LMFDB - Embedded newform 168.10.q.a.121.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("168.25");

S:= CuspForms(chi, 10);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$86.5260204755$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 18660372 x^{14} - 3458782984 x^{13} + 143123973101310 x^{12} + \cdots + 50\!\cdots\!97 $ x^16 - 18660372x^14 - 3458782984x^13 + 143123973101310x^12 + 48872053104519072x^11 - 578734520240854872804x^10 - 270042331189660586117064x^9 + 1317886932354758017843044627x^8 + 736352464217070243815722699808x^7 - 1673655580870016843616444263455812x^6 - 1020366474648370188416591376208359000x^5 + 1103325097889381327876435664684216492298x^4 + 649904594580478032138799228161609255854736x^3 - 331733869309416601638773034133657913991577560x^2 - 128648437449643500784782173321364984397814457552x + 50061741614807564009961574976312432423689032591597
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{38}\cdot 3^{5}\cdot 5^{2}\cdot 7^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.5
Root			$397.617 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	168.121
Dual form			168.10.q.a.25.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 - 70.1481i) q^{3} +(186.559 - 323.129i) q^{5} +(6073.24 + 1862.63i) q^{7} +(-3280.50 + 5681.99i) q^{9} +O(q^{10})$	\(q+(-40.5000 - 70.1481i) q^{3} +(186.559 - 323.129i) q^{5} +(6073.24 + 1862.63i) q^{7} +(-3280.50 + 5681.99i) q^{9} +(-1817.25 - 3147.57i) q^{11} +198558. q^{13} -30222.5 q^{15} +(-60321.3 - 104480. i) q^{17} +(-181012. + 313522. i) q^{19} +(-115306. - 501462. i) q^{21} +(30453.8 - 52747.5i) q^{23} +(906954. + 1.57089e6i) q^{25} +531441. q^{27} -1.46655e6 q^{29} +(2.44417e6 + 4.23343e6i) q^{31} +(-147197. + 254953. i) q^{33} +(1.73489e6 - 1.61495e6i) q^{35} +(-7.85252e6 + 1.36010e7i) q^{37} +(-8.04159e6 - 1.39284e7i) q^{39} -2.52357e7 q^{41} -1.07026e7 q^{43} +(1.22401e6 + 2.12005e6i) q^{45} +(7.55016e6 - 1.30773e7i) q^{47} +(3.34148e7 + 2.26244e7i) q^{49} +(-4.88603e6 + 8.46285e6i) q^{51} +(-1.29621e6 - 2.24510e6i) q^{53} -1.35609e6 q^{55} +2.93239e7 q^{57} +(7.47262e7 + 1.29430e8i) q^{59} +(7.16932e7 - 1.24176e8i) q^{61} +(-3.05067e7 + 2.83977e7i) q^{63} +(3.70427e7 - 6.41598e7i) q^{65} +(3.68848e7 + 6.38864e7i) q^{67} -4.93351e6 q^{69} -2.94801e8 q^{71} +(5.41887e7 + 9.38576e7i) q^{73} +(7.34633e7 - 1.27242e8i) q^{75} +(-5.17383e6 - 2.25008e7i) q^{77} +(-6.08547e7 + 1.05403e8i) q^{79} +(-2.15234e7 - 3.72796e7i) q^{81} +5.78171e8 q^{83} -4.50139e7 q^{85} +(5.93952e7 + 1.02876e8i) q^{87} +(2.47300e8 - 4.28336e8i) q^{89} +(1.20589e9 + 3.69840e8i) q^{91} +(1.97978e8 - 3.42908e8i) q^{93} +(6.75387e7 + 1.16980e8i) q^{95} +3.41998e8 q^{97} +2.38459e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) $ 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9	$ 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9} + 32460 q^{11} + 119048 q^{13} + 31752 q^{15} + 208352 q^{17} + 914588 q^{19} - 428652 q^{21} + 460920 q^{23} - 3040180 q^{25} + 8503056 q^{27} - 16376136 q^{29} - 944064 q^{31} + 2629260 q^{33} - 15546664 q^{35} - 9826516 q^{37} - 4821444 q^{39} + 11449216 q^{41} - 6933624 q^{43} - 1285956 q^{45} + 26549360 q^{47} + 83657504 q^{49} + 16876512 q^{51} - 15354476 q^{53} + 134121944 q^{55} - 148163256 q^{57} + 18404996 q^{59} - 260632792 q^{61} + 35823060 q^{63} + 191461840 q^{65} + 53879788 q^{67} - 74669040 q^{69} - 164207456 q^{71} + 248475540 q^{73} - 246254580 q^{75} + 670121788 q^{77} + 16631256 q^{79} - 344373768 q^{81} - 1138943272 q^{83} - 1690136272 q^{85} + 663233508 q^{87} + 236796360 q^{89} - 1455575212 q^{91} - 76469184 q^{93} + 182450488 q^{95} + 1339799464 q^{97} - 425940120 q^{99}+O(q^{100}) $ 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9 + 32460 * q^11 + 119048 * q^13 + 31752 * q^15 + 208352 * q^17 + 914588 * q^19 - 428652 * q^21 + 460920 * q^23 - 3040180 * q^25 + 8503056 * q^27 - 16376136 * q^29 - 944064 * q^31 + 2629260 * q^33 - 15546664 * q^35 - 9826516 * q^37 - 4821444 * q^39 + 11449216 * q^41 - 6933624 * q^43 - 1285956 * q^45 + 26549360 * q^47 + 83657504 * q^49 + 16876512 * q^51 - 15354476 * q^53 + 134121944 * q^55 - 148163256 * q^57 + 18404996 * q^59 - 260632792 * q^61 + 35823060 * q^63 + 191461840 * q^65 + 53879788 * q^67 - 74669040 * q^69 - 164207456 * q^71 + 248475540 * q^73 - 246254580 * q^75 + 670121788 * q^77 + 16631256 * q^79 - 344373768 * q^81 - 1138943272 * q^83 - 1690136272 * q^85 + 663233508 * q^87 + 236796360 * q^89 - 1455575212 * q^91 - 76469184 * q^93 + 182450488 * q^95 + 1339799464 * q^97 - 425940120 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−40.5000	−	70.1481i	−0.288675	−	0.500000i
$3$	−40.5000	−	70.1481i	−0.288675	−	0.500000i
$4$		0			0
$5$	186.559	−	323.129i	0.133491	−	0.231212i	−0.791529	−	0.611131i	$-0.790715\pi$
$5$	186.559	−	323.129i	0.133491	−	0.231212i	0.925020	+	0.379919i	$0.124048\pi$
$6$		0			0
$7$	6073.24	+	1862.63i	0.956047	+	0.293215i
$7$	6073.24	+	1862.63i	0.956047	+	0.293215i
$8$		0			0
$9$	−3280.50	+	5681.99i	−0.166667	+	0.288675i
$10$		0			0
$11$	−1817.25	−	3147.57i	−0.0374238	−	0.0648199i	0.846707	−	0.532060i	$-0.178582\pi$
$11$	−1817.25	−	3147.57i	−0.0374238	−	0.0648199i	−0.884131	+	0.467240i	$0.845248\pi$
$12$		0			0
$13$	198558.			1.92815			0.964077	−	0.265623i	$-0.0855778\pi$
$13$	198558.			1.92815			0.964077	+	0.265623i	$0.0855778\pi$
$14$		0			0
$15$	−30222.5			−0.154142
$16$		0			0
$17$	−60321.3	−	104480.i	−0.175166	−	0.303397i	0.765052	−	0.643968i	$-0.222713\pi$
$17$	−60321.3	−	104480.i	−0.175166	−	0.303397i	−0.940219	+	0.340571i	$0.889380\pi$
$18$		0			0
$19$	−181012.	+	313522.i	−0.318652	+	0.551921i	−0.980207	−	0.197976i	$-0.936563\pi$
$19$	−181012.	+	313522.i	−0.318652	+	0.551921i	0.661555	+	0.749896i	$0.269897\pi$
$20$		0			0
$21$	−115306.	−	501462.i	−0.129380	−	0.562667i
$22$		0			0
$23$	30453.8	−	52747.5i	0.0226916	−	0.0393031i	−0.854457	−	0.519523i	$-0.826110\pi$
$23$	30453.8	−	52747.5i	0.0226916	−	0.0393031i	0.877148	+	0.480220i	$0.159443\pi$
$24$		0			0
$25$	906954.	+	1.57089e6i	0.464361	+	0.804296i
$26$		0			0
$27$	531441.			0.192450
$28$		0			0
$29$	−1.46655e6			−0.385040			−0.192520	−	0.981293i	$-0.561666\pi$
$29$	−1.46655e6			−0.385040			−0.192520	+	0.981293i	$0.561666\pi$
$30$		0			0
$31$	2.44417e6	+	4.23343e6i	0.475339	+	0.823312i	0.999601	−	0.0282453i	$-0.00899194\pi$
$31$	2.44417e6	+	4.23343e6i	0.475339	+	0.823312i	−0.524262	+	0.851557i	$0.675659\pi$
$32$		0			0
$33$	−147197.	+	254953.i	−0.0216066	+	0.0374238i
$34$		0			0
$35$	1.73489e6	−	1.61495e6i	0.195418	−	0.181908i
$36$		0			0
$37$	−7.85252e6	+	1.36010e7i	−0.688813	+	1.19306i	0.283409	+	0.958999i	$0.408535\pi$
$37$	−7.85252e6	+	1.36010e7i	−0.688813	+	1.19306i	−0.972222	+	0.234060i	$0.924799\pi$
$38$		0			0
$39$	−8.04159e6	−	1.39284e7i	−0.556610	−	0.964077i
$40$		0			0
$41$	−2.52357e7			−1.39472			−0.697360	−	0.716721i	$-0.745642\pi$
$41$	−2.52357e7			−1.39472			−0.697360	+	0.716721i	$0.745642\pi$
$42$		0			0
$43$	−1.07026e7			−0.477399			−0.238700	−	0.971093i	$-0.576721\pi$
$43$	−1.07026e7			−0.477399			−0.238700	+	0.971093i	$0.576721\pi$
$44$		0			0
$45$	1.22401e6	+	2.12005e6i	0.0444968	+	0.0770708i
$46$		0			0
$47$	7.55016e6	−	1.30773e7i	0.225692	−	0.390910i	−0.730835	−	0.682554i	$-0.760869\pi$
$47$	7.55016e6	−	1.30773e7i	0.225692	−	0.390910i	0.956527	+	0.291644i	$0.0942024\pi$
$48$		0			0
$49$	3.34148e7	+	2.26244e7i	0.828050	+	0.560654i
$50$		0			0
$51$	−4.88603e6	+	8.46285e6i	−0.101132	+	0.175166i
$52$		0			0
$53$	−1.29621e6	−	2.24510e6i	−0.0225649	−	0.0390836i	0.854522	−	0.519415i	$-0.173850\pi$
$53$	−1.29621e6	−	2.24510e6i	−0.0225649	−	0.0390836i	−0.877087	+	0.480331i	$0.840517\pi$
$54$		0			0
$55$	−1.35609e6			−0.0199829
$56$		0			0
$57$	2.93239e7			0.367947
$58$		0			0
$59$	7.47262e7	+	1.29430e8i	0.802858	+	1.39059i	0.917728	+	0.397210i	$0.130022\pi$
$59$	7.47262e7	+	1.29430e8i	0.802858	+	1.39059i	−0.114870	+	0.993381i	$0.536645\pi$
$60$		0			0
$61$	7.16932e7	−	1.24176e8i	0.662970	−	1.14830i	−0.316862	−	0.948472i	$-0.602629\pi$
$61$	7.16932e7	−	1.24176e8i	0.662970	−	1.14830i	0.979831	−	0.199826i	$-0.0640375\pi$
$62$		0			0
$63$	−3.05067e7	+	2.83977e7i	−0.243985	+	0.227118i
$64$		0			0
$65$	3.70427e7	−	6.41598e7i	0.257390	−	0.445813i
$66$		0			0
$67$	3.68848e7	+	6.38864e7i	0.223620	+	0.387322i	0.955905	−	0.293677i	$-0.0948791\pi$
$67$	3.68848e7	+	6.38864e7i	0.223620	+	0.387322i	−0.732284	+	0.680999i	$0.761546\pi$
$68$		0			0
$69$	−4.93351e6			−0.0262020
$70$		0			0
$71$	−2.94801e8			−1.37679			−0.688393	−	0.725338i	$-0.741683\pi$
$71$	−2.94801e8			−1.37679			−0.688393	+	0.725338i	$0.741683\pi$
$72$		0			0
$73$	5.41887e7	+	9.38576e7i	0.223335	+	0.386827i	0.955819	−	0.293957i	$-0.0949724\pi$
$73$	5.41887e7	+	9.38576e7i	0.223335	+	0.386827i	−0.732484	+	0.680784i	$0.761639\pi$
$74$		0			0
$75$	7.34633e7	−	1.27242e8i	0.268099	−	0.464361i
$76$		0			0
$77$	−5.17383e6	−	2.25008e7i	−0.0167727	−	0.0729440i
$78$		0			0
$79$	−6.08547e7	+	1.05403e8i	−0.175781	+	0.304462i	−0.940431	−	0.339984i	$-0.889578\pi$
$79$	−6.08547e7	+	1.05403e8i	−0.175781	+	0.304462i	0.764650	+	0.644446i	$0.222912\pi$
$80$		0			0
$81$	−2.15234e7	−	3.72796e7i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	5.78171e8			1.33723			0.668613	−	0.743611i	$-0.266888\pi$
$83$	5.78171e8			1.33723			0.668613	+	0.743611i	$0.266888\pi$
$84$		0			0
$85$	−4.50139e7			−0.0935322
$86$		0			0
$87$	5.93952e7	+	1.02876e8i	0.111151	+	0.192520i
$88$		0			0
$89$	2.47300e8	−	4.28336e8i	0.417800	−	0.723651i	−0.577918	−	0.816095i	$-0.696135\pi$
$89$	2.47300e8	−	4.28336e8i	0.417800	−	0.723651i	0.995718	+	0.0924441i	$0.0294679\pi$
$90$		0			0
$91$	1.20589e9	+	3.69840e8i	1.84340	+	0.565363i
$92$		0			0
$93$	1.97978e8	−	3.42908e8i	0.274437	−	0.475339i
$94$		0			0
$95$	6.75387e7	+	1.16980e8i	0.0850739	+	0.147352i
$96$		0			0
$97$	3.41998e8			0.392239			0.196120	−	0.980580i	$-0.437166\pi$
$97$	3.41998e8			0.392239			0.196120	+	0.980580i	$0.437166\pi$
$98$		0			0
$99$	2.38459e7			0.0249492
$100$		0			0
$101$	5.18105e8	+	8.97384e8i	0.495418	+	0.858089i	0.999986	−	0.00528306i	$-0.00168166\pi$
$101$	5.18105e8	+	8.97384e8i	0.495418	+	0.858089i	−0.504568	+	0.863372i	$0.668348\pi$
$102$		0			0
$103$	1.49565e7	−	2.59054e7i	0.0130937	−	0.0226790i	−0.859404	−	0.511297i	$-0.829165\pi$
$103$	1.49565e7	−	2.59054e7i	0.0130937	−	0.0226790i	0.872498	+	0.488618i	$0.162499\pi$
$104$		0			0
$105$	−1.83548e8	−	5.62934e7i	−0.147367	−	0.0451965i
$106$		0			0
$107$	−2.76128e8	+	4.78267e8i	−0.203649	+	0.352731i	−0.949702	−	0.313156i	$-0.898614\pi$
$107$	−2.76128e8	+	4.78267e8i	−0.203649	+	0.352731i	0.746052	+	0.665888i	$0.231947\pi$
$108$		0			0
$109$	−3.06953e8	−	5.31658e8i	−0.208282	−	0.360755i	0.742891	−	0.669412i	$-0.233454\pi$
$109$	−3.06953e8	−	5.31658e8i	−0.208282	−	0.360755i	−0.951174	+	0.308657i	$0.900121\pi$
$110$		0			0
$111$	1.27211e9			0.795373
$112$		0			0
$113$	2.75987e9			1.59234			0.796169	−	0.605075i	$-0.206857\pi$
$113$	2.75987e9			1.59234			0.796169	+	0.605075i	$0.206857\pi$
$114$		0			0
$115$	−1.13628e7	−	1.96810e7i	−0.00605824	−	0.0104932i
$116$		0			0
$117$	−6.51369e8	+	1.12820e9i	−0.321359	+	0.556610i
$118$		0			0
$119$	−1.71739e8	−	7.46886e8i	−0.0785068	−	0.341423i
$120$		0			0
$121$	1.17237e9	−	2.03060e9i	0.497199	−	0.861174i
$122$		0			0
$123$	1.02204e9	+	1.77023e9i	0.402621	+	0.697360i
$124$		0			0
$125$	1.40555e9			0.514932
$126$		0			0
$127$	2.63101e9			0.897441			0.448721	−	0.893672i	$-0.351880\pi$
$127$	2.63101e9			0.897441			0.448721	+	0.893672i	$0.351880\pi$
$128$		0			0
$129$	4.33456e8	+	7.50767e8i	0.137813	+	0.238700i
$130$		0			0
$131$	−2.00326e9	+	3.46975e9i	−0.594315	+	1.02938i	0.399329	+	0.916808i	$0.369243\pi$
$131$	−2.00326e9	+	3.46975e9i	−0.594315	+	1.02938i	−0.993643	+	0.112575i	$0.964090\pi$
$132$		0			0
$133$	−1.68330e9	+	1.56693e9i	−0.466477	+	0.434229i
$134$		0			0
$135$	9.91449e7	−	1.71724e8i	0.0256903	−	0.0444968i
$136$		0			0
$137$	3.02079e9	+	5.23215e9i	0.732617	+	1.26893i	0.955761	+	0.294145i	$0.0950349\pi$
$137$	3.02079e9	+	5.23215e9i	0.732617	+	1.26893i	−0.223143	+	0.974786i	$0.571632\pi$
$138$		0			0
$139$	1.24005e9			0.281755			0.140878	−	0.990027i	$-0.455008\pi$
$139$	1.24005e9			0.281755			0.140878	+	0.990027i	$0.455008\pi$
$140$		0			0
$141$	−1.22313e9			−0.260607
$142$		0			0
$143$	−3.60829e8	−	6.24974e8i	−0.0721588	−	0.124983i
$144$		0			0
$145$	−2.73597e8	+	4.73884e8i	−0.0513991	+	0.0890259i
$146$		0			0
$147$	2.33757e8	−	3.26027e9i	0.0412892	−	0.575872i
$148$		0			0
$149$	3.88630e9	−	6.73127e9i	0.645949	−	1.11882i	−0.338133	−	0.941098i	$-0.609795\pi$
$149$	3.88630e9	−	6.73127e9i	0.645949	−	1.11882i	0.984081	−	0.177718i	$-0.0568714\pi$
$150$		0			0
$151$	2.14201e9	+	3.71007e9i	0.335294	+	0.580745i	0.983541	−	0.180684i	$-0.0578312\pi$
$151$	2.14201e9	+	3.71007e9i	0.335294	+	0.580745i	−0.648248	+	0.761430i	$0.724498\pi$
$152$		0			0
$153$	7.91537e8			0.116778
$154$		0			0
$155$	1.82392e9			0.253813
$156$		0			0
$157$	3.85338e9	+	6.67425e9i	0.506166	+	0.876706i	0.999975	+	0.00713480i	$0.00227110\pi$
$157$	3.85338e9	+	6.67425e9i	0.506166	+	0.876706i	−0.493808	+	0.869571i	$0.664396\pi$
$158$		0			0
$159$	−1.04993e8	+	1.81853e8i	−0.0130279	+	0.0225649i
$160$		0			0
$161$	2.83202e8	−	2.63624e8i	0.0332185	−	0.0309220i
$162$		0			0
$163$	−5.42114e9	+	9.38969e9i	−0.601515	+	1.04185i	0.391077	+	0.920358i	$0.372103\pi$
$163$	−5.42114e9	+	9.38969e9i	−0.601515	+	1.04185i	−0.992592	+	0.121497i	$0.961231\pi$
$164$		0			0
$165$	5.49218e7	+	9.51274e7i	0.00576856	+	0.00999143i
$166$		0			0
$167$	1.23016e10			1.22388			0.611939	−	0.790905i	$-0.290390\pi$
$167$	1.23016e10			1.22388			0.611939	+	0.790905i	$0.290390\pi$
$168$		0			0
$169$	2.88207e10			2.71778
$170$		0			0
$171$	−1.18762e9	−	2.05702e9i	−0.106217	−	0.183974i
$172$		0			0
$173$	4.85754e9	−	8.41350e9i	0.412295	−	0.714117i	−0.582845	−	0.812583i	$-0.698060\pi$
$173$	4.85754e9	−	8.41350e9i	0.412295	−	0.714117i	0.995140	+	0.0984667i	$0.0313938\pi$
$174$		0			0
$175$	2.58216e9	+	1.12297e10i	0.208119	+	0.905102i
$176$		0			0
$177$	6.05282e9	−	1.04838e10i	0.463530	−	0.802858i
$178$		0			0
$179$	−5.50133e9	−	9.52859e9i	−0.400525	−	0.693729i	0.593265	−	0.805007i	$-0.297839\pi$
$179$	−5.50133e9	−	9.52859e9i	−0.400525	−	0.693729i	−0.993789	+	0.111279i	$0.964505\pi$
$180$		0			0
$181$	−3.70666e8			−0.0256702			−0.0128351	−	0.999918i	$-0.504086\pi$
$181$	−3.70666e8			−0.0256702			−0.0128351	+	0.999918i	$0.504086\pi$
$182$		0			0
$183$	−1.16143e10			−0.765532
$184$		0			0
$185$	2.92991e9	+	5.07476e9i	0.183900	+	0.318524i
$186$		0			0
$187$	−2.19238e8	+	3.79731e8i	−0.0131108	+	0.0227085i
$188$		0			0
$189$	3.22757e9	+	9.89878e8i	0.183991	+	0.0564292i
$190$		0			0
$191$	1.15104e9	−	1.99366e9i	0.0625807	−	0.108393i	−0.833038	−	0.553216i	$-0.813400\pi$
$191$	1.15104e9	−	1.99366e9i	0.0625807	−	0.108393i	0.895618	+	0.444824i	$0.146734\pi$
$192$		0			0
$193$	−8.92969e9	−	1.54667e10i	−0.463264	−	0.802397i	0.535857	−	0.844309i	$-0.319988\pi$
$193$	−8.92969e9	−	1.54667e10i	−0.463264	−	0.802397i	−0.999121	+	0.0419118i	$0.986655\pi$
$194$		0			0
$195$	−6.00091e9			−0.297209
$196$		0			0
$197$	−2.48616e10			−1.17606			−0.588031	−	0.808838i	$-0.700097\pi$
$197$	−2.48616e10			−1.17606			−0.588031	+	0.808838i	$0.700097\pi$
$198$		0			0
$199$	−8.61380e9	−	1.49195e10i	−0.389364	−	0.674398i	0.603000	−	0.797741i	$-0.293972\pi$
$199$	−8.61380e9	−	1.49195e10i	−0.389364	−	0.674398i	−0.992364	+	0.123343i	$0.960639\pi$
$200$		0			0
$201$	2.98767e9	−	5.17480e9i	0.129107	−	0.223620i
$202$		0			0
$203$	−8.90670e9	−	2.73164e9i	−0.368116	−	0.112899i
$204$		0			0
$205$	−4.70793e9	+	8.15437e9i	−0.186182	+	0.322477i
$206$		0			0
$207$	1.99807e8	+	3.46076e8i	0.00756388	+	0.0131010i
$208$		0			0
$209$	1.31578e9			0.0477006
$210$		0			0
$211$	4.39354e9			0.152596			0.0762981	−	0.997085i	$-0.475690\pi$
$211$	4.39354e9			0.152596			0.0762981	+	0.997085i	$0.475690\pi$
$212$		0			0
$213$	1.19394e10	+	2.06797e10i	0.397444	+	0.688393i
$214$		0			0
$215$	−1.99666e9	+	3.45832e9i	−0.0637282	+	0.110381i
$216$		0			0
$217$	6.95872e9	+	3.02632e10i	0.213040	+	0.926501i
$218$		0			0
$219$	4.38929e9	−	7.60247e9i	0.128942	−	0.223335i
$220$		0			0
$221$	−1.19773e10	−	2.07452e10i	−0.337748	−	0.584996i
$222$		0			0
$223$	−4.24445e10			−1.14934			−0.574671	−	0.818385i	$-0.694870\pi$
$223$	−4.24445e10			−1.14934			−0.574671	+	0.818385i	$0.694870\pi$
$224$		0			0
$225$	−1.19011e10			−0.309574
$226$		0			0
$227$	3.76829e10	+	6.52688e10i	0.941951	+	1.63151i	0.761744	+	0.647879i	$0.224344\pi$
$227$	3.76829e10	+	6.52688e10i	0.941951	+	1.63151i	0.180208	+	0.983629i	$0.442323\pi$
$228$		0			0
$229$	1.63700e10	−	2.83537e10i	0.393360	−	0.681319i	−0.599531	−	0.800352i	$-0.704646\pi$
$229$	1.63700e10	−	2.83537e10i	0.393360	−	0.681319i	0.992890	+	0.119033i	$0.0379794\pi$
$230$		0			0
$231$	−1.36885e9	+	1.27422e9i	−0.0316301	+	0.0294435i
$232$		0			0
$233$	1.55298e10	−	2.68984e10i	0.345195	−	0.597895i	−0.640194	−	0.768213i	$-0.721146\pi$
$233$	1.55298e10	−	2.68984e10i	0.345195	−	0.597895i	0.985389	+	0.170318i	$0.0544794\pi$
$234$		0			0
$235$	−2.81710e9	−	4.87936e9i	−0.0602555	−	0.104366i
$236$		0			0
$237$	9.85846e9			0.202974
$238$		0			0
$239$	−6.71347e10			−1.33093			−0.665467	−	0.746427i	$-0.731768\pi$
$239$	−6.71347e10			−1.33093			−0.665467	+	0.746427i	$0.731768\pi$
$240$		0			0
$241$	−3.83926e10	−	6.64979e10i	−0.733113	−	1.26979i	−0.955546	−	0.294841i	$-0.904733\pi$
$241$	−3.83926e10	−	6.64979e10i	−0.733113	−	1.26979i	0.222433	−	0.974948i	$-0.428600\pi$
$242$		0			0
$243$	−1.74339e9	+	3.01964e9i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	1.35444e10	−	6.57652e9i	0.240167	−	0.116614i
$246$		0			0
$247$	−3.59413e10	+	6.22522e10i	−0.614409	+	1.06419i
$248$		0			0
$249$	−2.34159e10	−	4.05575e10i	−0.386024	−	0.668613i
$250$		0			0
$251$	−1.01009e11			−1.60630			−0.803152	−	0.595774i	$-0.796846\pi$
$251$	−1.01009e11			−1.60630			−0.803152	+	0.595774i	$0.796846\pi$
$252$		0			0
$253$	−2.21368e8			−0.00339683
$254$		0			0
$255$	1.82306e9	+	3.15764e9i	0.0270004	+	0.0467661i
$256$		0			0
$257$	−8.85199e9	+	1.53321e10i	−0.126573	+	0.219231i	−0.922347	−	0.386363i	$-0.873731\pi$
$257$	−8.85199e9	+	1.53321e10i	−0.126573	+	0.219231i	0.795774	+	0.605594i	$0.207065\pi$
$258$		0			0
$259$	−7.30238e10	+	6.79756e10i	−1.00836	+	0.938651i
$260$		0			0
$261$	4.81101e9	−	8.33292e9i	0.0641733	−	0.111151i
$262$		0			0
$263$	−1.28499e10	−	2.22567e10i	−0.165615	−	0.286853i	0.771259	−	0.636522i	$-0.219627\pi$
$263$	−1.28499e10	−	2.22567e10i	−0.165615	−	0.286853i	−0.936873	+	0.349669i	$0.886294\pi$
$264$		0			0
$265$	−9.67278e8			−0.0120488
$266$		0			0
$267$	−4.00625e10			−0.482434
$268$		0			0
$269$	−8.28206e10	−	1.43450e11i	−0.964391	−	1.67037i	−0.711242	−	0.702947i	$-0.751867\pi$
$269$	−8.28206e10	−	1.43450e11i	−0.964391	−	1.67037i	−0.253149	−	0.967427i	$-0.581466\pi$
$270$		0			0
$271$	−3.91976e10	+	6.78923e10i	−0.441467	+	0.764642i	−0.997799	−	0.0663176i	$-0.978875\pi$
$271$	−3.91976e10	+	6.78923e10i	−0.441467	+	0.764642i	0.556332	+	0.830960i	$0.312208\pi$
$272$		0			0
$273$	−2.28949e10	−	9.95692e10i	−0.249464	−	1.08491i
$274$		0			0
$275$	3.29632e9	−	5.70940e9i	0.0347562	−	0.0601996i
$276$		0			0
$277$	5.39874e10	+	9.35090e10i	0.550977	+	0.954321i	0.998204	+	0.0599001i	$0.0190782\pi$
$277$	5.39874e10	+	9.35090e10i	0.550977	+	0.954321i	−0.447227	+	0.894420i	$0.647588\pi$
$278$		0			0
$279$	−3.20724e10			−0.316893
$280$		0			0
$281$	7.38673e10			0.706763			0.353381	−	0.935479i	$-0.385032\pi$
$281$	7.38673e10			0.706763			0.353381	+	0.935479i	$0.385032\pi$
$282$		0			0
$283$	5.55275e10	+	9.61764e10i	0.514599	+	0.891312i	0.999857	+	0.0169403i	$0.00539252\pi$
$283$	5.55275e10	+	9.61764e10i	0.514599	+	0.891312i	−0.485258	+	0.874371i	$0.661274\pi$
$284$		0			0
$285$	5.47063e9	−	9.47542e9i	0.0491174	−	0.0850739i
$286$		0			0
$287$	−1.53262e11	−	4.70047e10i	−1.33342	−	0.408952i
$288$		0			0
$289$	5.20166e10	−	9.00954e10i	0.438633	−	0.759735i
$290$		0			0
$291$	−1.38509e10	−	2.39905e10i	−0.113230	−	0.196120i
$292$		0			0
$293$	−2.55504e10			−0.202532			−0.101266	−	0.994859i	$-0.532289\pi$
$293$	−2.55504e10			−0.202532			−0.101266	+	0.994859i	$0.532289\pi$
$294$		0			0
$295$	5.57633e10			0.428696
$296$		0			0
$297$	−9.65761e8	−	1.67275e9i	−0.00720221	−	0.0124746i
$298$		0			0
$299$	6.04683e9	−	1.04734e10i	0.0437530	−	0.0757823i
$300$		0			0
$301$	−6.49995e10	−	1.99350e10i	−0.456416	−	0.139980i
$302$		0			0
$303$	4.19665e10	−	7.26881e10i	0.286030	−	0.495418i
$304$		0			0
$305$	−2.67500e10	−	4.63323e10i	−0.177000	−	0.306574i
$306$		0			0
$307$	−1.72368e11			−1.10748			−0.553738	−	0.832691i	$-0.686799\pi$
$307$	−1.72368e11			−1.10748			−0.553738	+	0.832691i	$0.686799\pi$
$308$		0			0
$309$	−2.42295e9			−0.0151193
$310$		0			0
$311$	2.00567e10	+	3.47392e10i	0.121573	+	0.210571i	0.920388	−	0.391006i	$-0.127873\pi$
$311$	2.00567e10	+	3.47392e10i	0.121573	+	0.210571i	−0.798815	+	0.601577i	$0.794539\pi$
$312$		0			0
$313$	−4.13044e10	+	7.15413e10i	−0.243247	+	0.421315i	−0.961637	−	0.274324i	$-0.911546\pi$
$313$	−4.13044e10	+	7.15413e10i	−0.243247	+	0.421315i	0.718391	+	0.695640i	$0.244879\pi$
$314$		0			0
$315$	3.48484e9	+	1.51554e10i	0.0199428	+	0.0867304i
$316$		0			0
$317$	1.27450e11	−	2.20750e11i	0.708882	−	1.22782i	−0.256390	−	0.966573i	$-0.582533\pi$
$317$	1.27450e11	−	2.20750e11i	0.708882	−	1.22782i	0.965272	−	0.261246i	$-0.0841335\pi$
$318$		0			0
$319$	2.66508e9	+	4.61606e9i	0.0144096	+	0.0249582i
$320$		0			0
$321$	4.47327e10			0.235154
$322$		0			0
$323$	4.36755e10			0.223268
$324$		0			0
$325$	1.80083e11	+	3.11912e11i	0.895359	+	1.55081i
$326$		0			0
$327$	−2.48632e10	+	4.30643e10i	−0.120252	+	0.208282i
$328$		0			0
$329$	7.02121e10	−	6.53582e10i	0.330393	−	0.307552i
$330$		0			0
$331$	2.98795e10	−	5.17528e10i	0.136819	−	0.236978i	−0.789472	−	0.613787i	$-0.789645\pi$
$331$	2.98795e10	−	5.17528e10i	0.136819	−	0.236978i	0.926291	+	0.376809i	$0.122979\pi$
$332$		0			0
$333$	−5.15204e10	−	8.92360e10i	−0.229604	−	0.397686i
$334$		0			0
$335$	2.75248e10			0.119405
$336$		0			0
$337$	1.18092e11			0.498753			0.249376	−	0.968407i	$-0.419774\pi$
$337$	1.18092e11			0.498753			0.249376	+	0.968407i	$0.419774\pi$
$338$		0			0
$339$	−1.11775e11	−	1.93599e11i	−0.459668	−	0.796169i
$340$		0			0
$341$	8.88333e9	−	1.53864e10i	0.0355780	−	0.0616229i
$342$		0			0
$343$	1.60795e11	+	1.99643e11i	0.627263	+	0.778807i
$344$		0			0
$345$	−9.20389e8	+	1.59416e9i	−0.00349772	+	0.00605824i
$346$		0			0
$347$	1.26489e10	+	2.19085e10i	0.0468349	+	0.0811204i	0.888493	−	0.458891i	$-0.151753\pi$
$347$	1.26489e10	+	2.19085e10i	0.0468349	+	0.0811204i	−0.841658	+	0.540012i	$0.818420\pi$
$348$		0			0
$349$	2.82417e11			1.01900			0.509502	−	0.860470i	$-0.329830\pi$
$349$	2.82417e11			1.01900			0.509502	+	0.860470i	$0.329830\pi$
$350$		0			0
$351$	1.05522e11			0.371073
$352$		0			0
$353$	−1.44064e10	−	2.49527e10i	−0.0493822	−	0.0855325i	0.840278	−	0.542156i	$-0.182392\pi$
$353$	−1.44064e10	−	2.49527e10i	−0.0493822	−	0.0855325i	−0.889660	+	0.456624i	$0.849059\pi$
$354$		0			0
$355$	−5.49977e10	+	9.52587e10i	−0.183788	+	0.318330i
$356$		0			0
$357$	−4.54372e10	+	4.22960e10i	−0.148049	+	0.137814i
$358$		0			0
$359$	−3.61451e10	+	6.26052e10i	−0.114848	+	0.198923i	−0.917719	−	0.397230i	$-0.869972\pi$
$359$	−3.61451e10	+	6.26052e10i	−0.114848	+	0.198923i	0.802871	+	0.596153i	$0.203305\pi$
$360$		0			0
$361$	9.58132e10	+	1.65953e11i	0.296922	+	0.514285i
$362$		0			0
$363$	−1.89924e11			−0.574116
$364$		0			0
$365$	4.04375e10			0.119252
$366$		0			0
$367$	2.38707e11	+	4.13453e11i	0.686860	+	1.18968i	0.972848	+	0.231444i	$0.0743449\pi$
$367$	2.38707e11	+	4.13453e11i	0.686860	+	1.18968i	−0.285988	+	0.958233i	$0.592322\pi$
$368$		0			0
$369$	8.27856e10	−	1.43389e11i	0.232453	−	0.402621i
$370$		0			0
$371$	−3.69040e9	−	1.60494e10i	−0.0101133	−	0.0439821i
$372$		0			0
$373$	−2.89519e11	+	5.01462e11i	−0.774440	+	1.34137i	0.160669	+	0.987008i	$0.448635\pi$
$373$	−2.89519e11	+	5.01462e11i	−0.774440	+	1.34137i	−0.935109	+	0.354360i	$0.884699\pi$
$374$		0			0
$375$	−5.69246e10	−	9.85963e10i	−0.148648	−	0.257466i
$376$		0			0
$377$	−2.91195e11			−0.742416
$378$		0			0
$379$	−2.72101e11			−0.677414			−0.338707	−	0.940892i	$-0.609990\pi$
$379$	−2.72101e11			−0.677414			−0.338707	+	0.940892i	$0.609990\pi$
$380$		0			0
$381$	−1.06556e11	−	1.84560e11i	−0.259069	−	0.448721i
$382$		0			0
$383$	2.48603e11	−	4.30593e11i	0.590354	−	1.02252i	−0.403831	−	0.914834i	$-0.632322\pi$
$383$	2.48603e11	−	4.30593e11i	0.590354	−	1.02252i	0.994185	−	0.107689i	$-0.0343450\pi$
$384$		0			0
$385$	−8.23588e9	−	2.52590e9i	−0.0191046	−	0.00585927i
$386$		0			0
$387$	3.51099e10	−	6.08121e10i	0.0795665	−	0.137813i
$388$		0			0
$389$	5.80678e10	+	1.00576e11i	0.128577	+	0.222701i	0.923125	−	0.384499i	$-0.125626\pi$
$389$	5.80678e10	+	1.00576e11i	0.128577	+	0.222701i	−0.794549	+	0.607200i	$0.792292\pi$
$390$		0			0
$391$	−7.34805e9			−0.0158992
$392$		0			0
$393$	3.24528e11			0.686255
$394$		0			0
$395$	2.27059e10	+	3.93278e10i	0.0469302	+	0.0812855i
$396$		0			0
$397$	9.89740e10	−	1.71428e11i	0.199969	−	0.346357i	−0.748549	−	0.663080i	$-0.769249\pi$
$397$	9.89740e10	−	1.71428e11i	0.199969	−	0.346357i	0.948518	+	0.316722i	$0.102582\pi$
$398$		0			0
$399$	1.78091e11	+	5.46196e10i	0.351775	+	0.107887i
$400$		0			0
$401$	4.10669e11	−	7.11300e11i	0.793127	−	1.37374i	−0.130895	−	0.991396i	$-0.541785\pi$
$401$	4.10669e11	−	7.11300e11i	0.793127	−	1.37374i	0.924022	−	0.382340i	$-0.124882\pi$
$402$		0			0
$403$	4.85309e11	+	8.40580e11i	0.916527	+	1.58747i
$404$		0			0
$405$	−1.60615e10			−0.0296646
$406$		0			0
$407$	5.70800e10			0.103112
$408$		0			0
$409$	5.09639e11	+	8.82721e11i	0.900550	+	1.55980i	0.826781	+	0.562524i	$0.190169\pi$
$409$	5.09639e11	+	8.82721e11i	0.900550	+	1.55980i	0.0737692	+	0.997275i	$0.476497\pi$
$410$		0			0
$411$	2.44684e11	−	4.23804e11i	0.422977	−	0.732617i
$412$		0			0
$413$	2.12750e11	+	9.25244e11i	0.359828	+	1.56488i
$414$		0			0
$415$	1.07863e11	−	1.86824e11i	0.178507	−	0.309183i
$416$		0			0
$417$	−5.02220e10	−	8.69871e10i	−0.0813358	−	0.140878i
$418$		0			0
$419$	5.90256e11			0.935572			0.467786	−	0.883842i	$-0.345052\pi$
$419$	5.90256e11			0.935572			0.467786	+	0.883842i	$0.345052\pi$
$420$		0			0
$421$	−2.14811e11			−0.333262			−0.166631	−	0.986019i	$-0.553289\pi$
$421$	−2.14811e11			−0.333262			−0.166631	+	0.986019i	$0.553289\pi$
$422$		0			0
$423$	4.95366e10	+	8.58000e10i	0.0752307	+	0.130303i
$424$		0			0
$425$	1.09417e11	−	1.89516e11i	0.162681	−	0.281771i
$426$		0			0
$427$	6.66704e11	−	6.20614e11i	0.970527	−	0.903433i
$428$		0			0
$429$	−2.92271e10	+	5.06229e10i	−0.0416609	+	0.0721588i
$430$		0			0
$431$	−8.64350e10	−	1.49710e11i	−0.120654	−	0.208979i	0.799372	−	0.600837i	$-0.205166\pi$
$431$	−8.64350e10	−	1.49710e11i	−0.120654	−	0.208979i	−0.920026	+	0.391858i	$0.871833\pi$
$432$		0			0
$433$	6.76144e11			0.924365			0.462183	−	0.886785i	$-0.347066\pi$
$433$	6.76144e11			0.924365			0.462183	+	0.886785i	$0.347066\pi$
$434$		0			0
$435$	4.43228e10			0.0593506
$436$		0			0
$437$	1.10250e10	+	1.90958e10i	0.0144614	+	0.0250480i
$438$		0			0
$439$	−3.93516e11	+	6.81590e11i	−0.505676	+	0.875856i	0.494303	+	0.869290i	$0.335423\pi$
$439$	−3.93516e11	+	6.81590e11i	−0.505676	+	0.875856i	−0.999978	+	0.00656614i	$0.997910\pi$
$440$		0			0
$441$	−2.38169e11	+	1.15643e11i	−0.299855	+	0.145595i
$442$		0			0
$443$	−3.66968e11	+	6.35607e11i	−0.452701	+	0.784100i	−0.998553	−	0.0537811i	$-0.982873\pi$
$443$	−3.66968e11	+	6.35607e11i	−0.452701	+	0.784100i	0.545852	+	0.837881i	$0.316206\pi$
$444$		0			0
$445$	−9.22718e10	−	1.59819e11i	−0.111545	−	0.193201i
$446$		0			0
$447$	−6.29581e11			−0.745878
$448$		0			0
$449$	4.03061e11			0.468017			0.234009	−	0.972235i	$-0.424816\pi$
$449$	4.03061e11			0.468017			0.234009	+	0.972235i	$0.424816\pi$
$450$		0			0
$451$	4.58595e10	+	7.94309e10i	0.0521957	+	0.0904056i
$452$		0			0
$453$	1.73503e11	−	3.00516e11i	0.193582	−	0.335294i
$454$		0			0
$455$	3.44475e11	−	3.20661e11i	0.376796	−	0.350747i
$456$		0			0
$457$	3.98668e11	−	6.90513e11i	0.427551	−	0.740540i	−0.569104	−	0.822266i	$-0.692710\pi$
$457$	3.98668e11	−	6.90513e11i	0.427551	−	0.740540i	0.996655	+	0.0817254i	$0.0260431\pi$
$458$		0			0
$459$	−3.20572e10	−	5.55248e10i	−0.0337108	−	0.0583888i
$460$		0			0
$461$	−1.90428e12			−1.96370			−0.981852	−	0.189649i	$-0.939265\pi$
$461$	−1.90428e12			−1.96370			−0.981852	+	0.189649i	$0.939265\pi$
$462$		0			0
$463$	7.39584e11			0.747950			0.373975	−	0.927439i	$-0.377995\pi$
$463$	7.39584e11			0.747950			0.373975	+	0.927439i	$0.377995\pi$
$464$		0			0
$465$	−7.38689e10	−	1.27945e11i	−0.0732696	−	0.126907i
$466$		0			0
$467$	5.34486e11	−	9.25757e11i	0.520009	−	0.900681i	−0.479721	−	0.877421i	$-0.659262\pi$
$467$	5.34486e11	−	9.25757e11i	0.520009	−	0.900681i	0.999729	−	0.0232601i	$-0.00740460\pi$
$468$		0			0
$469$	1.05014e11	+	4.56700e11i	0.100223	+	0.435867i
$470$		0			0
$471$	3.12124e11	−	5.40614e11i	0.292235	−	0.506166i
$472$		0			0
$473$	1.94493e10	+	3.36872e10i	0.0178661	+	0.0309449i
$474$		0			0
$475$	−6.56678e11			−0.591877
$476$		0			0
$477$	1.70089e10			0.0150433
$478$		0			0
$479$	−7.00835e11	−	1.21388e12i	−0.608283	−	1.05358i	−0.991523	−	0.129929i	$-0.958525\pi$
$479$	−7.00835e11	−	1.21388e12i	−0.608283	−	1.05358i	0.383240	−	0.923649i	$-0.374808\pi$
$480$		0			0
$481$	−1.55918e12	+	2.70058e12i	−1.32814	+	2.30040i
$482$		0			0
$483$	−2.99624e10	−	9.18931e9i	−0.0250504	−	0.00768282i
$484$		0			0
$485$	6.38027e10	−	1.10510e11i	0.0523602	−	0.0906905i
$486$		0			0
$487$	5.81393e11	+	1.00700e12i	0.468370	+	0.811241i	0.999347	−	0.0361456i	$-0.0115080\pi$
$487$	5.81393e11	+	1.00700e12i	0.468370	+	0.811241i	−0.530976	+	0.847387i	$0.678175\pi$
$488$		0			0
$489$	8.78225e11			0.694570
$490$		0			0
$491$	1.39088e12			1.08000			0.539998	−	0.841667i	$-0.318425\pi$
$491$	1.39088e12			1.08000			0.539998	+	0.841667i	$0.318425\pi$
$492$		0			0
$493$	8.84642e10	+	1.53224e11i	0.0674460	+	0.116820i
$494$		0			0
$495$	4.44867e9	−	7.70532e9i	0.00333048	−	0.00576856i
$496$		0			0
$497$	−1.79040e12	−	5.49105e11i	−1.31627	−	0.403693i
$498$		0			0
$499$	4.96139e11	−	8.59338e11i	0.358221	−	0.620457i	−0.629443	−	0.777047i	$-0.716717\pi$
$499$	4.96139e11	−	8.59338e11i	0.358221	−	0.620457i	0.987664	+	0.156590i	$0.0500502\pi$
$500$		0			0
$501$	−4.98215e11	−	8.62934e11i	−0.353303	−	0.611939i
$502$		0			0
$503$	2.19351e12			1.52786			0.763930	−	0.645299i	$-0.223267\pi$
$503$	2.19351e12			1.52786			0.763930	+	0.645299i	$0.223267\pi$
$504$		0			0
$505$	3.86628e11			0.264534
$506$		0			0
$507$	−1.16724e12	−	2.02171e12i	−0.784555	−	1.35889i
$508$		0			0
$509$	−4.52823e11	+	7.84312e11i	−0.299019	+	0.517916i	−0.975912	−	0.218166i	$-0.929993\pi$
$509$	−4.52823e11	+	7.84312e11i	−0.299019	+	0.517916i	0.676893	+	0.736081i	$0.263326\pi$
$510$		0			0
$511$	1.54279e11	+	6.70953e11i	0.100095	+	0.435310i
$512$		0			0
$513$	−9.61972e10	+	1.66618e11i	−0.0613245	+	0.106217i
$514$		0			0
$515$	−5.58053e9	−	9.66576e9i	−0.00349577	−	0.00605485i
$516$		0			0
$517$	−5.48821e10			−0.0337850
$518$		0			0
$519$	−7.86921e11			−0.476078
$520$		0			0
$521$	−1.65454e12	−	2.86574e12i	−0.983800	−	1.70399i	−0.647154	−	0.762359i	$-0.724041\pi$
$521$	−1.65454e12	−	2.86574e12i	−0.983800	−	1.70399i	−0.336645	−	0.941631i	$-0.609292\pi$
$522$		0			0
$523$	−4.47732e11	+	7.75495e11i	−0.261674	+	0.453233i	−0.966687	−	0.255962i	$-0.917608\pi$
$523$	−4.47732e11	+	7.75495e11i	−0.261674	+	0.453233i	0.705013	+	0.709195i	$0.250941\pi$
$524$		0			0
$525$	6.83165e11	−	6.35937e11i	0.392472	−	0.365340i
$526$		0			0
$527$	2.94871e11	−	5.10732e11i	0.166527	−	0.288433i
$528$		0			0
$529$	8.98721e11	+	1.55663e12i	0.498970	+	0.864242i
$530$		0			0
$531$	−9.80557e11			−0.535239
$532$		0			0
$533$	−5.01073e12			−2.68924
$534$		0			0
$535$	1.03028e11	+	1.78450e11i	0.0543705	+	0.0941725i
$536$		0			0
$537$	−4.45608e11	+	7.71815e11i	−0.231243	+	0.400525i
$538$		0			0
$539$	1.04888e10	−	1.46290e11i	0.00535272	−	0.0746559i
$540$		0			0
$541$	−6.64984e11	+	1.15179e12i	−0.333752	+	0.578075i	−0.983244	−	0.182293i	$-0.941648\pi$
$541$	−6.64984e11	+	1.15179e12i	−0.333752	+	0.578075i	0.649493	+	0.760368i	$0.274981\pi$
$542$		0			0
$543$	1.50120e10	+	2.60015e10i	0.00741034	+	0.0128351i
$544$		0			0
$545$	−2.29059e11			−0.111215
$546$		0			0
$547$	−2.41556e12			−1.15365			−0.576826	−	0.816867i	$-0.695709\pi$
$547$	−2.41556e12			−1.15365			−0.576826	+	0.816867i	$0.695709\pi$
$548$		0			0
$549$	4.70379e11	+	8.14720e11i	0.220990	+	0.382766i
$550$		0			0
$551$	2.65463e11	−	4.59795e11i	0.122693	−	0.212511i
$552$		0			0
$553$	−5.65913e11	+	5.26790e11i	−0.257328	+	0.239538i
$554$		0			0
$555$	2.37323e11	−	4.11055e11i	0.106175	−	0.183900i
$556$		0			0
$557$	−3.62634e10	−	6.28100e10i	−0.0159632	−	0.0276491i	0.857933	−	0.513761i	$-0.171748\pi$
$557$	−3.62634e10	−	6.28100e10i	−0.0159632	−	0.0276491i	−0.873897	+	0.486112i	$0.838415\pi$
$558$		0			0
$559$	−2.12509e12			−0.920499
$560$		0			0
$561$	3.55165e10			0.0151390
$562$		0			0
$563$	1.15205e12	+	1.99542e12i	0.483265	+	0.837039i	0.999815	−	0.0192176i	$-0.00611753\pi$
$563$	1.15205e12	+	1.99542e12i	0.483265	+	0.837039i	−0.516551	+	0.856257i	$0.672784\pi$
$564$		0			0
$565$	5.14877e11	−	8.91793e11i	0.212562	−	0.368168i
$566$		0			0
$567$	−6.12785e10	−	2.66498e11i	−0.0248991	−	0.108285i
$568$		0			0
$569$	1.90378e11	−	3.29744e11i	0.0761396	−	0.131878i	−0.825442	−	0.564487i	$-0.809074\pi$
$569$	1.90378e11	−	3.29744e11i	0.0761396	−	0.131878i	0.901581	+	0.432610i	$0.142407\pi$
$570$		0			0
$571$	−1.27704e12	−	2.21190e12i	−0.502739	−	0.870769i	−0.999995	−	0.00316548i	$-0.998992\pi$
$571$	−1.27704e12	−	2.21190e12i	−0.502739	−	0.870769i	0.497256	−	0.867604i	$-0.334341\pi$
$572$		0			0
$573$	−1.86468e11			−0.0722619
$574$		0			0
$575$	1.10481e11			0.0421484
$576$		0			0
$577$	−2.17071e12	−	3.75978e12i	−0.815288	−	1.41212i	−0.909121	−	0.416532i	$-0.863245\pi$
$577$	−2.17071e12	−	3.75978e12i	−0.815288	−	1.41212i	0.0938334	−	0.995588i	$-0.470088\pi$
$578$		0			0
$579$	−7.23305e11	+	1.25280e12i	−0.267466	+	0.463264i
$580$		0			0
$581$	3.51137e12	+	1.07692e12i	1.27845	+	0.392094i
$582$		0			0
$583$	−4.71108e9	+	8.15983e9i	−0.00168893	+	0.00292531i
$584$		0			0
$585$	2.43037e11	+	4.20952e11i	0.0857967	+	0.148604i
$586$		0			0
$587$	−4.41993e12			−1.53654			−0.768269	−	0.640127i	$-0.778882\pi$
$587$	−4.41993e12			−1.53654			−0.768269	+	0.640127i	$0.778882\pi$
$588$		0			0
$589$	−1.76970e12			−0.605870
$590$		0			0
$591$	1.00689e12	+	1.74399e12i	0.339500	+	0.588031i
$592$		0			0
$593$	−2.45303e12	+	4.24878e12i	−0.814624	+	1.41097i	0.0949741	+	0.995480i	$0.469723\pi$
$593$	−2.45303e12	+	4.24878e12i	−0.814624	+	1.41097i	−0.909598	+	0.415490i	$0.863610\pi$
$594$		0			0
$595$	−2.73380e11	−	8.38442e10i	−0.0894212	−	0.0274250i
$596$		0			0
$597$	−6.97718e11	+	1.20848e12i	−0.224799	+	0.389364i
$598$		0			0
$599$	6.28774e11	+	1.08907e12i	0.199560	+	0.345649i	0.948386	−	0.317118i	$-0.102715\pi$
$599$	6.28774e11	+	1.08907e12i	0.199560	+	0.345649i	−0.748826	+	0.662767i	$0.769382\pi$
$600$		0			0
$601$	−2.24527e12			−0.701994			−0.350997	−	0.936377i	$-0.614157\pi$
$601$	−2.24527e12			−0.701994			−0.350997	+	0.936377i	$0.614157\pi$
$602$		0			0
$603$	−4.84003e11			−0.149080
$604$		0			0
$605$	−4.37431e11	−	7.57653e11i	−0.132743	−	0.229917i
$606$		0			0
$607$	3.16769e12	−	5.48660e12i	0.947094	−	1.64042i	0.195592	−	0.980685i	$-0.437337\pi$
$607$	3.16769e12	−	5.48660e12i	0.947094	−	1.64042i	0.751503	−	0.659730i	$-0.229329\pi$
$608$		0			0
$609$	1.69102e11	+	7.35419e11i	0.0498163	+	0.216649i
$610$		0			0
$611$	1.49914e12	−	2.59659e12i	0.435169	−	0.753734i
$612$		0			0
$613$	−1.24375e12	−	2.15424e12i	−0.355763	−	0.616199i	0.631486	−	0.775388i	$-0.282446\pi$
$613$	−1.24375e12	−	2.15424e12i	−0.355763	−	0.616199i	−0.987248	+	0.159189i	$0.949112\pi$
$614$		0			0
$615$	7.62685e11			0.214984
$616$		0			0
$617$	−4.28380e12			−1.19000			−0.594999	−	0.803727i	$-0.702847\pi$
$617$	−4.28380e12			−1.19000			−0.594999	+	0.803727i	$0.702847\pi$
$618$		0			0
$619$	−2.29967e12	−	3.98315e12i	−0.629590	−	1.09048i	−0.987634	−	0.156778i	$-0.949889\pi$
$619$	−2.29967e12	−	3.98315e12i	−0.629590	−	1.09048i	0.358044	−	0.933705i	$-0.383444\pi$
$620$		0			0
$621$	1.61844e10	−	2.80322e10i	0.00436701	−	0.00756388i
$622$		0			0
$623$	2.29974e12	−	2.14076e12i	0.611621	−	0.569339i
$624$		0			0
$625$	−1.50918e12	+	2.61397e12i	−0.395622	+	0.685237i
$626$		0			0
$627$	−5.32889e10	−	9.22991e10i	−0.0137700	−	0.0238503i
$628$		0			0
$629$	1.89470e12			0.482628
$630$		0			0
$631$	1.93528e12			0.485972			0.242986	−	0.970030i	$-0.421873\pi$
$631$	1.93528e12			0.485972			0.242986	+	0.970030i	$0.421873\pi$
$632$		0			0
$633$	−1.77938e11	−	3.08198e11i	−0.0440507	−	0.0762981i
$634$		0			0
$635$	4.90838e11	−	8.50156e11i	0.119800	−	0.207500i
$636$		0			0
$637$	6.63477e12	+	4.49225e12i	1.59661	+	1.08103i
$638$		0			0
$639$	9.67094e11	−	1.67506e12i	0.229464	−	0.397444i
$640$		0			0
$641$	1.58911e12	+	2.75241e12i	0.371785	+	0.643950i	0.989840	−	0.142184i	$-0.0454125\pi$
$641$	1.58911e12	+	2.75241e12i	0.371785	+	0.643950i	−0.618055	+	0.786135i	$0.712079\pi$
$642$		0			0
$643$	3.65901e11			0.0844138			0.0422069	−	0.999109i	$-0.486561\pi$
$643$	3.65901e11			0.0844138			0.0422069	+	0.999109i	$0.486561\pi$
$644$		0			0
$645$	3.23460e11			0.0735870
$646$		0			0
$647$	3.18183e11	+	5.51109e11i	0.0713851	+	0.123643i	0.899509	−	0.436903i	$-0.143925\pi$
$647$	3.18183e11	+	5.51109e11i	0.0713851	+	0.123643i	−0.828123	+	0.560546i	$0.810591\pi$
$648$		0			0
$649$	2.71592e11	−	4.70412e11i	0.0600919	−	0.104082i
$650$		0			0
$651$	1.84108e12	−	1.71380e12i	0.401751	−	0.373978i
$652$		0			0
$653$	1.73047e12	−	2.99727e12i	0.372439	−	0.645084i	−0.617501	−	0.786570i	$-0.711855\pi$
$653$	1.73047e12	−	2.99727e12i	0.372439	−	0.645084i	0.989940	+	0.141486i	$0.0451881\pi$
$654$		0			0
$655$	7.47451e11	+	1.29462e12i	0.158671	+	0.274826i
$656$		0			0
$657$	−7.11064e11			−0.148890
$658$		0			0
$659$	5.64045e11			0.116501			0.0582505	−	0.998302i	$-0.481448\pi$
$659$	5.64045e11			0.116501			0.0582505	+	0.998302i	$0.481448\pi$
$660$		0			0
$661$	−8.14145e11	−	1.41014e12i	−0.165881	−	0.287313i	0.771087	−	0.636730i	$-0.219713\pi$
$661$	−8.14145e11	−	1.41014e12i	−0.165881	−	0.287313i	−0.936968	+	0.349416i	$0.886380\pi$
$662$		0			0
$663$	−9.70159e11	+	1.68036e12i	−0.194999	+	0.337748i
$664$		0			0
$665$	1.92287e11	+	8.36250e11i	0.0381288	+	0.165821i
$666$		0			0
$667$	−4.46619e10	+	7.73567e10i	−0.00873718	+	0.0151332i
$668$		0			0
$669$	1.71900e12	+	2.97740e12i	0.331786	+	0.574671i
$670$		0			0
$671$	−5.21138e11			−0.0992433
$672$		0			0
$673$	3.38859e12			0.636723			0.318362	−	0.947969i	$-0.396867\pi$
$673$	3.38859e12			0.636723			0.318362	+	0.947969i	$0.396867\pi$
$674$		0			0
$675$	4.81993e11	+	8.34836e11i	0.0893662	+	0.154787i
$676$		0			0
$677$	2.09227e12	−	3.62392e12i	0.382797	−	0.663024i	−0.608664	−	0.793428i	$-0.708294\pi$
$677$	2.09227e12	−	3.62392e12i	0.382797	−	0.663024i	0.991461	+	0.130404i	$0.0416275\pi$
$678$		0			0
$679$	2.07704e12	+	6.37016e11i	0.374999	+	0.115010i
$680$		0			0
$681$	3.05232e12	−	5.28677e12i	0.543836	−	0.941951i
$682$		0			0
$683$	3.43518e12	+	5.94991e12i	0.604027	+	1.04621i	0.992204	+	0.124621i	$0.0397714\pi$
$683$	3.43518e12	+	5.94991e12i	0.604027	+	1.04621i	−0.388178	+	0.921585i	$0.626895\pi$
$684$		0			0
$685$	2.25421e12			0.391190
$686$		0			0
$687$	−2.65194e12			−0.454213
$688$		0			0
$689$	−2.57373e11	−	4.45783e11i	−0.0435087	−	0.0753593i
$690$		0			0
$691$	−2.20179e12	+	3.81361e12i	−0.367388	+	0.636334i	−0.989156	−	0.146867i	$-0.953081\pi$
$691$	−2.20179e12	+	3.81361e12i	−0.367388	+	0.636334i	0.621769	+	0.783201i	$0.286414\pi$
$692$		0			0
$693$	1.44822e11	+	4.44162e10i	0.0238526	+	0.00731546i
$694$		0			0
$695$	2.31342e11	−	4.00696e11i	0.0376117	−	0.0651453i
$696$		0			0
$697$	1.52225e12	+	2.63661e12i	0.244308	+	0.423154i
$698$		0			0
$699$	−2.51583e12			−0.398597
$700$		0			0
$701$	−2.64318e12			−0.413424			−0.206712	−	0.978402i	$-0.566276\pi$
$701$	−2.64318e12			−0.413424			−0.206712	+	0.978402i	$0.566276\pi$
$702$		0			0
$703$	−2.84280e12	−	4.92388e12i	−0.438983	−	0.760340i
$704$		0			0
$705$	−2.28185e11	+	3.95228e11i	−0.0347885	+	0.0602555i
$706$		0			0
$707$	1.47508e12	+	6.41507e12i	0.222038	+	0.965637i
$708$		0			0
$709$	5.44880e12	−	9.43759e12i	0.809828	−	1.40266i	−0.103156	−	0.994665i	$-0.532894\pi$
$709$	5.44880e12	−	9.43759e12i	0.809828	−	1.40266i	0.912983	−	0.407997i	$-0.133773\pi$
$710$		0			0
$711$	−3.99268e11	−	6.91552e11i	−0.0585937	−	0.101487i
$712$		0			0
$713$	2.97737e11			0.0431449
$714$		0			0
$715$	−2.69263e11			−0.0385300
$716$		0			0
$717$	2.71896e12	+	4.70937e12i	0.384207	+	0.665467i
$718$		0			0
$719$	−4.72223e12	+	8.17914e12i	−0.658972	+	1.14137i	0.321911	+	0.946770i	$0.395675\pi$
$719$	−4.72223e12	+	8.17914e12i	−0.658972	+	1.14137i	−0.980882	+	0.194602i	$0.937658\pi$
$720$		0			0
$721$	1.39087e11	−	1.29471e11i	0.0191680	−	0.0178429i
$722$		0			0
$723$	−3.10980e12	+	5.38633e12i	−0.423263	+	0.733113i
$724$		0			0
$725$	−1.33009e12	−	2.30379e12i	−0.178797	−	0.309686i
$726$		0			0
$727$	−6.89077e12			−0.914878			−0.457439	−	0.889241i	$-0.651233\pi$
$727$	−6.89077e12			−0.914878			−0.457439	+	0.889241i	$0.651233\pi$
$728$		0			0
$729$	2.82430e11			0.0370370
$730$		0			0
$731$	6.45596e11	+	1.11820e12i	0.0836243	+	0.144841i
$732$		0			0
$733$	2.60147e12	−	4.50588e12i	0.332852	−	0.576517i	−0.650218	−	0.759748i	$-0.725322\pi$
$733$	2.60147e12	−	4.50588e12i	0.332852	−	0.576517i	0.983070	+	0.183231i	$0.0586557\pi$
$734$		0			0
$735$	−1.00988e12	−	6.83766e11i	−0.127637	−	0.0864200i
$736$		0			0
$737$	1.34058e11	−	2.32195e11i	0.0167374	−	0.0289901i
$738$		0			0
$739$	1.17611e12	+	2.03709e12i	0.145060	+	0.251252i	0.929395	−	0.369085i	$-0.120329\pi$
$739$	1.17611e12	+	2.03709e12i	0.145060	+	0.251252i	−0.784335	+	0.620337i	$0.786996\pi$
$740$		0			0
$741$	5.82249e12			0.709459
$742$		0			0
$743$	1.60187e13			1.92831			0.964154	−	0.265342i	$-0.0854848\pi$
$743$	1.60187e13			1.92831			0.964154	+	0.265342i	$0.0854848\pi$
$744$		0			0
$745$	−1.45005e12	−	2.51155e12i	−0.172456	−	0.298703i
$746$		0			0
$747$	−1.89669e12	+	3.28516e12i	−0.222871	+	0.386024i
$748$		0			0
$749$	−2.56783e12	+	2.39031e12i	−0.298124	+	0.277515i
$750$		0			0
$751$	2.15824e12	−	3.73819e12i	0.247583	−	0.428826i	−0.715272	−	0.698847i	$-0.753697\pi$
$751$	2.15824e12	−	3.73819e12i	0.247583	−	0.428826i	0.962855	+	0.270020i	$0.0870304\pi$
$752$		0			0
$753$	4.09086e12	+	7.08558e12i	0.463700	+	0.803152i
$754$		0			0
$755$	1.59844e12			0.179034
$756$		0			0
$757$	1.16619e13			1.29074			0.645368	−	0.763872i	$-0.276704\pi$
$757$	1.16619e13			1.29074			0.645368	+	0.763872i	$0.276704\pi$
$758$		0			0
$759$	8.96542e9	+	1.55286e10i	0.000980579	+	0.00169841i
$760$		0			0
$761$	−3.27414e12	+	5.67098e12i	−0.353889	+	0.612953i	−0.986927	−	0.161167i	$-0.948474\pi$
$761$	−3.27414e12	+	5.67098e12i	−0.353889	+	0.612953i	0.633038	+	0.774120i	$0.281807\pi$
$762$		0			0
$763$	−8.73915e11	−	3.80062e12i	−0.0933488	−	0.405970i
$764$		0			0
$765$	1.47668e11	−	2.55768e11i	0.0155887	−	0.0270004i
$766$		0			0
$767$	1.48375e13	+	2.56992e13i	1.54803	+	2.68127i
$768$		0			0
$769$	2.41690e12			0.249224			0.124612	−	0.992206i	$-0.460231\pi$
$769$	2.41690e12			0.249224			0.124612	+	0.992206i	$0.460231\pi$
$770$		0			0
$771$	1.43402e12			0.146154
$772$		0			0
$773$	−3.89226e12	−	6.74160e12i	−0.392098	−	0.679133i	0.600628	−	0.799528i	$-0.294917\pi$
$773$	−3.89226e12	−	6.74160e12i	−0.392098	−	0.679133i	−0.992726	+	0.120395i	$0.961584\pi$
$774$		0			0
$775$	−4.43350e12	+	7.67905e12i	−0.441458	+	0.764627i
$776$		0			0
$777$	7.72582e12	+	2.36947e12i	0.760414	+	0.233215i
$778$		0			0
$779$	4.56795e12	−	7.91193e12i	0.444430	−	0.769775i
$780$		0			0
$781$	5.35727e11	+	9.27906e11i	0.0515245	+	0.0892430i
$782$		0			0
$783$	−7.79384e11			−0.0741009
$784$		0			0
$785$	2.87552e12			0.270274
$786$		0			0
$787$	−6.09901e12	−	1.05638e13i	−0.566726	−	0.981598i	−0.996887	−	0.0788459i	$-0.974876\pi$
$787$	−6.09901e12	−	1.05638e13i	−0.566726	−	0.981598i	0.430161	−	0.902752i	$-0.358457\pi$
$788$		0			0
$789$	−1.04084e12	+	1.80279e12i	−0.0956176	+	0.165615i
$790$		0			0
$791$	1.67613e13	+	5.14061e12i	1.52235	+	0.466897i
$792$		0			0
$793$	1.42352e13	−	2.46562e13i	1.27831	−	2.21409i
$794$		0			0
$795$	3.91747e10	+	6.78526e10i	0.00347820	+	0.00602441i
$796$		0			0
$797$	−1.47337e13			−1.29345			−0.646724	−	0.762724i	$-0.723861\pi$
$797$	−1.47337e13			−1.29345			−0.646724	+	0.762724i	$0.723861\pi$
$798$		0			0
$799$	−1.82174e12			−0.158135
$800$		0			0
$801$	1.62253e12	+	2.81031e12i	0.139267	+	0.241217i
$802$		0			0
$803$	1.96949e11	−	3.41125e11i	0.0167160	−	0.0289530i
$804$		0			0
$805$	−3.23507e10	−	1.40692e11i	−0.00271521	−	0.0118083i
$806$		0			0
$807$	−6.70847e12	+	1.16194e13i	−0.556791	+	0.964391i
$808$		0			0
$809$	−2.21030e12	−	3.82836e12i	−0.181419	−	0.314227i	0.760945	−	0.648817i	$-0.224736\pi$
$809$	−2.21030e12	−	3.82836e12i	−0.181419	−	0.314227i	−0.942364	+	0.334589i	$0.891402\pi$
$810$		0			0
$811$	−2.37745e13			−1.92983			−0.964913	−	0.262572i	$-0.915429\pi$
$811$	−2.37745e13			−1.92983			−0.964913	+	0.262572i	$0.915429\pi$
$812$		0			0
$813$	6.35001e12			0.509762
$814$		0			0
$815$	2.02272e12	+	3.50346e12i	0.160593	+	0.278155i
$816$		0			0
$817$	1.93730e12	−	3.35550e12i	0.152124	−	0.263486i
$818$		0			0
$819$	−6.05734e12	+	5.63859e12i	−0.470440	+	0.437918i
$820$		0			0
$821$	1.07776e12	−	1.86673e12i	0.0827900	−	0.143396i	−0.821657	−	0.569982i	$-0.806950\pi$
$821$	1.07776e12	−	1.86673e12i	0.0827900	−	0.143396i	0.904447	+	0.426585i	$0.140284\pi$
$822$		0			0
$823$	−1.38500e12	−	2.39889e12i	−0.105233	−	0.182269i	0.808600	−	0.588358i	$-0.200225\pi$
$823$	−1.38500e12	−	2.39889e12i	−0.105233	−	0.182269i	−0.913833	+	0.406090i	$0.866892\pi$
$824$		0			0
$825$	−5.34005e11			−0.0401330
$826$		0			0
$827$	−2.19727e13			−1.63346			−0.816732	−	0.577017i	$-0.804217\pi$
$827$	−2.19727e13			−1.63346			−0.816732	+	0.577017i	$0.804217\pi$
$828$		0			0
$829$	−1.21089e12	−	2.09733e12i	−0.0890453	−	0.154231i	0.818063	−	0.575129i	$-0.195048\pi$
$829$	−1.21089e12	−	2.09733e12i	−0.0890453	−	0.154231i	−0.907108	+	0.420898i	$0.861715\pi$
$830$		0			0
$831$	4.37298e12	−	7.57423e12i	0.318107	−	0.550977i
$832$		0			0
$833$	3.48161e11	−	4.85590e12i	0.0250540	−	0.349436i
$834$		0			0
$835$	2.29497e12	−	3.97501e12i	0.163376	−	0.282976i
$836$		0			0
$837$	1.29893e12	+	2.24982e12i	0.0914791	+	0.158446i
$838$		0			0
$839$	2.46172e12			0.171518			0.0857592	−	0.996316i	$-0.472668\pi$
$839$	2.46172e12			0.171518			0.0857592	+	0.996316i	$0.472668\pi$
$840$		0			0
$841$	−1.23564e13			−0.851744
$842$		0			0
$843$	−2.99162e12	−	5.18165e12i	−0.204025	−	0.353381i
$844$		0			0
$845$	5.37674e12	−	9.31279e12i	0.362797	−	0.628384i
$846$		0			0
$847$	1.09023e13	−	1.01486e13i	0.727854	−	0.677536i
$848$		0			0
$849$	4.49773e12	−	7.79029e12i	0.297104	−	0.514599i
$850$		0			0
$851$	4.78278e11	+	8.28402e11i	0.0312606	+	0.0541449i
$852$		0			0
$853$	−1.04960e13			−0.678818			−0.339409	−	0.940639i	$-0.610227\pi$
$853$	−1.04960e13			−0.678818			−0.339409	+	0.940639i	$0.610227\pi$
$854$		0			0
$855$	−8.86243e11			−0.0567159
$856$		0			0
$857$	7.13519e12	+	1.23585e13i	0.451847	+	0.782622i	0.998501	−	0.0547365i	$-0.0174319\pi$
$857$	7.13519e12	+	1.23585e13i	0.451847	+	0.782622i	−0.546654	+	0.837359i	$0.684099\pi$
$858$		0			0
$859$	−1.75787e12	+	3.04471e12i	−0.110158	+	0.190799i	−0.915834	−	0.401557i	$-0.868469\pi$
$859$	−1.75787e12	+	3.04471e12i	−0.110158	+	0.190799i	0.805676	+	0.592357i	$0.201802\pi$
$860$		0			0
$861$	2.90983e12	+	1.26547e13i	0.180448	+	0.784763i
$862$		0			0
$863$	2.95332e12	−	5.11531e12i	0.181244	−	0.313923i	−0.761061	−	0.648681i	$-0.775321\pi$
$863$	2.95332e12	−	5.11531e12i	0.181244	−	0.313923i	0.942304	+	0.334758i	$0.108654\pi$
$864$		0			0
$865$	−1.81243e12	−	3.13922e12i	−0.110075	−	0.190656i
$866$		0			0
$867$	−8.42669e12			−0.506490
$868$		0			0
$869$	4.42353e11			0.0263136
$870$		0			0
$871$	7.32377e12	+	1.26851e13i	0.431174	+	0.746816i
$872$		0			0
$873$	−1.12193e12	+	1.94323e12i	−0.0653732	+	0.113230i
$874$		0			0
$875$	8.53621e12	+	2.61801e12i	0.492299	+	0.150986i
$876$		0			0
$877$	1.20663e13	−	2.08994e13i	0.688770	−	1.19298i	−0.283466	−	0.958982i	$-0.591484\pi$
$877$	1.20663e13	−	2.08994e13i	0.688770	−	1.19298i	0.972236	−	0.234002i	$-0.0751823\pi$
$878$		0			0
$879$	1.03479e12	+	1.79231e12i	0.0584659	+	0.101266i
$880$		0			0
$881$	−1.02731e13			−0.574529			−0.287264	−	0.957851i	$-0.592746\pi$
$881$	−1.02731e13			−0.574529			−0.287264	+	0.957851i	$0.592746\pi$
$882$		0			0
$883$	−3.02488e13			−1.67450			−0.837250	−	0.546821i	$-0.815838\pi$
$883$	−3.02488e13			−1.67450			−0.837250	+	0.546821i	$0.815838\pi$
$884$		0			0
$885$	−2.25841e12	−	3.91168e12i	−0.123754	−	0.214348i
$886$		0			0
$887$	−3.59290e12	+	6.22309e12i	−0.194890	+	0.337559i	−0.946864	−	0.321633i	$-0.895768\pi$
$887$	−3.59290e12	+	6.22309e12i	−0.194890	+	0.337559i	0.751975	+	0.659192i	$0.229102\pi$
$888$		0			0
$889$	1.59788e13	+	4.90060e12i	0.857996	+	0.263143i
$890$		0			0
$891$	−7.82266e10	+	1.35492e11i	−0.00415820	+	0.00720221i
$892$		0			0
$893$	2.73334e12	+	4.73428e12i	0.143834	+	0.249128i
$894$		0			0
$895$	−4.10528e12			−0.213865
$896$		0			0
$897$	−9.79587e11			−0.0505216
$898$		0			0
$899$	−3.58449e12	−	6.20853e12i	−0.183025	−	0.317008i
$900$		0			0
$901$	−1.56378e11	+	2.70855e11i	−0.00790524	+	0.0136923i
$902$		0			0
$903$	1.23408e12	+	5.36695e12i	0.0617657	+	0.268617i
$904$		0			0
$905$	−6.91509e10	+	1.19773e11i	−0.00342672	+	0.00593526i
$906$		0			0
$907$	−1.29213e13	−	2.23804e13i	−0.633979	−	1.09808i	−0.986730	−	0.162367i	$-0.948087\pi$
$907$	−1.29213e13	−	2.23804e13i	−0.633979	−	1.09808i	0.352751	−	0.935717i	$-0.385246\pi$
$908$		0			0
$909$	−6.79857e12			−0.330279
$910$		0			0
$911$	2.09028e13			1.00548			0.502739	−	0.864438i	$-0.332326\pi$
$911$	2.09028e13			1.00548			0.502739	+	0.864438i	$0.332326\pi$
$912$		0			0
$913$	−1.05068e12	−	1.81983e12i	−0.0500440	−	0.0866788i
$914$		0			0
$915$	−2.16675e12	+	3.75292e12i	−0.102191	+	0.177000i
$916$		0			0
$917$	−1.86291e13	+	1.73413e13i	−0.870022	+	0.809876i
$918$		0			0
$919$	1.27996e13	−	2.21695e13i	0.591938	−	1.02527i	−0.402033	−	0.915625i	$-0.631696\pi$
$919$	1.27996e13	−	2.21695e13i	0.591938	−	1.02527i	0.993971	−	0.109642i	$-0.0349703\pi$
$920$		0			0
$921$	6.98091e12	+	1.20913e13i	0.319701	+	0.553738i
$922$		0			0
$923$	−5.85350e13			−2.65465
$924$		0			0
$925$	−2.84875e13			−1.27943
$926$		0			0
$927$	9.81296e10	+	1.69965e11i	0.00436457	+	0.00755965i
$928$		0			0
$929$	5.22678e12	−	9.05305e12i	0.230231	−	0.398771i	−0.727645	−	0.685954i	$-0.759385\pi$
$929$	5.22678e12	−	9.05305e12i	0.230231	−	0.398771i	0.957876	+	0.287182i	$0.0927186\pi$
$930$		0			0
$931$	−1.31417e13	+	6.38099e12i	−0.573296	+	0.278365i
$932$		0			0
$933$	1.62459e12	−	2.81387e12i	0.0701902	−	0.121573i
$934$		0			0
$935$	8.18014e10	+	1.41684e11i	0.00350033	+	0.00606275i
$936$		0			0
$937$	−8.68000e12			−0.367868			−0.183934	−	0.982939i	$-0.558883\pi$
$937$	−8.68000e12			−0.367868			−0.183934	+	0.982939i	$0.558883\pi$
$938$		0			0
$939$	6.69131e12			0.280877
$940$		0			0
$941$	−9.88557e12	−	1.71223e13i	−0.411006	−	0.711883i	0.583994	−	0.811758i	$-0.301489\pi$
$941$	−9.88557e12	−	1.71223e13i	−0.411006	−	0.711883i	−0.995000	+	0.0998746i	$0.968156\pi$
$942$		0			0
$943$	−7.68521e11	+	1.33112e12i	−0.0316485	+	0.0548168i
$944$		0			0
$945$	9.21989e11	−	8.58251e11i	0.0376082	−	0.0350083i
$946$		0			0
$947$	1.53711e13	−	2.66235e13i	0.621055	−	1.07570i	−0.368234	−	0.929733i	$-0.620037\pi$
$947$	1.53711e13	−	2.66235e13i	0.621055	−	1.07570i	0.989290	−	0.145967i	$-0.0466292\pi$
$948$		0			0
$949$	1.07596e13	+	1.86362e13i	0.430623	+	0.745862i
$950$		0			0
$951$	−2.06469e13			−0.818546
$952$		0			0
$953$	−1.25269e13			−0.491955			−0.245978	−	0.969276i	$-0.579109\pi$
$953$	−1.25269e13			−0.491955			−0.245978	+	0.969276i	$0.579109\pi$
$954$		0			0
$955$	−4.29473e11	−	7.43869e11i	−0.0167078	−	0.0289388i
$956$		0			0
$957$	2.15872e11	−	3.73901e11i	0.00831941	−	0.0144096i
$958$		0			0
$959$	8.60038e12	+	3.74027e13i	0.328348	+	1.42797i
$960$		0			0
$961$	1.27188e12	−	2.20295e12i	0.0481049	−	0.0833202i
$962$		0			0
$963$	−1.81167e12	−	3.13791e12i	−0.0678831	−	0.117577i
$964$		0			0
$965$	−6.66365e12			−0.247365
$966$		0			0
$967$	−2.52911e13			−0.930140			−0.465070	−	0.885274i	$-0.653971\pi$
$967$	−2.52911e13			−0.930140			−0.465070	+	0.885274i	$0.653971\pi$
$968$		0			0
$969$	−1.76886e12	−	3.06375e12i	−0.0644520	−	0.111634i
$970$		0			0
$971$	−1.28733e13	+	2.22972e13i	−0.464733	+	0.804942i	−0.999189	−	0.0402543i	$-0.987183\pi$
$971$	−1.28733e13	+	2.22972e13i	−0.464733	+	0.804942i	0.534456	+	0.845196i	$0.320517\pi$
$972$		0			0
$973$	7.53111e12	+	2.30975e12i	0.269371	+	0.0826148i
$974$		0			0
$975$	1.45867e13	−	2.52649e13i	0.516936	−	0.895359i
$976$		0			0
$977$	−1.24465e13	−	2.15579e13i	−0.437039	−	0.756973i	0.560421	−	0.828208i	$-0.310639\pi$
$977$	−1.24465e13	−	2.15579e13i	−0.437039	−	0.756973i	−0.997460	+	0.0712346i	$0.977306\pi$
$978$		0			0
$979$	−1.79762e12			−0.0625426
$980$		0			0
$981$	4.02783e12			0.138855
$982$		0			0
$983$	−4.52601e11	−	7.83928e11i	−0.0154605	−	0.0267784i	0.858192	−	0.513329i	$-0.171588\pi$
$983$	−4.52601e11	−	7.83928e11i	−0.0154605	−	0.0267784i	−0.873652	+	0.486551i	$0.838255\pi$
$984$		0			0
$985$	−4.63814e12	+	8.03349e12i	−0.156993	+	0.271920i
$986$		0			0
$987$	−7.42834e12	−	2.27823e12i	−0.249152	−	0.0764136i
$988$		0			0
$989$	−3.25935e11	+	5.64535e11i	−0.0108330	+	0.0187632i
$990$		0			0
$991$	1.04370e13	+	1.80774e13i	0.343751	+	0.595394i	0.985126	−	0.171833i	$-0.0549690\pi$
$991$	1.04370e13	+	1.80774e13i	0.343751	+	0.595394i	−0.641375	+	0.767228i	$0.721636\pi$
$992$		0			0
$993$	−4.84048e12			−0.157985
$994$		0			0
$995$	−6.42791e12			−0.207906
$996$		0			0
$997$	−9.58430e12	−	1.66005e13i	−0.307208	−	0.532099i	0.670543	−	0.741871i	$-0.266061\pi$
$997$	−9.58430e12	−	1.66005e13i	−0.307208	−	0.532099i	−0.977750	+	0.209772i	$0.932728\pi$
$998$		0			0
$999$	−4.17315e12	+	7.22811e12i	−0.132562	+	0.229604i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.10.q.a.121.5	yes	16
7.4	even	3	inner	168.10.q.a.25.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.10.q.a.25.5	✓	16	7.4	even	3	inner
168.10.q.a.121.5	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-40.5000 - 70.1481i) q^{3} +(186.559 - 323.129i) q^{5} +(6073.24 + 1862.63i) q^{7} +(-3280.50 + 5681.99i) q^{9} +O(q^{10})\)	\(q+(-40.5000 - 70.1481i) q^{3} +(186.559 - 323.129i) q^{5} +(6073.24 + 1862.63i) q^{7} +(-3280.50 + 5681.99i) q^{9} +(-1817.25 - 3147.57i) q^{11} +198558. q^{13} -30222.5 q^{15} +(-60321.3 - 104480. i) q^{17} +(-181012. + 313522. i) q^{19} +(-115306. - 501462. i) q^{21} +(30453.8 - 52747.5i) q^{23} +(906954. + 1.57089e6i) q^{25} +531441. q^{27} -1.46655e6 q^{29} +(2.44417e6 + 4.23343e6i) q^{31} +(-147197. + 254953. i) q^{33} +(1.73489e6 - 1.61495e6i) q^{35} +(-7.85252e6 + 1.36010e7i) q^{37} +(-8.04159e6 - 1.39284e7i) q^{39} -2.52357e7 q^{41} -1.07026e7 q^{43} +(1.22401e6 + 2.12005e6i) q^{45} +(7.55016e6 - 1.30773e7i) q^{47} +(3.34148e7 + 2.26244e7i) q^{49} +(-4.88603e6 + 8.46285e6i) q^{51} +(-1.29621e6 - 2.24510e6i) q^{53} -1.35609e6 q^{55} +2.93239e7 q^{57} +(7.47262e7 + 1.29430e8i) q^{59} +(7.16932e7 - 1.24176e8i) q^{61} +(-3.05067e7 + 2.83977e7i) q^{63} +(3.70427e7 - 6.41598e7i) q^{65} +(3.68848e7 + 6.38864e7i) q^{67} -4.93351e6 q^{69} -2.94801e8 q^{71} +(5.41887e7 + 9.38576e7i) q^{73} +(7.34633e7 - 1.27242e8i) q^{75} +(-5.17383e6 - 2.25008e7i) q^{77} +(-6.08547e7 + 1.05403e8i) q^{79} +(-2.15234e7 - 3.72796e7i) q^{81} +5.78171e8 q^{83} -4.50139e7 q^{85} +(5.93952e7 + 1.02876e8i) q^{87} +(2.47300e8 - 4.28336e8i) q^{89} +(1.20589e9 + 3.69840e8i) q^{91} +(1.97978e8 - 3.42908e8i) q^{93} +(6.75387e7 + 1.16980e8i) q^{95} +3.41998e8 q^{97} +2.38459e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) \) 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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