LMFDB - Embedded newform 168.10.q.a.121.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,10,Mod(25,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("168.25");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	168.q (of order $3$, 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:	$86.5260204755$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 18660372 x^{14} - 3458782984 x^{13} + 143123973101310 x^{12} + \cdots + 50\!\cdots\!97 $ x^16 - 18660372x^14 - 3458782984x^13 + 143123973101310x^12 + 48872053104519072x^11 - 578734520240854872804x^10 - 270042331189660586117064x^9 + 1317886932354758017843044627x^8 + 736352464217070243815722699808x^7 - 1673655580870016843616444263455812x^6 - 1020366474648370188416591376208359000x^5 + 1103325097889381327876435664684216492298x^4 + 649904594580478032138799228161609255854736x^3 - 331733869309416601638773034133657913991577560x^2 - 128648437449643500784782173321364984397814457552x + 50061741614807564009961574976312432423689032591597
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{38}\cdot 3^{5}\cdot 5^{2}\cdot 7^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.1
Root			$-1807.20 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	168.121
Dual form			168.10.q.a.25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 - 70.1481i) q^{3} +(-915.848 + 1586.30i) q^{5} +(5121.10 + 3758.71i) q^{7} +(-3280.50 + 5681.99i) q^{9} +O(q^{10})$	\(q+(-40.5000 - 70.1481i) q^{3} +(-915.848 + 1586.30i) q^{5} +(5121.10 + 3758.71i) q^{7} +(-3280.50 + 5681.99i) q^{9} +(-28896.4 - 50050.1i) q^{11} -149327. q^{13} +148367. q^{15} +(32340.7 + 56015.8i) q^{17} +(288535. - 499757. i) q^{19} +(56261.9 - 511463. i) q^{21} +(583685. - 1.01097e6i) q^{23} +(-700994. - 1.21416e6i) q^{25} +531441. q^{27} -3.62578e6 q^{29} +(-1.45477e6 - 2.51973e6i) q^{31} +(-2.34061e6 + 4.05406e6i) q^{33} +(-1.06526e7 + 4.68117e6i) q^{35} +(-4.80887e6 + 8.32920e6i) q^{37} +(6.04776e6 + 1.04750e7i) q^{39} -1.92089e6 q^{41} +2.02853e7 q^{43} +(-6.00888e6 - 1.04077e7i) q^{45} +(2.37967e6 - 4.12171e6i) q^{47} +(1.20977e7 + 3.84975e7i) q^{49} +(2.61960e6 - 4.53728e6i) q^{51} +(1.74116e7 + 3.01578e7i) q^{53} +1.05859e8 q^{55} -4.67426e7 q^{57} +(-2.15924e7 - 3.73991e7i) q^{59} +(-8.05090e7 + 1.39446e8i) q^{61} +(-3.81568e7 + 1.67676e7i) q^{63} +(1.36761e8 - 2.36877e8i) q^{65} +(1.41353e8 + 2.44831e8i) q^{67} -9.45569e7 q^{69} +1.95838e8 q^{71} +(-1.14150e8 - 1.97714e8i) q^{73} +(-5.67805e7 + 9.83468e7i) q^{75} +(4.01424e7 - 3.64925e8i) q^{77} +(1.24132e8 - 2.15002e8i) q^{79} +(-2.15234e7 - 3.72796e7i) q^{81} -5.69193e7 q^{83} -1.18477e8 q^{85} +(1.46844e8 + 2.54342e8i) q^{87} +(5.58308e8 - 9.67018e8i) q^{89} +(-7.64720e8 - 5.61279e8i) q^{91} +(-1.17836e8 + 2.04099e8i) q^{93} +(5.28508e8 + 9.15403e8i) q^{95} +1.19652e9 q^{97} +3.79179e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) $ 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9	$ 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9} + 32460 q^{11} + 119048 q^{13} + 31752 q^{15} + 208352 q^{17} + 914588 q^{19} - 428652 q^{21} + 460920 q^{23} - 3040180 q^{25} + 8503056 q^{27} - 16376136 q^{29} - 944064 q^{31} + 2629260 q^{33} - 15546664 q^{35} - 9826516 q^{37} - 4821444 q^{39} + 11449216 q^{41} - 6933624 q^{43} - 1285956 q^{45} + 26549360 q^{47} + 83657504 q^{49} + 16876512 q^{51} - 15354476 q^{53} + 134121944 q^{55} - 148163256 q^{57} + 18404996 q^{59} - 260632792 q^{61} + 35823060 q^{63} + 191461840 q^{65} + 53879788 q^{67} - 74669040 q^{69} - 164207456 q^{71} + 248475540 q^{73} - 246254580 q^{75} + 670121788 q^{77} + 16631256 q^{79} - 344373768 q^{81} - 1138943272 q^{83} - 1690136272 q^{85} + 663233508 q^{87} + 236796360 q^{89} - 1455575212 q^{91} - 76469184 q^{93} + 182450488 q^{95} + 1339799464 q^{97} - 425940120 q^{99}+O(q^{100}) $ 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9 + 32460 * q^11 + 119048 * q^13 + 31752 * q^15 + 208352 * q^17 + 914588 * q^19 - 428652 * q^21 + 460920 * q^23 - 3040180 * q^25 + 8503056 * q^27 - 16376136 * q^29 - 944064 * q^31 + 2629260 * q^33 - 15546664 * q^35 - 9826516 * q^37 - 4821444 * q^39 + 11449216 * q^41 - 6933624 * q^43 - 1285956 * q^45 + 26549360 * q^47 + 83657504 * q^49 + 16876512 * q^51 - 15354476 * q^53 + 134121944 * q^55 - 148163256 * q^57 + 18404996 * q^59 - 260632792 * q^61 + 35823060 * q^63 + 191461840 * q^65 + 53879788 * q^67 - 74669040 * q^69 - 164207456 * q^71 + 248475540 * q^73 - 246254580 * q^75 + 670121788 * q^77 + 16631256 * q^79 - 344373768 * q^81 - 1138943272 * q^83 - 1690136272 * q^85 + 663233508 * q^87 + 236796360 * q^89 - 1455575212 * q^91 - 76469184 * q^93 + 182450488 * q^95 + 1339799464 * q^97 - 425940120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−40.5000	−	70.1481i	−0.288675	−	0.500000i
$3$	−40.5000	−	70.1481i	−0.288675	−	0.500000i
$4$		0			0
$5$	−915.848	+	1586.30i	−0.655328	+	1.13506i	0.326484	+	0.945203i	$0.394136\pi$
$5$	−915.848	+	1586.30i	−0.655328	+	1.13506i	−0.981812	+	0.189858i	$0.939197\pi$
$6$		0			0
$7$	5121.10	+	3758.71i	0.806162	+	0.591695i
$7$	5121.10	+	3758.71i	0.806162	+	0.591695i
$8$		0			0
$9$	−3280.50	+	5681.99i	−0.166667	+	0.288675i
$10$		0			0
$11$	−28896.4	−	50050.1i	−0.595082	−	1.03071i	−0.993535	−	0.113524i	$-0.963786\pi$
$11$	−28896.4	−	50050.1i	−0.595082	−	1.03071i	0.398453	−	0.917189i	$-0.369547\pi$
$12$		0			0
$13$	−149327.			−1.45009			−0.725044	−	0.688703i	$-0.758180\pi$
$13$	−149327.			−1.45009			−0.725044	+	0.688703i	$0.758180\pi$
$14$		0			0
$15$	148367.			0.756707
$16$		0			0
$17$	32340.7	+	56015.8i	0.0939139	+	0.162664i	0.909155	−	0.416458i	$-0.136729\pi$
$17$	32340.7	+	56015.8i	0.0939139	+	0.162664i	−0.815241	+	0.579122i	$0.803396\pi$
$18$		0			0
$19$	288535.	−	499757.i	0.507933	−	0.879766i	−0.492024	−	0.870581i	$-0.663743\pi$
$19$	288535.	−	499757.i	0.507933	−	0.879766i	0.999958	−	0.00918497i	$-0.00292371\pi$
$20$		0			0
$21$	56261.9	−	511463.i	0.0631288	−	0.573889i
$22$		0			0
$23$	583685.	−	1.01097e6i	0.434914	−	0.753292i	−0.562375	−	0.826882i	$-0.690112\pi$
$23$	583685.	−	1.01097e6i	0.434914	−	0.753292i	0.997289	+	0.0735900i	$0.0234456\pi$
$24$		0			0
$25$	−700994.	−	1.21416e6i	−0.358909	−	0.621649i
$26$		0			0
$27$	531441.			0.192450
$28$		0			0
$29$	−3.62578e6			−0.951943			−0.475971	−	0.879461i	$-0.657903\pi$
$29$	−3.62578e6			−0.951943			−0.475971	+	0.879461i	$0.657903\pi$
$30$		0			0
$31$	−1.45477e6	−	2.51973e6i	−0.282922	−	0.490035i	0.689181	−	0.724589i	$-0.257970\pi$
$31$	−1.45477e6	−	2.51973e6i	−0.282922	−	0.490035i	−0.972103	+	0.234554i	$0.924637\pi$
$32$		0			0
$33$	−2.34061e6	+	4.05406e6i	−0.343571	+	0.595082i
$34$		0			0
$35$	−1.06526e7	+	4.68117e6i	−1.19991	+	0.527288i
$36$		0			0
$37$	−4.80887e6	+	8.32920e6i	−0.421828	+	0.730627i	−0.996118	−	0.0880247i	$-0.971945\pi$
$37$	−4.80887e6	+	8.32920e6i	−0.421828	+	0.730627i	0.574291	+	0.818651i	$0.305278\pi$
$38$		0			0
$39$	6.04776e6	+	1.04750e7i	0.418604	+	0.725044i
$40$		0			0
$41$	−1.92089e6			−0.106164			−0.0530818	−	0.998590i	$-0.516904\pi$
$41$	−1.92089e6			−0.106164			−0.0530818	+	0.998590i	$0.516904\pi$
$42$		0			0
$43$	2.02853e7			0.904844			0.452422	−	0.891804i	$-0.350560\pi$
$43$	2.02853e7			0.904844			0.452422	+	0.891804i	$0.350560\pi$
$44$		0			0
$45$	−6.00888e6	−	1.04077e7i	−0.218443	−	0.378354i
$46$		0			0
$47$	2.37967e6	−	4.12171e6i	0.0711338	−	0.123207i	−0.828265	−	0.560337i	$-0.810672\pi$
$47$	2.37967e6	−	4.12171e6i	0.0711338	−	0.123207i	0.899398	+	0.437130i	$0.144005\pi$
$48$		0			0
$49$	1.20977e7	+	3.84975e7i	0.299793	+	0.954004i
$50$		0			0
$51$	2.61960e6	−	4.53728e6i	0.0542212	−	0.0939139i
$52$		0			0
$53$	1.74116e7	+	3.01578e7i	0.303108	+	0.524998i	0.976838	−	0.213979i	$-0.0686425\pi$
$53$	1.74116e7	+	3.01578e7i	0.303108	+	0.524998i	−0.673731	+	0.738977i	$0.735309\pi$
$54$		0			0
$55$	1.05859e8			1.55990
$56$		0			0
$57$	−4.67426e7			−0.586511
$58$		0			0
$59$	−2.15924e7	−	3.73991e7i	−0.231989	−	0.401816i	0.726405	−	0.687267i	$-0.241190\pi$
$59$	−2.15924e7	−	3.73991e7i	−0.231989	−	0.401816i	−0.958393	+	0.285451i	$0.907857\pi$
$60$		0			0
$61$	−8.05090e7	+	1.39446e8i	−0.744492	+	1.28950i	0.205940	+	0.978565i	$0.433975\pi$
$61$	−8.05090e7	+	1.39446e8i	−0.744492	+	1.28950i	−0.950432	+	0.310933i	$0.899358\pi$
$62$		0			0
$63$	−3.81568e7	+	1.67676e7i	−0.305168	+	0.134103i
$64$		0			0
$65$	1.36761e8	−	2.36877e8i	0.950282	−	1.64594i
$66$		0			0
$67$	1.41353e8	+	2.44831e8i	0.856978	+	1.48433i	0.874797	+	0.484489i	$0.160994\pi$
$67$	1.41353e8	+	2.44831e8i	0.856978	+	1.48433i	−0.0178188	+	0.999841i	$0.505672\pi$
$68$		0			0
$69$	−9.45569e7			−0.502195
$70$		0			0
$71$	1.95838e8			0.914607			0.457303	−	0.889311i	$-0.348815\pi$
$71$	1.95838e8			0.914607			0.457303	+	0.889311i	$0.348815\pi$
$72$		0			0
$73$	−1.14150e8	−	1.97714e8i	−0.470462	−	0.814865i	0.528967	−	0.848642i	$-0.322580\pi$
$73$	−1.14150e8	−	1.97714e8i	−0.470462	−	0.814865i	−0.999429	+	0.0337776i	$0.989246\pi$
$74$		0			0
$75$	−5.67805e7	+	9.83468e7i	−0.207216	+	0.358909i
$76$		0			0
$77$	4.01424e7	−	3.64925e8i	0.130135	−	1.18303i
$78$		0			0
$79$	1.24132e8	−	2.15002e8i	0.358559	−	0.621043i	−0.629161	−	0.777275i	$-0.716601\pi$
$79$	1.24132e8	−	2.15002e8i	0.358559	−	0.621043i	0.987720	+	0.156232i	$0.0499348\pi$
$80$		0			0
$81$	−2.15234e7	−	3.72796e7i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−5.69193e7			−0.131646			−0.0658231	−	0.997831i	$-0.520967\pi$
$83$	−5.69193e7			−0.131646			−0.0658231	+	0.997831i	$0.520967\pi$
$84$		0			0
$85$	−1.18477e8			−0.246177
$86$		0			0
$87$	1.46844e8	+	2.54342e8i	0.274802	+	0.475971i
$88$		0			0
$89$	5.58308e8	−	9.67018e8i	0.943233	−	1.63373i	0.183982	−	0.982930i	$-0.441101\pi$
$89$	5.58308e8	−	9.67018e8i	0.943233	−	1.63373i	0.759251	−	0.650798i	$-0.225566\pi$
$90$		0			0
$91$	−7.64720e8	−	5.61279e8i	−1.16900	−	0.858010i
$92$		0			0
$93$	−1.17836e8	+	2.04099e8i	−0.163345	+	0.282922i
$94$		0			0
$95$	5.28508e8	+	9.15403e8i	0.665726	+	1.15307i
$96$		0			0
$97$	1.19652e9			1.37229			0.686145	−	0.727465i	$-0.259302\pi$
$97$	1.19652e9			1.37229			0.686145	+	0.727465i	$0.259302\pi$
$98$		0			0
$99$	3.79179e8			0.396722
$100$		0			0
$101$	2.11876e8	+	3.66980e8i	0.202598	+	0.350910i	0.949365	−	0.314176i	$-0.101728\pi$
$101$	2.11876e8	+	3.66980e8i	0.202598	+	0.350910i	−0.746767	+	0.665086i	$0.768395\pi$
$102$		0			0
$103$	5.91265e8	−	1.02410e9i	0.517624	−	0.896551i	−0.482167	−	0.876080i	$-0.660150\pi$
$103$	5.91265e8	−	1.02410e9i	0.517624	−	0.896551i	0.999790	−	0.0204713i	$-0.00651668\pi$
$104$		0			0
$105$	7.59805e8	+	5.57671e8i	0.610029	+	0.447740i
$106$		0			0
$107$	3.93801e8	−	6.82083e8i	0.290435	−	0.503049i	−0.683477	−	0.729972i	$-0.739533\pi$
$107$	3.93801e8	−	6.82083e8i	0.290435	−	0.503049i	0.973913	+	0.226923i	$0.0728665\pi$
$108$		0			0
$109$	1.06252e9	+	1.84035e9i	0.720974	+	1.24876i	0.960610	+	0.277901i	$0.0896388\pi$
$109$	1.06252e9	+	1.84035e9i	0.720974	+	1.24876i	−0.239635	+	0.970863i	$0.577028\pi$
$110$		0			0
$111$	7.79036e8			0.487084
$112$		0			0
$113$	2.83527e9			1.63585			0.817923	−	0.575328i	$-0.195126\pi$
$113$	2.83527e9			1.63585			0.817923	+	0.575328i	$0.195126\pi$
$114$		0			0
$115$	1.06913e9	+	1.85179e9i	0.570022	+	0.987307i
$116$		0			0
$117$	4.89868e8	−	8.48477e8i	0.241681	−	0.418604i
$118$		0			0
$119$	−4.49272e7	+	4.08422e8i	−0.0205375	+	0.186702i
$120$		0			0
$121$	−4.91033e8	+	8.50494e8i	−0.208246	+	0.360692i
$122$		0			0
$123$	7.77962e7	+	1.34747e8i	0.0306468	+	0.0530818i
$124$		0			0
$125$	−1.00952e9			−0.369843
$126$		0			0
$127$	−8.07210e8			−0.275340			−0.137670	−	0.990478i	$-0.543961\pi$
$127$	−8.07210e8			−0.275340			−0.137670	+	0.990478i	$0.543961\pi$
$128$		0			0
$129$	−8.21555e8	−	1.42298e9i	−0.261206	−	0.452422i
$130$		0			0
$131$	1.43217e8	−	2.48059e8i	0.0424888	−	0.0735927i	−0.843999	−	0.536345i	$-0.819805\pi$
$131$	1.43217e8	−	2.48059e8i	0.0424888	−	0.0735927i	0.886488	+	0.462752i	$0.153138\pi$
$132$		0			0
$133$	3.35606e9	−	1.47479e9i	0.930030	−	0.408692i
$134$		0			0
$135$	−4.86719e8	+	8.43023e8i	−0.126118	+	0.218443i
$136$		0			0
$137$	−8.41169e8	−	1.45695e9i	−0.204005	−	0.353347i	0.745810	−	0.666158i	$-0.232063\pi$
$137$	−8.41169e8	−	1.45695e9i	−0.204005	−	0.353347i	−0.949815	+	0.312811i	$0.898729\pi$
$138$		0			0
$139$	−6.04548e7			−0.0137361			−0.00686806	−	0.999976i	$-0.502186\pi$
$139$	−6.04548e7			−0.0137361			−0.00686806	+	0.999976i	$0.502186\pi$
$140$		0			0
$141$	−3.85506e8			−0.0821383
$142$		0			0
$143$	4.31502e9	+	7.47384e9i	0.862921	+	1.49462i
$144$		0			0
$145$	3.32067e9	−	5.75157e9i	0.623835	−	1.08051i
$146$		0			0
$147$	2.21057e9	−	2.40778e9i	0.390459	−	0.425294i
$148$		0			0
$149$	−1.39678e9	+	2.41929e9i	−0.232161	+	0.402114i	−0.958444	−	0.285282i	$-0.907913\pi$
$149$	−1.39678e9	+	2.41929e9i	−0.232161	+	0.402114i	0.726283	+	0.687396i	$0.241246\pi$
$150$		0			0
$151$	1.31123e9	+	2.27112e9i	0.205250	+	0.355504i	0.950212	−	0.311603i	$-0.100866\pi$
$151$	1.31123e9	+	2.27112e9i	0.205250	+	0.355504i	−0.744962	+	0.667107i	$0.767533\pi$
$152$		0			0
$153$	−4.24375e8			−0.0626092
$154$		0			0
$155$	5.32939e9			0.741626
$156$		0			0
$157$	2.76906e9	+	4.79616e9i	0.363735	+	0.630007i	0.988572	−	0.150748i	$-0.0481682\pi$
$157$	2.76906e9	+	4.79616e9i	0.363735	+	0.630007i	−0.624838	+	0.780755i	$0.714835\pi$
$158$		0			0
$159$	1.41034e9	−	2.44278e9i	0.174999	−	0.303108i
$160$		0			0
$161$	6.78906e9	−	2.98338e9i	0.796330	−	0.349939i
$162$		0			0
$163$	5.62560e9	−	9.74383e9i	0.624202	−	1.08115i	−0.364493	−	0.931206i	$-0.618758\pi$
$163$	5.62560e9	−	9.74383e9i	0.624202	−	1.08115i	0.988695	−	0.149943i	$-0.0479090\pi$
$164$		0			0
$165$	−4.28729e9	−	7.42580e9i	−0.450303	−	0.779948i
$166$		0			0
$167$	−7.86750e9			−0.782731			−0.391366	−	0.920235i	$-0.627997\pi$
$167$	−7.86750e9			−0.782731			−0.391366	+	0.920235i	$0.627997\pi$
$168$		0			0
$169$	1.16941e10			1.10275
$170$		0			0
$171$	1.89308e9	+	3.27890e9i	0.169311	+	0.293255i
$172$		0			0
$173$	−3.21154e8	+	5.56255e8i	−0.0272587	+	0.0472135i	−0.879333	−	0.476207i	$-0.842011\pi$
$173$	−3.21154e8	+	5.56255e8i	−0.0272587	+	0.0472135i	0.852074	+	0.523421i	$0.175344\pi$
$174$		0			0
$175$	9.73809e8	−	8.85266e9i	0.0784878	−	0.713514i
$176$		0			0
$177$	−1.74898e9	+	3.02933e9i	−0.133939	+	0.231989i
$178$		0			0
$179$	5.53364e9	+	9.58454e9i	0.402877	+	0.697803i	0.994072	−	0.108726i	$-0.0346771\pi$
$179$	5.53364e9	+	9.58454e9i	0.402877	+	0.697803i	−0.591195	+	0.806528i	$0.701344\pi$
$180$		0			0
$181$	−2.33974e9			−0.162037			−0.0810184	−	0.996713i	$-0.525817\pi$
$181$	−2.33974e9			−0.162037			−0.0810184	+	0.996713i	$0.525817\pi$
$182$		0			0
$183$	1.30425e10			0.859665
$184$		0			0
$185$	−8.80838e9	−	1.52566e10i	−0.552871	−	0.957600i
$186$		0			0
$187$	1.86906e9	−	3.23731e9i	0.111773	−	0.193596i
$188$		0			0
$189$	2.72156e9	+	1.99753e9i	0.155146	+	0.113872i
$190$		0			0
$191$	−4.06408e9	+	7.03919e9i	−0.220959	+	0.382712i	−0.955099	−	0.296285i	$-0.904252\pi$
$191$	−4.06408e9	+	7.03919e9i	−0.220959	+	0.382712i	0.734140	+	0.678998i	$0.237585\pi$
$192$		0			0
$193$	1.70512e10	+	2.95335e10i	0.884598	+	1.53217i	0.846174	+	0.532907i	$0.178901\pi$
$193$	1.70512e10	+	2.95335e10i	0.884598	+	1.53217i	0.0384247	+	0.999262i	$0.487766\pi$
$194$		0			0
$195$	−2.21553e10			−1.09729
$196$		0			0
$197$	3.33776e10			1.57891			0.789454	−	0.613810i	$-0.210364\pi$
$197$	3.33776e10			1.57891			0.789454	+	0.613810i	$0.210364\pi$
$198$		0			0
$199$	1.99292e10	+	3.45184e10i	0.900846	+	1.56031i	0.826398	+	0.563086i	$0.190386\pi$
$199$	1.99292e10	+	3.45184e10i	0.900846	+	1.56031i	0.0744479	+	0.997225i	$0.476281\pi$
$200$		0			0
$201$	1.14496e10	−	1.98313e10i	0.494777	−	0.856978i
$202$		0			0
$203$	−1.85680e10	−	1.36283e10i	−0.767420	−	0.563260i
$204$		0			0
$205$	1.75925e9	−	3.04710e9i	0.0695720	−	0.120502i
$206$		0			0
$207$	3.82955e9	+	6.63298e9i	0.144971	+	0.251097i
$208$		0			0
$209$	−3.33505e10			−1.20905
$210$		0			0
$211$	−5.05184e10			−1.75460			−0.877301	−	0.479941i	$-0.840658\pi$
$211$	−5.05184e10			−1.75460			−0.877301	+	0.479941i	$0.840658\pi$
$212$		0			0
$213$	−7.93144e9	−	1.37377e10i	−0.264024	−	0.457303i
$214$		0			0
$215$	−1.85783e10	+	3.21785e10i	−0.592969	+	1.02705i
$216$		0			0
$217$	2.02094e9	−	1.83719e10i	0.0618706	−	0.562451i
$218$		0			0
$219$	−9.24619e9	+	1.60149e10i	−0.271622	+	0.470462i
$220$		0			0
$221$	−4.82935e9	−	8.36469e9i	−0.136183	−	0.235876i
$222$		0			0
$223$	−3.55775e10			−0.963392			−0.481696	−	0.876338i	$-0.659979\pi$
$223$	−3.55775e10			−0.963392			−0.481696	+	0.876338i	$0.659979\pi$
$224$		0			0
$225$	9.19845e9			0.239273
$226$		0			0
$227$	−3.89954e10	−	6.75420e10i	−0.974758	−	1.68833i	−0.680730	−	0.732534i	$-0.738337\pi$
$227$	−3.89954e10	−	6.75420e10i	−0.974758	−	1.68833i	−0.294028	−	0.955797i	$-0.594996\pi$
$228$		0			0
$229$	1.12505e10	−	1.94865e10i	0.270341	−	0.468245i	−0.698608	−	0.715505i	$-0.746197\pi$
$229$	1.12505e10	−	1.94865e10i	0.270341	−	0.468245i	0.968949	+	0.247260i	$0.0795301\pi$
$230$		0			0
$231$	−2.72245e10	+	1.19635e10i	−0.629081	+	0.276443i
$232$		0			0
$233$	2.77074e10	−	4.79906e10i	0.615877	−	1.06673i	−0.374353	−	0.927286i	$-0.622135\pi$
$233$	2.77074e10	−	4.79906e10i	0.615877	−	1.06673i	0.990230	−	0.139444i	$-0.0445314\pi$
$234$		0			0
$235$	4.35883e9	+	7.54972e9i	0.0932320	+	0.161482i
$236$		0			0
$237$	−2.01093e10			−0.414028
$238$		0			0
$239$	7.63227e10			1.51308			0.756542	−	0.653945i	$-0.226887\pi$
$239$	7.63227e10			1.51308			0.756542	+	0.653945i	$0.226887\pi$
$240$		0			0
$241$	1.22809e10	+	2.12711e10i	0.234506	+	0.406176i	0.959129	−	0.282970i	$-0.0913195\pi$
$241$	1.22809e10	+	2.12711e10i	0.234506	+	0.406176i	−0.724623	+	0.689145i	$0.757986\pi$
$242$		0			0
$243$	−1.74339e9	+	3.01964e9i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	−7.21481e10	−	1.60673e10i	−1.27932	−	0.284902i
$246$		0			0
$247$	−4.30861e10	+	7.46273e10i	−0.736548	+	1.27574i
$248$		0			0
$249$	2.30523e9	+	3.99278e9i	0.0380030	+	0.0658231i
$250$		0			0
$251$	−2.98221e10			−0.474248			−0.237124	−	0.971479i	$-0.576205\pi$
$251$	−2.98221e10			−0.474248			−0.237124	+	0.971479i	$0.576205\pi$
$252$		0			0
$253$	−6.74656e10			−1.03524
$254$		0			0
$255$	4.79831e9	+	8.31092e9i	0.0710653	+	0.123089i
$256$		0			0
$257$	4.31293e10	−	7.47022e10i	0.616700	−	1.06816i	−0.373384	−	0.927677i	$-0.621803\pi$
$257$	4.31293e10	−	7.47022e10i	0.616700	−	1.06816i	0.990084	−	0.140478i	$-0.0448641\pi$
$258$		0			0
$259$	−5.59338e10	+	2.45795e10i	−0.772370	+	0.339410i
$260$		0			0
$261$	1.18944e10	−	2.06017e10i	0.158657	−	0.274802i
$262$		0			0
$263$	2.92653e10	+	5.06890e10i	0.377183	+	0.653300i	0.990651	−	0.136419i	$-0.0435595\pi$
$263$	2.92653e10	+	5.06890e10i	0.377183	+	0.653300i	−0.613468	+	0.789719i	$0.710226\pi$
$264$		0			0
$265$	−6.37855e10			−0.794540
$266$		0			0
$267$	−9.04459e10			−1.08915
$268$		0			0
$269$	3.36288e10	+	5.82468e10i	0.391585	+	0.678245i	0.992659	−	0.120949i	$-0.0385936\pi$
$269$	3.36288e10	+	5.82468e10i	0.391585	+	0.678245i	−0.601074	+	0.799193i	$0.705260\pi$
$270$		0			0
$271$	−7.22808e10	+	1.25194e11i	−0.814069	+	1.41001i	0.0959250	+	0.995389i	$0.469419\pi$
$271$	−7.22808e10	+	1.25194e11i	−0.814069	+	1.41001i	−0.909994	+	0.414621i	$0.863914\pi$
$272$		0			0
$273$	−8.40143e9	+	7.63754e10i	−0.0915422	+	0.832188i
$274$		0			0
$275$	−4.05125e10	+	7.01696e10i	−0.427161	+	0.739864i
$276$		0			0
$277$	−4.51107e10	−	7.81340e10i	−0.460384	−	0.797408i	0.538596	−	0.842564i	$-0.318955\pi$
$277$	−4.51107e10	−	7.81340e10i	−0.460384	−	0.797408i	−0.998980	+	0.0451557i	$0.985622\pi$
$278$		0			0
$279$	1.90895e10			0.188615
$280$		0			0
$281$	−1.85971e11			−1.77937			−0.889684	−	0.456577i	$-0.849075\pi$
$281$	−1.85971e11			−1.77937			−0.889684	+	0.456577i	$0.849075\pi$
$282$		0			0
$283$	1.01150e11	+	1.75196e11i	0.937400	+	1.62363i	0.770297	+	0.637686i	$0.220108\pi$
$283$	1.01150e11	+	1.75196e11i	0.937400	+	1.62363i	0.167104	+	0.985939i	$0.446559\pi$
$284$		0			0
$285$	4.28091e10	−	7.41476e10i	0.384357	−	0.665726i
$286$		0			0
$287$	−9.83709e9	−	7.22009e9i	−0.0855851	−	0.0628165i
$288$		0			0
$289$	5.72021e10	−	9.90769e10i	0.482360	−	0.835473i
$290$		0			0
$291$	−4.84589e10	−	8.39333e10i	−0.396146	−	0.686145i
$292$		0			0
$293$	3.74292e10			0.296692			0.148346	−	0.988935i	$-0.452605\pi$
$293$	3.74292e10			0.296692			0.148346	+	0.988935i	$0.452605\pi$
$294$		0			0
$295$	7.91015e10			0.608115
$296$		0			0
$297$	−1.53567e10	−	2.65987e10i	−0.114524	−	0.198361i
$298$		0			0
$299$	−8.71600e10	+	1.50966e11i	−0.630663	+	1.09234i
$300$		0			0
$301$	1.03883e11	+	7.62467e10i	0.729451	+	0.535392i
$302$		0			0
$303$	1.71619e10	−	2.97254e10i	0.116970	−	0.202598i
$304$		0			0
$305$	−1.47468e11	−	2.55422e11i	−0.975772	−	1.69009i
$306$		0			0
$307$	1.88435e11			1.21071			0.605354	−	0.795956i	$-0.293031\pi$
$307$	1.88435e11			1.21071			0.605354	+	0.795956i	$0.293031\pi$
$308$		0			0
$309$	−9.57849e10			−0.597701
$310$		0			0
$311$	−1.15884e11	−	2.00717e11i	−0.702428	−	1.21664i	−0.967612	−	0.252443i	$-0.918766\pi$
$311$	−1.15884e11	−	2.00717e11i	−0.702428	−	1.21664i	0.265184	−	0.964198i	$-0.414567\pi$
$312$		0			0
$313$	1.07730e11	−	1.86594e11i	0.634434	−	1.09887i	−0.352200	−	0.935925i	$-0.614566\pi$
$313$	1.07730e11	−	1.86594e11i	0.634434	−	1.09887i	0.986635	−	0.162948i	$-0.0521003\pi$
$314$		0			0
$315$	8.34743e9	−	7.58845e10i	0.0477700	−	0.434266i
$316$		0			0
$317$	4.72901e10	−	8.19089e10i	0.263029	−	0.455580i	−0.704016	−	0.710184i	$-0.748612\pi$
$317$	4.72901e10	−	8.19089e10i	0.263029	−	0.455580i	0.967045	+	0.254604i	$0.0819451\pi$
$318$		0			0
$319$	1.04772e11	+	1.81471e11i	0.566484	+	0.981180i
$320$		0			0
$321$	−6.37957e10			−0.335366
$322$		0			0
$323$	3.73257e10			0.190808
$324$		0			0
$325$	1.04678e11	+	1.81307e11i	0.520449	+	0.901445i
$326$		0			0
$327$	8.60645e10	−	1.49068e11i	0.416255	−	0.720974i
$328$		0			0
$329$	2.76788e10	−	1.21632e10i	0.130247	−	0.0572356i
$330$		0			0
$331$	7.41949e10	−	1.28509e11i	0.339741	−	0.588449i	−0.644643	−	0.764484i	$-0.722994\pi$
$331$	7.41949e10	−	1.28509e11i	0.339741	−	0.588449i	0.984384	+	0.176035i	$0.0563272\pi$
$332$		0			0
$333$	−3.15510e10	−	5.46479e10i	−0.140609	−	0.243542i
$334$		0			0
$335$	−5.17833e11			−2.24641
$336$		0			0
$337$	2.34876e10			0.0991984			0.0495992	−	0.998769i	$-0.484206\pi$
$337$	2.34876e10			0.0991984			0.0495992	+	0.998769i	$0.484206\pi$
$338$		0			0
$339$	−1.14829e11	−	1.98889e11i	−0.472228	−	0.817923i
$340$		0			0
$341$	−8.40753e10	+	1.45623e11i	−0.336724	+	0.583222i
$342$		0			0
$343$	−8.27473e10	+	2.42622e11i	−0.322798	+	0.946468i
$344$		0			0
$345$	8.65998e10	−	1.49995e11i	0.329102	−	0.570022i
$346$		0			0
$347$	−3.68664e10	−	6.38545e10i	−0.136505	−	0.236433i	0.789666	−	0.613536i	$-0.210254\pi$
$347$	−3.68664e10	−	6.38545e10i	−0.136505	−	0.236433i	−0.926171	+	0.377103i	$0.876920\pi$
$348$		0			0
$349$	1.66879e11			0.602125			0.301062	−	0.953604i	$-0.402659\pi$
$349$	1.66879e11			0.602125			0.301062	+	0.953604i	$0.402659\pi$
$350$		0			0
$351$	−7.93586e10			−0.279069
$352$		0			0
$353$	2.20407e11	+	3.81756e11i	0.755509	+	1.30858i	0.945121	+	0.326720i	$0.105944\pi$
$353$	2.20407e11	+	3.81756e11i	0.755509	+	1.30858i	−0.189612	+	0.981859i	$0.560723\pi$
$354$		0			0
$355$	−1.79358e11	+	3.10657e11i	−0.599367	+	1.03813i
$356$		0			0
$357$	3.04696e10	−	1.33895e10i	0.0992794	−	0.0436273i
$358$		0			0
$359$	2.23474e11	−	3.87068e11i	0.710070	−	1.22988i	−0.254760	−	0.967004i	$-0.581997\pi$
$359$	2.23474e11	−	3.87068e11i	0.710070	−	1.22988i	0.964830	−	0.262873i	$-0.0846701\pi$
$360$		0			0
$361$	−5.16059e9	−	8.93841e9i	−0.0159925	−	0.0276999i
$362$		0			0
$363$	7.95473e10			0.240462
$364$		0			0
$365$	4.18178e11			1.23323
$366$		0			0
$367$	−6.29622e10	−	1.09054e11i	−0.181168	−	0.313793i	0.761110	−	0.648622i	$-0.224655\pi$
$367$	−6.29622e10	−	1.09054e11i	−0.181168	−	0.313793i	−0.942279	+	0.334830i	$0.891321\pi$
$368$		0			0
$369$	6.30149e9	−	1.09145e10i	0.0176939	−	0.0306468i
$370$		0			0
$371$	−2.41879e10	+	2.19886e11i	−0.0662850	+	0.602581i
$372$		0			0
$373$	−4.87723e10	+	8.44761e10i	−0.130462	+	0.225966i	−0.923855	−	0.382743i	$-0.874979\pi$
$373$	−4.87723e10	+	8.44761e10i	−0.130462	+	0.225966i	0.793393	+	0.608710i	$0.208313\pi$
$374$		0			0
$375$	4.08854e10	+	7.08155e10i	0.106765	+	0.184922i
$376$		0			0
$377$	5.41428e11			1.38040
$378$		0			0
$379$	7.63006e11			1.89955			0.949777	−	0.312928i	$-0.101310\pi$
$379$	7.63006e11			1.89955			0.949777	+	0.312928i	$0.101310\pi$
$380$		0			0
$381$	3.26920e10	+	5.66242e10i	0.0794839	+	0.137670i
$382$		0			0
$383$	−2.05441e11	+	3.55833e11i	−0.487856	+	0.844991i	−0.999902	−	0.0139664i	$-0.995554\pi$
$383$	−2.05441e11	+	3.55833e11i	−0.487856	+	0.844991i	0.512046	+	0.858958i	$0.328888\pi$
$384$		0			0
$385$	5.42115e11	+	3.97894e11i	1.25753	+	0.922983i
$386$		0			0
$387$	−6.65460e10	+	1.15261e11i	−0.150807	+	0.261206i
$388$		0			0
$389$	−3.32175e11	−	5.75344e11i	−0.735518	−	1.27396i	−0.954495	−	0.298225i	$-0.903605\pi$
$389$	−3.32175e11	−	5.75344e11i	−0.735518	−	1.27396i	0.218977	−	0.975730i	$-0.429728\pi$
$390$		0			0
$391$	7.55071e10			0.163378
$392$		0			0
$393$	−2.32012e10			−0.0490618
$394$		0			0
$395$	2.27372e11	+	3.93819e11i	0.469948	+	0.813973i
$396$		0			0
$397$	−2.81285e11	+	4.87200e11i	−0.568316	+	0.984352i	0.428417	+	0.903581i	$0.359071\pi$
$397$	−2.81285e11	+	4.87200e11i	−0.568316	+	0.984352i	−0.996733	+	0.0807704i	$0.974262\pi$
$398$		0			0
$399$	−2.39374e11	−	1.75692e11i	−0.472823	−	0.347036i
$400$		0			0
$401$	1.03728e11	−	1.79662e11i	0.200330	−	0.346981i	−0.748305	−	0.663355i	$-0.769132\pi$
$401$	1.03728e11	−	1.79662e11i	0.200330	−	0.346981i	0.948635	+	0.316374i	$0.102465\pi$
$402$		0			0
$403$	2.17237e11	+	3.76265e11i	0.410261	+	0.710594i
$404$		0			0
$405$	7.88485e10			0.145628
$406$		0			0
$407$	5.55836e11			1.00409
$408$		0			0
$409$	−3.15965e10	−	5.47268e10i	−0.0558322	−	0.0967041i	0.836758	−	0.547572i	$-0.184448\pi$
$409$	−3.15965e10	−	5.47268e10i	−0.0558322	−	0.0967041i	−0.892591	+	0.450868i	$0.851115\pi$
$410$		0			0
$411$	−6.81347e10	+	1.18013e11i	−0.117782	+	0.204005i
$412$		0			0
$413$	2.99958e10	−	2.72684e11i	0.0507323	−	0.461195i
$414$		0			0
$415$	5.21294e10	−	9.02909e10i	0.0862714	−	0.149426i
$416$		0			0
$417$	2.44842e9	+	4.24078e9i	0.00396528	+	0.00686806i
$418$		0			0
$419$	8.74112e11			1.38549			0.692746	−	0.721182i	$-0.256401\pi$
$419$	8.74112e11			1.38549			0.692746	+	0.721182i	$0.256401\pi$
$420$		0			0
$421$	−6.60063e8			−0.00102404			−0.000512019	−	1.00000i	$-0.500163\pi$
$421$	−6.60063e8			−0.00102404			−0.000512019		1.00000i	$0.500163\pi$
$422$		0			0
$423$	1.56130e10	+	2.70425e10i	0.0237113	+	0.0410691i
$424$		0			0
$425$	4.53413e10	−	7.85335e10i	0.0674131	−	0.116763i
$426$		0			0
$427$	−9.36431e11	+	4.11505e11i	−1.36317	+	0.599031i
$428$		0			0
$429$	3.49517e11	−	6.05381e11i	0.498208	−	0.862921i
$430$		0			0
$431$	−5.33311e11	−	9.23722e11i	−0.744446	−	1.28942i	−0.950453	−	0.310867i	$-0.899381\pi$
$431$	−5.33311e11	−	9.23722e11i	−0.744446	−	1.28942i	0.206008	−	0.978550i	$-0.433953\pi$
$432$		0			0
$433$	1.37083e12			1.87408			0.937042	−	0.349216i	$-0.113552\pi$
$433$	1.37083e12			1.87408			0.937042	+	0.349216i	$0.113552\pi$
$434$		0			0
$435$	−5.37948e11			−0.720342
$436$		0			0
$437$	−3.36826e11	−	5.83400e11i	−0.441814	−	0.765245i
$438$		0			0
$439$	5.60927e11	−	9.71555e11i	0.720802	−	1.24847i	−0.239876	−	0.970804i	$-0.577107\pi$
$439$	5.60927e11	−	9.71555e11i	0.720802	−	1.24847i	0.960679	−	0.277663i	$-0.0895598\pi$
$440$		0			0
$441$	−2.58429e11	−	5.75518e10i	−0.325363	−	0.0724578i
$442$		0			0
$443$	7.59131e11	−	1.31485e12i	0.936484	−	1.62204i	0.164517	−	0.986374i	$-0.447394\pi$
$443$	7.59131e11	−	1.31485e12i	0.936484	−	1.62204i	0.771967	−	0.635663i	$-0.219273\pi$
$444$		0			0
$445$	1.02265e12	+	1.77128e12i	1.23625	+	2.14125i
$446$		0			0
$447$	2.26278e11			0.268076
$448$		0			0
$449$	−1.56604e12			−1.81842			−0.909211	−	0.416335i	$-0.863314\pi$
$449$	−1.56604e12			−1.81842			−0.909211	+	0.416335i	$0.863314\pi$
$450$		0			0
$451$	5.55069e10	+	9.61408e10i	0.0631761	+	0.109424i
$452$		0			0
$453$	1.06210e11	−	1.83961e11i	0.118501	−	0.205250i
$454$		0			0
$455$	1.59072e12	−	6.99027e11i	1.73997	−	0.764614i
$456$		0			0
$457$	−3.98322e11	+	6.89914e11i	−0.427180	+	0.739898i	−0.996621	−	0.0821343i	$-0.973826\pi$
$457$	−3.98322e11	+	6.89914e11i	−0.427180	+	0.739898i	0.569441	+	0.822032i	$0.307160\pi$
$458$		0			0
$459$	1.71872e10	+	2.97691e10i	0.0180737	+	0.0313046i
$460$		0			0
$461$	−1.71941e12			−1.77307			−0.886535	−	0.462661i	$-0.846895\pi$
$461$	−1.71941e12			−1.77307			−0.886535	+	0.462661i	$0.846895\pi$
$462$		0			0
$463$	1.00265e12			1.01399			0.506995	−	0.861949i	$-0.330756\pi$
$463$	1.00265e12			1.01399			0.506995	+	0.861949i	$0.330756\pi$
$464$		0			0
$465$	−2.15840e11	−	3.73847e11i	−0.214089	−	0.370813i
$466$		0			0
$467$	2.77084e11	−	4.79923e11i	0.269579	−	0.466924i	−0.699174	−	0.714951i	$-0.746449\pi$
$467$	2.77084e11	−	4.79923e11i	0.269579	−	0.466924i	0.968753	+	0.248027i	$0.0797823\pi$
$468$		0			0
$469$	−1.96366e11	+	1.78511e12i	−0.187408	+	1.70368i
$470$		0			0
$471$	2.24294e11	−	3.88489e11i	0.210002	−	0.363735i
$472$		0			0
$473$	−5.86173e11	−	1.01528e12i	−0.538457	−	0.932634i
$474$		0			0
$475$	−8.09044e11			−0.729207
$476$		0			0
$477$	−2.28475e11			−0.202072
$478$		0			0
$479$	−3.01266e11	−	5.21807e11i	−0.261481	−	0.452898i	0.705155	−	0.709053i	$-0.250877\pi$
$479$	−3.01266e11	−	5.21807e11i	−0.261481	−	0.452898i	−0.966636	+	0.256155i	$0.917544\pi$
$480$		0			0
$481$	7.18095e11	−	1.24378e12i	0.611687	−	1.05947i
$482$		0			0
$483$	−4.84235e11	−	3.55412e11i	−0.404850	−	0.297146i
$484$		0			0
$485$	−1.09583e12	+	1.89803e12i	−0.899299	+	1.55763i
$486$		0			0
$487$	8.80537e11	+	1.52513e12i	0.709361	+	1.22865i	0.965095	+	0.261902i	$0.0843497\pi$
$487$	8.80537e11	+	1.52513e12i	0.709361	+	1.22865i	−0.255734	+	0.966747i	$0.582317\pi$
$488$		0			0
$489$	−9.11348e11			−0.720766
$490$		0			0
$491$	1.45110e11			0.112676			0.0563379	−	0.998412i	$-0.482058\pi$
$491$	1.45110e11			0.112676			0.0563379	+	0.998412i	$0.482058\pi$
$492$		0			0
$493$	−1.17260e11	−	2.03101e11i	−0.0894006	−	0.154846i
$494$		0			0
$495$	−3.47270e11	+	6.01490e11i	−0.259983	+	0.450303i
$496$		0			0
$497$	1.00291e12	+	7.36099e11i	0.737321	+	0.541168i
$498$		0			0
$499$	−1.02325e11	+	1.77232e11i	−0.0738803	+	0.127964i	−0.900599	−	0.434651i	$-0.856872\pi$
$499$	−1.02325e11	+	1.77232e11i	−0.0738803	+	0.127964i	0.826719	+	0.562616i	$0.190205\pi$
$500$		0			0
$501$	3.18634e11	+	5.51890e11i	0.225955	+	0.391366i
$502$		0			0
$503$	−1.68542e12			−1.17396			−0.586980	−	0.809602i	$-0.699683\pi$
$503$	−1.68542e12			−1.17396			−0.586980	+	0.809602i	$0.699683\pi$
$504$		0			0
$505$	−7.76184e11			−0.531072
$506$		0			0
$507$	−4.73613e11	−	8.20321e11i	−0.318337	−	0.551376i
$508$		0			0
$509$	2.09022e11	−	3.62037e11i	0.138027	−	0.239069i	−0.788723	−	0.614749i	$-0.789257\pi$
$509$	2.09022e11	−	3.62037e11i	0.138027	−	0.239069i	0.926750	+	0.375680i	$0.122591\pi$
$510$		0			0
$511$	1.58576e11	−	1.44157e12i	0.102883	−	0.935283i
$512$		0			0
$513$	1.53339e11	−	2.65591e11i	0.0977518	−	0.169311i
$514$		0			0
$515$	1.08302e12	+	1.87584e12i	0.678427	+	1.17507i
$516$		0			0
$517$	−2.75056e11			−0.169322
$518$		0			0
$519$	5.20269e10			0.0314757
$520$		0			0
$521$	7.78332e11	+	1.34811e12i	0.462802	+	0.801597i	0.999099	−	0.0424325i	$-0.0135107\pi$
$521$	7.78332e11	+	1.34811e12i	0.462802	+	0.801597i	−0.536297	+	0.844029i	$0.680177\pi$
$522$		0			0
$523$	1.32534e11	−	2.29556e11i	0.0774587	−	0.134162i	−0.824694	−	0.565579i	$-0.808653\pi$
$523$	1.32534e11	−	2.29556e11i	0.0774587	−	0.134162i	0.902153	+	0.431417i	$0.141986\pi$
$524$		0			0
$525$	−6.60436e11	+	2.90222e11i	−0.379415	+	0.166730i
$526$		0			0
$527$	9.40966e10	−	1.62980e11i	0.0531406	−	0.0920422i
$528$		0			0
$529$	2.19201e11	+	3.79667e11i	0.121700	+	0.210791i
$530$		0			0
$531$	2.83335e11			0.154659
$532$		0			0
$533$	2.86842e11			0.153947
$534$		0			0
$535$	7.21324e11	+	1.24937e12i	0.380661	+	0.659324i
$536$		0			0
$537$	4.48225e11	−	7.76348e11i	0.232601	−	0.402877i
$538$		0			0
$539$	1.57722e12	−	1.71793e12i	0.804902	−	0.876712i
$540$		0			0
$541$	2.41567e11	−	4.18406e11i	0.121241	−	0.209996i	−0.799016	−	0.601309i	$-0.794646\pi$
$541$	2.41567e11	−	4.18406e11i	0.121241	−	0.209996i	0.920257	+	0.391314i	$0.127979\pi$
$542$		0			0
$543$	9.47594e10	+	1.64128e11i	0.0467760	+	0.0810184i
$544$		0			0
$545$	−3.89245e12			−1.88990
$546$		0			0
$547$	−2.87051e12			−1.37093			−0.685467	−	0.728104i	$-0.740402\pi$
$547$	−2.87051e12			−1.37093			−0.685467	+	0.728104i	$0.740402\pi$
$548$		0			0
$549$	−5.28219e11	−	9.14903e11i	−0.248164	−	0.429833i
$550$		0			0
$551$	−1.04616e12	+	1.81201e12i	−0.483523	+	0.837487i
$552$		0			0
$553$	1.44382e12	−	6.34474e11i	0.656525	−	0.288503i
$554$		0			0
$555$	−7.13479e11	+	1.23578e12i	−0.319200	+	0.552871i
$556$		0			0
$557$	−1.06457e12	−	1.84388e12i	−0.468623	−	0.811680i	0.530733	−	0.847539i	$-0.321917\pi$
$557$	−1.06457e12	−	1.84388e12i	−0.468623	−	0.811680i	−0.999357	+	0.0358592i	$0.988583\pi$
$558$		0			0
$559$	−3.02915e12			−1.31210
$560$		0			0
$561$	−3.02788e11			−0.129064
$562$		0			0
$563$	−4.91680e11	−	8.51615e11i	−0.206250	−	0.357236i	0.744280	−	0.667868i	$-0.232793\pi$
$563$	−4.91680e11	−	8.51615e11i	−0.206250	−	0.357236i	−0.950530	+	0.310632i	$0.899459\pi$
$564$		0			0
$565$	−2.59668e12	+	4.49759e12i	−1.07201	+	1.85678i
$566$		0			0
$567$	2.98999e10	−	2.71813e11i	0.0121491	−	0.110445i
$568$		0			0
$569$	−1.93460e12	+	3.35083e12i	−0.773725	+	1.34013i	0.161784	+	0.986826i	$0.448275\pi$
$569$	−1.93460e12	+	3.35083e12i	−0.773725	+	1.34013i	−0.935508	+	0.353304i	$0.885058\pi$
$570$		0			0
$571$	−9.22174e11	−	1.59725e12i	−0.363037	−	0.628798i	0.625422	−	0.780286i	$-0.284927\pi$
$571$	−9.22174e11	−	1.59725e12i	−0.363037	−	0.628798i	−0.988459	+	0.151488i	$0.951593\pi$
$572$		0			0
$573$	6.58380e11			0.255142
$574$		0			0
$575$	−1.63664e12			−0.624378
$576$		0			0
$577$	−5.86327e11	−	1.01555e12i	−0.220216	−	0.381425i	0.734657	−	0.678438i	$-0.237343\pi$
$577$	−5.86327e11	−	1.01555e12i	−0.220216	−	0.381425i	−0.954873	+	0.297013i	$0.904010\pi$
$578$		0			0
$579$	1.38114e12	−	2.39221e12i	0.510723	−	0.884598i
$580$		0			0
$581$	−2.91489e11	−	2.13943e11i	−0.106128	−	0.0778944i
$582$		0			0
$583$	1.00627e12	−	1.74290e12i	0.360748	−	0.624834i
$584$		0			0
$585$	8.97290e11	+	1.55415e12i	0.316761	+	0.548646i
$586$		0			0
$587$	−4.28345e12			−1.48910			−0.744548	−	0.667569i	$-0.767335\pi$
$587$	−4.28345e12			−1.48910			−0.744548	+	0.667569i	$0.767335\pi$
$588$		0			0
$589$	−1.67901e12			−0.574822
$590$		0			0
$591$	−1.35179e12	−	2.34137e12i	−0.455791	−	0.789454i
$592$		0			0
$593$	−2.22161e12	+	3.84794e12i	−0.737771	+	1.27786i	0.215726	+	0.976454i	$0.430788\pi$
$593$	−2.22161e12	+	3.84794e12i	−0.737771	+	1.27786i	−0.953497	+	0.301402i	$0.902545\pi$
$594$		0			0
$595$	−6.06732e11	−	4.45320e11i	−0.198459	−	0.145662i
$596$		0			0
$597$	1.61426e12	−	2.79599e12i	0.520104	−	0.900846i
$598$		0			0
$599$	3.33200e11	+	5.77120e11i	0.105751	+	0.183166i	0.914045	−	0.405613i	$-0.132942\pi$
$599$	3.33200e11	+	5.77120e11i	0.105751	+	0.183166i	−0.808294	+	0.588779i	$0.799609\pi$
$600$		0			0
$601$	−7.35151e11			−0.229848			−0.114924	−	0.993374i	$-0.536663\pi$
$601$	−7.35151e11			−0.229848			−0.114924	+	0.993374i	$0.536663\pi$
$602$		0			0
$603$	−1.85484e12			−0.571319
$604$		0			0
$605$	−8.99423e11	−	1.55785e12i	−0.272939	−	0.472744i
$606$		0			0
$607$	−3.03729e11	+	5.26074e11i	−0.0908107	+	0.157289i	−0.907852	−	0.419290i	$-0.862279\pi$
$607$	−3.03729e11	+	5.26074e11i	−0.0908107	+	0.157289i	0.817042	+	0.576578i	$0.195612\pi$
$608$		0			0
$609$	−2.03993e11	+	1.85445e12i	−0.0600950	+	0.546309i
$610$		0			0
$611$	−3.55350e11	+	6.15483e11i	−0.103150	+	0.178661i
$612$		0			0
$613$	3.12876e12	+	5.41916e12i	0.894951	+	1.55010i	0.833865	+	0.551969i	$0.186123\pi$
$613$	3.12876e12	+	5.41916e12i	0.894951	+	1.55010i	0.0610865	+	0.998132i	$0.480543\pi$
$614$		0			0
$615$	−2.84998e11			−0.0803348
$616$		0			0
$617$	1.58842e12			0.441248			0.220624	−	0.975359i	$-0.429191\pi$
$617$	1.58842e12			0.441248			0.220624	+	0.975359i	$0.429191\pi$
$618$		0			0
$619$	−3.32393e12	−	5.75721e12i	−0.910004	−	1.57617i	−0.814055	−	0.580788i	$-0.802745\pi$
$619$	−3.32393e12	−	5.75721e12i	−0.910004	−	1.57617i	−0.0959493	−	0.995386i	$-0.530589\pi$
$620$		0			0
$621$	3.10194e11	−	5.37272e11i	0.0836991	−	0.144971i
$622$		0			0
$623$	6.49390e12	−	2.85368e12i	1.72707	−	0.758942i
$624$		0			0
$625$	2.29369e12	−	3.97279e12i	0.601278	−	1.04144i
$626$		0			0
$627$	1.35069e12	+	2.33947e12i	0.349022	+	0.604524i
$628$		0			0
$629$	−6.22089e11			−0.158462
$630$		0			0
$631$	5.09775e12			1.28011			0.640053	−	0.768331i	$-0.278912\pi$
$631$	5.09775e12			1.28011			0.640053	+	0.768331i	$0.278912\pi$
$632$		0			0
$633$	2.04599e12	+	3.54377e12i	0.506510	+	0.877301i
$634$		0			0
$635$	7.39282e11	−	1.28047e12i	0.180438	−	0.312528i
$636$		0			0
$637$	−1.80652e12	−	5.74873e12i	−0.434727	−	1.38339i
$638$		0			0
$639$	−6.42446e11	+	1.11275e12i	−0.152434	+	0.264024i
$640$		0			0
$641$	−1.25266e12	−	2.16968e12i	−0.293072	−	0.507615i	0.681463	−	0.731853i	$-0.261344\pi$
$641$	−1.25266e12	−	2.16968e12i	−0.293072	−	0.507615i	−0.974534	+	0.224238i	$0.928011\pi$
$642$		0			0
$643$	−3.20633e11			−0.0739705			−0.0369852	−	0.999316i	$-0.511775\pi$
$643$	−3.20633e11			−0.0739705			−0.0369852	+	0.999316i	$0.511775\pi$
$644$		0			0
$645$	3.00968e12			0.684702
$646$		0			0
$647$	1.27172e12	+	2.20268e12i	0.285313	+	0.494176i	0.972685	−	0.232129i	$-0.0745692\pi$
$647$	1.27172e12	+	2.20268e12i	0.285313	+	0.494176i	−0.687372	+	0.726305i	$0.741236\pi$
$648$		0			0
$649$	−1.24789e12	+	2.16140e12i	−0.276105	+	0.478227i
$650$		0			0
$651$	−1.37060e12	+	6.02296e11i	−0.299086	+	0.131430i
$652$		0			0
$653$	3.68527e12	−	6.38308e12i	0.793159	−	1.37379i	−0.130842	−	0.991403i	$-0.541768\pi$
$653$	3.68527e12	−	6.38308e12i	0.793159	−	1.37379i	0.924002	−	0.382389i	$-0.124898\pi$
$654$		0			0
$655$	2.62330e11	+	4.54370e11i	0.0556881	+	0.0964547i
$656$		0			0
$657$	1.49788e12			0.313642
$658$		0			0
$659$	6.35541e12			1.31268			0.656341	−	0.754464i	$-0.272103\pi$
$659$	6.35541e12			1.31268			0.656341	+	0.754464i	$0.272103\pi$
$660$		0			0
$661$	−1.68992e12	−	2.92703e12i	−0.344318	−	0.596376i	0.640912	−	0.767615i	$-0.278556\pi$
$661$	−1.68992e12	−	2.92703e12i	−0.344318	−	0.596376i	−0.985230	+	0.171239i	$0.945223\pi$
$662$		0			0
$663$	−3.91178e11	+	6.77540e11i	−0.0786254	+	0.136183i
$664$		0			0
$665$	−7.34194e11	+	6.67438e12i	−0.145584	+	1.32347i
$666$		0			0
$667$	−2.11631e12	+	3.66556e12i	−0.414013	+	0.717091i
$668$		0			0
$669$	1.44089e12	+	2.49569e12i	0.278107	+	0.481696i
$670$		0			0
$671$	9.30568e12			1.77214
$672$		0			0
$673$	−7.27688e12			−1.36734			−0.683671	−	0.729790i	$-0.739618\pi$
$673$	−7.27688e12			−1.36734			−0.683671	+	0.729790i	$0.739618\pi$
$674$		0			0
$675$	−3.72537e11	−	6.45253e11i	−0.0690721	−	0.119636i
$676$		0			0
$677$	−3.16042e12	+	5.47402e12i	−0.578224	+	1.00151i	0.417459	+	0.908696i	$0.362921\pi$
$677$	−3.16042e12	+	5.47402e12i	−0.578224	+	1.00151i	−0.995683	+	0.0928182i	$0.970412\pi$
$678$		0			0
$679$	6.12748e12	+	4.49736e12i	1.10629	+	0.811977i
$680$		0			0
$681$	−3.15863e12	+	5.47090e12i	−0.562777	+	0.974758i
$682$		0			0
$683$	−7.47846e11	−	1.29531e12i	−0.131498	−	0.227761i	0.792756	−	0.609539i	$-0.208645\pi$
$683$	−7.47846e11	−	1.29531e12i	−0.131498	−	0.227761i	−0.924254	+	0.381778i	$0.875312\pi$
$684$		0			0
$685$	3.08153e12			0.534760
$686$		0			0
$687$	−1.82258e12			−0.312163
$688$		0			0
$689$	−2.60003e12	−	4.50338e12i	−0.439533	−	0.761293i
$690$		0			0
$691$	3.96649e12	−	6.87016e12i	0.661843	−	1.14635i	−0.318288	−	0.947994i	$-0.603108\pi$
$691$	3.96649e12	−	6.87016e12i	0.661843	−	1.14635i	0.980131	−	0.198352i	$-0.0635588\pi$
$692$		0			0
$693$	1.94181e12	+	1.42522e12i	0.319822	+	0.234738i
$694$		0			0
$695$	5.53674e10	−	9.58992e10i	0.00900166	−	0.0155913i
$696$		0			0
$697$	−6.21231e10	−	1.07600e11i	−0.00997024	−	0.0172690i
$698$		0			0
$699$	−4.48860e12			−0.711153
$700$		0			0
$701$	8.22869e12			1.28706			0.643531	−	0.765420i	$-0.277469\pi$
$701$	8.22869e12			1.28706			0.643531	+	0.765420i	$0.277469\pi$
$702$		0			0
$703$	2.77505e12	+	4.80652e12i	0.428520	+	0.742219i
$704$		0			0
$705$	3.53065e11	−	6.11527e11i	0.0538275	−	0.0932320i
$706$		0			0
$707$	−2.94334e11	+	2.67572e12i	−0.0443050	+	0.402767i
$708$		0			0
$709$	4.30296e12	−	7.45295e12i	0.639528	−	1.10770i	−0.346008	−	0.938231i	$-0.612463\pi$
$709$	4.30296e12	−	7.45295e12i	0.639528	−	1.10770i	0.985536	−	0.169464i	$-0.0542035\pi$
$710$		0			0
$711$	8.14428e11	+	1.41063e12i	0.119520	+	0.207014i
$712$		0			0
$713$	−3.39651e12			−0.492186
$714$		0			0
$715$	−1.58076e13			−2.26198
$716$		0			0
$717$	−3.09107e12	−	5.35389e12i	−0.436790	−	0.756542i
$718$		0			0
$719$	2.88894e12	−	5.00379e12i	0.403143	−	0.698263i	−0.590961	−	0.806700i	$-0.701251\pi$
$719$	2.88894e12	−	5.00379e12i	0.403143	−	0.698263i	0.994103	+	0.108437i	$0.0345845\pi$
$720$		0			0
$721$	6.87723e12	−	3.02213e12i	0.947773	−	0.416489i
$722$		0			0
$723$	9.94753e11	−	1.72296e12i	0.135392	−	0.234506i
$724$		0			0
$725$	2.54165e12	+	4.40227e12i	0.341661	+	0.591774i
$726$		0			0
$727$	−9.42101e12			−1.25081			−0.625407	−	0.780299i	$-0.715067\pi$
$727$	−9.42101e12			−1.25081			−0.625407	+	0.780299i	$0.715067\pi$
$728$		0			0
$729$	2.82430e11			0.0370370
$730$		0			0
$731$	6.56042e11	+	1.13630e12i	0.0849774	+	0.147185i
$732$		0			0
$733$	3.80139e12	−	6.58421e12i	0.486379	−	0.842434i	−0.513498	−	0.858091i	$-0.671651\pi$
$733$	3.80139e12	−	6.58421e12i	0.486379	−	0.842434i	0.999877	+	0.0156570i	$0.00498399\pi$
$734$		0			0
$735$	1.79491e12	+	5.71178e12i	0.226856	+	0.721902i
$736$		0			0
$737$	8.16922e12	−	1.41495e13i	1.01995	−	1.76660i
$738$		0			0
$739$	4.88859e12	+	8.46729e12i	0.602954	+	1.04435i	0.992371	+	0.123285i	$0.0393430\pi$
$739$	4.88859e12	+	8.46729e12i	0.602954	+	1.04435i	−0.389417	+	0.921061i	$0.627324\pi$
$740$		0			0
$741$	6.97995e12			0.850492
$742$		0			0
$743$	5.98220e12			0.720131			0.360065	−	0.932927i	$-0.382754\pi$
$743$	5.98220e12			0.720131			0.360065	+	0.932927i	$0.382754\pi$
$744$		0			0
$745$	−2.55847e12	−	4.43140e12i	−0.304283	−	0.527033i
$746$		0			0
$747$	1.86724e11	−	3.23415e11i	0.0219410	−	0.0380030i
$748$		0			0
$749$	4.58045e12	−	2.01283e12i	0.531789	−	0.233689i
$750$		0			0
$751$	3.93896e12	−	6.82247e12i	0.451858	−	0.782640i	−0.546644	−	0.837365i	$-0.684095\pi$
$751$	3.93896e12	−	6.82247e12i	0.451858	−	0.782640i	0.998501	+	0.0547250i	$0.0174282\pi$
$752$		0			0
$753$	1.20779e12	+	2.09196e12i	0.136904	+	0.237124i
$754$		0			0
$755$	−4.80356e12			−0.538025
$756$		0			0
$757$	−9.36850e11			−0.103690			−0.0518452	−	0.998655i	$-0.516510\pi$
$757$	−9.36850e11			−0.103690			−0.0518452	+	0.998655i	$0.516510\pi$
$758$		0			0
$759$	2.73236e12	+	4.73258e12i	0.298847	+	0.517619i
$760$		0			0
$761$	−6.59680e12	+	1.14260e13i	−0.713022	+	1.23499i	0.250696	+	0.968066i	$0.419341\pi$
$761$	−6.59680e12	+	1.14260e13i	−0.713022	+	1.23499i	−0.963718	+	0.266924i	$0.913993\pi$
$762$		0			0
$763$	−1.47604e12	+	1.34183e13i	−0.157666	+	1.43330i
$764$		0			0
$765$	3.88663e11	−	6.73184e11i	0.0410296	−	0.0710653i
$766$		0			0
$767$	3.22433e12	+	5.58471e12i	0.336404	+	0.582669i
$768$		0			0
$769$	1.80879e13			1.86518			0.932588	−	0.360942i	$-0.117545\pi$
$769$	1.80879e13			1.86518			0.932588	+	0.360942i	$0.117545\pi$
$770$		0			0
$771$	−6.98695e12			−0.712104
$772$		0			0
$773$	−4.87668e12	−	8.44665e12i	−0.491265	−	0.850897i	0.508684	−	0.860953i	$-0.330132\pi$
$773$	−4.87668e12	−	8.44665e12i	−0.491265	−	0.850897i	−0.999949	+	0.0100567i	$0.996799\pi$
$774$		0			0
$775$	−2.03957e12	+	3.53264e12i	−0.203086	+	0.351756i
$776$		0			0
$777$	3.98952e12	+	2.92817e12i	0.392669	+	0.288206i
$778$		0			0
$779$	−5.54244e11	+	9.59979e11i	−0.0539241	+	0.0933992i
$780$		0			0
$781$	−5.65902e12	−	9.80171e12i	−0.544266	−	0.942697i
$782$		0			0
$783$	−1.92689e12			−0.183201
$784$		0			0
$785$	−1.01442e13			−0.953462
$786$		0			0
$787$	−4.23171e12	−	7.32953e12i	−0.393214	−	0.681067i	0.599657	−	0.800257i	$-0.295304\pi$
$787$	−4.23171e12	−	7.32953e12i	−0.393214	−	0.681067i	−0.992871	+	0.119190i	$0.961970\pi$
$788$		0			0
$789$	2.37049e12	−	4.10581e12i	0.217767	−	0.377183i
$790$		0			0
$791$	1.45197e13	+	1.06570e13i	1.31876	+	0.967922i
$792$		0			0
$793$	1.20222e13	−	2.08230e13i	1.07958	−	1.86988i
$794$		0			0
$795$	2.58331e12	+	4.47443e12i	0.229364	+	0.397270i
$796$		0			0
$797$	1.22846e11			0.0107845			0.00539225	−	0.999985i	$-0.498284\pi$
$797$	1.22846e11			0.0107845			0.00539225	+	0.999985i	$0.498284\pi$
$798$		0			0
$799$	3.07841e11			0.0267218
$800$		0			0
$801$	3.66306e12	+	6.34461e12i	0.314411	+	0.544576i
$802$		0			0
$803$	−6.59708e12	+	1.14265e13i	−0.559928	+	0.969823i
$804$		0			0
$805$	−1.48522e12	+	1.35018e13i	−0.124655	+	1.13321i
$806$		0			0
$807$	2.72393e12	−	4.71799e12i	0.226082	−	0.391585i
$808$		0			0
$809$	7.08064e12	+	1.22640e13i	0.581171	+	1.00662i	0.995341	+	0.0964180i	$0.0307386\pi$
$809$	7.08064e12	+	1.22640e13i	0.581171	+	1.00662i	−0.414170	+	0.910200i	$0.635928\pi$
$810$		0			0
$811$	−1.61464e13			−1.31064			−0.655319	−	0.755352i	$-0.727466\pi$
$811$	−1.61464e13			−1.31064			−0.655319	+	0.755352i	$0.727466\pi$
$812$		0			0
$813$	1.17095e13			0.940006
$814$		0			0
$815$	1.03044e13	+	1.78477e13i	0.818114	+	1.41701i
$816$		0			0
$817$	5.85302e12	−	1.01377e13i	0.459600	−	0.796051i
$818$		0			0
$819$	5.69785e12	−	2.50386e12i	0.442520	−	0.194461i
$820$		0			0
$821$	−2.17303e12	+	3.76380e12i	−0.166925	+	0.289123i	−0.937337	−	0.348423i	$-0.886717\pi$
$821$	−2.17303e12	+	3.76380e12i	−0.166925	+	0.289123i	0.770412	+	0.637546i	$0.220050\pi$
$822$		0			0
$823$	−1.25123e13	−	2.16719e13i	−0.950688	−	1.64664i	−0.743941	−	0.668245i	$-0.767046\pi$
$823$	−1.25123e13	−	2.16719e13i	−0.950688	−	1.64664i	−0.206747	−	0.978395i	$-0.566288\pi$
$824$		0			0
$825$	6.56302e12			0.493243
$826$		0			0
$827$	2.97569e12			0.221214			0.110607	−	0.993864i	$-0.464720\pi$
$827$	2.97569e12			0.221214			0.110607	+	0.993864i	$0.464720\pi$
$828$		0			0
$829$	−1.83910e12	−	3.18541e12i	−0.135241	−	0.234245i	0.790448	−	0.612529i	$-0.209848\pi$
$829$	−1.83910e12	−	3.18541e12i	−0.135241	−	0.234245i	−0.925690	+	0.378284i	$0.876514\pi$
$830$		0			0
$831$	−3.65396e12	+	6.32885e12i	−0.265803	+	0.460384i
$832$		0			0
$833$	−1.76522e12	+	1.92270e12i	−0.127027	+	0.138360i
$834$		0			0
$835$	7.20544e12	−	1.24802e13i	0.512945	−	0.888448i
$836$		0			0
$837$	−7.73124e11	−	1.33909e12i	−0.0544484	−	0.0943073i
$838$		0			0
$839$	1.05368e13			0.734139			0.367069	−	0.930194i	$-0.380361\pi$
$839$	1.05368e13			0.734139			0.367069	+	0.930194i	$0.380361\pi$
$840$		0			0
$841$	−1.36084e12			−0.0938048
$842$		0			0
$843$	7.53181e12	+	1.30455e13i	0.513659	+	0.889684i
$844$		0			0
$845$	−1.07101e13	+	1.85504e13i	−0.722665	+	1.25169i
$846$		0			0
$847$	−5.71139e12	+	2.50981e12i	−0.381300	+	0.167558i
$848$		0			0
$849$	8.19312e12	−	1.41909e13i	0.541208	−	0.937400i
$850$		0			0
$851$	5.61372e12	+	9.72325e12i	0.366917	+	0.635519i
$852$		0			0
$853$	−1.91258e13			−1.23694			−0.618470	−	0.785809i	$-0.712247\pi$
$853$	−1.91258e13			−1.23694			−0.618470	+	0.785809i	$0.712247\pi$
$854$		0			0
$855$	−6.93508e12			−0.443817
$856$		0			0
$857$	−1.06570e13	−	1.84584e13i	−0.674870	−	1.16891i	−0.976507	−	0.215487i	$-0.930866\pi$
$857$	−1.06570e13	−	1.84584e13i	−0.674870	−	1.16891i	0.301636	−	0.953423i	$-0.402467\pi$
$858$		0			0
$859$	−5.98869e12	+	1.03727e13i	−0.375286	+	0.650015i	−0.990370	−	0.138447i	$-0.955789\pi$
$859$	−5.98869e12	+	1.03727e13i	−0.375286	+	0.650015i	0.615083	+	0.788462i	$0.289122\pi$
$860$		0			0
$861$	−1.08073e11	+	9.82466e11i	−0.00670198	+	0.0609261i
$862$		0			0
$863$	1.39466e13	−	2.41563e13i	0.855896	−	1.48246i	−0.0199152	−	0.999802i	$-0.506340\pi$
$863$	1.39466e13	−	2.41563e13i	0.855896	−	1.48246i	0.875811	−	0.482654i	$-0.160327\pi$
$864$		0			0
$865$	−5.88257e11	−	1.01889e12i	−0.0357268	−	0.0618807i
$866$		0			0
$867$	−9.26674e12			−0.556982
$868$		0			0
$869$	−1.43479e13			−0.853489
$870$		0			0
$871$	−2.11079e13	−	3.65600e13i	−1.24269	−	2.15241i
$872$		0			0
$873$	−3.92517e12	+	6.79860e12i	−0.228715	+	0.396146i
$874$		0			0
$875$	−5.16983e12	−	3.79448e12i	−0.298154	−	0.218835i
$876$		0			0
$877$	1.10049e12	−	1.90611e12i	0.0628188	−	0.108805i	−0.832906	−	0.553415i	$-0.813324\pi$
$877$	1.10049e12	−	1.90611e12i	0.0628188	−	0.108805i	0.895724	+	0.444610i	$0.146658\pi$
$878$		0			0
$879$	−1.51588e12	−	2.62559e12i	−0.0856477	−	0.148346i
$880$		0			0
$881$	−1.56831e13			−0.877085			−0.438542	−	0.898711i	$-0.644505\pi$
$881$	−1.56831e13			−0.877085			−0.438542	+	0.898711i	$0.644505\pi$
$882$		0			0
$883$	−2.46959e12			−0.136710			−0.0683551	−	0.997661i	$-0.521775\pi$
$883$	−2.46959e12			−0.136710			−0.0683551	+	0.997661i	$0.521775\pi$
$884$		0			0
$885$	−3.20361e12	−	5.54881e12i	−0.175548	−	0.304057i
$886$		0			0
$887$	−9.38597e12	+	1.62570e13i	−0.509123	+	0.881827i	0.490821	+	0.871260i	$0.336697\pi$
$887$	−9.38597e12	+	1.62570e13i	−0.509123	+	0.881827i	−0.999944	+	0.0105666i	$0.996636\pi$
$888$		0			0
$889$	−4.13381e12	−	3.03407e12i	−0.221969	−	0.162918i
$890$		0			0
$891$	−1.24390e12	+	2.15449e12i	−0.0661203	+	0.114524i
$892$		0			0
$893$	−1.37323e12	−	2.37851e12i	−0.0722625	−	0.125162i
$894$		0			0
$895$	−2.02719e13			−1.05606
$896$		0			0
$897$	1.41199e13			0.728226
$898$		0			0
$899$	5.27468e12	+	9.13601e12i	0.269326	+	0.466485i
$900$		0			0
$901$	−1.12621e12	+	1.95065e12i	−0.0569320	+	0.0986092i
$902$		0			0
$903$	1.14129e12	−	1.03752e13i	0.0571217	−	0.519280i
$904$		0			0
$905$	2.14284e12	−	3.71152e12i	0.106187	−	0.183922i
$906$		0			0
$907$	−4.33166e12	−	7.50266e12i	−0.212531	−	0.368114i	0.739975	−	0.672634i	$-0.234837\pi$
$907$	−4.33166e12	−	7.50266e12i	−0.212531	−	0.368114i	−0.952506	+	0.304520i	$0.901504\pi$
$908$		0			0
$909$	−2.78023e12			−0.135065
$910$		0			0
$911$	2.29858e13			1.10568			0.552838	−	0.833289i	$-0.313545\pi$
$911$	2.29858e13			1.10568			0.552838	+	0.833289i	$0.313545\pi$
$912$		0			0
$913$	1.64476e12	+	2.84882e12i	0.0783403	+	0.135689i
$914$		0			0
$915$	−1.19449e13	+	2.06892e13i	−0.563362	+	0.975772i
$916$		0			0
$917$	1.66581e12	−	7.32025e11i	0.0777973	−	0.0341872i
$918$		0			0
$919$	−8.46776e12	+	1.46666e13i	−0.391605	+	0.678281i	−0.992661	−	0.120926i	$-0.961414\pi$
$919$	−8.46776e12	+	1.46666e13i	−0.391605	+	0.678281i	0.601056	+	0.799207i	$0.294747\pi$
$920$		0			0
$921$	−7.63163e12	−	1.32184e13i	−0.349502	−	0.605354i
$922$		0			0
$923$	−2.92440e13			−1.32626
$924$		0			0
$925$	1.34839e13			0.605591
$926$		0			0
$927$	3.87929e12	+	6.71912e12i	0.172541	+	0.298850i
$928$		0			0
$929$	1.37714e13	−	2.38527e13i	0.606606	−	1.05067i	−0.385189	−	0.922838i	$-0.625864\pi$
$929$	1.37714e13	−	2.38527e13i	0.606606	−	1.05067i	0.991795	−	0.127835i	$-0.0408028\pi$
$930$		0			0
$931$	2.27300e13	+	5.06193e12i	0.991576	+	0.220822i
$932$		0			0
$933$	−9.38661e12	+	1.62581e13i	−0.405547	+	0.702428i
$934$		0			0
$935$	3.42356e12	+	5.92977e12i	0.146496	+	0.253738i
$936$		0			0
$937$	−1.42275e13			−0.602976			−0.301488	−	0.953470i	$-0.597483\pi$
$937$	−1.42275e13			−0.602976			−0.301488	+	0.953470i	$0.597483\pi$
$938$		0			0
$939$	−1.74522e13			−0.732582
$940$		0			0
$941$	−4.10843e12	−	7.11602e12i	−0.170814	−	0.295858i	0.767891	−	0.640581i	$-0.221306\pi$
$941$	−4.10843e12	−	7.11602e12i	−0.170814	−	0.295858i	−0.938705	+	0.344723i	$0.887973\pi$
$942$		0			0
$943$	−1.12120e12	+	1.94197e12i	−0.0461720	+	0.0799723i
$944$		0			0
$945$	−5.66122e12	+	2.48777e12i	−0.230923	+	0.101477i
$946$		0			0
$947$	7.34464e12	−	1.27213e13i	0.296753	−	0.513992i	−0.678638	−	0.734473i	$-0.737429\pi$
$947$	7.34464e12	−	1.27213e13i	0.296753	−	0.513992i	0.975391	+	0.220481i	$0.0707628\pi$
$948$		0			0
$949$	1.70458e13	+	2.95242e13i	0.682211	+	1.18162i
$950$		0			0
$951$	−7.66100e12			−0.303720
$952$		0			0
$953$	−3.08625e13			−1.21203			−0.606014	−	0.795454i	$-0.707233\pi$
$953$	−3.08625e13			−1.21203			−0.606014	+	0.795454i	$0.707233\pi$
$954$		0			0
$955$	−7.44416e12	−	1.28937e13i	−0.289601	−	0.501604i
$956$		0			0
$957$	8.48655e12	−	1.46991e13i	0.327060	−	0.566484i
$958$		0			0
$959$	1.16854e12	−	1.06229e13i	0.0446127	−	0.405563i
$960$		0			0
$961$	8.98710e12	−	1.55661e13i	0.339910	−	0.588742i
$962$		0			0
$963$	2.58373e12	+	4.47515e12i	0.0968118	+	0.167683i
$964$		0			0
$965$	−6.24651e13			−2.31881
$966$		0			0
$967$	1.91266e13			0.703426			0.351713	−	0.936108i	$-0.385599\pi$
$967$	1.91266e13			0.703426			0.351713	+	0.936108i	$0.385599\pi$
$968$		0			0
$969$	−1.51169e12	−	2.61832e12i	−0.0550815	−	0.0954040i
$970$		0			0
$971$	4.37603e12	−	7.57951e12i	0.157977	−	0.273624i	−0.776162	−	0.630534i	$-0.782836\pi$
$971$	4.37603e12	−	7.57951e12i	0.157977	−	0.273624i	0.934139	+	0.356909i	$0.116169\pi$
$972$		0			0
$973$	−3.09595e11	−	2.27232e11i	−0.0110735	−	0.00812760i
$974$		0			0
$975$	8.47888e12	−	1.46859e13i	0.300482	−	0.520449i
$976$		0			0
$977$	1.32885e13	+	2.30163e13i	0.466605	+	0.808183i	0.999272	−	0.0381416i	$-0.0121438\pi$
$977$	1.32885e13	+	2.30163e13i	0.466605	+	0.808183i	−0.532668	+	0.846324i	$0.678810\pi$
$978$		0			0
$979$	−6.45325e13			−2.24521
$980$		0			0
$981$	−1.39424e13			−0.480649
$982$		0			0
$983$	7.18518e12	+	1.24451e13i	0.245441	+	0.425116i	0.962255	−	0.272148i	$-0.0877340\pi$
$983$	7.18518e12	+	1.24451e13i	0.245441	+	0.425116i	−0.716815	+	0.697264i	$0.754401\pi$
$984$		0			0
$985$	−3.05688e13	+	5.29467e13i	−1.03470	+	1.79216i
$986$		0			0
$987$	−1.97422e12	−	1.44901e12i	−0.0662167	−	0.0486008i
$988$		0			0
$989$	1.18402e13	−	2.05079e13i	0.393529	−	0.681612i
$990$		0			0
$991$	2.05922e13	+	3.56668e13i	0.678222	+	1.17472i	0.975516	+	0.219929i	$0.0705827\pi$
$991$	2.05922e13	+	3.56668e13i	0.678222	+	1.17472i	−0.297293	+	0.954786i	$0.596084\pi$
$992$		0			0
$993$	−1.20196e13			−0.392299
$994$		0			0
$995$	−7.30084e13			−2.36140
$996$		0			0
$997$	2.79283e13	+	4.83733e13i	0.895193	+	1.55052i	0.833565	+	0.552421i	$0.186296\pi$
$997$	2.79283e13	+	4.83733e13i	0.895193	+	1.55052i	0.0616285	+	0.998099i	$0.480371\pi$
$998$		0			0
$999$	−2.55563e12	+	4.42648e12i	−0.0811807	+	0.140609i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.10.q.a.121.1	yes	16
7.4	even	3	inner	168.10.q.a.25.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.10.q.a.25.1	✓	16	7.4	even	3	inner
168.10.q.a.121.1	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	168.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-40.5000 - 70.1481i) q^{3} +(-915.848 + 1586.30i) q^{5} +(5121.10 + 3758.71i) q^{7} +(-3280.50 + 5681.99i) q^{9} +O(q^{10})\)	\(q+(-40.5000 - 70.1481i) q^{3} +(-915.848 + 1586.30i) q^{5} +(5121.10 + 3758.71i) q^{7} +(-3280.50 + 5681.99i) q^{9} +(-28896.4 - 50050.1i) q^{11} -149327. q^{13} +148367. q^{15} +(32340.7 + 56015.8i) q^{17} +(288535. - 499757. i) q^{19} +(56261.9 - 511463. i) q^{21} +(583685. - 1.01097e6i) q^{23} +(-700994. - 1.21416e6i) q^{25} +531441. q^{27} -3.62578e6 q^{29} +(-1.45477e6 - 2.51973e6i) q^{31} +(-2.34061e6 + 4.05406e6i) q^{33} +(-1.06526e7 + 4.68117e6i) q^{35} +(-4.80887e6 + 8.32920e6i) q^{37} +(6.04776e6 + 1.04750e7i) q^{39} -1.92089e6 q^{41} +2.02853e7 q^{43} +(-6.00888e6 - 1.04077e7i) q^{45} +(2.37967e6 - 4.12171e6i) q^{47} +(1.20977e7 + 3.84975e7i) q^{49} +(2.61960e6 - 4.53728e6i) q^{51} +(1.74116e7 + 3.01578e7i) q^{53} +1.05859e8 q^{55} -4.67426e7 q^{57} +(-2.15924e7 - 3.73991e7i) q^{59} +(-8.05090e7 + 1.39446e8i) q^{61} +(-3.81568e7 + 1.67676e7i) q^{63} +(1.36761e8 - 2.36877e8i) q^{65} +(1.41353e8 + 2.44831e8i) q^{67} -9.45569e7 q^{69} +1.95838e8 q^{71} +(-1.14150e8 - 1.97714e8i) q^{73} +(-5.67805e7 + 9.83468e7i) q^{75} +(4.01424e7 - 3.64925e8i) q^{77} +(1.24132e8 - 2.15002e8i) q^{79} +(-2.15234e7 - 3.72796e7i) q^{81} -5.69193e7 q^{83} -1.18477e8 q^{85} +(1.46844e8 + 2.54342e8i) q^{87} +(5.58308e8 - 9.67018e8i) q^{89} +(-7.64720e8 - 5.61279e8i) q^{91} +(-1.17836e8 + 2.04099e8i) q^{93} +(5.28508e8 + 9.15403e8i) q^{95} +1.19652e9 q^{97} +3.79179e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) \) 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9} + 32460 q^{11} + 119048 q^{13} + 31752 q^{15} + 208352 q^{17} + 914588 q^{19} - 428652 q^{21} + 460920 q^{23} - 3040180 q^{25} + 8503056 q^{27} - 16376136 q^{29} - 944064 q^{31} + 2629260 q^{33} - 15546664 q^{35} - 9826516 q^{37} - 4821444 q^{39} + 11449216 q^{41} - 6933624 q^{43} - 1285956 q^{45} + 26549360 q^{47} + 83657504 q^{49} + 16876512 q^{51} - 15354476 q^{53} + 134121944 q^{55} - 148163256 q^{57} + 18404996 q^{59} - 260632792 q^{61} + 35823060 q^{63} + 191461840 q^{65} + 53879788 q^{67} - 74669040 q^{69} - 164207456 q^{71} + 248475540 q^{73} - 246254580 q^{75} + 670121788 q^{77} + 16631256 q^{79} - 344373768 q^{81} - 1138943272 q^{83} - 1690136272 q^{85} + 663233508 q^{87} + 236796360 q^{89} - 1455575212 q^{91} - 76469184 q^{93} + 182450488 q^{95} + 1339799464 q^{97} - 425940120 q^{99}+O(q^{100}) \) 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9 + 32460 * q^11 + 119048 * q^13 + 31752 * q^15 + 208352 * q^17 + 914588 * q^19 - 428652 * q^21 + 460920 * q^23 - 3040180 * q^25 + 8503056 * q^27 - 16376136 * q^29 - 944064 * q^31 + 2629260 * q^33 - 15546664 * q^35 - 9826516 * q^37 - 4821444 * q^39 + 11449216 * q^41 - 6933624 * q^43 - 1285956 * q^45 + 26549360 * q^47 + 83657504 * q^49 + 16876512 * q^51 - 15354476 * q^53 + 134121944 * q^55 - 148163256 * q^57 + 18404996 * q^59 - 260632792 * q^61 + 35823060 * q^63 + 191461840 * q^65 + 53879788 * q^67 - 74669040 * q^69 - 164207456 * q^71 + 248475540 * q^73 - 246254580 * q^75 + 670121788 * q^77 + 16631256 * q^79 - 344373768 * q^81 - 1138943272 * q^83 - 1690136272 * q^85 + 663233508 * q^87 + 236796360 * q^89 - 1455575212 * q^91 - 76469184 * q^93 + 182450488 * q^95 + 1339799464 * q^97 - 425940120 * q^99

Introduction

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