LMFDB - Embedded newform 975.2.i.i.451.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,2,Mod(451,975)] 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("975.451"); 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, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.i (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,1,-2,0,-2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			451.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	975.451
Dual form			975.2.i.i.601.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.73205i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-1.00000 - 1.73205i) q^{6} +(2.50000 + 4.33013i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} -2.00000 q^{12} +(2.50000 - 2.59808i) q^{13} +10.0000 q^{14} +(2.00000 - 3.46410i) q^{16} +(1.00000 + 1.73205i) q^{17} -2.00000 q^{18} +5.00000 q^{21} +(2.00000 + 3.46410i) q^{22} +(3.00000 - 5.19615i) q^{23} +(-2.00000 - 6.92820i) q^{26} -1.00000 q^{27} +(5.00000 - 8.66025i) q^{28} +(2.00000 - 3.46410i) q^{29} -7.00000 q^{31} +(-4.00000 - 6.92820i) q^{32} +(1.00000 + 1.73205i) q^{33} +4.00000 q^{34} +(-1.00000 + 1.73205i) q^{36} +(-1.00000 + 1.73205i) q^{37} +(-1.00000 - 3.46410i) q^{39} +(-3.00000 + 5.19615i) q^{41} +(5.00000 - 8.66025i) q^{42} +(0.500000 + 0.866025i) q^{43} +4.00000 q^{44} +(-6.00000 - 10.3923i) q^{46} +8.00000 q^{47} +(-2.00000 - 3.46410i) q^{48} +(-9.00000 + 15.5885i) q^{49} +2.00000 q^{51} +(-7.00000 - 1.73205i) q^{52} +4.00000 q^{53} +(-1.00000 + 1.73205i) q^{54} +(-4.00000 - 6.92820i) q^{58} +(-6.00000 - 10.3923i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-7.00000 + 12.1244i) q^{62} +(2.50000 - 4.33013i) q^{63} -8.00000 q^{64} +4.00000 q^{66} +(-3.50000 + 6.06218i) q^{67} +(2.00000 - 3.46410i) q^{68} +(-3.00000 - 5.19615i) q^{69} +(-6.00000 - 10.3923i) q^{71} -15.0000 q^{73} +(2.00000 + 3.46410i) q^{74} -10.0000 q^{77} +(-7.00000 - 1.73205i) q^{78} +3.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} +(6.00000 + 10.3923i) q^{82} -8.00000 q^{83} +(-5.00000 - 8.66025i) q^{84} +2.00000 q^{86} +(-2.00000 - 3.46410i) q^{87} +(-7.00000 + 12.1244i) q^{89} +(17.5000 + 4.33013i) q^{91} -12.0000 q^{92} +(-3.50000 + 6.06218i) q^{93} +(8.00000 - 13.8564i) q^{94} -8.00000 q^{96} +(-2.50000 - 4.33013i) q^{97} +(18.0000 + 31.1769i) q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + q^{3} - 2 q^{4} - 2 q^{6} + 5 q^{7} - q^{9} - 2 q^{11} - 4 q^{12} + 5 q^{13} + 20 q^{14} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 10 q^{21} + 4 q^{22} + 6 q^{23} - 4 q^{26} - 2 q^{27} + 10 q^{28}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + q^3 - 2 * q^4 - 2 * q^6 + 5 * q^7 - q^9 - 2 * q^11 - 4 * q^12 + 5 * q^13 + 20 * q^14 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 10 * q^21 + 4 * q^22 + 6 * q^23 - 4 * q^26 - 2 * q^27 + 10 * q^28 + 4 * q^29 - 14 * q^31 - 8 * q^32 + 2 * q^33 + 8 * q^34 - 2 * q^36 - 2 * q^37 - 2 * q^39 - 6 * q^41 + 10 * q^42 + q^43 + 8 * q^44 - 12 * q^46 + 16 * q^47 - 4 * q^48 - 18 * q^49 + 4 * q^51 - 14 * q^52 + 8 * q^53 - 2 * q^54 - 8 * q^58 - 12 * q^59 + 13 * q^61 - 14 * q^62 + 5 * q^63 - 16 * q^64 + 8 * q^66 - 7 * q^67 + 4 * q^68 - 6 * q^69 - 12 * q^71 - 30 * q^73 + 4 * q^74 - 20 * q^77 - 14 * q^78 + 6 * q^79 - q^81 + 12 * q^82 - 16 * q^83 - 10 * q^84 + 4 * q^86 - 4 * q^87 - 14 * q^89 + 35 * q^91 - 24 * q^92 - 7 * q^93 + 16 * q^94 - 16 * q^96 - 5 * q^97 + 36 * q^98 + 4 * q^99

Character values

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

$n$	$301$	$326$	$352$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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 $

$2$	1.00000	−	1.73205i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	1.00000	−	1.73205i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$	−1.00000	−	1.73205i	−0.408248	−	0.707107i
$7$	2.50000	+	4.33013i	0.944911	+	1.63663i	0.755929	+	0.654654i	$0.227186\pi$
$7$	2.50000	+	4.33013i	0.944911	+	1.63663i	0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$	−2.00000			−0.577350
$13$	2.50000	−	2.59808i	0.693375	−	0.720577i
$13$	2.50000	−	2.59808i	0.693375	−	0.720577i
$14$	10.0000			2.67261
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	0.961436	−	0.275029i	$-0.0886875\pi$
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	−0.718900	+	0.695113i	$0.755354\pi$
$18$	−2.00000			−0.471405
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$	5.00000			1.09109
$22$	2.00000	+	3.46410i	0.426401	+	0.738549i
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$		0			0
$26$	−2.00000	−	6.92820i	−0.392232	−	1.35873i
$27$	−1.00000			−0.192450
$28$	5.00000	−	8.66025i	0.944911	−	1.63663i
$29$	2.00000	−	3.46410i	0.371391	−	0.643268i	−0.618389	−	0.785872i	$-0.712214\pi$
$29$	2.00000	−	3.46410i	0.371391	−	0.643268i	0.989780	+	0.142605i	$0.0455477\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$	−4.00000	−	6.92820i	−0.707107	−	1.22474i
$33$	1.00000	+	1.73205i	0.174078	+	0.301511i
$34$	4.00000			0.685994
$35$		0			0
$36$	−1.00000	+	1.73205i	−0.166667	+	0.288675i
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	−0.936442	−	0.350823i	$-0.885902\pi$
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	0.772043	+	0.635571i	$0.219235\pi$
$38$		0			0
$39$	−1.00000	−	3.46410i	−0.160128	−	0.554700i
$40$		0			0
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	0.530831	+	0.847477i	$0.321880\pi$
$42$	5.00000	−	8.66025i	0.771517	−	1.33631i
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$	4.00000			0.603023
$45$		0			0
$46$	−6.00000	−	10.3923i	−0.884652	−	1.53226i
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$	−2.00000	−	3.46410i	−0.288675	−	0.500000i
$49$	−9.00000	+	15.5885i	−1.28571	+	2.22692i
$50$		0			0
$51$	2.00000			0.280056
$52$	−7.00000	−	1.73205i	−0.970725	−	0.240192i
$53$	4.00000			0.549442			0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000			0.549442			0.274721	+	0.961524i	$0.411414\pi$
$54$	−1.00000	+	1.73205i	−0.136083	+	0.235702i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−4.00000	−	6.92820i	−0.525226	−	0.909718i
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	−0.931282	−	0.364299i	$-0.881308\pi$
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	0.150148	−	0.988663i	$-0.452025\pi$
$60$		0			0
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−7.00000	+	12.1244i	−0.889001	+	1.53979i
$63$	2.50000	−	4.33013i	0.314970	−	0.545545i
$64$	−8.00000			−1.00000
$65$		0			0
$66$	4.00000			0.492366
$67$	−3.50000	+	6.06218i	−0.427593	+	0.740613i	−0.996659	−	0.0816792i	$-0.973972\pi$
$67$	−3.50000	+	6.06218i	−0.427593	+	0.740613i	0.569066	+	0.822292i	$0.307305\pi$
$68$	2.00000	−	3.46410i	0.242536	−	0.420084i
$69$	−3.00000	−	5.19615i	−0.361158	−	0.625543i
$70$		0			0
$71$	−6.00000	−	10.3923i	−0.712069	−	1.23334i	−0.964079	−	0.265615i	$-0.914425\pi$
$71$	−6.00000	−	10.3923i	−0.712069	−	1.23334i	0.252010	−	0.967725i	$-0.418908\pi$
$72$		0			0
$73$	−15.0000			−1.75562			−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000			−1.75562			−0.877809	+	0.479012i	$0.840995\pi$
$74$	2.00000	+	3.46410i	0.232495	+	0.402694i
$75$		0			0
$76$		0			0
$77$	−10.0000			−1.13961
$78$	−7.00000	−	1.73205i	−0.792594	−	0.196116i
$79$	3.00000			0.337526			0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000			0.337526			0.168763	+	0.985657i	$0.446023\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$	6.00000	+	10.3923i	0.662589	+	1.14764i
$83$	−8.00000			−0.878114			−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000			−0.878114			−0.439057	+	0.898459i	$0.644687\pi$
$84$	−5.00000	−	8.66025i	−0.545545	−	0.944911i
$85$		0			0
$86$	2.00000			0.215666
$87$	−2.00000	−	3.46410i	−0.214423	−	0.371391i
$88$		0			0
$89$	−7.00000	+	12.1244i	−0.741999	+	1.28518i	0.209585	+	0.977790i	$0.432789\pi$
$89$	−7.00000	+	12.1244i	−0.741999	+	1.28518i	−0.951584	+	0.307389i	$0.900545\pi$
$90$		0			0
$91$	17.5000	+	4.33013i	1.83450	+	0.453921i
$92$	−12.0000			−1.25109
$93$	−3.50000	+	6.06218i	−0.362933	+	0.628619i
$94$	8.00000	−	13.8564i	0.825137	−	1.42918i
$95$		0			0
$96$	−8.00000			−0.816497
$97$	−2.50000	−	4.33013i	−0.253837	−	0.439658i	0.710742	−	0.703452i	$-0.248359\pi$
$97$	−2.50000	−	4.33013i	−0.253837	−	0.439658i	−0.964579	+	0.263795i	$0.915026\pi$
$98$	18.0000	+	31.1769i	1.81827	+	3.14934i
$99$	2.00000			0.201008
$100$		0			0
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.0623905	−	0.998052i	$-0.480128\pi$
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.833143	−	0.553058i	$-0.186539\pi$
$102$	2.00000	−	3.46410i	0.198030	−	0.342997i
$103$	7.00000			0.689730			0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000			0.689730			0.344865	+	0.938652i	$0.387925\pi$
$104$		0			0
$105$		0			0
$106$	4.00000	−	6.92820i	0.388514	−	0.672927i
$107$	2.00000	−	3.46410i	0.193347	−	0.334887i	−0.753010	−	0.658009i	$-0.771399\pi$
$107$	2.00000	−	3.46410i	0.193347	−	0.334887i	0.946357	+	0.323122i	$0.104732\pi$
$108$	1.00000	+	1.73205i	0.0962250	+	0.166667i
$109$	−11.0000			−1.05361			−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000			−1.05361			−0.526804	+	0.849987i	$0.676610\pi$
$110$		0			0
$111$	1.00000	+	1.73205i	0.0949158	+	0.164399i
$112$	20.0000			1.88982
$113$	−1.00000	−	1.73205i	−0.0940721	−	0.162938i	0.815149	−	0.579252i	$-0.196655\pi$
$113$	−1.00000	−	1.73205i	−0.0940721	−	0.162938i	−0.909221	+	0.416314i	$0.863322\pi$
$114$		0			0
$115$		0			0
$116$	−8.00000			−0.742781
$117$	−3.50000	−	0.866025i	−0.323575	−	0.0800641i
$118$	−24.0000			−2.20938
$119$	−5.00000	+	8.66025i	−0.458349	+	0.793884i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$	26.0000			2.35393
$123$	3.00000	+	5.19615i	0.270501	+	0.468521i
$124$	7.00000	+	12.1244i	0.628619	+	1.08880i
$125$		0			0
$126$	−5.00000	−	8.66025i	−0.445435	−	0.771517i
$127$	−5.50000	+	9.52628i	−0.488046	+	0.845321i	−0.999905	−	0.0137486i	$-0.995624\pi$
$127$	−5.50000	+	9.52628i	−0.488046	+	0.845321i	0.511859	+	0.859069i	$0.328957\pi$
$128$		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	$0.555909\pi$
$132$	2.00000	−	3.46410i	0.174078	−	0.301511i
$133$		0			0
$134$	7.00000	+	12.1244i	0.604708	+	1.04738i
$135$		0			0
$136$		0			0
$137$	−1.00000	−	1.73205i	−0.0854358	−	0.147979i	0.820141	−	0.572161i	$-0.193895\pi$
$137$	−1.00000	−	1.73205i	−0.0854358	−	0.147979i	−0.905577	+	0.424182i	$0.860562\pi$
$138$	−12.0000			−1.02151
$139$	1.50000	+	2.59808i	0.127228	+	0.220366i	0.922602	−	0.385754i	$-0.126059\pi$
$139$	1.50000	+	2.59808i	0.127228	+	0.220366i	−0.795373	+	0.606120i	$0.792725\pi$
$140$		0			0
$141$	4.00000	−	6.92820i	0.336861	−	0.583460i
$142$	−24.0000			−2.01404
$143$	2.00000	+	6.92820i	0.167248	+	0.579365i
$144$	−4.00000			−0.333333
$145$		0			0
$146$	−15.0000	+	25.9808i	−1.24141	+	2.15018i
$147$	9.00000	+	15.5885i	0.742307	+	1.28571i
$148$	4.00000			0.328798
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	0.999953	−	0.00974235i	$-0.00310113\pi$
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	−0.508413	+	0.861113i	$0.669768\pi$
$150$		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
$153$	1.00000	−	1.73205i	0.0808452	−	0.140028i
$154$	−10.0000	+	17.3205i	−0.805823	+	1.39573i
$155$		0			0
$156$	−5.00000	+	5.19615i	−0.400320	+	0.416025i
$157$	15.0000			1.19713			0.598565	−	0.801074i	$-0.295738\pi$
$157$	15.0000			1.19713			0.598565	+	0.801074i	$0.295738\pi$
$158$	3.00000	−	5.19615i	0.238667	−	0.413384i
$159$	2.00000	−	3.46410i	0.158610	−	0.274721i
$160$		0			0
$161$	30.0000			2.36433
$162$	1.00000	+	1.73205i	0.0785674	+	0.136083i
$163$	−7.50000	−	12.9904i	−0.587445	−	1.01749i	−0.994566	−	0.104111i	$-0.966800\pi$
$163$	−7.50000	−	12.9904i	−0.587445	−	1.01749i	0.407120	−	0.913375i	$-0.366533\pi$
$164$	12.0000			0.937043
$165$		0			0
$166$	−8.00000	+	13.8564i	−0.620920	+	1.07547i
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	−0.500000	−	12.9904i	−0.0384615	−	0.999260i
$170$		0			0
$171$		0			0
$172$	1.00000	−	1.73205i	0.0762493	−	0.132068i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$	−8.00000			−0.606478
$175$		0			0
$176$	4.00000	+	6.92820i	0.301511	+	0.522233i
$177$	−12.0000			−0.901975
$178$	14.0000	+	24.2487i	1.04934	+	1.81752i
$179$	3.00000	−	5.19615i	0.224231	−	0.388379i	−0.731858	−	0.681457i	$-0.761346\pi$
$179$	3.00000	−	5.19615i	0.224231	−	0.388379i	0.956088	+	0.293079i	$0.0946798\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$	25.0000	−	25.9808i	1.85312	−	1.92582i
$183$	13.0000			0.960988
$184$		0			0
$185$		0			0
$186$	7.00000	+	12.1244i	0.513265	+	0.889001i
$187$	−4.00000			−0.292509
$188$	−8.00000	−	13.8564i	−0.583460	−	1.01058i
$189$	−2.50000	−	4.33013i	−0.181848	−	0.314970i
$190$		0			0
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	0.563081	−	0.826402i	$-0.309616\pi$
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	−0.997225	+	0.0744412i	$0.976283\pi$
$192$	−4.00000	+	6.92820i	−0.288675	+	0.500000i
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	−0.993215	−	0.116289i	$-0.962900\pi$
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	0.597317	+	0.802005i	$0.296234\pi$
$194$	−10.0000			−0.717958
$195$		0			0
$196$	36.0000			2.57143
$197$	−6.00000	+	10.3923i	−0.427482	+	0.740421i	−0.996649	−	0.0818013i	$-0.973933\pi$
$197$	−6.00000	+	10.3923i	−0.427482	+	0.740421i	0.569166	+	0.822222i	$0.307266\pi$
$198$	2.00000	−	3.46410i	0.142134	−	0.246183i
$199$	−8.50000	−	14.7224i	−0.602549	−	1.04365i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−8.50000	−	14.7224i	−0.602549	−	1.04365i	0.389885	−	0.920864i	$-0.372515\pi$
$200$		0			0
$201$	3.50000	+	6.06218i	0.246871	+	0.427593i
$202$	−18.0000	−	31.1769i	−1.26648	−	2.19360i
$203$	20.0000			1.40372
$204$	−2.00000	−	3.46410i	−0.140028	−	0.242536i
$205$		0			0
$206$	7.00000	−	12.1244i	0.487713	−	0.844744i
$207$	−6.00000			−0.417029
$208$	−4.00000	−	13.8564i	−0.277350	−	0.960769i
$209$		0			0
$210$		0			0
$211$	−7.50000	+	12.9904i	−0.516321	+	0.894295i	0.483499	+	0.875345i	$0.339366\pi$
$211$	−7.50000	+	12.9904i	−0.516321	+	0.894295i	−0.999820	+	0.0189499i	$0.993968\pi$
$212$	−4.00000	−	6.92820i	−0.274721	−	0.475831i
$213$	−12.0000			−0.822226
$214$	−4.00000	−	6.92820i	−0.273434	−	0.473602i
$215$		0			0
$216$		0			0
$217$	−17.5000	−	30.3109i	−1.18798	−	2.05764i
$218$	−11.0000	+	19.0526i	−0.745014	+	1.29040i
$219$	−7.50000	+	12.9904i	−0.506803	+	0.877809i
$220$		0			0
$221$	7.00000	+	1.73205i	0.470871	+	0.116510i
$222$	4.00000			0.268462
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$	20.0000	−	34.6410i	1.33631	−	2.31455i
$225$		0			0
$226$	−4.00000			−0.266076
$227$	5.00000	+	8.66025i	0.331862	+	0.574801i	0.982877	−	0.184263i	$-0.0589899\pi$
$227$	5.00000	+	8.66025i	0.331862	+	0.574801i	−0.651015	+	0.759065i	$0.725657\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$		0			0
$231$	−5.00000	+	8.66025i	−0.328976	+	0.569803i
$232$		0			0
$233$	14.0000			0.917170			0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000			0.917170			0.458585	+	0.888650i	$0.348356\pi$
$234$	−5.00000	+	5.19615i	−0.326860	+	0.339683i
$235$		0			0
$236$	−12.0000	+	20.7846i	−0.781133	+	1.35296i
$237$	1.50000	−	2.59808i	0.0974355	−	0.168763i
$238$	10.0000	+	17.3205i	0.648204	+	1.12272i
$239$	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
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	0.658838	−	0.752285i	$-0.271048\pi$
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	−0.980917	+	0.194429i	$0.937715\pi$
$242$	14.0000			0.899954
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$	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	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$	−10.0000			−0.629941
$253$	6.00000	+	10.3923i	0.377217	+	0.653359i
$254$	11.0000	+	19.0526i	0.690201	+	1.19546i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−11.0000	+	19.0526i	−0.686161	+	1.18847i	0.286909	+	0.957958i	$0.407372\pi$
$257$	−11.0000	+	19.0526i	−0.686161	+	1.18847i	−0.973070	+	0.230508i	$0.925961\pi$
$258$	1.00000	−	1.73205i	0.0622573	−	0.107833i
$259$	−10.0000			−0.621370
$260$		0			0
$261$	−4.00000			−0.247594
$262$	−4.00000	+	6.92820i	−0.247121	+	0.428026i
$263$	−5.00000	+	8.66025i	−0.308313	+	0.534014i	−0.977993	−	0.208635i	$-0.933098\pi$
$263$	−5.00000	+	8.66025i	−0.308313	+	0.534014i	0.669680	+	0.742650i	$0.266431\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	7.00000	+	12.1244i	0.428393	+	0.741999i
$268$	14.0000			0.855186
$269$	3.00000	+	5.19615i	0.182913	+	0.316815i	0.942871	−	0.333157i	$-0.108114\pi$
$269$	3.00000	+	5.19615i	0.182913	+	0.316815i	−0.759958	+	0.649972i	$0.774781\pi$
$270$		0			0
$271$	−14.5000	+	25.1147i	−0.880812	+	1.52561i	−0.0303728	+	0.999539i	$0.509669\pi$
$271$	−14.5000	+	25.1147i	−0.880812	+	1.52561i	−0.850439	+	0.526073i	$0.823664\pi$
$272$	8.00000			0.485071
$273$	12.5000	−	12.9904i	0.756534	−	0.786214i
$274$	−4.00000			−0.241649
$275$		0			0
$276$	−6.00000	+	10.3923i	−0.361158	+	0.625543i
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$	6.00000			0.359856
$279$	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	$0.616517\pi$
$282$	−8.00000	−	13.8564i	−0.476393	−	0.825137i
$283$	2.50000	−	4.33013i	0.148610	−	0.257399i	−0.782104	−	0.623148i	$-0.785854\pi$
$283$	2.50000	−	4.33013i	0.148610	−	0.257399i	0.930714	+	0.365748i	$0.119187\pi$
$284$	−12.0000	+	20.7846i	−0.712069	+	1.23334i
$285$		0			0
$286$	14.0000	+	3.46410i	0.827837	+	0.204837i
$287$	−30.0000			−1.77084
$288$	−4.00000	+	6.92820i	−0.235702	+	0.408248i
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$	−5.00000			−0.293105
$292$	15.0000	+	25.9808i	0.877809	+	1.52041i
$293$	8.00000	+	13.8564i	0.467365	+	0.809500i	0.999305	−	0.0372823i	$-0.0118701\pi$
$293$	8.00000	+	13.8564i	0.467365	+	0.809500i	−0.531940	+	0.846782i	$0.678537\pi$
$294$	36.0000			2.09956
$295$		0			0
$296$		0			0
$297$	1.00000	−	1.73205i	0.0580259	−	0.100504i
$298$	24.0000			1.39028
$299$	−6.00000	−	20.7846i	−0.346989	−	1.20201i
$300$		0			0
$301$	−2.50000	+	4.33013i	−0.144098	+	0.249584i
$302$	−8.00000	+	13.8564i	−0.460348	+	0.797347i
$303$	−9.00000	−	15.5885i	−0.517036	−	0.895533i
$304$		0			0
$305$		0			0
$306$	−2.00000	−	3.46410i	−0.114332	−	0.198030i
$307$	31.0000			1.76926			0.884632	−	0.466290i	$-0.154410\pi$
$307$	31.0000			1.76926			0.884632	+	0.466290i	$0.154410\pi$
$308$	10.0000	+	17.3205i	0.569803	+	0.986928i
$309$	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
$313$	31.0000			1.75222			0.876112	−	0.482108i	$-0.160129\pi$
$313$	31.0000			1.75222			0.876112	+	0.482108i	$0.160129\pi$
$314$	15.0000	−	25.9808i	0.846499	−	1.46618i
$315$		0			0
$316$	−3.00000	−	5.19615i	−0.168763	−	0.292306i
$317$	−12.0000			−0.673987			−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000			−0.673987			−0.336994	+	0.941507i	$0.609410\pi$
$318$	−4.00000	−	6.92820i	−0.224309	−	0.388514i
$319$	4.00000	+	6.92820i	0.223957	+	0.387905i
$320$		0			0
$321$	−2.00000	−	3.46410i	−0.111629	−	0.193347i
$322$	30.0000	−	51.9615i	1.67183	−	2.89570i
$323$		0			0
$324$	2.00000			0.111111
$325$		0			0
$326$	−30.0000			−1.66155
$327$	−5.50000	+	9.52628i	−0.304151	+	0.526804i
$328$		0			0
$329$	20.0000	+	34.6410i	1.10264	+	1.90982i
$330$		0			0
$331$	−4.50000	−	7.79423i	−0.247342	−	0.428410i	0.715445	−	0.698669i	$-0.246224\pi$
$331$	−4.50000	−	7.79423i	−0.247342	−	0.428410i	−0.962788	+	0.270259i	$0.912891\pi$
$332$	8.00000	+	13.8564i	0.439057	+	0.760469i
$333$	2.00000			0.109599
$334$	−12.0000	−	20.7846i	−0.656611	−	1.13728i
$335$		0			0
$336$	10.0000	−	17.3205i	0.545545	−	0.944911i
$337$	1.00000			0.0544735			0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000			0.0544735			0.0272367	+	0.999629i	$0.491329\pi$
$338$	−23.0000	−	12.1244i	−1.25104	−	0.659478i
$339$	−2.00000			−0.108625
$340$		0			0
$341$	7.00000	−	12.1244i	0.379071	−	0.656571i
$342$		0			0
$343$	−55.0000			−2.96972
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.00000	−	13.8564i	−0.429463	−	0.743851i	0.567363	−	0.823468i	$-0.307964\pi$
$347$	−8.00000	−	13.8564i	−0.429463	−	0.743851i	−0.996826	+	0.0796169i	$0.974630\pi$
$348$	−4.00000	+	6.92820i	−0.214423	+	0.371391i
$349$	−1.50000	+	2.59808i	−0.0802932	+	0.139072i	−0.903376	−	0.428850i	$-0.858919\pi$
$349$	−1.50000	+	2.59808i	−0.0802932	+	0.139072i	0.823083	+	0.567922i	$0.192252\pi$
$350$		0			0
$351$	−2.50000	+	2.59808i	−0.133440	+	0.138675i
$352$	16.0000			0.852803
$353$	3.00000	−	5.19615i	0.159674	−	0.276563i	−0.775077	−	0.631867i	$-0.782289\pi$
$353$	3.00000	−	5.19615i	0.159674	−	0.276563i	0.934751	+	0.355303i	$0.115622\pi$
$354$	−12.0000	+	20.7846i	−0.637793	+	1.10469i
$355$		0			0
$356$	28.0000			1.48400
$357$	5.00000	+	8.66025i	0.264628	+	0.458349i
$358$	−6.00000	−	10.3923i	−0.317110	−	0.549250i
$359$	2.00000			0.105556			0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000			0.105556			0.0527780	+	0.998606i	$0.483192\pi$
$360$		0			0
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$	−22.0000	+	38.1051i	−1.15629	+	2.00276i
$363$	7.00000			0.367405
$364$	−10.0000	−	34.6410i	−0.524142	−	1.81568i
$365$		0			0
$366$	13.0000	−	22.5167i	0.679521	−	1.17696i
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	−0.760100	−	0.649806i	$-0.774850\pi$
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	0.942799	+	0.333363i	$0.108183\pi$
$368$	−12.0000	−	20.7846i	−0.625543	−	1.08347i
$369$	6.00000			0.312348
$370$		0			0
$371$	10.0000	+	17.3205i	0.519174	+	0.899236i
$372$	14.0000			0.725866
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	−0.647225	+	0.762299i	$0.724071\pi$
$374$	−4.00000	+	6.92820i	−0.206835	+	0.358249i
$375$		0			0
$376$		0			0
$377$	−4.00000	−	13.8564i	−0.206010	−	0.713641i
$378$	−10.0000			−0.514344
$379$	2.50000	−	4.33013i	0.128416	−	0.222424i	−0.794647	−	0.607072i	$-0.792344\pi$
$379$	2.50000	−	4.33013i	0.128416	−	0.222424i	0.923063	+	0.384648i	$0.125677\pi$
$380$		0			0
$381$	5.50000	+	9.52628i	0.281774	+	0.488046i
$382$	−24.0000			−1.22795
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	0.539076	−	0.842257i	$-0.318774\pi$
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	−0.998954	+	0.0457244i	$0.985440\pi$
$384$		0			0
$385$		0			0
$386$	11.0000	+	19.0526i	0.559885	+	0.969750i
$387$	0.500000	−	0.866025i	0.0254164	−	0.0440225i
$388$	−5.00000	+	8.66025i	−0.253837	+	0.439658i
$389$	8.00000			0.405616			0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000			0.405616			0.202808	+	0.979219i	$0.434993\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$	−2.00000	+	3.46410i	−0.100887	+	0.174741i
$394$	12.0000	+	20.7846i	0.604551	+	1.04711i
$395$		0			0
$396$	−2.00000	−	3.46410i	−0.100504	−	0.174078i
$397$	−7.50000	−	12.9904i	−0.376414	−	0.651969i	0.614123	−	0.789210i	$-0.289510\pi$
$397$	−7.50000	−	12.9904i	−0.376414	−	0.651969i	−0.990538	+	0.137241i	$0.956176\pi$
$398$	−34.0000			−1.70427
$399$		0			0
$400$		0			0
$401$	−8.00000	+	13.8564i	−0.399501	+	0.691956i	−0.993664	−	0.112388i	$-0.964150\pi$
$401$	−8.00000	+	13.8564i	−0.399501	+	0.691956i	0.594163	+	0.804344i	$0.297483\pi$
$402$	14.0000			0.698257
$403$	−17.5000	+	18.1865i	−0.871737	+	0.905936i
$404$	−36.0000			−1.79107
$405$		0			0
$406$	20.0000	−	34.6410i	0.992583	−	1.71920i
$407$	−2.00000	−	3.46410i	−0.0991363	−	0.171709i
$408$		0			0
$409$	7.50000	+	12.9904i	0.370851	+	0.642333i	0.989697	−	0.143180i	$-0.0457327\pi$
$409$	7.50000	+	12.9904i	0.370851	+	0.642333i	−0.618846	+	0.785513i	$0.712399\pi$
$410$		0			0
$411$	−2.00000			−0.0986527
$412$	−7.00000	−	12.1244i	−0.344865	−	0.597324i
$413$	30.0000	−	51.9615i	1.47620	−	2.55686i
$414$	−6.00000	+	10.3923i	−0.294884	+	0.510754i
$415$		0			0
$416$	−28.0000	−	6.92820i	−1.37281	−	0.339683i
$417$	3.00000			0.146911
$418$		0			0
$419$	−19.0000	+	32.9090i	−0.928211	+	1.60771i	−0.141896	+	0.989882i	$0.545320\pi$
$419$	−19.0000	+	32.9090i	−0.928211	+	1.60771i	−0.786314	+	0.617827i	$0.788013\pi$
$420$		0			0
$421$	−23.0000			−1.12095			−0.560476	−	0.828171i	$-0.689382\pi$
$421$	−23.0000			−1.12095			−0.560476	+	0.828171i	$0.689382\pi$
$422$	15.0000	+	25.9808i	0.730189	+	1.26472i
$423$	−4.00000	−	6.92820i	−0.194487	−	0.336861i
$424$		0			0
$425$		0			0
$426$	−12.0000	+	20.7846i	−0.581402	+	1.00702i
$427$	−32.5000	+	56.2917i	−1.57279	+	2.72414i
$428$	−8.00000			−0.386695
$429$	7.00000	+	1.73205i	0.337963	+	0.0836242i
$430$		0			0
$431$	14.0000	−	24.2487i	0.674356	−	1.16802i	−0.302300	−	0.953213i	$-0.597755\pi$
$431$	14.0000	−	24.2487i	0.674356	−	1.16802i	0.976657	−	0.214807i	$-0.0689121\pi$
$432$	−2.00000	+	3.46410i	−0.0962250	+	0.166667i
$433$	0.500000	+	0.866025i	0.0240285	+	0.0416185i	0.877790	−	0.479046i	$-0.159017\pi$
$433$	0.500000	+	0.866025i	0.0240285	+	0.0416185i	−0.853761	+	0.520665i	$0.825684\pi$
$434$	−70.0000			−3.36011
$435$		0			0
$436$	11.0000	+	19.0526i	0.526804	+	0.912452i
$437$		0			0
$438$	15.0000	+	25.9808i	0.716728	+	1.24141i
$439$	−7.50000	+	12.9904i	−0.357955	+	0.619997i	−0.987619	−	0.156871i	$-0.949859\pi$
$439$	−7.50000	+	12.9904i	−0.357955	+	0.619997i	0.629664	+	0.776868i	$0.283193\pi$
$440$		0			0
$441$	18.0000			0.857143
$442$	10.0000	−	10.3923i	0.475651	−	0.494312i
$443$	−26.0000			−1.23530			−0.617649	−	0.786454i	$-0.711915\pi$
$443$	−26.0000			−1.23530			−0.617649	+	0.786454i	$0.711915\pi$
$444$	2.00000	−	3.46410i	0.0949158	−	0.164399i
$445$		0			0
$446$	8.00000	+	13.8564i	0.378811	+	0.656120i
$447$	12.0000			0.567581
$448$	−20.0000	−	34.6410i	−0.944911	−	1.63663i
$449$	−9.00000	−	15.5885i	−0.424736	−	0.735665i	0.571660	−	0.820491i	$-0.306300\pi$
$449$	−9.00000	−	15.5885i	−0.424736	−	0.735665i	−0.996396	+	0.0848262i	$0.972967\pi$
$450$		0			0
$451$	−6.00000	−	10.3923i	−0.282529	−	0.489355i
$452$	−2.00000	+	3.46410i	−0.0940721	+	0.162938i
$453$	−4.00000	+	6.92820i	−0.187936	+	0.325515i
$454$	20.0000			0.938647
$455$		0			0
$456$		0			0
$457$	17.5000	−	30.3109i	0.818615	−	1.41788i	−0.0880870	−	0.996113i	$-0.528075\pi$
$457$	17.5000	−	30.3109i	0.818615	−	1.41788i	0.906702	−	0.421771i	$-0.138591\pi$
$458$	14.0000	−	24.2487i	0.654177	−	1.13307i
$459$	−1.00000	−	1.73205i	−0.0466760	−	0.0808452i
$460$		0			0
$461$	1.00000	+	1.73205i	0.0465746	+	0.0806696i	0.888373	−	0.459123i	$-0.151836\pi$
$461$	1.00000	+	1.73205i	0.0465746	+	0.0806696i	−0.841798	+	0.539792i	$0.818503\pi$
$462$	10.0000	+	17.3205i	0.465242	+	0.805823i
$463$	−3.00000			−0.139422			−0.0697109	−	0.997567i	$-0.522208\pi$
$463$	−3.00000			−0.139422			−0.0697109	+	0.997567i	$0.522208\pi$
$464$	−8.00000	−	13.8564i	−0.371391	−	0.643268i
$465$		0			0
$466$	14.0000	−	24.2487i	0.648537	−	1.12330i
$467$	4.00000			0.185098			0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000			0.185098			0.0925490	+	0.995708i	$0.470499\pi$
$468$	2.00000	+	6.92820i	0.0924500	+	0.320256i
$469$	−35.0000			−1.61615
$470$		0			0
$471$	7.50000	−	12.9904i	0.345582	−	0.598565i
$472$		0			0
$473$	−2.00000			−0.0919601
$474$	−3.00000	−	5.19615i	−0.137795	−	0.238667i
$475$		0			0
$476$	20.0000			0.916698
$477$	−2.00000	−	3.46410i	−0.0915737	−	0.158610i
$478$	12.0000	−	20.7846i	0.548867	−	0.950666i
$479$	−21.0000	+	36.3731i	−0.959514	+	1.66193i	−0.235833	+	0.971794i	$0.575782\pi$
$479$	−21.0000	+	36.3731i	−0.959514	+	1.66193i	−0.723681	+	0.690134i	$0.757551\pi$
$480$		0			0
$481$	2.00000	+	6.92820i	0.0911922	+	0.315899i
$482$	−20.0000			−0.910975
$483$	15.0000	−	25.9808i	0.682524	−	1.18217i
$484$	7.00000	−	12.1244i	0.318182	−	0.551107i
$485$		0			0
$486$	2.00000			0.0907218
$487$	−14.0000	−	24.2487i	−0.634401	−	1.09881i	−0.986642	−	0.162905i	$-0.947914\pi$
$487$	−14.0000	−	24.2487i	−0.634401	−	1.09881i	0.352241	−	0.935909i	$-0.385420\pi$
$488$		0			0
$489$	−15.0000			−0.678323
$490$		0			0
$491$	12.0000	−	20.7846i	0.541552	−	0.937996i	−0.457263	−	0.889332i	$-0.651170\pi$
$491$	12.0000	−	20.7846i	0.541552	−	0.937996i	0.998815	−	0.0486647i	$-0.0154966\pi$
$492$	6.00000	−	10.3923i	0.270501	−	0.468521i
$493$	8.00000			0.360302
$494$		0			0
$495$		0			0
$496$	−14.0000	+	24.2487i	−0.628619	+	1.08880i
$497$	30.0000	−	51.9615i	1.34568	−	2.33079i
$498$	8.00000	+	13.8564i	0.358489	+	0.620920i
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$	−6.00000	−	10.3923i	−0.268060	−	0.464294i
$502$		0			0
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	0.791361	−	0.611349i	$-0.209373\pi$
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	−0.925124	+	0.379664i	$0.876040\pi$
$504$		0			0
$505$		0			0
$506$	24.0000			1.06693
$507$	−11.5000	−	6.06218i	−0.510733	−	0.269231i
$508$	22.0000			0.976092
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	−0.314459	−	0.949271i	$-0.601823\pi$
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	0.979322	−	0.202306i	$-0.0648436\pi$
$510$		0			0
$511$	−37.5000	−	64.9519i	−1.65890	−	2.87330i
$512$	−32.0000			−1.41421
$513$		0			0
$514$	22.0000	+	38.1051i	0.970378	+	1.68074i
$515$		0			0
$516$	−1.00000	−	1.73205i	−0.0440225	−	0.0762493i
$517$	−8.00000	+	13.8564i	−0.351840	+	0.609404i
$518$	−10.0000	+	17.3205i	−0.439375	+	0.761019i
$519$		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	$0.728243\pi$
$522$	−4.00000	+	6.92820i	−0.175075	+	0.303239i
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$	4.00000	+	6.92820i	0.174741	+	0.302660i
$525$		0			0
$526$	10.0000	+	17.3205i	0.436021	+	0.755210i
$527$	−7.00000	−	12.1244i	−0.304925	−	0.528145i
$528$	8.00000			0.348155
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$	−6.00000	+	10.3923i	−0.260378	+	0.450988i
$532$		0			0
$533$	6.00000	+	20.7846i	0.259889	+	0.900281i
$534$	28.0000			1.21168
$535$		0			0
$536$		0			0
$537$	−3.00000	−	5.19615i	−0.129460	−	0.224231i
$538$	12.0000			0.517357
$539$	−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	$0.285749\pi$
$542$	29.0000	+	50.2295i	1.24566	+	2.15754i
$543$	−11.0000	+	19.0526i	−0.472055	+	0.817624i
$544$	8.00000	−	13.8564i	0.342997	−	0.594089i
$545$		0			0
$546$	−10.0000	−	34.6410i	−0.427960	−	1.48250i
$547$	9.00000			0.384812			0.192406	−	0.981315i	$-0.438371\pi$
$547$	9.00000			0.384812			0.192406	+	0.981315i	$0.438371\pi$
$548$	−2.00000	+	3.46410i	−0.0854358	+	0.147979i
$549$	6.50000	−	11.2583i	0.277413	−	0.480494i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	7.50000	+	12.9904i	0.318932	+	0.552407i
$554$	20.0000			0.849719
$555$		0			0
$556$	3.00000	−	5.19615i	0.127228	−	0.220366i
$557$	−10.0000	+	17.3205i	−0.423714	+	0.733893i	−0.996299	−	0.0859514i	$-0.972607\pi$
$557$	−10.0000	+	17.3205i	−0.423714	+	0.733893i	0.572586	+	0.819845i	$0.305940\pi$
$558$	14.0000			0.592667
$559$	3.50000	+	0.866025i	0.148034	+	0.0366290i
$560$		0			0
$561$	−2.00000	+	3.46410i	−0.0844401	+	0.146254i
$562$	−12.0000	+	20.7846i	−0.506189	+	0.876746i
$563$	13.0000	+	22.5167i	0.547885	+	0.948964i	0.998419	+	0.0562051i	$0.0179001\pi$
$563$	13.0000	+	22.5167i	0.547885	+	0.948964i	−0.450535	+	0.892759i	$0.648767\pi$
$564$	−16.0000			−0.673722
$565$		0			0
$566$	−5.00000	−	8.66025i	−0.210166	−	0.364018i
$567$	−5.00000			−0.209980
$568$		0			0
$569$	10.0000	−	17.3205i	0.419222	−	0.726113i	−0.576640	−	0.816999i	$-0.695636\pi$
$569$	10.0000	−	17.3205i	0.419222	−	0.726113i	0.995861	+	0.0908852i	$0.0289696\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$	10.0000	−	10.3923i	0.418121	−	0.434524i
$573$	−12.0000			−0.501307
$574$	−30.0000	+	51.9615i	−1.25218	+	2.16883i
$575$		0			0
$576$	4.00000	+	6.92820i	0.166667	+	0.288675i
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−13.0000	−	22.5167i	−0.540729	−	0.936570i
$579$	5.50000	+	9.52628i	0.228572	+	0.395899i
$580$		0			0
$581$	−20.0000	−	34.6410i	−0.829740	−	1.43715i
$582$	−5.00000	+	8.66025i	−0.207257	+	0.358979i
$583$	−4.00000	+	6.92820i	−0.165663	+	0.286937i
$584$		0			0
$585$		0			0
$586$	32.0000			1.32191
$587$	14.0000	−	24.2487i	0.577842	−	1.00085i	−0.417885	−	0.908500i	$-0.637228\pi$
$587$	14.0000	−	24.2487i	0.577842	−	1.00085i	0.995726	−	0.0923513i	$-0.0294383\pi$
$588$	18.0000	−	31.1769i	0.742307	−	1.28571i
$589$		0			0
$590$		0			0
$591$	6.00000	+	10.3923i	0.246807	+	0.427482i
$592$	4.00000	+	6.92820i	0.164399	+	0.284747i
$593$	10.0000			0.410651			0.205325	−	0.978694i	$-0.434175\pi$
$593$	10.0000			0.410651			0.205325	+	0.978694i	$0.434175\pi$
$594$	−2.00000	−	3.46410i	−0.0820610	−	0.142134i
$595$		0			0
$596$	12.0000	−	20.7846i	0.491539	−	0.851371i
$597$	−17.0000			−0.695764
$598$	−42.0000	−	10.3923i	−1.71751	−	0.424973i
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	−11.0000	+	19.0526i	−0.448699	+	0.777170i	−0.998302	−	0.0582563i	$-0.981446\pi$
$601$	−11.0000	+	19.0526i	−0.448699	+	0.777170i	0.549602	+	0.835426i	$0.314779\pi$
$602$	5.00000	+	8.66025i	0.203785	+	0.352966i
$603$	7.00000			0.285062
$604$	8.00000	+	13.8564i	0.325515	+	0.563809i
$605$		0			0
$606$	−36.0000			−1.46240
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	0.981454	−	0.191700i	$-0.0614000\pi$
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	−0.656744	+	0.754114i	$0.728067\pi$
$608$		0			0
$609$	10.0000	−	17.3205i	0.405220	−	0.701862i
$610$		0			0
$611$	20.0000	−	20.7846i	0.809113	−	0.840855i
$612$	−4.00000			−0.161690
$613$	−7.50000	+	12.9904i	−0.302922	+	0.524677i	−0.976797	−	0.214169i	$-0.931296\pi$
$613$	−7.50000	+	12.9904i	−0.302922	+	0.524677i	0.673874	+	0.738846i	$0.264629\pi$
$614$	31.0000	−	53.6936i	1.25106	−	2.16690i
$615$		0			0
$616$		0			0
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	0.920074	−	0.391745i	$-0.128129\pi$
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	−0.799298	+	0.600935i	$0.794795\pi$
$618$	−7.00000	−	12.1244i	−0.281581	−	0.487713i
$619$	37.0000			1.48716			0.743578	−	0.668649i	$-0.233127\pi$
$619$	37.0000			1.48716			0.743578	+	0.668649i	$0.233127\pi$
$620$		0			0
$621$	−3.00000	+	5.19615i	−0.120386	+	0.208514i
$622$	−22.0000	+	38.1051i	−0.882120	+	1.52788i
$623$	−70.0000			−2.80449
$624$	−14.0000	−	3.46410i	−0.560449	−	0.138675i
$625$		0			0
$626$	31.0000	−	53.6936i	1.23901	−	2.14603i
$627$		0			0
$628$	−15.0000	−	25.9808i	−0.598565	−	1.03675i
$629$	−4.00000			−0.159490
$630$		0			0
$631$	−3.50000	−	6.06218i	−0.139333	−	0.241331i	0.787911	−	0.615789i	$-0.211162\pi$
$631$	−3.50000	−	6.06218i	−0.139333	−	0.241331i	−0.927244	+	0.374457i	$0.877829\pi$
$632$		0			0
$633$	7.50000	+	12.9904i	0.298098	+	0.516321i
$634$	−12.0000	+	20.7846i	−0.476581	+	0.825462i
$635$		0			0
$636$	−8.00000			−0.317221
$637$	18.0000	+	62.3538i	0.713186	+	2.47055i
$638$	16.0000			0.633446
$639$	−6.00000	+	10.3923i	−0.237356	+	0.411113i
$640$		0			0
$641$	−1.00000	−	1.73205i	−0.0394976	−	0.0684119i	0.845601	−	0.533816i	$-0.179242\pi$
$641$	−1.00000	−	1.73205i	−0.0394976	−	0.0684119i	−0.885098	+	0.465404i	$0.845909\pi$
$642$	−8.00000			−0.315735
$643$	−9.50000	−	16.4545i	−0.374643	−	0.648901i	0.615630	−	0.788035i	$-0.288902\pi$
$643$	−9.50000	−	16.4545i	−0.374643	−	0.648901i	−0.990274	+	0.139134i	$0.955568\pi$
$644$	−30.0000	−	51.9615i	−1.18217	−	2.04757i
$645$		0			0
$646$		0			0
$647$	19.0000	−	32.9090i	0.746967	−	1.29378i	−0.202303	−	0.979323i	$-0.564843\pi$
$647$	19.0000	−	32.9090i	0.746967	−	1.29378i	0.949270	−	0.314462i	$-0.101824\pi$
$648$		0			0
$649$	24.0000			0.942082
$650$		0			0
$651$	−35.0000			−1.37176
$652$	−15.0000	+	25.9808i	−0.587445	+	1.01749i
$653$	21.0000	−	36.3731i	0.821794	−	1.42339i	−0.0825519	−	0.996587i	$-0.526307\pi$
$653$	21.0000	−	36.3731i	0.821794	−	1.42339i	0.904345	−	0.426801i	$-0.140360\pi$
$654$	11.0000	+	19.0526i	0.430134	+	0.745014i
$655$		0			0
$656$	12.0000	+	20.7846i	0.468521	+	0.811503i
$657$	7.50000	+	12.9904i	0.292603	+	0.506803i
$658$	80.0000			3.11872
$659$	12.0000	+	20.7846i	0.467454	+	0.809653i	0.999309	−	0.0371821i	$-0.0118382\pi$
$659$	12.0000	+	20.7846i	0.467454	+	0.809653i	−0.531855	+	0.846836i	$0.678505\pi$
$660$		0			0
$661$	−17.5000	+	30.3109i	−0.680671	+	1.17896i	0.294105	+	0.955773i	$0.404978\pi$
$661$	−17.5000	+	30.3109i	−0.680671	+	1.17896i	−0.974776	+	0.223184i	$0.928355\pi$
$662$	−18.0000			−0.699590
$663$	5.00000	−	5.19615i	0.194184	−	0.201802i
$664$		0			0
$665$		0			0
$666$	2.00000	−	3.46410i	0.0774984	−	0.134231i
$667$	−12.0000	−	20.7846i	−0.464642	−	0.804783i
$668$	−24.0000			−0.928588
$669$	4.00000	+	6.92820i	0.154649	+	0.267860i
$670$		0			0
$671$	−26.0000			−1.00372
$672$	−20.0000	−	34.6410i	−0.771517	−	1.33631i
$673$	16.5000	−	28.5788i	0.636028	−	1.10163i	−0.350268	−	0.936650i	$-0.613909\pi$
$673$	16.5000	−	28.5788i	0.636028	−	1.10163i	0.986296	−	0.164984i	$-0.0527572\pi$
$674$	1.00000	−	1.73205i	0.0385186	−	0.0667161i
$675$		0			0
$676$	−22.0000	+	13.8564i	−0.846154	+	0.532939i
$677$	12.0000			0.461197			0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000			0.461197			0.230599	+	0.973049i	$0.425932\pi$
$678$	−2.00000	+	3.46410i	−0.0768095	+	0.133038i
$679$	12.5000	−	21.6506i	0.479706	−	0.830875i
$680$		0			0
$681$	10.0000			0.383201
$682$	−14.0000	−	24.2487i	−0.536088	−	0.928531i
$683$	−10.0000	−	17.3205i	−0.382639	−	0.662751i	0.608799	−	0.793324i	$-0.291651\pi$
$683$	−10.0000	−	17.3205i	−0.382639	−	0.662751i	−0.991439	+	0.130573i	$0.958318\pi$
$684$		0			0
$685$		0			0
$686$	−55.0000	+	95.2628i	−2.09991	+	3.63715i
$687$	7.00000	−	12.1244i	0.267067	−	0.462573i
$688$	4.00000			0.152499
$689$	10.0000	−	10.3923i	0.380970	−	0.395915i
$690$		0			0
$691$	−18.5000	+	32.0429i	−0.703773	+	1.21897i	0.263359	+	0.964698i	$0.415170\pi$
$691$	−18.5000	+	32.0429i	−0.703773	+	1.21897i	−0.967132	+	0.254273i	$0.918164\pi$
$692$		0			0
$693$	5.00000	+	8.66025i	0.189934	+	0.328976i
$694$	−32.0000			−1.21470
$695$		0			0
$696$		0			0
$697$	−12.0000			−0.454532
$698$	3.00000	+	5.19615i	0.113552	+	0.196677i
$699$	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	$0.227448\pi$
$702$	2.00000	+	6.92820i	0.0754851	+	0.261488i
$703$		0			0
$704$	8.00000	−	13.8564i	0.301511	−	0.522233i
$705$		0			0
$706$	−6.00000	−	10.3923i	−0.225813	−	0.391120i
$707$	90.0000			3.38480
$708$	12.0000	+	20.7846i	0.450988	+	0.781133i
$709$	11.5000	+	19.9186i	0.431892	+	0.748058i	0.997036	−	0.0769337i	$-0.0245130\pi$
$709$	11.5000	+	19.9186i	0.431892	+	0.748058i	−0.565145	+	0.824992i	$0.691180\pi$
$710$		0			0
$711$	−1.50000	−	2.59808i	−0.0562544	−	0.0974355i
$712$		0			0
$713$	−21.0000	+	36.3731i	−0.786456	+	1.36218i
$714$	20.0000			0.748481
$715$		0			0
$716$	−12.0000			−0.448461
$717$	6.00000	−	10.3923i	0.224074	−	0.388108i
$718$	2.00000	−	3.46410i	0.0746393	−	0.129279i
$719$	−20.0000	−	34.6410i	−0.745874	−	1.29189i	−0.949785	−	0.312903i	$-0.898699\pi$
$719$	−20.0000	−	34.6410i	−0.745874	−	1.29189i	0.203911	−	0.978989i	$-0.434635\pi$
$720$		0			0
$721$	17.5000	+	30.3109i	0.651734	+	1.12884i
$722$	−19.0000	−	32.9090i	−0.707107	−	1.22474i
$723$	−10.0000			−0.371904
$724$	22.0000	+	38.1051i	0.817624	+	1.41617i
$725$		0			0
$726$	7.00000	−	12.1244i	0.259794	−	0.449977i
$727$	−9.00000			−0.333792			−0.166896	−	0.985975i	$-0.553374\pi$
$727$	−9.00000			−0.333792			−0.166896	+	0.985975i	$0.553374\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−1.00000	+	1.73205i	−0.0369863	+	0.0640622i
$732$	−13.0000	−	22.5167i	−0.480494	−	0.832240i
$733$	−7.00000			−0.258551			−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000			−0.258551			−0.129275	+	0.991609i	$0.541265\pi$
$734$	−7.00000	−	12.1244i	−0.258375	−	0.447518i
$735$		0			0
$736$	−48.0000			−1.76930
$737$	−7.00000	−	12.1244i	−0.257848	−	0.446606i
$738$	6.00000	−	10.3923i	0.220863	−	0.382546i
$739$	18.0000	−	31.1769i	0.662141	−	1.14686i	−0.317911	−	0.948120i	$-0.602981\pi$
$739$	18.0000	−	31.1769i	0.662141	−	1.14686i	0.980052	−	0.198741i	$-0.0636852\pi$
$740$		0			0
$741$		0			0
$742$	40.0000			1.46845
$743$	12.0000	−	20.7846i	0.440237	−	0.762513i	−0.557470	−	0.830197i	$-0.688228\pi$
$743$	12.0000	−	20.7846i	0.440237	−	0.762513i	0.997707	+	0.0676840i	$0.0215610\pi$
$744$		0			0
$745$		0			0
$746$	26.0000			0.951928
$747$	4.00000	+	6.92820i	0.146352	+	0.253490i
$748$	4.00000	+	6.92820i	0.146254	+	0.253320i
$749$	20.0000			0.730784
$750$		0			0
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	−0.489053	−	0.872254i	$-0.662658\pi$
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	0.999921	−	0.0125942i	$-0.00400897\pi$
$752$	16.0000	−	27.7128i	0.583460	−	1.01058i
$753$		0			0
$754$	−28.0000	−	6.92820i	−1.01970	−	0.252310i
$755$		0			0
$756$	−5.00000	+	8.66025i	−0.181848	+	0.314970i
$757$	1.00000	−	1.73205i	0.0363456	−	0.0629525i	−0.847280	−	0.531146i	$-0.821762\pi$
$757$	1.00000	−	1.73205i	0.0363456	−	0.0629525i	0.883626	+	0.468193i	$0.155095\pi$
$758$	−5.00000	−	8.66025i	−0.181608	−	0.314555i
$759$	12.0000			0.435572
$760$		0			0
$761$	18.0000	+	31.1769i	0.652499	+	1.13016i	0.982514	+	0.186187i	$0.0596129\pi$
$761$	18.0000	+	31.1769i	0.652499	+	1.13016i	−0.330015	+	0.943976i	$0.607054\pi$
$762$	22.0000			0.796976
$763$	−27.5000	−	47.6314i	−0.995567	−	1.72437i
$764$	−12.0000	+	20.7846i	−0.434145	+	0.751961i
$765$		0			0
$766$	−36.0000			−1.30073
$767$	−42.0000	−	10.3923i	−1.51653	−	0.375244i
$768$	−16.0000			−0.577350
$769$	17.0000	−	29.4449i	0.613036	−	1.06181i	−0.377690	−	0.925932i	$-0.623282\pi$
$769$	17.0000	−	29.4449i	0.613036	−	1.06181i	0.990726	−	0.135877i	$-0.0433852\pi$
$770$		0			0
$771$	11.0000	+	19.0526i	0.396155	+	0.686161i
$772$	22.0000			0.791797
$773$	23.0000	+	39.8372i	0.827253	+	1.43284i	0.900186	+	0.435507i	$0.143431\pi$
$773$	23.0000	+	39.8372i	0.827253	+	1.43284i	−0.0729331	+	0.997337i	$0.523236\pi$
$774$	−1.00000	−	1.73205i	−0.0359443	−	0.0622573i
$775$		0			0
$776$		0			0
$777$	−5.00000	+	8.66025i	−0.179374	+	0.310685i
$778$	8.00000	−	13.8564i	0.286814	−	0.496776i
$779$		0			0
$780$		0			0
$781$	24.0000			0.858788
$782$	12.0000	−	20.7846i	0.429119	−	0.743256i
$783$	−2.00000	+	3.46410i	−0.0714742	+	0.123797i
$784$	36.0000	+	62.3538i	1.28571	+	2.22692i
$785$		0			0
$786$	4.00000	+	6.92820i	0.142675	+	0.247121i
$787$	8.50000	+	14.7224i	0.302992	+	0.524798i	0.976812	−	0.214097i	$-0.0686810\pi$
$787$	8.50000	+	14.7224i	0.302992	+	0.524798i	−0.673820	+	0.738896i	$0.735348\pi$
$788$	24.0000			0.854965
$789$	5.00000	+	8.66025i	0.178005	+	0.308313i
$790$		0			0
$791$	5.00000	−	8.66025i	0.177780	−	0.307923i
$792$		0			0
$793$	45.5000	+	11.2583i	1.61575	+	0.399795i
$794$	−30.0000			−1.06466
$795$		0			0
$796$	−17.0000	+	29.4449i	−0.602549	+	1.04365i
$797$	15.0000	+	25.9808i	0.531327	+	0.920286i	0.999331	+	0.0365596i	$0.0116399\pi$
$797$	15.0000	+	25.9808i	0.531327	+	0.920286i	−0.468004	+	0.883726i	$0.655027\pi$
$798$		0			0
$799$	8.00000	+	13.8564i	0.283020	+	0.490204i
$800$		0			0
$801$	14.0000			0.494666
$802$	16.0000	+	27.7128i	0.564980	+	0.978573i
$803$	15.0000	−	25.9808i	0.529339	−	0.916841i
$804$	7.00000	−	12.1244i	0.246871	−	0.427593i
$805$		0			0
$806$	14.0000	+	48.4974i	0.493129	+	1.70825i
$807$	6.00000			0.211210
$808$		0			0
$809$	−2.00000	+	3.46410i	−0.0703163	+	0.121791i	−0.899040	−	0.437867i	$-0.855734\pi$
$809$	−2.00000	+	3.46410i	−0.0703163	+	0.121791i	0.828724	+	0.559658i	$0.189068\pi$
$810$		0			0
$811$	45.0000			1.58016			0.790082	−	0.613001i	$-0.210038\pi$
$811$	45.0000			1.58016			0.790082	+	0.613001i	$0.210038\pi$
$812$	−20.0000	−	34.6410i	−0.701862	−	1.21566i
$813$	14.5000	+	25.1147i	0.508537	+	0.880812i
$814$	−8.00000			−0.280400
$815$		0			0
$816$	4.00000	−	6.92820i	0.140028	−	0.242536i
$817$		0			0
$818$	30.0000			1.04893
$819$	−5.00000	−	17.3205i	−0.174714	−	0.605228i
$820$		0			0
$821$	11.0000	−	19.0526i	0.383903	−	0.664939i	−0.607714	−	0.794156i	$-0.707913\pi$
$821$	11.0000	−	19.0526i	0.383903	−	0.664939i	0.991616	+	0.129217i	$0.0412465\pi$
$822$	−2.00000	+	3.46410i	−0.0697580	+	0.120824i
$823$	10.0000	+	17.3205i	0.348578	+	0.603755i	0.985997	−	0.166762i	$-0.0533313\pi$
$823$	10.0000	+	17.3205i	0.348578	+	0.603755i	−0.637419	+	0.770517i	$0.719998\pi$
$824$		0			0
$825$		0			0
$826$	−60.0000	−	103.923i	−2.08767	−	3.61595i
$827$	−46.0000			−1.59958			−0.799788	−	0.600282i	$-0.795055\pi$
$827$	−46.0000			−1.59958			−0.799788	+	0.600282i	$0.795055\pi$
$828$	6.00000	+	10.3923i	0.208514	+	0.361158i
$829$	5.50000	−	9.52628i	0.191023	−	0.330861i	−0.754567	−	0.656223i	$-0.772153\pi$
$829$	5.50000	−	9.52628i	0.191023	−	0.330861i	0.945589	+	0.325362i	$0.105486\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$	−20.0000	+	20.7846i	−0.693375	+	0.720577i
$833$	−36.0000			−1.24733
$834$	3.00000	−	5.19615i	0.103882	−	0.179928i
$835$		0			0
$836$		0			0
$837$	7.00000			0.241955
$838$	38.0000	+	65.8179i	1.31269	+	2.27364i
$839$	17.0000	+	29.4449i	0.586905	+	1.01655i	0.994635	+	0.103447i	$0.0329872\pi$
$839$	17.0000	+	29.4449i	0.586905	+	1.01655i	−0.407730	+	0.913103i	$0.633679\pi$
$840$		0			0
$841$	6.50000	+	11.2583i	0.224138	+	0.388218i
$842$	−23.0000	+	39.8372i	−0.792632	+	1.37288i
$843$	−6.00000	+	10.3923i	−0.206651	+	0.357930i
$844$	30.0000			1.03264
$845$		0			0
$846$	−16.0000			−0.550091
$847$	−17.5000	+	30.3109i	−0.601307	+	1.04149i
$848$	8.00000	−	13.8564i	0.274721	−	0.475831i
$849$	−2.50000	−	4.33013i	−0.0857998	−	0.148610i
$850$		0			0
$851$	6.00000	+	10.3923i	0.205677	+	0.356244i
$852$	12.0000	+	20.7846i	0.411113	+	0.712069i
$853$	−9.00000			−0.308154			−0.154077	−	0.988059i	$-0.549240\pi$
$853$	−9.00000			−0.308154			−0.154077	+	0.988059i	$0.549240\pi$
$854$	65.0000	+	112.583i	2.22425	+	3.85252i
$855$		0			0
$856$		0			0
$857$	12.0000			0.409912			0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000			0.409912			0.204956	+	0.978771i	$0.434295\pi$
$858$	10.0000	−	10.3923i	0.341394	−	0.354787i
$859$	43.0000			1.46714			0.733571	−	0.679613i	$-0.237852\pi$
$859$	43.0000			1.46714			0.733571	+	0.679613i	$0.237852\pi$
$860$		0			0
$861$	−15.0000	+	25.9808i	−0.511199	+	0.885422i
$862$	−28.0000	−	48.4974i	−0.953684	−	1.65183i
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$	4.00000	+	6.92820i	0.136083	+	0.235702i
$865$		0			0
$866$	2.00000			0.0679628
$867$	−6.50000	−	11.2583i	−0.220752	−	0.382353i
$868$	−35.0000	+	60.6218i	−1.18798	+	2.05764i
$869$	−3.00000	+	5.19615i	−0.101768	+	0.176267i
$870$		0			0
$871$	7.00000	+	24.2487i	0.237186	+	0.821636i
$872$		0			0
$873$	−2.50000	+	4.33013i	−0.0846122	+	0.146553i
$874$		0			0
$875$		0			0
$876$	30.0000			1.01361
$877$	3.00000	+	5.19615i	0.101303	+	0.175462i	0.912222	−	0.409697i	$-0.134366\pi$
$877$	3.00000	+	5.19615i	0.101303	+	0.175462i	−0.810919	+	0.585159i	$0.801032\pi$
$878$	15.0000	+	25.9808i	0.506225	+	0.876808i
$879$	16.0000			0.539667
$880$		0			0
$881$	−10.0000	+	17.3205i	−0.336909	+	0.583543i	−0.983850	−	0.178997i	$-0.942715\pi$
$881$	−10.0000	+	17.3205i	−0.336909	+	0.583543i	0.646941	+	0.762540i	$0.276048\pi$
$882$	18.0000	−	31.1769i	0.606092	−	1.04978i
$883$	25.0000			0.841317			0.420658	−	0.907219i	$-0.361799\pi$
$883$	25.0000			0.841317			0.420658	+	0.907219i	$0.361799\pi$
$884$	−4.00000	−	13.8564i	−0.134535	−	0.466041i
$885$		0			0
$886$	−26.0000	+	45.0333i	−0.873487	+	1.51292i
$887$	−22.0000	+	38.1051i	−0.738688	+	1.27944i	0.214399	+	0.976746i	$0.431221\pi$
$887$	−22.0000	+	38.1051i	−0.738688	+	1.27944i	−0.953086	+	0.302698i	$0.902113\pi$
$888$		0			0
$889$	−55.0000			−1.84464
$890$		0			0
$891$	−1.00000	−	1.73205i	−0.0335013	−	0.0580259i
$892$	16.0000			0.535720
$893$		0			0
$894$	12.0000	−	20.7846i	0.401340	−	0.695141i
$895$		0			0
$896$		0			0
$897$	−21.0000	−	5.19615i	−0.701170	−	0.173494i
$898$	−36.0000			−1.20134
$899$	−14.0000	+	24.2487i	−0.466926	+	0.808740i
$900$		0			0
$901$	4.00000	+	6.92820i	0.133259	+	0.230812i
$902$	−24.0000			−0.799113
$903$	2.50000	+	4.33013i	0.0831948	+	0.144098i
$904$		0			0
$905$		0			0
$906$	8.00000	+	13.8564i	0.265782	+	0.460348i
$907$	10.0000	−	17.3205i	0.332045	−	0.575118i	−0.650868	−	0.759191i	$-0.725595\pi$
$907$	10.0000	−	17.3205i	0.332045	−	0.575118i	0.982913	+	0.184073i	$0.0589282\pi$
$908$	10.0000	−	17.3205i	0.331862	−	0.574801i
$909$	−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
$913$	8.00000	−	13.8564i	0.264761	−	0.458580i
$914$	−35.0000	−	60.6218i	−1.15770	−	2.00519i
$915$		0			0
$916$	−14.0000	−	24.2487i	−0.462573	−	0.801200i
$917$	−10.0000	−	17.3205i	−0.330229	−	0.571974i
$918$	−4.00000			−0.132020
$919$	−4.00000	−	6.92820i	−0.131948	−	0.228540i	0.792480	−	0.609898i	$-0.208790\pi$
$919$	−4.00000	−	6.92820i	−0.131948	−	0.228540i	−0.924427	+	0.381358i	$0.875456\pi$
$920$		0			0
$921$	15.5000	−	26.8468i	0.510742	−	0.884632i
$922$	4.00000			0.131733
$923$	−42.0000	−	10.3923i	−1.38245	−	0.342067i
$924$	20.0000			0.657952
$925$		0			0
$926$	−3.00000	+	5.19615i	−0.0985861	+	0.170756i
$927$	−3.50000	−	6.06218i	−0.114955	−	0.199108i
$928$	−32.0000			−1.05045
$929$	−26.0000	−	45.0333i	−0.853032	−	1.47750i	−0.878459	−	0.477819i	$-0.841428\pi$
$929$	−26.0000	−	45.0333i	−0.853032	−	1.47750i	0.0254262	−	0.999677i	$-0.491906\pi$
$930$		0			0
$931$		0			0
$932$	−14.0000	−	24.2487i	−0.458585	−	0.794293i
$933$	−11.0000	+	19.0526i	−0.360124	+	0.623753i
$934$	4.00000	−	6.92820i	0.130884	−	0.226698i
$935$		0			0
$936$		0			0
$937$	30.0000			0.980057			0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000			0.980057			0.490029	+	0.871706i	$0.336986\pi$
$938$	−35.0000	+	60.6218i	−1.14279	+	1.97937i
$939$	15.5000	−	26.8468i	0.505823	−	0.876112i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$	−15.0000	−	25.9808i	−0.488726	−	0.846499i
$943$	18.0000	+	31.1769i	0.586161	+	1.01526i
$944$	−48.0000			−1.56227
$945$		0			0
$946$	−2.00000	+	3.46410i	−0.0650256	+	0.112628i
$947$	9.00000	−	15.5885i	0.292461	−	0.506557i	−0.681930	−	0.731417i	$-0.738859\pi$
$947$	9.00000	−	15.5885i	0.292461	−	0.506557i	0.974391	+	0.224860i	$0.0721926\pi$
$948$	−6.00000			−0.194871
$949$	−37.5000	+	38.9711i	−1.21730	+	1.26506i
$950$		0			0
$951$	−6.00000	+	10.3923i	−0.194563	+	0.336994i
$952$		0			0
$953$	3.00000	+	5.19615i	0.0971795	+	0.168320i	0.910516	−	0.413473i	$-0.135685\pi$
$953$	3.00000	+	5.19615i	0.0971795	+	0.168320i	−0.813337	+	0.581793i	$0.802351\pi$
$954$	−8.00000			−0.259010
$955$		0			0
$956$	−12.0000	−	20.7846i	−0.388108	−	0.672222i
$957$	8.00000			0.258603
$958$	42.0000	+	72.7461i	1.35696	+	2.35032i
$959$	5.00000	−	8.66025i	0.161458	−	0.279654i
$960$		0			0
$961$	18.0000			0.580645
$962$	14.0000	+	3.46410i	0.451378	+	0.111687i
$963$	−4.00000			−0.128898
$964$	−10.0000	+	17.3205i	−0.322078	+	0.557856i
$965$		0			0
$966$	−30.0000	−	51.9615i	−0.965234	−	1.67183i
$967$	56.0000			1.80084			0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000			1.80084			0.900419	+	0.435023i	$0.143260\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.0000	−	17.3205i	−0.320915	−	0.555842i	0.659762	−	0.751475i	$-0.270657\pi$
$971$	−10.0000	−	17.3205i	−0.320915	−	0.555842i	−0.980677	+	0.195633i	$0.937324\pi$
$972$	1.00000	−	1.73205i	0.0320750	−	0.0555556i
$973$	−7.50000	+	12.9904i	−0.240439	+	0.416452i
$974$	−56.0000			−1.79436
$975$		0			0
$976$	52.0000			1.66448
$977$	30.0000	−	51.9615i	0.959785	−	1.66240i	0.236768	−	0.971566i	$-0.423912\pi$
$977$	30.0000	−	51.9615i	0.959785	−	1.66240i	0.723017	−	0.690830i	$-0.242755\pi$
$978$	−15.0000	+	25.9808i	−0.479647	+	0.830773i
$979$	−14.0000	−	24.2487i	−0.447442	−	0.774992i
$980$		0			0
$981$	5.50000	+	9.52628i	0.175601	+	0.304151i
$982$	−24.0000	−	41.5692i	−0.765871	−	1.32653i
$983$	−38.0000			−1.21201			−0.606006	−	0.795460i	$-0.707229\pi$
$983$	−38.0000			−1.21201			−0.606006	+	0.795460i	$0.707229\pi$
$984$		0			0
$985$		0			0
$986$	8.00000	−	13.8564i	0.254772	−	0.441278i
$987$	40.0000			1.27321
$988$		0			0
$989$	6.00000			0.190789
$990$		0			0
$991$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$991$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$992$	28.0000	+	48.4974i	0.889001	+	1.53979i
$993$	−9.00000			−0.285606
$994$	−60.0000	−	103.923i	−1.90308	−	3.29624i
$995$		0			0
$996$	16.0000			0.506979
$997$	14.5000	+	25.1147i	0.459220	+	0.795392i	0.998920	−	0.0464655i	$-0.0147958\pi$
$997$	14.5000	+	25.1147i	0.459220	+	0.795392i	−0.539700	+	0.841857i	$0.681462\pi$
$998$	4.00000	−	6.92820i	0.126618	−	0.219308i
$999$	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.i.i.451.1		2
5.2	odd	4		975.2.bb.f.724.2		4
5.3	odd	4		975.2.bb.f.724.1		4
5.4	even	2		195.2.i.a.61.1	yes	2
13.3	even	3	inner	975.2.i.i.601.1		2
15.14	odd	2		585.2.j.b.451.1		2
65.3	odd	12		975.2.bb.f.874.2		4
65.4	even	6		2535.2.a.c.1.1		1
65.9	even	6		2535.2.a.m.1.1		1
65.29	even	6		195.2.i.a.16.1	✓	2
65.42	odd	12		975.2.bb.f.874.1		4
195.29	odd	6		585.2.j.b.406.1		2
195.74	odd	6		7605.2.a.a.1.1		1
195.134	odd	6		7605.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.i.a.16.1	✓	2	65.29	even	6
195.2.i.a.61.1	yes	2	5.4	even	2
585.2.j.b.406.1		2	195.29	odd	6
585.2.j.b.451.1		2	15.14	odd	2
975.2.i.i.451.1		2	1.1	even	1	trivial
975.2.i.i.601.1		2	13.3	even	3	inner
975.2.bb.f.724.1		4	5.3	odd	4
975.2.bb.f.724.2		4	5.2	odd	4
975.2.bb.f.874.1		4	65.42	odd	12
975.2.bb.f.874.2		4	65.3	odd	12
2535.2.a.c.1.1		1	65.4	even	6
2535.2.a.m.1.1		1	65.9	even	6
7605.2.a.a.1.1		1	195.74	odd	6
7605.2.a.s.1.1		1	195.134	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-1.00000 - 1.73205i) q^{6} +(2.50000 + 4.33013i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} -2.00000 q^{12} +(2.50000 - 2.59808i) q^{13} +10.0000 q^{14} +(2.00000 - 3.46410i) q^{16} +(1.00000 + 1.73205i) q^{17} -2.00000 q^{18} +5.00000 q^{21} +(2.00000 + 3.46410i) q^{22} +(3.00000 - 5.19615i) q^{23} +(-2.00000 - 6.92820i) q^{26} -1.00000 q^{27} +(5.00000 - 8.66025i) q^{28} +(2.00000 - 3.46410i) q^{29} -7.00000 q^{31} +(-4.00000 - 6.92820i) q^{32} +(1.00000 + 1.73205i) q^{33} +4.00000 q^{34} +(-1.00000 + 1.73205i) q^{36} +(-1.00000 + 1.73205i) q^{37} +(-1.00000 - 3.46410i) q^{39} +(-3.00000 + 5.19615i) q^{41} +(5.00000 - 8.66025i) q^{42} +(0.500000 + 0.866025i) q^{43} +4.00000 q^{44} +(-6.00000 - 10.3923i) q^{46} +8.00000 q^{47} +(-2.00000 - 3.46410i) q^{48} +(-9.00000 + 15.5885i) q^{49} +2.00000 q^{51} +(-7.00000 - 1.73205i) q^{52} +4.00000 q^{53} +(-1.00000 + 1.73205i) q^{54} +(-4.00000 - 6.92820i) q^{58} +(-6.00000 - 10.3923i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-7.00000 + 12.1244i) q^{62} +(2.50000 - 4.33013i) q^{63} -8.00000 q^{64} +4.00000 q^{66} +(-3.50000 + 6.06218i) q^{67} +(2.00000 - 3.46410i) q^{68} +(-3.00000 - 5.19615i) q^{69} +(-6.00000 - 10.3923i) q^{71} -15.0000 q^{73} +(2.00000 + 3.46410i) q^{74} -10.0000 q^{77} +(-7.00000 - 1.73205i) q^{78} +3.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} +(6.00000 + 10.3923i) q^{82} -8.00000 q^{83} +(-5.00000 - 8.66025i) q^{84} +2.00000 q^{86} +(-2.00000 - 3.46410i) q^{87} +(-7.00000 + 12.1244i) q^{89} +(17.5000 + 4.33013i) q^{91} -12.0000 q^{92} +(-3.50000 + 6.06218i) q^{93} +(8.00000 - 13.8564i) q^{94} -8.00000 q^{96} +(-2.50000 - 4.33013i) q^{97} +(18.0000 + 31.1769i) q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + q^{3} - 2 q^{4} - 2 q^{6} + 5 q^{7} - q^{9} - 2 q^{11} - 4 q^{12} + 5 q^{13} + 20 q^{14} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 10 q^{21} + 4 q^{22} + 6 q^{23} - 4 q^{26} - 2 q^{27} + 10 q^{28}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + q^3 - 2 * q^4 - 2 * q^6 + 5 * q^7 - q^9 - 2 * q^11 - 4 * q^12 + 5 * q^13 + 20 * q^14 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 10 * q^21 + 4 * q^22 + 6 * q^23 - 4 * q^26 - 2 * q^27 + 10 * q^28 + 4 * q^29 - 14 * q^31 - 8 * q^32 + 2 * q^33 + 8 * q^34 - 2 * q^36 - 2 * q^37 - 2 * q^39 - 6 * q^41 + 10 * q^42 + q^43 + 8 * q^44 - 12 * q^46 + 16 * q^47 - 4 * q^48 - 18 * q^49 + 4 * q^51 - 14 * q^52 + 8 * q^53 - 2 * q^54 - 8 * q^58 - 12 * q^59 + 13 * q^61 - 14 * q^62 + 5 * q^63 - 16 * q^64 + 8 * q^66 - 7 * q^67 + 4 * q^68 - 6 * q^69 - 12 * q^71 - 30 * q^73 + 4 * q^74 - 20 * q^77 - 14 * q^78 + 6 * q^79 - q^81 + 12 * q^82 - 16 * q^83 - 10 * q^84 + 4 * q^86 - 4 * q^87 - 14 * q^89 + 35 * q^91 - 24 * q^92 - 7 * q^93 + 16 * q^94 - 16 * q^96 - 5 * q^97 + 36 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more