LMFDB - Embedded newform 168.10.q.a.25.4

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			25.4
Root			$-1013.60 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	168.25
Dual form			168.10.q.a.121.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 + 70.1481i) q^{3} +(-519.049 - 899.019i) q^{5} +(-5773.43 + 2649.73i) q^{7} +(-3280.50 - 5681.99i) q^{9} +O(q^{10})$	\(q+(-40.5000 + 70.1481i) q^{3} +(-519.049 - 899.019i) q^{5} +(-5773.43 + 2649.73i) q^{7} +(-3280.50 - 5681.99i) q^{9} +(-24724.5 + 42824.0i) q^{11} +134578. q^{13} +84085.9 q^{15} +(253394. - 438891. i) q^{17} +(475233. + 823127. i) q^{19} +(47950.4 - 512309. i) q^{21} +(-436083. - 755318. i) q^{23} +(437739. - 758186. i) q^{25} +531441. q^{27} -1.12503e6 q^{29} +(-885328. + 1.53343e6i) q^{31} +(-2.00268e6 - 3.46874e6i) q^{33} +(5.37886e6 + 3.81509e6i) q^{35} +(2.51463e6 + 4.35547e6i) q^{37} +(-5.45039e6 + 9.44036e6i) q^{39} +2.26196e7 q^{41} -2.75814e7 q^{43} +(-3.40548e6 + 5.89847e6i) q^{45} +(-8.13204e6 - 1.40851e7i) q^{47} +(2.63114e7 - 3.05961e7i) q^{49} +(2.05249e7 + 3.55501e7i) q^{51} +(-1.96236e7 + 3.39891e7i) q^{53} +5.13328e7 q^{55} -7.69877e7 q^{57} +(-7.10315e7 + 1.23030e8i) q^{59} +(8.11499e7 + 1.40556e8i) q^{61} +(3.39955e7 + 2.41122e7i) q^{63} +(-6.98524e7 - 1.20988e8i) q^{65} +(8.79884e7 - 1.52400e8i) q^{67} +7.06455e7 q^{69} -3.10676e8 q^{71} +(7.28376e7 - 1.26158e8i) q^{73} +(3.54568e7 + 6.14130e7i) q^{75} +(2.92728e7 - 3.12755e8i) q^{77} +(-7.60290e7 - 1.31686e8i) q^{79} +(-2.15234e7 + 3.72796e7i) q^{81} -7.68636e8 q^{83} -5.26095e8 q^{85} +(4.55637e7 - 7.89187e7i) q^{87} +(3.06231e7 + 5.30407e7i) q^{89} +(-7.76975e8 + 3.56595e8i) q^{91} +(-7.17116e7 - 1.24208e8i) q^{93} +(4.93338e8 - 8.54487e8i) q^{95} -1.09136e9 q^{97} +3.24434e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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
$5$	−519.049	−	899.019i	−0.371401	−	0.643286i	0.618380	−	0.785879i	$-0.287789\pi$
$5$	−519.049	−	899.019i	−0.371401	−	0.643286i	−0.989781	+	0.142593i	$0.954456\pi$
$6$		0			0
$7$	−5773.43	+	2649.73i	−0.908851	+	0.417120i
$7$	−5773.43	+	2649.73i	−0.908851	+	0.417120i
$8$		0			0
$9$	−3280.50	−	5681.99i	−0.166667	−	0.288675i
$10$		0			0
$11$	−24724.5	+	42824.0i	−0.509166	+	0.881902i	0.490777	+	0.871285i	$0.336713\pi$
$11$	−24724.5	+	42824.0i	−0.509166	+	0.881902i	−0.999944	+	0.0106168i	$0.996620\pi$
$12$		0			0
$13$	134578.			1.30686			0.653428	−	0.756989i	$-0.273330\pi$
$13$	134578.			1.30686			0.653428	+	0.756989i	$0.273330\pi$
$14$		0			0
$15$	84085.9			0.428857
$16$		0			0
$17$	253394.	−	438891.i	0.735827	−	1.27449i	−0.218533	−	0.975830i	$-0.570127\pi$
$17$	253394.	−	438891.i	0.735827	−	1.27449i	0.954360	−	0.298659i	$-0.0965395\pi$
$18$		0			0
$19$	475233.	+	823127.i	0.836595	+	1.44902i	0.892725	+	0.450601i	$0.148790\pi$
$19$	475233.	+	823127.i	0.836595	+	1.44902i	−0.0561303	+	0.998423i	$0.517876\pi$
$20$		0			0
$21$	47950.4	−	512309.i	0.0538029	−	0.574838i
$22$		0			0
$23$	−436083.	−	755318.i	−0.324933	−	0.562801i	0.656566	−	0.754269i	$-0.272008\pi$
$23$	−436083.	−	755318.i	−0.324933	−	0.562801i	−0.981499	+	0.191468i	$0.938675\pi$
$24$		0			0
$25$	437739.	−	758186.i	0.224122	−	0.388191i
$26$		0			0
$27$	531441.			0.192450
$28$		0			0
$29$	−1.12503e6			−0.295375			−0.147687	−	0.989034i	$-0.547183\pi$
$29$	−1.12503e6			−0.295375			−0.147687	+	0.989034i	$0.547183\pi$
$30$		0			0
$31$	−885328.	+	1.53343e6i	−0.172178	+	0.298220i	−0.939181	−	0.343423i	$-0.888414\pi$
$31$	−885328.	+	1.53343e6i	−0.172178	+	0.298220i	0.767003	+	0.641643i	$0.221747\pi$
$32$		0			0
$33$	−2.00268e6	−	3.46874e6i	−0.293967	−	0.509166i
$34$		0			0
$35$	5.37886e6	+	3.81509e6i	0.605876	+	0.429732i
$36$		0			0
$37$	2.51463e6	+	4.35547e6i	0.220580	+	0.382057i	0.954984	−	0.296656i	$-0.0958715\pi$
$37$	2.51463e6	+	4.35547e6i	0.220580	+	0.382057i	−0.734404	+	0.678713i	$0.762538\pi$
$38$		0			0
$39$	−5.45039e6	+	9.44036e6i	−0.377257	+	0.653428i
$40$		0			0
$41$	2.26196e7			1.25014			0.625068	−	0.780570i	$-0.285071\pi$
$41$	2.26196e7			1.25014			0.625068	+	0.780570i	$0.285071\pi$
$42$		0			0
$43$	−2.75814e7			−1.23029			−0.615147	−	0.788412i	$-0.710903\pi$
$43$	−2.75814e7			−1.23029			−0.615147	+	0.788412i	$0.710903\pi$
$44$		0			0
$45$	−3.40548e6	+	5.89847e6i	−0.123800	+	0.214429i
$46$		0			0
$47$	−8.13204e6	−	1.40851e7i	−0.243086	−	0.421037i	0.718506	−	0.695521i	$-0.244826\pi$
$47$	−8.13204e6	−	1.40851e7i	−0.243086	−	0.421037i	−0.961592	+	0.274484i	$0.911493\pi$
$48$		0			0
$49$	2.63114e7	−	3.05961e7i	0.652022	−	0.758200i
$50$		0			0
$51$	2.05249e7	+	3.55501e7i	0.424830	+	0.735827i
$52$		0			0
$53$	−1.96236e7	+	3.39891e7i	−0.341615	+	0.591695i	−0.984733	−	0.174073i	$-0.944307\pi$
$53$	−1.96236e7	+	3.39891e7i	−0.341615	+	0.591695i	0.643118	+	0.765767i	$0.277641\pi$
$54$		0			0
$55$	5.13328e7			0.756420
$56$		0			0
$57$	−7.69877e7			−0.966017
$58$		0			0
$59$	−7.10315e7	+	1.23030e8i	−0.763162	+	1.32184i	0.178051	+	0.984021i	$0.443021\pi$
$59$	−7.10315e7	+	1.23030e8i	−0.763162	+	1.32184i	−0.941213	+	0.337814i	$0.890312\pi$
$60$		0			0
$61$	8.11499e7	+	1.40556e8i	0.750419	+	1.29976i	0.947620	+	0.319401i	$0.103482\pi$
$61$	8.11499e7	+	1.40556e8i	0.750419	+	1.29976i	−0.197200	+	0.980363i	$0.563185\pi$
$62$		0			0
$63$	3.39955e7	+	2.41122e7i	0.271887	+	0.192843i
$64$		0			0
$65$	−6.98524e7	−	1.20988e8i	−0.485368	−	0.840682i
$66$		0			0
$67$	8.79884e7	−	1.52400e8i	0.533444	−	0.923952i	−0.465793	−	0.884894i	$-0.654231\pi$
$67$	8.79884e7	−	1.52400e8i	0.533444	−	0.923952i	0.999237	−	0.0390582i	$-0.0124358\pi$
$68$		0			0
$69$	7.06455e7			0.375200
$70$		0			0
$71$	−3.10676e8			−1.45092			−0.725462	−	0.688262i	$-0.758374\pi$
$71$	−3.10676e8			−1.45092			−0.725462	+	0.688262i	$0.758374\pi$
$72$		0			0
$73$	7.28376e7	−	1.26158e8i	0.300195	−	0.519952i	−0.675985	−	0.736915i	$-0.736282\pi$
$73$	7.28376e7	−	1.26158e8i	0.300195	−	0.519952i	0.976180	+	0.216963i	$0.0696150\pi$
$74$		0			0
$75$	3.54568e7	+	6.14130e7i	0.129397	+	0.224122i
$76$		0			0
$77$	2.92728e7	−	3.12755e8i	0.0948978	−	1.01390i
$78$		0			0
$79$	−7.60290e7	−	1.31686e8i	−0.219613	−	0.380381i	0.735077	−	0.677984i	$-0.237146\pi$
$79$	−7.60290e7	−	1.31686e8i	−0.219613	−	0.380381i	−0.954690	+	0.297603i	$0.903813\pi$
$80$		0			0
$81$	−2.15234e7	+	3.72796e7i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−7.68636e8			−1.77774			−0.888872	−	0.458155i	$-0.848510\pi$
$83$	−7.68636e8			−1.77774			−0.888872	+	0.458155i	$0.848510\pi$
$84$		0			0
$85$	−5.26095e8			−1.09315
$86$		0			0
$87$	4.55637e7	−	7.89187e7i	0.0852673	−	0.147687i
$88$		0			0
$89$	3.06231e7	+	5.30407e7i	0.0517361	+	0.0896095i	0.890734	−	0.454526i	$-0.150191\pi$
$89$	3.06231e7	+	5.30407e7i	0.0517361	+	0.0896095i	−0.838998	+	0.544135i	$0.816858\pi$
$90$		0			0
$91$	−7.76975e8	+	3.56595e8i	−1.18774	+	0.545115i
$92$		0			0
$93$	−7.17116e7	−	1.24208e8i	−0.0994067	−	0.172178i
$94$		0			0
$95$	4.93338e8	−	8.54487e8i	0.621425	−	1.07634i
$96$		0			0
$97$	−1.09136e9			−1.25168			−0.625841	−	0.779950i	$-0.715244\pi$
$97$	−1.09136e9			−1.25168			−0.625841	+	0.779950i	$0.715244\pi$
$98$		0			0
$99$	3.24434e8			0.339444
$100$		0			0
$101$	2.08842e8	−	3.61725e8i	0.199697	−	0.345885i	−0.748733	−	0.662871i	$-0.769338\pi$
$101$	2.08842e8	−	3.61725e8i	0.199697	−	0.345885i	0.948430	+	0.316986i	$0.102671\pi$
$102$		0			0
$103$	7.58943e8	+	1.31453e9i	0.664418	+	1.15081i	0.979443	+	0.201723i	$0.0646540\pi$
$103$	7.58943e8	+	1.31453e9i	0.664418	+	1.15081i	−0.315024	+	0.949084i	$0.602013\pi$
$104$		0			0
$105$	−4.85465e8	+	2.22805e8i	−0.389768	+	0.178885i
$106$		0			0
$107$	2.40188e8	+	4.16017e8i	0.177143	+	0.306820i	0.940901	−	0.338682i	$-0.109981\pi$
$107$	2.40188e8	+	4.16017e8i	0.177143	+	0.306820i	−0.763758	+	0.645503i	$0.776648\pi$
$108$		0			0
$109$	−6.61807e8	+	1.14628e9i	−0.449068	+	0.777809i	−0.998326	−	0.0578444i	$-0.981577\pi$
$109$	−6.61807e8	+	1.14628e9i	−0.449068	+	0.777809i	0.549257	+	0.835653i	$0.314911\pi$
$110$		0			0
$111$	−4.07371e8			−0.254704
$112$		0			0
$113$	−1.06336e9			−0.613518			−0.306759	−	0.951787i	$-0.599244\pi$
$113$	−1.06336e9			−0.613518			−0.306759	+	0.951787i	$0.599244\pi$
$114$		0			0
$115$	−4.52697e8	+	7.84094e8i	−0.241361	+	0.418050i
$116$		0			0
$117$	−4.41482e8	−	7.64669e8i	−0.217809	−	0.377257i
$118$		0			0
$119$	−3.00008e8	+	3.20533e9i	−0.137142	+	1.46525i
$120$		0			0
$121$	−4.36234e7	−	7.55580e7i	−0.0185006	−	0.0320440i
$122$		0			0
$123$	−9.16094e8	+	1.58672e9i	−0.360883	+	0.625068i
$124$		0			0
$125$	−2.93637e9			−1.07576
$126$		0			0
$127$	−3.20516e9			−1.09328			−0.546642	−	0.837366i	$-0.684094\pi$
$127$	−3.20516e9			−1.09328			−0.546642	+	0.837366i	$0.684094\pi$
$128$		0			0
$129$	1.11705e9	−	1.93478e9i	0.355155	−	0.615147i
$130$		0			0
$131$	−2.59841e9	−	4.50058e9i	−0.770880	−	1.33520i	−0.937081	−	0.349112i	$-0.886483\pi$
$131$	−2.59841e9	−	4.50058e9i	−0.770880	−	1.33520i	0.166201	−	0.986092i	$-0.446850\pi$
$132$		0			0
$133$	−4.92479e9	−	3.49303e9i	−1.36476	−	0.967988i
$134$		0			0
$135$	−2.75844e8	−	4.77776e8i	−0.0714762	−	0.123800i
$136$		0			0
$137$	−6.84710e8	+	1.18595e9i	−0.166060	+	0.287624i	−0.937031	−	0.349246i	$-0.886438\pi$
$137$	−6.84710e8	+	1.18595e9i	−0.166060	+	0.287624i	0.770971	+	0.636870i	$0.219771\pi$
$138$		0			0
$139$	1.61762e9			0.367546			0.183773	−	0.982969i	$-0.441169\pi$
$139$	1.61762e9			0.367546			0.183773	+	0.982969i	$0.441169\pi$
$140$		0			0
$141$	1.31739e9			0.280691
$142$		0			0
$143$	−3.32736e9	+	5.76315e9i	−0.665407	+	1.15252i
$144$		0			0
$145$	5.83946e8	+	1.01142e9i	0.109703	+	0.190010i
$146$		0			0
$147$	1.08064e9	+	3.08484e9i	0.190877	+	0.544885i
$148$		0			0
$149$	1.95200e9	+	3.38097e9i	0.324446	+	0.561957i	0.981400	−	0.191974i	$-0.0614889\pi$
$149$	1.95200e9	+	3.38097e9i	0.324446	+	0.561957i	−0.656954	+	0.753930i	$0.728156\pi$
$150$		0			0
$151$	1.92279e9	−	3.33038e9i	0.300979	−	0.521311i	−0.675379	−	0.737471i	$-0.736020\pi$
$151$	1.92279e9	−	3.33038e9i	0.300979	−	0.521311i	0.976358	+	0.216160i	$0.0693532\pi$
$152$		0			0
$153$	−3.32503e9			−0.490551
$154$		0			0
$155$	1.83811e9			0.255788
$156$		0			0
$157$	−5.42366e9	+	9.39405e9i	−0.712433	+	1.23397i	0.251509	+	0.967855i	$0.419073\pi$
$157$	−5.42366e9	+	9.39405e9i	−0.712433	+	1.23397i	−0.963941	+	0.266115i	$0.914260\pi$
$158$		0			0
$159$	−1.58951e9	−	2.75311e9i	−0.197232	−	0.341615i
$160$		0			0
$161$	4.51909e9	+	3.20527e9i	0.530071	+	0.375966i
$162$		0			0
$163$	−6.03248e9	−	1.04486e10i	−0.669348	−	1.15934i	−0.978087	−	0.208198i	$-0.933240\pi$
$163$	−6.03248e9	−	1.04486e10i	−0.669348	−	1.15934i	0.308739	−	0.951147i	$-0.400093\pi$
$164$		0			0
$165$	−2.07898e9	+	3.60090e9i	−0.218360	+	0.378210i
$166$		0			0
$167$	−8.63043e9			−0.858634			−0.429317	−	0.903154i	$-0.641246\pi$
$167$	−8.63043e9			−0.858634			−0.429317	+	0.903154i	$0.641246\pi$
$168$		0			0
$169$	7.50663e9			0.707872
$170$		0			0
$171$	3.11800e9	−	5.40054e9i	0.278865	−	0.483008i
$172$		0			0
$173$	−8.50302e9	−	1.47277e10i	−0.721715	−	1.25005i	−0.960312	−	0.278928i	$-0.910021\pi$
$173$	−8.50302e9	−	1.47277e10i	−0.721715	−	1.25005i	0.238597	−	0.971119i	$-0.423313\pi$
$174$		0			0
$175$	−5.18266e8	+	5.53722e9i	−0.0417716	+	0.446294i
$176$		0			0
$177$	−5.75355e9	−	9.96544e9i	−0.440612	−	0.763162i
$178$		0			0
$179$	−1.60389e9	+	2.77801e9i	−0.116771	+	0.202253i	−0.918486	−	0.395453i	$-0.870588\pi$
$179$	−1.60389e9	+	2.77801e9i	−0.116771	+	0.202253i	0.801715	+	0.597706i	$0.203921\pi$
$180$		0			0
$181$	−2.55012e10			−1.76606			−0.883032	−	0.469313i	$-0.844502\pi$
$181$	−2.55012e10			−1.76606			−0.883032	+	0.469313i	$0.844502\pi$
$182$		0			0
$183$	−1.31463e10			−0.866510
$184$		0			0
$185$	2.61044e9	−	4.52141e9i	0.163848	−	0.283793i
$186$		0			0
$187$	1.25300e10	+	2.17027e10i	0.749316	+	1.29785i
$188$		0			0
$189$	−3.06824e9	+	1.40818e9i	−0.174909	+	0.0802748i
$190$		0			0
$191$	7.85171e9	+	1.35996e10i	0.426888	+	0.739392i	0.996595	−	0.0824561i	$-0.0262764\pi$
$191$	7.85171e9	+	1.35996e10i	0.426888	+	0.739392i	−0.569706	+	0.821848i	$0.692943\pi$
$192$		0			0
$193$	−4.23964e8	+	7.34327e8i	−0.0219948	+	0.0380962i	−0.876813	−	0.480831i	$-0.840335\pi$
$193$	−4.23964e8	+	7.34327e8i	−0.0219948	+	0.0380962i	0.854818	+	0.518927i	$0.173668\pi$
$194$		0			0
$195$	1.13161e10			0.560455
$196$		0			0
$197$	1.62315e9			0.0767820			0.0383910	−	0.999263i	$-0.487777\pi$
$197$	1.62315e9			0.0767820			0.0383910	+	0.999263i	$0.487777\pi$
$198$		0			0
$199$	5.40108e9	−	9.35495e9i	0.244142	−	0.422866i	−0.717748	−	0.696303i	$-0.754827\pi$
$199$	5.40108e9	−	9.35495e9i	0.244142	−	0.422866i	0.961890	+	0.273437i	$0.0881605\pi$
$200$		0			0
$201$	7.12706e9	+	1.23444e10i	0.307984	+	0.533444i
$202$		0			0
$203$	6.49529e9	−	2.98103e9i	0.268452	−	0.123207i
$204$		0			0
$205$	−1.17407e10	−	2.03355e10i	−0.464302	−	0.804195i
$206$		0			0
$207$	−2.86114e9	+	4.95564e9i	−0.108311	+	0.187600i
$208$		0			0
$209$	−4.69995e10			−1.70386
$210$		0			0
$211$	1.60897e10			0.558825			0.279413	−	0.960171i	$-0.409860\pi$
$211$	1.60897e10			0.558825			0.279413	+	0.960171i	$0.409860\pi$
$212$		0			0
$213$	1.25824e10	−	2.17933e10i	0.418846	−	0.725462i
$214$		0			0
$215$	1.43161e10	+	2.47962e10i	0.456933	+	0.791431i
$216$		0			0
$217$	1.04819e9	−	1.11991e10i	0.0320902	−	0.342857i
$218$		0			0
$219$	5.89985e9	+	1.02188e10i	0.173317	+	0.300195i
$220$		0			0
$221$	3.41011e10	−	5.90648e10i	0.961619	−	1.66557i
$222$		0			0
$223$	−1.48095e10			−0.401023			−0.200511	−	0.979691i	$-0.564260\pi$
$223$	−1.48095e10			−0.401023			−0.200511	+	0.979691i	$0.564260\pi$
$224$		0			0
$225$	−5.74401e9			−0.149415
$226$		0			0
$227$	2.06791e10	−	3.58173e10i	0.516911	−	0.895315i	−0.482897	−	0.875677i	$-0.660415\pi$
$227$	2.06791e10	−	3.58173e10i	0.516911	−	0.895315i	0.999807	−	0.0196379i	$-0.00625135\pi$
$228$		0			0
$229$	2.83112e10	+	4.90365e10i	0.680297	+	1.17831i	0.974890	+	0.222687i	$0.0714827\pi$
$229$	2.83112e10	+	4.90365e10i	0.680297	+	1.17831i	−0.294593	+	0.955623i	$0.595184\pi$
$230$		0			0
$231$	2.07536e10	+	1.47200e10i	0.479556	+	0.340137i
$232$		0			0
$233$	3.42454e9	+	5.93147e9i	0.0761202	+	0.131844i	0.901573	−	0.432627i	$-0.142413\pi$
$233$	3.42454e9	+	5.93147e9i	0.0761202	+	0.131844i	−0.825453	+	0.564471i	$0.809080\pi$
$234$		0			0
$235$	−8.44186e9	+	1.46217e10i	−0.180565	+	0.312747i
$236$		0			0
$237$	1.23167e10			0.253587
$238$		0			0
$239$	8.65885e10			1.71660			0.858301	−	0.513147i	$-0.171520\pi$
$239$	8.65885e10			1.71660			0.858301	+	0.513147i	$0.171520\pi$
$240$		0			0
$241$	4.77458e9	−	8.26981e9i	0.0911713	−	0.157913i	−0.816833	−	0.576874i	$-0.804272\pi$
$241$	4.77458e9	−	8.26981e9i	0.0911713	−	0.157913i	0.908004	+	0.418961i	$0.137606\pi$
$242$		0			0
$243$	−1.74339e9	−	3.01964e9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	−4.11634e10	−	7.77362e9i	−0.729901	−	0.137840i
$246$		0			0
$247$	6.39557e10	+	1.10775e11i	1.09331	+	1.89367i
$248$		0			0
$249$	3.11298e10	−	5.39183e10i	0.513191	−	0.888872i
$250$		0			0
$251$	9.30391e10			1.47956			0.739782	−	0.672847i	$-0.234929\pi$
$251$	9.30391e10			1.47956			0.739782	+	0.672847i	$0.234929\pi$
$252$		0			0
$253$	4.31277e10			0.661780
$254$		0			0
$255$	2.13068e10	−	3.69045e10i	0.315565	−	0.546574i
$256$		0			0
$257$	−2.50007e10	−	4.33024e10i	−0.357481	−	0.619174i	0.630059	−	0.776548i	$-0.283031\pi$
$257$	−2.50007e10	−	4.33024e10i	−0.357481	−	0.619174i	−0.987539	+	0.157373i	$0.949697\pi$
$258$		0			0
$259$	−2.60589e10	−	1.84829e10i	−0.359838	−	0.255224i
$260$		0			0
$261$	3.69066e9	+	6.39241e9i	0.0492291	+	0.0852673i
$262$		0			0
$263$	−2.15709e9	+	3.73619e9i	−0.0278014	+	0.0481535i	−0.879591	−	0.475730i	$-0.842184\pi$
$263$	−2.15709e9	+	3.73619e9i	−0.0278014	+	0.0481535i	0.851790	+	0.523883i	$0.175517\pi$
$264$		0			0
$265$	4.07424e10			0.507505
$266$		0			0
$267$	−4.96094e9			−0.0597397
$268$		0			0
$269$	5.45359e10	−	9.44590e10i	0.635035	−	1.09991i	−0.351473	−	0.936198i	$-0.614319\pi$
$269$	5.45359e10	−	9.44590e10i	0.635035	−	1.09991i	0.986508	−	0.163715i	$-0.0523476\pi$
$270$		0			0
$271$	4.70061e10	+	8.14170e10i	0.529411	+	0.916966i	0.999412	+	0.0343004i	$0.0109203\pi$
$271$	4.70061e10	+	8.14170e10i	0.529411	+	0.916966i	−0.470001	+	0.882666i	$0.655746\pi$
$272$		0			0
$273$	6.45305e9	−	6.89453e10i	0.0703126	−	0.751230i
$274$		0			0
$275$	2.16457e10	+	3.74914e10i	0.228231	+	0.395308i
$276$		0			0
$277$	−8.32512e10	+	1.44195e11i	−0.849633	+	1.47161i	0.0319023	+	0.999491i	$0.489843\pi$
$277$	−8.32512e10	+	1.44195e11i	−0.849633	+	1.47161i	−0.881536	+	0.472117i	$0.843490\pi$
$278$		0			0
$279$	1.16173e10			0.114785
$280$		0			0
$281$	−1.12828e11			−1.07954			−0.539769	−	0.841813i	$-0.681488\pi$
$281$	−1.12828e11			−1.07954			−0.539769	+	0.841813i	$0.681488\pi$
$282$		0			0
$283$	−7.48259e10	+	1.29602e11i	−0.693447	+	1.20108i	0.277255	+	0.960796i	$0.410575\pi$
$283$	−7.48259e10	+	1.29602e11i	−0.693447	+	1.20108i	−0.970702	+	0.240288i	$0.922758\pi$
$284$		0			0
$285$	3.99604e10	+	6.92134e10i	0.358780	+	0.621425i
$286$		0			0
$287$	−1.30593e11	+	5.99359e10i	−1.13619	+	0.521457i
$288$		0			0
$289$	−6.91227e10	−	1.19724e11i	−0.582881	−	1.00958i
$290$		0			0
$291$	4.42000e10	−	7.65566e10i	0.361330	−	0.625841i
$292$		0			0
$293$	1.78947e11			1.41847			0.709234	−	0.704973i	$-0.249041\pi$
$293$	1.78947e11			1.41847			0.709234	+	0.704973i	$0.249041\pi$
$294$		0			0
$295$	1.47475e11			1.13376
$296$		0			0
$297$	−1.31396e10	+	2.27584e10i	−0.0979891	+	0.169722i
$298$		0			0
$299$	−5.86870e10	−	1.01649e11i	−0.424641	−	0.735499i
$300$		0			0
$301$	1.59240e11	−	7.30834e10i	1.11815	−	0.513180i
$302$		0			0
$303$	1.69162e10	+	2.92997e10i	0.115295	+	0.199697i
$304$		0			0
$305$	8.42416e10	−	1.45911e11i	0.557413	−	0.965468i
$306$		0			0
$307$	−5.88269e10			−0.377966			−0.188983	−	0.981980i	$-0.560519\pi$
$307$	−5.88269e10			−0.377966			−0.188983	+	0.981980i	$0.560519\pi$
$308$		0			0
$309$	−1.22949e11			−0.767204
$310$		0			0
$311$	−2.48734e10	+	4.30820e10i	−0.150769	+	0.261140i	−0.931511	−	0.363714i	$-0.881508\pi$
$311$	−2.48734e10	+	4.30820e10i	−0.150769	+	0.261140i	0.780741	+	0.624855i	$0.214842\pi$
$312$		0			0
$313$	−1.63749e11	−	2.83622e11i	−0.964340	−	1.67029i	−0.711378	−	0.702810i	$-0.751928\pi$
$313$	−1.63749e11	−	2.83622e11i	−0.964340	−	1.67029i	−0.252962	−	0.967476i	$-0.581405\pi$
$314$		0			0
$315$	4.03196e9	−	4.30780e10i	0.0230738	−	0.246523i
$316$		0			0
$317$	6.64540e10	+	1.15102e11i	0.369619	+	0.640199i	0.989506	−	0.144492i	$-0.0461548\pi$
$317$	6.64540e10	+	1.15102e11i	0.369619	+	0.640199i	−0.619887	+	0.784691i	$0.712821\pi$
$318$		0			0
$319$	2.78158e10	−	4.81783e10i	0.150395	−	0.260491i
$320$		0			0
$321$	−3.89104e10			−0.204547
$322$		0			0
$323$	4.81684e11			2.46236
$324$		0			0
$325$	5.89098e10	−	1.02035e11i	0.292895	−	0.507310i
$326$		0			0
$327$	−5.36064e10	−	9.28490e10i	−0.259270	−	0.449068i
$328$		0			0
$329$	8.42716e10	+	5.97717e10i	0.396551	+	0.281264i
$330$		0			0
$331$	1.84955e11	+	3.20352e11i	0.846916	+	1.46690i	0.883947	+	0.467587i	$0.154877\pi$
$331$	1.84955e11	+	3.20352e11i	0.846916	+	1.46690i	−0.0370314	+	0.999314i	$0.511790\pi$
$332$		0			0
$333$	1.64985e10	−	2.85763e10i	0.0735268	−	0.127352i
$334$		0			0
$335$	−1.82681e11			−0.792487
$336$		0			0
$337$	−3.07721e11			−1.29964			−0.649820	−	0.760088i	$-0.725156\pi$
$337$	−3.07721e11			−1.29964			−0.649820	+	0.760088i	$0.725156\pi$
$338$		0			0
$339$	4.30660e10	−	7.45926e10i	0.177107	−	0.306759i
$340$		0			0
$341$	−4.37785e10	−	7.58266e10i	−0.175334	−	0.303687i
$342$		0			0
$343$	−7.08358e10	+	2.46363e11i	−0.276331	+	0.961063i
$344$		0			0
$345$	−3.66685e10	−	6.35116e10i	−0.139350	−	0.241361i
$346$		0			0
$347$	−1.41570e11	+	2.45206e11i	−0.524188	+	0.907921i	0.475415	+	0.879762i	$0.342298\pi$
$347$	−1.41570e11	+	2.45206e11i	−0.524188	+	0.907921i	−0.999603	+	0.0281590i	$0.991036\pi$
$348$		0			0
$349$	−3.90446e11			−1.40879			−0.704395	−	0.709808i	$-0.748782\pi$
$349$	−3.90446e11			−1.40879			−0.704395	+	0.709808i	$0.748782\pi$
$350$		0			0
$351$	7.15200e10			0.251505
$352$		0			0
$353$	1.07431e11	−	1.86075e11i	0.368249	−	0.637826i	−0.621043	−	0.783777i	$-0.713291\pi$
$353$	1.07431e11	−	1.86075e11i	0.368249	−	0.637826i	0.989292	+	0.145951i	$0.0466241\pi$
$354$		0			0
$355$	1.61256e11	+	2.79303e11i	0.538875	+	0.933359i
$356$		0			0
$357$	−2.12697e11	−	1.50861e11i	−0.693035	−	0.491552i
$358$		0			0
$359$	−2.69412e11	−	4.66635e11i	−0.856036	−	1.48270i	−0.875681	−	0.482890i	$-0.839587\pi$
$359$	−2.69412e11	−	4.66635e11i	−0.856036	−	1.48270i	0.0196454	−	0.999807i	$-0.493746\pi$
$360$		0			0
$361$	−2.90349e11	+	5.02899e11i	−0.899782	+	1.55847i
$362$		0			0
$363$	7.06700e9			0.0213626
$364$		0			0
$365$	−1.51225e11			−0.445971
$366$		0			0
$367$	7.21377e10	−	1.24946e11i	0.207570	−	0.359522i	−0.743378	−	0.668871i	$-0.766778\pi$
$367$	7.21377e10	−	1.24946e11i	0.207570	−	0.359522i	0.950949	+	0.309349i	$0.100111\pi$
$368$		0			0
$369$	−7.42036e10	−	1.28524e11i	−0.208356	−	0.360883i
$370$		0			0
$371$	2.32336e10	−	2.48231e11i	0.0636698	−	0.680257i
$372$		0			0
$373$	2.91705e11	+	5.05248e11i	0.780286	+	1.35149i	0.931775	+	0.363036i	$0.118260\pi$
$373$	2.91705e11	+	5.05248e11i	0.780286	+	1.35149i	−0.151489	+	0.988459i	$0.548407\pi$
$374$		0			0
$375$	1.18923e11	−	2.05980e11i	0.310545	−	0.537880i
$376$		0			0
$377$	−1.51404e11			−0.386012
$378$		0			0
$379$	−7.00848e11			−1.74481			−0.872403	−	0.488787i	$-0.837440\pi$
$379$	−7.00848e11			−1.74481			−0.872403	+	0.488787i	$0.837440\pi$
$380$		0			0
$381$	1.29809e11	−	2.24836e11i	0.315604	−	0.546642i
$382$		0			0
$383$	−5.87597e10	−	1.01775e11i	−0.139536	−	0.241683i	0.787785	−	0.615950i	$-0.211228\pi$
$383$	−5.87597e10	−	1.01775e11i	−0.139536	−	0.241683i	−0.927321	+	0.374267i	$0.877894\pi$
$384$		0			0
$385$	−2.96367e11	+	1.36018e11i	−0.687473	+	0.315518i
$386$		0			0
$387$	9.04809e10	+	1.56718e11i	0.205049	+	0.355155i
$388$		0			0
$389$	−3.68579e11	+	6.38398e11i	−0.816127	+	1.41357i	0.0923897	+	0.995723i	$0.470549\pi$
$389$	−3.68579e11	+	6.38398e11i	−0.816127	+	1.41357i	−0.908516	+	0.417850i	$0.862784\pi$
$390$		0			0
$391$	−4.42003e11			−0.956378
$392$		0			0
$393$	4.20942e11			0.890136
$394$		0			0
$395$	−7.89256e10	+	1.36703e11i	−0.163129	+	0.282548i
$396$		0			0
$397$	3.52771e11	+	6.11018e11i	0.712748	+	1.23452i	0.963822	+	0.266547i	$0.0858828\pi$
$397$	3.52771e11	+	6.11018e11i	0.712748	+	1.23452i	−0.251074	+	0.967968i	$0.580784\pi$
$398$		0			0
$399$	4.44483e11	−	2.03997e11i	0.877966	−	0.402945i
$400$		0			0
$401$	1.10857e11	+	1.92010e11i	0.214099	+	0.370830i	0.952993	−	0.302991i	$-0.0979851\pi$
$401$	1.10857e11	+	1.92010e11i	0.214099	+	0.370830i	−0.738894	+	0.673821i	$0.764652\pi$
$402$		0			0
$403$	−1.19145e11	+	2.06366e11i	−0.225011	+	0.389731i
$404$		0			0
$405$	4.46867e10			0.0825336
$406$		0			0
$407$	−2.48692e11			−0.449248
$408$		0			0
$409$	−4.75289e11	+	8.23224e11i	−0.839852	+	1.45467i	0.0501664	+	0.998741i	$0.484025\pi$
$409$	−4.75289e11	+	8.23224e11i	−0.839852	+	1.45467i	−0.890018	+	0.455925i	$0.849309\pi$
$410$		0			0
$411$	−5.54615e10	−	9.60621e10i	−0.0958746	−	0.166060i
$412$		0			0
$413$	8.40985e10	−	8.98521e11i	0.142237	−	1.51968i
$414$		0			0
$415$	3.98960e11	+	6.91019e11i	0.660257	+	1.14360i
$416$		0			0
$417$	−6.55138e10	+	1.13473e11i	−0.106101	+	0.183773i
$418$		0			0
$419$	2.43950e11			0.386667			0.193334	−	0.981133i	$-0.438070\pi$
$419$	2.43950e11			0.386667			0.193334	+	0.981133i	$0.438070\pi$
$420$		0			0
$421$	−6.27899e11			−0.974138			−0.487069	−	0.873364i	$-0.661934\pi$
$421$	−6.27899e11			−0.974138			−0.487069	+	0.873364i	$0.661934\pi$
$422$		0			0
$423$	−5.33543e10	+	9.24124e10i	−0.0810285	+	0.140346i
$424$		0			0
$425$	−2.21840e11	−	3.84239e11i	−0.329830	−	0.571282i
$426$		0			0
$427$	−8.40949e11	−	5.96464e11i	−1.22418	−	0.868278i
$428$		0			0
$429$	−2.69516e11	−	4.66815e11i	−0.384173	−	0.665407i
$430$		0			0
$431$	−7.99682e10	+	1.38509e11i	−0.111627	+	0.193344i	−0.916426	−	0.400203i	$-0.868940\pi$
$431$	−7.99682e10	+	1.38509e11i	−0.111627	+	0.193344i	0.804799	+	0.593547i	$0.202273\pi$
$432$		0			0
$433$	6.60121e11			0.902460			0.451230	−	0.892408i	$-0.350985\pi$
$433$	6.60121e11			0.902460			0.451230	+	0.892408i	$0.350985\pi$
$434$		0			0
$435$	−9.45992e10			−0.126674
$436$		0			0
$437$	4.14482e11	−	7.17904e11i	0.543675	−	0.941672i
$438$		0			0
$439$	−2.82347e11	−	4.89039e11i	−0.362821	−	0.628424i	0.625603	−	0.780142i	$-0.284853\pi$
$439$	−2.82347e11	−	4.89039e11i	−0.362821	−	0.628424i	−0.988424	+	0.151717i	$0.951520\pi$
$440$		0			0
$441$	−2.60162e11	−	4.91309e10i	−0.327544	−	0.0618559i
$442$		0			0
$443$	−4.36081e11	−	7.55314e11i	−0.537960	−	0.931774i	−0.999014	−	0.0444020i	$-0.985862\pi$
$443$	−4.36081e11	−	7.55314e11i	−0.537960	−	0.931774i	0.461054	−	0.887372i	$-0.347472\pi$
$444$		0			0
$445$	3.17897e10	−	5.50615e10i	0.0384297	−	0.0665622i
$446$		0			0
$447$	−3.16224e11			−0.374638
$448$		0			0
$449$	5.49539e11			0.638102			0.319051	−	0.947738i	$-0.396636\pi$
$449$	5.49539e11			0.638102			0.319051	+	0.947738i	$0.396636\pi$
$450$		0			0
$451$	−5.59257e11	+	9.68662e11i	−0.636527	+	1.10250i
$452$		0			0
$453$	1.55746e11	+	2.69760e11i	0.173770	+	0.300979i
$454$		0			0
$455$	7.23873e11	+	5.13425e11i	0.791792	+	0.561598i
$456$		0			0
$457$	−1.93268e11	−	3.34749e11i	−0.207270	−	0.359002i	0.743584	−	0.668643i	$-0.233124\pi$
$457$	−1.93268e11	−	3.34749e11i	−0.207270	−	0.359002i	−0.950854	+	0.309641i	$0.899791\pi$
$458$		0			0
$459$	1.34664e11	−	2.33244e11i	0.141610	−	0.245276i
$460$		0			0
$461$	−1.45095e12			−1.49623			−0.748115	−	0.663569i	$-0.769041\pi$
$461$	−1.45095e12			−1.49623			−0.748115	+	0.663569i	$0.769041\pi$
$462$		0			0
$463$	−5.07128e11			−0.512865			−0.256433	−	0.966562i	$-0.582547\pi$
$463$	−5.07128e11			−0.512865			−0.256433	+	0.966562i	$0.582547\pi$
$464$		0			0
$465$	−7.44436e10	+	1.28940e11i	−0.0738396	+	0.127894i
$466$		0			0
$467$	−5.65226e11	−	9.79000e11i	−0.549915	−	0.952481i	−0.998280	−	0.0586305i	$-0.981327\pi$
$467$	−5.65226e11	−	9.79000e11i	−0.549915	−	0.952481i	0.448364	−	0.893851i	$-0.352007\pi$
$468$		0			0
$469$	−1.04175e11	+	1.11302e12i	−0.0994226	+	1.06224i
$470$		0			0
$471$	−4.39316e11	−	7.60918e11i	−0.411323	−	0.712433i
$472$		0			0
$473$	6.81936e11	−	1.18115e12i	0.626424	−	1.08500i
$474$		0			0
$475$	8.32111e11			0.749998
$476$		0			0
$477$	2.57501e11			0.227743
$478$		0			0
$479$	−6.65347e10	+	1.15242e11i	−0.0577482	+	0.100023i	−0.893454	−	0.449154i	$-0.851725\pi$
$479$	−6.65347e10	+	1.15242e11i	−0.0577482	+	0.100023i	0.835706	+	0.549177i	$0.185059\pi$
$480$		0			0
$481$	3.38413e11	+	5.86149e11i	0.288267	+	0.499293i
$482$		0			0
$483$	−4.07867e11	+	1.87192e11i	−0.341001	+	0.156504i
$484$		0			0
$485$	5.66468e11	+	9.81152e11i	0.464877	+	0.805190i
$486$		0			0
$487$	7.06495e11	−	1.22369e12i	0.569153	−	0.985801i	−0.427497	−	0.904017i	$-0.640605\pi$
$487$	7.06495e11	−	1.22369e12i	0.569153	−	0.985801i	0.996650	−	0.0817847i	$-0.0260620\pi$
$488$		0			0
$489$	9.77262e11			0.772897
$490$		0			0
$491$	−1.17252e12			−0.910445			−0.455222	−	0.890378i	$-0.650440\pi$
$491$	−1.17252e12			−0.910445			−0.455222	+	0.890378i	$0.650440\pi$
$492$		0			0
$493$	−2.85075e11	+	4.93765e11i	−0.217344	+	0.376452i
$494$		0			0
$495$	−1.68397e11	−	2.91673e11i	−0.126070	−	0.218360i
$496$		0			0
$497$	1.79366e12	−	8.23207e11i	1.31867	−	0.605209i
$498$		0			0
$499$	4.04221e11	+	7.00131e11i	0.291854	+	0.505507i	0.974248	−	0.225478i	$-0.0723944\pi$
$499$	4.04221e11	+	7.00131e11i	0.291854	+	0.505507i	−0.682394	+	0.730985i	$0.739061\pi$
$500$		0			0
$501$	3.49532e11	−	6.05408e11i	0.247866	−	0.429317i
$502$		0			0
$503$	−2.47583e12			−1.72451			−0.862254	−	0.506476i	$-0.830948\pi$
$503$	−2.47583e12			−1.72451			−0.862254	+	0.506476i	$0.830948\pi$
$504$		0			0
$505$	−4.33596e11			−0.296671
$506$		0			0
$507$	−3.04018e11	+	5.26575e11i	−0.204345	+	0.353936i
$508$		0			0
$509$	8.17143e11	+	1.41533e12i	0.539595	+	0.934606i	0.998926	+	0.0463409i	$0.0147560\pi$
$509$	8.17143e11	+	1.41533e12i	0.539595	+	0.934606i	−0.459330	+	0.888265i	$0.651911\pi$
$510$		0			0
$511$	−8.62369e10	+	9.21368e11i	−0.0559499	+	0.597777i
$512$		0			0
$513$	2.52558e11	+	4.37444e11i	0.161003	+	0.278865i
$514$		0			0
$515$	7.87857e11	−	1.36461e12i	0.493532	−	0.854822i
$516$		0			0
$517$	8.04241e11			0.495084
$518$		0			0
$519$	1.37749e12			0.833365
$520$		0			0
$521$	−6.32541e11	+	1.09559e12i	−0.376113	+	0.651448i	−0.990493	−	0.137563i	$-0.956073\pi$
$521$	−6.32541e11	+	1.09559e12i	−0.376113	+	0.651448i	0.614380	+	0.789011i	$0.289406\pi$
$522$		0			0
$523$	1.53136e11	+	2.65240e11i	0.0894995	+	0.155018i	0.907300	−	0.420485i	$-0.138140\pi$
$523$	1.53136e11	+	2.65240e11i	0.0894995	+	0.155018i	−0.817800	+	0.575502i	$0.804807\pi$
$524$		0			0
$525$	−3.67436e11	−	2.60613e11i	−0.211088	−	0.149720i
$526$		0			0
$527$	4.48673e11	+	7.77124e11i	0.253386	+	0.438877i
$528$		0			0
$529$	5.20239e11	−	9.01081e11i	0.288837	−	0.500280i
$530$		0			0
$531$	9.32075e11			0.508775
$532$		0			0
$533$	3.04409e12			1.63375
$534$		0			0
$535$	2.49338e11	−	4.31867e11i	0.131582	−	0.227907i
$536$		0			0
$537$	−1.29915e11	−	2.25019e11i	−0.0674178	−	0.116771i
$538$		0			0
$539$	6.59712e11	+	1.88323e12i	0.336670	+	0.961069i
$540$		0			0
$541$	5.91627e11	+	1.02473e12i	0.296934	+	0.514305i	0.975433	−	0.220296i	$-0.0707024\pi$
$541$	5.91627e11	+	1.02473e12i	0.296934	+	0.514305i	−0.678499	+	0.734602i	$0.737369\pi$
$542$		0			0
$543$	1.03280e12	−	1.78886e12i	0.509819	−	0.883032i
$544$		0			0
$545$	1.37404e12			0.667138
$546$		0			0
$547$	3.91387e12			1.86923			0.934616	−	0.355658i	$-0.115743\pi$
$547$	3.91387e12			1.86923			0.934616	+	0.355658i	$0.115743\pi$
$548$		0			0
$549$	5.32425e11	−	9.22187e11i	0.250140	−	0.433255i
$550$		0			0
$551$	−5.34651e11	−	9.26043e11i	−0.247109	−	0.428005i
$552$		0			0
$553$	7.87882e11	+	5.58825e11i	0.358260	+	0.254105i
$554$		0			0
$555$	2.11445e11	+	3.66234e11i	0.0945975	+	0.163848i
$556$		0			0
$557$	−5.35882e11	+	9.28175e11i	−0.235896	+	0.408584i	−0.959533	−	0.281597i	$-0.909136\pi$
$557$	−5.35882e11	+	9.28175e11i	−0.235896	+	0.408584i	0.723637	+	0.690181i	$0.242469\pi$
$558$		0			0
$559$	−3.71184e12			−1.60782
$560$		0			0
$561$	−2.02987e12			−0.865236
$562$		0			0
$563$	2.13997e12	−	3.70654e12i	0.897676	−	1.55482i	0.0672193	−	0.997738i	$-0.478587\pi$
$563$	2.13997e12	−	3.70654e12i	0.897676	−	1.55482i	0.830457	−	0.557083i	$-0.188079\pi$
$564$		0			0
$565$	5.51936e11	+	9.55980e11i	0.227861	+	0.394667i
$566$		0			0
$567$	2.54828e10	−	2.72262e11i	0.0103544	−	0.110628i
$568$		0			0
$569$	1.16890e12	+	2.02460e12i	0.467491	+	0.809718i	0.999310	−	0.0371398i	$-0.0118247\pi$
$569$	1.16890e12	+	2.02460e12i	0.467491	+	0.809718i	−0.531819	+	0.846858i	$0.678491\pi$
$570$		0			0
$571$	9.62062e11	−	1.66634e12i	0.378739	−	0.655996i	−0.612140	−	0.790750i	$-0.709691\pi$
$571$	9.62062e11	−	1.66634e12i	0.378739	−	0.655996i	0.990879	+	0.134754i	$0.0430243\pi$
$572$		0			0
$573$	−1.27198e12			−0.492928
$574$		0			0
$575$	−7.63562e11			−0.291299
$576$		0			0
$577$	−1.46472e12	+	2.53697e12i	−0.550128	+	0.952850i	0.448136	+	0.893965i	$0.352088\pi$
$577$	−1.46472e12	+	2.53697e12i	−0.550128	+	0.952850i	−0.998265	+	0.0588851i	$0.981245\pi$
$578$		0			0
$579$	−3.43411e10	−	5.94805e10i	−0.0126987	−	0.0219948i
$580$		0			0
$581$	4.43767e12	−	2.03668e12i	1.61571	−	0.741532i
$582$		0			0
$583$	−9.70365e11	−	1.68072e12i	−0.347878	−	0.602542i
$584$		0			0
$585$	−4.58301e11	+	7.93801e11i	−0.161789	+	0.280227i
$586$		0			0
$587$	3.58679e11			0.124691			0.0623454	−	0.998055i	$-0.480142\pi$
$587$	3.58679e11			0.124691			0.0623454	+	0.998055i	$0.480142\pi$
$588$		0			0
$589$	−1.68295e12			−0.576171
$590$		0			0
$591$	−6.57374e10	+	1.13861e11i	−0.0221651	+	0.0383910i
$592$		0			0
$593$	−4.43154e11	−	7.67565e11i	−0.147166	−	0.254900i	0.783013	−	0.622006i	$-0.213682\pi$
$593$	−4.43154e11	−	7.67565e11i	−0.147166	−	0.254900i	−0.930179	+	0.367106i	$0.880349\pi$
$594$		0			0
$595$	3.03737e12	−	1.39401e12i	0.993509	−	0.455974i
$596$		0			0
$597$	4.37488e11	+	7.57751e11i	0.140955	+	0.244142i
$598$		0			0
$599$	−9.52272e11	+	1.64938e12i	−0.302232	+	0.523481i	−0.976641	−	0.214877i	$-0.931065\pi$
$599$	−9.52272e11	+	1.64938e12i	−0.302232	+	0.523481i	0.674409	+	0.738358i	$0.264398\pi$
$600$		0			0
$601$	1.29260e12			0.404136			0.202068	−	0.979372i	$-0.435234\pi$
$601$	1.29260e12			0.404136			0.202068	+	0.979372i	$0.435234\pi$
$602$		0			0
$603$	−1.15458e12			−0.355629
$604$		0			0
$605$	−4.52854e10	+	7.84366e10i	−0.0137423	+	0.0238023i
$606$		0			0
$607$	−4.42039e11	−	7.65635e11i	−0.132164	−	0.228914i	0.792347	−	0.610071i	$-0.208859\pi$
$607$	−4.42039e11	−	7.65635e11i	−0.132164	−	0.228914i	−0.924510	+	0.381157i	$0.875526\pi$
$608$		0			0
$609$	−5.39457e10	+	5.76363e11i	−0.0158920	+	0.169793i
$610$		0			0
$611$	−1.09439e12	−	1.89554e12i	−0.317678	−	0.550234i
$612$		0			0
$613$	−4.59745e11	+	7.96301e11i	−0.131506	+	0.227775i	−0.924257	−	0.381770i	$-0.875315\pi$
$613$	−4.59745e11	+	7.96301e11i	−0.131506	+	0.227775i	0.792751	+	0.609545i	$0.208648\pi$
$614$		0			0
$615$	1.90199e12			0.536130
$616$		0			0
$617$	2.79767e12			0.777165			0.388582	−	0.921414i	$-0.372965\pi$
$617$	2.79767e12			0.777165			0.388582	+	0.921414i	$0.372965\pi$
$618$		0			0
$619$	1.76934e12	−	3.06458e12i	0.484398	−	0.839002i	−0.515441	−	0.856925i	$-0.672372\pi$
$619$	1.76934e12	−	3.06458e12i	0.484398	−	0.839002i	0.999839	+	0.0179230i	$0.00570537\pi$
$620$		0			0
$621$	−2.31752e11	−	4.01407e11i	−0.0625334	−	0.108311i
$622$		0			0
$623$	−3.17344e11	−	2.25084e11i	−0.0843983	−	0.0598616i
$624$		0			0
$625$	6.69160e11	+	1.15902e12i	0.175416	+	0.303830i
$626$		0			0
$627$	1.90348e12	−	3.29692e12i	0.491863	−	0.851932i
$628$		0			0
$629$	2.54877e12			0.649236
$630$		0			0
$631$	1.69697e12			0.426129			0.213065	−	0.977038i	$-0.431656\pi$
$631$	1.69697e12			0.426129			0.213065	+	0.977038i	$0.431656\pi$
$632$		0			0
$633$	−6.51632e11	+	1.12866e12i	−0.161319	+	0.279413i
$634$		0			0
$635$	1.66364e12	+	2.88150e12i	0.406047	+	0.703294i
$636$		0			0
$637$	3.54093e12	−	4.11755e12i	0.852099	−	0.990858i
$638$		0			0
$639$	1.01917e12	+	1.76526e12i	0.241821	+	0.418846i
$640$		0			0
$641$	4.92328e11	−	8.52737e11i	0.115184	−	0.199505i	−0.802669	−	0.596425i	$-0.796587\pi$
$641$	4.92328e11	−	8.52737e11i	0.115184	−	0.199505i	0.917853	+	0.396919i	$0.129921\pi$
$642$		0			0
$643$	−6.77513e12			−1.56303			−0.781516	−	0.623885i	$-0.785554\pi$
$643$	−6.77513e12			−1.56303			−0.781516	+	0.623885i	$0.785554\pi$
$644$		0			0
$645$	−2.31921e12			−0.527621
$646$		0			0
$647$	−2.19919e12	+	3.80911e12i	−0.493394	+	0.854583i	−0.999971	−	0.00761138i	$-0.997577\pi$
$647$	−2.19919e12	+	3.80911e12i	−0.493394	+	0.854583i	0.506577	+	0.862195i	$0.330911\pi$
$648$		0			0
$649$	−3.51243e12	−	6.08371e12i	−0.777153	−	1.34607i
$650$		0			0
$651$	7.43140e11	+	5.27090e11i	0.162165	+	0.115019i
$652$		0			0
$653$	−9.03157e11	−	1.56431e12i	−0.194381	−	0.336678i	0.752316	−	0.658802i	$-0.228936\pi$
$653$	−9.03157e11	−	1.56431e12i	−0.194381	−	0.336678i	−0.946697	+	0.322124i	$0.895603\pi$
$654$		0			0
$655$	−2.69741e12	+	4.67204e12i	−0.572612	+	0.991793i
$656$		0			0
$657$	−9.55775e11			−0.200130
$658$		0			0
$659$	−8.35283e12			−1.72524			−0.862619	−	0.505854i	$-0.831177\pi$
$659$	−8.35283e12			−1.72524			−0.862619	+	0.505854i	$0.831177\pi$
$660$		0			0
$661$	−2.42835e12	+	4.20602e12i	−0.494771	+	0.856968i	−0.999982	−	0.00602747i	$-0.998081\pi$
$661$	−2.42835e12	+	4.20602e12i	−0.494771	+	0.856968i	0.505211	+	0.862996i	$0.331415\pi$
$662$		0			0
$663$	2.76219e12	+	4.78425e12i	0.555191	+	0.961619i
$664$		0			0
$665$	−5.84093e11	+	6.24054e12i	−0.115820	+	1.23744i
$666$		0			0
$667$	4.90607e11	+	8.49756e11i	0.0959770	+	0.166237i
$668$		0			0
$669$	5.99786e11	−	1.03886e12i	0.115765	−	0.200511i
$670$		0			0
$671$	−8.02555e12			−1.52835
$672$		0			0
$673$	−2.92327e12			−0.549289			−0.274644	−	0.961546i	$-0.588560\pi$
$673$	−2.92327e12			−0.549289			−0.274644	+	0.961546i	$0.588560\pi$
$674$		0			0
$675$	2.32632e11	−	4.02931e11i	0.0431323	−	0.0747074i
$676$		0			0
$677$	−3.36385e12	−	5.82636e12i	−0.615443	−	1.06598i	−0.990307	−	0.138898i	$-0.955644\pi$
$677$	−3.36385e12	−	5.82636e12i	−0.615443	−	1.06598i	0.374864	−	0.927080i	$-0.377689\pi$
$678$		0			0
$679$	6.30088e12	−	2.89181e12i	1.13759	−	0.522102i
$680$		0			0
$681$	1.67501e12	+	2.90120e12i	0.298438	+	0.516911i
$682$		0			0
$683$	2.72209e12	−	4.71480e12i	0.478640	−	0.829029i	−0.521060	−	0.853520i	$-0.674463\pi$
$683$	2.72209e12	−	4.71480e12i	0.478640	−	0.829029i	0.999700	+	0.0244911i	$0.00779654\pi$
$684$		0			0
$685$	1.42159e12			0.246699
$686$		0			0
$687$	−4.58642e12			−0.785540
$688$		0			0
$689$	−2.64090e12	+	4.57416e12i	−0.446442	+	0.773260i
$690$		0			0
$691$	2.77826e12	+	4.81209e12i	0.463577	+	0.802938i	0.999136	−	0.0415591i	$-0.0132325\pi$
$691$	2.77826e12	+	4.81209e12i	0.463577	+	0.802938i	−0.535559	+	0.844498i	$0.679899\pi$
$692$		0			0
$693$	−1.87310e12	+	8.59664e11i	−0.308504	+	0.141589i
$694$		0			0
$695$	−8.39627e11	−	1.45428e12i	−0.136507	−	0.236437i
$696$		0			0
$697$	5.73166e12	−	9.92753e12i	0.919884	−	1.59329i
$698$		0			0
$699$	−5.54775e11			−0.0878961
$700$		0			0
$701$	−3.79560e12			−0.593676			−0.296838	−	0.954928i	$-0.595932\pi$
$701$	−3.79560e12			−0.593676			−0.296838	+	0.954928i	$0.595932\pi$
$702$		0			0
$703$	−2.39007e12	+	4.13973e12i	−0.369073	+	0.639253i
$704$		0			0
$705$	−6.83790e11	−	1.18436e12i	−0.104249	−	0.180565i
$706$		0			0
$707$	−2.47261e11	+	2.64177e12i	−0.0372192	+	0.397656i
$708$		0			0
$709$	−1.27068e12	−	2.20088e12i	−0.188854	−	0.327105i	0.756014	−	0.654555i	$-0.227144\pi$
$709$	−1.27068e12	−	2.20088e12i	−0.188854	−	0.327105i	−0.944869	+	0.327450i	$0.893811\pi$
$710$		0			0
$711$	−4.98827e11	+	8.63993e11i	−0.0732043	+	0.126794i
$712$		0			0
$713$	1.54431e12			0.223785
$714$		0			0
$715$	6.90825e12			0.988532
$716$		0			0
$717$	−3.50683e12	+	6.07401e12i	−0.495540	+	0.858301i
$718$		0			0
$719$	−6.19073e12	−	1.07227e13i	−0.863897	−	1.49631i	−0.868138	−	0.496322i	$-0.834684\pi$
$719$	−6.19073e12	−	1.07227e13i	−0.863897	−	1.49631i	0.00424172	−	0.999991i	$-0.498650\pi$
$720$		0			0
$721$	−7.86485e12	−	5.57834e12i	−1.08388	−	0.768770i
$722$		0			0
$723$	3.86741e11	+	6.69855e11i	0.0526378	+	0.0911713i
$724$		0			0
$725$	−4.92469e11	+	8.52982e11i	−0.0662000	+	0.114662i
$726$		0			0
$727$	2.85958e12			0.379663			0.189831	−	0.981817i	$-0.439206\pi$
$727$	2.85958e12			0.379663			0.189831	+	0.981817i	$0.439206\pi$
$728$		0			0
$729$	2.82430e11			0.0370370
$730$		0			0
$731$	−6.98896e12	+	1.21052e13i	−0.905283	+	1.56800i
$732$		0			0
$733$	−1.64705e12	−	2.85278e12i	−0.210736	−	0.365006i	0.741209	−	0.671275i	$-0.234253\pi$
$733$	−1.64705e12	−	2.85278e12i	−0.210736	−	0.365006i	−0.951945	+	0.306268i	$0.900920\pi$
$734$		0			0
$735$	2.21242e12	−	2.57270e12i	0.279624	−	0.325160i
$736$		0			0
$737$	4.35093e12	+	7.53603e12i	0.543223	+	0.940890i
$738$		0			0
$739$	7.26731e12	−	1.25873e13i	0.896341	−	1.55251i	0.0642060	−	0.997937i	$-0.479549\pi$
$739$	7.26731e12	−	1.25873e13i	0.896341	−	1.55251i	0.832136	−	0.554572i	$-0.187118\pi$
$740$		0			0
$741$	−1.03608e13			−1.26244
$742$		0			0
$743$	−1.78133e12			−0.214434			−0.107217	−	0.994236i	$-0.534194\pi$
$743$	−1.78133e12			−0.214434			−0.107217	+	0.994236i	$0.534194\pi$
$744$		0			0
$745$	2.02637e12	−	3.50978e12i	0.240999	−	0.417423i
$746$		0			0
$747$	2.52151e12	+	4.36738e12i	0.296291	+	0.513191i
$748$		0			0
$749$	−2.48904e12	−	1.76541e12i	−0.288977	−	0.204964i
$750$		0			0
$751$	−4.36499e11	−	7.56039e11i	−0.0500730	−	0.0867290i	0.839903	−	0.542737i	$-0.182612\pi$
$751$	−4.36499e11	−	7.56039e11i	−0.0500730	−	0.0867290i	−0.889976	+	0.456008i	$0.849279\pi$
$752$		0			0
$753$	−3.76808e12	+	6.52651e12i	−0.427113	+	0.739782i
$754$		0			0
$755$	−3.99210e12			−0.447136
$756$		0			0
$757$	−1.13153e13			−1.25238			−0.626189	−	0.779671i	$-0.715386\pi$
$757$	−1.13153e13			−1.25238			−0.626189	+	0.779671i	$0.715386\pi$
$758$		0			0
$759$	−1.74667e12	+	3.02532e12i	−0.191039	+	0.330890i
$760$		0			0
$761$	−1.19543e12	−	2.07054e12i	−0.129209	−	0.223796i	0.794161	−	0.607707i	$-0.207910\pi$
$761$	−1.19543e12	−	2.07054e12i	−0.129209	−	0.223796i	−0.923370	+	0.383911i	$0.874577\pi$
$762$		0			0
$763$	7.83554e11	−	8.37160e12i	0.0836967	−	0.894228i
$764$		0			0
$765$	1.72585e12	+	2.98927e12i	0.182191	+	0.315565i
$766$		0			0
$767$	−9.55924e12	+	1.65571e13i	−0.997343	+	1.72745i
$768$		0			0
$769$	−1.49493e13			−1.54153			−0.770766	−	0.637118i	$-0.780126\pi$
$769$	−1.49493e13			−1.54153			−0.770766	+	0.637118i	$0.780126\pi$
$770$		0			0
$771$	4.05011e12			0.412783
$772$		0			0
$773$	3.67321e12	−	6.36219e12i	0.370031	−	0.640913i	−0.619539	−	0.784966i	$-0.712680\pi$
$773$	3.67321e12	−	6.36219e12i	0.370031	−	0.640913i	0.989570	+	0.144053i	$0.0460137\pi$
$774$		0			0
$775$	7.75084e11	+	1.34249e12i	0.0771776	+	0.133676i
$776$		0			0
$777$	2.35193e12	−	1.07942e12i	0.231488	−	0.106242i
$778$		0			0
$779$	1.07496e13	+	1.86188e13i	1.04586	+	1.81148i
$780$		0			0
$781$	7.68129e12	−	1.33044e13i	0.738761	−	1.27957i
$782$		0			0
$783$	−5.97887e11			−0.0568449
$784$		0			0
$785$	1.12606e13			1.05839
$786$		0			0
$787$	−2.49809e12	+	4.32681e12i	−0.232125	+	0.402051i	−0.958433	−	0.285317i	$-0.907901\pi$
$787$	−2.49809e12	+	4.32681e12i	−0.232125	+	0.402051i	0.726309	+	0.687369i	$0.241234\pi$
$788$		0			0
$789$	−1.74724e11	−	3.02631e11i	−0.0160512	−	0.0278014i
$790$		0			0
$791$	6.13923e12	−	2.81762e12i	0.557596	−	0.255910i
$792$		0			0
$793$	1.09210e13	+	1.89157e13i	0.980690	+	1.69860i
$794$		0			0
$795$	−1.65007e12	+	2.85800e12i	−0.146504	+	0.253753i
$796$		0			0
$797$	−6.96597e12			−0.611532			−0.305766	−	0.952107i	$-0.598912\pi$
$797$	−6.96597e12			−0.611532			−0.305766	+	0.952107i	$0.598912\pi$
$798$		0			0
$799$	−8.24243e12			−0.715475
$800$		0			0
$801$	2.00918e11	−	3.48000e11i	0.0172454	−	0.0298698i
$802$		0			0
$803$	3.60174e12	+	6.23840e12i	0.305698	+	0.529484i
$804$		0			0
$805$	5.35976e11	−	5.72644e12i	0.0449846	−	0.480622i
$806$		0			0
$807$	4.41741e12	+	7.65118e12i	0.366638	+	0.635035i
$808$		0			0
$809$	1.89371e12	−	3.28000e12i	0.155434	−	0.269219i	−0.777783	−	0.628533i	$-0.783656\pi$
$809$	1.89371e12	−	3.28000e12i	0.155434	−	0.269219i	0.933217	+	0.359314i	$0.116989\pi$
$810$		0			0
$811$	1.17809e13			0.956277			0.478138	−	0.878285i	$-0.341312\pi$
$811$	1.17809e13			0.956277			0.478138	+	0.878285i	$0.341312\pi$
$812$		0			0
$813$	−7.61499e12			−0.611311
$814$		0			0
$815$	−6.26231e12	+	1.08466e13i	−0.497193	+	0.861164i
$816$		0			0
$817$	−1.31076e13	−	2.27030e13i	−1.02926	−	1.78273i
$818$		0			0
$819$	4.57503e12	+	3.24496e12i	0.355318	+	0.252018i
$820$		0			0
$821$	7.78784e10	+	1.34889e11i	0.00598236	+	0.0103618i	0.869001	−	0.494810i	$-0.164762\pi$
$821$	7.78784e10	+	1.34889e11i	0.00598236	+	0.0103618i	−0.863019	+	0.505172i	$0.831429\pi$
$822$		0			0
$823$	−3.44272e11	+	5.96297e11i	−0.0261579	+	0.0453068i	−0.878808	−	0.477176i	$-0.841661\pi$
$823$	−3.44272e11	+	5.96297e11i	−0.0261579	+	0.0453068i	0.852650	+	0.522482i	$0.174994\pi$
$824$		0			0
$825$	−3.50660e12			−0.263538
$826$		0			0
$827$	5.36499e12			0.398836			0.199418	−	0.979915i	$-0.436095\pi$
$827$	5.36499e12			0.398836			0.199418	+	0.979915i	$0.436095\pi$
$828$		0			0
$829$	1.69230e11	−	2.93116e11i	0.0124447	−	0.0215548i	−0.859736	−	0.510739i	$-0.829372\pi$
$829$	1.69230e11	−	2.93116e11i	0.0124447	−	0.0215548i	0.872181	+	0.489184i	$0.162705\pi$
$830$		0			0
$831$	−6.74335e12	−	1.16798e13i	−0.490536	−	0.849633i
$832$		0			0
$833$	−6.76119e12	−	1.93007e13i	−0.486542	−	1.38890i
$834$		0			0
$835$	4.47962e12	+	7.75892e12i	0.318898	+	0.552347i
$836$		0			0
$837$	−4.70500e11	+	8.14929e11i	−0.0331356	+	0.0573925i
$838$		0			0
$839$	−2.54839e13			−1.77557			−0.887784	−	0.460261i	$-0.847756\pi$
$839$	−2.54839e13			−1.77557			−0.887784	+	0.460261i	$0.847756\pi$
$840$		0			0
$841$	−1.32415e13			−0.912754
$842$		0			0
$843$	4.56953e12	−	7.91466e12i	0.311636	−	0.539769i
$844$		0			0
$845$	−3.89631e12	−	6.74860e12i	−0.262905	−	0.455364i
$846$		0			0
$847$	4.52065e11	+	3.20639e11i	0.0301805	+	0.0214062i
$848$		0			0
$849$	−6.06090e12	−	1.04978e13i	−0.400362	−	0.693447i
$850$		0			0
$851$	2.19318e12	−	3.79870e12i	0.143348	−	0.248286i
$852$		0			0
$853$	−3.46282e12			−0.223954			−0.111977	−	0.993711i	$-0.535718\pi$
$853$	−3.46282e12			−0.223954			−0.111977	+	0.993711i	$0.535718\pi$
$854$		0			0
$855$	−6.47359e12			−0.414283
$856$		0			0
$857$	1.36561e11	−	2.36530e11i	0.00864794	−	0.0149787i	−0.861669	−	0.507471i	$-0.830581\pi$
$857$	1.36561e11	−	2.36530e11i	0.00864794	−	0.0149787i	0.870317	+	0.492492i	$0.163914\pi$
$858$		0			0
$859$	−2.60376e12	−	4.50984e12i	−0.163167	−	0.282613i	0.772836	−	0.634606i	$-0.218837\pi$
$859$	−2.60376e12	−	4.50984e12i	−0.163167	−	0.282613i	−0.936003	+	0.351993i	$0.885504\pi$
$860$		0			0
$861$	1.08462e12	−	1.15882e13i	0.0672610	−	0.718626i
$862$		0			0
$863$	1.31133e13	+	2.27129e13i	0.804753	+	1.39387i	0.916458	+	0.400131i	$0.131035\pi$
$863$	1.31133e13	+	2.27129e13i	0.804753	+	1.39387i	−0.111705	+	0.993741i	$0.535631\pi$
$864$		0			0
$865$	−8.82697e12	+	1.52888e13i	−0.536092	+	0.928538i
$866$		0			0
$867$	1.11979e13			0.673053
$868$		0			0
$869$	7.51911e12			0.447278
$870$		0			0
$871$	1.18413e13	−	2.05097e13i	0.697134	−	1.20747i
$872$		0			0
$873$	3.58020e12	+	6.20109e12i	0.208614	+	0.361330i
$874$		0			0
$875$	1.69529e13	−	7.78059e12i	0.977706	−	0.448721i
$876$		0			0
$877$	−1.03572e12	−	1.79392e12i	−0.0591214	−	0.102401i	0.834950	−	0.550326i	$-0.185497\pi$
$877$	−1.03572e12	−	1.79392e12i	−0.0591214	−	0.102401i	−0.894071	+	0.447925i	$0.852163\pi$
$878$		0			0
$879$	−7.24734e12	+	1.25528e13i	−0.409476	+	0.709234i
$880$		0			0
$881$	1.33748e13			0.747990			0.373995	−	0.927431i	$-0.377988\pi$
$881$	1.33748e13			0.747990			0.373995	+	0.927431i	$0.377988\pi$
$882$		0			0
$883$	5.92266e12			0.327864			0.163932	−	0.986472i	$-0.447582\pi$
$883$	5.92266e12			0.327864			0.163932	+	0.986472i	$0.447582\pi$
$884$		0			0
$885$	−5.97275e12	+	1.03451e13i	−0.327288	+	0.566879i
$886$		0			0
$887$	−6.65416e12	−	1.15254e13i	−0.360942	−	0.625170i	0.627174	−	0.778879i	$-0.284211\pi$
$887$	−6.65416e12	−	1.15254e13i	−0.360942	−	0.625170i	−0.988116	+	0.153709i	$0.950878\pi$
$888$		0			0
$889$	1.85048e13	−	8.49282e12i	0.993633	−	0.456030i
$890$		0			0
$891$	−1.06431e12	−	1.84343e12i	−0.0565740	−	0.0979891i
$892$		0			0
$893$	7.72923e12	−	1.33874e13i	0.406728	−	0.704474i
$894$		0			0
$895$	3.32998e12			0.173476
$896$		0			0
$897$	9.50730e12			0.490333
$898$		0			0
$899$	9.96021e11	−	1.72516e12i	0.0508569	−	0.0880867i
$900$		0			0
$901$	9.94498e12	+	1.72252e13i	0.502739	+	0.870769i
$902$		0			0
$903$	−1.32254e12	+	1.41302e13i	−0.0661934	+	0.707220i
$904$		0			0
$905$	1.32364e13	+	2.29260e13i	0.655918	+	1.13608i
$906$		0			0
$907$	1.67762e13	−	2.90573e13i	0.823117	−	1.42568i	−0.0802334	−	0.996776i	$-0.525567\pi$
$907$	1.67762e13	−	2.90573e13i	0.823117	−	1.42568i	0.903350	−	0.428904i	$-0.141100\pi$
$908$		0			0
$909$	−2.74042e12			−0.133131
$910$		0			0
$911$	1.32379e12			0.0636774			0.0318387	−	0.999493i	$-0.489864\pi$
$911$	1.32379e12			0.0636774			0.0318387	+	0.999493i	$0.489864\pi$
$912$		0			0
$913$	1.90041e13	−	3.29161e13i	0.905167	−	1.56780i
$914$		0			0
$915$	6.82357e12	+	1.18188e13i	0.321823	+	0.557413i
$916$		0			0
$917$	2.69271e13	+	1.90987e13i	1.25756	+	0.891953i
$918$		0			0
$919$	−8.08614e11	−	1.40056e12i	−0.0373957	−	0.0647712i	0.846722	−	0.532036i	$-0.178573\pi$
$919$	−8.08614e11	−	1.40056e12i	−0.0373957	−	0.0647712i	−0.884117	+	0.467265i	$0.845240\pi$
$920$		0			0
$921$	2.38249e12	−	4.12659e12i	0.109109	−	0.188983i
$922$		0			0
$923$	−4.18100e13			−1.89615
$924$		0			0
$925$	4.40301e12			0.197748
$926$		0			0
$927$	4.97942e12	−	8.62462e12i	0.221473	−	0.383602i
$928$		0			0
$929$	1.37761e13	+	2.38610e13i	0.606816	+	1.05104i	0.991762	+	0.128096i	$0.0408867\pi$
$929$	1.37761e13	+	2.38610e13i	0.606816	+	1.05104i	−0.384946	+	0.922939i	$0.625780\pi$
$930$		0			0
$931$	3.76886e13	+	7.11740e12i	1.64413	+	0.310490i
$932$		0			0
$933$	−2.01475e12	−	3.48964e12i	−0.0870468	−	0.150769i
$934$		0			0
$935$	1.30074e13	−	2.25295e13i	0.556594	−	0.964049i
$936$		0			0
$937$	−4.44740e12			−0.188485			−0.0942427	−	0.995549i	$-0.530043\pi$
$937$	−4.44740e12			−0.188485			−0.0942427	+	0.995549i	$0.530043\pi$
$938$		0			0
$939$	2.65274e13			1.11352
$940$		0			0
$941$	−8.53074e12	+	1.47757e13i	−0.354677	+	0.614319i	−0.987063	−	0.160335i	$-0.948743\pi$
$941$	−8.53074e12	+	1.47757e13i	−0.354677	+	0.614319i	0.632385	+	0.774654i	$0.282076\pi$
$942$		0			0
$943$	−9.86402e12	−	1.70850e13i	−0.406211	−	0.703578i
$944$		0			0
$945$	2.85854e12	+	2.02749e12i	0.116601	+	0.0827021i
$946$		0			0
$947$	2.84288e12	+	4.92401e12i	0.114864	+	0.198950i	0.917725	−	0.397216i	$-0.130023\pi$
$947$	2.84288e12	+	4.92401e12i	0.114864	+	0.198950i	−0.802861	+	0.596166i	$0.796690\pi$
$948$		0			0
$949$	9.80231e12	−	1.69781e13i	0.392311	−	0.679503i
$950$		0			0
$951$	−1.07655e13			−0.426799
$952$		0			0
$953$	−3.42353e13			−1.34449			−0.672243	−	0.740330i	$-0.734669\pi$
$953$	−3.42353e13			−1.34449			−0.672243	+	0.740330i	$0.734669\pi$
$954$		0			0
$955$	8.15085e12	−	1.41177e13i	0.317094	−	0.549222i
$956$		0			0
$957$	2.25308e12	+	3.90244e12i	0.0868305	+	0.150395i
$958$		0			0
$959$	8.10670e11	−	8.66131e12i	0.0309500	−	0.330674i
$960$		0			0
$961$	1.16522e13	+	2.01822e13i	0.440710	+	0.763332i
$962$		0			0
$963$	1.57587e12	−	2.72949e12i	0.0590476	−	0.102273i
$964$		0			0
$965$	8.80232e11			0.0326757
$966$		0			0
$967$	1.15872e13			0.426146			0.213073	−	0.977036i	$-0.431653\pi$
$967$	1.15872e13			0.426146			0.213073	+	0.977036i	$0.431653\pi$
$968$		0			0
$969$	−1.95082e13	+	3.37892e13i	−0.710821	+	1.23118i
$970$		0			0
$971$	6.46160e12	+	1.11918e13i	0.233267	+	0.404030i	0.958768	−	0.284191i	$-0.0917251\pi$
$971$	6.46160e12	+	1.11918e13i	0.233267	+	0.404030i	−0.725501	+	0.688222i	$0.758392\pi$
$972$		0			0
$973$	−9.33925e12	+	4.28627e12i	−0.334044	+	0.153311i
$974$		0			0
$975$	4.77169e12	+	8.26482e12i	0.169103	+	0.292895i
$976$		0			0
$977$	2.68350e13	−	4.64796e13i	0.942273	−	1.63206i	0.181151	−	0.983455i	$-0.442018\pi$
$977$	2.68350e13	−	4.64796e13i	0.942273	−	1.63206i	0.761122	−	0.648609i	$-0.224649\pi$
$978$		0			0
$979$	−3.02855e12			−0.105369
$980$		0			0
$981$	8.68423e12			0.299379
$982$		0			0
$983$	−6.98492e12	+	1.20982e13i	−0.238600	+	0.413268i	−0.960313	−	0.278925i	$-0.910022\pi$
$983$	−6.98492e12	+	1.20982e13i	−0.238600	+	0.413268i	0.721713	+	0.692193i	$0.243355\pi$
$984$		0			0
$985$	−8.42492e11	−	1.45924e12i	−0.0285169	−	0.0493928i
$986$		0			0
$987$	−7.60587e12	+	3.49073e12i	−0.255106	+	0.117082i
$988$		0			0
$989$	1.20278e13	+	2.08328e13i	0.399763	+	0.692410i
$990$		0			0
$991$	2.30467e12	−	3.99180e12i	0.0759062	−	0.131473i	−0.825574	−	0.564294i	$-0.809148\pi$
$991$	2.30467e12	−	3.99180e12i	0.0759062	−	0.131473i	0.901480	+	0.432821i	$0.142482\pi$
$992$		0			0
$993$	−2.99627e13			−0.977934
$994$		0			0
$995$	−1.12137e13			−0.362698
$996$		0			0
$997$	−1.66463e13	+	2.88322e13i	−0.533567	+	0.924165i	0.465664	+	0.884961i	$0.345815\pi$
$997$	−1.66463e13	+	2.88322e13i	−0.533567	+	0.924165i	−0.999231	+	0.0392035i	$0.987518\pi$
$998$		0			0
$999$	1.33638e12	+	2.31468e12i	0.0424507	+	0.0735268i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.10.q.a.25.4	✓	16
7.2	even	3	inner	168.10.q.a.121.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.10.q.a.25.4	✓	16	1.1	even	1	trivial
168.10.q.a.121.4	yes	16	7.2	even	3	inner

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)

\(f(q)\)	\(=\)	\(q+(-40.5000 + 70.1481i) q^{3} +(-519.049 - 899.019i) q^{5} +(-5773.43 + 2649.73i) q^{7} +(-3280.50 - 5681.99i) q^{9} +O(q^{10})\)	\(q+(-40.5000 + 70.1481i) q^{3} +(-519.049 - 899.019i) q^{5} +(-5773.43 + 2649.73i) q^{7} +(-3280.50 - 5681.99i) q^{9} +(-24724.5 + 42824.0i) q^{11} +134578. q^{13} +84085.9 q^{15} +(253394. - 438891. i) q^{17} +(475233. + 823127. i) q^{19} +(47950.4 - 512309. i) q^{21} +(-436083. - 755318. i) q^{23} +(437739. - 758186. i) q^{25} +531441. q^{27} -1.12503e6 q^{29} +(-885328. + 1.53343e6i) q^{31} +(-2.00268e6 - 3.46874e6i) q^{33} +(5.37886e6 + 3.81509e6i) q^{35} +(2.51463e6 + 4.35547e6i) q^{37} +(-5.45039e6 + 9.44036e6i) q^{39} +2.26196e7 q^{41} -2.75814e7 q^{43} +(-3.40548e6 + 5.89847e6i) q^{45} +(-8.13204e6 - 1.40851e7i) q^{47} +(2.63114e7 - 3.05961e7i) q^{49} +(2.05249e7 + 3.55501e7i) q^{51} +(-1.96236e7 + 3.39891e7i) q^{53} +5.13328e7 q^{55} -7.69877e7 q^{57} +(-7.10315e7 + 1.23030e8i) q^{59} +(8.11499e7 + 1.40556e8i) q^{61} +(3.39955e7 + 2.41122e7i) q^{63} +(-6.98524e7 - 1.20988e8i) q^{65} +(8.79884e7 - 1.52400e8i) q^{67} +7.06455e7 q^{69} -3.10676e8 q^{71} +(7.28376e7 - 1.26158e8i) q^{73} +(3.54568e7 + 6.14130e7i) q^{75} +(2.92728e7 - 3.12755e8i) q^{77} +(-7.60290e7 - 1.31686e8i) q^{79} +(-2.15234e7 + 3.72796e7i) q^{81} -7.68636e8 q^{83} -5.26095e8 q^{85} +(4.55637e7 - 7.89187e7i) q^{87} +(3.06231e7 + 5.30407e7i) q^{89} +(-7.76975e8 + 3.56595e8i) q^{91} +(-7.17116e7 - 1.24208e8i) q^{93} +(4.93338e8 - 8.54487e8i) q^{95} -1.09136e9 q^{97} +3.24434e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

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Groups

Database

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