LMFDB - Embedded newform 168.10.q.a.25.7

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			25.7
Root			$1765.22 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	168.25
Dual form			168.10.q.a.121.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 + 70.1481i) q^{3} +(870.360 + 1507.51i) q^{5} +(3862.22 + 5043.50i) q^{7} +(-3280.50 - 5681.99i) q^{9} +O(q^{10})$	\(q+(-40.5000 + 70.1481i) q^{3} +(870.360 + 1507.51i) q^{5} +(3862.22 + 5043.50i) q^{7} +(-3280.50 - 5681.99i) q^{9} +(-4034.49 + 6987.94i) q^{11} -101548. q^{13} -140998. q^{15} +(147922. - 256209. i) q^{17} +(-186677. - 323334. i) q^{19} +(-510211. + 66665.3i) q^{21} +(-1.19734e6 - 2.07385e6i) q^{23} +(-538490. + 932692. i) q^{25} +531441. q^{27} -5.40640e6 q^{29} +(75620.0 - 130978. i) q^{31} +(-326794. - 566023. i) q^{33} +(-4.24159e6 + 1.02120e7i) q^{35} +(78963.8 + 136769. i) q^{37} +(4.11269e6 - 7.12338e6i) q^{39} +1.33473e7 q^{41} -1.32297e7 q^{43} +(5.71043e6 - 9.89076e6i) q^{45} +(4.26385e6 + 7.38520e6i) q^{47} +(-1.05202e7 + 3.89582e7i) q^{49} +(1.19817e7 + 2.07529e7i) q^{51} +(1.08313e7 - 1.87603e7i) q^{53} -1.40458e7 q^{55} +3.02416e7 q^{57} +(-5.99453e7 + 1.03828e8i) q^{59} +(-6.30819e7 - 1.09261e8i) q^{61} +(1.59871e7 - 3.84903e7i) q^{63} +(-8.83831e7 - 1.53084e8i) q^{65} +(3.07186e7 - 5.32062e7i) q^{67} +1.93969e8 q^{69} +2.83255e7 q^{71} +(3.90888e7 - 6.77038e7i) q^{73} +(-4.36177e7 - 7.55481e7i) q^{75} +(-5.08258e7 + 6.64101e6i) q^{77} +(-2.51276e7 - 4.35223e7i) q^{79} +(-2.15234e7 + 3.72796e7i) q^{81} -3.60147e8 q^{83} +5.14983e8 q^{85} +(2.18959e8 - 3.79248e8i) q^{87} +(-2.00857e8 - 3.47894e8i) q^{89} +(-3.92200e8 - 5.12156e8i) q^{91} +(6.12522e6 + 1.06092e7i) q^{93} +(3.24952e8 - 5.62833e8i) q^{95} +1.53266e9 q^{97} +5.29406e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	870.360	+	1507.51i	0.622779	+	1.07868i	0.988966	+	0.148143i	$0.0473297\pi$
$5$	870.360	+	1507.51i	0.622779	+	1.07868i	−0.366187	+	0.930541i	$0.619337\pi$
$6$		0			0
$7$	3862.22	+	5043.50i	0.607989	+	0.793946i
$7$	3862.22	+	5043.50i	0.607989	+	0.793946i
$8$		0			0
$9$	−3280.50	−	5681.99i	−0.166667	−	0.288675i
$10$		0			0
$11$	−4034.49	+	6987.94i	−0.0830848	+	0.143907i	−0.904574	−	0.426318i	$-0.859811\pi$
$11$	−4034.49	+	6987.94i	−0.0830848	+	0.143907i	0.821489	+	0.570225i	$0.193144\pi$
$12$		0			0
$13$	−101548.			−0.986110			−0.493055	−	0.869998i	$-0.664120\pi$
$13$	−101548.			−0.986110			−0.493055	+	0.869998i	$0.664120\pi$
$14$		0			0
$15$	−140998.			−0.719123
$16$		0			0
$17$	147922.	−	256209.i	0.429550	−	0.744002i	−0.567283	−	0.823523i	$-0.692006\pi$
$17$	147922.	−	256209.i	0.429550	−	0.744002i	0.996833	+	0.0795205i	$0.0253389\pi$
$18$		0			0
$19$	−186677.	−	323334.i	−0.328624	−	0.569193i	0.653615	−	0.756827i	$-0.273251\pi$
$19$	−186677.	−	323334.i	−0.328624	−	0.569193i	−0.982239	+	0.187634i	$0.939918\pi$
$20$		0			0
$21$	−510211.	+	66665.3i	−0.572484	+	0.0748020i
$22$		0			0
$23$	−1.19734e6	−	2.07385e6i	−0.892159	−	1.54526i	−0.837282	−	0.546771i	$-0.815857\pi$
$23$	−1.19734e6	−	2.07385e6i	−0.892159	−	1.54526i	−0.0548765	−	0.998493i	$-0.517477\pi$
$24$		0			0
$25$	−538490.	+	932692.i	−0.275707	+	0.477539i
$26$		0			0
$27$	531441.			0.192450
$28$		0			0
$29$	−5.40640e6			−1.41944			−0.709720	−	0.704484i	$-0.751179\pi$
$29$	−5.40640e6			−1.41944			−0.709720	+	0.704484i	$0.751179\pi$
$30$		0			0
$31$	75620.0	−	130978.i	0.0147065	−	0.0254724i	−0.858579	−	0.512682i	$-0.828652\pi$
$31$	75620.0	−	130978.i	0.0147065	−	0.0254724i	0.873285	+	0.487210i	$0.161985\pi$
$32$		0			0
$33$	−326794.	−	566023.i	−0.0479690	−	0.0830848i
$34$		0			0
$35$	−4.24159e6	+	1.02120e7i	−0.477774	+	1.15028i
$36$		0			0
$37$	78963.8	+	136769.i	0.00692660	+	0.0119972i	0.869468	−	0.493989i	$-0.164462\pi$
$37$	78963.8	+	136769.i	0.00692660	+	0.0119972i	−0.862541	+	0.505987i	$0.831129\pi$
$38$		0			0
$39$	4.11269e6	−	7.12338e6i	0.284665	−	0.493055i
$40$		0			0
$41$	1.33473e7			0.737678			0.368839	−	0.929493i	$-0.379755\pi$
$41$	1.33473e7			0.737678			0.368839	+	0.929493i	$0.379755\pi$
$42$		0			0
$43$	−1.32297e7			−0.590123			−0.295061	−	0.955478i	$-0.595340\pi$
$43$	−1.32297e7			−0.590123			−0.295061	+	0.955478i	$0.595340\pi$
$44$		0			0
$45$	5.71043e6	−	9.89076e6i	0.207593	−	0.359562i
$46$		0			0
$47$	4.26385e6	+	7.38520e6i	0.127456	+	0.220761i	0.922690	−	0.385542i	$-0.125985\pi$
$47$	4.26385e6	+	7.38520e6i	0.127456	+	0.220761i	−0.795234	+	0.606302i	$0.792652\pi$
$48$		0			0
$49$	−1.05202e7	+	3.89582e7i	−0.260699	+	0.965420i
$50$		0			0
$51$	1.19817e7	+	2.07529e7i	0.248001	+	0.429550i
$52$		0			0
$53$	1.08313e7	−	1.87603e7i	0.188555	−	0.326587i	−0.756214	−	0.654325i	$-0.772953\pi$
$53$	1.08313e7	−	1.87603e7i	0.188555	−	0.326587i	0.944769	+	0.327738i	$0.106286\pi$
$54$		0			0
$55$	−1.40458e7			−0.206974
$56$		0			0
$57$	3.02416e7			0.379462
$58$		0			0
$59$	−5.99453e7	+	1.03828e8i	−0.644052	+	1.11553i	0.340467	+	0.940256i	$0.389415\pi$
$59$	−5.99453e7	+	1.03828e8i	−0.644052	+	1.11553i	−0.984520	+	0.175275i	$0.943919\pi$
$60$		0			0
$61$	−6.30819e7	−	1.09261e8i	−0.583338	−	1.01037i	−0.995080	−	0.0990708i	$-0.968413\pi$
$61$	−6.30819e7	−	1.09261e8i	−0.583338	−	1.01037i	0.411742	−	0.911300i	$-0.364920\pi$
$62$		0			0
$63$	1.59871e7	−	3.84903e7i	0.127861	−	0.307836i
$64$		0			0
$65$	−8.83831e7	−	1.53084e8i	−0.614128	−	1.06370i
$66$		0			0
$67$	3.07186e7	−	5.32062e7i	0.186237	−	0.322571i	−0.757756	−	0.652538i	$-0.773704\pi$
$67$	3.07186e7	−	5.32062e7i	0.186237	−	0.322571i	0.943993	+	0.329967i	$0.107038\pi$
$68$		0			0
$69$	1.93969e8			1.03018
$70$		0			0
$71$	2.83255e7			0.132287			0.0661433	−	0.997810i	$-0.478931\pi$
$71$	2.83255e7			0.132287			0.0661433	+	0.997810i	$0.478931\pi$
$72$		0			0
$73$	3.90888e7	−	6.77038e7i	0.161102	−	0.279036i	−0.774162	−	0.632987i	$-0.781829\pi$
$73$	3.90888e7	−	6.77038e7i	0.161102	−	0.279036i	0.935264	+	0.353951i	$0.115162\pi$
$74$		0			0
$75$	−4.36177e7	−	7.55481e7i	−0.159180	−	0.275707i
$76$		0			0
$77$	−5.08258e7	+	6.64101e6i	−0.164769	+	0.0215291i
$78$		0			0
$79$	−2.51276e7	−	4.35223e7i	−0.0725821	−	0.125716i	0.827450	−	0.561539i	$-0.189791\pi$
$79$	−2.51276e7	−	4.35223e7i	−0.0725821	−	0.125716i	−0.900032	+	0.435823i	$0.856457\pi$
$80$		0			0
$81$	−2.15234e7	+	3.72796e7i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−3.60147e8			−0.832968			−0.416484	−	0.909143i	$-0.636738\pi$
$83$	−3.60147e8			−0.832968			−0.416484	+	0.909143i	$0.636738\pi$
$84$		0			0
$85$	5.14983e8			1.07006
$86$		0			0
$87$	2.18959e8	−	3.79248e8i	0.409757	−	0.709720i
$88$		0			0
$89$	−2.00857e8	−	3.47894e8i	−0.339337	−	0.587749i	0.644971	−	0.764207i	$-0.276869\pi$
$89$	−2.00857e8	−	3.47894e8i	−0.339337	−	0.587749i	−0.984308	+	0.176458i	$0.943536\pi$
$90$		0			0
$91$	−3.92200e8	−	5.12156e8i	−0.599544	−	0.782918i
$92$		0			0
$93$	6.12522e6	+	1.06092e7i	0.00849079	+	0.0147065i
$94$		0			0
$95$	3.24952e8	−	5.62833e8i	0.409320	−	0.708963i
$96$		0			0
$97$	1.53266e9			1.75781			0.878907	−	0.476993i	$-0.158273\pi$
$97$	1.53266e9			1.75781			0.878907	+	0.476993i	$0.158273\pi$
$98$		0			0
$99$	5.29406e7			0.0553899
$100$		0			0
$101$	−1.57903e8	+	2.73496e8i	−0.150989	+	0.261520i	−0.931591	−	0.363508i	$-0.881579\pi$
$101$	−1.57903e8	+	2.73496e8i	−0.150989	+	0.261520i	0.780603	+	0.625028i	$0.214912\pi$
$102$		0			0
$103$	−6.69659e8	−	1.15988e9i	−0.586255	−	1.01542i	−0.994718	−	0.102648i	$-0.967268\pi$
$103$	−6.69659e8	−	1.15988e9i	−0.586255	−	1.01542i	0.408463	−	0.912775i	$-0.366065\pi$
$104$		0			0
$105$	−5.44566e8	−	7.11125e8i	−0.437219	−	0.570945i
$106$		0			0
$107$	−1.45331e8	−	2.51720e8i	−0.107184	−	0.185649i	0.807444	−	0.589944i	$-0.200850\pi$
$107$	−1.45331e8	−	2.51720e8i	−0.107184	−	0.185649i	−0.914629	+	0.404295i	$0.867517\pi$
$108$		0			0
$109$	1.18214e8	−	2.04752e8i	0.0802136	−	0.138934i	−0.823128	−	0.567856i	$-0.807773\pi$
$109$	1.18214e8	−	2.04752e8i	0.0802136	−	0.138934i	0.903342	+	0.428922i	$0.141106\pi$
$110$		0			0
$111$	−1.27921e7			−0.00799815
$112$		0			0
$113$	−2.33497e9			−1.34719			−0.673594	−	0.739102i	$-0.735250\pi$
$113$	−2.33497e9			−1.34719			−0.673594	+	0.739102i	$0.735250\pi$
$114$		0			0
$115$	2.08423e9	−	3.61000e9i	1.11124	−	1.92472i
$116$		0			0
$117$	3.33127e8	+	5.76994e8i	0.164352	+	0.284665i
$118$		0			0
$119$	1.86350e9	−	2.43489e8i	0.851859	−	0.111306i
$120$		0			0
$121$	1.14642e9	+	1.98566e9i	0.486194	+	0.842112i
$122$		0			0
$123$	−5.40566e8	+	9.36288e8i	−0.212949	+	0.368839i
$124$		0			0
$125$	1.52512e9			0.558740
$126$		0			0
$127$	−2.47087e9			−0.842818			−0.421409	−	0.906871i	$-0.638464\pi$
$127$	−2.47087e9			−0.842818			−0.421409	+	0.906871i	$0.638464\pi$
$128$		0			0
$129$	5.35804e8	−	9.28039e8i	0.170354	−	0.295061i
$130$		0			0
$131$	−1.21015e9	−	2.09603e9i	−0.359018	−	0.621838i	0.628779	−	0.777584i	$-0.283555\pi$
$131$	−1.21015e9	−	2.09603e9i	−0.359018	−	0.621838i	−0.987797	+	0.155746i	$0.950222\pi$
$132$		0			0
$133$	9.09746e8	−	2.19029e9i	0.252109	−	0.606972i
$134$		0			0
$135$	4.62545e8	+	8.01151e8i	0.119854	+	0.207593i
$136$		0			0
$137$	3.09180e9	−	5.35515e9i	0.749840	−	1.29876i	−0.198059	−	0.980190i	$-0.563464\pi$
$137$	3.09180e9	−	5.35515e9i	0.749840	−	1.29876i	0.947899	−	0.318571i	$-0.103203\pi$
$138$		0			0
$139$	−4.40604e9			−1.00111			−0.500555	−	0.865705i	$-0.666871\pi$
$139$	−4.40604e9			−1.00111			−0.500555	+	0.865705i	$0.666871\pi$
$140$		0			0
$141$	−6.90743e8			−0.147174
$142$		0			0
$143$	4.09694e8	−	7.09610e8i	0.0819308	−	0.141908i
$144$		0			0
$145$	−4.70551e9	−	8.15019e9i	−0.883997	−	1.53113i
$146$		0			0
$147$	−2.30677e9	−	2.31577e9i	−0.407453	−	0.409042i
$148$		0			0
$149$	−2.50780e9	−	4.34365e9i	−0.416827	−	0.721965i	0.578792	−	0.815476i	$-0.303524\pi$
$149$	−2.50780e9	−	4.34365e9i	−0.416827	−	0.721965i	−0.995618	+	0.0935105i	$0.970191\pi$
$150$		0			0
$151$	−3.89899e9	+	6.75325e9i	−0.610318	+	1.05710i	0.380869	+	0.924629i	$0.375625\pi$
$151$	−3.89899e9	+	6.75325e9i	−0.610318	+	1.05710i	−0.991187	+	0.132473i	$0.957708\pi$
$152$		0			0
$153$	−1.94104e9			−0.286367
$154$		0			0
$155$	2.63266e8			0.0366356
$156$		0			0
$157$	8.52391e8	−	1.47638e9i	0.111967	−	0.193933i	−0.804596	−	0.593822i	$-0.797618\pi$
$157$	8.52391e8	−	1.47638e9i	0.111967	−	0.193933i	0.916563	+	0.399890i	$0.130952\pi$
$158$		0			0
$159$	8.77333e8	+	1.51959e9i	0.108862	+	0.188555i
$160$		0			0
$161$	5.83509e9	−	1.40485e10i	0.684433	−	1.64783i
$162$		0			0
$163$	−6.09839e9	−	1.05627e10i	−0.676661	−	1.17201i	−0.975980	−	0.217858i	$-0.930093\pi$
$163$	−6.09839e9	−	1.05627e10i	−0.676661	−	1.17201i	0.299320	−	0.954153i	$-0.403240\pi$
$164$		0			0
$165$	5.68856e8	−	9.85288e8i	0.0597482	−	0.103487i
$166$		0			0
$167$	1.59715e10			1.58899			0.794497	−	0.607268i	$-0.207735\pi$
$167$	1.59715e10			1.58899			0.794497	+	0.607268i	$0.207735\pi$
$168$		0			0
$169$	−2.92548e8			−0.0275872
$170$		0			0
$171$	−1.22479e9	+	2.12139e9i	−0.109541	+	0.189731i
$172$		0			0
$173$	−2.76506e9	−	4.78922e9i	−0.234691	−	0.406497i	0.724492	−	0.689284i	$-0.242075\pi$
$173$	−2.76506e9	−	4.78922e9i	−0.234691	−	0.406497i	−0.959183	+	0.282786i	$0.908741\pi$
$174$		0			0
$175$	−6.78380e9	+	8.86386e8i	−0.546766	+	0.0714417i
$176$		0			0
$177$	−4.85557e9	−	8.41009e9i	−0.371844	−	0.644052i
$178$		0			0
$179$	−5.38489e9	+	9.32690e9i	−0.392047	+	0.679046i	−0.992719	−	0.120450i	$-0.961566\pi$
$179$	−5.38489e9	+	9.32690e9i	−0.392047	+	0.679046i	0.600672	+	0.799495i	$0.294900\pi$
$180$		0			0
$181$	2.16188e10			1.49719			0.748596	−	0.663026i	$-0.230728\pi$
$181$	2.16188e10			1.49719			0.748596	+	0.663026i	$0.230728\pi$
$182$		0			0
$183$	1.02193e10			0.673581
$184$		0			0
$185$	−1.37454e8	+	2.38077e8i	−0.00862748	+	0.0149432i
$186$		0			0
$187$	1.19358e9	+	2.06735e9i	0.0713781	+	0.123631i
$188$		0			0
$189$	2.05254e9	+	2.68032e9i	0.117007	+	0.152795i
$190$		0			0
$191$	1.44717e10	+	2.50658e10i	0.786810	+	1.36280i	0.927912	+	0.372800i	$0.121602\pi$
$191$	1.44717e10	+	2.50658e10i	0.786810	+	1.36280i	−0.141101	+	0.989995i	$0.545064\pi$
$192$		0			0
$193$	−2.75358e9	+	4.76934e9i	−0.142853	+	0.247429i	−0.928570	−	0.371158i	$-0.878961\pi$
$193$	−2.75358e9	+	4.76934e9i	−0.142853	+	0.247429i	0.785717	+	0.618586i	$0.212294\pi$
$194$		0			0
$195$	1.43181e10			0.709134
$196$		0			0
$197$	2.79298e10			1.32120			0.660601	−	0.750737i	$-0.270302\pi$
$197$	2.79298e10			1.32120			0.660601	+	0.750737i	$0.270302\pi$
$198$		0			0
$199$	7.62692e9	−	1.32102e10i	0.344755	−	0.597133i	−0.640554	−	0.767913i	$-0.721295\pi$
$199$	7.62692e9	−	1.32102e10i	0.344755	−	0.597133i	0.985309	+	0.170780i	$0.0546287\pi$
$200$		0			0
$201$	2.48821e9	+	4.30970e9i	0.107524	+	0.186237i
$202$		0			0
$203$	−2.08807e10	−	2.72672e10i	−0.863004	−	1.12696i
$204$		0			0
$205$	1.16170e10	+	2.01212e10i	0.459410	+	0.795721i
$206$		0			0
$207$	−7.85575e9	+	1.36066e10i	−0.297386	+	0.515088i
$208$		0			0
$209$	3.01258e9			0.109215
$210$		0			0
$211$	−4.37026e10			−1.51788			−0.758939	−	0.651162i	$-0.774282\pi$
$211$	−4.37026e10			−1.51788			−0.758939	+	0.651162i	$0.774282\pi$
$212$		0			0
$213$	−1.14718e9	+	1.98698e9i	−0.0381878	+	0.0661433i
$214$		0			0
$215$	−1.15146e10	−	1.99439e10i	−0.367516	−	0.636557i
$216$		0			0
$217$	9.52646e8	−	1.24475e8i	0.0291651	−	0.00381077i
$218$		0			0
$219$	3.16619e9	+	5.48401e9i	0.0930120	+	0.161102i
$220$		0			0
$221$	−1.50212e10	+	2.60175e10i	−0.423583	+	0.733668i
$222$		0			0
$223$	−1.83190e10			−0.496056			−0.248028	−	0.968753i	$-0.579783\pi$
$223$	−1.83190e10			−0.496056			−0.248028	+	0.968753i	$0.579783\pi$
$224$		0			0
$225$	7.06607e9			0.183805
$226$		0			0
$227$	−2.36748e10	+	4.10060e10i	−0.591793	+	1.02502i	0.402197	+	0.915553i	$0.368247\pi$
$227$	−2.36748e10	+	4.10060e10i	−0.591793	+	1.02502i	−0.993991	+	0.109463i	$0.965087\pi$
$228$		0			0
$229$	−2.94494e10	−	5.10079e10i	−0.707648	−	1.22568i	−0.965728	−	0.259558i	$-0.916423\pi$
$229$	−2.94494e10	−	5.10079e10i	−0.707648	−	1.22568i	0.258080	−	0.966124i	$-0.416910\pi$
$230$		0			0
$231$	1.59259e9	−	3.83429e9i	0.0368002	−	0.0885995i
$232$		0			0
$233$	4.02159e10	+	6.96560e10i	0.893915	+	1.54831i	0.835142	+	0.550034i	$0.185385\pi$
$233$	4.02159e10	+	6.96560e10i	0.893915	+	1.54831i	0.0587724	+	0.998271i	$0.481281\pi$
$234$		0			0
$235$	−7.42216e9	+	1.28556e10i	−0.158754	+	0.274970i
$236$		0			0
$237$	4.07067e9			0.0838106
$238$		0			0
$239$	−4.92303e10			−0.975983			−0.487991	−	0.872848i	$-0.662270\pi$
$239$	−4.92303e10			−0.975983			−0.487991	+	0.872848i	$0.662270\pi$
$240$		0			0
$241$	1.06660e10	−	1.84740e10i	0.203668	−	0.352764i	−0.746039	−	0.665902i	$-0.768047\pi$
$241$	1.06660e10	−	1.84740e10i	0.203668	−	0.352764i	0.949708	+	0.313138i	$0.101380\pi$
$242$		0			0
$243$	−1.74339e9	−	3.01964e9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	−6.78861e10	+	1.80484e10i	−1.20374	+	0.320031i
$246$		0			0
$247$	1.89566e10	+	3.28338e10i	0.324059	+	0.561287i
$248$		0			0
$249$	1.45860e10	−	2.52636e10i	0.240457	−	0.416484i
$250$		0			0
$251$	−8.96706e10			−1.42600			−0.712998	−	0.701166i	$-0.752663\pi$
$251$	−8.96706e10			−1.42600			−0.712998	+	0.701166i	$0.752663\pi$
$252$		0			0
$253$	1.93226e10			0.296499
$254$		0			0
$255$	−2.08568e10	+	3.61250e10i	−0.308899	+	0.535029i
$256$		0			0
$257$	4.97451e10	+	8.61611e10i	0.711298	+	1.23200i	0.964370	+	0.264556i	$0.0852255\pi$
$257$	4.97451e10	+	8.61611e10i	0.711298	+	1.23200i	−0.253073	+	0.967447i	$0.581441\pi$
$258$		0			0
$259$	−3.84820e8	+	9.26487e8i	−0.00531385	+	0.0127935i
$260$		0			0
$261$	1.77357e10	+	3.07191e10i	0.236573	+	0.409757i
$262$		0			0
$263$	−5.17652e10	+	8.96599e10i	−0.667170	+	1.15557i	0.311522	+	0.950239i	$0.399161\pi$
$263$	−5.17652e10	+	8.96599e10i	−0.667170	+	1.15557i	−0.978692	+	0.205334i	$0.934172\pi$
$264$		0			0
$265$	3.77084e10			0.469712
$266$		0			0
$267$	3.25388e10			0.391833
$268$		0			0
$269$	2.37440e10	−	4.11258e10i	0.276483	−	0.478883i	−0.694025	−	0.719951i	$-0.744164\pi$
$269$	2.37440e10	−	4.11258e10i	0.276483	−	0.478883i	0.970508	+	0.241068i	$0.0774977\pi$
$270$		0			0
$271$	5.84156e10	+	1.01179e11i	0.657911	+	1.13954i	0.981155	+	0.193220i	$0.0618932\pi$
$271$	5.84156e10	+	1.01179e11i	0.657911	+	1.13954i	−0.323244	+	0.946316i	$0.604773\pi$
$272$		0			0
$273$	5.18108e10	−	6.76972e9i	0.564532	−	0.0737630i
$274$		0			0
$275$	−4.34507e9	−	7.52588e9i	−0.0458141	−	0.0793524i
$276$		0			0
$277$	−2.45348e10	+	4.24955e10i	−0.250393	+	0.433694i	−0.963634	−	0.267225i	$-0.913893\pi$
$277$	−2.45348e10	+	4.24955e10i	−0.250393	+	0.433694i	0.713241	+	0.700919i	$0.247227\pi$
$278$		0			0
$279$	−9.92285e8			−0.00980432
$280$		0			0
$281$	1.91091e11			1.82836			0.914180	−	0.405309i	$-0.132836\pi$
$281$	1.91091e11			1.82836			0.914180	+	0.405309i	$0.132836\pi$
$282$		0			0
$283$	−4.18856e10	+	7.25481e10i	−0.388174	+	0.672337i	−0.992204	−	0.124625i	$-0.960227\pi$
$283$	−4.18856e10	+	7.25481e10i	−0.388174	+	0.672337i	0.604030	+	0.796961i	$0.293561\pi$
$284$		0			0
$285$	2.63211e10	+	4.55895e10i	0.236321	+	0.409320i
$286$		0			0
$287$	5.15502e10	+	6.73172e10i	0.448500	+	0.585676i
$288$		0			0
$289$	1.55319e10	+	2.69021e10i	0.130974	+	0.226853i
$290$		0			0
$291$	−6.20727e10	+	1.07513e11i	−0.507437	+	0.878907i
$292$		0			0
$293$	4.77398e10			0.378422			0.189211	−	0.981936i	$-0.439407\pi$
$293$	4.77398e10			0.378422			0.189211	+	0.981936i	$0.439407\pi$
$294$		0			0
$295$	−2.08696e11			−1.60441
$296$		0			0
$297$	−2.14409e9	+	3.71368e9i	−0.0159897	+	0.0276949i
$298$		0			0
$299$	1.21587e11	+	2.10595e11i	0.879767	+	1.52380i
$300$		0			0
$301$	−5.10960e10	−	6.67241e10i	−0.358788	−	0.468526i
$302$		0			0
$303$	−1.27901e10	−	2.21532e10i	−0.0871733	−	0.150989i
$304$		0			0
$305$	1.09808e11	−	1.90193e11i	0.726581	−	1.25848i
$306$		0			0
$307$	−2.73316e11			−1.75607			−0.878037	−	0.478593i	$-0.841147\pi$
$307$	−2.73316e11			−1.75607			−0.878037	+	0.478593i	$0.841147\pi$
$308$		0			0
$309$	1.08485e11			0.676949
$310$		0			0
$311$	8.09282e10	−	1.40172e11i	0.490544	−	0.849647i	−0.509397	−	0.860532i	$-0.670131\pi$
$311$	8.09282e10	−	1.40172e11i	0.490544	−	0.849647i	0.999941	+	0.0108847i	$0.00346478\pi$
$312$		0			0
$313$	−1.40147e11	−	2.42741e11i	−0.825341	−	1.42953i	−0.901658	−	0.432450i	$-0.857649\pi$
$313$	−1.40147e11	−	2.42741e11i	−0.825341	−	1.42953i	0.0763168	−	0.997084i	$-0.475684\pi$
$314$		0			0
$315$	7.19390e10	−	9.39970e9i	0.411686	−	0.0537919i
$316$		0			0
$317$	2.54780e10	+	4.41293e10i	0.141710	+	0.245448i	0.928141	−	0.372230i	$-0.121407\pi$
$317$	2.54780e10	+	4.41293e10i	0.141710	+	0.245448i	−0.786431	+	0.617678i	$0.788073\pi$
$318$		0			0
$319$	2.18121e10	−	3.77796e10i	0.117934	−	0.204268i
$320$		0			0
$321$	2.35436e10			0.123766
$322$		0			0
$323$	−1.10455e11			−0.564641
$324$		0			0
$325$	5.46825e10	−	9.47128e10i	0.271877	−	0.470906i
$326$		0			0
$327$	9.57530e9	+	1.65849e10i	0.0463113	+	0.0802136i
$328$		0			0
$329$	−2.07793e10	+	5.00279e10i	−0.0977800	+	0.235413i
$330$		0			0
$331$	−1.07934e11	−	1.86947e11i	−0.494232	−	0.856036i	0.505745	−	0.862683i	$-0.331217\pi$
$331$	−1.07934e11	−	1.86947e11i	−0.494232	−	0.856036i	−0.999978	+	0.00664705i	$0.997884\pi$
$332$		0			0
$333$	5.18081e8	−	8.97343e8i	0.00230887	−	0.00399908i
$334$		0			0
$335$	1.06945e11			0.463937
$336$		0			0
$337$	−8.47199e10			−0.357809			−0.178904	−	0.983866i	$-0.557255\pi$
$337$	−8.47199e10			−0.357809			−0.178904	+	0.983866i	$0.557255\pi$
$338$		0			0
$339$	9.45662e10	−	1.63793e11i	0.388899	−	0.673594i
$340$		0			0
$341$	6.10176e8	+	1.05686e9i	0.00244377	+	0.00423274i
$342$		0			0
$343$	−2.37117e11	+	9.74066e10i	−0.924993	+	0.379983i
$344$		0			0
$345$	1.68823e11	+	2.92410e11i	0.641572	+	1.11124i
$346$		0			0
$347$	1.37707e11	−	2.38516e11i	0.509887	−	0.883150i	−0.490048	−	0.871696i	$-0.663021\pi$
$347$	1.37707e11	−	2.38516e11i	0.509887	−	0.883150i	0.999934	−	0.0114542i	$-0.00364606\pi$
$348$		0			0
$349$	−5.02551e11			−1.81328			−0.906642	−	0.421902i	$-0.861363\pi$
$349$	−5.02551e11			−1.81328			−0.906642	+	0.421902i	$0.861363\pi$
$350$		0			0
$351$	−5.39667e10			−0.189777
$352$		0			0
$353$	−3.20587e10	+	5.55274e10i	−0.109891	+	0.190336i	−0.915726	−	0.401804i	$-0.868383\pi$
$353$	−3.20587e10	+	5.55274e10i	−0.109891	+	0.190336i	0.805835	+	0.592140i	$0.201717\pi$
$354$		0			0
$355$	2.46534e10	+	4.27010e10i	0.0823852	+	0.142695i
$356$		0			0
$357$	−5.83914e10	+	1.40582e11i	−0.190258	+	0.458061i
$358$		0			0
$359$	1.50375e10	+	2.60457e10i	0.0477805	+	0.0827583i	0.888927	−	0.458050i	$-0.151452\pi$
$359$	1.50375e10	+	2.60457e10i	0.0477805	+	0.0827583i	−0.841146	+	0.540808i	$0.818119\pi$
$360$		0			0
$361$	9.16475e10	−	1.58738e11i	0.284013	−	0.491925i
$362$		0			0
$363$	−1.85720e11			−0.561408
$364$		0			0
$365$	1.36085e11			0.401322
$366$		0			0
$367$	2.38727e11	−	4.13487e11i	0.686917	−	1.18978i	−0.285913	−	0.958256i	$-0.592297\pi$
$367$	2.38727e11	−	4.13487e11i	0.686917	−	1.18978i	0.972830	−	0.231520i	$-0.0743698\pi$
$368$		0			0
$369$	−4.37859e10	−	7.58393e10i	−0.122946	−	0.212949i
$370$		0			0
$371$	1.36450e11	−	1.78289e10i	0.373932	−	0.0488587i
$372$		0			0
$373$	−6.14295e10	−	1.06399e11i	−0.164319	−	0.284608i	0.772094	−	0.635508i	$-0.219209\pi$
$373$	−6.14295e10	−	1.06399e11i	−0.164319	−	0.284608i	−0.936413	+	0.350899i	$0.885876\pi$
$374$		0			0
$375$	−6.17674e10	+	1.06984e11i	−0.161294	+	0.279370i
$376$		0			0
$377$	5.49008e11			1.39972
$378$		0			0
$379$	5.44230e11			1.35490			0.677448	−	0.735571i	$-0.263086\pi$
$379$	5.44230e11			1.35490			0.677448	+	0.735571i	$0.263086\pi$
$380$		0			0
$381$	1.00070e11	−	1.73327e11i	0.243300	−	0.421409i
$382$		0			0
$383$	3.36799e10	+	5.83354e10i	0.0799791	+	0.138528i	0.903241	−	0.429134i	$-0.141181\pi$
$383$	3.36799e10	+	5.83354e10i	0.0799791	+	0.138528i	−0.823262	+	0.567662i	$0.807848\pi$
$384$		0			0
$385$	−5.42481e10	−	7.08402e10i	−0.125838	−	0.164326i
$386$		0			0
$387$	4.34001e10	+	7.51712e10i	0.0983538	+	0.170354i
$388$		0			0
$389$	7.92837e10	−	1.37323e11i	0.175554	−	0.304068i	−0.764799	−	0.644269i	$-0.777162\pi$
$389$	7.92837e10	−	1.37323e11i	0.175554	−	0.304068i	0.940353	+	0.340201i	$0.110495\pi$
$390$		0			0
$391$	−7.08453e11			−1.53291
$392$		0			0
$393$	1.96044e11			0.414559
$394$		0			0
$395$	4.37401e10	−	7.57602e10i	0.0904052	−	0.156586i
$396$		0			0
$397$	−3.13065e11	−	5.42244e11i	−0.632524	−	1.09556i	−0.987034	−	0.160512i	$-0.948686\pi$
$397$	−3.13065e11	−	5.42244e11i	−0.632524	−	1.09556i	0.354510	−	0.935052i	$-0.384648\pi$
$398$		0			0
$399$	1.16800e11	+	1.52524e11i	0.230709	+	0.301272i
$400$		0			0
$401$	−1.99233e11	−	3.45082e11i	−0.384780	−	0.666458i	0.606959	−	0.794733i	$-0.292389\pi$
$401$	−1.99233e11	−	3.45082e11i	−0.384780	−	0.666458i	−0.991739	+	0.128275i	$0.959056\pi$
$402$		0			0
$403$	−7.67904e9	+	1.33005e10i	−0.0145022	+	0.0251186i
$404$		0			0
$405$	−7.49323e10			−0.138395
$406$		0			0
$407$	−1.27431e9			−0.00230198
$408$		0			0
$409$	6.25040e10	−	1.08260e11i	0.110447	−	0.191300i	−0.805504	−	0.592591i	$-0.798105\pi$
$409$	6.25040e10	−	1.08260e11i	0.110447	−	0.191300i	0.915951	+	0.401291i	$0.131438\pi$
$410$		0			0
$411$	2.50436e11	+	4.33767e11i	0.432920	+	0.749840i
$412$		0			0
$413$	−7.55180e11	+	9.86734e10i	−1.27725	+	0.166888i
$414$		0			0
$415$	−3.13457e11	−	5.42924e11i	−0.518755	−	0.898510i
$416$		0			0
$417$	1.78445e11	−	3.09075e11i	0.288996	−	0.500555i
$418$		0			0
$419$	−9.13685e11			−1.44822			−0.724108	−	0.689687i	$-0.757748\pi$
$419$	−9.13685e11			−1.44822			−0.724108	+	0.689687i	$0.757748\pi$
$420$		0			0
$421$	6.16305e11			0.956150			0.478075	−	0.878319i	$-0.341335\pi$
$421$	6.16305e11			0.956150			0.478075	+	0.878319i	$0.341335\pi$
$422$		0			0
$423$	2.79751e10	−	4.84543e10i	0.0424854	−	0.0735869i
$424$		0			0
$425$	1.59309e11	+	2.75932e11i	0.236860	+	0.410253i
$426$		0			0
$427$	3.07422e11	−	7.40143e11i	0.447517	−	1.07743i
$428$		0			0
$429$	3.31852e10	+	5.74784e10i	0.0473028	+	0.0819308i
$430$		0			0
$431$	5.74428e11	−	9.94938e11i	0.801840	−	1.38883i	−0.116564	−	0.993183i	$-0.537188\pi$
$431$	5.74428e11	−	9.94938e11i	0.801840	−	1.38883i	0.918404	−	0.395644i	$-0.129479\pi$
$432$		0			0
$433$	−9.49214e11			−1.29768			−0.648842	−	0.760923i	$-0.724746\pi$
$433$	−9.49214e11			−1.29768			−0.648842	+	0.760923i	$0.724746\pi$
$434$		0			0
$435$	7.62293e11			1.02075
$436$		0			0
$437$	−4.47031e11	+	7.74280e11i	−0.586369	+	1.01562i
$438$		0			0
$439$	−4.95644e11	−	8.58481e11i	−0.636912	−	1.10316i	−0.986107	−	0.166114i	$-0.946878\pi$
$439$	−4.95644e11	−	8.58481e11i	−0.636912	−	1.10316i	0.349194	−	0.937050i	$-0.386455\pi$
$440$		0			0
$441$	2.55871e11	−	6.80269e10i	0.322143	−	0.0856459i
$442$		0			0
$443$	5.83796e11	+	1.01116e12i	0.720185	+	1.24740i	0.960925	+	0.276808i	$0.0892765\pi$
$443$	5.83796e11	+	1.01116e12i	0.720185	+	1.24740i	−0.240740	+	0.970590i	$0.577390\pi$
$444$		0			0
$445$	3.49635e11	−	6.05586e11i	0.422664	−	0.732075i
$446$		0			0
$447$	4.06264e11			0.481310
$448$		0			0
$449$	−8.40277e11			−0.975695			−0.487847	−	0.872929i	$-0.662218\pi$
$449$	−8.40277e11			−0.975695			−0.487847	+	0.872929i	$0.662218\pi$
$450$		0			0
$451$	−5.38496e10	+	9.32703e10i	−0.0612898	+	0.106157i
$452$		0			0
$453$	−3.15818e11	−	5.47013e11i	−0.352367	−	0.610318i
$454$		0			0
$455$	4.30724e11	−	1.03700e12i	0.471138	−	1.13430i
$456$		0			0
$457$	−7.76611e11	−	1.34513e12i	−0.832876	−	1.44258i	−0.895748	−	0.444563i	$-0.853359\pi$
$457$	−7.76611e11	−	1.34513e12i	−0.832876	−	1.44258i	0.0628714	−	0.998022i	$-0.479974\pi$
$458$		0			0
$459$	7.86120e10	−	1.36160e11i	0.0826669	−	0.143183i
$460$		0			0
$461$	5.58274e11			0.575696			0.287848	−	0.957676i	$-0.407060\pi$
$461$	5.58274e11			0.575696			0.287848	+	0.957676i	$0.407060\pi$
$462$		0			0
$463$	−1.16047e12			−1.17359			−0.586797	−	0.809734i	$-0.699611\pi$
$463$	−1.16047e12			−1.17359			−0.586797	+	0.809734i	$0.699611\pi$
$464$		0			0
$465$	−1.06623e10	+	1.84676e10i	−0.0105758	+	0.0183178i
$466$		0			0
$467$	4.99089e11	+	8.64448e11i	0.485571	+	0.841033i	0.999863	−	0.0165822i	$-0.00527852\pi$
$467$	4.99089e11	+	8.64448e11i	0.485571	+	0.841033i	−0.514292	+	0.857615i	$0.671945\pi$
$468$		0			0
$469$	3.86987e11	−	5.05646e10i	0.369334	−	0.0482580i
$470$		0			0
$471$	6.90436e10	+	1.19587e11i	0.0646442	+	0.111967i
$472$		0			0
$473$	5.33752e10	−	9.24485e10i	0.0490303	−	0.0849229i
$474$		0			0
$475$	4.02094e11			0.362415
$476$		0			0
$477$	−1.42128e11			−0.125703
$478$		0			0
$479$	4.77744e11	−	8.27476e11i	0.414653	−	0.718200i	−0.580739	−	0.814090i	$-0.697236\pi$
$479$	4.77744e11	−	8.27476e11i	0.414653	−	0.718200i	0.995392	+	0.0958896i	$0.0305696\pi$
$480$		0			0
$481$	−8.01860e9	−	1.38886e10i	−0.00683039	−	0.0118306i
$482$		0			0
$483$	7.49151e11	+	9.78283e11i	0.626336	+	0.817904i
$484$		0			0
$485$	1.33397e12	+	2.31050e12i	1.09473	+	1.89613i
$486$		0			0
$487$	−5.58057e11	+	9.66584e11i	−0.449571	+	0.778680i	−0.998358	−	0.0572820i	$-0.981757\pi$
$487$	−5.58057e11	+	9.66584e11i	−0.449571	+	0.778680i	0.548787	+	0.835962i	$0.315090\pi$
$488$		0			0
$489$	9.87939e11			0.781341
$490$		0			0
$491$	−7.05975e11			−0.548179			−0.274090	−	0.961704i	$-0.588376\pi$
$491$	−7.05975e11			−0.548179			−0.274090	+	0.961704i	$0.588376\pi$
$492$		0			0
$493$	−7.99727e11	+	1.38517e12i	−0.609720	+	1.05607i
$494$		0			0
$495$	4.60774e10	+	7.98084e10i	0.0344956	+	0.0597482i
$496$		0			0
$497$	1.09399e11	+	1.42860e11i	0.0804287	+	0.105028i
$498$		0			0
$499$	1.98891e11	+	3.44490e11i	0.143603	+	0.248727i	0.928851	−	0.370454i	$-0.120798\pi$
$499$	1.98891e11	+	3.44490e11i	0.143603	+	0.248727i	−0.785248	+	0.619181i	$0.787465\pi$
$500$		0			0
$501$	−6.46847e11	+	1.12037e12i	−0.458703	+	0.794497i
$502$		0			0
$503$	4.78950e11			0.333606			0.166803	−	0.985990i	$-0.446656\pi$
$503$	4.78950e11			0.333606			0.166803	+	0.985990i	$0.446656\pi$
$504$		0			0
$505$	−5.49729e11			−0.376130
$506$		0			0
$507$	1.18482e10	−	2.05217e10i	0.00796373	−	0.0137936i
$508$		0			0
$509$	2.36631e11	+	4.09857e11i	0.156258	+	0.270646i	0.933516	−	0.358535i	$-0.116724\pi$
$509$	2.36631e11	+	4.09857e11i	0.156258	+	0.270646i	−0.777259	+	0.629181i	$0.783390\pi$
$510$		0			0
$511$	4.92434e11	−	6.43424e10i	0.319487	−	0.0417449i
$512$		0			0
$513$	−9.92077e10	−	1.71833e11i	−0.0632437	−	0.109541i
$514$		0			0
$515$	1.16569e12	−	2.01903e12i	0.730214	−	1.26477i
$516$		0			0
$517$	−6.88098e10			−0.0423587
$518$		0			0
$519$	4.47939e11			0.270998
$520$		0			0
$521$	−3.96295e11	+	6.86403e11i	−0.235640	+	0.408140i	−0.959458	−	0.281850i	$-0.909052\pi$
$521$	−3.96295e11	+	6.86403e11i	−0.235640	+	0.408140i	0.723819	+	0.689990i	$0.242385\pi$
$522$		0			0
$523$	−1.77672e11	−	3.07738e11i	−0.103839	−	0.179855i	0.809424	−	0.587225i	$-0.199779\pi$
$523$	−1.77672e11	−	3.07738e11i	−0.103839	−	0.179855i	−0.913263	+	0.407369i	$0.866446\pi$
$524$		0			0
$525$	2.12566e11	−	5.11769e11i	0.122117	−	0.294007i
$526$		0			0
$527$	−2.23718e10	−	3.87490e10i	−0.0126343	−	0.0218833i
$528$		0			0
$529$	−1.96667e12	+	3.40637e12i	−1.09189	+	1.89122i
$530$		0			0
$531$	7.86602e11			0.429368
$532$		0			0
$533$	−1.35539e12			−0.727431
$534$		0			0
$535$	2.52980e11	−	4.38175e11i	0.133504	−	0.231236i
$536$		0			0
$537$	−4.36176e11	−	7.55479e11i	−0.226349	−	0.392047i
$538$		0			0
$539$	−2.29794e11	−	2.30691e11i	−0.117271	−	0.117728i
$540$		0			0
$541$	9.06573e11	+	1.57023e12i	0.455004	+	0.788090i	0.998688	−	0.0511998i	$-0.0163045\pi$
$541$	9.06573e11	+	1.57023e12i	0.455004	+	0.788090i	−0.543685	+	0.839290i	$0.682971\pi$
$542$		0			0
$543$	−8.75560e11	+	1.51652e12i	−0.432202	+	0.748596i
$544$		0			0
$545$	4.11553e11			0.199821
$546$		0			0
$547$	−2.35693e12			−1.12565			−0.562826	−	0.826576i	$-0.690286\pi$
$547$	−2.35693e12			−1.12565			−0.562826	+	0.826576i	$0.690286\pi$
$548$		0			0
$549$	−4.13880e11	+	7.16861e11i	−0.194446	+	0.336790i
$550$		0			0
$551$	1.00925e12	+	1.74807e12i	0.466462	+	0.807935i
$552$		0			0
$553$	1.22456e11	−	2.94824e11i	0.0556825	−	0.134060i
$554$		0			0
$555$	−1.11338e10	−	1.92842e10i	−0.00498108	−	0.00862748i
$556$		0			0
$557$	−1.08370e12	+	1.87703e12i	−0.477048	+	0.826271i	−0.999654	−	0.0263031i	$-0.991627\pi$
$557$	−1.08370e12	+	1.87703e12i	−0.477048	+	0.826271i	0.522606	+	0.852574i	$0.324960\pi$
$558$		0			0
$559$	1.34345e12			0.581926
$560$		0			0
$561$	−1.93360e11			−0.0824204
$562$		0			0
$563$	−1.03154e12	+	1.78667e12i	−0.432709	+	0.749475i	−0.997106	−	0.0760295i	$-0.975776\pi$
$563$	−1.03154e12	+	1.78667e12i	−0.432709	+	0.749475i	0.564396	+	0.825504i	$0.309109\pi$
$564$		0			0
$565$	−2.03226e12	−	3.51998e12i	−0.839000	−	1.45319i
$566$		0			0
$567$	−2.71147e11	+	3.54287e10i	−0.110175	+	0.0143957i
$568$		0			0
$569$	1.39414e12	+	2.41472e12i	0.557572	+	0.965743i	0.997698	+	0.0678074i	$0.0216004\pi$
$569$	1.39414e12	+	2.41472e12i	0.557572	+	0.965743i	−0.440126	+	0.897936i	$0.645066\pi$
$570$		0			0
$571$	−3.98091e11	+	6.89514e11i	−0.156718	+	0.271444i	−0.933683	−	0.358100i	$-0.883425\pi$
$571$	−3.98091e11	+	6.89514e11i	−0.156718	+	0.271444i	0.776965	+	0.629544i	$0.216758\pi$
$572$		0			0
$573$	−2.34442e12			−0.908530
$574$		0			0
$575$	2.57902e12			0.983898
$576$		0			0
$577$	−1.51207e12	+	2.61898e12i	−0.567910	+	0.983649i	0.428862	+	0.903370i	$0.358915\pi$
$577$	−1.51207e12	+	2.61898e12i	−0.567910	+	0.983649i	−0.996772	+	0.0802795i	$0.974419\pi$
$578$		0			0
$579$	−2.23040e11	−	3.86316e11i	−0.0824763	−	0.142853i
$580$		0			0
$581$	−1.39097e12	−	1.81640e12i	−0.506435	−	0.661331i
$582$		0			0
$583$	8.73974e10	+	1.51377e11i	0.0313321	+	0.0542688i
$584$		0			0
$585$	−5.79882e11	+	1.00438e12i	−0.204709	+	0.354567i
$586$		0			0
$587$	1.63282e12			0.567632			0.283816	−	0.958879i	$-0.408400\pi$
$587$	1.63282e12			0.567632			0.283816	+	0.958879i	$0.408400\pi$
$588$		0			0
$589$	−5.64659e10			−0.0193316
$590$		0			0
$591$	−1.13116e12	+	1.95922e12i	−0.381398	+	0.660601i
$592$		0			0
$593$	9.81415e11	+	1.69986e12i	0.325917	+	0.564504i	0.981697	−	0.190447i	$-0.0609938\pi$
$593$	9.81415e11	+	1.69986e12i	0.325917	+	0.564504i	−0.655781	+	0.754951i	$0.727660\pi$
$594$		0			0
$595$	1.98897e12	+	2.59731e12i	0.650583	+	0.849568i
$596$		0			0
$597$	6.17781e11	+	1.07003e12i	0.199044	+	0.344755i
$598$		0			0
$599$	2.10553e12	−	3.64688e12i	0.668251	−	1.15744i	−0.310142	−	0.950690i	$-0.600376\pi$
$599$	2.10553e12	−	3.64688e12i	0.668251	−	1.15744i	0.978393	−	0.206755i	$-0.0662902\pi$
$600$		0			0
$601$	−1.30078e12			−0.406696			−0.203348	−	0.979107i	$-0.565182\pi$
$601$	−1.30078e12			−0.406696			−0.203348	+	0.979107i	$0.565182\pi$
$602$		0			0
$603$	−4.03090e11			−0.124158
$604$		0			0
$605$	−1.99560e12	+	3.45647e12i	−0.605582	+	1.04890i
$606$		0			0
$607$	3.00598e11	+	5.20651e11i	0.0898746	+	0.155667i	0.907458	−	0.420143i	$-0.138020\pi$
$607$	3.00598e11	+	5.20651e11i	0.0898746	+	0.155667i	−0.817583	+	0.575810i	$0.804687\pi$
$608$		0			0
$609$	2.75841e12	−	3.60419e11i	0.812607	−	0.106177i
$610$		0			0
$611$	−4.32984e11	−	7.49951e11i	−0.125686	−	0.217694i
$612$		0			0
$613$	2.25042e12	−	3.89785e12i	0.643713	−	1.11494i	−0.340885	−	0.940105i	$-0.610727\pi$
$613$	2.25042e12	−	3.89785e12i	0.643713	−	1.11494i	0.984597	−	0.174838i	$-0.0559401\pi$
$614$		0			0
$615$	−1.88195e12			−0.530481
$616$		0			0
$617$	−4.81479e12			−1.33750			−0.668750	−	0.743487i	$-0.733171\pi$
$617$	−4.81479e12			−1.33750			−0.668750	+	0.743487i	$0.733171\pi$
$618$		0			0
$619$	−3.67799e11	+	6.37047e11i	−0.100694	+	0.174407i	−0.911971	−	0.410255i	$-0.865440\pi$
$619$	−3.67799e11	+	6.37047e11i	−0.100694	+	0.174407i	0.811277	+	0.584662i	$0.198773\pi$
$620$		0			0
$621$	−6.36315e11	−	1.10213e12i	−0.171696	−	0.297386i
$622$		0			0
$623$	9.78851e11	−	2.35666e12i	0.260328	−	0.626760i
$624$		0			0
$625$	2.37914e12	+	4.12080e12i	0.623678	+	1.08024i
$626$		0			0
$627$	−1.22010e11	+	2.11327e11i	−0.0315275	+	0.0546073i
$628$		0			0
$629$	4.67220e10			0.0119013
$630$		0			0
$631$	6.36224e12			1.59764			0.798818	−	0.601572i	$-0.205459\pi$
$631$	6.36224e12			1.59764			0.798818	+	0.601572i	$0.205459\pi$
$632$		0			0
$633$	1.76996e12	−	3.06566e12i	0.438173	−	0.758939i
$634$		0			0
$635$	−2.15055e12	−	3.72486e12i	−0.524889	−	0.909134i
$636$		0			0
$637$	1.06830e12	−	3.95612e12i	0.257078	−	0.952010i
$638$		0			0
$639$	−9.29219e10	−	1.60945e11i	−0.0220478	−	0.0381878i
$640$		0			0
$641$	2.46882e11	−	4.27613e11i	0.0577603	−	0.100044i	−0.835699	−	0.549187i	$-0.814937\pi$
$641$	2.46882e11	−	4.27613e11i	0.0577603	−	0.100044i	0.893460	+	0.449143i	$0.148271\pi$
$642$		0			0
$643$	−2.95245e12			−0.681136			−0.340568	−	0.940220i	$-0.610619\pi$
$643$	−2.95245e12			−0.681136			−0.340568	+	0.940220i	$0.610619\pi$
$644$		0			0
$645$	1.86537e12			0.424371
$646$		0			0
$647$	−2.52816e12	+	4.37890e12i	−0.567199	+	0.982417i	0.429643	+	0.902999i	$0.358639\pi$
$647$	−2.52816e12	+	4.37890e12i	−0.567199	+	0.982417i	−0.996841	+	0.0794178i	$0.974694\pi$
$648$		0			0
$649$	−4.83698e11	−	8.37789e11i	−0.107022	−	0.185367i
$650$		0			0
$651$	−2.98505e10	+	7.18675e10i	−0.00651384	+	0.0156826i
$652$		0			0
$653$	5.34283e11	+	9.25405e11i	0.114990	+	0.199169i	0.917776	−	0.397099i	$-0.129983\pi$
$653$	5.34283e11	+	9.25405e11i	0.114990	+	0.199169i	−0.802785	+	0.596268i	$0.796650\pi$
$654$		0			0
$655$	2.10652e12	−	3.64861e12i	0.447178	−	0.774535i
$656$		0			0
$657$	−5.12923e11			−0.107401
$658$		0			0
$659$	−2.91991e12			−0.603093			−0.301547	−	0.953451i	$-0.597503\pi$
$659$	−2.91991e12			−0.603093			−0.301547	+	0.953451i	$0.597503\pi$
$660$		0			0
$661$	9.11416e11	−	1.57862e12i	0.185699	−	0.321641i	−0.758113	−	0.652124i	$-0.773878\pi$
$661$	9.11416e11	−	1.57862e12i	0.185699	−	0.321641i	0.943812	+	0.330483i	$0.107212\pi$
$662$		0			0
$663$	−1.21672e12	−	2.10741e12i	−0.244556	−	0.423583i
$664$		0			0
$665$	4.09368e12	−	5.34890e11i	0.811740	−	0.106064i
$666$		0			0
$667$	6.47330e12	+	1.12121e13i	1.26637	+	2.19341i
$668$		0			0
$669$	7.41921e11	−	1.28504e12i	0.143199	−	0.248028i
$670$		0			0
$671$	1.01801e12			0.193866
$672$		0			0
$673$	−7.99201e12			−1.50172			−0.750859	−	0.660462i	$-0.770360\pi$
$673$	−7.99201e12			−1.50172			−0.750859	+	0.660462i	$0.770360\pi$
$674$		0			0
$675$	−2.86176e11	+	4.95671e11i	−0.0530598	+	0.0919023i
$676$		0			0
$677$	2.89945e11	+	5.02199e11i	0.0530477	+	0.0918812i	0.891330	−	0.453355i	$-0.149773\pi$
$677$	2.89945e11	+	5.02199e11i	0.0530477	+	0.0918812i	−0.838282	+	0.545237i	$0.816440\pi$
$678$		0			0
$679$	5.91947e12	+	7.72997e12i	1.06873	+	1.39561i
$680$		0			0
$681$	−1.91766e12	−	3.32148e12i	−0.341672	−	0.591793i
$682$		0			0
$683$	3.25877e12	−	5.64435e12i	0.573008	−	0.992478i	−0.423247	−	0.906014i	$-0.639110\pi$
$683$	3.25877e12	−	5.64435e12i	0.573008	−	0.992478i	0.996255	−	0.0864640i	$-0.0275568\pi$
$684$		0			0
$685$	1.07639e13			1.86794
$686$		0			0
$687$	4.77081e12			0.817121
$688$		0			0
$689$	−1.09989e12	+	1.90507e12i	−0.185936	+	0.322051i
$690$		0			0
$691$	−2.26473e12	−	3.92263e12i	−0.377890	−	0.654525i	0.612865	−	0.790188i	$-0.290017\pi$
$691$	−2.26473e12	−	3.92263e12i	−0.377890	−	0.654525i	−0.990755	+	0.135663i	$0.956684\pi$
$692$		0			0
$693$	2.04468e11	+	2.67006e11i	0.0336764	+	0.0439766i
$694$		0			0
$695$	−3.83484e12	−	6.64214e12i	−0.623470	−	1.07988i
$696$		0			0
$697$	1.97437e12	−	3.41970e12i	0.316869	−	0.548834i
$698$		0			0
$699$	−6.51497e12			−1.03220
$700$		0			0
$701$	−6.10147e12			−0.954341			−0.477170	−	0.878811i	$-0.658338\pi$
$701$	−6.10147e12			−0.954341			−0.477170	+	0.878811i	$0.658338\pi$
$702$		0			0
$703$	2.94814e10	−	5.10633e10i	0.00455249	−	0.00788515i
$704$		0			0
$705$	−6.01195e11	−	1.04130e12i	−0.0916567	−	0.158754i
$706$		0			0
$707$	−1.98923e12	+	2.59917e11i	−0.299432	+	0.0391244i
$708$		0			0
$709$	−3.96451e12	−	6.86673e12i	−0.589225	−	1.02057i	−0.994334	−	0.106300i	$-0.966100\pi$
$709$	−3.96451e12	−	6.86673e12i	−0.589225	−	1.02057i	0.405109	−	0.914269i	$-0.367234\pi$
$710$		0			0
$711$	−1.64862e11	+	2.85550e11i	−0.0241940	+	0.0419053i
$712$		0			0
$713$	−3.62171e11			−0.0524821
$714$		0			0
$715$	1.42632e12			0.204099
$716$		0			0
$717$	1.99383e12	−	3.45341e12i	0.281742	−	0.487991i
$718$		0			0
$719$	−5.25744e12	−	9.10615e12i	−0.733659	−	1.27073i	−0.955309	−	0.295609i	$-0.904478\pi$
$719$	−5.25744e12	−	9.10615e12i	−0.733659	−	1.27073i	0.221650	−	0.975126i	$-0.428856\pi$
$720$		0			0
$721$	3.26350e12	−	7.85715e12i	0.449754	−	1.08282i
$722$		0			0
$723$	8.63942e11	+	1.49639e12i	0.117588	+	0.203668i
$724$		0			0
$725$	2.91129e12	−	5.04251e12i	0.391350	−	0.677837i
$726$		0			0
$727$	−1.21471e13			−1.61275			−0.806377	−	0.591402i	$-0.798575\pi$
$727$	−1.21471e13			−1.61275			−0.806377	+	0.591402i	$0.798575\pi$
$728$		0			0
$729$	2.82430e11			0.0370370
$730$		0			0
$731$	−1.95697e12	+	3.38957e12i	−0.253487	+	0.439053i
$732$		0			0
$733$	4.47508e12	+	7.75106e12i	0.572575	+	0.991730i	0.996300	+	0.0859388i	$0.0273889\pi$
$733$	4.47508e12	+	7.75106e12i	0.572575	+	0.991730i	−0.423725	+	0.905791i	$0.639278\pi$
$734$		0			0
$735$	1.48332e12	−	5.49304e12i	0.187475	−	0.694256i
$736$		0			0
$737$	2.47868e11	+	4.29320e11i	0.0309469	+	0.0536015i
$738$		0			0
$739$	−1.64851e12	+	2.85531e12i	−0.203326	+	0.352170i	−0.949598	−	0.313470i	$-0.898508\pi$
$739$	−1.64851e12	+	2.85531e12i	−0.203326	+	0.352170i	0.746272	+	0.665641i	$0.231842\pi$
$740$		0			0
$741$	−3.07097e12			−0.374191
$742$		0			0
$743$	−6.46630e12			−0.778406			−0.389203	−	0.921152i	$-0.627249\pi$
$743$	−6.46630e12			−0.778406			−0.389203	+	0.921152i	$0.627249\pi$
$744$		0			0
$745$	4.36539e12	−	7.56107e12i	0.519182	−	0.899249i
$746$		0			0
$747$	1.18146e12	+	2.04635e12i	0.138828	+	0.240457i
$748$		0			0
$749$	7.08252e11	−	1.70518e12i	0.0822280	−	0.197971i
$750$		0			0
$751$	−9.42161e11	−	1.63187e12i	−0.108080	−	0.187200i	0.806912	−	0.590671i	$-0.201137\pi$
$751$	−9.42161e11	−	1.63187e12i	−0.108080	−	0.187200i	−0.914992	+	0.403471i	$0.867804\pi$
$752$		0			0
$753$	3.63166e12	−	6.29022e12i	0.411650	−	0.712998i
$754$		0			0
$755$	−1.35741e13			−1.52037
$756$		0			0
$757$	1.35983e13			1.50506			0.752528	−	0.658561i	$-0.228834\pi$
$757$	1.35983e13			1.50506			0.752528	+	0.658561i	$0.228834\pi$
$758$		0			0
$759$	−7.82566e11	+	1.35544e12i	−0.0855920	+	0.148250i
$760$		0			0
$761$	5.71562e12	+	9.89974e12i	0.617777	+	1.07002i	0.989890	+	0.141834i	$0.0453001\pi$
$761$	5.71562e12	+	9.89974e12i	0.617777	+	1.07002i	−0.372113	+	0.928187i	$0.621367\pi$
$762$		0			0
$763$	1.48923e12	−	1.94586e11i	0.159075	−	0.0207851i
$764$		0			0
$765$	−1.68940e12	−	2.92613e12i	−0.178343	−	0.308899i
$766$		0			0
$767$	6.08731e12	−	1.05435e13i	0.635106	−	1.10004i
$768$		0			0
$769$	−1.88997e13			−1.94888			−0.974441	−	0.224642i	$-0.927879\pi$
$769$	−1.88997e13			−1.94888			−0.974441	+	0.224642i	$0.927879\pi$
$770$		0			0
$771$	−8.05871e12			−0.821336
$772$		0			0
$773$	1.03674e12	−	1.79568e12i	0.104438	−	0.180893i	−0.809070	−	0.587712i	$-0.800029\pi$
$773$	1.03674e12	−	1.79568e12i	0.104438	−	0.180893i	0.913509	+	0.406819i	$0.133362\pi$
$774$		0			0
$775$	8.14412e10	+	1.41060e11i	0.00810936	+	0.0140458i
$776$		0			0
$777$	−4.94060e10	−	6.45171e10i	−0.00486279	−	0.00635010i
$778$		0			0
$779$	−2.49163e12	−	4.31564e12i	−0.242418	−	0.419881i
$780$		0			0
$781$	−1.14279e11	+	1.97937e11i	−0.0109910	+	0.0190370i
$782$		0			0
$783$	−2.87318e12			−0.273171
$784$		0			0
$785$	2.96755e12			0.278923
$786$		0			0
$787$	−2.70087e12	+	4.67804e12i	−0.250967	+	0.434688i	−0.963792	−	0.266654i	$-0.914082\pi$
$787$	−2.70087e12	+	4.67804e12i	−0.250967	+	0.434688i	0.712825	+	0.701342i	$0.247415\pi$
$788$		0			0
$789$	−4.19298e12	−	7.26245e12i	−0.385191	−	0.667170i
$790$		0			0
$791$	−9.01815e12	−	1.17764e13i	−0.819075	−	1.06959i
$792$		0			0
$793$	6.40582e12	+	1.10952e13i	0.575235	+	0.996337i
$794$		0			0
$795$	−1.52719e12	+	2.64517e12i	−0.135594	+	0.234856i
$796$		0			0
$797$	−1.02998e13			−0.904202			−0.452101	−	0.891967i	$-0.649325\pi$
$797$	−1.02998e13			−0.904202			−0.452101	+	0.891967i	$0.649325\pi$
$798$		0			0
$799$	2.52287e12			0.218995
$800$		0			0
$801$	−1.31782e12	+	2.28253e12i	−0.113112	+	0.195916i
$802$		0			0
$803$	3.15407e11	+	5.46301e11i	0.0267702	+	0.0463673i
$804$		0			0
$805$	2.62568e13	−	3.43077e12i	2.20374	−	0.287945i
$806$		0			0
$807$	1.92326e12	+	3.33119e12i	0.159628	+	0.276483i
$808$		0			0
$809$	7.34763e12	−	1.27265e13i	0.603085	−	1.04457i	−0.389266	−	0.921126i	$-0.627271\pi$
$809$	7.34763e12	−	1.27265e13i	0.603085	−	1.04457i	0.992351	−	0.123449i	$-0.0393955\pi$
$810$		0			0
$811$	1.39350e12			0.113113			0.0565564	−	0.998399i	$-0.481988\pi$
$811$	1.39350e12			0.113113			0.0565564	+	0.998399i	$0.481988\pi$
$812$		0			0
$813$	−9.46333e12			−0.759691
$814$		0			0
$815$	1.06156e13	−	1.83867e13i	0.842820	−	1.45981i
$816$		0			0
$817$	2.46968e12	+	4.27761e12i	0.193928	+	0.335894i
$818$		0			0
$819$	−1.62346e12	+	3.90860e12i	−0.126085	+	0.303560i
$820$		0			0
$821$	1.14467e13	+	1.98262e13i	0.879295	+	1.52298i	0.852116	+	0.523354i	$0.175319\pi$
$821$	1.14467e13	+	1.98262e13i	0.879295	+	1.52298i	0.0271799	+	0.999631i	$0.491347\pi$
$822$		0			0
$823$	1.08645e13	−	1.88179e13i	0.825489	−	1.42979i	−0.0760568	−	0.997103i	$-0.524233\pi$
$823$	1.08645e13	−	1.88179e13i	0.825489	−	1.42979i	0.901545	−	0.432685i	$-0.142434\pi$
$824$		0			0
$825$	7.03901e11			0.0529016
$826$		0			0
$827$	2.24595e13			1.66965			0.834826	−	0.550515i	$-0.185568\pi$
$827$	2.24595e13			1.66965			0.834826	+	0.550515i	$0.185568\pi$
$828$		0			0
$829$	−6.01849e12	+	1.04243e13i	−0.442580	+	0.766571i	−0.997880	−	0.0650790i	$-0.979270\pi$
$829$	−6.01849e12	+	1.04243e13i	−0.442580	+	0.766571i	0.555300	+	0.831650i	$0.312603\pi$
$830$		0			0
$831$	−1.98732e12	−	3.44213e12i	−0.144565	−	0.250393i
$832$		0			0
$833$	8.42527e12	+	8.45814e12i	0.606291	+	0.608657i
$834$		0			0
$835$	1.39010e13	+	2.40772e13i	0.989592	+	1.71402i
$836$		0			0
$837$	4.01876e10	−	6.96069e10i	0.00283026	−	0.00490216i
$838$		0			0
$839$	−2.59484e13			−1.80793			−0.903965	−	0.427606i	$-0.859357\pi$
$839$	−2.59484e13			−1.80793			−0.903965	+	0.427606i	$0.859357\pi$
$840$		0			0
$841$	1.47220e13			1.01481
$842$		0			0
$843$	−7.73918e12	+	1.34047e13i	−0.527802	+	0.914180i
$844$		0			0
$845$	−2.54622e11	−	4.41018e11i	−0.0171807	−	0.0297578i
$846$		0			0
$847$	−5.58694e12	+	1.34510e13i	−0.372991	+	0.898006i
$848$		0			0
$849$	−3.39274e12	−	5.87639e12i	−0.224112	−	0.388174i
$850$		0			0
$851$	1.89093e11	−	3.27519e11i	0.0123593	−	0.0214069i
$852$		0			0
$853$	2.21826e13			1.43464			0.717320	−	0.696744i	$-0.245369\pi$
$853$	2.21826e13			1.43464			0.717320	+	0.696744i	$0.245369\pi$
$854$		0			0
$855$	−4.26402e12			−0.272880
$856$		0			0
$857$	−6.28917e10	+	1.08932e11i	−0.00398272	+	0.00689827i	−0.868010	−	0.496547i	$-0.834601\pi$
$857$	−6.28917e10	+	1.08932e11i	−0.00398272	+	0.00689827i	0.864027	+	0.503445i	$0.167934\pi$
$858$		0			0
$859$	6.88940e12	+	1.19328e13i	0.431730	+	0.747778i	0.997022	−	0.0771125i	$-0.0245701\pi$
$859$	6.88940e12	+	1.19328e13i	0.431730	+	0.747778i	−0.565293	+	0.824890i	$0.691237\pi$
$860$		0			0
$861$	−6.80995e12	+	8.89803e11i	−0.422309	+	0.0551798i
$862$		0			0
$863$	6.05047e12	+	1.04797e13i	0.371313	+	0.643133i	0.989768	−	0.142687i	$-0.0455743\pi$
$863$	6.05047e12	+	1.04797e13i	0.371313	+	0.643133i	−0.618455	+	0.785820i	$0.712241\pi$
$864$		0			0
$865$	4.81319e12	−	8.33669e12i	0.292321	−	0.506315i
$866$		0			0
$867$	−2.51617e12			−0.151236
$868$		0			0
$869$	4.05509e11			0.0241219
$870$		0			0
$871$	−3.11941e12	+	5.40297e12i	−0.183650	+	0.318091i
$872$		0			0
$873$	−5.02789e12	−	8.70856e12i	−0.292969	−	0.507437i
$874$		0			0
$875$	5.89035e12	+	7.69195e12i	0.339707	+	0.443609i
$876$		0			0
$877$	−9.94447e11	−	1.72243e12i	−0.0567654	−	0.0983205i	0.836246	−	0.548354i	$-0.184745\pi$
$877$	−9.94447e11	−	1.72243e12i	−0.0567654	−	0.0983205i	−0.893012	+	0.450034i	$0.851412\pi$
$878$		0			0
$879$	−1.93346e12	+	3.34886e12i	−0.109241	+	0.189211i
$880$		0			0
$881$	−2.57886e13			−1.44224			−0.721118	−	0.692812i	$-0.756371\pi$
$881$	−2.57886e13			−1.44224			−0.721118	+	0.692812i	$0.756371\pi$
$882$		0			0
$883$	−1.32697e13			−0.734579			−0.367290	−	0.930107i	$-0.619714\pi$
$883$	−1.32697e13			−0.734579			−0.367290	+	0.930107i	$0.619714\pi$
$884$		0			0
$885$	8.45218e12	−	1.46396e13i	0.463153	−	0.802204i
$886$		0			0
$887$	−1.02177e13	−	1.76975e13i	−0.554237	−	0.959967i	−0.997962	−	0.0638044i	$-0.979677\pi$
$887$	−1.02177e13	−	1.76975e13i	−0.554237	−	0.959967i	0.443725	−	0.896163i	$-0.353657\pi$
$888$		0			0
$889$	−9.54305e12	−	1.24618e13i	−0.512424	−	0.669151i
$890$		0			0
$891$	−1.73672e11	−	3.00808e11i	−0.00923165	−	0.0159897i
$892$		0			0
$893$	1.59192e12	−	2.75729e12i	0.0837703	−	0.145094i
$894$		0			0
$895$	−1.87472e13			−0.976635
$896$		0			0
$897$	−1.96971e13			−1.01587
$898$		0			0
$899$	−4.08832e11	+	7.08117e11i	−0.0208750	+	0.0361565i
$900$		0			0
$901$	−3.20438e12	−	5.55014e12i	−0.161988	−	0.280571i
$902$		0			0
$903$	6.74995e12	−	8.81964e11i	0.337836	−	0.0441424i
$904$		0			0
$905$	1.88161e13	+	3.25905e13i	0.932420	+	1.61500i
$906$		0			0
$907$	−1.51357e13	+	2.62157e13i	−0.742623	+	1.28626i	0.208674	+	0.977985i	$0.433085\pi$
$907$	−1.51357e13	+	2.62157e13i	−0.742623	+	1.28626i	−0.951297	+	0.308275i	$0.900248\pi$
$908$		0			0
$909$	2.07200e12			0.100659
$910$		0			0
$911$	−2.45921e13			−1.18294			−0.591471	−	0.806326i	$-0.701453\pi$
$911$	−2.45921e13			−1.18294			−0.591471	+	0.806326i	$0.701453\pi$
$912$		0			0
$913$	1.45301e12	−	2.51669e12i	0.0692070	−	0.119870i
$914$		0			0
$915$	8.89444e12	+	1.54056e13i	0.419492	+	0.726581i
$916$		0			0
$917$	5.89750e12	−	1.41987e13i	0.275426	−	0.663112i
$918$		0			0
$919$	1.42008e13	+	2.45965e13i	0.656738	+	1.13750i	0.981455	+	0.191692i	$0.0613976\pi$
$919$	1.42008e13	+	2.45965e13i	0.656738	+	1.13750i	−0.324717	+	0.945811i	$0.605269\pi$
$920$		0			0
$921$	1.10693e13	−	1.91726e13i	0.506935	−	0.878037i
$922$		0			0
$923$	−2.87640e12			−0.130449
$924$		0			0
$925$	−1.70085e11			−0.00763885
$926$		0			0
$927$	−4.39363e12	+	7.61000e12i	−0.195418	+	0.338474i
$928$		0			0
$929$	−1.69602e13	−	2.93759e13i	−0.747068	−	1.29396i	−0.949222	−	0.314606i	$-0.898127\pi$
$929$	−1.69602e13	−	2.93759e13i	−0.747068	−	1.29396i	0.202154	−	0.979354i	$-0.435206\pi$
$930$		0			0
$931$	1.45604e13	−	3.87107e12i	0.635182	−	0.168872i
$932$		0			0
$933$	6.55518e12	+	1.13539e13i	0.283216	+	0.490544i
$934$		0			0
$935$	−2.07769e12	+	3.59867e12i	−0.0889056	+	0.153989i
$936$		0			0
$937$	−9.56440e12			−0.405349			−0.202675	−	0.979246i	$-0.564963\pi$
$937$	−9.56440e12			−0.405349			−0.202675	+	0.979246i	$0.564963\pi$
$938$		0			0
$939$	2.27038e13			0.953022
$940$		0			0
$941$	−1.92096e13	+	3.32720e13i	−0.798665	+	1.38333i	0.121820	+	0.992552i	$0.461127\pi$
$941$	−1.92096e13	+	3.32720e13i	−0.798665	+	1.38333i	−0.920485	+	0.390777i	$0.872206\pi$
$942$		0			0
$943$	−1.59813e13	−	2.76804e13i	−0.658125	−	1.13991i
$944$		0			0
$945$	−2.25416e12	+	5.42707e12i	−0.0919477	+	0.221372i
$946$		0			0
$947$	1.59403e13	+	2.76095e13i	0.644054	+	1.11553i	0.984519	+	0.175278i	$0.0560823\pi$
$947$	1.59403e13	+	2.76095e13i	0.644054	+	1.11553i	−0.340465	+	0.940257i	$0.610584\pi$
$948$		0			0
$949$	−3.96938e12	+	6.87517e12i	−0.158864	+	0.275160i
$950$		0			0
$951$	−4.12744e12			−0.163632
$952$		0			0
$953$	−2.16214e13			−0.849113			−0.424557	−	0.905401i	$-0.639570\pi$
$953$	−2.16214e13			−0.849113			−0.424557	+	0.905401i	$0.639570\pi$
$954$		0			0
$955$	−2.51912e13	+	4.36325e13i	−0.980018	+	1.69744i
$956$		0			0
$957$	1.76678e12	+	3.06015e12i	0.0680892	+	0.117934i
$958$		0			0
$959$	3.89499e13	−	5.08928e12i	1.48704	−	0.194300i
$960$		0			0
$961$	1.32084e13	+	2.28776e13i	0.499567	+	0.865276i
$962$		0			0
$963$	−9.53516e11	+	1.65154e12i	−0.0357281	+	0.0618828i
$964$		0			0
$965$	−9.58642e12			−0.355863
$966$		0			0
$967$	−8.23195e12			−0.302749			−0.151375	−	0.988476i	$-0.548370\pi$
$967$	−8.23195e12			−0.302749			−0.151375	+	0.988476i	$0.548370\pi$
$968$		0			0
$969$	4.47341e12	−	7.74818e12i	0.162998	−	0.282321i
$970$		0			0
$971$	6.04185e12	+	1.04648e13i	0.218114	+	0.377784i	0.954231	−	0.299069i	$-0.0966762\pi$
$971$	6.04185e12	+	1.04648e13i	0.218114	+	0.377784i	−0.736117	+	0.676854i	$0.763343\pi$
$972$		0			0
$973$	−1.70171e13	−	2.22219e13i	−0.608664	−	0.794827i
$974$		0			0
$975$	4.42928e12	+	7.67174e12i	0.156969	+	0.271877i
$976$		0			0
$977$	−1.81791e13	+	3.14872e13i	−0.638334	+	1.10563i	0.347464	+	0.937693i	$0.387043\pi$
$977$	−1.81791e13	+	3.14872e13i	−0.638334	+	1.10563i	−0.985798	+	0.167934i	$0.946291\pi$
$978$		0			0
$979$	3.24142e12			0.112775
$980$		0			0
$981$	−1.55120e12			−0.0534757
$982$		0			0
$983$	2.63725e12	−	4.56786e12i	0.0900868	−	0.156035i	−0.817461	−	0.575984i	$-0.804619\pi$
$983$	2.63725e12	−	4.56786e12i	0.0900868	−	0.156035i	0.907547	+	0.419950i	$0.137952\pi$
$984$		0			0
$985$	2.43089e13	+	4.21043e13i	0.822816	+	1.42516i
$986$		0			0
$987$	−2.66780e12	−	3.48376e12i	−0.0894800	−	0.116848i
$988$		0			0
$989$	1.58405e13	+	2.74365e13i	0.526483	+	0.911896i
$990$		0			0
$991$	1.12135e13	−	1.94223e13i	0.369324	−	0.639689i	−0.620136	−	0.784495i	$-0.712922\pi$
$991$	1.12135e13	−	1.94223e13i	0.369324	−	0.639689i	0.989460	+	0.144806i	$0.0462558\pi$
$992$		0			0
$993$	1.74853e13			0.570690
$994$		0			0
$995$	2.65527e13			0.858824
$996$		0			0
$997$	−2.17710e13	+	3.77085e13i	−0.697831	+	1.20868i	0.271386	+	0.962470i	$0.412518\pi$
$997$	−2.17710e13	+	3.77085e13i	−0.697831	+	1.20868i	−0.969217	+	0.246208i	$0.920815\pi$
$998$		0			0
$999$	4.19646e10	+	7.26848e10i	0.00133303	+	0.00230887i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.10.q.a.25.7	✓	16
7.2	even	3	inner	168.10.q.a.121.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.10.q.a.25.7	✓	16	1.1	even	1	trivial
168.10.q.a.121.7	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)

\(f(q)\)	\(=\)	\(q+(-40.5000 + 70.1481i) q^{3} +(870.360 + 1507.51i) q^{5} +(3862.22 + 5043.50i) q^{7} +(-3280.50 - 5681.99i) q^{9} +O(q^{10})\)	\(q+(-40.5000 + 70.1481i) q^{3} +(870.360 + 1507.51i) q^{5} +(3862.22 + 5043.50i) q^{7} +(-3280.50 - 5681.99i) q^{9} +(-4034.49 + 6987.94i) q^{11} -101548. q^{13} -140998. q^{15} +(147922. - 256209. i) q^{17} +(-186677. - 323334. i) q^{19} +(-510211. + 66665.3i) q^{21} +(-1.19734e6 - 2.07385e6i) q^{23} +(-538490. + 932692. i) q^{25} +531441. q^{27} -5.40640e6 q^{29} +(75620.0 - 130978. i) q^{31} +(-326794. - 566023. i) q^{33} +(-4.24159e6 + 1.02120e7i) q^{35} +(78963.8 + 136769. i) q^{37} +(4.11269e6 - 7.12338e6i) q^{39} +1.33473e7 q^{41} -1.32297e7 q^{43} +(5.71043e6 - 9.89076e6i) q^{45} +(4.26385e6 + 7.38520e6i) q^{47} +(-1.05202e7 + 3.89582e7i) q^{49} +(1.19817e7 + 2.07529e7i) q^{51} +(1.08313e7 - 1.87603e7i) q^{53} -1.40458e7 q^{55} +3.02416e7 q^{57} +(-5.99453e7 + 1.03828e8i) q^{59} +(-6.30819e7 - 1.09261e8i) q^{61} +(1.59871e7 - 3.84903e7i) q^{63} +(-8.83831e7 - 1.53084e8i) q^{65} +(3.07186e7 - 5.32062e7i) q^{67} +1.93969e8 q^{69} +2.83255e7 q^{71} +(3.90888e7 - 6.77038e7i) q^{73} +(-4.36177e7 - 7.55481e7i) q^{75} +(-5.08258e7 + 6.64101e6i) q^{77} +(-2.51276e7 - 4.35223e7i) q^{79} +(-2.15234e7 + 3.72796e7i) q^{81} -3.60147e8 q^{83} +5.14983e8 q^{85} +(2.18959e8 - 3.79248e8i) q^{87} +(-2.00857e8 - 3.47894e8i) q^{89} +(-3.92200e8 - 5.12156e8i) q^{91} +(6.12522e6 + 1.06092e7i) q^{93} +(3.24952e8 - 5.62833e8i) q^{95} +1.53266e9 q^{97} +5.29406e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) \) 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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Representations

Groups

Database

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