		2
117.83	even	12		3042.2.b.a.1351.1		2
117.88	even	6		338.2.c.a.191.1		2
117.97	odd	12		338.2.e.a.23.2		4
117.106	odd	12		338.2.e.a.147.1		4
117.115	odd	12		338.2.e.a.147.2		4
144.43	odd	12		3328.2.b.j.1665.2		2
144.61	even	12		3328.2.b.m.1665.2		2
144.115	odd	12		3328.2.b.j.1665.1		2
144.133	even	12		3328.2.b.m.1665.1		2
153.16	even	6		7514.2.a.c.1.1		1
171.151	odd	6		9386.2.a.j.1.1		1
180.79	odd	6		5200.2.a.x.1.1		1
468.151	even	12		2704.2.f.d.337.1		2
468.187	even	12		2704.2.f.d.337.2		2
468.259	odd	6		2704.2.a.f.1.1		1
585.259	even	6		8450.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.a.1.1	✓	1	9.7	even	3
208.2.a.a.1.1		1	36.7	odd	6
234.2.a.e.1.1		1	9.2	odd	6
338.2.a.f.1.1		1	117.25	even	6
338.2.b.c.337.1		2	117.34	odd	12
338.2.b.c.337.2		2	117.70	odd	12
338.2.c.a.191.1		2	117.88	even	6
338.2.c.a.315.1		2	117.43	even	6
338.2.c.d.191.1		2	117.16	even	3
338.2.c.d.315.1		2	117.61	even	3
338.2.e.a.23.1		4	117.7	odd	12
338.2.e.a.23.2		4	117.97	odd	12
338.2.e.a.147.1		4	117.106	odd	12
338.2.e.a.147.2		4	117.115	odd	12
650.2.a.j.1.1		1	45.34	even	6
650.2.b.d.599.1		2	45.7	odd	12
650.2.b.d.599.2		2	45.43	odd	12
832.2.a.d.1.1		1	72.61	even	6
832.2.a.i.1.1		1	72.43	odd	6
1274.2.a.d.1.1		1	63.34	odd	6
1274.2.f.p.79.1		2	63.25	even	3
1274.2.f.p.1145.1		2	63.16	even	3
1274.2.f.r.79.1		2	63.52	odd	6
1274.2.f.r.1145.1		2	63.61	odd	6
1872.2.a.q.1.1		1	36.11	even	6
2106.2.e.b.703.1		2	3.2	odd	2
2106.2.e.b.1405.1		2	9.5	odd	6
2106.2.e.ba.703.1		2	1.1	even	1	trivial
2106.2.e.ba.1405.1		2	9.4	even	3	inner
2704.2.a.f.1.1		1	468.259	odd	6
2704.2.f.d.337.1		2	468.151	even	12
2704.2.f.d.337.2		2	468.187	even	12
3042.2.a.a.1.1		1	117.38	odd	6
3042.2.b.a.1351.1		2	117.83	even	12
3042.2.b.a.1351.2		2	117.47	even	12
3146.2.a.n.1.1		1	99.43	odd	6
3328.2.b.j.1665.1		2	144.115	odd	12
3328.2.b.j.1665.2		2	144.43	odd	12
3328.2.b.m.1665.1		2	144.133	even	12
3328.2.b.m.1665.2		2	144.61	even	12
5200.2.a.x.1.1		1	180.79	odd	6
5850.2.a.p.1.1		1	45.29	odd	6
5850.2.e.a.5149.1		2	45.38	even	12
5850.2.e.a.5149.2		2	45.2	even	12
7488.2.a.g.1.1		1	72.29	odd	6
7488.2.a.h.1.1		1	72.11	even	6
7514.2.a.c.1.1		1	153.16	even	6
8450.2.a.c.1.1		1	585.259	even	6
9386.2.a.j.1.1		1	171.151	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)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.50000 + 2.59808i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.50000 + 2.59808i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +3.00000 q^{10} +(-3.00000 + 5.19615i) q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -3.00000 q^{17} +2.00000 q^{19} +(1.50000 - 2.59808i) q^{20} +(3.00000 + 5.19615i) q^{22} +(-2.00000 + 3.46410i) q^{25} -1.00000 q^{26} -1.00000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(2.00000 + 3.46410i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-1.50000 + 2.59808i) q^{34} +3.00000 q^{35} -7.00000 q^{37} +(1.00000 - 1.73205i) q^{38} +(-1.50000 - 2.59808i) q^{40} +(0.500000 - 0.866025i) q^{43} +6.00000 q^{44} +(-1.50000 + 2.59808i) q^{47} +(3.00000 + 5.19615i) q^{49} +(2.00000 + 3.46410i) q^{50} +(-0.500000 + 0.866025i) q^{52} -18.0000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(3.00000 + 5.19615i) q^{58} +(3.00000 + 5.19615i) q^{59} +(-4.00000 + 6.92820i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(1.50000 - 2.59808i) q^{65} +(-7.00000 - 12.1244i) q^{67} +(1.50000 + 2.59808i) q^{68} +(1.50000 - 2.59808i) q^{70} -3.00000 q^{71} +2.00000 q^{73} +(-3.50000 + 6.06218i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(3.00000 + 5.19615i) q^{77} +(-4.00000 + 6.92820i) q^{79} -3.00000 q^{80} +(-6.00000 + 10.3923i) q^{83} +(-4.50000 - 7.79423i) q^{85} +(-0.500000 - 0.866025i) q^{86} +(3.00000 - 5.19615i) q^{88} -6.00000 q^{89} -1.00000 q^{91} +(1.50000 + 2.59808i) q^{94} +(3.00000 + 5.19615i) q^{95} +(5.00000 - 8.66025i) q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 3 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 + 3 * q^5 + q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} + 3 q^{5} + q^{7} - 2 q^{8} + 6 q^{10} - 6 q^{11} - q^{13} - q^{14} - q^{16} - 6 q^{17} + 4 q^{19} + 3 q^{20} + 6 q^{22} - 4 q^{25} - 2 q^{26} - 2 q^{28} - 6 q^{29} + 4 q^{31} + q^{32} - 3 q^{34} + 6 q^{35} - 14 q^{37} + 2 q^{38} - 3 q^{40} + q^{43} + 12 q^{44} - 3 q^{47} + 6 q^{49} + 4 q^{50} - q^{52} - 36 q^{55} - q^{56} + 6 q^{58} + 6 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} + 3 q^{65} - 14 q^{67} + 3 q^{68} + 3 q^{70} - 6 q^{71} + 4 q^{73} - 7 q^{74} - 2 q^{76} + 6 q^{77} - 8 q^{79} - 6 q^{80} - 12 q^{83} - 9 q^{85} - q^{86} + 6 q^{88} - 12 q^{89} - 2 q^{91} + 3 q^{94} + 6 q^{95} + 10 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 3 * q^5 + q^7 - 2 * q^8 + 6 * q^10 - 6 * q^11 - q^13 - q^14 - q^16 - 6 * q^17 + 4 * q^19 + 3 * q^20 + 6 * q^22 - 4 * q^25 - 2 * q^26 - 2 * q^28 - 6 * q^29 + 4 * q^31 + q^32 - 3 * q^34 + 6 * q^35 - 14 * q^37 + 2 * q^38 - 3 * q^40 + q^43 + 12 * q^44 - 3 * q^47 + 6 * q^49 + 4 * q^50 - q^52 - 36 * q^55 - q^56 + 6 * q^58 + 6 * q^59 - 8 * q^61 + 8 * q^62 + 2 * q^64 + 3 * q^65 - 14 * q^67 + 3 * q^68 + 3 * q^70 - 6 * q^71 + 4 * q^73 - 7 * q^74 - 2 * q^76 + 6 * q^77 - 8 * q^79 - 6 * q^80 - 12 * q^83 - 9 * q^85 - q^86 + 6 * q^88 - 12 * q^89 - 2 * q^91 + 3 * q^94 + 6 * q^95 + 10 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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