LMFDB - Embedded newform 126.3.s.b.107.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,3,Mod(53,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("126.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	126.s (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.43325133094$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			107.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	126.107
Dual form			126.3.s.b.53.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(1.22474 + 0.707107i) q^{5} +(6.50000 + 2.59808i) q^{7} +2.82843i q^{8} +O(q^{10})$	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(1.22474 + 0.707107i) q^{5} +(6.50000 + 2.59808i) q^{7} +2.82843i q^{8} +(1.00000 + 1.73205i) q^{10} +(-6.12372 + 3.53553i) q^{11} +15.0000 q^{13} +(6.12372 + 7.77817i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(-9.79796 + 5.65685i) q^{17} +(6.50000 - 11.2583i) q^{19} +2.82843i q^{20} -10.0000 q^{22} +(-19.5959 - 11.3137i) q^{23} +(-11.5000 - 19.9186i) q^{25} +(18.3712 + 10.6066i) q^{26} +(2.00000 + 13.8564i) q^{28} -22.6274i q^{29} +(-1.50000 - 2.59808i) q^{31} +(-4.89898 + 2.82843i) q^{32} -16.0000 q^{34} +(6.12372 + 7.77817i) q^{35} +(-8.50000 + 14.7224i) q^{37} +(15.9217 - 9.19239i) q^{38} +(-2.00000 + 3.46410i) q^{40} -80.6102i q^{41} -85.0000 q^{43} +(-12.2474 - 7.07107i) q^{44} +(-16.0000 - 27.7128i) q^{46} +(62.4620 + 36.0624i) q^{47} +(35.5000 + 33.7750i) q^{49} -32.5269i q^{50} +(15.0000 + 25.9808i) q^{52} +(29.3939 - 16.9706i) q^{53} -10.0000 q^{55} +(-7.34847 + 18.3848i) q^{56} +(16.0000 - 27.7128i) q^{58} +(-78.3837 + 45.2548i) q^{59} +(36.0000 - 62.3538i) q^{61} -4.24264i q^{62} -8.00000 q^{64} +(18.3712 + 10.6066i) q^{65} +(-21.5000 - 37.2391i) q^{67} +(-19.5959 - 11.3137i) q^{68} +(2.00000 + 13.8564i) q^{70} +52.3259i q^{71} +(47.5000 + 82.2724i) q^{73} +(-20.8207 + 12.0208i) q^{74} +26.0000 q^{76} +(-48.9898 + 7.07107i) q^{77} +(-34.5000 + 59.7558i) q^{79} +(-4.89898 + 2.82843i) q^{80} +(57.0000 - 98.7269i) q^{82} +60.8112i q^{83} -16.0000 q^{85} +(-104.103 - 60.1041i) q^{86} +(-10.0000 - 17.3205i) q^{88} +(117.576 + 67.8823i) q^{89} +(97.5000 + 38.9711i) q^{91} -45.2548i q^{92} +(51.0000 + 88.3346i) q^{94} +(15.9217 - 9.19239i) q^{95} +16.0000 q^{97} +(19.5959 + 66.4680i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 26 q^{7}+O(q^{10}) $ 4 * q + 4 * q^4 + 26 * q^7	$ 4 q + 4 q^{4} + 26 q^{7} + 4 q^{10} + 60 q^{13} - 8 q^{16} + 26 q^{19} - 40 q^{22} - 46 q^{25} + 8 q^{28} - 6 q^{31} - 64 q^{34} - 34 q^{37} - 8 q^{40} - 340 q^{43} - 64 q^{46} + 142 q^{49} + 60 q^{52} - 40 q^{55} + 64 q^{58} + 144 q^{61} - 32 q^{64} - 86 q^{67} + 8 q^{70} + 190 q^{73} + 104 q^{76} - 138 q^{79} + 228 q^{82} - 64 q^{85} - 40 q^{88} + 390 q^{91} + 204 q^{94} + 64 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 26 * q^7 + 4 * q^10 + 60 * q^13 - 8 * q^16 + 26 * q^19 - 40 * q^22 - 46 * q^25 + 8 * q^28 - 6 * q^31 - 64 * q^34 - 34 * q^37 - 8 * q^40 - 340 * q^43 - 64 * q^46 + 142 * q^49 + 60 * q^52 - 40 * q^55 + 64 * q^58 + 144 * q^61 - 32 * q^64 - 86 * q^67 + 8 * q^70 + 190 * q^73 + 104 * q^76 - 138 * q^79 + 228 * q^82 - 64 * q^85 - 40 * q^88 + 390 * q^91 + 204 * q^94 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$.

$n$	$29$	$73$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	1.22474	+	0.707107i	0.612372	+	0.353553i
$2$	1.22474	+	0.707107i	0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	1.00000	+	1.73205i	0.250000	+	0.433013i
$5$	1.22474	+	0.707107i	0.244949	+	0.141421i	0.617449	−	0.786611i	$-0.288166\pi$
$5$	1.22474	+	0.707107i	0.244949	+	0.141421i	−0.372500	+	0.928032i	$0.621499\pi$
$6$		0			0
$7$	6.50000	+	2.59808i	0.928571	+	0.371154i
$7$	6.50000	+	2.59808i	0.928571	+	0.371154i
$8$			2.82843i			0.353553i
$9$		0			0
$10$	1.00000	+	1.73205i	0.100000	+	0.173205i
$11$	−6.12372	+	3.53553i	−0.556702	+	0.321412i	−0.751821	−	0.659367i	$-0.770824\pi$
$11$	−6.12372	+	3.53553i	−0.556702	+	0.321412i	0.195119	+	0.980780i	$0.437491\pi$
$12$		0			0
$13$	15.0000			1.15385			0.576923	−	0.816798i	$-0.304253\pi$
$13$	15.0000			1.15385			0.576923	+	0.816798i	$0.304253\pi$
$14$	6.12372	+	7.77817i	0.437409	+	0.555584i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.125000	+	0.216506i
$17$	−9.79796	+	5.65685i	−0.576351	+	0.332756i	−0.759682	−	0.650295i	$-0.774645\pi$
$17$	−9.79796	+	5.65685i	−0.576351	+	0.332756i	0.183331	+	0.983051i	$0.441312\pi$
$18$		0			0
$19$	6.50000	−	11.2583i	0.342105	−	0.592544i	−0.642718	−	0.766103i	$-0.722193\pi$
$19$	6.50000	−	11.2583i	0.342105	−	0.592544i	0.984823	+	0.173559i	$0.0555267\pi$
$20$			2.82843i			0.141421i
$21$		0			0
$22$	−10.0000			−0.454545
$23$	−19.5959	−	11.3137i	−0.851996	−	0.491900i	0.00932753	−	0.999956i	$-0.497031\pi$
$23$	−19.5959	−	11.3137i	−0.851996	−	0.491900i	−0.861324	+	0.508056i	$0.830364\pi$
$24$		0			0
$25$	−11.5000	−	19.9186i	−0.460000	−	0.796743i
$26$	18.3712	+	10.6066i	0.706584	+	0.407946i
$27$		0			0
$28$	2.00000	+	13.8564i	0.0714286	+	0.494872i
$29$		−	22.6274i		−	0.780256i	−0.920761	−	0.390128i	$-0.872431\pi$
$29$		−	22.6274i		−	0.780256i	0.920761	−	0.390128i	$-0.127569\pi$
$30$		0			0
$31$	−1.50000	−	2.59808i	−0.0483871	−	0.0838089i	0.840817	−	0.541319i	$-0.182075\pi$
$31$	−1.50000	−	2.59808i	−0.0483871	−	0.0838089i	−0.889205	+	0.457510i	$0.848741\pi$
$32$	−4.89898	+	2.82843i	−0.153093	+	0.0883883i
$33$		0			0
$34$	−16.0000			−0.470588
$35$	6.12372	+	7.77817i	0.174964	+	0.222234i
$36$		0			0
$37$	−8.50000	+	14.7224i	−0.229730	+	0.397904i	−0.957728	−	0.287675i	$-0.907118\pi$
$37$	−8.50000	+	14.7224i	−0.229730	+	0.397904i	0.727998	+	0.685579i	$0.240451\pi$
$38$	15.9217	−	9.19239i	0.418992	−	0.241905i
$39$		0			0
$40$	−2.00000	+	3.46410i	−0.0500000	+	0.0866025i
$41$		−	80.6102i		−	1.96610i	−0.183333	−	0.983051i	$-0.558689\pi$
$41$		−	80.6102i		−	1.96610i	0.183333	−	0.983051i	$-0.441311\pi$
$42$		0			0
$43$	−85.0000			−1.97674			−0.988372	−	0.152055i	$-0.951411\pi$
$43$	−85.0000			−1.97674			−0.988372	+	0.152055i	$0.951411\pi$
$44$	−12.2474	−	7.07107i	−0.278351	−	0.160706i
$45$		0			0
$46$	−16.0000	−	27.7128i	−0.347826	−	0.602452i
$47$	62.4620	+	36.0624i	1.32898	+	0.767286i	0.985142	−	0.171743i	$-0.0549400\pi$
$47$	62.4620	+	36.0624i	1.32898	+	0.767286i	0.343837	+	0.939029i	$0.388273\pi$
$48$		0			0
$49$	35.5000	+	33.7750i	0.724490	+	0.689286i
$50$		−	32.5269i		−	0.650538i
$51$		0			0
$52$	15.0000	+	25.9808i	0.288462	+	0.499630i
$53$	29.3939	−	16.9706i	0.554601	−	0.320199i	−0.196374	−	0.980529i	$-0.562917\pi$
$53$	29.3939	−	16.9706i	0.554601	−	0.320199i	0.750976	+	0.660330i	$0.229583\pi$
$54$		0			0
$55$	−10.0000			−0.181818
$56$	−7.34847	+	18.3848i	−0.131223	+	0.328300i
$57$		0			0
$58$	16.0000	−	27.7128i	0.275862	−	0.477807i
$59$	−78.3837	+	45.2548i	−1.32854	+	0.767031i	−0.985073	−	0.172135i	$-0.944933\pi$
$59$	−78.3837	+	45.2548i	−1.32854	+	0.767031i	−0.343463	+	0.939166i	$0.611600\pi$
$60$		0			0
$61$	36.0000	−	62.3538i	0.590164	−	1.02219i	−0.404046	−	0.914739i	$-0.632396\pi$
$61$	36.0000	−	62.3538i	0.590164	−	1.02219i	0.994210	−	0.107455i	$-0.0342702\pi$
$62$		−	4.24264i		−	0.0684297i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	18.3712	+	10.6066i	0.282633	+	0.163178i
$66$		0			0
$67$	−21.5000	−	37.2391i	−0.320896	−	0.555807i	0.659778	−	0.751461i	$-0.270651\pi$
$67$	−21.5000	−	37.2391i	−0.320896	−	0.555807i	−0.980673	+	0.195654i	$0.937317\pi$
$68$	−19.5959	−	11.3137i	−0.288175	−	0.166378i
$69$		0			0
$70$	2.00000	+	13.8564i	0.0285714	+	0.197949i
$71$			52.3259i			0.736985i	0.929631	+	0.368492i	$0.120126\pi$
$71$			52.3259i			0.736985i	−0.929631	+	0.368492i	$0.879874\pi$
$72$		0			0
$73$	47.5000	+	82.2724i	0.650685	+	1.12702i	0.982957	+	0.183836i	$0.0588515\pi$
$73$	47.5000	+	82.2724i	0.650685	+	1.12702i	−0.332272	+	0.943184i	$0.607815\pi$
$74$	−20.8207	+	12.0208i	−0.281360	+	0.162443i
$75$		0			0
$76$	26.0000			0.342105
$77$	−48.9898	+	7.07107i	−0.636231	+	0.0918320i
$78$		0			0
$79$	−34.5000	+	59.7558i	−0.436709	+	0.756402i	−0.997433	−	0.0716005i	$-0.977189\pi$
$79$	−34.5000	+	59.7558i	−0.436709	+	0.756402i	0.560725	+	0.828002i	$0.310523\pi$
$80$	−4.89898	+	2.82843i	−0.0612372	+	0.0353553i
$81$		0			0
$82$	57.0000	−	98.7269i	0.695122	−	1.20399i
$83$			60.8112i			0.732665i	0.930484	+	0.366332i	$0.119387\pi$
$83$			60.8112i			0.732665i	−0.930484	+	0.366332i	$0.880613\pi$
$84$		0			0
$85$	−16.0000			−0.188235
$86$	−104.103	−	60.1041i	−1.21050	−	0.698885i
$87$		0			0
$88$	−10.0000	−	17.3205i	−0.113636	−	0.196824i
$89$	117.576	+	67.8823i	1.32107	+	0.762722i	0.983900	−	0.178721i	$-0.0571958\pi$
$89$	117.576	+	67.8823i	1.32107	+	0.762722i	0.337173	+	0.941443i	$0.390529\pi$
$90$		0			0
$91$	97.5000	+	38.9711i	1.07143	+	0.428254i
$92$		−	45.2548i		−	0.491900i
$93$		0			0
$94$	51.0000	+	88.3346i	0.542553	+	0.939730i
$95$	15.9217	−	9.19239i	0.167597	−	0.0967620i
$96$		0			0
$97$	16.0000			0.164948			0.0824742	−	0.996593i	$-0.473718\pi$
$97$	16.0000			0.164948			0.0824742	+	0.996593i	$0.473718\pi$
$98$	19.5959	+	66.4680i	0.199958	+	0.678245i
$99$		0			0
$100$	23.0000	−	39.8372i	0.230000	−	0.398372i
$101$	−60.0125	+	34.6482i	−0.594183	+	0.343052i	−0.766750	−	0.641946i	$-0.778127\pi$
$101$	−60.0125	+	34.6482i	−0.594183	+	0.343052i	0.172567	+	0.984998i	$0.444794\pi$
$102$		0			0
$103$	30.5000	−	52.8275i	0.296117	−	0.512889i	−0.679127	−	0.734020i	$-0.737642\pi$
$103$	30.5000	−	52.8275i	0.296117	−	0.512889i	0.975244	+	0.221131i	$0.0709750\pi$
$104$			42.4264i			0.407946i
$105$		0			0
$106$	48.0000			0.452830
$107$	146.969	+	84.8528i	1.37355	+	0.793017i	0.991373	−	0.131074i	$-0.0418425\pi$
$107$	146.969	+	84.8528i	1.37355	+	0.793017i	0.382173	+	0.924091i	$0.375176\pi$
$108$		0			0
$109$	32.5000	+	56.2917i	0.298165	+	0.516437i	0.975716	−	0.219039i	$-0.0702921\pi$
$109$	32.5000	+	56.2917i	0.298165	+	0.516437i	−0.677551	+	0.735476i	$0.736959\pi$
$110$	−12.2474	−	7.07107i	−0.111340	−	0.0642824i
$111$		0			0
$112$	−22.0000	+	17.3205i	−0.196429	+	0.154647i
$113$		−	137.179i		−	1.21397i	−0.794713	−	0.606985i	$-0.792379\pi$
$113$		−	137.179i		−	1.21397i	0.794713	−	0.606985i	$-0.207621\pi$
$114$		0			0
$115$	−16.0000	−	27.7128i	−0.139130	−	0.240981i
$116$	39.1918	−	22.6274i	0.337861	−	0.195064i
$117$		0			0
$118$	−128.000			−1.08475
$119$	−78.3837	+	11.3137i	−0.658686	+	0.0950732i
$120$		0			0
$121$	−35.5000	+	61.4878i	−0.293388	+	0.508164i
$122$	88.1816	−	50.9117i	0.722800	−	0.417309i
$123$		0			0
$124$	3.00000	−	5.19615i	0.0241935	−	0.0419045i
$125$		−	67.8823i		−	0.543058i
$126$		0			0
$127$	−171.000			−1.34646			−0.673228	−	0.739435i	$-0.735093\pi$
$127$	−171.000			−1.34646			−0.673228	+	0.739435i	$0.735093\pi$
$128$	−9.79796	−	5.65685i	−0.0765466	−	0.0441942i
$129$		0			0
$130$	15.0000	+	25.9808i	0.115385	+	0.199852i
$131$	−101.654	−	58.6899i	−0.775983	−	0.448014i	0.0590215	−	0.998257i	$-0.481202\pi$
$131$	−101.654	−	58.6899i	−0.775983	−	0.448014i	−0.835005	+	0.550242i	$0.814535\pi$
$132$		0			0
$133$	71.5000	−	56.2917i	0.537594	−	0.423245i
$134$		−	60.8112i		−	0.453815i
$135$		0			0
$136$	−16.0000	−	27.7128i	−0.117647	−	0.203771i
$137$	186.161	−	107.480i	1.35884	−	0.784527i	0.369373	−	0.929281i	$-0.379572\pi$
$137$	186.161	−	107.480i	1.35884	−	0.784527i	0.989468	+	0.144754i	$0.0462390\pi$
$138$		0			0
$139$	−83.0000			−0.597122			−0.298561	−	0.954391i	$-0.596507\pi$
$139$	−83.0000			−0.597122			−0.298561	+	0.954391i	$0.596507\pi$
$140$	−7.34847	+	18.3848i	−0.0524891	+	0.131320i
$141$		0			0
$142$	−37.0000	+	64.0859i	−0.260563	+	0.451309i
$143$	−91.8559	+	53.0330i	−0.642349	+	0.370860i
$144$		0			0
$145$	16.0000	−	27.7128i	0.110345	−	0.191123i
$146$			134.350i			0.920207i
$147$		0			0
$148$	−34.0000			−0.229730
$149$	−88.1816	−	50.9117i	−0.591823	−	0.341689i	0.173995	−	0.984747i	$-0.444332\pi$
$149$	−88.1816	−	50.9117i	−0.591823	−	0.341689i	−0.765818	+	0.643057i	$0.777666\pi$
$150$		0			0
$151$	−20.0000	−	34.6410i	−0.132450	−	0.229411i	0.792170	−	0.610300i	$-0.208951\pi$
$151$	−20.0000	−	34.6410i	−0.132450	−	0.229411i	−0.924621	+	0.380889i	$0.875618\pi$
$152$	31.8434	+	18.3848i	0.209496	+	0.120952i
$153$		0			0
$154$	−65.0000	−	25.9808i	−0.422078	−	0.168706i
$155$		−	4.24264i		−	0.0273719i
$156$		0			0
$157$	−148.000	−	256.344i	−0.942675	−	1.63276i	−0.760340	−	0.649525i	$-0.774968\pi$
$157$	−148.000	−	256.344i	−0.942675	−	1.63276i	−0.182335	−	0.983237i	$-0.558365\pi$
$158$	−84.5074	+	48.7904i	−0.534857	+	0.308800i
$159$		0			0
$160$	−8.00000			−0.0500000
$161$	−97.9796	−	124.451i	−0.608569	−	0.772986i
$162$		0			0
$163$	−64.0000	+	110.851i	−0.392638	+	0.680069i	−0.992797	−	0.119812i	$-0.961771\pi$
$163$	−64.0000	+	110.851i	−0.392638	+	0.680069i	0.600159	+	0.799881i	$0.295104\pi$
$164$	139.621	−	80.6102i	0.851347	−	0.491525i
$165$		0			0
$166$	−43.0000	+	74.4782i	−0.259036	+	0.448664i
$167$		−	60.8112i		−	0.364139i	−0.983286	−	0.182069i	$-0.941720\pi$
$167$		−	60.8112i		−	0.364139i	0.983286	−	0.182069i	$-0.0582796\pi$
$168$		0			0
$169$	56.0000			0.331361
$170$	−19.5959	−	11.3137i	−0.115270	−	0.0665512i
$171$		0			0
$172$	−85.0000	−	147.224i	−0.494186	−	0.855955i
$173$	−97.9796	−	56.5685i	−0.566356	−	0.326986i	0.189337	−	0.981912i	$-0.439366\pi$
$173$	−97.9796	−	56.5685i	−0.566356	−	0.326986i	−0.755693	+	0.654926i	$0.772700\pi$
$174$		0			0
$175$	−23.0000	−	159.349i	−0.131429	−	0.910564i
$176$		−	28.2843i		−	0.160706i
$177$		0			0
$178$	96.0000	+	166.277i	0.539326	+	0.934140i
$179$	−25.7196	+	14.8492i	−0.143685	+	0.0829567i	−0.570119	−	0.821562i	$-0.693103\pi$
$179$	−25.7196	+	14.8492i	−0.143685	+	0.0829567i	0.426434	+	0.904519i	$0.359770\pi$
$180$		0			0
$181$	81.0000			0.447514			0.223757	−	0.974645i	$-0.428168\pi$
$181$	81.0000			0.447514			0.223757	+	0.974645i	$0.428168\pi$
$182$	91.8559	+	116.673i	0.504703	+	0.641058i
$183$		0			0
$184$	32.0000	−	55.4256i	0.173913	−	0.301226i
$185$	−20.8207	+	12.0208i	−0.112544	+	0.0649774i
$186$		0			0
$187$	40.0000	−	69.2820i	0.213904	−	0.370492i
$188$			144.250i			0.767286i
$189$		0			0
$190$	26.0000			0.136842
$191$	55.1135	+	31.8198i	0.288552	+	0.166596i	0.637289	−	0.770625i	$-0.280056\pi$
$191$	55.1135	+	31.8198i	0.288552	+	0.166596i	−0.348736	+	0.937221i	$0.613389\pi$
$192$		0			0
$193$	111.500	+	193.124i	0.577720	+	1.00064i	0.995740	+	0.0922029i	$0.0293909\pi$
$193$	111.500	+	193.124i	0.577720	+	1.00064i	−0.418020	+	0.908438i	$0.637276\pi$
$194$	19.5959	+	11.3137i	0.101010	+	0.0583181i
$195$		0			0
$196$	−23.0000	+	95.2628i	−0.117347	+	0.486035i
$197$			158.392i			0.804020i	0.915635	+	0.402010i	$0.131688\pi$
$197$			158.392i			0.804020i	−0.915635	+	0.402010i	$0.868312\pi$
$198$		0			0
$199$	68.0000	+	117.779i	0.341709	+	0.591857i	0.984750	−	0.173975i	$-0.0556611\pi$
$199$	68.0000	+	117.779i	0.341709	+	0.591857i	−0.643042	+	0.765831i	$0.722328\pi$
$200$	56.3383	−	32.5269i	0.281691	−	0.162635i
$201$		0			0
$202$	−98.0000			−0.485149
$203$	58.7878	−	147.078i	0.289595	−	0.724523i
$204$		0			0
$205$	57.0000	−	98.7269i	0.278049	−	0.481595i
$206$	74.7094	−	43.1335i	0.362667	−	0.209386i
$207$		0			0
$208$	−30.0000	+	51.9615i	−0.144231	+	0.249815i
$209$			91.9239i			0.439827i
$210$		0			0
$211$	272.000			1.28910			0.644550	−	0.764562i	$-0.277045\pi$
$211$	272.000			1.28910			0.644550	+	0.764562i	$0.277045\pi$
$212$	58.7878	+	33.9411i	0.277301	+	0.160100i
$213$		0			0
$214$	120.000	+	207.846i	0.560748	+	0.971243i
$215$	−104.103	−	60.1041i	−0.484201	−	0.279554i
$216$		0			0
$217$	−3.00000	−	20.7846i	−0.0138249	−	0.0957816i
$218$			91.9239i			0.421669i
$219$		0			0
$220$	−10.0000	−	17.3205i	−0.0454545	−	0.0787296i
$221$	−146.969	+	84.8528i	−0.665020	+	0.383949i
$222$		0			0
$223$	248.000			1.11211			0.556054	−	0.831146i	$-0.312315\pi$
$223$	248.000			1.11211			0.556054	+	0.831146i	$0.312315\pi$
$224$	−39.1918	+	5.65685i	−0.174964	+	0.0252538i
$225$		0			0
$226$	97.0000	−	168.009i	0.429204	−	0.743402i
$227$	143.295	−	82.7315i	0.631256	−	0.364456i	−0.149982	−	0.988689i	$-0.547922\pi$
$227$	143.295	−	82.7315i	0.631256	−	0.364456i	0.781238	+	0.624233i	$0.214588\pi$
$228$		0			0
$229$	−216.500	+	374.989i	−0.945415	+	1.63751i	−0.190496	+	0.981688i	$0.561010\pi$
$229$	−216.500	+	374.989i	−0.945415	+	1.63751i	−0.754918	+	0.655819i	$0.772324\pi$
$230$		−	45.2548i		−	0.196760i
$231$		0			0
$232$	64.0000			0.275862
$233$	197.184	+	113.844i	0.846283	+	0.488602i	0.859395	−	0.511312i	$-0.170840\pi$
$233$	197.184	+	113.844i	0.846283	+	0.488602i	−0.0131120	+	0.999914i	$0.504174\pi$
$234$		0			0
$235$	51.0000	+	88.3346i	0.217021	+	0.375892i
$236$	−156.767	−	90.5097i	−0.664268	−	0.383516i
$237$		0			0
$238$	−104.000	−	41.5692i	−0.436975	−	0.174661i
$239$			343.654i			1.43788i	0.695071	+	0.718941i	$0.255373\pi$
$239$			343.654i			1.43788i	−0.695071	+	0.718941i	$0.744627\pi$
$240$		0			0
$241$	−79.0000	−	136.832i	−0.327801	−	0.567768i	0.654274	−	0.756257i	$-0.272974\pi$
$241$	−79.0000	−	136.832i	−0.327801	−	0.567768i	−0.982075	+	0.188490i	$0.939641\pi$
$242$	−86.9569	+	50.2046i	−0.359326	+	0.207457i
$243$		0			0
$244$	144.000			0.590164
$245$	19.5959	+	66.4680i	0.0799833	+	0.271298i
$246$		0			0
$247$	97.5000	−	168.875i	0.394737	−	0.683704i
$248$	7.34847	−	4.24264i	0.0296309	−	0.0171074i
$249$		0			0
$250$	48.0000	−	83.1384i	0.192000	−	0.332554i
$251$		−	237.588i		−	0.946565i	−0.880911	−	0.473283i	$-0.843069\pi$
$251$		−	237.588i		−	0.946565i	0.880911	−	0.473283i	$-0.156931\pi$
$252$		0			0
$253$	160.000			0.632411
$254$	−209.431	−	120.915i	−0.824533	−	0.476044i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.0312500	−	0.0541266i
$257$	275.568	+	159.099i	1.07225	+	0.619062i	0.928795	−	0.370595i	$-0.120846\pi$
$257$	275.568	+	159.099i	1.07225	+	0.619062i	0.143453	+	0.989657i	$0.454179\pi$
$258$		0			0
$259$	−93.5000	+	73.6122i	−0.361004	+	0.284217i
$260$			42.4264i			0.163178i
$261$		0			0
$262$	−83.0000	−	143.760i	−0.316794	−	0.548703i
$263$	−146.969	+	84.8528i	−0.558819	+	0.322634i	−0.752671	−	0.658396i	$-0.771235\pi$
$263$	−146.969	+	84.8528i	−0.558819	+	0.322634i	0.193852	+	0.981031i	$0.437902\pi$
$264$		0			0
$265$	48.0000			0.181132
$266$	127.373	−	18.3848i	0.478848	−	0.0691157i
$267$		0			0
$268$	43.0000	−	74.4782i	0.160448	−	0.277904i
$269$	−420.087	+	242.538i	−1.56166	+	0.901627i	−0.564574	+	0.825382i	$0.690960\pi$
$269$	−420.087	+	242.538i	−1.56166	+	0.901627i	−0.997089	+	0.0762447i	$0.975707\pi$
$270$		0			0
$271$	−108.000	+	187.061i	−0.398524	+	0.690264i	−0.993544	−	0.113447i	$-0.963811\pi$
$271$	−108.000	+	187.061i	−0.398524	+	0.690264i	0.595020	+	0.803711i	$0.297144\pi$
$272$		−	45.2548i		−	0.166378i
$273$		0			0
$274$	304.000			1.10949
$275$	140.846	+	81.3173i	0.512166	+	0.295699i
$276$		0			0
$277$	−152.500	−	264.138i	−0.550542	−	0.953566i	−0.998236	−	0.0593790i	$-0.981088\pi$
$277$	−152.500	−	264.138i	−0.550542	−	0.953566i	0.447694	−	0.894187i	$-0.352245\pi$
$278$	−101.654	−	58.6899i	−0.365661	−	0.211115i
$279$		0			0
$280$	−22.0000	+	17.3205i	−0.0785714	+	0.0618590i
$281$			22.6274i			0.0805246i	0.999189	+	0.0402623i	$0.0128194\pi$
$281$			22.6274i			0.0805246i	−0.999189	+	0.0402623i	$0.987181\pi$
$282$		0			0
$283$	−78.5000	−	135.966i	−0.277385	−	0.480445i	0.693349	−	0.720602i	$-0.256134\pi$
$283$	−78.5000	−	135.966i	−0.277385	−	0.480445i	−0.970734	+	0.240157i	$0.922801\pi$
$284$	−90.6311	+	52.3259i	−0.319124	+	0.184246i
$285$		0			0
$286$	−150.000			−0.524476
$287$	209.431	−	523.966i	0.729726	−	1.82567i
$288$		0			0
$289$	−80.5000	+	139.430i	−0.278547	+	0.482457i
$290$	39.1918	−	22.6274i	0.135144	−	0.0780256i
$291$		0			0
$292$	−95.0000	+	164.545i	−0.325342	+	0.563510i
$293$			101.823i			0.347520i	0.984788	+	0.173760i	$0.0555917\pi$
$293$			101.823i			0.347520i	−0.984788	+	0.173760i	$0.944408\pi$
$294$		0			0
$295$	−128.000			−0.433898
$296$	−41.6413	−	24.0416i	−0.140680	−	0.0812217i
$297$		0			0
$298$	−72.0000	−	124.708i	−0.241611	−	0.418482i
$299$	−293.939	−	169.706i	−0.983073	−	0.567577i
$300$		0			0
$301$	−552.500	−	220.836i	−1.83555	−	0.733676i
$302$		−	56.5685i		−	0.187313i
$303$		0			0
$304$	26.0000	+	45.0333i	0.0855263	+	0.148136i
$305$	88.1816	−	50.9117i	0.289120	−	0.166924i
$306$		0			0
$307$	−11.0000			−0.0358306			−0.0179153	−	0.999840i	$-0.505703\pi$
$307$	−11.0000			−0.0358306			−0.0179153	+	0.999840i	$0.505703\pi$
$308$	−61.2372	−	77.7817i	−0.198822	−	0.252538i
$309$		0			0
$310$	3.00000	−	5.19615i	0.00967742	−	0.0167618i
$311$	454.380	−	262.337i	1.46103	−	0.843526i	0.461971	−	0.886895i	$-0.347142\pi$
$311$	454.380	−	262.337i	1.46103	−	0.843526i	0.999059	+	0.0433690i	$0.0138091\pi$
$312$		0			0
$313$	−276.500	+	478.912i	−0.883387	+	1.53007i	−0.0358348	+	0.999358i	$0.511409\pi$
$313$	−276.500	+	478.912i	−0.883387	+	1.53007i	−0.847552	+	0.530713i	$0.821924\pi$
$314$		−	418.607i		−	1.33314i
$315$		0			0
$316$	−138.000			−0.436709
$317$	−117.576	−	67.8823i	−0.370901	−	0.214140i	0.302951	−	0.953006i	$-0.402028\pi$
$317$	−117.576	−	67.8823i	−0.370901	−	0.214140i	−0.673852	+	0.738866i	$0.735361\pi$
$318$		0			0
$319$	80.0000	+	138.564i	0.250784	+	0.434370i
$320$	−9.79796	−	5.65685i	−0.0306186	−	0.0176777i
$321$		0			0
$322$	−32.0000	−	221.703i	−0.0993789	−	0.688517i
$323$			147.078i			0.455350i
$324$		0			0
$325$	−172.500	−	298.779i	−0.530769	−	0.919319i
$326$	−156.767	+	90.5097i	−0.480881	+	0.277637i
$327$		0			0
$328$	228.000			0.695122
$329$	312.310	+	396.687i	0.949270	+	1.20574i
$330$		0			0
$331$	30.5000	−	52.8275i	0.0921450	−	0.159600i	−0.816268	−	0.577673i	$-0.803961\pi$
$331$	30.5000	−	52.8275i	0.0921450	−	0.159600i	0.908413	+	0.418073i	$0.137294\pi$
$332$	−105.328	+	60.8112i	−0.317253	+	0.183166i
$333$		0			0
$334$	43.0000	−	74.4782i	0.128743	−	0.222989i
$335$		−	60.8112i		−	0.181526i
$336$		0			0
$337$	135.000			0.400593			0.200297	−	0.979735i	$-0.435809\pi$
$337$	135.000			0.400593			0.200297	+	0.979735i	$0.435809\pi$
$338$	68.5857	+	39.5980i	0.202916	+	0.117154i
$339$		0			0
$340$	−16.0000	−	27.7128i	−0.0470588	−	0.0815083i
$341$	18.3712	+	10.6066i	0.0538744	+	0.0311044i
$342$		0			0
$343$	143.000	+	311.769i	0.416910	+	0.908948i
$344$		−	240.416i		−	0.698885i
$345$		0			0
$346$	−80.0000	−	138.564i	−0.231214	−	0.400474i
$347$	88.1816	−	50.9117i	0.254126	−	0.146720i	−0.367526	−	0.930013i	$-0.619795\pi$
$347$	88.1816	−	50.9117i	0.254126	−	0.146720i	0.621652	+	0.783294i	$0.286462\pi$
$348$		0			0
$349$	−152.000			−0.435530			−0.217765	−	0.976001i	$-0.569877\pi$
$349$	−152.000			−0.435530			−0.217765	+	0.976001i	$0.569877\pi$
$350$	84.5074	−	211.425i	0.241450	−	0.604071i
$351$		0			0
$352$	20.0000	−	34.6410i	0.0568182	−	0.0984120i
$353$	334.355	−	193.040i	0.947182	−	0.546856i	0.0549778	−	0.998488i	$-0.482491\pi$
$353$	334.355	−	193.040i	0.947182	−	0.546856i	0.892205	+	0.451632i	$0.149158\pi$
$354$		0			0
$355$	−37.0000	+	64.0859i	−0.104225	+	0.180524i
$356$			271.529i			0.762722i
$357$		0			0
$358$	−42.0000			−0.117318
$359$	−39.1918	−	22.6274i	−0.109169	−	0.0630290i	0.444421	−	0.895818i	$-0.353409\pi$
$359$	−39.1918	−	22.6274i	−0.109169	−	0.0630290i	−0.553591	+	0.832789i	$0.686743\pi$
$360$		0			0
$361$	96.0000	+	166.277i	0.265928	+	0.460601i
$362$	99.2043	+	57.2756i	0.274045	+	0.158220i
$363$		0			0
$364$	30.0000	+	207.846i	0.0824176	+	0.571006i
$365$			134.350i			0.368083i
$366$		0			0
$367$	−50.5000	−	87.4686i	−0.137602	−	0.238334i	0.788986	−	0.614411i	$-0.210606\pi$
$367$	−50.5000	−	87.4686i	−0.137602	−	0.238334i	−0.926588	+	0.376077i	$0.877273\pi$
$368$	78.3837	−	45.2548i	0.212999	−	0.122975i
$369$		0			0
$370$	−34.0000			−0.0918919
$371$	235.151	−	33.9411i	0.633830	−	0.0914855i
$372$		0			0
$373$	155.500	−	269.334i	0.416890	−	0.722075i	−0.578735	−	0.815516i	$-0.696453\pi$
$373$	155.500	−	269.334i	0.416890	−	0.722075i	0.995625	+	0.0934411i	$0.0297867\pi$
$374$	97.9796	−	56.5685i	0.261978	−	0.151253i
$375$		0			0
$376$	−102.000	+	176.669i	−0.271277	+	0.469865i
$377$		−	339.411i		−	0.900295i
$378$		0			0
$379$	91.0000			0.240106			0.120053	−	0.992768i	$-0.461694\pi$
$379$	91.0000			0.240106			0.120053	+	0.992768i	$0.461694\pi$
$380$	31.8434	+	18.3848i	0.0837983	+	0.0483810i
$381$		0			0
$382$	45.0000	+	77.9423i	0.117801	+	0.204037i
$383$	−284.141	−	164.049i	−0.741882	−	0.428326i	0.0808712	−	0.996725i	$-0.474230\pi$
$383$	−284.141	−	164.049i	−0.741882	−	0.428326i	−0.822753	+	0.568399i	$0.807563\pi$
$384$		0			0
$385$	−65.0000	−	25.9808i	−0.168831	−	0.0674825i
$386$			315.370i			0.817020i
$387$		0			0
$388$	16.0000	+	27.7128i	0.0412371	+	0.0714248i
$389$	−596.451	+	344.361i	−1.53329	+	0.885247i	−0.534085	+	0.845431i	$0.679344\pi$
$389$	−596.451	+	344.361i	−1.53329	+	0.885247i	−0.999207	+	0.0398163i	$0.987323\pi$
$390$		0			0
$391$	256.000			0.654731
$392$	−95.5301	+	100.409i	−0.243699	+	0.256146i
$393$		0			0
$394$	−112.000	+	193.990i	−0.284264	+	0.492360i
$395$	−84.5074	+	48.7904i	−0.213943	+	0.123520i
$396$		0			0
$397$	139.500	−	241.621i	0.351385	−	0.608617i	−0.635107	−	0.772424i	$-0.719044\pi$
$397$	139.500	−	241.621i	0.351385	−	0.608617i	0.986492	+	0.163807i	$0.0523774\pi$
$398$			192.333i			0.483249i
$399$		0			0
$400$	92.0000			0.230000
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−22.5000	−	38.9711i	−0.0558313	−	0.0967026i
$404$	−120.025	−	69.2965i	−0.297092	−	0.171526i
$405$		0			0
$406$	176.000	−	138.564i	0.433498	−	0.341291i
$407$		−	120.208i		−	0.295352i
$408$		0			0
$409$	111.500	+	193.124i	0.272616	+	0.472185i	0.969531	−	0.244969i	$-0.0787778\pi$
$409$	111.500	+	193.124i	0.272616	+	0.472185i	−0.696915	+	0.717154i	$0.745444\pi$
$410$	139.621	−	80.6102i	0.340539	−	0.196610i
$411$		0			0
$412$	122.000			0.296117
$413$	−627.069	+	90.5097i	−1.51833	+	0.219152i
$414$		0			0
$415$	−43.0000	+	74.4782i	−0.103614	+	0.179466i
$416$	−73.4847	+	42.4264i	−0.176646	+	0.101987i
$417$		0			0
$418$	−65.0000	+	112.583i	−0.155502	+	0.269338i
$419$			581.242i			1.38721i	0.720355	+	0.693606i	$0.243979\pi$
$419$			581.242i			1.38721i	−0.720355	+	0.693606i	$0.756021\pi$
$420$		0			0
$421$	153.000			0.363420			0.181710	−	0.983352i	$-0.441837\pi$
$421$	153.000			0.363420			0.181710	+	0.983352i	$0.441837\pi$
$422$	333.131	+	192.333i	0.789409	+	0.455766i
$423$		0			0
$424$	48.0000	+	83.1384i	0.113208	+	0.196081i
$425$	225.353	+	130.108i	0.530242	+	0.306136i
$426$		0			0
$427$	396.000	−	311.769i	0.927400	−	0.730139i
$428$			339.411i			0.793017i
$429$		0			0
$430$	−85.0000	−	147.224i	−0.197674	−	0.342382i
$431$	−121.250	+	70.0036i	−0.281322	+	0.162421i	−0.634022	−	0.773315i	$-0.718597\pi$
$431$	−121.250	+	70.0036i	−0.281322	+	0.162421i	0.352700	+	0.935737i	$0.385264\pi$
$432$		0			0
$433$	137.000			0.316397			0.158199	−	0.987407i	$-0.449431\pi$
$433$	137.000			0.316397			0.158199	+	0.987407i	$0.449431\pi$
$434$	11.0227	−	27.5772i	0.0253979	−	0.0635419i
$435$		0			0
$436$	−65.0000	+	112.583i	−0.149083	+	0.258219i
$437$	−254.747	+	147.078i	−0.582945	+	0.336563i
$438$		0			0
$439$	300.000	−	519.615i	0.683371	−	1.18363i	−0.290574	−	0.956852i	$-0.593846\pi$
$439$	300.000	−	519.615i	0.683371	−	1.18363i	0.973946	−	0.226781i	$-0.0728203\pi$
$440$		−	28.2843i		−	0.0642824i
$441$		0			0
$442$	−240.000			−0.542986
$443$	−548.686	−	316.784i	−1.23857	−	0.715088i	−0.269767	−	0.962926i	$-0.586947\pi$
$443$	−548.686	−	316.784i	−1.23857	−	0.715088i	−0.968801	+	0.247838i	$0.920280\pi$
$444$		0			0
$445$	96.0000	+	166.277i	0.215730	+	0.373656i
$446$	303.737	+	175.362i	0.681024	+	0.393189i
$447$		0			0
$448$	−52.0000	−	20.7846i	−0.116071	−	0.0463942i
$449$		−	383.252i		−	0.853568i	−0.904354	−	0.426784i	$-0.859647\pi$
$449$		−	383.252i		−	0.853568i	0.904354	−	0.426784i	$-0.140353\pi$
$450$		0			0
$451$	285.000	+	493.634i	0.631929	+	1.09453i
$452$	237.601	−	137.179i	0.525665	−	0.303493i
$453$		0			0
$454$	234.000			0.515419
$455$	91.8559	+	116.673i	0.201881	+	0.256423i
$456$		0			0
$457$	119.500	−	206.980i	0.261488	−	0.452910i	−0.705150	−	0.709059i	$-0.749120\pi$
$457$	119.500	−	206.980i	0.261488	−	0.452910i	0.966638	+	0.256148i	$0.0824535\pi$
$458$	−530.315	+	306.177i	−1.15789	+	0.668509i
$459$		0			0
$460$	32.0000	−	55.4256i	0.0695652	−	0.120490i
$461$			452.548i			0.981667i	0.871253	+	0.490833i	$0.163308\pi$
$461$			452.548i			0.981667i	−0.871253	+	0.490833i	$0.836692\pi$
$462$		0			0
$463$	211.000			0.455724			0.227862	−	0.973693i	$-0.426827\pi$
$463$	211.000			0.455724			0.227862	+	0.973693i	$0.426827\pi$
$464$	78.3837	+	45.2548i	0.168930	+	0.0975320i
$465$		0			0
$466$	161.000	+	278.860i	0.345494	+	0.598412i
$467$	−270.669	−	156.271i	−0.579590	−	0.334627i	0.181380	−	0.983413i	$-0.441944\pi$
$467$	−270.669	−	156.271i	−0.579590	−	0.334627i	−0.760971	+	0.648786i	$0.775277\pi$
$468$		0			0
$469$	−43.0000	−	297.913i	−0.0916844	−	0.635208i
$470$			144.250i			0.306914i
$471$		0			0
$472$	−128.000	−	221.703i	−0.271186	−	0.469709i
$473$	520.517	−	300.520i	1.10046	−	0.635350i
$474$		0			0
$475$	−299.000			−0.629474
$476$	−97.9796	−	124.451i	−0.205839	−	0.261451i
$477$		0			0
$478$	−243.000	+	420.888i	−0.508368	+	0.880520i
$479$	607.473	−	350.725i	1.26821	−	0.732202i	0.293562	−	0.955940i	$-0.405159\pi$
$479$	607.473	−	350.725i	1.26821	−	0.732202i	0.974649	+	0.223737i	$0.0718258\pi$
$480$		0			0
$481$	−127.500	+	220.836i	−0.265073	+	0.459119i
$482$		−	223.446i		−	0.463580i
$483$		0			0
$484$	−142.000			−0.293388
$485$	19.5959	+	11.3137i	0.0404040	+	0.0233272i
$486$		0			0
$487$	−209.500	−	362.865i	−0.430185	−	0.745102i	0.566704	−	0.823921i	$-0.308218\pi$
$487$	−209.500	−	362.865i	−0.430185	−	0.745102i	−0.996889	+	0.0788195i	$0.974885\pi$
$488$	176.363	+	101.823i	0.361400	+	0.208654i
$489$		0			0
$490$	−23.0000	+	95.2628i	−0.0469388	+	0.194414i
$491$		−	169.706i		−	0.345633i	−0.984954	−	0.172816i	$-0.944713\pi$
$491$		−	169.706i		−	0.345633i	0.984954	−	0.172816i	$-0.0552867\pi$
$492$		0			0
$493$	128.000	+	221.703i	0.259635	+	0.449701i
$494$	238.825	−	137.886i	0.483452	−	0.279121i
$495$		0			0
$496$	12.0000			0.0241935
$497$	−135.947	+	340.118i	−0.273535	+	0.684343i
$498$		0			0
$499$	−93.5000	+	161.947i	−0.187375	+	0.324543i	−0.944374	−	0.328873i	$-0.893331\pi$
$499$	−93.5000	+	161.947i	−0.187375	+	0.324543i	0.756999	+	0.653416i	$0.226665\pi$
$500$	117.576	−	67.8823i	0.235151	−	0.135765i
$501$		0			0
$502$	168.000	−	290.985i	0.334661	−	0.579650i
$503$		−	173.948i		−	0.345822i	−0.984937	−	0.172911i	$-0.944683\pi$
$503$		−	173.948i		−	0.345822i	0.984937	−	0.172911i	$-0.0553172\pi$
$504$		0			0
$505$	−98.0000			−0.194059
$506$	195.959	+	113.137i	0.387271	+	0.223591i
$507$		0			0
$508$	−171.000	−	296.181i	−0.336614	−	0.583033i
$509$	224.128	+	129.401i	0.440331	+	0.254225i	0.703738	−	0.710460i	$-0.251513\pi$
$509$	224.128	+	129.401i	0.440331	+	0.254225i	−0.263407	+	0.964685i	$0.584846\pi$
$510$		0			0
$511$	95.0000	+	658.179i	0.185910	+	1.28802i
$512$		−	22.6274i		−	0.0441942i
$513$		0			0
$514$	225.000	+	389.711i	0.437743	+	0.758193i
$515$	74.7094	−	43.1335i	0.145067	−	0.0837544i
$516$		0			0
$517$	−510.000			−0.986460
$518$	−166.565	+	24.0416i	−0.321555	+	0.0464124i
$519$		0			0
$520$	−30.0000	+	51.9615i	−0.0576923	+	0.0999260i
$521$	−352.727	+	203.647i	−0.677018	+	0.390877i	−0.798731	−	0.601689i	$-0.794495\pi$
$521$	−352.727	+	203.647i	−0.677018	+	0.390877i	0.121712	+	0.992565i	$0.461161\pi$
$522$		0			0
$523$	405.500	−	702.347i	0.775335	−	1.34292i	−0.159272	−	0.987235i	$-0.550915\pi$
$523$	405.500	−	702.347i	0.775335	−	1.34292i	0.934606	−	0.355684i	$-0.115752\pi$
$524$		−	234.759i		−	0.448014i
$525$		0			0
$526$	−240.000			−0.456274
$527$	29.3939	+	16.9706i	0.0557759	+	0.0322022i
$528$		0			0
$529$	−8.50000	−	14.7224i	−0.0160681	−	0.0278307i
$530$	58.7878	+	33.9411i	0.110920	+	0.0640399i
$531$		0			0
$532$	169.000	+	67.5500i	0.317669	+	0.126974i
$533$		−	1209.15i		−	2.26858i
$534$		0			0
$535$	120.000	+	207.846i	0.224299	+	0.388497i
$536$	105.328	−	60.8112i	0.196508	−	0.113454i
$537$		0			0
$538$	−686.000			−1.27509
$539$	−336.805	−	81.3173i	−0.624870	−	0.150867i
$540$		0			0
$541$	172.500	−	298.779i	0.318854	−	0.552271i	−0.661395	−	0.750038i	$-0.730035\pi$
$541$	172.500	−	298.779i	0.318854	−	0.552271i	0.980249	+	0.197766i	$0.0633687\pi$
$542$	−264.545	+	152.735i	−0.488090	+	0.281799i
$543$		0			0
$544$	32.0000	−	55.4256i	0.0588235	−	0.101885i
$545$			91.9239i			0.168668i
$546$		0			0
$547$	−864.000			−1.57952			−0.789762	−	0.613413i	$-0.789796\pi$
$547$	−864.000			−1.57952			−0.789762	+	0.613413i	$0.789796\pi$
$548$	372.322	+	214.960i	0.679421	+	0.392264i
$549$		0			0
$550$	115.000	+	199.186i	0.209091	+	0.362156i
$551$	−254.747	−	147.078i	−0.462336	−	0.266930i
$552$		0			0
$553$	−379.500	+	298.779i	−0.686257	+	0.540287i
$554$		−	431.335i		−	0.778583i
$555$		0			0
$556$	−83.0000	−	143.760i	−0.149281	−	0.258562i
$557$	821.804	−	474.469i	1.47541	−	0.851829i	0.475795	−	0.879556i	$-0.342160\pi$
$557$	821.804	−	474.469i	1.47541	−	0.851829i	0.999616	+	0.0277272i	$0.00882699\pi$
$558$		0			0
$559$	−1275.00			−2.28086
$560$	−39.1918	+	5.65685i	−0.0699854	+	0.0101015i
$561$		0			0
$562$	−16.0000	+	27.7128i	−0.0284698	+	0.0493111i
$563$	640.542	−	369.817i	1.13773	−	0.656868i	0.191862	−	0.981422i	$-0.438547\pi$
$563$	640.542	−	369.817i	1.13773	−	0.656868i	0.945867	+	0.324554i	$0.105214\pi$
$564$		0			0
$565$	97.0000	−	168.009i	0.171681	−	0.297361i
$566$		−	222.032i		−	0.392282i
$567$		0			0
$568$	−148.000			−0.260563
$569$	−665.036	−	383.959i	−1.16878	−	0.674796i	−0.215388	−	0.976528i	$-0.569102\pi$
$569$	−665.036	−	383.959i	−1.16878	−	0.674796i	−0.953393	+	0.301732i	$0.902435\pi$
$570$		0			0
$571$	−238.500	−	413.094i	−0.417688	−	0.723457i	0.578018	−	0.816024i	$-0.303826\pi$
$571$	−238.500	−	413.094i	−0.417688	−	0.723457i	−0.995706	+	0.0925665i	$0.970493\pi$
$572$	−183.712	−	106.066i	−0.321174	−	0.185430i
$573$		0			0
$574$	627.000	−	493.634i	1.09233	−	0.859990i
$575$			520.431i			0.905097i
$576$		0			0
$577$	131.500	+	227.765i	0.227903	+	0.394739i	0.957186	−	0.289472i	$-0.0934798\pi$
$577$	131.500	+	227.765i	0.227903	+	0.394739i	−0.729283	+	0.684212i	$0.760146\pi$
$578$	−197.184	+	113.844i	−0.341149	+	0.196962i
$579$		0			0
$580$	64.0000			0.110345
$581$	−157.992	+	395.273i	−0.271931	+	0.680332i
$582$		0			0
$583$	−120.000	+	207.846i	−0.205832	+	0.356511i
$584$	−232.702	+	134.350i	−0.398462	+	0.230052i
$585$		0			0
$586$	−72.0000	+	124.708i	−0.122867	+	0.212812i
$587$			531.744i			0.905868i	0.891544	+	0.452934i	$0.149623\pi$
$587$			531.744i			0.905868i	−0.891544	+	0.452934i	$0.850377\pi$
$588$		0			0
$589$	−39.0000			−0.0662139
$590$	−156.767	−	90.5097i	−0.265707	−	0.153406i
$591$		0			0
$592$	−34.0000	−	58.8897i	−0.0574324	−	0.0994759i
$593$	694.430	+	400.930i	1.17105	+	0.676104i	0.953926	−	0.300041i	$-0.0970004\pi$
$593$	694.430	+	400.930i	1.17105	+	0.676104i	0.217120	+	0.976145i	$0.430334\pi$
$594$		0			0
$595$	−104.000	−	41.5692i	−0.174790	−	0.0698642i
$596$		−	203.647i		−	0.341689i
$597$		0			0
$598$	−240.000	−	415.692i	−0.401338	−	0.695137i
$599$	−676.059	+	390.323i	−1.12865	+	0.651624i	−0.943594	−	0.331104i	$-0.892579\pi$
$599$	−676.059	+	390.323i	−1.12865	+	0.651624i	−0.185052	+	0.982729i	$0.559245\pi$
$600$		0			0
$601$	383.000			0.637271			0.318636	−	0.947877i	$-0.396775\pi$
$601$	383.000			0.637271			0.318636	+	0.947877i	$0.396775\pi$
$602$	−520.517	−	661.145i	−0.864645	−	1.09825i
$603$		0			0
$604$	40.0000	−	69.2820i	0.0662252	−	0.114705i
$605$	−86.9569	+	50.2046i	−0.143730	+	0.0829828i
$606$		0			0
$607$	382.500	−	662.509i	0.630148	−	1.09145i	−0.357373	−	0.933962i	$-0.616327\pi$
$607$	382.500	−	662.509i	0.630148	−	1.09145i	0.987521	−	0.157487i	$-0.0503392\pi$
$608$			73.5391i			0.120952i
$609$		0			0
$610$	144.000			0.236066
$611$	936.930	+	540.937i	1.53344	+	0.885330i
$612$		0			0
$613$	−204.000	−	353.338i	−0.332790	−	0.576408i	0.650268	−	0.759705i	$-0.274657\pi$
$613$	−204.000	−	353.338i	−0.332790	−	0.576408i	−0.983058	+	0.183296i	$0.941323\pi$
$614$	−13.4722	−	7.77817i	−0.0219417	−	0.0126680i
$615$		0			0
$616$	−20.0000	−	138.564i	−0.0324675	−	0.224942i
$617$		−	914.996i		−	1.48298i	−0.670966	−	0.741488i	$-0.734120\pi$
$617$		−	914.996i		−	1.48298i	0.670966	−	0.741488i	$-0.265880\pi$
$618$		0			0
$619$	473.500	+	820.126i	0.764943	+	1.32492i	0.940276	+	0.340412i	$0.110566\pi$
$619$	473.500	+	820.126i	0.764943	+	1.32492i	−0.175333	+	0.984509i	$0.556100\pi$
$620$	7.34847	−	4.24264i	0.0118524	−	0.00684297i
$621$		0			0
$622$	742.000			1.19293
$623$	587.878	+	746.705i	0.943624	+	1.19856i
$624$		0			0
$625$	−239.500	+	414.826i	−0.383200	+	0.663722i
$626$	−677.284	+	391.030i	−1.08192	+	0.624649i
$627$		0			0
$628$	296.000	−	512.687i	0.471338	−	0.816381i
$629$		−	192.333i		−	0.305776i
$630$		0			0
$631$	−760.000			−1.20444			−0.602219	−	0.798331i	$-0.705716\pi$
$631$	−760.000			−1.20444			−0.602219	+	0.798331i	$0.705716\pi$
$632$	−169.015	−	97.5807i	−0.267428	−	0.154400i
$633$		0			0
$634$	−96.0000	−	166.277i	−0.151420	−	0.262266i
$635$	−209.431	−	120.915i	−0.329813	−	0.190418i
$636$		0			0
$637$	532.500	+	506.625i	0.835950	+	0.795329i
$638$			226.274i			0.354662i
$639$		0			0
$640$	−8.00000	−	13.8564i	−0.0125000	−	0.0216506i
$641$	401.716	−	231.931i	0.626703	−	0.361827i	−0.152771	−	0.988262i	$-0.548820\pi$
$641$	401.716	−	231.931i	0.626703	−	0.361827i	0.779474	+	0.626435i	$0.215486\pi$
$642$		0			0
$643$	−19.0000			−0.0295490			−0.0147745	−	0.999891i	$-0.504703\pi$
$643$	−19.0000			−0.0295490			−0.0147745	+	0.999891i	$0.504703\pi$
$644$	117.576	−	294.156i	0.182571	−	0.456765i
$645$		0			0
$646$	−104.000	+	180.133i	−0.160991	+	0.278844i
$647$	−1003.07	+	579.120i	−1.55033	+	0.895086i	−0.552220	+	0.833698i	$0.686219\pi$
$647$	−1003.07	+	579.120i	−1.55033	+	0.895086i	−0.998114	+	0.0613872i	$0.980448\pi$
$648$		0			0
$649$	320.000	−	554.256i	0.493066	−	0.854016i
$650$		−	487.904i		−	0.750621i
$651$		0			0
$652$	−256.000			−0.392638
$653$	−892.839	−	515.481i	−1.36729	−	0.789404i	−0.376707	−	0.926332i	$-0.622944\pi$
$653$	−892.839	−	515.481i	−1.36729	−	0.789404i	−0.990581	+	0.136928i	$0.956277\pi$
$654$		0			0
$655$	−83.0000	−	143.760i	−0.126718	−	0.219481i
$656$	279.242	+	161.220i	0.425674	+	0.245763i
$657$		0			0
$658$	102.000	+	706.677i	0.155015	+	1.07398i
$659$			972.979i			1.47645i	0.674556	+	0.738224i	$0.264335\pi$
$659$			972.979i			1.47645i	−0.674556	+	0.738224i	$0.735665\pi$
$660$		0			0
$661$	200.500	+	347.276i	0.303328	+	0.525380i	0.976888	−	0.213753i	$-0.0685688\pi$
$661$	200.500	+	347.276i	0.303328	+	0.525380i	−0.673559	+	0.739133i	$0.735235\pi$
$662$	74.7094	−	43.1335i	0.112854	−	0.0651564i
$663$		0			0
$664$	−172.000			−0.259036
$665$	127.373	−	18.3848i	0.191539	−	0.0276463i
$666$		0			0
$667$	−256.000	+	443.405i	−0.383808	+	0.664775i
$668$	105.328	−	60.8112i	0.157677	−	0.0910347i
$669$		0			0
$670$	43.0000	−	74.4782i	0.0641791	−	0.111161i
$671$			509.117i			0.758743i
$672$		0			0
$673$	665.000			0.988113			0.494056	−	0.869430i	$-0.335514\pi$
$673$	665.000			0.988113			0.494056	+	0.869430i	$0.335514\pi$
$674$	165.341	+	95.4594i	0.245312	+	0.141631i
$675$		0			0
$676$	56.0000	+	96.9948i	0.0828402	+	0.143483i
$677$	793.635	+	458.205i	1.17228	+	0.676817i	0.954216	−	0.299117i	$-0.0966921\pi$
$677$	793.635	+	458.205i	1.17228	+	0.676817i	0.218065	+	0.975934i	$0.430025\pi$
$678$		0			0
$679$	104.000	+	41.5692i	0.153166	+	0.0612212i
$680$		−	45.2548i		−	0.0665512i
$681$		0			0
$682$	15.0000	+	25.9808i	0.0219941	+	0.0380950i
$683$	117.576	−	67.8823i	0.172146	−	0.0993884i	−0.411452	−	0.911432i	$-0.634978\pi$
$683$	117.576	−	67.8823i	0.172146	−	0.0993884i	0.583597	+	0.812043i	$0.301645\pi$
$684$		0			0
$685$	304.000			0.443796
$686$	−45.3156	+	482.954i	−0.0660577	+	0.704014i
$687$		0			0
$688$	170.000	−	294.449i	0.247093	−	0.427978i
$689$	440.908	−	254.558i	0.639925	−	0.369461i
$690$		0			0
$691$	−110.500	+	191.392i	−0.159913	+	0.276978i	−0.934837	−	0.355077i	$-0.884455\pi$
$691$	−110.500	+	191.392i	−0.159913	+	0.276978i	0.774924	+	0.632054i	$0.217788\pi$
$692$		−	226.274i		−	0.326986i
$693$		0			0
$694$	144.000			0.207493
$695$	−101.654	−	58.6899i	−0.146264	−	0.0844458i
$696$		0			0
$697$	456.000	+	789.815i	0.654232	+	1.13316i
$698$	−186.161	−	107.480i	−0.266707	−	0.153983i
$699$		0			0
$700$	253.000	−	199.186i	0.361429	−	0.284551i
$701$		−	192.333i		−	0.274370i	−0.990545	−	0.137185i	$-0.956195\pi$
$701$		−	192.333i		−	0.274370i	0.990545	−	0.137185i	$-0.0438054\pi$
$702$		0			0
$703$	110.500	+	191.392i	0.157183	+	0.272250i
$704$	48.9898	−	28.2843i	0.0695878	−	0.0401765i
$705$		0			0
$706$	546.000			0.773371
$707$	−480.100	+	69.2965i	−0.679066	+	0.0980148i
$708$		0			0
$709$	164.000	−	284.056i	0.231312	−	0.400644i	−0.726883	−	0.686762i	$-0.759032\pi$
$709$	164.000	−	284.056i	0.231312	−	0.400644i	0.958194	+	0.286118i	$0.0923649\pi$
$710$	−90.6311	+	52.3259i	−0.127649	+	0.0736985i
$711$		0			0
$712$	−192.000	+	332.554i	−0.269663	+	0.467070i
$713$			67.8823i			0.0952065i
$714$		0			0
$715$	−150.000			−0.209790
$716$	−51.4393	−	29.6985i	−0.0718426	−	0.0414783i
$717$		0			0
$718$	−32.0000	−	55.4256i	−0.0445682	−	0.0771945i
$719$	878.142	+	506.996i	1.22134	+	0.705140i	0.965203	−	0.261501i	$-0.0842175\pi$
$719$	878.142	+	506.996i	1.22134	+	0.705140i	0.256135	+	0.966641i	$0.417551\pi$
$720$		0			0
$721$	335.500	−	264.138i	0.465326	−	0.366349i
$722$			271.529i			0.376079i
$723$		0			0
$724$	81.0000	+	140.296i	0.111878	+	0.193779i
$725$	−450.706	+	260.215i	−0.621664	+	0.358918i
$726$		0			0
$727$	1069.00			1.47043			0.735213	−	0.677836i	$-0.237082\pi$
$727$	1069.00			1.47043			0.735213	+	0.677836i	$0.237082\pi$
$728$	−110.227	+	275.772i	−0.151411	+	0.378807i
$729$		0			0
$730$	−95.0000	+	164.545i	−0.130137	+	0.225404i
$731$	832.827	−	480.833i	1.13930	−	0.657774i
$732$		0			0
$733$	−247.500	+	428.683i	−0.337653	+	0.584833i	−0.983991	−	0.178219i	$-0.942967\pi$
$733$	−247.500	+	428.683i	−0.337653	+	0.584833i	0.646337	+	0.763052i	$0.276300\pi$
$734$		−	142.836i		−	0.194599i
$735$		0			0
$736$	128.000			0.173913
$737$	263.320	+	152.028i	0.357286	+	0.206279i
$738$		0			0
$739$	−226.500	−	392.310i	−0.306495	−	0.530865i	0.671098	−	0.741369i	$-0.265823\pi$
$739$	−226.500	−	392.310i	−0.306495	−	0.530865i	−0.977593	+	0.210503i	$0.932490\pi$
$740$	−41.6413	−	24.0416i	−0.0562721	−	0.0324887i
$741$		0			0
$742$	312.000	+	124.708i	0.420485	+	0.168070i
$743$		−	538.815i		−	0.725189i	−0.931947	−	0.362594i	$-0.881891\pi$
$743$		−	538.815i		−	0.725189i	0.931947	−	0.362594i	$-0.118109\pi$
$744$		0			0
$745$	−72.0000	−	124.708i	−0.0966443	−	0.167393i
$746$	380.896	−	219.910i	0.510584	−	0.294786i
$747$		0			0
$748$	160.000			0.213904
$749$	734.847	+	933.381i	0.981104	+	1.24617i
$750$		0			0
$751$	169.500	−	293.583i	0.225699	−	0.390922i	−0.730830	−	0.682560i	$-0.760867\pi$
$751$	169.500	−	293.583i	0.225699	−	0.390922i	0.956529	+	0.291637i	$0.0942001\pi$
$752$	−249.848	+	144.250i	−0.332245	+	0.191822i
$753$		0			0
$754$	240.000	−	415.692i	0.318302	−	0.551316i
$755$		−	56.5685i		−	0.0749252i
$756$		0			0
$757$	198.000			0.261559			0.130779	−	0.991411i	$-0.458252\pi$
$757$	198.000			0.261559			0.130779	+	0.991411i	$0.458252\pi$
$758$	111.452	+	64.3467i	0.147034	+	0.0848901i
$759$		0			0
$760$	26.0000	+	45.0333i	0.0342105	+	0.0592544i
$761$	−313.535	−	181.019i	−0.412004	−	0.237870i	0.279647	−	0.960103i	$-0.409783\pi$
$761$	−313.535	−	181.019i	−0.412004	−	0.237870i	−0.691650	+	0.722233i	$0.743116\pi$
$762$		0			0
$763$	65.0000	+	450.333i	0.0851900	+	0.590214i
$764$			127.279i			0.166596i
$765$		0			0
$766$	−232.000	−	401.836i	−0.302872	−	0.524590i
$767$	−1175.76	+	678.823i	−1.53293	+	0.885036i
$768$		0			0
$769$	−929.000			−1.20806			−0.604031	−	0.796961i	$-0.706440\pi$
$769$	−929.000			−1.20806			−0.604031	+	0.796961i	$0.706440\pi$
$770$	−61.2372	−	77.7817i	−0.0795289	−	0.101015i
$771$		0			0
$772$	−223.000	+	386.247i	−0.288860	+	0.500320i
$773$	−148.194	+	85.5599i	−0.191713	+	0.110686i	−0.592784	−	0.805361i	$-0.701971\pi$
$773$	−148.194	+	85.5599i	−0.191713	+	0.110686i	0.401071	+	0.916047i	$0.368638\pi$
$774$		0			0
$775$	−34.5000	+	59.7558i	−0.0445161	+	0.0771042i
$776$			45.2548i			0.0583181i
$777$		0			0
$778$	−974.000			−1.25193
$779$	−907.536	−	523.966i	−1.16500	−	0.672614i
$780$		0			0
$781$	−185.000	−	320.429i	−0.236876	−	0.410281i
$782$	313.535	+	181.019i	0.400939	+	0.231483i
$783$		0			0
$784$	−188.000	+	55.4256i	−0.239796	+	0.0706960i
$785$		−	418.607i		−	0.533258i
$786$		0			0
$787$	−45.0000	−	77.9423i	−0.0571792	−	0.0990372i	0.836019	−	0.548701i	$-0.184877\pi$
$787$	−45.0000	−	77.9423i	−0.0571792	−	0.0990372i	−0.893198	+	0.449663i	$0.851544\pi$
$788$	−274.343	+	158.392i	−0.348151	+	0.201005i
$789$		0			0
$790$	−138.000			−0.174684
$791$	356.401	−	891.662i	0.450570	−	1.12726i
$792$		0			0
$793$	540.000	−	935.307i	0.680958	−	1.17945i
$794$	341.704	−	197.283i	0.430357	−	0.248467i
$795$		0			0
$796$	−136.000	+	235.559i	−0.170854	+	0.295928i
$797$		−	1154.00i		−	1.44793i	−0.689838	−	0.723964i	$-0.742318\pi$
$797$		−	1154.00i		−	1.44793i	0.689838	−	0.723964i	$-0.257682\pi$
$798$		0			0
$799$	−816.000			−1.02128
$800$	112.677	+	65.0538i	0.140846	+	0.0813173i
$801$		0			0
$802$		0			0
$803$	−581.754	−	335.876i	−0.724475	−	0.418276i
$804$		0			0
$805$	−32.0000	−	221.703i	−0.0397516	−	0.275407i
$806$		−	63.6396i		−	0.0789573i
$807$		0			0
$808$	−98.0000	−	169.741i	−0.121287	−	0.210075i
$809$	−148.194	+	85.5599i	−0.183182	+	0.105760i	−0.588787	−	0.808288i	$-0.700394\pi$
$809$	−148.194	+	85.5599i	−0.183182	+	0.105760i	0.405605	+	0.914049i	$0.367061\pi$
$810$		0			0
$811$	−554.000			−0.683107			−0.341554	−	0.939862i	$-0.610953\pi$
$811$	−554.000			−0.683107			−0.341554	+	0.939862i	$0.610953\pi$
$812$	313.535	−	45.2548i	0.386126	−	0.0557326i
$813$		0			0
$814$	85.0000	−	147.224i	0.104423	−	0.180865i
$815$	−156.767	+	90.5097i	−0.192353	+	0.111055i
$816$		0			0
$817$	−552.500	+	956.958i	−0.676255	+	1.17131i
$818$			315.370i			0.385537i
$819$		0			0
$820$	228.000			0.278049
$821$	−919.783	−	531.037i	−1.12032	−	0.646818i	−0.178838	−	0.983879i	$-0.557234\pi$
$821$	−919.783	−	531.037i	−1.12032	−	0.646818i	−0.941483	+	0.337061i	$0.890567\pi$
$822$		0			0
$823$	−428.000	−	741.318i	−0.520049	−	0.900751i	−0.999728	−	0.0233070i	$-0.992580\pi$
$823$	−428.000	−	741.318i	−0.520049	−	0.900751i	0.479680	−	0.877444i	$-0.340753\pi$
$824$	149.419	+	86.2670i	0.181334	+	0.104693i
$825$		0			0
$826$	−832.000	−	332.554i	−1.00726	−	0.402607i
$827$			1022.48i			1.23637i	0.786033	+	0.618184i	$0.212131\pi$
$827$			1022.48i			1.23637i	−0.786033	+	0.618184i	$0.787869\pi$
$828$		0			0
$829$	−519.500	−	899.800i	−0.626659	−	1.08540i	−0.988218	−	0.153055i	$-0.951089\pi$
$829$	−519.500	−	899.800i	−0.626659	−	1.08540i	0.361559	−	0.932349i	$-0.382245\pi$
$830$	−105.328	+	60.8112i	−0.126901	+	0.0732665i
$831$		0			0
$832$	−120.000			−0.144231
$833$	−538.888	−	130.108i	−0.646924	−	0.156192i
$834$		0			0
$835$	43.0000	−	74.4782i	0.0514970	−	0.0891954i
$836$	−159.217	+	91.9239i	−0.190451	+	0.109957i
$837$		0			0
$838$	−411.000	+	711.873i	−0.490453	+	0.849490i
$839$		−	656.195i		−	0.782116i	−0.920366	−	0.391058i	$-0.872109\pi$
$839$		−	656.195i		−	0.782116i	0.920366	−	0.391058i	$-0.127891\pi$
$840$		0			0
$841$	329.000			0.391201
$842$	187.386	+	108.187i	0.222549	+	0.128489i
$843$		0			0
$844$	272.000	+	471.118i	0.322275	+	0.558196i
$845$	68.5857	+	39.5980i	0.0811665	+	0.0468615i
$846$		0			0
$847$	−390.500	+	307.439i	−0.461039	+	0.362974i
$848$			135.765i			0.160100i
$849$		0			0
$850$	184.000	+	318.697i	0.216471	+	0.374938i
$851$	333.131	−	192.333i	0.391458	−	0.226008i
$852$		0			0
$853$	1463.00			1.71512			0.857562	−	0.514381i	$-0.171978\pi$
$853$	1463.00			1.71512			0.857562	+	0.514381i	$0.171978\pi$
$854$	705.453	−	101.823i	0.826057	−	0.119231i
$855$		0			0
$856$	−240.000	+	415.692i	−0.280374	+	0.485622i
$857$	−58.7878	+	33.9411i	−0.0685971	+	0.0396046i	−0.533906	−	0.845544i	$-0.679276\pi$
$857$	−58.7878	+	33.9411i	−0.0685971	+	0.0396046i	0.465309	+	0.885148i	$0.345943\pi$
$858$		0			0
$859$	731.000	−	1266.13i	0.850990	−	1.47396i	−0.0293268	−	0.999570i	$-0.509336\pi$
$859$	731.000	−	1266.13i	0.850990	−	1.47396i	0.880316	−	0.474387i	$-0.157330\pi$
$860$		−	240.416i		−	0.279554i
$861$		0			0
$862$	−198.000			−0.229698
$863$	699.329	+	403.758i	0.810347	+	0.467854i	0.847076	−	0.531471i	$-0.178361\pi$
$863$	699.329	+	403.758i	0.810347	+	0.467854i	−0.0367295	+	0.999325i	$0.511694\pi$
$864$		0			0
$865$	−80.0000	−	138.564i	−0.0924855	−	0.160190i
$866$	167.790	+	96.8736i	0.193753	+	0.111863i
$867$		0			0
$868$	33.0000	−	25.9808i	0.0380184	−	0.0299318i
$869$		−	487.904i		−	0.561454i
$870$		0			0
$871$	−322.500	−	558.586i	−0.370264	−	0.641316i
$872$	−159.217	+	91.9239i	−0.182588	+	0.105417i
$873$		0			0
$874$	−416.000			−0.475973
$875$	176.363	−	441.235i	0.201558	−	0.504268i
$876$		0			0
$877$	−740.000	+	1281.72i	−0.843786	+	1.46148i	0.0428860	+	0.999080i	$0.486345\pi$
$877$	−740.000	+	1281.72i	−0.843786	+	1.46148i	−0.886672	+	0.462400i	$0.846989\pi$
$878$	734.847	−	424.264i	0.836955	−	0.483216i
$879$		0			0
$880$	20.0000	−	34.6410i	0.0227273	−	0.0393648i
$881$		−	712.764i		−	0.809039i	−0.914529	−	0.404520i	$-0.867439\pi$
$881$		−	712.764i		−	0.809039i	0.914529	−	0.404520i	$-0.132561\pi$
$882$		0			0
$883$	−115.000			−0.130238			−0.0651189	−	0.997878i	$-0.520743\pi$
$883$	−115.000			−0.130238			−0.0651189	+	0.997878i	$0.520743\pi$
$884$	−293.939	−	169.706i	−0.332510	−	0.191975i
$885$		0			0
$886$	−448.000	−	775.959i	−0.505643	−	0.875800i
$887$	−1032.46	−	596.091i	−1.16399	−	0.672030i	−0.211734	−	0.977327i	$-0.567911\pi$
$887$	−1032.46	−	596.091i	−1.16399	−	0.672030i	−0.952257	+	0.305297i	$0.901244\pi$
$888$		0			0
$889$	−1111.50	−	444.271i	−1.25028	−	0.499742i
$890$			271.529i			0.305089i
$891$		0			0
$892$	248.000	+	429.549i	0.278027	+	0.481557i
$893$	812.006	−	468.812i	0.909301	−	0.524985i
$894$		0			0
$895$	−42.0000			−0.0469274
$896$	−48.9898	−	62.2254i	−0.0546761	−	0.0694480i
$897$		0			0
$898$	271.000	−	469.386i	0.301782	−	0.522701i
$899$	−58.7878	+	33.9411i	−0.0653924	+	0.0377543i
$900$		0			0
$901$	−192.000	+	332.554i	−0.213097	+	0.369094i
$902$			806.102i			0.893683i
$903$		0			0
$904$	388.000			0.429204
$905$	99.2043	+	57.2756i	0.109618	+	0.0632880i
$906$		0			0
$907$	250.500	+	433.879i	0.276185	+	0.478367i	0.970433	−	0.241369i	$-0.0775963\pi$
$907$	250.500	+	433.879i	0.276185	+	0.478367i	−0.694248	+	0.719736i	$0.744263\pi$
$908$	286.590	+	165.463i	0.315628	+	0.182228i
$909$		0			0
$910$	30.0000	+	207.846i	0.0329670	+	0.228402i
$911$			203.647i			0.223542i	0.993734	+	0.111771i	$0.0356523\pi$
$911$			203.647i			0.223542i	−0.993734	+	0.111771i	$0.964348\pi$
$912$		0			0
$913$	−215.000	−	372.391i	−0.235487	−	0.407876i
$914$	292.714	−	168.999i	0.320256	−	0.184900i
$915$		0			0
$916$	−866.000			−0.945415
$917$	−508.269	−	645.588i	−0.554274	−	0.704022i
$918$		0			0
$919$	326.500	−	565.515i	0.355277	−	0.615359i	−0.631888	−	0.775060i	$-0.717720\pi$
$919$	326.500	−	565.515i	0.355277	−	0.615359i	0.987165	+	0.159701i	$0.0510530\pi$
$920$	78.3837	−	45.2548i	0.0851996	−	0.0491900i
$921$		0			0
$922$	−320.000	+	554.256i	−0.347072	+	0.601146i
$923$			784.889i			0.850367i
$924$		0			0
$925$	391.000			0.422703
$926$	258.421	+	149.200i	0.279073	+	0.161123i
$927$		0			0
$928$	64.0000	+	110.851i	0.0689655	+	0.119452i
$929$	−11.0227	−	6.36396i	−0.0118651	−	0.00685033i	0.494056	−	0.869430i	$-0.335514\pi$
$929$	−11.0227	−	6.36396i	−0.0118651	−	0.00685033i	−0.505921	+	0.862580i	$0.668847\pi$
$930$		0			0
$931$	611.000	−	180.133i	0.656284	−	0.193484i
$932$			455.377i			0.488602i
$933$		0			0
$934$	−221.000	−	382.783i	−0.236617	−	0.409832i
$935$	97.9796	−	56.5685i	0.104791	−	0.0605011i
$936$		0			0
$937$	761.000			0.812166			0.406083	−	0.913836i	$-0.366894\pi$
$937$	761.000			0.812166			0.406083	+	0.913836i	$0.366894\pi$
$938$	157.992	−	395.273i	0.168435	−	0.421399i
$939$		0			0
$940$	−102.000	+	176.669i	−0.108511	+	0.187946i
$941$	244.949	−	141.421i	0.260307	−	0.150288i	−0.364168	−	0.931333i	$-0.618646\pi$
$941$	244.949	−	141.421i	0.260307	−	0.150288i	0.624475	+	0.781045i	$0.285313\pi$
$942$		0			0
$943$	−912.000	+	1579.63i	−0.967126	+	1.67511i
$944$		−	362.039i		−	0.383516i
$945$		0			0
$946$	850.000			0.898520
$947$	934.480	+	539.522i	0.986780	+	0.569718i	0.904310	−	0.426876i	$-0.140386\pi$
$947$	934.480	+	539.522i	0.986780	+	0.569718i	0.0824695	+	0.996594i	$0.473719\pi$
$948$		0			0
$949$	712.500	+	1234.09i	0.750790	+	1.30041i
$950$	−366.199	−	211.425i	−0.385472	−	0.222553i
$951$		0			0
$952$	−32.0000	−	221.703i	−0.0336134	−	0.232881i
$953$			1074.80i			1.12781i	0.825840	+	0.563905i	$0.190701\pi$
$953$			1074.80i			1.12781i	−0.825840	+	0.563905i	$0.809299\pi$
$954$		0			0
$955$	45.0000	+	77.9423i	0.0471204	+	0.0816150i
$956$	−595.226	+	343.654i	−0.622621	+	0.359471i
$957$		0			0
$958$	992.000			1.03549
$959$	1489.29	−	214.960i	1.55296	−	0.224151i
$960$		0			0
$961$	476.000	−	824.456i	0.495317	−	0.857915i
$962$	−312.310	+	180.312i	−0.324647	+	0.187435i
$963$		0			0
$964$	158.000	−	273.664i	0.163900	−	0.283884i
$965$			315.370i			0.326808i
$966$		0			0
$967$	1163.00			1.20269			0.601344	−	0.798990i	$-0.294632\pi$
$967$	1163.00			1.20269			0.601344	+	0.798990i	$0.294632\pi$
$968$	−173.914	−	100.409i	−0.179663	−	0.103728i
$969$		0			0
$970$	16.0000	+	27.7128i	0.0164948	+	0.0285699i
$971$	−1352.12	−	780.646i	−1.39250	−	0.803961i	−0.398909	−	0.916990i	$-0.630611\pi$
$971$	−1352.12	−	780.646i	−1.39250	−	0.803961i	−0.993592	+	0.113030i	$0.963944\pi$
$972$		0			0
$973$	−539.500	−	215.640i	−0.554471	−	0.221624i
$974$		−	592.555i		−	0.608373i
$975$		0			0
$976$	144.000	+	249.415i	0.147541	+	0.255548i
$977$	−981.021	+	566.393i	−1.00412	+	0.579726i	−0.909463	−	0.415784i	$-0.863507\pi$
$977$	−981.021	+	566.393i	−1.00412	+	0.579726i	−0.0946520	+	0.995510i	$0.530174\pi$
$978$		0			0
$979$	−960.000			−0.980592
$980$	−95.5301	+	100.409i	−0.0974797	+	0.102458i
$981$		0			0
$982$	120.000	−	207.846i	0.122200	−	0.211656i
$983$	264.545	−	152.735i	0.269120	−	0.155376i	−0.359368	−	0.933196i	$-0.617008\pi$
$983$	264.545	−	152.735i	0.269120	−	0.155376i	0.628488	+	0.777820i	$0.283674\pi$
$984$		0			0
$985$	−112.000	+	193.990i	−0.113706	+	0.196944i
$986$			362.039i			0.367179i
$987$		0			0
$988$	390.000			0.394737
$989$	1665.65	+	961.665i	1.68418	+	0.972361i
$990$		0			0
$991$	486.500	+	842.643i	0.490918	+	0.850295i	0.999945	−	0.0104551i	$-0.00332803\pi$
$991$	486.500	+	842.643i	0.490918	+	0.850295i	−0.509027	+	0.860751i	$0.669995\pi$
$992$	14.6969	+	8.48528i	0.0148155	+	0.00855371i
$993$		0			0
$994$	−407.000	+	320.429i	−0.409457	+	0.322364i
$995$			192.333i			0.193300i
$996$		0			0
$997$	39.5000	+	68.4160i	0.0396189	+	0.0686219i	0.885155	−	0.465296i	$-0.154052\pi$
$997$	39.5000	+	68.4160i	0.0396189	+	0.0686219i	−0.845536	+	0.533918i	$0.820719\pi$
$998$	−229.027	+	132.229i	−0.229486	+	0.132494i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.s.b.107.2	yes	4
3.2	odd	2	inner	126.3.s.b.107.1	yes	4
4.3	odd	2		1008.3.dc.a.737.2		4
7.2	even	3		882.3.b.c.197.1		2
7.3	odd	6		882.3.s.c.557.1		4
7.4	even	3	inner	126.3.s.b.53.1	✓	4
7.5	odd	6		882.3.b.d.197.1		2
7.6	odd	2		882.3.s.c.863.2		4
12.11	even	2		1008.3.dc.a.737.1		4
21.2	odd	6		882.3.b.c.197.2		2
21.5	even	6		882.3.b.d.197.2		2
21.11	odd	6	inner	126.3.s.b.53.2	yes	4
21.17	even	6		882.3.s.c.557.2		4
21.20	even	2		882.3.s.c.863.1		4
28.11	odd	6		1008.3.dc.a.305.1		4
84.11	even	6		1008.3.dc.a.305.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.s.b.53.1	✓	4	7.4	even	3	inner
126.3.s.b.53.2	yes	4	21.11	odd	6	inner
126.3.s.b.107.1	yes	4	3.2	odd	2	inner
126.3.s.b.107.2	yes	4	1.1	even	1	trivial
882.3.b.c.197.1		2	7.2	even	3
882.3.b.c.197.2		2	21.2	odd	6
882.3.b.d.197.1		2	7.5	odd	6
882.3.b.d.197.2		2	21.5	even	6
882.3.s.c.557.1		4	7.3	odd	6
882.3.s.c.557.2		4	21.17	even	6
882.3.s.c.863.1		4	21.20	even	2
882.3.s.c.863.2		4	7.6	odd	2
1008.3.dc.a.305.1		4	28.11	odd	6
1008.3.dc.a.305.2		4	84.11	even	6
1008.3.dc.a.737.1		4	12.11	even	2
1008.3.dc.a.737.2		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	126.s (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(1.22474 + 0.707107i) q^{5} +(6.50000 + 2.59808i) q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(1.22474 + 0.707107i) q^{5} +(6.50000 + 2.59808i) q^{7} +2.82843i q^{8} +(1.00000 + 1.73205i) q^{10} +(-6.12372 + 3.53553i) q^{11} +15.0000 q^{13} +(6.12372 + 7.77817i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(-9.79796 + 5.65685i) q^{17} +(6.50000 - 11.2583i) q^{19} +2.82843i q^{20} -10.0000 q^{22} +(-19.5959 - 11.3137i) q^{23} +(-11.5000 - 19.9186i) q^{25} +(18.3712 + 10.6066i) q^{26} +(2.00000 + 13.8564i) q^{28} -22.6274i q^{29} +(-1.50000 - 2.59808i) q^{31} +(-4.89898 + 2.82843i) q^{32} -16.0000 q^{34} +(6.12372 + 7.77817i) q^{35} +(-8.50000 + 14.7224i) q^{37} +(15.9217 - 9.19239i) q^{38} +(-2.00000 + 3.46410i) q^{40} -80.6102i q^{41} -85.0000 q^{43} +(-12.2474 - 7.07107i) q^{44} +(-16.0000 - 27.7128i) q^{46} +(62.4620 + 36.0624i) q^{47} +(35.5000 + 33.7750i) q^{49} -32.5269i q^{50} +(15.0000 + 25.9808i) q^{52} +(29.3939 - 16.9706i) q^{53} -10.0000 q^{55} +(-7.34847 + 18.3848i) q^{56} +(16.0000 - 27.7128i) q^{58} +(-78.3837 + 45.2548i) q^{59} +(36.0000 - 62.3538i) q^{61} -4.24264i q^{62} -8.00000 q^{64} +(18.3712 + 10.6066i) q^{65} +(-21.5000 - 37.2391i) q^{67} +(-19.5959 - 11.3137i) q^{68} +(2.00000 + 13.8564i) q^{70} +52.3259i q^{71} +(47.5000 + 82.2724i) q^{73} +(-20.8207 + 12.0208i) q^{74} +26.0000 q^{76} +(-48.9898 + 7.07107i) q^{77} +(-34.5000 + 59.7558i) q^{79} +(-4.89898 + 2.82843i) q^{80} +(57.0000 - 98.7269i) q^{82} +60.8112i q^{83} -16.0000 q^{85} +(-104.103 - 60.1041i) q^{86} +(-10.0000 - 17.3205i) q^{88} +(117.576 + 67.8823i) q^{89} +(97.5000 + 38.9711i) q^{91} -45.2548i q^{92} +(51.0000 + 88.3346i) q^{94} +(15.9217 - 9.19239i) q^{95} +16.0000 q^{97} +(19.5959 + 66.4680i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 26 q^{7}+O(q^{10}) \) 4 * q + 4 * q^4 + 26 * q^7	\( 4 q + 4 q^{4} + 26 q^{7} + 4 q^{10} + 60 q^{13} - 8 q^{16} + 26 q^{19} - 40 q^{22} - 46 q^{25} + 8 q^{28} - 6 q^{31} - 64 q^{34} - 34 q^{37} - 8 q^{40} - 340 q^{43} - 64 q^{46} + 142 q^{49} + 60 q^{52} - 40 q^{55} + 64 q^{58} + 144 q^{61} - 32 q^{64} - 86 q^{67} + 8 q^{70} + 190 q^{73} + 104 q^{76} - 138 q^{79} + 228 q^{82} - 64 q^{85} - 40 q^{88} + 390 q^{91} + 204 q^{94} + 64 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 26 * q^7 + 4 * q^10 + 60 * q^13 - 8 * q^16 + 26 * q^19 - 40 * q^22 - 46 * q^25 + 8 * q^28 - 6 * q^31 - 64 * q^34 - 34 * q^37 - 8 * q^40 - 340 * q^43 - 64 * q^46 + 142 * q^49 + 60 * q^52 - 40 * q^55 + 64 * q^58 + 144 * q^61 - 32 * q^64 - 86 * q^67 + 8 * q^70 + 190 * q^73 + 104 * q^76 - 138 * q^79 + 228 * q^82 - 64 * q^85 - 40 * q^88 + 390 * q^91 + 204 * q^94 + 64 * q^97

Introduction

L-functions

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