LMFDB - Embedded newform 975.2.bb.f.874.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,2,Mod(724,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.724"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.bb (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,4,0,-4,0,0,2,0,-4,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$7.78541419707$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			874.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	975.874
Dual form			975.2.bb.f.724.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.73205 + 1.00000i) q^{2} +(0.866025 - 0.500000i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.00000 + 1.73205i) q^{6} +(4.33013 + 2.50000i) q^{7} +(0.500000 - 0.866025i) q^{9} +(-1.00000 - 1.73205i) q^{11} -2.00000i q^{12} +(2.59808 - 2.50000i) q^{13} -10.0000 q^{14} +(2.00000 + 3.46410i) q^{16} +(1.73205 + 1.00000i) q^{17} +2.00000i q^{18} +5.00000 q^{21} +(3.46410 + 2.00000i) q^{22} +(5.19615 - 3.00000i) q^{23} +(-2.00000 + 6.92820i) q^{26} -1.00000i q^{27} +(8.66025 - 5.00000i) q^{28} +(-2.00000 - 3.46410i) q^{29} -7.00000 q^{31} +(-6.92820 - 4.00000i) q^{32} +(-1.73205 - 1.00000i) q^{33} -4.00000 q^{34} +(-1.00000 - 1.73205i) q^{36} +(1.73205 - 1.00000i) q^{37} +(1.00000 - 3.46410i) q^{39} +(-3.00000 - 5.19615i) q^{41} +(-8.66025 + 5.00000i) q^{42} +(-0.866025 - 0.500000i) q^{43} -4.00000 q^{44} +(-6.00000 + 10.3923i) q^{46} +8.00000i q^{47} +(3.46410 + 2.00000i) q^{48} +(9.00000 + 15.5885i) q^{49} +2.00000 q^{51} +(-1.73205 - 7.00000i) q^{52} -4.00000i q^{53} +(1.00000 + 1.73205i) q^{54} +(6.92820 + 4.00000i) q^{58} +(6.00000 - 10.3923i) q^{59} +(6.50000 - 11.2583i) q^{61} +(12.1244 - 7.00000i) q^{62} +(4.33013 - 2.50000i) q^{63} +8.00000 q^{64} +4.00000 q^{66} +(6.06218 - 3.50000i) q^{67} +(3.46410 - 2.00000i) q^{68} +(3.00000 - 5.19615i) q^{69} +(-6.00000 + 10.3923i) q^{71} +15.0000i q^{73} +(-2.00000 + 3.46410i) q^{74} -10.0000i q^{77} +(1.73205 + 7.00000i) q^{78} -3.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} +(10.3923 + 6.00000i) q^{82} +8.00000i q^{83} +(5.00000 - 8.66025i) q^{84} +2.00000 q^{86} +(-3.46410 - 2.00000i) q^{87} +(7.00000 + 12.1244i) q^{89} +(17.5000 - 4.33013i) q^{91} -12.0000i q^{92} +(-6.06218 + 3.50000i) q^{93} +(-8.00000 - 13.8564i) q^{94} -8.00000 q^{96} +(-4.33013 - 2.50000i) q^{97} +(-31.1769 - 18.0000i) q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{6} + 2 q^{9} - 4 q^{11} - 40 q^{14} + 8 q^{16} + 20 q^{21} - 8 q^{26} - 8 q^{29} - 28 q^{31} - 16 q^{34} - 4 q^{36} + 4 q^{39} - 12 q^{41} - 16 q^{44} - 24 q^{46} + 36 q^{49} + 8 q^{51}+ \cdots - 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^6 + 2 * q^9 - 4 * q^11 - 40 * q^14 + 8 * q^16 + 20 * q^21 - 8 * q^26 - 8 * q^29 - 28 * q^31 - 16 * q^34 - 4 * q^36 + 4 * q^39 - 12 * q^41 - 16 * q^44 - 24 * q^46 + 36 * q^49 + 8 * q^51 + 4 * q^54 + 24 * q^59 + 26 * q^61 + 32 * q^64 + 16 * q^66 + 12 * q^69 - 24 * q^71 - 8 * q^74 - 12 * q^79 - 2 * q^81 + 20 * q^84 + 8 * q^86 + 28 * q^89 + 70 * q^91 - 32 * q^94 - 32 * q^96 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$	−1.73205	+	1.00000i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$2$	−1.73205	+	1.00000i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$	−1.00000	+	1.73205i	−0.408248	+	0.707107i
$7$	4.33013	+	2.50000i	1.63663	+	0.944911i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	4.33013	+	2.50000i	1.63663	+	0.944911i	0.654654	+	0.755929i	$0.272814\pi$
$8$		0			0
$9$	0.500000	−	0.866025i	0.166667	−	0.288675i
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		−	2.00000i		−	0.577350i
$13$	2.59808	−	2.50000i	0.720577	−	0.693375i
$13$	2.59808	−	2.50000i	0.720577	−	0.693375i
$14$	−10.0000			−2.67261
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$	1.73205	+	1.00000i	0.420084	+	0.242536i	0.695113	−	0.718900i	$-0.255354\pi$
$17$	1.73205	+	1.00000i	0.420084	+	0.242536i	−0.275029	+	0.961436i	$0.588688\pi$
$18$			2.00000i			0.471405i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$		0			0
$21$	5.00000			1.09109
$22$	3.46410	+	2.00000i	0.738549	+	0.426401i
$23$	5.19615	−	3.00000i	1.08347	−	0.625543i	0.151642	−	0.988436i	$-0.451544\pi$
$23$	5.19615	−	3.00000i	1.08347	−	0.625543i	0.931831	+	0.362892i	$0.118211\pi$
$24$		0			0
$25$		0			0
$26$	−2.00000	+	6.92820i	−0.392232	+	1.35873i
$27$		−	1.00000i		−	0.192450i
$28$	8.66025	−	5.00000i	1.63663	−	0.944911i
$29$	−2.00000	−	3.46410i	−0.371391	−	0.643268i	0.618389	−	0.785872i	$-0.287786\pi$
$29$	−2.00000	−	3.46410i	−0.371391	−	0.643268i	−0.989780	+	0.142605i	$0.954452\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$	−6.92820	−	4.00000i	−1.22474	−	0.707107i
$33$	−1.73205	−	1.00000i	−0.301511	−	0.174078i
$34$	−4.00000			−0.685994
$35$		0			0
$36$	−1.00000	−	1.73205i	−0.166667	−	0.288675i
$37$	1.73205	−	1.00000i	0.284747	−	0.164399i	−0.350823	−	0.936442i	$-0.614098\pi$
$37$	1.73205	−	1.00000i	0.284747	−	0.164399i	0.635571	+	0.772043i	$0.280765\pi$
$38$		0			0
$39$	1.00000	−	3.46410i	0.160128	−	0.554700i
$40$		0			0
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	0.530831	−	0.847477i	$-0.321880\pi$
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$	−8.66025	+	5.00000i	−1.33631	+	0.771517i
$43$	−0.866025	−	0.500000i	−0.132068	−	0.0762493i	0.432511	−	0.901629i	$-0.357628\pi$
$43$	−0.866025	−	0.500000i	−0.132068	−	0.0762493i	−0.564578	+	0.825380i	$0.690961\pi$
$44$	−4.00000			−0.603023
$45$		0			0
$46$	−6.00000	+	10.3923i	−0.884652	+	1.53226i
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	3.46410	+	2.00000i	0.500000	+	0.288675i
$49$	9.00000	+	15.5885i	1.28571	+	2.22692i
$50$		0			0
$51$	2.00000			0.280056
$52$	−1.73205	−	7.00000i	−0.240192	−	0.970725i
$53$		−	4.00000i		−	0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$		−	4.00000i		−	0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	1.00000	+	1.73205i	0.136083	+	0.235702i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	6.92820	+	4.00000i	0.909718	+	0.525226i
$59$	6.00000	−	10.3923i	0.781133	−	1.35296i	−0.150148	−	0.988663i	$-0.547975\pi$
$59$	6.00000	−	10.3923i	0.781133	−	1.35296i	0.931282	−	0.364299i	$-0.118692\pi$
$60$		0			0
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	0.896258	−	0.443533i	$-0.146275\pi$
$62$	12.1244	−	7.00000i	1.53979	−	0.889001i
$63$	4.33013	−	2.50000i	0.545545	−	0.314970i
$64$	8.00000			1.00000
$65$		0			0
$66$	4.00000			0.492366
$67$	6.06218	−	3.50000i	0.740613	−	0.427593i	−0.0816792	−	0.996659i	$-0.526028\pi$
$67$	6.06218	−	3.50000i	0.740613	−	0.427593i	0.822292	+	0.569066i	$0.192695\pi$
$68$	3.46410	−	2.00000i	0.420084	−	0.242536i
$69$	3.00000	−	5.19615i	0.361158	−	0.625543i
$70$		0			0
$71$	−6.00000	+	10.3923i	−0.712069	+	1.23334i	0.252010	+	0.967725i	$0.418908\pi$
$71$	−6.00000	+	10.3923i	−0.712069	+	1.23334i	−0.964079	+	0.265615i	$0.914425\pi$
$72$		0			0
$73$			15.0000i			1.75562i	0.479012	+	0.877809i	$0.340995\pi$
$73$			15.0000i			1.75562i	−0.479012	+	0.877809i	$0.659005\pi$
$74$	−2.00000	+	3.46410i	−0.232495	+	0.402694i
$75$		0			0
$76$		0			0
$77$		−	10.0000i		−	1.13961i
$78$	1.73205	+	7.00000i	0.196116	+	0.792594i
$79$	−3.00000			−0.337526			−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000			−0.337526			−0.168763	+	0.985657i	$0.553977\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$	10.3923	+	6.00000i	1.14764	+	0.662589i
$83$			8.00000i			0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$			8.00000i			0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	5.00000	−	8.66025i	0.545545	−	0.944911i
$85$		0			0
$86$	2.00000			0.215666
$87$	−3.46410	−	2.00000i	−0.371391	−	0.214423i
$88$		0			0
$89$	7.00000	+	12.1244i	0.741999	+	1.28518i	0.951584	+	0.307389i	$0.0994552\pi$
$89$	7.00000	+	12.1244i	0.741999	+	1.28518i	−0.209585	+	0.977790i	$0.567211\pi$
$90$		0			0
$91$	17.5000	−	4.33013i	1.83450	−	0.453921i
$92$		−	12.0000i		−	1.25109i
$93$	−6.06218	+	3.50000i	−0.628619	+	0.362933i
$94$	−8.00000	−	13.8564i	−0.825137	−	1.42918i
$95$		0			0
$96$	−8.00000			−0.816497
$97$	−4.33013	−	2.50000i	−0.439658	−	0.253837i	0.263795	−	0.964579i	$-0.415026\pi$
$97$	−4.33013	−	2.50000i	−0.439658	−	0.253837i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	−31.1769	−	18.0000i	−3.14934	−	1.81827i
$99$	−2.00000			−0.201008
$100$		0			0
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.833143	+	0.553058i	$0.186539\pi$
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.0623905	+	0.998052i	$0.480128\pi$
$102$	−3.46410	+	2.00000i	−0.342997	+	0.198030i
$103$		−	7.00000i		−	0.689730i	−0.938652	−	0.344865i	$-0.887925\pi$
$103$		−	7.00000i		−	0.689730i	0.938652	−	0.344865i	$-0.112075\pi$
$104$		0			0
$105$		0			0
$106$	4.00000	+	6.92820i	0.388514	+	0.672927i
$107$	−3.46410	+	2.00000i	−0.334887	+	0.193347i	−0.658009	−	0.753010i	$-0.728601\pi$
$107$	−3.46410	+	2.00000i	−0.334887	+	0.193347i	0.323122	+	0.946357i	$0.395268\pi$
$108$	−1.73205	−	1.00000i	−0.166667	−	0.0962250i
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$		0			0
$111$	1.00000	−	1.73205i	0.0949158	−	0.164399i
$112$			20.0000i			1.88982i
$113$	1.73205	+	1.00000i	0.162938	+	0.0940721i	0.579252	−	0.815149i	$-0.303345\pi$
$113$	1.73205	+	1.00000i	0.162938	+	0.0940721i	−0.416314	+	0.909221i	$0.636678\pi$
$114$		0			0
$115$		0			0
$116$	−8.00000			−0.742781
$117$	−0.866025	−	3.50000i	−0.0800641	−	0.323575i
$118$			24.0000i			2.20938i
$119$	5.00000	+	8.66025i	0.458349	+	0.793884i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$			26.0000i			2.35393i
$123$	−5.19615	−	3.00000i	−0.468521	−	0.270501i
$124$	−7.00000	+	12.1244i	−0.628619	+	1.08880i
$125$		0			0
$126$	−5.00000	+	8.66025i	−0.445435	+	0.771517i
$127$	9.52628	−	5.50000i	0.845321	−	0.488046i	−0.0137486	−	0.999905i	$-0.504376\pi$
$127$	9.52628	−	5.50000i	0.845321	−	0.488046i	0.859069	+	0.511859i	$0.171043\pi$
$128$		0			0
$129$	−1.00000			−0.0880451
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$	−3.46410	+	2.00000i	−0.301511	+	0.174078i
$133$		0			0
$134$	−7.00000	+	12.1244i	−0.604708	+	1.04738i
$135$		0			0
$136$		0			0
$137$	−1.73205	−	1.00000i	−0.147979	−	0.0854358i	0.424182	−	0.905577i	$-0.360562\pi$
$137$	−1.73205	−	1.00000i	−0.147979	−	0.0854358i	−0.572161	+	0.820141i	$0.693895\pi$
$138$			12.0000i			1.02151i
$139$	−1.50000	+	2.59808i	−0.127228	+	0.220366i	−0.922602	−	0.385754i	$-0.873941\pi$
$139$	−1.50000	+	2.59808i	−0.127228	+	0.220366i	0.795373	+	0.606120i	$0.207275\pi$
$140$		0			0
$141$	4.00000	+	6.92820i	0.336861	+	0.583460i
$142$		−	24.0000i		−	2.01404i
$143$	−6.92820	−	2.00000i	−0.579365	−	0.167248i
$144$	4.00000			0.333333
$145$		0			0
$146$	−15.0000	−	25.9808i	−1.24141	−	2.15018i
$147$	15.5885	+	9.00000i	1.28571	+	0.742307i
$148$		−	4.00000i		−	0.328798i
$149$	−6.00000	+	10.3923i	−0.491539	+	0.851371i	−0.999953	−	0.00974235i	$-0.996899\pi$
$149$	−6.00000	+	10.3923i	−0.491539	+	0.851371i	0.508413	+	0.861113i	$0.330232\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$	1.73205	−	1.00000i	0.140028	−	0.0808452i
$154$	10.0000	+	17.3205i	0.805823	+	1.39573i
$155$		0			0
$156$	−5.00000	−	5.19615i	−0.400320	−	0.416025i
$157$			15.0000i			1.19713i	0.801074	+	0.598565i	$0.204262\pi$
$157$			15.0000i			1.19713i	−0.801074	+	0.598565i	$0.795738\pi$
$158$	5.19615	−	3.00000i	0.413384	−	0.238667i
$159$	−2.00000	−	3.46410i	−0.158610	−	0.274721i
$160$		0			0
$161$	30.0000			2.36433
$162$	1.73205	+	1.00000i	0.136083	+	0.0785674i
$163$	12.9904	+	7.50000i	1.01749	+	0.587445i	0.913375	−	0.407120i	$-0.133467\pi$
$163$	12.9904	+	7.50000i	1.01749	+	0.587445i	0.104111	+	0.994566i	$0.466800\pi$
$164$	−12.0000			−0.937043
$165$		0			0
$166$	−8.00000	−	13.8564i	−0.620920	−	1.07547i
$167$	−10.3923	+	6.00000i	−0.804181	+	0.464294i	−0.844931	−	0.534875i	$-0.820359\pi$
$167$	−10.3923	+	6.00000i	−0.804181	+	0.464294i	0.0407502	+	0.999169i	$0.487025\pi$
$168$		0			0
$169$	0.500000	−	12.9904i	0.0384615	−	0.999260i
$170$		0			0
$171$		0			0
$172$	−1.73205	+	1.00000i	−0.132068	+	0.0762493i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$	8.00000			0.606478
$175$		0			0
$176$	4.00000	−	6.92820i	0.301511	−	0.522233i
$177$		−	12.0000i		−	0.901975i
$178$	−24.2487	−	14.0000i	−1.81752	−	1.04934i
$179$	−3.00000	−	5.19615i	−0.224231	−	0.388379i	0.731858	−	0.681457i	$-0.238654\pi$
$179$	−3.00000	−	5.19615i	−0.224231	−	0.388379i	−0.956088	+	0.293079i	$0.905320\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$	−25.9808	+	25.0000i	−1.92582	+	1.85312i
$183$		−	13.0000i		−	0.960988i
$184$		0			0
$185$		0			0
$186$	7.00000	−	12.1244i	0.513265	−	0.889001i
$187$		−	4.00000i		−	0.292509i
$188$	13.8564	+	8.00000i	1.01058	+	0.583460i
$189$	2.50000	−	4.33013i	0.181848	−	0.314970i
$190$		0			0
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	−0.997225	−	0.0744412i	$-0.976283\pi$
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	0.563081	+	0.826402i	$0.309616\pi$
$192$	6.92820	−	4.00000i	0.500000	−	0.288675i
$193$	−9.52628	+	5.50000i	−0.685717	+	0.395899i	−0.802005	−	0.597317i	$-0.796234\pi$
$193$	−9.52628	+	5.50000i	−0.685717	+	0.395899i	0.116289	+	0.993215i	$0.462900\pi$
$194$	10.0000			0.717958
$195$		0			0
$196$	36.0000			2.57143
$197$	10.3923	−	6.00000i	0.740421	−	0.427482i	−0.0818013	−	0.996649i	$-0.526067\pi$
$197$	10.3923	−	6.00000i	0.740421	−	0.427482i	0.822222	+	0.569166i	$0.192734\pi$
$198$	3.46410	−	2.00000i	0.246183	−	0.142134i
$199$	8.50000	−	14.7224i	0.602549	−	1.04365i	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	8.50000	−	14.7224i	0.602549	−	1.04365i	0.992434	−	0.122782i	$-0.0391815\pi$
$200$		0			0
$201$	3.50000	−	6.06218i	0.246871	−	0.427593i
$202$	−31.1769	−	18.0000i	−2.19360	−	1.26648i
$203$		−	20.0000i		−	1.40372i
$204$	2.00000	−	3.46410i	0.140028	−	0.242536i
$205$		0			0
$206$	7.00000	+	12.1244i	0.487713	+	0.844744i
$207$		−	6.00000i		−	0.417029i
$208$	13.8564	+	4.00000i	0.960769	+	0.277350i
$209$		0			0
$210$		0			0
$211$	−7.50000	−	12.9904i	−0.516321	−	0.894295i	−0.999820	−	0.0189499i	$-0.993968\pi$
$211$	−7.50000	−	12.9904i	−0.516321	−	0.894295i	0.483499	−	0.875345i	$-0.339366\pi$
$212$	−6.92820	−	4.00000i	−0.475831	−	0.274721i
$213$			12.0000i			0.822226i
$214$	4.00000	−	6.92820i	0.273434	−	0.473602i
$215$		0			0
$216$		0			0
$217$	−30.3109	−	17.5000i	−2.05764	−	1.18798i
$218$	−19.0526	+	11.0000i	−1.29040	+	0.745014i
$219$	7.50000	+	12.9904i	0.506803	+	0.877809i
$220$		0			0
$221$	7.00000	−	1.73205i	0.470871	−	0.116510i
$222$			4.00000i			0.268462i
$223$	−6.92820	+	4.00000i	−0.463947	+	0.267860i	−0.713702	−	0.700449i	$-0.752983\pi$
$223$	−6.92820	+	4.00000i	−0.463947	+	0.267860i	0.249756	+	0.968309i	$0.419650\pi$
$224$	−20.0000	−	34.6410i	−1.33631	−	2.31455i
$225$		0			0
$226$	−4.00000			−0.266076
$227$	8.66025	+	5.00000i	0.574801	+	0.331862i	0.759065	−	0.651015i	$-0.225657\pi$
$227$	8.66025	+	5.00000i	0.574801	+	0.331862i	−0.184263	+	0.982877i	$0.558990\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$	−5.00000	−	8.66025i	−0.328976	−	0.569803i
$232$		0			0
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	5.00000	+	5.19615i	0.326860	+	0.339683i
$235$		0			0
$236$	−12.0000	−	20.7846i	−0.781133	−	1.35296i
$237$	−2.59808	+	1.50000i	−0.168763	+	0.0974355i
$238$	−17.3205	−	10.0000i	−1.12272	−	0.648204i
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−5.00000	+	8.66025i	−0.322078	+	0.557856i	−0.980917	−	0.194429i	$-0.937715\pi$
$241$	−5.00000	+	8.66025i	−0.322078	+	0.557856i	0.658838	+	0.752285i	$0.271048\pi$
$242$			14.0000i			0.899954i
$243$	−0.866025	−	0.500000i	−0.0555556	−	0.0320750i
$244$	−13.0000	−	22.5167i	−0.832240	−	1.44148i
$245$		0			0
$246$	12.0000			0.765092
$247$		0			0
$248$		0			0
$249$	4.00000	+	6.92820i	0.253490	+	0.439057i
$250$		0			0
$251$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$251$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$252$		−	10.0000i		−	0.629941i
$253$	−10.3923	−	6.00000i	−0.653359	−	0.377217i
$254$	−11.0000	+	19.0526i	−0.690201	+	1.19546i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	19.0526	−	11.0000i	1.18847	−	0.686161i	0.230508	−	0.973070i	$-0.425961\pi$
$257$	19.0526	−	11.0000i	1.18847	−	0.686161i	0.957958	+	0.286909i	$0.0926278\pi$
$258$	1.73205	−	1.00000i	0.107833	−	0.0622573i
$259$	10.0000			0.621370
$260$		0			0
$261$	−4.00000			−0.247594
$262$	6.92820	−	4.00000i	0.428026	−	0.247121i
$263$	−8.66025	+	5.00000i	−0.534014	+	0.308313i	−0.742650	−	0.669680i	$-0.766431\pi$
$263$	−8.66025	+	5.00000i	−0.534014	+	0.308313i	0.208635	+	0.977993i	$0.433098\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	12.1244	+	7.00000i	0.741999	+	0.428393i
$268$		−	14.0000i		−	0.855186i
$269$	−3.00000	+	5.19615i	−0.182913	+	0.316815i	−0.942871	−	0.333157i	$-0.891886\pi$
$269$	−3.00000	+	5.19615i	−0.182913	+	0.316815i	0.759958	+	0.649972i	$0.225219\pi$
$270$		0			0
$271$	−14.5000	−	25.1147i	−0.880812	−	1.52561i	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−14.5000	−	25.1147i	−0.880812	−	1.52561i	−0.0303728	−	0.999539i	$-0.509669\pi$
$272$			8.00000i			0.485071i
$273$	12.9904	−	12.5000i	0.786214	−	0.756534i
$274$	4.00000			0.241649
$275$		0			0
$276$	−6.00000	−	10.3923i	−0.361158	−	0.625543i
$277$	8.66025	+	5.00000i	0.520344	+	0.300421i	0.737075	−	0.675810i	$-0.236206\pi$
$277$	8.66025	+	5.00000i	0.520344	+	0.300421i	−0.216731	+	0.976231i	$0.569540\pi$
$278$		−	6.00000i		−	0.359856i
$279$	−3.50000	+	6.06218i	−0.209540	+	0.362933i
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$	−13.8564	−	8.00000i	−0.825137	−	0.476393i
$283$	4.33013	−	2.50000i	0.257399	−	0.148610i	−0.365748	−	0.930714i	$-0.619187\pi$
$283$	4.33013	−	2.50000i	0.257399	−	0.148610i	0.623148	+	0.782104i	$0.285854\pi$
$284$	12.0000	+	20.7846i	0.712069	+	1.23334i
$285$		0			0
$286$	14.0000	−	3.46410i	0.827837	−	0.204837i
$287$		−	30.0000i		−	1.77084i
$288$	−6.92820	+	4.00000i	−0.408248	+	0.235702i
$289$	−6.50000	−	11.2583i	−0.382353	−	0.662255i
$290$		0			0
$291$	−5.00000			−0.293105
$292$	25.9808	+	15.0000i	1.52041	+	0.877809i
$293$	−13.8564	−	8.00000i	−0.809500	−	0.467365i	0.0372823	−	0.999305i	$-0.488130\pi$
$293$	−13.8564	−	8.00000i	−0.809500	−	0.467365i	−0.846782	+	0.531940i	$0.821463\pi$
$294$	−36.0000			−2.09956
$295$		0			0
$296$		0			0
$297$	−1.73205	+	1.00000i	−0.100504	+	0.0580259i
$298$		−	24.0000i		−	1.39028i
$299$	6.00000	−	20.7846i	0.346989	−	1.20201i
$300$		0			0
$301$	−2.50000	−	4.33013i	−0.144098	−	0.249584i
$302$	13.8564	−	8.00000i	0.797347	−	0.460348i
$303$	15.5885	+	9.00000i	0.895533	+	0.517036i
$304$		0			0
$305$		0			0
$306$	−2.00000	+	3.46410i	−0.114332	+	0.198030i
$307$			31.0000i			1.76926i	0.466290	+	0.884632i	$0.345590\pi$
$307$			31.0000i			1.76926i	−0.466290	+	0.884632i	$0.654410\pi$
$308$	−17.3205	−	10.0000i	−0.986928	−	0.569803i
$309$	−3.50000	−	6.06218i	−0.199108	−	0.344865i
$310$		0			0
$311$	−22.0000			−1.24751			−0.623753	−	0.781622i	$-0.714393\pi$
$311$	−22.0000			−1.24751			−0.623753	+	0.781622i	$0.714393\pi$
$312$		0			0
$313$		−	31.0000i		−	1.75222i	−0.482108	−	0.876112i	$-0.660129\pi$
$313$		−	31.0000i		−	1.75222i	0.482108	−	0.876112i	$-0.339871\pi$
$314$	−15.0000	−	25.9808i	−0.846499	−	1.46618i
$315$		0			0
$316$	−3.00000	+	5.19615i	−0.168763	+	0.292306i
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	6.92820	+	4.00000i	0.388514	+	0.224309i
$319$	−4.00000	+	6.92820i	−0.223957	+	0.387905i
$320$		0			0
$321$	−2.00000	+	3.46410i	−0.111629	+	0.193347i
$322$	−51.9615	+	30.0000i	−2.89570	+	1.67183i
$323$		0			0
$324$	−2.00000			−0.111111
$325$		0			0
$326$	−30.0000			−1.66155
$327$	9.52628	−	5.50000i	0.526804	−	0.304151i
$328$		0			0
$329$	−20.0000	+	34.6410i	−1.10264	+	1.90982i
$330$		0			0
$331$	−4.50000	+	7.79423i	−0.247342	+	0.428410i	−0.962788	−	0.270259i	$-0.912891\pi$
$331$	−4.50000	+	7.79423i	−0.247342	+	0.428410i	0.715445	+	0.698669i	$0.246224\pi$
$332$	13.8564	+	8.00000i	0.760469	+	0.439057i
$333$		−	2.00000i		−	0.109599i
$334$	12.0000	−	20.7846i	0.656611	−	1.13728i
$335$		0			0
$336$	10.0000	+	17.3205i	0.545545	+	0.944911i
$337$			1.00000i			0.0544735i	0.999629	+	0.0272367i	$0.00867079\pi$
$337$			1.00000i			0.0544735i	−0.999629	+	0.0272367i	$0.991329\pi$
$338$	12.1244	+	23.0000i	0.659478	+	1.25104i
$339$	2.00000			0.108625
$340$		0			0
$341$	7.00000	+	12.1244i	0.379071	+	0.656571i
$342$		0			0
$343$			55.0000i			2.96972i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.8564	−	8.00000i	−0.743851	−	0.429463i	0.0796169	−	0.996826i	$-0.474630\pi$
$347$	−13.8564	−	8.00000i	−0.743851	−	0.429463i	−0.823468	+	0.567363i	$0.807964\pi$
$348$	−6.92820	+	4.00000i	−0.371391	+	0.214423i
$349$	1.50000	+	2.59808i	0.0802932	+	0.139072i	0.903376	−	0.428850i	$-0.141081\pi$
$349$	1.50000	+	2.59808i	0.0802932	+	0.139072i	−0.823083	+	0.567922i	$0.807748\pi$
$350$		0			0
$351$	−2.50000	−	2.59808i	−0.133440	−	0.138675i
$352$			16.0000i			0.852803i
$353$	5.19615	−	3.00000i	0.276563	−	0.159674i	−0.355303	−	0.934751i	$-0.615622\pi$
$353$	5.19615	−	3.00000i	0.276563	−	0.159674i	0.631867	+	0.775077i	$0.282289\pi$
$354$	12.0000	+	20.7846i	0.637793	+	1.10469i
$355$		0			0
$356$	28.0000			1.48400
$357$	8.66025	+	5.00000i	0.458349	+	0.264628i
$358$	10.3923	+	6.00000i	0.549250	+	0.317110i
$359$	−2.00000			−0.105556			−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000			−0.105556			−0.0527780	+	0.998606i	$0.516808\pi$
$360$		0			0
$361$	9.50000	+	16.4545i	0.500000	+	0.866025i
$362$	38.1051	−	22.0000i	2.00276	−	1.15629i
$363$		−	7.00000i		−	0.367405i
$364$	10.0000	−	34.6410i	0.524142	−	1.81568i
$365$		0			0
$366$	13.0000	+	22.5167i	0.679521	+	1.17696i
$367$	−6.06218	+	3.50000i	−0.316443	+	0.182699i	−0.649806	−	0.760100i	$-0.725150\pi$
$367$	−6.06218	+	3.50000i	−0.316443	+	0.182699i	0.333363	+	0.942799i	$0.391817\pi$
$368$	20.7846	+	12.0000i	1.08347	+	0.625543i
$369$	−6.00000			−0.312348
$370$		0			0
$371$	10.0000	−	17.3205i	0.519174	−	0.899236i
$372$			14.0000i			0.725866i
$373$	−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	$0.775929\pi$
$374$	4.00000	+	6.92820i	0.206835	+	0.358249i
$375$		0			0
$376$		0			0
$377$	−13.8564	−	4.00000i	−0.713641	−	0.206010i
$378$			10.0000i			0.514344i
$379$	−2.50000	−	4.33013i	−0.128416	−	0.222424i	0.794647	−	0.607072i	$-0.207656\pi$
$379$	−2.50000	−	4.33013i	−0.128416	−	0.222424i	−0.923063	+	0.384648i	$0.874323\pi$
$380$		0			0
$381$	5.50000	−	9.52628i	0.281774	−	0.488046i
$382$		−	24.0000i		−	1.22795i
$383$	15.5885	+	9.00000i	0.796533	+	0.459879i	0.842257	−	0.539076i	$-0.181226\pi$
$383$	15.5885	+	9.00000i	0.796533	+	0.459879i	−0.0457244	+	0.998954i	$0.514560\pi$
$384$		0			0
$385$		0			0
$386$	11.0000	−	19.0526i	0.559885	−	0.969750i
$387$	−0.866025	+	0.500000i	−0.0440225	+	0.0254164i
$388$	−8.66025	+	5.00000i	−0.439658	+	0.253837i
$389$	−8.00000			−0.405616			−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000			−0.405616			−0.202808	+	0.979219i	$0.565007\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$	−3.46410	+	2.00000i	−0.174741	+	0.100887i
$394$	−12.0000	+	20.7846i	−0.604551	+	1.04711i
$395$		0			0
$396$	−2.00000	+	3.46410i	−0.100504	+	0.174078i
$397$	−12.9904	−	7.50000i	−0.651969	−	0.376414i	0.137241	−	0.990538i	$-0.456176\pi$
$397$	−12.9904	−	7.50000i	−0.651969	−	0.376414i	−0.789210	+	0.614123i	$0.789510\pi$
$398$			34.0000i			1.70427i
$399$		0			0
$400$		0			0
$401$	−8.00000	−	13.8564i	−0.399501	−	0.691956i	0.594163	−	0.804344i	$-0.297483\pi$
$401$	−8.00000	−	13.8564i	−0.399501	−	0.691956i	−0.993664	+	0.112388i	$0.964150\pi$
$402$			14.0000i			0.698257i
$403$	−18.1865	+	17.5000i	−0.905936	+	0.871737i
$404$	36.0000			1.79107
$405$		0			0
$406$	20.0000	+	34.6410i	0.992583	+	1.71920i
$407$	−3.46410	−	2.00000i	−0.171709	−	0.0991363i
$408$		0			0
$409$	−7.50000	+	12.9904i	−0.370851	+	0.642333i	−0.989697	−	0.143180i	$-0.954267\pi$
$409$	−7.50000	+	12.9904i	−0.370851	+	0.642333i	0.618846	+	0.785513i	$0.287601\pi$
$410$		0			0
$411$	−2.00000			−0.0986527
$412$	−12.1244	−	7.00000i	−0.597324	−	0.344865i
$413$	51.9615	−	30.0000i	2.55686	−	1.47620i
$414$	6.00000	+	10.3923i	0.294884	+	0.510754i
$415$		0			0
$416$	−28.0000	+	6.92820i	−1.37281	+	0.339683i
$417$			3.00000i			0.146911i
$418$		0			0
$419$	19.0000	+	32.9090i	0.928211	+	1.60771i	0.786314	+	0.617827i	$0.211987\pi$
$419$	19.0000	+	32.9090i	0.928211	+	1.60771i	0.141896	+	0.989882i	$0.454680\pi$
$420$		0			0
$421$	−23.0000			−1.12095			−0.560476	−	0.828171i	$-0.689382\pi$
$421$	−23.0000			−1.12095			−0.560476	+	0.828171i	$0.689382\pi$
$422$	25.9808	+	15.0000i	1.26472	+	0.730189i
$423$	6.92820	+	4.00000i	0.336861	+	0.194487i
$424$		0			0
$425$		0			0
$426$	−12.0000	−	20.7846i	−0.581402	−	1.00702i
$427$	56.2917	−	32.5000i	2.72414	−	1.57279i
$428$			8.00000i			0.386695i
$429$	−7.00000	+	1.73205i	−0.337963	+	0.0836242i
$430$		0			0
$431$	14.0000	+	24.2487i	0.674356	+	1.16802i	0.976657	+	0.214807i	$0.0689121\pi$
$431$	14.0000	+	24.2487i	0.674356	+	1.16802i	−0.302300	+	0.953213i	$0.597755\pi$
$432$	3.46410	−	2.00000i	0.166667	−	0.0962250i
$433$	−0.866025	−	0.500000i	−0.0416185	−	0.0240285i	0.479046	−	0.877790i	$-0.340983\pi$
$433$	−0.866025	−	0.500000i	−0.0416185	−	0.0240285i	−0.520665	+	0.853761i	$0.674316\pi$
$434$	70.0000			3.36011
$435$		0			0
$436$	11.0000	−	19.0526i	0.526804	−	0.912452i
$437$		0			0
$438$	−25.9808	−	15.0000i	−1.24141	−	0.716728i
$439$	7.50000	+	12.9904i	0.357955	+	0.619997i	0.987619	−	0.156871i	$-0.0501406\pi$
$439$	7.50000	+	12.9904i	0.357955	+	0.619997i	−0.629664	+	0.776868i	$0.716807\pi$
$440$		0			0
$441$	18.0000			0.857143
$442$	−10.3923	+	10.0000i	−0.494312	+	0.475651i
$443$			26.0000i			1.23530i	0.786454	+	0.617649i	$0.211915\pi$
$443$			26.0000i			1.23530i	−0.786454	+	0.617649i	$0.788085\pi$
$444$	−2.00000	−	3.46410i	−0.0949158	−	0.164399i
$445$		0			0
$446$	8.00000	−	13.8564i	0.378811	−	0.656120i
$447$			12.0000i			0.567581i
$448$	34.6410	+	20.0000i	1.63663	+	0.944911i
$449$	9.00000	−	15.5885i	0.424736	−	0.735665i	−0.571660	−	0.820491i	$-0.693700\pi$
$449$	9.00000	−	15.5885i	0.424736	−	0.735665i	0.996396	+	0.0848262i	$0.0270335\pi$
$450$		0			0
$451$	−6.00000	+	10.3923i	−0.282529	+	0.489355i
$452$	3.46410	−	2.00000i	0.162938	−	0.0940721i
$453$	−6.92820	+	4.00000i	−0.325515	+	0.187936i
$454$	−20.0000			−0.938647
$455$		0			0
$456$		0			0
$457$	−30.3109	+	17.5000i	−1.41788	+	0.818615i	−0.996113	−	0.0880870i	$-0.971925\pi$
$457$	−30.3109	+	17.5000i	−1.41788	+	0.818615i	−0.421771	+	0.906702i	$0.638591\pi$
$458$	24.2487	−	14.0000i	1.13307	−	0.654177i
$459$	1.00000	−	1.73205i	0.0466760	−	0.0808452i
$460$		0			0
$461$	1.00000	−	1.73205i	0.0465746	−	0.0806696i	−0.841798	−	0.539792i	$-0.818503\pi$
$461$	1.00000	−	1.73205i	0.0465746	−	0.0806696i	0.888373	+	0.459123i	$0.151836\pi$
$462$	17.3205	+	10.0000i	0.805823	+	0.465242i
$463$			3.00000i			0.139422i	0.997567	+	0.0697109i	$0.0222077\pi$
$463$			3.00000i			0.139422i	−0.997567	+	0.0697109i	$0.977792\pi$
$464$	8.00000	−	13.8564i	0.371391	−	0.643268i
$465$		0			0
$466$	14.0000	+	24.2487i	0.648537	+	1.12330i
$467$			4.00000i			0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$			4.00000i			0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$	−6.92820	−	2.00000i	−0.320256	−	0.0924500i
$469$	35.0000			1.61615
$470$		0			0
$471$	7.50000	+	12.9904i	0.345582	+	0.598565i
$472$		0			0
$473$			2.00000i			0.0919601i
$474$	3.00000	−	5.19615i	0.137795	−	0.238667i
$475$		0			0
$476$	20.0000			0.916698
$477$	−3.46410	−	2.00000i	−0.158610	−	0.0915737i
$478$	20.7846	−	12.0000i	0.950666	−	0.548867i
$479$	21.0000	+	36.3731i	0.959514	+	1.66193i	0.723681	+	0.690134i	$0.242449\pi$
$479$	21.0000	+	36.3731i	0.959514	+	1.66193i	0.235833	+	0.971794i	$0.424218\pi$
$480$		0			0
$481$	2.00000	−	6.92820i	0.0911922	−	0.315899i
$482$		−	20.0000i		−	0.910975i
$483$	25.9808	−	15.0000i	1.18217	−	0.682524i
$484$	−7.00000	−	12.1244i	−0.318182	−	0.551107i
$485$		0			0
$486$	2.00000			0.0907218
$487$	−24.2487	−	14.0000i	−1.09881	−	0.634401i	−0.162905	−	0.986642i	$-0.552086\pi$
$487$	−24.2487	−	14.0000i	−1.09881	−	0.634401i	−0.935909	+	0.352241i	$0.885420\pi$
$488$		0			0
$489$	15.0000			0.678323
$490$		0			0
$491$	12.0000	+	20.7846i	0.541552	+	0.937996i	0.998815	+	0.0486647i	$0.0154966\pi$
$491$	12.0000	+	20.7846i	0.541552	+	0.937996i	−0.457263	+	0.889332i	$0.651170\pi$
$492$	−10.3923	+	6.00000i	−0.468521	+	0.270501i
$493$		−	8.00000i		−	0.360302i
$494$		0			0
$495$		0			0
$496$	−14.0000	−	24.2487i	−0.628619	−	1.08880i
$497$	−51.9615	+	30.0000i	−2.33079	+	1.34568i
$498$	−13.8564	−	8.00000i	−0.620920	−	0.358489i
$499$	−4.00000			−0.179065			−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000			−0.179065			−0.0895323	+	0.995984i	$0.528537\pi$
$500$		0			0
$501$	−6.00000	+	10.3923i	−0.268060	+	0.464294i
$502$		0			0
$503$	5.19615	+	3.00000i	0.231685	+	0.133763i	0.611349	−	0.791361i	$-0.290627\pi$
$503$	5.19615	+	3.00000i	0.231685	+	0.133763i	−0.379664	+	0.925124i	$0.623960\pi$
$504$		0			0
$505$		0			0
$506$	24.0000			1.06693
$507$	−6.06218	−	11.5000i	−0.269231	−	0.510733i
$508$		−	22.0000i		−	0.976092i
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	−0.979322	−	0.202306i	$-0.935156\pi$
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	0.314459	−	0.949271i	$-0.398177\pi$
$510$		0			0
$511$	−37.5000	+	64.9519i	−1.65890	+	2.87330i
$512$		−	32.0000i		−	1.41421i
$513$		0			0
$514$	−22.0000	+	38.1051i	−0.970378	+	1.68074i
$515$		0			0
$516$	−1.00000	+	1.73205i	−0.0440225	+	0.0762493i
$517$	13.8564	−	8.00000i	0.609404	−	0.351840i
$518$	−17.3205	+	10.0000i	−0.761019	+	0.439375i
$519$		0			0
$520$		0			0
$521$	−30.0000			−1.31432			−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000			−1.31432			−0.657162	+	0.753749i	$0.728243\pi$
$522$	6.92820	−	4.00000i	0.303239	−	0.175075i
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$	−4.00000	+	6.92820i	−0.174741	+	0.302660i
$525$		0			0
$526$	10.0000	−	17.3205i	0.436021	−	0.755210i
$527$	−12.1244	−	7.00000i	−0.528145	−	0.304925i
$528$		−	8.00000i		−	0.348155i
$529$	6.50000	−	11.2583i	0.282609	−	0.489493i
$530$		0			0
$531$	−6.00000	−	10.3923i	−0.260378	−	0.450988i
$532$		0			0
$533$	−20.7846	−	6.00000i	−0.900281	−	0.259889i
$534$	−28.0000			−1.21168
$535$		0			0
$536$		0			0
$537$	−5.19615	−	3.00000i	−0.224231	−	0.129460i
$538$		−	12.0000i		−	0.517357i
$539$	18.0000	−	31.1769i	0.775315	−	1.34288i
$540$		0			0
$541$	29.0000			1.24681			0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000			1.24681			0.623404	+	0.781900i	$0.285749\pi$
$542$	50.2295	+	29.0000i	2.15754	+	1.24566i
$543$	−19.0526	+	11.0000i	−0.817624	+	0.472055i
$544$	−8.00000	−	13.8564i	−0.342997	−	0.594089i
$545$		0			0
$546$	−10.0000	+	34.6410i	−0.427960	+	1.48250i
$547$			9.00000i			0.384812i	0.981315	+	0.192406i	$0.0616291\pi$
$547$			9.00000i			0.384812i	−0.981315	+	0.192406i	$0.938371\pi$
$548$	−3.46410	+	2.00000i	−0.147979	+	0.0854358i
$549$	−6.50000	−	11.2583i	−0.277413	−	0.480494i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−12.9904	−	7.50000i	−0.552407	−	0.318932i
$554$	−20.0000			−0.849719
$555$		0			0
$556$	3.00000	+	5.19615i	0.127228	+	0.220366i
$557$	17.3205	−	10.0000i	0.733893	−	0.423714i	−0.0859514	−	0.996299i	$-0.527393\pi$
$557$	17.3205	−	10.0000i	0.733893	−	0.423714i	0.819845	+	0.572586i	$0.194060\pi$
$558$		−	14.0000i		−	0.592667i
$559$	−3.50000	+	0.866025i	−0.148034	+	0.0366290i
$560$		0			0
$561$	−2.00000	−	3.46410i	−0.0844401	−	0.146254i
$562$	20.7846	−	12.0000i	0.876746	−	0.506189i
$563$	−22.5167	−	13.0000i	−0.948964	−	0.547885i	−0.0562051	−	0.998419i	$-0.517900\pi$
$563$	−22.5167	−	13.0000i	−0.948964	−	0.547885i	−0.892759	+	0.450535i	$0.851233\pi$
$564$	16.0000			0.673722
$565$		0			0
$566$	−5.00000	+	8.66025i	−0.210166	+	0.364018i
$567$		−	5.00000i		−	0.209980i
$568$		0			0
$569$	−10.0000	−	17.3205i	−0.419222	−	0.726113i	0.576640	−	0.816999i	$-0.304364\pi$
$569$	−10.0000	−	17.3205i	−0.419222	−	0.726113i	−0.995861	+	0.0908852i	$0.971030\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$	−10.3923	+	10.0000i	−0.434524	+	0.418121i
$573$			12.0000i			0.501307i
$574$	30.0000	+	51.9615i	1.25218	+	2.16883i
$575$		0			0
$576$	4.00000	−	6.92820i	0.166667	−	0.288675i
$577$		−	2.00000i		−	0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$		−	2.00000i		−	0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	22.5167	+	13.0000i	0.936570	+	0.540729i
$579$	−5.50000	+	9.52628i	−0.228572	+	0.395899i
$580$		0			0
$581$	−20.0000	+	34.6410i	−0.829740	+	1.43715i
$582$	8.66025	−	5.00000i	0.358979	−	0.207257i
$583$	−6.92820	+	4.00000i	−0.286937	+	0.165663i
$584$		0			0
$585$		0			0
$586$	32.0000			1.32191
$587$	−24.2487	+	14.0000i	−1.00085	+	0.577842i	−0.908500	−	0.417885i	$-0.862772\pi$
$587$	−24.2487	+	14.0000i	−1.00085	+	0.577842i	−0.0923513	+	0.995726i	$0.529438\pi$
$588$	31.1769	−	18.0000i	1.28571	−	0.742307i
$589$		0			0
$590$		0			0
$591$	6.00000	−	10.3923i	0.246807	−	0.427482i
$592$	6.92820	+	4.00000i	0.284747	+	0.164399i
$593$		−	10.0000i		−	0.410651i	−0.978694	−	0.205325i	$-0.934175\pi$
$593$		−	10.0000i		−	0.410651i	0.978694	−	0.205325i	$-0.0658253\pi$
$594$	2.00000	−	3.46410i	0.0820610	−	0.142134i
$595$		0			0
$596$	12.0000	+	20.7846i	0.491539	+	0.851371i
$597$		−	17.0000i		−	0.695764i
$598$	10.3923	+	42.0000i	0.424973	+	1.71751i
$599$	16.0000			0.653742			0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000			0.653742			0.326871	+	0.945069i	$0.394006\pi$
$600$		0			0
$601$	−11.0000	−	19.0526i	−0.448699	−	0.777170i	0.549602	−	0.835426i	$-0.314779\pi$
$601$	−11.0000	−	19.0526i	−0.448699	−	0.777170i	−0.998302	+	0.0582563i	$0.981446\pi$
$602$	8.66025	+	5.00000i	0.352966	+	0.203785i
$603$		−	7.00000i		−	0.285062i
$604$	−8.00000	+	13.8564i	−0.325515	+	0.563809i
$605$		0			0
$606$	−36.0000			−1.46240
$607$	13.8564	+	8.00000i	0.562414	+	0.324710i	0.754114	−	0.656744i	$-0.228067\pi$
$607$	13.8564	+	8.00000i	0.562414	+	0.324710i	−0.191700	+	0.981454i	$0.561400\pi$
$608$		0			0
$609$	−10.0000	−	17.3205i	−0.405220	−	0.701862i
$610$		0			0
$611$	20.0000	+	20.7846i	0.809113	+	0.840855i
$612$		−	4.00000i		−	0.161690i
$613$	−12.9904	+	7.50000i	−0.524677	+	0.302922i	−0.738846	−	0.673874i	$-0.764629\pi$
$613$	−12.9904	+	7.50000i	−0.524677	+	0.302922i	0.214169	+	0.976797i	$0.431296\pi$
$614$	−31.0000	−	53.6936i	−1.25106	−	2.16690i
$615$		0			0
$616$		0			0
$617$	5.19615	+	3.00000i	0.209189	+	0.120775i	0.600935	−	0.799298i	$-0.294795\pi$
$617$	5.19615	+	3.00000i	0.209189	+	0.120775i	−0.391745	+	0.920074i	$0.628129\pi$
$618$	12.1244	+	7.00000i	0.487713	+	0.281581i
$619$	−37.0000			−1.48716			−0.743578	−	0.668649i	$-0.766873\pi$
$619$	−37.0000			−1.48716			−0.743578	+	0.668649i	$0.766873\pi$
$620$		0			0
$621$	−3.00000	−	5.19615i	−0.120386	−	0.208514i
$622$	38.1051	−	22.0000i	1.52788	−	0.882120i
$623$			70.0000i			2.80449i
$624$	14.0000	−	3.46410i	0.560449	−	0.138675i
$625$		0			0
$626$	31.0000	+	53.6936i	1.23901	+	2.14603i
$627$		0			0
$628$	25.9808	+	15.0000i	1.03675	+	0.598565i
$629$	4.00000			0.159490
$630$		0			0
$631$	−3.50000	+	6.06218i	−0.139333	+	0.241331i	−0.927244	−	0.374457i	$-0.877829\pi$
$631$	−3.50000	+	6.06218i	−0.139333	+	0.241331i	0.787911	+	0.615789i	$0.211162\pi$
$632$		0			0
$633$	−12.9904	−	7.50000i	−0.516321	−	0.298098i
$634$	12.0000	+	20.7846i	0.476581	+	0.825462i
$635$		0			0
$636$	−8.00000			−0.317221
$637$	62.3538	+	18.0000i	2.47055	+	0.713186i
$638$		−	16.0000i		−	0.633446i
$639$	6.00000	+	10.3923i	0.237356	+	0.411113i
$640$		0			0
$641$	−1.00000	+	1.73205i	−0.0394976	+	0.0684119i	−0.885098	−	0.465404i	$-0.845909\pi$
$641$	−1.00000	+	1.73205i	−0.0394976	+	0.0684119i	0.845601	+	0.533816i	$0.179242\pi$
$642$		−	8.00000i		−	0.315735i
$643$	16.4545	+	9.50000i	0.648901	+	0.374643i	0.788035	−	0.615630i	$-0.211098\pi$
$643$	16.4545	+	9.50000i	0.648901	+	0.374643i	−0.139134	+	0.990274i	$0.544432\pi$
$644$	30.0000	−	51.9615i	1.18217	−	2.04757i
$645$		0			0
$646$		0			0
$647$	−32.9090	+	19.0000i	−1.29378	+	0.746967i	−0.979323	−	0.202303i	$-0.935157\pi$
$647$	−32.9090	+	19.0000i	−1.29378	+	0.746967i	−0.314462	+	0.949270i	$0.601824\pi$
$648$		0			0
$649$	−24.0000			−0.942082
$650$		0			0
$651$	−35.0000			−1.37176
$652$	25.9808	−	15.0000i	1.01749	−	0.587445i
$653$	36.3731	−	21.0000i	1.42339	−	0.821794i	0.426801	−	0.904345i	$-0.359640\pi$
$653$	36.3731	−	21.0000i	1.42339	−	0.821794i	0.996587	+	0.0825519i	$0.0263070\pi$
$654$	−11.0000	+	19.0526i	−0.430134	+	0.745014i
$655$		0			0
$656$	12.0000	−	20.7846i	0.468521	−	0.811503i
$657$	12.9904	+	7.50000i	0.506803	+	0.292603i
$658$		−	80.0000i		−	3.11872i
$659$	−12.0000	+	20.7846i	−0.467454	+	0.809653i	−0.999309	−	0.0371821i	$-0.988162\pi$
$659$	−12.0000	+	20.7846i	−0.467454	+	0.809653i	0.531855	+	0.846836i	$0.321495\pi$
$660$		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	$-0.404978\pi$
$662$		−	18.0000i		−	0.699590i
$663$	5.19615	−	5.00000i	0.201802	−	0.194184i
$664$		0			0
$665$		0			0
$666$	2.00000	+	3.46410i	0.0774984	+	0.134231i
$667$	−20.7846	−	12.0000i	−0.804783	−	0.464642i
$668$			24.0000i			0.928588i
$669$	−4.00000	+	6.92820i	−0.154649	+	0.267860i
$670$		0			0
$671$	−26.0000			−1.00372
$672$	−34.6410	−	20.0000i	−1.33631	−	0.771517i
$673$	28.5788	−	16.5000i	1.10163	−	0.636028i	0.164984	−	0.986296i	$-0.447243\pi$
$673$	28.5788	−	16.5000i	1.10163	−	0.636028i	0.936650	+	0.350268i	$0.113909\pi$
$674$	−1.00000	−	1.73205i	−0.0385186	−	0.0667161i
$675$		0			0
$676$	−22.0000	−	13.8564i	−0.846154	−	0.532939i
$677$			12.0000i			0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$			12.0000i			0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	−3.46410	+	2.00000i	−0.133038	+	0.0768095i
$679$	−12.5000	−	21.6506i	−0.479706	−	0.830875i
$680$		0			0
$681$	10.0000			0.383201
$682$	−24.2487	−	14.0000i	−0.928531	−	0.536088i
$683$	17.3205	+	10.0000i	0.662751	+	0.382639i	0.793324	−	0.608799i	$-0.208349\pi$
$683$	17.3205	+	10.0000i	0.662751	+	0.382639i	−0.130573	+	0.991439i	$0.541682\pi$
$684$		0			0
$685$		0			0
$686$	−55.0000	−	95.2628i	−2.09991	−	3.63715i
$687$	−12.1244	+	7.00000i	−0.462573	+	0.267067i
$688$		−	4.00000i		−	0.152499i
$689$	−10.0000	−	10.3923i	−0.380970	−	0.395915i
$690$		0			0
$691$	−18.5000	−	32.0429i	−0.703773	−	1.21897i	−0.967132	−	0.254273i	$-0.918164\pi$
$691$	−18.5000	−	32.0429i	−0.703773	−	1.21897i	0.263359	−	0.964698i	$-0.415170\pi$
$692$		0			0
$693$	−8.66025	−	5.00000i	−0.328976	−	0.189934i
$694$	32.0000			1.21470
$695$		0			0
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$	−5.19615	−	3.00000i	−0.196677	−	0.113552i
$699$	−7.00000	−	12.1244i	−0.264764	−	0.458585i
$700$		0			0
$701$	40.0000			1.51078			0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000			1.51078			0.755390	+	0.655276i	$0.227448\pi$
$702$	6.92820	+	2.00000i	0.261488	+	0.0754851i
$703$		0			0
$704$	−8.00000	−	13.8564i	−0.301511	−	0.522233i
$705$		0			0
$706$	−6.00000	+	10.3923i	−0.225813	+	0.391120i
$707$			90.0000i			3.38480i
$708$	−20.7846	−	12.0000i	−0.781133	−	0.450988i
$709$	−11.5000	+	19.9186i	−0.431892	+	0.748058i	−0.997036	−	0.0769337i	$-0.975487\pi$
$709$	−11.5000	+	19.9186i	−0.431892	+	0.748058i	0.565145	+	0.824992i	$0.308820\pi$
$710$		0			0
$711$	−1.50000	+	2.59808i	−0.0562544	+	0.0974355i
$712$		0			0
$713$	−36.3731	+	21.0000i	−1.36218	+	0.786456i
$714$	−20.0000			−0.748481
$715$		0			0
$716$	−12.0000			−0.448461
$717$	−10.3923	+	6.00000i	−0.388108	+	0.224074i
$718$	3.46410	−	2.00000i	0.129279	−	0.0746393i
$719$	20.0000	−	34.6410i	0.745874	−	1.29189i	−0.203911	−	0.978989i	$-0.565365\pi$
$719$	20.0000	−	34.6410i	0.745874	−	1.29189i	0.949785	−	0.312903i	$-0.101301\pi$
$720$		0			0
$721$	17.5000	−	30.3109i	0.651734	−	1.12884i
$722$	−32.9090	−	19.0000i	−1.22474	−	0.707107i
$723$			10.0000i			0.371904i
$724$	−22.0000	+	38.1051i	−0.817624	+	1.41617i
$725$		0			0
$726$	7.00000	+	12.1244i	0.259794	+	0.449977i
$727$		−	9.00000i		−	0.333792i	−0.985975	−	0.166896i	$-0.946626\pi$
$727$		−	9.00000i		−	0.333792i	0.985975	−	0.166896i	$-0.0533743\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−1.00000	−	1.73205i	−0.0369863	−	0.0640622i
$732$	−22.5167	−	13.0000i	−0.832240	−	0.480494i
$733$			7.00000i			0.258551i	0.991609	+	0.129275i	$0.0412651\pi$
$733$			7.00000i			0.258551i	−0.991609	+	0.129275i	$0.958735\pi$
$734$	7.00000	−	12.1244i	0.258375	−	0.447518i
$735$		0			0
$736$	−48.0000			−1.76930
$737$	−12.1244	−	7.00000i	−0.446606	−	0.257848i
$738$	10.3923	−	6.00000i	0.382546	−	0.220863i
$739$	−18.0000	−	31.1769i	−0.662141	−	1.14686i	−0.980052	−	0.198741i	$-0.936315\pi$
$739$	−18.0000	−	31.1769i	−0.662141	−	1.14686i	0.317911	−	0.948120i	$-0.397019\pi$
$740$		0			0
$741$		0			0
$742$			40.0000i			1.46845i
$743$	20.7846	−	12.0000i	0.762513	−	0.440237i	−0.0676840	−	0.997707i	$-0.521561\pi$
$743$	20.7846	−	12.0000i	0.762513	−	0.440237i	0.830197	+	0.557470i	$0.188228\pi$
$744$		0			0
$745$		0			0
$746$	26.0000			0.951928
$747$	6.92820	+	4.00000i	0.253490	+	0.146352i
$748$	−6.92820	−	4.00000i	−0.253320	−	0.146254i
$749$	−20.0000			−0.730784
$750$		0			0
$751$	14.0000	+	24.2487i	0.510867	+	0.884848i	0.999921	+	0.0125942i	$0.00400897\pi$
$751$	14.0000	+	24.2487i	0.510867	+	0.884848i	−0.489053	+	0.872254i	$0.662658\pi$
$752$	−27.7128	+	16.0000i	−1.01058	+	0.583460i
$753$		0			0
$754$	28.0000	−	6.92820i	1.01970	−	0.252310i
$755$		0			0
$756$	−5.00000	−	8.66025i	−0.181848	−	0.314970i
$757$	−1.73205	+	1.00000i	−0.0629525	+	0.0363456i	−0.531146	−	0.847280i	$-0.678238\pi$
$757$	−1.73205	+	1.00000i	−0.0629525	+	0.0363456i	0.468193	+	0.883626i	$0.344905\pi$
$758$	8.66025	+	5.00000i	0.314555	+	0.181608i
$759$	−12.0000			−0.435572
$760$		0			0
$761$	18.0000	−	31.1769i	0.652499	−	1.13016i	−0.330015	−	0.943976i	$-0.607054\pi$
$761$	18.0000	−	31.1769i	0.652499	−	1.13016i	0.982514	−	0.186187i	$-0.0596129\pi$
$762$			22.0000i			0.796976i
$763$	47.6314	+	27.5000i	1.72437	+	0.995567i
$764$	12.0000	+	20.7846i	0.434145	+	0.751961i
$765$		0			0
$766$	−36.0000			−1.30073
$767$	−10.3923	−	42.0000i	−0.375244	−	1.51653i
$768$			16.0000i			0.577350i
$769$	−17.0000	−	29.4449i	−0.613036	−	1.06181i	−0.990726	−	0.135877i	$-0.956615\pi$
$769$	−17.0000	−	29.4449i	−0.613036	−	1.06181i	0.377690	−	0.925932i	$-0.376718\pi$
$770$		0			0
$771$	11.0000	−	19.0526i	0.396155	−	0.686161i
$772$			22.0000i			0.791797i
$773$	−39.8372	−	23.0000i	−1.43284	−	0.827253i	−0.435507	−	0.900186i	$-0.643431\pi$
$773$	−39.8372	−	23.0000i	−1.43284	−	0.827253i	−0.997337	+	0.0729331i	$0.976764\pi$
$774$	1.00000	−	1.73205i	0.0359443	−	0.0622573i
$775$		0			0
$776$		0			0
$777$	8.66025	−	5.00000i	0.310685	−	0.179374i
$778$	13.8564	−	8.00000i	0.496776	−	0.286814i
$779$		0			0
$780$		0			0
$781$	24.0000			0.858788
$782$	−20.7846	+	12.0000i	−0.743256	+	0.429119i
$783$	−3.46410	+	2.00000i	−0.123797	+	0.0714742i
$784$	−36.0000	+	62.3538i	−1.28571	+	2.22692i
$785$		0			0
$786$	4.00000	−	6.92820i	0.142675	−	0.247121i
$787$	14.7224	+	8.50000i	0.524798	+	0.302992i	0.738896	−	0.673820i	$-0.235348\pi$
$787$	14.7224	+	8.50000i	0.524798	+	0.302992i	−0.214097	+	0.976812i	$0.568681\pi$
$788$		−	24.0000i		−	0.854965i
$789$	−5.00000	+	8.66025i	−0.178005	+	0.308313i
$790$		0			0
$791$	5.00000	+	8.66025i	0.177780	+	0.307923i
$792$		0			0
$793$	−11.2583	−	45.5000i	−0.399795	−	1.61575i
$794$	30.0000			1.06466
$795$		0			0
$796$	−17.0000	−	29.4449i	−0.602549	−	1.04365i
$797$	25.9808	+	15.0000i	0.920286	+	0.531327i	0.883726	−	0.468004i	$-0.155027\pi$
$797$	25.9808	+	15.0000i	0.920286	+	0.531327i	0.0365596	+	0.999331i	$0.488360\pi$
$798$		0			0
$799$	−8.00000	+	13.8564i	−0.283020	+	0.490204i
$800$		0			0
$801$	14.0000			0.494666
$802$	27.7128	+	16.0000i	0.978573	+	0.564980i
$803$	25.9808	−	15.0000i	0.916841	−	0.529339i
$804$	−7.00000	−	12.1244i	−0.246871	−	0.427593i
$805$		0			0
$806$	14.0000	−	48.4974i	0.493129	−	1.70825i
$807$			6.00000i			0.211210i
$808$		0			0
$809$	2.00000	+	3.46410i	0.0703163	+	0.121791i	0.899040	−	0.437867i	$-0.144266\pi$
$809$	2.00000	+	3.46410i	0.0703163	+	0.121791i	−0.828724	+	0.559658i	$0.810932\pi$
$810$		0			0
$811$	45.0000			1.58016			0.790082	−	0.613001i	$-0.210038\pi$
$811$	45.0000			1.58016			0.790082	+	0.613001i	$0.210038\pi$
$812$	−34.6410	−	20.0000i	−1.21566	−	0.701862i
$813$	−25.1147	−	14.5000i	−0.880812	−	0.508537i
$814$	8.00000			0.280400
$815$		0			0
$816$	4.00000	+	6.92820i	0.140028	+	0.242536i
$817$		0			0
$818$		−	30.0000i		−	1.04893i
$819$	5.00000	−	17.3205i	0.174714	−	0.605228i
$820$		0			0
$821$	11.0000	+	19.0526i	0.383903	+	0.664939i	0.991616	−	0.129217i	$-0.0412465\pi$
$821$	11.0000	+	19.0526i	0.383903	+	0.664939i	−0.607714	+	0.794156i	$0.707913\pi$
$822$	3.46410	−	2.00000i	0.120824	−	0.0697580i
$823$	−17.3205	−	10.0000i	−0.603755	−	0.348578i	0.166762	−	0.985997i	$-0.446669\pi$
$823$	−17.3205	−	10.0000i	−0.603755	−	0.348578i	−0.770517	+	0.637419i	$0.780002\pi$
$824$		0			0
$825$		0			0
$826$	−60.0000	+	103.923i	−2.08767	+	3.61595i
$827$		−	46.0000i		−	1.59958i	−0.600282	−	0.799788i	$-0.704945\pi$
$827$		−	46.0000i		−	1.59958i	0.600282	−	0.799788i	$-0.295055\pi$
$828$	−10.3923	−	6.00000i	−0.361158	−	0.208514i
$829$	−5.50000	−	9.52628i	−0.191023	−	0.330861i	0.754567	−	0.656223i	$-0.227847\pi$
$829$	−5.50000	−	9.52628i	−0.191023	−	0.330861i	−0.945589	+	0.325362i	$0.894514\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$	20.7846	−	20.0000i	0.720577	−	0.693375i
$833$			36.0000i			1.24733i
$834$	−3.00000	−	5.19615i	−0.103882	−	0.179928i
$835$		0			0
$836$		0			0
$837$			7.00000i			0.241955i
$838$	−65.8179	−	38.0000i	−2.27364	−	1.31269i
$839$	−17.0000	+	29.4449i	−0.586905	+	1.01655i	0.407730	+	0.913103i	$0.366321\pi$
$839$	−17.0000	+	29.4449i	−0.586905	+	1.01655i	−0.994635	+	0.103447i	$0.967013\pi$
$840$		0			0
$841$	6.50000	−	11.2583i	0.224138	−	0.388218i
$842$	39.8372	−	23.0000i	1.37288	−	0.792632i
$843$	−10.3923	+	6.00000i	−0.357930	+	0.206651i
$844$	−30.0000			−1.03264
$845$		0			0
$846$	−16.0000			−0.550091
$847$	30.3109	−	17.5000i	1.04149	−	0.601307i
$848$	13.8564	−	8.00000i	0.475831	−	0.274721i
$849$	2.50000	−	4.33013i	0.0857998	−	0.148610i
$850$		0			0
$851$	6.00000	−	10.3923i	0.205677	−	0.356244i
$852$	20.7846	+	12.0000i	0.712069	+	0.411113i
$853$			9.00000i			0.308154i	0.988059	+	0.154077i	$0.0492404\pi$
$853$			9.00000i			0.308154i	−0.988059	+	0.154077i	$0.950760\pi$
$854$	−65.0000	+	112.583i	−2.22425	+	3.85252i
$855$		0			0
$856$		0			0
$857$			12.0000i			0.409912i	0.978771	+	0.204956i	$0.0657052\pi$
$857$			12.0000i			0.409912i	−0.978771	+	0.204956i	$0.934295\pi$
$858$	10.3923	−	10.0000i	0.354787	−	0.341394i
$859$	−43.0000			−1.46714			−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−43.0000			−1.46714			−0.733571	+	0.679613i	$0.762148\pi$
$860$		0			0
$861$	−15.0000	−	25.9808i	−0.511199	−	0.885422i
$862$	−48.4974	−	28.0000i	−1.65183	−	0.953684i
$863$			6.00000i			0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$			6.00000i			0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	−4.00000	+	6.92820i	−0.136083	+	0.235702i
$865$		0			0
$866$	2.00000			0.0679628
$867$	−11.2583	−	6.50000i	−0.382353	−	0.220752i
$868$	−60.6218	+	35.0000i	−2.05764	+	1.18798i
$869$	3.00000	+	5.19615i	0.101768	+	0.176267i
$870$		0			0
$871$	7.00000	−	24.2487i	0.237186	−	0.821636i
$872$		0			0
$873$	−4.33013	+	2.50000i	−0.146553	+	0.0846122i
$874$		0			0
$875$		0			0
$876$	30.0000			1.01361
$877$	5.19615	+	3.00000i	0.175462	+	0.101303i	0.585159	−	0.810919i	$-0.301032\pi$
$877$	5.19615	+	3.00000i	0.175462	+	0.101303i	−0.409697	+	0.912222i	$0.634366\pi$
$878$	−25.9808	−	15.0000i	−0.876808	−	0.506225i
$879$	−16.0000			−0.539667
$880$		0			0
$881$	−10.0000	−	17.3205i	−0.336909	−	0.583543i	0.646941	−	0.762540i	$-0.276048\pi$
$881$	−10.0000	−	17.3205i	−0.336909	−	0.583543i	−0.983850	+	0.178997i	$0.942715\pi$
$882$	−31.1769	+	18.0000i	−1.04978	+	0.606092i
$883$		−	25.0000i		−	0.841317i	−0.907219	−	0.420658i	$-0.861799\pi$
$883$		−	25.0000i		−	0.841317i	0.907219	−	0.420658i	$-0.138201\pi$
$884$	4.00000	−	13.8564i	0.134535	−	0.466041i
$885$		0			0
$886$	−26.0000	−	45.0333i	−0.873487	−	1.51292i
$887$	38.1051	−	22.0000i	1.27944	−	0.738688i	0.302698	−	0.953086i	$-0.402113\pi$
$887$	38.1051	−	22.0000i	1.27944	−	0.738688i	0.976746	+	0.214399i	$0.0687792\pi$
$888$		0			0
$889$	55.0000			1.84464
$890$		0			0
$891$	−1.00000	+	1.73205i	−0.0335013	+	0.0580259i
$892$			16.0000i			0.535720i
$893$		0			0
$894$	−12.0000	−	20.7846i	−0.401340	−	0.695141i
$895$		0			0
$896$		0			0
$897$	−5.19615	−	21.0000i	−0.173494	−	0.701170i
$898$			36.0000i			1.20134i
$899$	14.0000	+	24.2487i	0.466926	+	0.808740i
$900$		0			0
$901$	4.00000	−	6.92820i	0.133259	−	0.230812i
$902$		−	24.0000i		−	0.799113i
$903$	−4.33013	−	2.50000i	−0.144098	−	0.0831948i
$904$		0			0
$905$		0			0
$906$	8.00000	−	13.8564i	0.265782	−	0.460348i
$907$	−17.3205	+	10.0000i	−0.575118	+	0.332045i	−0.759191	−	0.650868i	$-0.774405\pi$
$907$	−17.3205	+	10.0000i	−0.575118	+	0.332045i	0.184073	+	0.982913i	$0.441072\pi$
$908$	17.3205	−	10.0000i	0.574801	−	0.331862i
$909$	18.0000			0.597022
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	13.8564	−	8.00000i	0.458580	−	0.264761i
$914$	35.0000	−	60.6218i	1.15770	−	2.00519i
$915$		0			0
$916$	−14.0000	+	24.2487i	−0.462573	+	0.801200i
$917$	−17.3205	−	10.0000i	−0.571974	−	0.330229i
$918$			4.00000i			0.132020i
$919$	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
$921$	15.5000	+	26.8468i	0.510742	+	0.884632i
$922$			4.00000i			0.131733i
$923$	10.3923	+	42.0000i	0.342067	+	1.38245i
$924$	−20.0000			−0.657952
$925$		0			0
$926$	−3.00000	−	5.19615i	−0.0985861	−	0.170756i
$927$	−6.06218	−	3.50000i	−0.199108	−	0.114955i
$928$			32.0000i			1.05045i
$929$	26.0000	−	45.0333i	0.853032	−	1.47750i	−0.0254262	−	0.999677i	$-0.508094\pi$
$929$	26.0000	−	45.0333i	0.853032	−	1.47750i	0.878459	−	0.477819i	$-0.158572\pi$
$930$		0			0
$931$		0			0
$932$	−24.2487	−	14.0000i	−0.794293	−	0.458585i
$933$	−19.0526	+	11.0000i	−0.623753	+	0.360124i
$934$	−4.00000	−	6.92820i	−0.130884	−	0.226698i
$935$		0			0
$936$		0			0
$937$			30.0000i			0.980057i	0.871706	+	0.490029i	$0.163014\pi$
$937$			30.0000i			0.980057i	−0.871706	+	0.490029i	$0.836986\pi$
$938$	−60.6218	+	35.0000i	−1.97937	+	1.14279i
$939$	−15.5000	−	26.8468i	−0.505823	−	0.876112i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$	−25.9808	−	15.0000i	−0.846499	−	0.488726i
$943$	−31.1769	−	18.0000i	−1.01526	−	0.586161i
$944$	48.0000			1.56227
$945$		0			0
$946$	−2.00000	−	3.46410i	−0.0650256	−	0.112628i
$947$	−15.5885	+	9.00000i	−0.506557	+	0.292461i	−0.731417	−	0.681930i	$-0.761141\pi$
$947$	−15.5885	+	9.00000i	−0.506557	+	0.292461i	0.224860	+	0.974391i	$0.427807\pi$
$948$			6.00000i			0.194871i
$949$	37.5000	+	38.9711i	1.21730	+	1.26506i
$950$		0			0
$951$	−6.00000	−	10.3923i	−0.194563	−	0.336994i
$952$		0			0
$953$	−5.19615	−	3.00000i	−0.168320	−	0.0971795i	0.413473	−	0.910516i	$-0.364315\pi$
$953$	−5.19615	−	3.00000i	−0.168320	−	0.0971795i	−0.581793	+	0.813337i	$0.697649\pi$
$954$	8.00000			0.259010
$955$		0			0
$956$	−12.0000	+	20.7846i	−0.388108	+	0.672222i
$957$			8.00000i			0.258603i
$958$	−72.7461	−	42.0000i	−2.35032	−	1.35696i
$959$	−5.00000	−	8.66025i	−0.161458	−	0.279654i
$960$		0			0
$961$	18.0000			0.580645
$962$	3.46410	+	14.0000i	0.111687	+	0.451378i
$963$			4.00000i			0.128898i
$964$	10.0000	+	17.3205i	0.322078	+	0.557856i
$965$		0			0
$966$	−30.0000	+	51.9615i	−0.965234	+	1.67183i
$967$			56.0000i			1.80084i	0.435023	+	0.900419i	$0.356740\pi$
$967$			56.0000i			1.80084i	−0.435023	+	0.900419i	$0.643260\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	−0.980677	−	0.195633i	$-0.937324\pi$
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	0.659762	+	0.751475i	$0.270657\pi$
$972$	−1.73205	+	1.00000i	−0.0555556	+	0.0320750i
$973$	−12.9904	+	7.50000i	−0.416452	+	0.240439i
$974$	56.0000			1.79436
$975$		0			0
$976$	52.0000			1.66448
$977$	−51.9615	+	30.0000i	−1.66240	+	0.959785i	−0.690830	+	0.723017i	$0.742755\pi$
$977$	−51.9615	+	30.0000i	−1.66240	+	0.959785i	−0.971566	+	0.236768i	$0.923912\pi$
$978$	−25.9808	+	15.0000i	−0.830773	+	0.479647i
$979$	14.0000	−	24.2487i	0.447442	−	0.774992i
$980$		0			0
$981$	5.50000	−	9.52628i	0.175601	−	0.304151i
$982$	−41.5692	−	24.0000i	−1.32653	−	0.765871i
$983$			38.0000i			1.21201i	0.795460	+	0.606006i	$0.207229\pi$
$983$			38.0000i			1.21201i	−0.795460	+	0.606006i	$0.792771\pi$
$984$		0			0
$985$		0			0
$986$	8.00000	+	13.8564i	0.254772	+	0.441278i
$987$			40.0000i			1.27321i
$988$		0			0
$989$	−6.00000			−0.190789
$990$		0			0
$991$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$991$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$992$	48.4974	+	28.0000i	1.53979	+	0.889001i
$993$			9.00000i			0.285606i
$994$	60.0000	−	103.923i	1.90308	−	3.29624i
$995$		0			0
$996$	16.0000			0.506979
$997$	25.1147	+	14.5000i	0.795392	+	0.459220i	0.841857	−	0.539700i	$-0.181462\pi$
$997$	25.1147	+	14.5000i	0.795392	+	0.459220i	−0.0464655	+	0.998920i	$0.514796\pi$
$998$	6.92820	−	4.00000i	0.219308	−	0.126618i
$999$	−1.00000	−	1.73205i	−0.0316386	−	0.0547997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.bb.f.874.1		4
5.2	odd	4		195.2.i.a.16.1	✓	2
5.3	odd	4		975.2.i.i.601.1		2
5.4	even	2	inner	975.2.bb.f.874.2		4
13.9	even	3	inner	975.2.bb.f.724.2		4
15.2	even	4		585.2.j.b.406.1		2
65.9	even	6	inner	975.2.bb.f.724.1		4
65.22	odd	12		195.2.i.a.61.1	yes	2
65.42	odd	12		2535.2.a.m.1.1		1
65.48	odd	12		975.2.i.i.451.1		2
65.62	odd	12		2535.2.a.c.1.1		1
195.62	even	12		7605.2.a.s.1.1		1
195.107	even	12		7605.2.a.a.1.1		1
195.152	even	12		585.2.j.b.451.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.i.a.16.1	✓	2	5.2	odd	4
195.2.i.a.61.1	yes	2	65.22	odd	12
585.2.j.b.406.1		2	15.2	even	4
585.2.j.b.451.1		2	195.152	even	12
975.2.i.i.451.1		2	65.48	odd	12
975.2.i.i.601.1		2	5.3	odd	4
975.2.bb.f.724.1		4	65.9	even	6	inner
975.2.bb.f.724.2		4	13.9	even	3	inner
975.2.bb.f.874.1		4	1.1	even	1	trivial
975.2.bb.f.874.2		4	5.4	even	2	inner
2535.2.a.c.1.1		1	65.62	odd	12
2535.2.a.m.1.1		1	65.42	odd	12
7605.2.a.a.1.1		1	195.107	even	12
7605.2.a.s.1.1		1	195.62	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.bb (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(0.866025 - 0.500000i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.00000 + 1.73205i) q^{6} +(4.33013 + 2.50000i) q^{7} +(0.500000 - 0.866025i) q^{9} +(-1.00000 - 1.73205i) q^{11} -2.00000i q^{12} +(2.59808 - 2.50000i) q^{13} -10.0000 q^{14} +(2.00000 + 3.46410i) q^{16} +(1.73205 + 1.00000i) q^{17} +2.00000i q^{18} +5.00000 q^{21} +(3.46410 + 2.00000i) q^{22} +(5.19615 - 3.00000i) q^{23} +(-2.00000 + 6.92820i) q^{26} -1.00000i q^{27} +(8.66025 - 5.00000i) q^{28} +(-2.00000 - 3.46410i) q^{29} -7.00000 q^{31} +(-6.92820 - 4.00000i) q^{32} +(-1.73205 - 1.00000i) q^{33} -4.00000 q^{34} +(-1.00000 - 1.73205i) q^{36} +(1.73205 - 1.00000i) q^{37} +(1.00000 - 3.46410i) q^{39} +(-3.00000 - 5.19615i) q^{41} +(-8.66025 + 5.00000i) q^{42} +(-0.866025 - 0.500000i) q^{43} -4.00000 q^{44} +(-6.00000 + 10.3923i) q^{46} +8.00000i q^{47} +(3.46410 + 2.00000i) q^{48} +(9.00000 + 15.5885i) q^{49} +2.00000 q^{51} +(-1.73205 - 7.00000i) q^{52} -4.00000i q^{53} +(1.00000 + 1.73205i) q^{54} +(6.92820 + 4.00000i) q^{58} +(6.00000 - 10.3923i) q^{59} +(6.50000 - 11.2583i) q^{61} +(12.1244 - 7.00000i) q^{62} +(4.33013 - 2.50000i) q^{63} +8.00000 q^{64} +4.00000 q^{66} +(6.06218 - 3.50000i) q^{67} +(3.46410 - 2.00000i) q^{68} +(3.00000 - 5.19615i) q^{69} +(-6.00000 + 10.3923i) q^{71} +15.0000i q^{73} +(-2.00000 + 3.46410i) q^{74} -10.0000i q^{77} +(1.73205 + 7.00000i) q^{78} -3.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} +(10.3923 + 6.00000i) q^{82} +8.00000i q^{83} +(5.00000 - 8.66025i) q^{84} +2.00000 q^{86} +(-3.46410 - 2.00000i) q^{87} +(7.00000 + 12.1244i) q^{89} +(17.5000 - 4.33013i) q^{91} -12.0000i q^{92} +(-6.06218 + 3.50000i) q^{93} +(-8.00000 - 13.8564i) q^{94} -8.00000 q^{96} +(-4.33013 - 2.50000i) q^{97} +(-31.1769 - 18.0000i) q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{6} + 2 q^{9} - 4 q^{11} - 40 q^{14} + 8 q^{16} + 20 q^{21} - 8 q^{26} - 8 q^{29} - 28 q^{31} - 16 q^{34} - 4 q^{36} + 4 q^{39} - 12 q^{41} - 16 q^{44} - 24 q^{46} + 36 q^{49} + 8 q^{51}+ \cdots - 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^6 + 2 * q^9 - 4 * q^11 - 40 * q^14 + 8 * q^16 + 20 * q^21 - 8 * q^26 - 8 * q^29 - 28 * q^31 - 16 * q^34 - 4 * q^36 + 4 * q^39 - 12 * q^41 - 16 * q^44 - 24 * q^46 + 36 * q^49 + 8 * q^51 + 4 * q^54 + 24 * q^59 + 26 * q^61 + 32 * q^64 + 16 * q^66 + 12 * q^69 - 24 * q^71 - 8 * q^74 - 12 * q^79 - 2 * q^81 + 20 * q^84 + 8 * q^86 + 28 * q^89 + 70 * q^91 - 32 * q^94 - 32 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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