LMFDB - Embedded newform 1050.2.o.d.499.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(499,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1050.499");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.o (of order $6$, 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:	$8.38429221223$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			499.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1050.499
Dual form			1050.2.o.d.949.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} -1.00000 q^{6} +(-2.59808 - 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} -1.00000 q^{6} +(-2.59808 - 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.866025 - 0.500000i) q^{12} -5.00000i q^{13} +(-2.00000 - 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-5.19615 + 3.00000i) q^{17} +(0.866025 - 0.500000i) q^{18} +(-3.50000 + 6.06218i) q^{19} +(2.50000 - 0.866025i) q^{21} +(-5.19615 - 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{24} +(2.50000 - 4.33013i) q^{26} +1.00000i q^{27} +(-0.866025 - 2.50000i) q^{28} +(-4.00000 - 6.92820i) q^{31} +(-0.866025 + 0.500000i) q^{32} -6.00000 q^{34} +1.00000 q^{36} +(0.866025 + 0.500000i) q^{37} +(-6.06218 + 3.50000i) q^{38} +(2.50000 + 4.33013i) q^{39} +(2.59808 + 0.500000i) q^{42} -8.00000i q^{43} +(-3.00000 - 5.19615i) q^{46} +(-5.19615 - 3.00000i) q^{47} -1.00000i q^{48} +(6.50000 + 2.59808i) q^{49} +(3.00000 - 5.19615i) q^{51} +(4.33013 - 2.50000i) q^{52} +(5.19615 - 3.00000i) q^{53} +(-0.500000 + 0.866025i) q^{54} +(0.500000 - 2.59808i) q^{56} -7.00000i q^{57} +(-3.00000 - 5.19615i) q^{59} +(0.500000 - 0.866025i) q^{61} -8.00000i q^{62} +(-1.73205 + 2.00000i) q^{63} -1.00000 q^{64} +(-11.2583 + 6.50000i) q^{67} +(-5.19615 - 3.00000i) q^{68} +6.00000 q^{69} +12.0000 q^{71} +(0.866025 + 0.500000i) q^{72} +(-4.33013 + 2.50000i) q^{73} +(0.500000 + 0.866025i) q^{74} -7.00000 q^{76} +5.00000i q^{78} +(-3.50000 + 6.06218i) q^{79} +(-0.500000 - 0.866025i) q^{81} +18.0000i q^{83} +(2.00000 + 1.73205i) q^{84} +(4.00000 - 6.92820i) q^{86} +(-3.00000 + 5.19615i) q^{89} +(-2.50000 + 12.9904i) q^{91} -6.00000i q^{92} +(6.92820 + 4.00000i) q^{93} +(-3.00000 - 5.19615i) q^{94} +(0.500000 - 0.866025i) q^{96} -7.00000i q^{97} +(4.33013 + 5.50000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 4 * q^6 + 2 * q^9	$ 4 q + 2 q^{4} - 4 q^{6} + 2 q^{9} - 8 q^{14} - 2 q^{16} - 14 q^{19} + 10 q^{21} - 2 q^{24} + 10 q^{26} - 16 q^{31} - 24 q^{34} + 4 q^{36} + 10 q^{39} - 12 q^{46} + 26 q^{49} + 12 q^{51} - 2 q^{54} + 2 q^{56} - 12 q^{59} + 2 q^{61} - 4 q^{64} + 24 q^{69} + 48 q^{71} + 2 q^{74} - 28 q^{76} - 14 q^{79} - 2 q^{81} + 8 q^{84} + 16 q^{86} - 12 q^{89} - 10 q^{91} - 12 q^{94} + 2 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 - 4 * q^6 + 2 * q^9 - 8 * q^14 - 2 * q^16 - 14 * q^19 + 10 * q^21 - 2 * q^24 + 10 * q^26 - 16 * q^31 - 24 * q^34 + 4 * q^36 + 10 * q^39 - 12 * q^46 + 26 * q^49 + 12 * q^51 - 2 * q^54 + 2 * q^56 - 12 * q^59 + 2 * q^61 - 4 * q^64 + 24 * q^69 + 48 * q^71 + 2 * q^74 - 28 * q^76 - 14 * q^79 - 2 * q^81 + 8 * q^84 + 16 * q^86 - 12 * q^89 - 10 * q^91 - 12 * q^94 + 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	0.866025	+	0.500000i	0.612372	+	0.353553i
$2$	0.866025	+	0.500000i	0.612372	+	0.353553i
$3$	−0.866025	+	0.500000i	−0.500000	+	0.288675i
$3$	−0.866025	+	0.500000i	−0.500000	+	0.288675i
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$		0			0
$5$		0			0
$6$	−1.00000			−0.408248
$7$	−2.59808	−	0.500000i	−0.981981	−	0.188982i
$7$	−2.59808	−	0.500000i	−0.981981	−	0.188982i
$8$			1.00000i			0.353553i
$9$	0.500000	−	0.866025i	0.166667	−	0.288675i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$	−0.866025	−	0.500000i	−0.250000	−	0.144338i
$13$		−	5.00000i		−	1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$		−	5.00000i		−	1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	−2.00000	−	1.73205i	−0.534522	−	0.462910i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−5.19615	+	3.00000i	−1.26025	+	0.727607i	−0.973123	−	0.230285i	$-0.926034\pi$
$17$	−5.19615	+	3.00000i	−1.26025	+	0.727607i	−0.287129	+	0.957892i	$0.592701\pi$
$18$	0.866025	−	0.500000i	0.204124	−	0.117851i
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	0.114708	+	0.993399i	$0.463407\pi$
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	2.50000	−	0.866025i	0.545545	−	0.188982i
$22$		0			0
$23$	−5.19615	−	3.00000i	−1.08347	−	0.625543i	−0.151642	−	0.988436i	$-0.548456\pi$
$23$	−5.19615	−	3.00000i	−1.08347	−	0.625543i	−0.931831	+	0.362892i	$0.881789\pi$
$24$	−0.500000	−	0.866025i	−0.102062	−	0.176777i
$25$		0			0
$26$	2.50000	−	4.33013i	0.490290	−	0.849208i
$27$			1.00000i			0.192450i
$28$	−0.866025	−	2.50000i	−0.163663	−	0.472456i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$	−0.866025	+	0.500000i	−0.153093	+	0.0883883i
$33$		0			0
$34$	−6.00000			−1.02899
$35$		0			0
$36$	1.00000			0.166667
$37$	0.866025	+	0.500000i	0.142374	+	0.0821995i	0.569495	−	0.821995i	$-0.307139\pi$
$37$	0.866025	+	0.500000i	0.142374	+	0.0821995i	−0.427121	+	0.904194i	$0.640472\pi$
$38$	−6.06218	+	3.50000i	−0.983415	+	0.567775i
$39$	2.50000	+	4.33013i	0.400320	+	0.693375i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	2.59808	+	0.500000i	0.400892	+	0.0771517i
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$		0			0
$45$		0			0
$46$	−3.00000	−	5.19615i	−0.442326	−	0.766131i
$47$	−5.19615	−	3.00000i	−0.757937	−	0.437595i	0.0706177	−	0.997503i	$-0.477503\pi$
$47$	−5.19615	−	3.00000i	−0.757937	−	0.437595i	−0.828554	+	0.559908i	$0.810836\pi$
$48$		−	1.00000i		−	0.144338i
$49$	6.50000	+	2.59808i	0.928571	+	0.371154i
$50$		0			0
$51$	3.00000	−	5.19615i	0.420084	−	0.727607i
$52$	4.33013	−	2.50000i	0.600481	−	0.346688i
$53$	5.19615	−	3.00000i	0.713746	−	0.412082i	−0.0987002	−	0.995117i	$-0.531468\pi$
$53$	5.19615	−	3.00000i	0.713746	−	0.412082i	0.812447	+	0.583036i	$0.198135\pi$
$54$	−0.500000	+	0.866025i	−0.0680414	+	0.117851i
$55$		0			0
$56$	0.500000	−	2.59808i	0.0668153	−	0.347183i
$57$		−	7.00000i		−	0.927173i
$58$		0			0
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	0.601958	−	0.798528i	$-0.294388\pi$
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	−0.992524	+	0.122047i	$0.961054\pi$
$60$		0			0
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	0.896258	+	0.443533i	$0.146275\pi$
$62$		−	8.00000i		−	1.01600i
$63$	−1.73205	+	2.00000i	−0.218218	+	0.251976i
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	−11.2583	+	6.50000i	−1.37542	+	0.794101i	−0.991605	−	0.129307i	$-0.958725\pi$
$67$	−11.2583	+	6.50000i	−1.37542	+	0.794101i	−0.383819	+	0.923408i	$0.625391\pi$
$68$	−5.19615	−	3.00000i	−0.630126	−	0.363803i
$69$	6.00000			0.722315
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$	0.866025	+	0.500000i	0.102062	+	0.0589256i
$73$	−4.33013	+	2.50000i	−0.506803	+	0.292603i	−0.731519	−	0.681822i	$-0.761188\pi$
$73$	−4.33013	+	2.50000i	−0.506803	+	0.292603i	0.224716	+	0.974424i	$0.427855\pi$
$74$	0.500000	+	0.866025i	0.0581238	+	0.100673i
$75$		0			0
$76$	−7.00000			−0.802955
$77$		0			0
$78$			5.00000i			0.566139i
$79$	−3.50000	+	6.06218i	−0.393781	+	0.682048i	−0.992945	−	0.118578i	$-0.962166\pi$
$79$	−3.50000	+	6.06218i	−0.393781	+	0.682048i	0.599164	+	0.800626i	$0.295500\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			18.0000i			1.97576i	0.155230	+	0.987878i	$0.450388\pi$
$83$			18.0000i			1.97576i	−0.155230	+	0.987878i	$0.549612\pi$
$84$	2.00000	+	1.73205i	0.218218	+	0.188982i
$85$		0			0
$86$	4.00000	−	6.92820i	0.431331	−	0.747087i
$87$		0			0
$88$		0			0
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	−0.980071	−	0.198650i	$-0.936344\pi$
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	0.662071	+	0.749441i	$0.269678\pi$
$90$		0			0
$91$	−2.50000	+	12.9904i	−0.262071	+	1.36176i
$92$		−	6.00000i		−	0.625543i
$93$	6.92820	+	4.00000i	0.718421	+	0.414781i
$94$	−3.00000	−	5.19615i	−0.309426	−	0.535942i
$95$		0			0
$96$	0.500000	−	0.866025i	0.0510310	−	0.0883883i
$97$		−	7.00000i		−	0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$		−	7.00000i		−	0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$	4.33013	+	5.50000i	0.437409	+	0.555584i
$99$		0			0
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$	5.19615	−	3.00000i	0.514496	−	0.297044i
$103$	−11.2583	−	6.50000i	−1.10932	−	0.640464i	−0.170664	−	0.985329i	$-0.554591\pi$
$103$	−11.2583	−	6.50000i	−1.10932	−	0.640464i	−0.938652	+	0.344865i	$0.887925\pi$
$104$	5.00000			0.490290
$105$		0			0
$106$	6.00000			0.582772
$107$	15.5885	+	9.00000i	1.50699	+	0.870063i	0.999967	+	0.00813215i	$0.00258857\pi$
$107$	15.5885	+	9.00000i	1.50699	+	0.870063i	0.507026	+	0.861931i	$0.330745\pi$
$108$	−0.866025	+	0.500000i	−0.0833333	+	0.0481125i
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	0.648292	−	0.761392i	$-0.275484\pi$
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	−0.983531	+	0.180741i	$0.942150\pi$
$110$		0			0
$111$	−1.00000			−0.0949158
$112$	1.73205	−	2.00000i	0.163663	−	0.188982i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	3.50000	−	6.06218i	0.327805	−	0.567775i
$115$		0			0
$116$		0			0
$117$	−4.33013	−	2.50000i	−0.400320	−	0.231125i
$118$		−	6.00000i		−	0.552345i
$119$	15.0000	−	5.19615i	1.37505	−	0.476331i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$	0.866025	−	0.500000i	0.0784063	−	0.0452679i
$123$		0			0
$124$	4.00000	−	6.92820i	0.359211	−	0.622171i
$125$		0			0
$126$	−2.50000	+	0.866025i	−0.222718	+	0.0771517i
$127$			11.0000i			0.976092i	0.872818	+	0.488046i	$0.162290\pi$
$127$			11.0000i			0.976092i	−0.872818	+	0.488046i	$0.837710\pi$
$128$	−0.866025	−	0.500000i	−0.0765466	−	0.0441942i
$129$	4.00000	+	6.92820i	0.352180	+	0.609994i
$130$		0			0
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	−0.966803	−	0.255524i	$-0.917752\pi$
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	0.704692	+	0.709514i	$0.251085\pi$
$132$		0			0
$133$	12.1244	−	14.0000i	1.05131	−	1.21395i
$134$	−13.0000			−1.12303
$135$		0			0
$136$	−3.00000	−	5.19615i	−0.257248	−	0.445566i
$137$	−15.5885	+	9.00000i	−1.33181	+	0.768922i	−0.985577	−	0.169226i	$-0.945873\pi$
$137$	−15.5885	+	9.00000i	−1.33181	+	0.768922i	−0.346235	+	0.938148i	$0.612540\pi$
$138$	5.19615	+	3.00000i	0.442326	+	0.255377i
$139$	−11.0000			−0.933008			−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000			−0.933008			−0.466504	+	0.884519i	$0.654487\pi$
$140$		0			0
$141$	6.00000			0.505291
$142$	10.3923	+	6.00000i	0.872103	+	0.503509i
$143$		0			0
$144$	0.500000	+	0.866025i	0.0416667	+	0.0721688i
$145$		0			0
$146$	−5.00000			−0.413803
$147$	−6.92820	+	1.00000i	−0.571429	+	0.0824786i
$148$			1.00000i			0.0821995i
$149$	6.00000	−	10.3923i	0.491539	−	0.851371i	−0.508413	−	0.861113i	$-0.669768\pi$
$149$	6.00000	−	10.3923i	0.491539	−	0.851371i	0.999953	+	0.00974235i	$0.00310113\pi$
$150$		0			0
$151$	0.500000	+	0.866025i	0.0406894	+	0.0704761i	0.885653	−	0.464348i	$-0.153711\pi$
$151$	0.500000	+	0.866025i	0.0406894	+	0.0704761i	−0.844963	+	0.534824i	$0.820378\pi$
$152$	−6.06218	−	3.50000i	−0.491708	−	0.283887i
$153$			6.00000i			0.485071i
$154$		0			0
$155$		0			0
$156$	−2.50000	+	4.33013i	−0.200160	+	0.346688i
$157$	−0.866025	+	0.500000i	−0.0691164	+	0.0399043i	−0.534160	−	0.845383i	$-0.679372\pi$
$157$	−0.866025	+	0.500000i	−0.0691164	+	0.0399043i	0.465044	+	0.885288i	$0.346039\pi$
$158$	−6.06218	+	3.50000i	−0.482281	+	0.278445i
$159$	−3.00000	+	5.19615i	−0.237915	+	0.412082i
$160$		0			0
$161$	12.0000	+	10.3923i	0.945732	+	0.819028i
$162$		−	1.00000i		−	0.0785674i
$163$	−0.866025	−	0.500000i	−0.0678323	−	0.0391630i	0.465700	−	0.884943i	$-0.345802\pi$
$163$	−0.866025	−	0.500000i	−0.0678323	−	0.0391630i	−0.533533	+	0.845780i	$0.679136\pi$
$164$		0			0
$165$		0			0
$166$	−9.00000	+	15.5885i	−0.698535	+	1.20990i
$167$		−	6.00000i		−	0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$		−	6.00000i		−	0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$	0.866025	+	2.50000i	0.0668153	+	0.192879i
$169$	−12.0000			−0.923077
$170$		0			0
$171$	3.50000	+	6.06218i	0.267652	+	0.463586i
$172$	6.92820	−	4.00000i	0.528271	−	0.304997i
$173$	15.5885	+	9.00000i	1.18517	+	0.684257i	0.957205	−	0.289412i	$-0.0934598\pi$
$173$	15.5885	+	9.00000i	1.18517	+	0.684257i	0.227964	+	0.973670i	$0.426793\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.19615	+	3.00000i	0.390567	+	0.225494i
$178$	−5.19615	+	3.00000i	−0.389468	+	0.224860i
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	0.998286	−	0.0585225i	$-0.0186389\pi$
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	−0.549825	+	0.835280i	$0.685306\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	−8.66025	+	10.0000i	−0.641941	+	0.741249i
$183$			1.00000i			0.0739221i
$184$	3.00000	−	5.19615i	0.221163	−	0.383065i
$185$		0			0
$186$	4.00000	+	6.92820i	0.293294	+	0.508001i
$187$		0			0
$188$		−	6.00000i		−	0.437595i
$189$	0.500000	−	2.59808i	0.0363696	−	0.188982i
$190$		0			0
$191$	−12.0000	+	20.7846i	−0.868290	+	1.50392i	−0.00454614	+	0.999990i	$0.501447\pi$
$191$	−12.0000	+	20.7846i	−0.868290	+	1.50392i	−0.863743	+	0.503932i	$0.831886\pi$
$192$	0.866025	−	0.500000i	0.0625000	−	0.0360844i
$193$	8.66025	−	5.00000i	0.623379	−	0.359908i	−0.154805	−	0.987945i	$-0.549475\pi$
$193$	8.66025	−	5.00000i	0.623379	−	0.359908i	0.778183	+	0.628037i	$0.216141\pi$
$194$	3.50000	−	6.06218i	0.251285	−	0.435239i
$195$		0			0
$196$	1.00000	+	6.92820i	0.0714286	+	0.494872i
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		0			0
$199$	5.50000	+	9.52628i	0.389885	+	0.675300i	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	5.50000	+	9.52628i	0.389885	+	0.675300i	−0.602549	+	0.798082i	$0.705848\pi$
$200$		0			0
$201$	6.50000	−	11.2583i	0.458475	−	0.794101i
$202$		0			0
$203$		0			0
$204$	6.00000			0.420084
$205$		0			0
$206$	−6.50000	−	11.2583i	−0.452876	−	0.784405i
$207$	−5.19615	+	3.00000i	−0.361158	+	0.208514i
$208$	4.33013	+	2.50000i	0.300240	+	0.173344i
$209$		0			0
$210$		0			0
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$	5.19615	+	3.00000i	0.356873	+	0.206041i
$213$	−10.3923	+	6.00000i	−0.712069	+	0.411113i
$214$	9.00000	+	15.5885i	0.615227	+	1.06561i
$215$		0			0
$216$	−1.00000			−0.0680414
$217$	6.92820	+	20.0000i	0.470317	+	1.35769i
$218$		−	7.00000i		−	0.474100i
$219$	2.50000	−	4.33013i	0.168934	−	0.292603i
$220$		0			0
$221$	15.0000	+	25.9808i	1.00901	+	1.74766i
$222$	−0.866025	−	0.500000i	−0.0581238	−	0.0335578i
$223$		−	17.0000i		−	1.13840i	−0.822198	−	0.569202i	$-0.807252\pi$
$223$		−	17.0000i		−	1.13840i	0.822198	−	0.569202i	$-0.192748\pi$
$224$	2.50000	−	0.866025i	0.167038	−	0.0578638i
$225$		0			0
$226$	−3.00000	+	5.19615i	−0.199557	+	0.345643i
$227$	10.3923	−	6.00000i	0.689761	−	0.398234i	−0.113761	−	0.993508i	$-0.536290\pi$
$227$	10.3923	−	6.00000i	0.689761	−	0.398234i	0.803523	+	0.595274i	$0.202957\pi$
$228$	6.06218	−	3.50000i	0.401478	−	0.231793i
$229$	2.50000	−	4.33013i	0.165205	−	0.286143i	−0.771523	−	0.636201i	$-0.780505\pi$
$229$	2.50000	−	4.33013i	0.165205	−	0.286143i	0.936728	+	0.350058i	$0.113838\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	20.7846	+	12.0000i	1.36165	+	0.786146i	0.989843	−	0.142166i	$-0.0454066\pi$
$233$	20.7846	+	12.0000i	1.36165	+	0.786146i	0.371802	+	0.928312i	$0.378740\pi$
$234$	−2.50000	−	4.33013i	−0.163430	−	0.283069i
$235$		0			0
$236$	3.00000	−	5.19615i	0.195283	−	0.338241i
$237$		−	7.00000i		−	0.454699i
$238$	15.5885	+	3.00000i	1.01045	+	0.194461i
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	0.774202	−	0.632938i	$-0.218151\pi$
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	−0.935242	+	0.354010i	$0.884818\pi$
$242$	9.52628	−	5.50000i	0.612372	−	0.353553i
$243$	0.866025	+	0.500000i	0.0555556	+	0.0320750i
$244$	1.00000			0.0640184
$245$		0			0
$246$		0			0
$247$	30.3109	+	17.5000i	1.92864	+	1.11350i
$248$	6.92820	−	4.00000i	0.439941	−	0.254000i
$249$	−9.00000	−	15.5885i	−0.570352	−	0.987878i
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$	−2.59808	−	0.500000i	−0.163663	−	0.0314970i
$253$		0			0
$254$	−5.50000	+	9.52628i	−0.345101	+	0.597732i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	20.7846	+	12.0000i	1.29651	+	0.748539i	0.979799	−	0.199983i	$-0.0640888\pi$
$257$	20.7846	+	12.0000i	1.29651	+	0.748539i	0.316709	+	0.948523i	$0.397422\pi$
$258$			8.00000i			0.498058i
$259$	−2.00000	−	1.73205i	−0.124274	−	0.107624i
$260$		0			0
$261$		0			0
$262$	−5.19615	+	3.00000i	−0.321019	+	0.185341i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$	17.5000	−	6.06218i	1.07299	−	0.371696i
$267$		−	6.00000i		−	0.367194i
$268$	−11.2583	−	6.50000i	−0.687712	−	0.397051i
$269$	−12.0000	−	20.7846i	−0.731653	−	1.26726i	−0.956176	−	0.292791i	$-0.905416\pi$
$269$	−12.0000	−	20.7846i	−0.731653	−	1.26726i	0.224523	−	0.974469i	$-0.427917\pi$
$270$		0			0
$271$	−10.0000	+	17.3205i	−0.607457	+	1.05215i	0.384201	+	0.923249i	$0.374477\pi$
$271$	−10.0000	+	17.3205i	−0.607457	+	1.05215i	−0.991658	+	0.128897i	$0.958856\pi$
$272$		−	6.00000i		−	0.363803i
$273$	−4.33013	−	12.5000i	−0.262071	−	0.756534i
$274$	−18.0000			−1.08742
$275$		0			0
$276$	3.00000	+	5.19615i	0.180579	+	0.312772i
$277$	−16.4545	+	9.50000i	−0.988654	+	0.570800i	−0.904872	−	0.425684i	$-0.860033\pi$
$277$	−16.4545	+	9.50000i	−0.988654	+	0.570800i	−0.0837823	+	0.996484i	$0.526700\pi$
$278$	−9.52628	−	5.50000i	−0.571348	−	0.329868i
$279$	−8.00000			−0.478947
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$	5.19615	+	3.00000i	0.309426	+	0.178647i
$283$	−4.33013	+	2.50000i	−0.257399	+	0.148610i	−0.623148	−	0.782104i	$-0.714146\pi$
$283$	−4.33013	+	2.50000i	−0.257399	+	0.148610i	0.365748	+	0.930714i	$0.380813\pi$
$284$	6.00000	+	10.3923i	0.356034	+	0.616670i
$285$		0			0
$286$		0			0
$287$		0			0
$288$			1.00000i			0.0589256i
$289$	9.50000	−	16.4545i	0.558824	−	0.967911i
$290$		0			0
$291$	3.50000	+	6.06218i	0.205174	+	0.355371i
$292$	−4.33013	−	2.50000i	−0.253402	−	0.146301i
$293$			24.0000i			1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$			24.0000i			1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$	−6.50000	−	2.59808i	−0.379088	−	0.151523i
$295$		0			0
$296$	−0.500000	+	0.866025i	−0.0290619	+	0.0503367i
$297$		0			0
$298$	10.3923	−	6.00000i	0.602010	−	0.347571i
$299$	−15.0000	+	25.9808i	−0.867472	+	1.50251i
$300$		0			0
$301$	−4.00000	+	20.7846i	−0.230556	+	1.19800i
$302$			1.00000i			0.0575435i
$303$		0			0
$304$	−3.50000	−	6.06218i	−0.200739	−	0.347690i
$305$		0			0
$306$	−3.00000	+	5.19615i	−0.171499	+	0.297044i
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$		0			0
$309$	13.0000			0.739544
$310$		0			0
$311$	−9.00000	−	15.5885i	−0.510343	−	0.883940i	−0.999928	−	0.0119847i	$-0.996185\pi$
$311$	−9.00000	−	15.5885i	−0.510343	−	0.883940i	0.489585	−	0.871956i	$-0.337148\pi$
$312$	−4.33013	+	2.50000i	−0.245145	+	0.141535i
$313$	12.1244	+	7.00000i	0.685309	+	0.395663i	0.801852	−	0.597522i	$-0.203848\pi$
$313$	12.1244	+	7.00000i	0.685309	+	0.395663i	−0.116543	+	0.993186i	$0.537181\pi$
$314$	−1.00000			−0.0564333
$315$		0			0
$316$	−7.00000			−0.393781
$317$	15.5885	+	9.00000i	0.875535	+	0.505490i	0.869184	−	0.494489i	$-0.164645\pi$
$317$	15.5885	+	9.00000i	0.875535	+	0.505490i	0.00635137	+	0.999980i	$0.497978\pi$
$318$	−5.19615	+	3.00000i	−0.291386	+	0.168232i
$319$		0			0
$320$		0			0
$321$	−18.0000			−1.00466
$322$	5.19615	+	15.0000i	0.289570	+	0.835917i
$323$		−	42.0000i		−	2.33694i
$324$	0.500000	−	0.866025i	0.0277778	−	0.0481125i
$325$		0			0
$326$	−0.500000	−	0.866025i	−0.0276924	−	0.0479647i
$327$	6.06218	+	3.50000i	0.335239	+	0.193550i
$328$		0			0
$329$	12.0000	+	10.3923i	0.661581	+	0.572946i
$330$		0			0
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	−0.477500	−	0.878632i	$-0.658457\pi$
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	0.999667	−	0.0257885i	$-0.00820965\pi$
$332$	−15.5885	+	9.00000i	−0.855528	+	0.493939i
$333$	0.866025	−	0.500000i	0.0474579	−	0.0273998i
$334$	3.00000	−	5.19615i	0.164153	−	0.284321i
$335$		0			0
$336$	−0.500000	+	2.59808i	−0.0272772	+	0.141737i
$337$		−	34.0000i		−	1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$		−	34.0000i		−	1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$	−10.3923	−	6.00000i	−0.565267	−	0.326357i
$339$	−3.00000	−	5.19615i	−0.162938	−	0.282216i
$340$		0			0
$341$		0			0
$342$			7.00000i			0.378517i
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$	8.00000			0.431331
$345$		0			0
$346$	9.00000	+	15.5885i	0.483843	+	0.838041i
$347$	15.5885	−	9.00000i	0.836832	−	0.483145i	−0.0193540	−	0.999813i	$-0.506161\pi$
$347$	15.5885	−	9.00000i	0.836832	−	0.483145i	0.856186	+	0.516667i	$0.172828\pi$
$348$		0			0
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$		0			0
$351$	5.00000			0.266880
$352$		0			0
$353$	20.7846	−	12.0000i	1.10625	−	0.638696i	0.168397	−	0.985719i	$-0.446141\pi$
$353$	20.7846	−	12.0000i	1.10625	−	0.638696i	0.937856	+	0.347024i	$0.112808\pi$
$354$	3.00000	+	5.19615i	0.159448	+	0.276172i
$355$		0			0
$356$	−6.00000			−0.317999
$357$	−10.3923	+	12.0000i	−0.550019	+	0.635107i
$358$			12.0000i			0.634220i
$359$	15.0000	−	25.9808i	0.791670	−	1.37121i	−0.133263	−	0.991081i	$-0.542545\pi$
$359$	15.0000	−	25.9808i	0.791670	−	1.37121i	0.924932	−	0.380131i	$-0.124121\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$	−8.66025	−	5.00000i	−0.455173	−	0.262794i
$363$			11.0000i			0.577350i
$364$	−12.5000	+	4.33013i	−0.655178	+	0.226960i
$365$		0			0
$366$	−0.500000	+	0.866025i	−0.0261354	+	0.0452679i
$367$	6.92820	−	4.00000i	0.361649	−	0.208798i	−0.308155	−	0.951336i	$-0.599711\pi$
$367$	6.92820	−	4.00000i	0.361649	−	0.208798i	0.669804	+	0.742538i	$0.266378\pi$
$368$	5.19615	−	3.00000i	0.270868	−	0.156386i
$369$		0			0
$370$		0			0
$371$	−15.0000	+	5.19615i	−0.778761	+	0.269771i
$372$			8.00000i			0.414781i
$373$	−11.2583	−	6.50000i	−0.582934	−	0.336557i	0.179364	−	0.983783i	$-0.442596\pi$
$373$	−11.2583	−	6.50000i	−0.582934	−	0.336557i	−0.762299	+	0.647225i	$0.775929\pi$
$374$		0			0
$375$		0			0
$376$	3.00000	−	5.19615i	0.154713	−	0.267971i
$377$		0			0
$378$	1.73205	−	2.00000i	0.0890871	−	0.102869i
$379$	−11.0000			−0.565032			−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000			−0.565032			−0.282516	+	0.959263i	$0.591169\pi$
$380$		0			0
$381$	−5.50000	−	9.52628i	−0.281774	−	0.488046i
$382$	−20.7846	+	12.0000i	−1.06343	+	0.613973i
$383$	−5.19615	−	3.00000i	−0.265511	−	0.153293i	0.361335	−	0.932436i	$-0.382321\pi$
$383$	−5.19615	−	3.00000i	−0.265511	−	0.153293i	−0.626846	+	0.779143i	$0.715654\pi$
$384$	1.00000			0.0510310
$385$		0			0
$386$	10.0000			0.508987
$387$	−6.92820	−	4.00000i	−0.352180	−	0.203331i
$388$	6.06218	−	3.50000i	0.307760	−	0.177686i
$389$	−6.00000	−	10.3923i	−0.304212	−	0.526911i	0.672874	−	0.739758i	$-0.265060\pi$
$389$	−6.00000	−	10.3923i	−0.304212	−	0.526911i	−0.977086	+	0.212847i	$0.931726\pi$
$390$		0			0
$391$	36.0000			1.82060
$392$	−2.59808	+	6.50000i	−0.131223	+	0.328300i
$393$		−	6.00000i		−	0.302660i
$394$	6.00000	−	10.3923i	0.302276	−	0.523557i
$395$		0			0
$396$		0			0
$397$	−12.1244	−	7.00000i	−0.608504	−	0.351320i	0.163876	−	0.986481i	$-0.447600\pi$
$397$	−12.1244	−	7.00000i	−0.608504	−	0.351320i	−0.772380	+	0.635161i	$0.780934\pi$
$398$			11.0000i			0.551380i
$399$	−3.50000	+	18.1865i	−0.175219	+	0.910465i
$400$		0			0
$401$	18.0000	−	31.1769i	0.898877	−	1.55690i	0.0699455	−	0.997551i	$-0.477717\pi$
$401$	18.0000	−	31.1769i	0.898877	−	1.55690i	0.828932	−	0.559350i	$-0.188949\pi$
$402$	11.2583	−	6.50000i	0.561514	−	0.324191i
$403$	−34.6410	+	20.0000i	−1.72559	+	0.996271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	5.19615	+	3.00000i	0.257248	+	0.148522i
$409$	2.50000	+	4.33013i	0.123617	+	0.214111i	0.921192	−	0.389109i	$-0.127217\pi$
$409$	2.50000	+	4.33013i	0.123617	+	0.214111i	−0.797574	+	0.603220i	$0.793884\pi$
$410$		0			0
$411$	9.00000	−	15.5885i	0.443937	−	0.768922i
$412$		−	13.0000i		−	0.640464i
$413$	5.19615	+	15.0000i	0.255686	+	0.738102i
$414$	−6.00000			−0.294884
$415$		0			0
$416$	2.50000	+	4.33013i	0.122573	+	0.212302i
$417$	9.52628	−	5.50000i	0.466504	−	0.269336i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	5.00000			0.243685			0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000			0.243685			0.121843	+	0.992549i	$0.461120\pi$
$422$	−11.2583	−	6.50000i	−0.548047	−	0.316415i
$423$	−5.19615	+	3.00000i	−0.252646	+	0.145865i
$424$	3.00000	+	5.19615i	0.145693	+	0.252347i
$425$		0			0
$426$	−12.0000			−0.581402
$427$	−1.73205	+	2.00000i	−0.0838198	+	0.0967868i
$428$			18.0000i			0.870063i
$429$		0			0
$430$		0			0
$431$	9.00000	+	15.5885i	0.433515	+	0.750870i	0.997173	−	0.0751385i	$-0.0239399\pi$
$431$	9.00000	+	15.5885i	0.433515	+	0.750870i	−0.563658	+	0.826008i	$0.690607\pi$
$432$	−0.866025	−	0.500000i	−0.0416667	−	0.0240563i
$433$			34.0000i			1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$			34.0000i			1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−4.00000	+	20.7846i	−0.192006	+	0.997693i
$435$		0			0
$436$	3.50000	−	6.06218i	0.167620	−	0.290326i
$437$	36.3731	−	21.0000i	1.73996	−	1.00457i
$438$	4.33013	−	2.50000i	0.206901	−	0.119455i
$439$	2.50000	−	4.33013i	0.119318	−	0.206666i	−0.800179	−	0.599761i	$-0.795262\pi$
$439$	2.50000	−	4.33013i	0.119318	−	0.206666i	0.919498	+	0.393095i	$0.128596\pi$
$440$		0			0
$441$	5.50000	−	4.33013i	0.261905	−	0.206197i
$442$			30.0000i			1.42695i
$443$	−36.3731	−	21.0000i	−1.72814	−	0.997740i	−0.897664	−	0.440681i	$-0.854737\pi$
$443$	−36.3731	−	21.0000i	−1.72814	−	0.997740i	−0.830473	−	0.557059i	$-0.811930\pi$
$444$	−0.500000	−	0.866025i	−0.0237289	−	0.0410997i
$445$		0			0
$446$	8.50000	−	14.7224i	0.402487	−	0.697127i
$447$			12.0000i			0.567581i
$448$	2.59808	+	0.500000i	0.122748	+	0.0236228i
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$		0			0
$452$	−5.19615	+	3.00000i	−0.244406	+	0.141108i
$453$	−0.866025	−	0.500000i	−0.0406894	−	0.0234920i
$454$	12.0000			0.563188
$455$		0			0
$456$	7.00000			0.327805
$457$	−25.1147	−	14.5000i	−1.17482	−	0.678281i	−0.220008	−	0.975498i	$-0.570608\pi$
$457$	−25.1147	−	14.5000i	−1.17482	−	0.678281i	−0.954810	+	0.297217i	$0.903942\pi$
$458$	4.33013	−	2.50000i	0.202334	−	0.116817i
$459$	−3.00000	−	5.19615i	−0.140028	−	0.242536i
$460$		0			0
$461$	6.00000			0.279448			0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000			0.279448			0.139724	+	0.990190i	$0.455378\pi$
$462$		0			0
$463$			31.0000i			1.44069i	0.693615	+	0.720346i	$0.256017\pi$
$463$			31.0000i			1.44069i	−0.693615	+	0.720346i	$0.743983\pi$
$464$		0			0
$465$		0			0
$466$	12.0000	+	20.7846i	0.555889	+	0.962828i
$467$	−25.9808	−	15.0000i	−1.20225	−	0.694117i	−0.241192	−	0.970477i	$-0.577538\pi$
$467$	−25.9808	−	15.0000i	−1.20225	−	0.694117i	−0.961054	+	0.276360i	$0.910872\pi$
$468$		−	5.00000i		−	0.231125i
$469$	32.5000	−	11.2583i	1.50071	−	0.519861i
$470$		0			0
$471$	0.500000	−	0.866025i	0.0230388	−	0.0399043i
$472$	5.19615	−	3.00000i	0.239172	−	0.138086i
$473$		0			0
$474$	3.50000	−	6.06218i	0.160760	−	0.278445i
$475$		0			0
$476$	12.0000	+	10.3923i	0.550019	+	0.476331i
$477$		−	6.00000i		−	0.274721i
$478$	10.3923	+	6.00000i	0.475333	+	0.274434i
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	0.995023	−	0.0996406i	$-0.0317693\pi$
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	−0.583803	+	0.811895i	$0.698436\pi$
$480$		0			0
$481$	2.50000	−	4.33013i	0.113990	−	0.197437i
$482$		−	5.00000i		−	0.227744i
$483$	−15.5885	−	3.00000i	−0.709299	−	0.136505i
$484$	11.0000			0.500000
$485$		0			0
$486$	0.500000	+	0.866025i	0.0226805	+	0.0392837i
$487$	−34.6410	+	20.0000i	−1.56973	+	0.906287i	−0.573535	+	0.819181i	$0.694428\pi$
$487$	−34.6410	+	20.0000i	−1.56973	+	0.906287i	−0.996199	+	0.0871056i	$0.972238\pi$
$488$	0.866025	+	0.500000i	0.0392031	+	0.0226339i
$489$	1.00000			0.0452216
$490$		0			0
$491$	−6.00000			−0.270776			−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000			−0.270776			−0.135388	+	0.990793i	$0.543228\pi$
$492$		0			0
$493$		0			0
$494$	17.5000	+	30.3109i	0.787362	+	1.36375i
$495$		0			0
$496$	8.00000			0.359211
$497$	−31.1769	−	6.00000i	−1.39848	−	0.269137i
$498$		−	18.0000i		−	0.806599i
$499$	2.50000	−	4.33013i	0.111915	−	0.193843i	−0.804627	−	0.593780i	$-0.797635\pi$
$499$	2.50000	−	4.33013i	0.111915	−	0.193843i	0.916542	+	0.399937i	$0.130968\pi$
$500$		0			0
$501$	3.00000	+	5.19615i	0.134030	+	0.232147i
$502$	−20.7846	−	12.0000i	−0.927663	−	0.535586i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	−2.00000	−	1.73205i	−0.0890871	−	0.0771517i
$505$		0			0
$506$		0			0
$507$	10.3923	−	6.00000i	0.461538	−	0.266469i
$508$	−9.52628	+	5.50000i	−0.422660	+	0.244023i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	12.5000	−	4.33013i	0.552967	−	0.191554i
$512$		−	1.00000i		−	0.0441942i
$513$	−6.06218	−	3.50000i	−0.267652	−	0.154529i
$514$	12.0000	+	20.7846i	0.529297	+	0.916770i
$515$		0			0
$516$	−4.00000	+	6.92820i	−0.176090	+	0.304997i
$517$		0			0
$518$	−0.866025	−	2.50000i	−0.0380510	−	0.109844i
$519$	−18.0000			−0.790112
$520$		0			0
$521$	−21.0000	−	36.3731i	−0.920027	−	1.59353i	−0.799370	−	0.600839i	$-0.794833\pi$
$521$	−21.0000	−	36.3731i	−0.920027	−	1.59353i	−0.120656	−	0.992694i	$-0.538500\pi$
$522$		0			0
$523$	27.7128	+	16.0000i	1.21180	+	0.699631i	0.963150	−	0.268963i	$-0.0866810\pi$
$523$	27.7128	+	16.0000i	1.21180	+	0.699631i	0.248646	+	0.968594i	$0.420014\pi$
$524$	−6.00000			−0.262111
$525$		0			0
$526$		0			0
$527$	41.5692	+	24.0000i	1.81078	+	1.04546i
$528$		0			0
$529$	6.50000	+	11.2583i	0.282609	+	0.489493i
$530$		0			0
$531$	−6.00000			−0.260378
$532$	18.1865	+	3.50000i	0.788486	+	0.151744i
$533$		0			0
$534$	3.00000	−	5.19615i	0.129823	−	0.224860i
$535$		0			0
$536$	−6.50000	−	11.2583i	−0.280757	−	0.486286i
$537$	−10.3923	−	6.00000i	−0.448461	−	0.258919i
$538$		−	24.0000i		−	1.03471i
$539$		0			0
$540$		0			0
$541$	21.5000	−	37.2391i	0.924357	−	1.60103i	0.131765	−	0.991281i	$-0.457935\pi$
$541$	21.5000	−	37.2391i	0.924357	−	1.60103i	0.792592	−	0.609753i	$-0.208731\pi$
$542$	−17.3205	+	10.0000i	−0.743980	+	0.429537i
$543$	8.66025	−	5.00000i	0.371647	−	0.214571i
$544$	3.00000	−	5.19615i	0.128624	−	0.222783i
$545$		0			0
$546$	2.50000	−	12.9904i	0.106990	−	0.555937i
$547$		−	28.0000i		−	1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$		−	28.0000i		−	1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	−15.5885	−	9.00000i	−0.665906	−	0.384461i
$549$	−0.500000	−	0.866025i	−0.0213395	−	0.0369611i
$550$		0			0
$551$		0			0
$552$			6.00000i			0.255377i
$553$	12.1244	−	14.0000i	0.515580	−	0.595341i
$554$	−19.0000			−0.807233
$555$		0			0
$556$	−5.50000	−	9.52628i	−0.233252	−	0.404004i
$557$	−10.3923	+	6.00000i	−0.440336	+	0.254228i	−0.703740	−	0.710457i	$-0.748488\pi$
$557$	−10.3923	+	6.00000i	−0.440336	+	0.254228i	0.263404	+	0.964686i	$0.415155\pi$
$558$	−6.92820	−	4.00000i	−0.293294	−	0.169334i
$559$	−40.0000			−1.69182
$560$		0			0
$561$		0			0
$562$	−15.5885	−	9.00000i	−0.657559	−	0.379642i
$563$	10.3923	−	6.00000i	0.437983	−	0.252870i	−0.264758	−	0.964315i	$-0.585292\pi$
$563$	10.3923	−	6.00000i	0.437983	−	0.252870i	0.702742	+	0.711445i	$0.251959\pi$
$564$	3.00000	+	5.19615i	0.126323	+	0.218797i
$565$		0			0
$566$	−5.00000			−0.210166
$567$	0.866025	+	2.50000i	0.0363696	+	0.104990i
$568$			12.0000i			0.503509i
$569$	−18.0000	+	31.1769i	−0.754599	+	1.30700i	0.190974	+	0.981595i	$0.438835\pi$
$569$	−18.0000	+	31.1769i	−0.754599	+	1.30700i	−0.945573	+	0.325409i	$0.894498\pi$
$570$		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	$0.866697\pi$
$572$		0			0
$573$		−	24.0000i		−	1.00261i
$574$		0			0
$575$		0			0
$576$	−0.500000	+	0.866025i	−0.0208333	+	0.0360844i
$577$	12.1244	−	7.00000i	0.504744	−	0.291414i	−0.225927	−	0.974144i	$-0.572541\pi$
$577$	12.1244	−	7.00000i	0.504744	−	0.291414i	0.730670	+	0.682730i	$0.239208\pi$
$578$	16.4545	−	9.50000i	0.684416	−	0.395148i
$579$	−5.00000	+	8.66025i	−0.207793	+	0.359908i
$580$		0			0
$581$	9.00000	−	46.7654i	0.373383	−	1.94015i
$582$			7.00000i			0.290159i
$583$		0			0
$584$	−2.50000	−	4.33013i	−0.103451	−	0.179182i
$585$		0			0
$586$	−12.0000	+	20.7846i	−0.495715	+	0.858604i
$587$		−	30.0000i		−	1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$		−	30.0000i		−	1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$	−4.33013	−	5.50000i	−0.178571	−	0.226816i
$589$	56.0000			2.30744
$590$		0			0
$591$	6.00000	+	10.3923i	0.246807	+	0.427482i
$592$	−0.866025	+	0.500000i	−0.0355934	+	0.0205499i
$593$	−5.19615	−	3.00000i	−0.213380	−	0.123195i	0.389501	−	0.921026i	$-0.372647\pi$
$593$	−5.19615	−	3.00000i	−0.213380	−	0.123195i	−0.602881	+	0.797831i	$0.705981\pi$
$594$		0			0
$595$		0			0
$596$	12.0000			0.491539
$597$	−9.52628	−	5.50000i	−0.389885	−	0.225100i
$598$	−25.9808	+	15.0000i	−1.06243	+	0.613396i
$599$	−15.0000	−	25.9808i	−0.612883	−	1.06155i	−0.990752	−	0.135686i	$-0.956676\pi$
$599$	−15.0000	−	25.9808i	−0.612883	−	1.06155i	0.377869	−	0.925859i	$-0.376657\pi$
$600$		0			0
$601$	35.0000			1.42768			0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000			1.42768			0.713840	+	0.700309i	$0.246954\pi$
$602$	−13.8564	+	16.0000i	−0.564745	+	0.652111i
$603$			13.0000i			0.529401i
$604$	−0.500000	+	0.866025i	−0.0203447	+	0.0352381i
$605$		0			0
$606$		0			0
$607$	37.2391	+	21.5000i	1.51149	+	0.872658i	0.999910	+	0.0134214i	$0.00427228\pi$
$607$	37.2391	+	21.5000i	1.51149	+	0.872658i	0.511578	+	0.859237i	$0.329061\pi$
$608$		−	7.00000i		−	0.283887i
$609$		0			0
$610$		0			0
$611$	−15.0000	+	25.9808i	−0.606835	+	1.05107i
$612$	−5.19615	+	3.00000i	−0.210042	+	0.121268i
$613$	−32.9090	+	19.0000i	−1.32918	+	0.767403i	−0.985173	−	0.171564i	$-0.945118\pi$
$613$	−32.9090	+	19.0000i	−1.32918	+	0.767403i	−0.344008	+	0.938967i	$0.611785\pi$
$614$	14.0000	−	24.2487i	0.564994	−	0.978598i
$615$		0			0
$616$		0			0
$617$			48.0000i			1.93241i	0.257780	+	0.966204i	$0.417009\pi$
$617$			48.0000i			1.93241i	−0.257780	+	0.966204i	$0.582991\pi$
$618$	11.2583	+	6.50000i	0.452876	+	0.261468i
$619$	10.0000	+	17.3205i	0.401934	+	0.696170i	0.993959	−	0.109749i	$-0.0350048\pi$
$619$	10.0000	+	17.3205i	0.401934	+	0.696170i	−0.592025	+	0.805919i	$0.701671\pi$
$620$		0			0
$621$	3.00000	−	5.19615i	0.120386	−	0.208514i
$622$		−	18.0000i		−	0.721734i
$623$	10.3923	−	12.0000i	0.416359	−	0.480770i
$624$	−5.00000			−0.200160
$625$		0			0
$626$	7.00000	+	12.1244i	0.279776	+	0.484587i
$627$		0			0
$628$	−0.866025	−	0.500000i	−0.0345582	−	0.0199522i
$629$	−6.00000			−0.239236
$630$		0			0
$631$	−19.0000			−0.756378			−0.378189	−	0.925728i	$-0.623453\pi$
$631$	−19.0000			−0.756378			−0.378189	+	0.925728i	$0.623453\pi$
$632$	−6.06218	−	3.50000i	−0.241140	−	0.139223i
$633$	11.2583	−	6.50000i	0.447478	−	0.258352i
$634$	9.00000	+	15.5885i	0.357436	+	0.619097i
$635$		0			0
$636$	−6.00000			−0.237915
$637$	12.9904	−	32.5000i	0.514698	−	1.28770i
$638$		0			0
$639$	6.00000	−	10.3923i	0.237356	−	0.411113i
$640$		0			0
$641$	12.0000	+	20.7846i	0.473972	+	0.820943i	0.999556	−	0.0297987i	$-0.00948663\pi$
$641$	12.0000	+	20.7846i	0.473972	+	0.820943i	−0.525584	+	0.850741i	$0.676153\pi$
$642$	−15.5885	−	9.00000i	−0.615227	−	0.355202i
$643$		−	11.0000i		−	0.433798i	−0.976194	−	0.216899i	$-0.930406\pi$
$643$		−	11.0000i		−	0.433798i	0.976194	−	0.216899i	$-0.0695942\pi$
$644$	−3.00000	+	15.5885i	−0.118217	+	0.614271i
$645$		0			0
$646$	21.0000	−	36.3731i	0.826234	−	1.43108i
$647$	−10.3923	+	6.00000i	−0.408564	+	0.235884i	−0.690172	−	0.723645i	$-0.742465\pi$
$647$	−10.3923	+	6.00000i	−0.408564	+	0.235884i	0.281609	+	0.959529i	$0.409132\pi$
$648$	0.866025	−	0.500000i	0.0340207	−	0.0196419i
$649$		0			0
$650$		0			0
$651$	−16.0000	−	13.8564i	−0.627089	−	0.543075i
$652$		−	1.00000i		−	0.0391630i
$653$	−31.1769	−	18.0000i	−1.22005	−	0.704394i	−0.255119	−	0.966910i	$-0.582115\pi$
$653$	−31.1769	−	18.0000i	−1.22005	−	0.704394i	−0.964928	+	0.262515i	$0.915448\pi$
$654$	3.50000	+	6.06218i	0.136861	+	0.237050i
$655$		0			0
$656$		0			0
$657$			5.00000i			0.195069i
$658$	5.19615	+	15.0000i	0.202567	+	0.584761i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−17.5000	−	30.3109i	−0.680671	−	1.17896i	−0.974776	−	0.223184i	$-0.928355\pi$
$661$	−17.5000	−	30.3109i	−0.680671	−	1.17896i	0.294105	−	0.955773i	$-0.404978\pi$
$662$	16.4545	−	9.50000i	0.639522	−	0.369228i
$663$	−25.9808	−	15.0000i	−1.00901	−	0.582552i
$664$	−18.0000			−0.698535
$665$		0			0
$666$	1.00000			0.0387492
$667$		0			0
$668$	5.19615	−	3.00000i	0.201045	−	0.116073i
$669$	8.50000	+	14.7224i	0.328629	+	0.569202i
$670$		0			0
$671$		0			0
$672$	−1.73205	+	2.00000i	−0.0668153	+	0.0771517i
$673$			1.00000i			0.0385472i	0.999814	+	0.0192736i	$0.00613535\pi$
$673$			1.00000i			0.0385472i	−0.999814	+	0.0192736i	$0.993865\pi$
$674$	17.0000	−	29.4449i	0.654816	−	1.13417i
$675$		0			0
$676$	−6.00000	−	10.3923i	−0.230769	−	0.399704i
$677$	−36.3731	−	21.0000i	−1.39793	−	0.807096i	−0.403755	−	0.914867i	$-0.632295\pi$
$677$	−36.3731	−	21.0000i	−1.39793	−	0.807096i	−0.994176	+	0.107772i	$0.965628\pi$
$678$		−	6.00000i		−	0.230429i
$679$	−3.50000	+	18.1865i	−0.134318	+	0.697935i
$680$		0			0
$681$	−6.00000	+	10.3923i	−0.229920	+	0.398234i
$682$		0			0
$683$	−20.7846	+	12.0000i	−0.795301	+	0.459167i	−0.841825	−	0.539750i	$-0.818519\pi$
$683$	−20.7846	+	12.0000i	−0.795301	+	0.459167i	0.0465244	+	0.998917i	$0.485185\pi$
$684$	−3.50000	+	6.06218i	−0.133826	+	0.231793i
$685$		0			0
$686$	−8.50000	−	16.4545i	−0.324532	−	0.628235i
$687$			5.00000i			0.190762i
$688$	6.92820	+	4.00000i	0.264135	+	0.152499i
$689$	−15.0000	−	25.9808i	−0.571454	−	0.989788i
$690$		0			0
$691$	3.50000	−	6.06218i	0.133146	−	0.230616i	−0.791742	−	0.610856i	$-0.790825\pi$
$691$	3.50000	−	6.06218i	0.133146	−	0.230616i	0.924888	+	0.380240i	$0.124159\pi$
$692$			18.0000i			0.684257i
$693$		0			0
$694$	18.0000			0.683271
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−22.5167	−	13.0000i	−0.852268	−	0.492057i
$699$	−24.0000			−0.907763
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$	4.33013	+	2.50000i	0.163430	+	0.0943564i
$703$	−6.06218	+	3.50000i	−0.228639	+	0.132005i
$704$		0			0
$705$		0			0
$706$	24.0000			0.903252
$707$		0			0
$708$			6.00000i			0.225494i
$709$	−3.50000	+	6.06218i	−0.131445	+	0.227670i	−0.924234	−	0.381827i	$-0.875295\pi$
$709$	−3.50000	+	6.06218i	−0.131445	+	0.227670i	0.792789	+	0.609497i	$0.208628\pi$
$710$		0			0
$711$	3.50000	+	6.06218i	0.131260	+	0.227349i
$712$	−5.19615	−	3.00000i	−0.194734	−	0.112430i
$713$			48.0000i			1.79761i
$714$	−15.0000	+	5.19615i	−0.561361	+	0.194461i
$715$		0			0
$716$	−6.00000	+	10.3923i	−0.224231	+	0.388379i
$717$	−10.3923	+	6.00000i	−0.388108	+	0.224074i
$718$	25.9808	−	15.0000i	0.969593	−	0.559795i
$719$	−18.0000	+	31.1769i	−0.671287	+	1.16270i	0.306253	+	0.951950i	$0.400925\pi$
$719$	−18.0000	+	31.1769i	−0.671287	+	1.16270i	−0.977539	+	0.210752i	$0.932409\pi$
$720$		0			0
$721$	26.0000	+	22.5167i	0.968291	+	0.838564i
$722$		−	30.0000i		−	1.11648i
$723$	4.33013	+	2.50000i	0.161039	+	0.0929760i
$724$	−5.00000	−	8.66025i	−0.185824	−	0.321856i
$725$		0			0
$726$	−5.50000	+	9.52628i	−0.204124	+	0.353553i
$727$		−	37.0000i		−	1.37225i	−0.727482	−	0.686127i	$-0.759309\pi$
$727$		−	37.0000i		−	1.37225i	0.727482	−	0.686127i	$-0.240691\pi$
$728$	−12.9904	−	2.50000i	−0.481456	−	0.0926562i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	24.0000	+	41.5692i	0.887672	+	1.53749i
$732$	−0.866025	+	0.500000i	−0.0320092	+	0.0184805i
$733$	35.5070	+	20.5000i	1.31148	+	0.757185i	0.982342	−	0.187096i	$-0.0599076\pi$
$733$	35.5070	+	20.5000i	1.31148	+	0.757185i	0.329141	+	0.944281i	$0.393241\pi$
$734$	8.00000			0.295285
$735$		0			0
$736$	6.00000			0.221163
$737$		0			0
$738$		0			0
$739$	−12.5000	−	21.6506i	−0.459820	−	0.796431i	0.539131	−	0.842222i	$-0.318753\pi$
$739$	−12.5000	−	21.6506i	−0.459820	−	0.796431i	−0.998951	+	0.0457903i	$0.985419\pi$
$740$		0			0
$741$	−35.0000			−1.28576
$742$	−15.5885	−	3.00000i	−0.572270	−	0.110133i
$743$		−	30.0000i		−	1.10059i	−0.834969	−	0.550297i	$-0.814515\pi$
$743$		−	30.0000i		−	1.10059i	0.834969	−	0.550297i	$-0.185485\pi$
$744$	−4.00000	+	6.92820i	−0.146647	+	0.254000i
$745$		0			0
$746$	−6.50000	−	11.2583i	−0.237982	−	0.412197i
$747$	15.5885	+	9.00000i	0.570352	+	0.329293i
$748$		0			0
$749$	−36.0000	−	31.1769i	−1.31541	−	1.13918i
$750$		0			0
$751$	12.5000	−	21.6506i	0.456131	−	0.790043i	−0.542621	−	0.839978i	$-0.682568\pi$
$751$	12.5000	−	21.6506i	0.456131	−	0.790043i	0.998752	+	0.0499348i	$0.0159013\pi$
$752$	5.19615	−	3.00000i	0.189484	−	0.109399i
$753$	20.7846	−	12.0000i	0.757433	−	0.437304i
$754$		0			0
$755$		0			0
$756$	2.50000	−	0.866025i	0.0909241	−	0.0314970i
$757$			29.0000i			1.05402i	0.849858	+	0.527011i	$0.176688\pi$
$757$			29.0000i			1.05402i	−0.849858	+	0.527011i	$0.823312\pi$
$758$	−9.52628	−	5.50000i	−0.346010	−	0.199769i
$759$		0			0
$760$		0			0
$761$	−9.00000	+	15.5885i	−0.326250	+	0.565081i	−0.981764	−	0.190101i	$-0.939118\pi$
$761$	−9.00000	+	15.5885i	−0.326250	+	0.565081i	0.655515	+	0.755182i	$0.272452\pi$
$762$		−	11.0000i		−	0.398488i
$763$	6.06218	+	17.5000i	0.219466	+	0.633543i
$764$	−24.0000			−0.868290
$765$		0			0
$766$	−3.00000	−	5.19615i	−0.108394	−	0.187745i
$767$	−25.9808	+	15.0000i	−0.938111	+	0.541619i
$768$	0.866025	+	0.500000i	0.0312500	+	0.0180422i
$769$	10.0000			0.360609			0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000			0.360609			0.180305	+	0.983611i	$0.442292\pi$
$770$		0			0
$771$	−24.0000			−0.864339
$772$	8.66025	+	5.00000i	0.311689	+	0.179954i
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$	−4.00000	−	6.92820i	−0.143777	−	0.249029i
$775$		0			0
$776$	7.00000			0.251285
$777$	2.59808	+	0.500000i	0.0932055	+	0.0179374i
$778$		−	12.0000i		−	0.430221i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	31.1769	+	18.0000i	1.11488	+	0.643679i
$783$		0			0
$784$	−5.50000	+	4.33013i	−0.196429	+	0.154647i
$785$		0			0
$786$	3.00000	−	5.19615i	0.107006	−	0.185341i
$787$	19.9186	−	11.5000i	0.710021	−	0.409931i	−0.101048	−	0.994882i	$-0.532220\pi$
$787$	19.9186	−	11.5000i	0.710021	−	0.409931i	0.811069	+	0.584951i	$0.198886\pi$
$788$	10.3923	−	6.00000i	0.370211	−	0.213741i
$789$		0			0
$790$		0			0
$791$	3.00000	−	15.5885i	0.106668	−	0.554262i
$792$		0			0
$793$	−4.33013	−	2.50000i	−0.153767	−	0.0887776i
$794$	−7.00000	−	12.1244i	−0.248421	−	0.430277i
$795$		0			0
$796$	−5.50000	+	9.52628i	−0.194942	+	0.337650i
$797$			54.0000i			1.91278i	0.292096	+	0.956389i	$0.405647\pi$
$797$			54.0000i			1.91278i	−0.292096	+	0.956389i	$0.594353\pi$
$798$	−12.1244	+	14.0000i	−0.429198	+	0.495595i
$799$	36.0000			1.27359
$800$		0			0
$801$	3.00000	+	5.19615i	0.106000	+	0.183597i
$802$	31.1769	−	18.0000i	1.10090	−	0.635602i
$803$		0			0
$804$	13.0000			0.458475
$805$		0			0
$806$	−40.0000			−1.40894
$807$	20.7846	+	12.0000i	0.731653	+	0.422420i
$808$		0			0
$809$	−15.0000	−	25.9808i	−0.527372	−	0.913435i	−0.999491	−	0.0319002i	$-0.989844\pi$
$809$	−15.0000	−	25.9808i	−0.527372	−	0.913435i	0.472119	−	0.881535i	$-0.343489\pi$
$810$		0			0
$811$	11.0000			0.386262			0.193131	−	0.981173i	$-0.438136\pi$
$811$	11.0000			0.386262			0.193131	+	0.981173i	$0.438136\pi$
$812$		0			0
$813$		−	20.0000i		−	0.701431i
$814$		0			0
$815$		0			0
$816$	3.00000	+	5.19615i	0.105021	+	0.181902i
$817$	48.4974	+	28.0000i	1.69671	+	0.979596i
$818$			5.00000i			0.174821i
$819$	10.0000	+	8.66025i	0.349428	+	0.302614i
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$	15.5885	−	9.00000i	0.543710	−	0.313911i
$823$	−4.33013	+	2.50000i	−0.150939	+	0.0871445i	−0.573567	−	0.819159i	$-0.694441\pi$
$823$	−4.33013	+	2.50000i	−0.150939	+	0.0871445i	0.422628	+	0.906303i	$0.361108\pi$
$824$	6.50000	−	11.2583i	0.226438	−	0.392203i
$825$		0			0
$826$	−3.00000	+	15.5885i	−0.104383	+	0.542392i
$827$		−	24.0000i		−	0.834562i	−0.908778	−	0.417281i	$-0.862983\pi$
$827$		−	24.0000i		−	0.834562i	0.908778	−	0.417281i	$-0.137017\pi$
$828$	−5.19615	−	3.00000i	−0.180579	−	0.104257i
$829$	−3.50000	−	6.06218i	−0.121560	−	0.210548i	0.798823	−	0.601566i	$-0.205456\pi$
$829$	−3.50000	−	6.06218i	−0.121560	−	0.210548i	−0.920383	+	0.391018i	$0.872123\pi$
$830$		0			0
$831$	9.50000	−	16.4545i	0.329551	−	0.570800i
$832$			5.00000i			0.173344i
$833$	−41.5692	+	6.00000i	−1.44029	+	0.207888i
$834$	11.0000			0.380899
$835$		0			0
$836$		0			0
$837$	6.92820	−	4.00000i	0.239474	−	0.138260i
$838$		0			0
$839$	18.0000			0.621429			0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000			0.621429			0.310715	+	0.950503i	$0.399432\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	4.33013	+	2.50000i	0.149226	+	0.0861557i
$843$	15.5885	−	9.00000i	0.536895	−	0.309976i
$844$	−6.50000	−	11.2583i	−0.223739	−	0.387528i
$845$		0			0
$846$	−6.00000			−0.206284
$847$	−19.0526	+	22.0000i	−0.654654	+	0.755929i
$848$			6.00000i			0.206041i
$849$	2.50000	−	4.33013i	0.0857998	−	0.148610i
$850$		0			0
$851$	−3.00000	−	5.19615i	−0.102839	−	0.178122i
$852$	−10.3923	−	6.00000i	−0.356034	−	0.205557i
$853$		−	26.0000i		−	0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$		−	26.0000i		−	0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	−2.50000	+	0.866025i	−0.0855482	+	0.0296348i
$855$		0			0
$856$	−9.00000	+	15.5885i	−0.307614	+	0.532803i
$857$	15.5885	−	9.00000i	0.532492	−	0.307434i	−0.209539	−	0.977800i	$-0.567196\pi$
$857$	15.5885	−	9.00000i	0.532492	−	0.307434i	0.742030	+	0.670366i	$0.233863\pi$
$858$		0			0
$859$	−2.00000	+	3.46410i	−0.0682391	+	0.118194i	−0.898126	−	0.439738i	$-0.855071\pi$
$859$	−2.00000	+	3.46410i	−0.0682391	+	0.118194i	0.829887	+	0.557931i	$0.188405\pi$
$860$		0			0
$861$		0			0
$862$			18.0000i			0.613082i
$863$	25.9808	+	15.0000i	0.884395	+	0.510606i	0.872105	−	0.489319i	$-0.162754\pi$
$863$	25.9808	+	15.0000i	0.884395	+	0.510606i	0.0122903	+	0.999924i	$0.496088\pi$
$864$	−0.500000	−	0.866025i	−0.0170103	−	0.0294628i
$865$		0			0
$866$	−17.0000	+	29.4449i	−0.577684	+	1.00058i
$867$			19.0000i			0.645274i
$868$	−13.8564	+	16.0000i	−0.470317	+	0.543075i
$869$		0			0
$870$		0			0
$871$	32.5000	+	56.2917i	1.10122	+	1.90737i
$872$	6.06218	−	3.50000i	0.205291	−	0.118525i
$873$	−6.06218	−	3.50000i	−0.205174	−	0.118457i
$874$	42.0000			1.42067
$875$		0			0
$876$	5.00000			0.168934
$877$	−40.7032	−	23.5000i	−1.37445	−	0.793539i	−0.382965	−	0.923763i	$-0.625097\pi$
$877$	−40.7032	−	23.5000i	−1.37445	−	0.793539i	−0.991485	+	0.130224i	$0.958430\pi$
$878$	4.33013	−	2.50000i	0.146135	−	0.0843709i
$879$	−12.0000	−	20.7846i	−0.404750	−	0.701047i
$880$		0			0
$881$	−12.0000			−0.404290			−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000			−0.404290			−0.202145	+	0.979356i	$0.564791\pi$
$882$	6.92820	−	1.00000i	0.233285	−	0.0336718i
$883$			25.0000i			0.841317i	0.907219	+	0.420658i	$0.138201\pi$
$883$			25.0000i			0.841317i	−0.907219	+	0.420658i	$0.861799\pi$
$884$	−15.0000	+	25.9808i	−0.504505	+	0.873828i
$885$		0			0
$886$	−21.0000	−	36.3731i	−0.705509	−	1.22198i
$887$	5.19615	+	3.00000i	0.174470	+	0.100730i	0.584692	−	0.811256i	$-0.301215\pi$
$887$	5.19615	+	3.00000i	0.174470	+	0.100730i	−0.410222	+	0.911986i	$0.634549\pi$
$888$		−	1.00000i		−	0.0335578i
$889$	5.50000	−	28.5788i	0.184464	−	0.958503i
$890$		0			0
$891$		0			0
$892$	14.7224	−	8.50000i	0.492943	−	0.284601i
$893$	36.3731	−	21.0000i	1.21718	−	0.702738i
$894$	−6.00000	+	10.3923i	−0.200670	+	0.347571i
$895$		0			0
$896$	2.00000	+	1.73205i	0.0668153	+	0.0578638i
$897$		−	30.0000i		−	1.00167i
$898$	−5.19615	−	3.00000i	−0.173398	−	0.100111i
$899$		0			0
$900$		0			0
$901$	−18.0000	+	31.1769i	−0.599667	+	1.03865i
$902$		0			0
$903$	−6.92820	−	20.0000i	−0.230556	−	0.665558i
$904$	−6.00000			−0.199557
$905$		0			0
$906$	−0.500000	−	0.866025i	−0.0166114	−	0.0287718i
$907$	−6.06218	+	3.50000i	−0.201291	+	0.116216i	−0.597258	−	0.802049i	$-0.703743\pi$
$907$	−6.06218	+	3.50000i	−0.201291	+	0.116216i	0.395966	+	0.918265i	$0.370410\pi$
$908$	10.3923	+	6.00000i	0.344881	+	0.199117i
$909$		0			0
$910$		0			0
$911$	30.0000			0.993944			0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000			0.993944			0.496972	+	0.867766i	$0.334445\pi$
$912$	6.06218	+	3.50000i	0.200739	+	0.115897i
$913$		0			0
$914$	−14.5000	−	25.1147i	−0.479617	−	0.830722i
$915$		0			0
$916$	5.00000			0.165205
$917$	10.3923	−	12.0000i	0.343184	−	0.396275i
$918$		−	6.00000i		−	0.198030i
$919$	4.00000	−	6.92820i	0.131948	−	0.228540i	−0.792480	−	0.609898i	$-0.791210\pi$
$919$	4.00000	−	6.92820i	0.131948	−	0.228540i	0.924427	+	0.381358i	$0.124544\pi$
$920$		0			0
$921$	14.0000	+	24.2487i	0.461316	+	0.799022i
$922$	5.19615	+	3.00000i	0.171126	+	0.0987997i
$923$		−	60.0000i		−	1.97492i
$924$		0			0
$925$		0			0
$926$	−15.5000	+	26.8468i	−0.509362	+	0.882240i
$927$	−11.2583	+	6.50000i	−0.369772	+	0.213488i
$928$		0			0
$929$	−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
$931$	−38.5000	+	30.3109i	−1.26179	+	0.993399i
$932$			24.0000i			0.786146i
$933$	15.5885	+	9.00000i	0.510343	+	0.294647i
$934$	−15.0000	−	25.9808i	−0.490815	−	0.850117i
$935$		0			0
$936$	2.50000	−	4.33013i	0.0817151	−	0.141535i
$937$		−	10.0000i		−	0.326686i	−0.986569	−	0.163343i	$-0.947772\pi$
$937$		−	10.0000i		−	0.326686i	0.986569	−	0.163343i	$-0.0522277\pi$
$938$	33.7750	+	6.50000i	1.10279	+	0.212233i
$939$	−14.0000			−0.456873
$940$		0			0
$941$	3.00000	+	5.19615i	0.0977972	+	0.169390i	0.910773	−	0.412908i	$-0.135487\pi$
$941$	3.00000	+	5.19615i	0.0977972	+	0.169390i	−0.812975	+	0.582298i	$0.802154\pi$
$942$	0.866025	−	0.500000i	0.0282166	−	0.0162909i
$943$		0			0
$944$	6.00000			0.195283
$945$		0			0
$946$		0			0
$947$	−36.3731	−	21.0000i	−1.18197	−	0.682408i	−0.225497	−	0.974244i	$-0.572401\pi$
$947$	−36.3731	−	21.0000i	−1.18197	−	0.682408i	−0.956469	+	0.291835i	$0.905734\pi$
$948$	6.06218	−	3.50000i	0.196890	−	0.113675i
$949$	12.5000	+	21.6506i	0.405767	+	0.702809i
$950$		0			0
$951$	−18.0000			−0.583690
$952$	5.19615	+	15.0000i	0.168408	+	0.486153i
$953$			12.0000i			0.388718i	0.980930	+	0.194359i	$0.0622627\pi$
$953$			12.0000i			0.388718i	−0.980930	+	0.194359i	$0.937737\pi$
$954$	3.00000	−	5.19615i	0.0971286	−	0.168232i
$955$		0			0
$956$	6.00000	+	10.3923i	0.194054	+	0.336111i
$957$		0			0
$958$			18.0000i			0.581554i
$959$	45.0000	−	15.5885i	1.45313	−	0.503378i
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$	4.33013	−	2.50000i	0.139609	−	0.0806032i
$963$	15.5885	−	9.00000i	0.502331	−	0.290021i
$964$	2.50000	−	4.33013i	0.0805196	−	0.139464i
$965$		0			0
$966$	−12.0000	−	10.3923i	−0.386094	−	0.334367i
$967$			41.0000i			1.31847i	0.751936	+	0.659236i	$0.229120\pi$
$967$			41.0000i			1.31847i	−0.751936	+	0.659236i	$0.770880\pi$
$968$	9.52628	+	5.50000i	0.306186	+	0.176777i
$969$	21.0000	+	36.3731i	0.674617	+	1.16847i
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$			1.00000i			0.0320750i
$973$	28.5788	+	5.50000i	0.916195	+	0.176322i
$974$	−40.0000			−1.28168
$975$		0			0
$976$	0.500000	+	0.866025i	0.0160046	+	0.0277208i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$	0.866025	+	0.500000i	0.0276924	+	0.0159882i
$979$		0			0
$980$		0			0
$981$	−7.00000			−0.223493
$982$	−5.19615	−	3.00000i	−0.165816	−	0.0957338i
$983$	25.9808	−	15.0000i	0.828658	−	0.478426i	−0.0247352	−	0.999694i	$-0.507874\pi$
$983$	25.9808	−	15.0000i	0.828658	−	0.478426i	0.853393	+	0.521268i	$0.174541\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−15.5885	−	3.00000i	−0.496186	−	0.0954911i
$988$			35.0000i			1.11350i
$989$	−24.0000	+	41.5692i	−0.763156	+	1.32182i
$990$		0			0
$991$	−16.0000	−	27.7128i	−0.508257	−	0.880327i	−0.999954	−	0.00956046i	$-0.996957\pi$
$991$	−16.0000	−	27.7128i	−0.508257	−	0.880327i	0.491698	−	0.870766i	$-0.336377\pi$
$992$	6.92820	+	4.00000i	0.219971	+	0.127000i
$993$			19.0000i			0.602947i
$994$	−24.0000	−	20.7846i	−0.761234	−	0.659248i
$995$		0			0
$996$	9.00000	−	15.5885i	0.285176	−	0.493939i
$997$	−11.2583	+	6.50000i	−0.356555	+	0.205857i	−0.667568	−	0.744548i	$-0.732665\pi$
$997$	−11.2583	+	6.50000i	−0.356555	+	0.205857i	0.311014	+	0.950405i	$0.399332\pi$
$998$	4.33013	−	2.50000i	0.137068	−	0.0791361i
$999$	−0.500000	+	0.866025i	−0.0158193	+	0.0273998i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.o.d.499.2		4
5.2	odd	4		1050.2.i.g.751.1	yes	2
5.3	odd	4		1050.2.i.n.751.1	yes	2
5.4	even	2	inner	1050.2.o.d.499.1		4
7.4	even	3	inner	1050.2.o.d.949.1		4
35.2	odd	12		7350.2.a.bv.1.1		1
35.4	even	6	inner	1050.2.o.d.949.2		4
35.12	even	12		7350.2.a.cv.1.1		1
35.18	odd	12		1050.2.i.n.151.1	yes	2
35.23	odd	12		7350.2.a.bf.1.1		1
35.32	odd	12		1050.2.i.g.151.1	✓	2
35.33	even	12		7350.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.i.g.151.1	✓	2	35.32	odd	12
1050.2.i.g.751.1	yes	2	5.2	odd	4
1050.2.i.n.151.1	yes	2	35.18	odd	12
1050.2.i.n.751.1	yes	2	5.3	odd	4
1050.2.o.d.499.1		4	5.4	even	2	inner
1050.2.o.d.499.2		4	1.1	even	1	trivial
1050.2.o.d.949.1		4	7.4	even	3	inner
1050.2.o.d.949.2		4	35.4	even	6	inner
7350.2.a.k.1.1		1	35.33	even	12
7350.2.a.bf.1.1		1	35.23	odd	12
7350.2.a.bv.1.1		1	35.2	odd	12
7350.2.a.cv.1.1		1	35.12	even	12

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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} -1.00000 q^{6} +(-2.59808 - 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} -1.00000 q^{6} +(-2.59808 - 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.866025 - 0.500000i) q^{12} -5.00000i q^{13} +(-2.00000 - 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-5.19615 + 3.00000i) q^{17} +(0.866025 - 0.500000i) q^{18} +(-3.50000 + 6.06218i) q^{19} +(2.50000 - 0.866025i) q^{21} +(-5.19615 - 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{24} +(2.50000 - 4.33013i) q^{26} +1.00000i q^{27} +(-0.866025 - 2.50000i) q^{28} +(-4.00000 - 6.92820i) q^{31} +(-0.866025 + 0.500000i) q^{32} -6.00000 q^{34} +1.00000 q^{36} +(0.866025 + 0.500000i) q^{37} +(-6.06218 + 3.50000i) q^{38} +(2.50000 + 4.33013i) q^{39} +(2.59808 + 0.500000i) q^{42} -8.00000i q^{43} +(-3.00000 - 5.19615i) q^{46} +(-5.19615 - 3.00000i) q^{47} -1.00000i q^{48} +(6.50000 + 2.59808i) q^{49} +(3.00000 - 5.19615i) q^{51} +(4.33013 - 2.50000i) q^{52} +(5.19615 - 3.00000i) q^{53} +(-0.500000 + 0.866025i) q^{54} +(0.500000 - 2.59808i) q^{56} -7.00000i q^{57} +(-3.00000 - 5.19615i) q^{59} +(0.500000 - 0.866025i) q^{61} -8.00000i q^{62} +(-1.73205 + 2.00000i) q^{63} -1.00000 q^{64} +(-11.2583 + 6.50000i) q^{67} +(-5.19615 - 3.00000i) q^{68} +6.00000 q^{69} +12.0000 q^{71} +(0.866025 + 0.500000i) q^{72} +(-4.33013 + 2.50000i) q^{73} +(0.500000 + 0.866025i) q^{74} -7.00000 q^{76} +5.00000i q^{78} +(-3.50000 + 6.06218i) q^{79} +(-0.500000 - 0.866025i) q^{81} +18.0000i q^{83} +(2.00000 + 1.73205i) q^{84} +(4.00000 - 6.92820i) q^{86} +(-3.00000 + 5.19615i) q^{89} +(-2.50000 + 12.9904i) q^{91} -6.00000i q^{92} +(6.92820 + 4.00000i) q^{93} +(-3.00000 - 5.19615i) q^{94} +(0.500000 - 0.866025i) q^{96} -7.00000i q^{97} +(4.33013 + 5.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 4 * q^6 + 2 * q^9	\( 4 q + 2 q^{4} - 4 q^{6} + 2 q^{9} - 8 q^{14} - 2 q^{16} - 14 q^{19} + 10 q^{21} - 2 q^{24} + 10 q^{26} - 16 q^{31} - 24 q^{34} + 4 q^{36} + 10 q^{39} - 12 q^{46} + 26 q^{49} + 12 q^{51} - 2 q^{54} + 2 q^{56} - 12 q^{59} + 2 q^{61} - 4 q^{64} + 24 q^{69} + 48 q^{71} + 2 q^{74} - 28 q^{76} - 14 q^{79} - 2 q^{81} + 8 q^{84} + 16 q^{86} - 12 q^{89} - 10 q^{91} - 12 q^{94} + 2 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 - 4 * q^6 + 2 * q^9 - 8 * q^14 - 2 * q^16 - 14 * q^19 + 10 * q^21 - 2 * q^24 + 10 * q^26 - 16 * q^31 - 24 * q^34 + 4 * q^36 + 10 * q^39 - 12 * q^46 + 26 * q^49 + 12 * q^51 - 2 * q^54 + 2 * q^56 - 12 * q^59 + 2 * q^61 - 4 * q^64 + 24 * q^69 + 48 * q^71 + 2 * q^74 - 28 * q^76 - 14 * q^79 - 2 * q^81 + 8 * q^84 + 16 * q^86 - 12 * q^89 - 10 * q^91 - 12 * q^94 + 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more