LMFDB - Embedded newform 13.14.b.a.12.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [13,14,Mod(12,13)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 14, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13.12"); S:= CuspForms(chi, 14); N := Newforms(S);

Level:	$ N $	$=$	$ 13 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	13.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$13.9400207637$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 81921 x^{12} + 2522899104 x^{10} + 37246030694192 x^{8} + \cdots + 56\!\cdots\!64 $ x^14 + 81921x^12 + 2522899104x^10 + 37246030694192x^8 + 270611051280428544x^6 + 843436691228915970048x^4 + 602910050859212473368576x^2 + 56765934420683632215588864
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{6}\cdot 13^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			12.5
Root			$-87.5010i$ of defining polynomial
Character	$\chi$	$=$	13.12
Dual form			13.14.b.a.12.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-87.5010i q^{2} +2407.03 q^{3} +535.569 q^{4} -31938.7i q^{5} -210617. i q^{6} +191671. i q^{7} -763671. i q^{8} +4.19945e6 q^{9} -2.79467e6 q^{10} +9.74341e6i q^{11} +1.28913e6 q^{12} +(-633744. - 1.73918e7i) q^{13} +1.67714e7 q^{14} -7.68772e7i q^{15} -6.24346e7 q^{16} -6.29773e7 q^{17} -3.67456e8i q^{18} +3.67427e7i q^{19} -1.71054e7i q^{20} +4.61357e8i q^{21} +8.52559e8 q^{22} -6.82777e8 q^{23} -1.83818e9i q^{24} +2.00624e8 q^{25} +(-1.52180e9 + 5.54533e7i) q^{26} +6.27061e9 q^{27} +1.02653e8i q^{28} -4.28310e9 q^{29} -6.72684e9 q^{30} +1.74048e8i q^{31} -7.92899e8i q^{32} +2.34526e10i q^{33} +5.51058e9i q^{34} +6.12171e9 q^{35} +2.24910e9 q^{36} -2.72739e7i q^{37} +3.21503e9 q^{38} +(-1.52544e9 - 4.18624e10i) q^{39} -2.43906e10 q^{40} +2.25984e10i q^{41} +4.03692e10 q^{42} +3.58063e10 q^{43} +5.21827e9i q^{44} -1.34125e11i q^{45} +5.97437e10i q^{46} +5.55708e10i q^{47} -1.50282e11 q^{48} +6.01513e10 q^{49} -1.75548e10i q^{50} -1.51588e11 q^{51} +(-3.39414e8 - 9.31449e9i) q^{52} -4.92496e10 q^{53} -5.48685e11i q^{54} +3.11192e11 q^{55} +1.46374e11 q^{56} +8.84407e10i q^{57} +3.74776e11i q^{58} +3.00035e11i q^{59} -4.11730e10i q^{60} -8.99954e10 q^{61} +1.52294e10 q^{62} +8.04912e11i q^{63} -5.80844e11 q^{64} +(-5.55470e11 + 2.02410e10i) q^{65} +2.05213e12 q^{66} +1.13281e12i q^{67} -3.37287e10 q^{68} -1.64346e12 q^{69} -5.35656e11i q^{70} +8.22933e11i q^{71} -3.20700e12i q^{72} -9.13149e11i q^{73} -2.38649e9 q^{74} +4.82908e11 q^{75} +1.96783e10i q^{76} -1.86753e12 q^{77} +(-3.66301e12 + 1.33477e11i) q^{78} +1.70734e12 q^{79} +1.99408e12i q^{80} +8.39824e12 q^{81} +1.97738e12 q^{82} -3.23695e12i q^{83} +2.47088e11i q^{84} +2.01141e12i q^{85} -3.13309e12i q^{86} -1.03095e13 q^{87} +7.44077e12 q^{88} -1.90362e12i q^{89} -1.17361e13 q^{90} +(3.33349e12 - 1.21470e11i) q^{91} -3.65674e11 q^{92} +4.18938e11i q^{93} +4.86250e12 q^{94} +1.17351e12 q^{95} -1.90853e12i q^{96} -8.72249e12i q^{97} -5.26330e12i q^{98} +4.09170e13i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 1456 q^{3} - 49154 q^{4} + 5149262 q^{9} - 3968358 q^{10} - 21959110 q^{12} - 14362478 q^{13} + 72843942 q^{14} + 243447170 q^{16} - 178512492 q^{17} + 1330288056 q^{22} - 1641693744 q^{23} + 1601386654 q^{25}+ \cdots + 11940434574336 q^{95}+O(q^{100}) $ 14 * q + 1456 * q^3 - 49154 * q^4 + 5149262 * q^9 - 3968358 * q^10 - 21959110 * q^12 - 14362478 * q^13 + 72843942 * q^14 + 243447170 * q^16 - 178512492 * q^17 + 1330288056 * q^22 - 1641693744 * q^23 + 1601386654 * q^25 + 138020766 * q^26 + 6360372232 * q^27 - 10348326348 * q^29 - 11230427010 * q^30 + 20242816056 * q^35 + 33181294876 * q^36 - 72234754500 * q^38 + 3102574280 * q^39 - 50528567766 * q^40 + 90123482634 * q^42 - 42288191696 * q^43 + 256465114846 * q^48 + 202681235282 * q^49 - 454251099624 * q^51 + 605927200580 * q^52 - 541269433788 * q^53 + 294791560608 * q^55 - 1027260370698 * q^56 + 1693279412900 * q^61 + 1808450592168 * q^62 - 4815047793506 * q^64 + 534977124552 * q^65 + 1417411440480 * q^66 + 65016763122 * q^68 - 4207322259360 * q^69 + 7595100503238 * q^74 + 1662071406584 * q^75 - 1391939668824 * q^77 - 361798757502 * q^78 - 5060278211056 * q^79 + 6637938210830 * q^81 + 507818992992 * q^82 - 13033010125344 * q^87 + 2586979258080 * q^88 - 4680167629284 * q^90 + 3450737986176 * q^91 + 7241299383420 * q^92 + 11219792119974 * q^94 + 11940434574336 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$		−	87.5010i		−	0.966759i	−0.875411	−	0.483379i	$-0.839409\pi$
$2$		−	87.5010i		−	0.966759i	0.875411	−	0.483379i	$-0.160591\pi$
$3$	2407.03			1.90631			0.953153	−	0.302489i	$-0.0978174\pi$
$3$	2407.03			1.90631			0.953153	+	0.302489i	$0.0978174\pi$
$4$	535.569			0.0653771
$5$		−	31938.7i		−	0.914138i	−0.889431	−	0.457069i	$-0.848899\pi$
$5$		−	31938.7i		−	0.914138i	0.889431	−	0.457069i	$-0.151101\pi$
$6$		−	210617.i		−	1.84294i
$7$			191671.i			0.615770i	0.951424	+	0.307885i	$0.0996213\pi$
$7$			191671.i			0.615770i	−0.951424	+	0.307885i	$0.900379\pi$
$8$		−	763671.i		−	1.02996i
$9$	4.19945e6			2.63400
$10$	−2.79467e6			−0.883751
$11$			9.74341e6i			1.65828i	0.559039	+	0.829142i	$0.311170\pi$
$11$			9.74341e6i			1.65828i	−0.559039	+	0.829142i	$0.688830\pi$
$12$	1.28913e6			0.124629
$13$	−633744.	−	1.73918e7i	−0.0364152	−	0.999337i
$13$	−633744.	−	1.73918e7i	−0.0364152	−	0.999337i
$14$	1.67714e7			0.595302
$15$		−	7.68772e7i		−	1.74263i
$16$	−6.24346e7			−0.930349
$17$	−6.29773e7			−0.632800			−0.316400	−	0.948626i	$-0.602474\pi$
$17$	−6.29773e7			−0.632800			−0.316400	+	0.948626i	$0.602474\pi$
$18$		−	3.67456e8i		−	2.54645i
$19$			3.67427e7i			0.179173i	0.995979	+	0.0895866i	$0.0285546\pi$
$19$			3.67427e7i			0.179173i	−0.995979	+	0.0895866i	$0.971445\pi$
$20$		−	1.71054e7i		−	0.0597637i
$21$			4.61357e8i			1.17385i
$22$	8.52559e8			1.60316
$23$	−6.82777e8			−0.961718			−0.480859	−	0.876798i	$-0.659675\pi$
$23$	−6.82777e8			−0.961718			−0.480859	+	0.876798i	$0.659675\pi$
$24$		−	1.83818e9i		−	1.96342i
$25$	2.00624e8			0.164351
$26$	−1.52180e9	+	5.54533e7i	−0.966118	+	0.0352047i
$27$	6.27061e9			3.11491
$28$			1.02653e8i			0.0402573i
$29$	−4.28310e9			−1.33712			−0.668562	−	0.743656i	$-0.733090\pi$
$29$	−4.28310e9			−1.33712			−0.668562	+	0.743656i	$0.733090\pi$
$30$	−6.72684e9			−1.68470
$31$			1.74048e8i			0.0352223i	0.999845	+	0.0176111i	$0.00560609\pi$
$31$			1.74048e8i			0.0352223i	−0.999845	+	0.0176111i	$0.994394\pi$
$32$		−	7.92899e8i		−	0.130540i
$33$			2.34526e10i			3.16120i
$34$			5.51058e9i			0.611765i
$35$	6.12171e9			0.562899
$36$	2.24910e9			0.172203
$37$		−	2.72739e7i		−	0.00174757i	−1.00000		0.000873787i	$-0.999722\pi$
$37$		−	2.72739e7i		−	0.00174757i	1.00000		0.000873787i	$-0.000278135\pi$
$38$	3.21503e9			0.173217
$39$	−1.52544e9	−	4.18624e10i	−0.0694184	−	1.90504i
$40$	−2.43906e10			−0.941528
$41$			2.25984e10i			0.742990i	0.928435	+	0.371495i	$0.121155\pi$
$41$			2.25984e10i			0.742990i	−0.928435	+	0.371495i	$0.878845\pi$
$42$	4.03692e10			1.13483
$43$	3.58063e10			0.863804			0.431902	−	0.901921i	$-0.357843\pi$
$43$	3.58063e10			0.863804			0.431902	+	0.901921i	$0.357843\pi$
$44$			5.21827e9i			0.108414i
$45$		−	1.34125e11i		−	2.40784i
$46$			5.97437e10i			0.929750i
$47$			5.55708e10i			0.751987i	0.926622	+	0.375993i	$0.122698\pi$
$47$			5.55708e10i			0.751987i	−0.926622	+	0.375993i	$0.877302\pi$
$48$	−1.50282e11			−1.77353
$49$	6.01513e10			0.620827
$50$		−	1.75548e10i		−	0.158888i
$51$	−1.51588e11			−1.20631
$52$	−3.39414e8	−	9.31449e9i	−0.00238072	−	0.0653337i
$53$	−4.92496e10			−0.305217			−0.152609	−	0.988287i	$-0.548767\pi$
$53$	−4.92496e10			−0.305217			−0.152609	+	0.988287i	$0.548767\pi$
$54$		−	5.48685e11i		−	3.01137i
$55$	3.11192e11			1.51590
$56$	1.46374e11			0.634221
$57$			8.84407e10i			0.341559i
$58$			3.74776e11i			1.29268i
$59$			3.00035e11i			0.926050i	0.886345	+	0.463025i	$0.153236\pi$
$59$			3.00035e11i			0.926050i	−0.886345	+	0.463025i	$0.846764\pi$
$60$		−	4.11730e10i		−	0.113928i
$61$	−8.99954e10			−0.223654			−0.111827	−	0.993728i	$-0.535670\pi$
$61$	−8.99954e10			−0.223654			−0.111827	+	0.993728i	$0.535670\pi$
$62$	1.52294e10			0.0340515
$63$			8.04912e11i			1.62194i
$64$	−5.80844e11			−1.05655
$65$	−5.55470e11	+	2.02410e10i	−0.913532	+	0.0332885i
$66$	2.05213e12			3.05611
$67$			1.13281e12i			1.52992i	0.644077	+	0.764960i	$0.277241\pi$
$67$			1.13281e12i			1.52992i	−0.644077	+	0.764960i	$0.722759\pi$
$68$	−3.37287e10			−0.0413706
$69$	−1.64346e12			−1.83333
$70$		−	5.35656e11i		−	0.544188i
$71$			8.22933e11i			0.762404i	0.924492	+	0.381202i	$0.124490\pi$
$71$			8.22933e11i			0.762404i	−0.924492	+	0.381202i	$0.875510\pi$
$72$		−	3.20700e12i		−	2.71292i
$73$		−	9.13149e11i		−	0.706225i	−0.935581	−	0.353113i	$-0.885123\pi$
$73$		−	9.13149e11i		−	0.706225i	0.935581	−	0.353113i	$-0.114877\pi$
$74$	−2.38649e9			−0.00168948
$75$	4.82908e11			0.313304
$76$			1.96783e10i			0.0117138i
$77$	−1.86753e12			−1.02112
$78$	−3.66301e12	+	1.33477e11i	−1.84172	+	0.0671109i
$79$	1.70734e12			0.790211			0.395106	−	0.918636i	$-0.370708\pi$
$79$	1.70734e12			0.790211			0.395106	+	0.918636i	$0.370708\pi$
$80$			1.99408e12i			0.850467i
$81$	8.39824e12			3.30397
$82$	1.97738e12			0.718292
$83$		−	3.23695e12i		−	1.08675i	−0.839491	−	0.543374i	$-0.817147\pi$
$83$		−	3.23695e12i		−	1.08675i	0.839491	−	0.543374i	$-0.182853\pi$
$84$			2.47088e11i			0.0767427i
$85$			2.01141e12i			0.578466i
$86$		−	3.13309e12i		−	0.835090i
$87$	−1.03095e13			−2.54897
$88$	7.44077e12			1.70797
$89$		−	1.90362e12i		−	0.406018i	−0.979177	−	0.203009i	$-0.934928\pi$
$89$		−	1.90362e12i		−	0.406018i	0.979177	−	0.203009i	$-0.0650722\pi$
$90$	−1.17361e13			−2.32780
$91$	3.33349e12	−	1.21470e11i	0.615362	−	0.0224234i
$92$	−3.65674e11			−0.0628743
$93$			4.18938e11i			0.0671445i
$94$	4.86250e12			0.726990
$95$	1.17351e12			0.163789
$96$		−	1.90853e12i		−	0.248849i
$97$		−	8.72249e12i		−	1.06322i	−0.846988	−	0.531612i	$-0.821587\pi$
$97$		−	8.72249e12i		−	1.06322i	0.846988	−	0.531612i	$-0.178413\pi$
$98$		−	5.26330e12i		−	0.600190i
$99$			4.09170e13i			4.36792i
$100$	1.07448e11			0.0107448
$101$	−9.35975e12			−0.877355			−0.438677	−	0.898645i	$-0.644553\pi$
$101$	−9.35975e12			−0.877355			−0.438677	+	0.898645i	$0.644553\pi$
$102$			1.32641e13i			1.16621i
$103$	−4.32981e12			−0.357295			−0.178648	−	0.983913i	$-0.557172\pi$
$103$	−4.32981e12			−0.357295			−0.178648	+	0.983913i	$0.557172\pi$
$104$	−1.32816e13	+	4.83972e11i	−1.02928	+	0.0375063i
$105$	1.47351e13			1.07306
$106$			4.30939e12i			0.295072i
$107$	2.03267e13			1.30940			0.654700	−	0.755889i	$-0.272795\pi$
$107$	2.03267e13			1.30940			0.654700	+	0.755889i	$0.272795\pi$
$108$	3.35834e12			0.203644
$109$		−	8.76206e12i		−	0.500419i	−0.968192	−	0.250210i	$-0.919500\pi$
$109$		−	8.76206e12i		−	0.500419i	0.968192	−	0.250210i	$-0.0804996\pi$
$110$		−	2.72296e13i		−	1.46551i
$111$		−	6.56489e10i		−	0.00333141i
$112$		−	1.19669e13i		−	0.572881i
$113$	8.52942e12			0.385398			0.192699	−	0.981258i	$-0.438276\pi$
$113$	8.52942e12			0.385398			0.192699	+	0.981258i	$0.438276\pi$
$114$	7.73866e12			0.330205
$115$			2.18070e13i			0.879143i
$116$	−2.29390e12			−0.0874172
$117$	−2.66138e12	−	7.30359e13i	−0.0959176	−	2.63226i
$118$	2.62534e13			0.895267
$119$		−	1.20709e13i		−	0.389659i
$120$	−5.87089e13			−1.79484
$121$	−6.04114e13			−1.74990
$122$			7.87469e12i			0.216219i
$123$			5.43949e13i			1.41637i
$124$			9.32146e10i			0.00230273i
$125$		−	4.53953e13i		−	1.06438i
$126$	7.04307e13			1.56803
$127$	−3.34991e13			−0.708449			−0.354224	−	0.935160i	$-0.615255\pi$
$127$	−3.34991e13			−0.708449			−0.354224	+	0.935160i	$0.615255\pi$
$128$			4.43290e13i			0.890889i
$129$	8.61868e13			1.64667
$130$	1.77110e12	+	4.86042e13i	0.0321819	+	0.883165i
$131$	−1.10053e14			−1.90256			−0.951280	−	0.308329i	$-0.900230\pi$
$131$	−1.10053e14			−1.90256			−0.951280	+	0.308329i	$0.900230\pi$
$132$			1.25605e13i			0.206670i
$133$	−7.04251e12			−0.110330
$134$	9.91216e13			1.47906
$135$		−	2.00275e14i		−	2.84746i
$136$			4.80940e13i			0.651760i
$137$		−	1.09480e14i		−	1.41466i	−0.706883	−	0.707330i	$-0.749899\pi$
$137$		−	1.09480e14i		−	1.41466i	0.706883	−	0.707330i	$-0.250101\pi$
$138$			1.43805e14i			1.77239i
$139$	−1.77596e13			−0.208851			−0.104425	−	0.994533i	$-0.533300\pi$
$139$	−1.77596e13			−0.208851			−0.104425	+	0.994533i	$0.533300\pi$
$140$	3.27860e12			0.0368007
$141$			1.33760e14i			1.43352i
$142$	7.20075e13			0.737061
$143$	1.69455e14	−	6.17483e12i	1.65718	−	0.0603867i
$144$	−2.62191e14			−2.45054
$145$			1.36797e14i			1.22232i
$146$	−7.99015e13			−0.682750
$147$	1.44786e14			1.18349
$148$		−	1.46070e10i		−	0.000114251i
$149$			1.57109e14i			1.17622i	0.808780	+	0.588111i	$0.200128\pi$
$149$			1.57109e14i			1.17622i	−0.808780	+	0.588111i	$0.799872\pi$
$150$		−	4.22549e13i		−	0.302890i
$151$		−	7.81355e13i		−	0.536412i	−0.963362	−	0.268206i	$-0.913569\pi$
$151$		−	7.81355e13i		−	0.536412i	0.963362	−	0.268206i	$-0.0864307\pi$
$152$	2.80594e13			0.184542
$153$	−2.64470e14			−1.66680
$154$			1.63411e14i			0.987179i
$155$	5.55886e12			0.0321980
$156$	−8.16978e11	−	2.24202e13i	−0.00453837	−	0.124546i
$157$	5.69426e12			0.0303451			0.0151726	−	0.999885i	$-0.495170\pi$
$157$	5.69426e12			0.0303451			0.0151726	+	0.999885i	$0.495170\pi$
$158$		−	1.49394e14i		−	0.763944i
$159$	−1.18545e14			−0.581838
$160$	−2.53241e13			−0.119331
$161$		−	1.30868e14i		−	0.592198i
$162$		−	7.34855e14i		−	3.19414i
$163$			1.35936e14i			0.567693i	0.958870	+	0.283847i	$0.0916106\pi$
$163$			1.35936e14i			0.567693i	−0.958870	+	0.283847i	$0.908389\pi$
$164$			1.21030e13i			0.0485745i
$165$	7.49046e14			2.88977
$166$	−2.83237e14			−1.05062
$167$		−	4.92949e14i		−	1.75851i	−0.476351	−	0.879255i	$-0.658041\pi$
$167$		−	4.92949e14i		−	1.75851i	0.476351	−	0.879255i	$-0.341959\pi$
$168$	3.52325e14			1.20902
$169$	−3.02072e14	+	2.20439e13i	−0.997348	+	0.0727820i
$170$	1.76001e14			0.559237
$171$			1.54299e14i			0.471943i
$172$	1.91768e13			0.0564729
$173$	−3.51977e14			−0.998194			−0.499097	−	0.866546i	$-0.666335\pi$
$173$	−3.51977e14			−0.998194			−0.499097	+	0.866546i	$0.666335\pi$
$174$			9.02096e14i			2.46424i
$175$			3.84538e13i			0.101203i
$176$		−	6.08327e14i		−	1.54278i
$177$			7.22193e14i			1.76533i
$178$	−1.66569e14			−0.392522
$179$	5.79869e14			1.31760			0.658802	−	0.752316i	$-0.271063\pi$
$179$	5.79869e14			1.31760			0.658802	+	0.752316i	$0.271063\pi$
$180$		−	7.18331e13i		−	0.157418i
$181$	−1.86793e14			−0.394867			−0.197433	−	0.980316i	$-0.563261\pi$
$181$	−1.86793e14			−0.394867			−0.197433	+	0.980316i	$0.563261\pi$
$182$	−1.06288e13	−	2.91684e14i	−0.0216780	−	0.594907i
$183$	−2.16621e14			−0.426353
$184$			5.21417e14i			0.990534i
$185$	−8.71091e11			−0.00159752
$186$	3.66575e13			0.0649125
$187$		−	6.13614e14i		−	1.04936i
$188$			2.97620e13i			0.0491627i
$189$			1.20189e15i			1.91807i
$190$		−	1.02684e14i		−	0.158345i
$191$	7.07758e13			0.105480			0.0527398	−	0.998608i	$-0.483205\pi$
$191$	7.07758e13			0.105480			0.0527398	+	0.998608i	$0.483205\pi$
$192$	−1.39811e15			−2.01411
$193$		−	4.06497e14i		−	0.566155i	−0.959097	−	0.283077i	$-0.908645\pi$
$193$		−	4.06497e14i		−	0.566155i	0.959097	−	0.283077i	$-0.0913553\pi$
$194$	−7.63227e14			−1.02788
$195$	−1.33703e15	+	4.87205e13i	−1.74147	+	0.0634581i
$196$	3.22152e13			0.0405878
$197$		−	1.43964e15i		−	1.75479i	−0.479770	−	0.877394i	$-0.659280\pi$
$197$		−	1.43964e15i		−	1.75479i	0.479770	−	0.877394i	$-0.340720\pi$
$198$	3.58028e15			4.22273
$199$	1.38293e15			1.57854			0.789269	−	0.614048i	$-0.210460\pi$
$199$	1.38293e15			1.57854			0.789269	+	0.614048i	$0.210460\pi$
$200$		−	1.53211e14i		−	0.169276i
$201$			2.72669e15i			2.91650i
$202$			8.18988e14i			0.848191i
$203$		−	8.20946e14i		−	0.823361i
$204$	−8.11858e13			−0.0788650
$205$	7.21763e14			0.679195
$206$			3.78863e14i			0.345418i
$207$	−2.86729e15			−2.53317
$208$	3.95676e13	+	1.08585e15i	0.0338788	+	0.929732i
$209$	−3.58000e14			−0.297120
$210$		−	1.28934e15i		−	1.03739i
$211$	1.51265e15			1.18006			0.590029	−	0.807382i	$-0.299116\pi$
$211$	1.51265e15			1.18006			0.590029	+	0.807382i	$0.299116\pi$
$212$	−2.63765e13			−0.0199542
$213$			1.98082e15i			1.45338i
$214$		−	1.77861e15i		−	1.26587i
$215$		−	1.14361e15i		−	0.789636i
$216$		−	4.78868e15i		−	3.20824i
$217$	−3.33599e13			−0.0216888
$218$	−7.66690e14			−0.483785
$219$		−	2.19797e15i		−	1.34628i
$220$	1.66665e14			0.0991051
$221$	3.99115e13	+	1.09529e15i	0.0230435	+	0.632380i
$222$	−5.74435e12			−0.00322067
$223$			2.47515e15i			1.34778i	0.738831	+	0.673891i	$0.235378\pi$
$223$			2.47515e15i			1.34778i	−0.738831	+	0.673891i	$0.764622\pi$
$224$	1.51976e14			0.0803826
$225$	8.42512e14			0.432902
$226$		−	7.46333e14i		−	0.372587i
$227$		−	1.78560e15i		−	0.866195i	−0.901347	−	0.433097i	$-0.857421\pi$
$227$		−	1.78560e15i		−	0.866195i	0.901347	−	0.433097i	$-0.142579\pi$
$228$			4.73661e13i			0.0223301i
$229$		−	1.48202e15i		−	0.679084i	−0.940591	−	0.339542i	$-0.889728\pi$
$229$		−	1.48202e15i		−	0.679084i	0.940591	−	0.339542i	$-0.110272\pi$
$230$	1.90813e15			0.849920
$231$	−4.49519e15			−1.94657
$232$			3.27088e15i			1.37719i
$233$	−1.08461e15			−0.444079			−0.222039	−	0.975038i	$-0.571271\pi$
$233$	−1.08461e15			−0.444079			−0.222039	+	0.975038i	$0.571271\pi$
$234$	−6.39071e15	+	2.32873e14i	−2.54476	+	0.0927292i
$235$	1.77486e15			0.687420
$236$			1.60690e14i			0.0605424i
$237$	4.10961e15			1.50638
$238$	−1.05622e15			−0.376707
$239$			2.75278e15i			0.955401i	0.878523	+	0.477700i	$0.158530\pi$
$239$			2.75278e15i			0.955401i	−0.878523	+	0.477700i	$0.841470\pi$
$240$			4.79980e15i			1.62125i
$241$			2.34617e15i			0.771345i	0.922636	+	0.385672i	$0.126031\pi$
$241$			2.34617e15i			0.771345i	−0.922636	+	0.385672i	$0.873969\pi$
$242$			5.28606e15i			1.69173i
$243$	1.02174e16			3.18346
$244$	−4.81988e13			−0.0146218
$245$		−	1.92115e15i		−	0.567521i
$246$	4.75961e15			1.36928
$247$	6.39021e14	−	2.32855e13i	0.179054	−	0.00652462i
$248$	1.32915e14			0.0362776
$249$		−	7.79143e15i		−	2.07167i
$250$	−3.97214e15			−1.02900
$251$	−5.69252e15			−1.43690			−0.718448	−	0.695581i	$-0.755147\pi$
$251$	−5.69252e15			−1.43690			−0.718448	+	0.695581i	$0.755147\pi$
$252$			4.31086e14i			0.106038i
$253$		−	6.65258e15i		−	1.59480i
$254$			2.93120e15i			0.684899i
$255$			4.84152e15i			1.10273i
$256$	−8.79439e14			−0.195275
$257$	−1.80324e15			−0.390381			−0.195190	−	0.980765i	$-0.562532\pi$
$257$	−1.80324e15			−0.390381			−0.195190	+	0.980765i	$0.562532\pi$
$258$		−	7.54143e15i		−	1.59194i
$259$	5.22760e12			0.00107610
$260$	−2.97492e14	+	1.08404e13i	−0.0597240	+	0.00217630i
$261$	−1.79867e16			−3.52199
$262$			9.62975e15i			1.83932i
$263$	4.61101e15			0.859178			0.429589	−	0.903024i	$-0.358658\pi$
$263$	4.61101e15			0.859178			0.429589	+	0.903024i	$0.358658\pi$
$264$	1.79101e16			3.25591
$265$			1.57297e15i			0.279011i
$266$			6.16227e14i			0.106662i
$267$		−	4.58207e15i		−	0.773995i
$268$			6.06695e14i			0.100022i
$269$	6.36121e15			1.02365			0.511823	−	0.859091i	$-0.328970\pi$
$269$	6.36121e15			1.02365			0.511823	+	0.859091i	$0.328970\pi$
$270$	−1.75243e16			−2.75280
$271$		−	9.80335e15i		−	1.50340i	−0.659506	−	0.751700i	$-0.729234\pi$
$271$		−	9.80335e15i		−	1.50340i	0.659506	−	0.751700i	$-0.270766\pi$
$272$	3.93197e15			0.588724
$273$	8.02381e15	−	2.92382e14i	1.17307	−	0.0427458i
$274$	−9.57964e15			−1.36764
$275$			1.95477e15i			0.272541i
$276$	−8.80187e14			−0.119858
$277$	7.93739e14			0.105574			0.0527872	−	0.998606i	$-0.483189\pi$
$277$	7.93739e14			0.105574			0.0527872	+	0.998606i	$0.483189\pi$
$278$			1.55398e15i			0.201909i
$279$			7.30905e14i			0.0927756i
$280$		−	4.67498e15i		−	0.579765i
$281$			4.13004e15i			0.500452i	0.968187	+	0.250226i	$0.0805050\pi$
$281$			4.13004e15i			0.500452i	−0.968187	+	0.250226i	$0.919495\pi$
$282$	1.17042e16			1.38587
$283$	−4.88199e15			−0.564917			−0.282459	−	0.959280i	$-0.591150\pi$
$283$	−4.88199e15			−0.564917			−0.282459	+	0.959280i	$0.591150\pi$
$284$			4.40738e14i			0.0498438i
$285$	2.82468e15			0.312232
$286$	−5.40304e14	−	1.48275e16i	−0.0583793	−	1.60210i
$287$	−4.33146e15			−0.457511
$288$		−	3.32974e15i		−	0.343842i
$289$	−5.93844e15			−0.599565
$290$	1.19698e16			1.18168
$291$		−	2.09953e16i		−	2.02683i
$292$		−	4.89054e14i		−	0.0461709i
$293$		−	7.21337e15i		−	0.666037i	−0.942920	−	0.333019i	$-0.891933\pi$
$293$		−	7.21337e15i		−	0.666037i	0.942920	−	0.333019i	$-0.108067\pi$
$294$		−	1.26689e16i		−	1.14415i
$295$	9.58273e15			0.846537
$296$	−2.08283e13			−0.00179994
$297$			6.10971e16i			5.16540i
$298$	1.37472e16			1.13712
$299$	4.32706e14	+	1.18747e16i	0.0350211	+	0.961080i
$300$	2.58630e14			0.0204829
$301$			6.86303e15i			0.531905i
$302$	−6.83694e15			−0.518581
$303$	−2.25292e16			−1.67251
$304$		−	2.29402e15i		−	0.166694i
$305$			2.87433e15i			0.204451i
$306$			2.31414e16i			1.61139i
$307$		−	6.03857e15i		−	0.411656i	−0.978588	−	0.205828i	$-0.934011\pi$
$307$		−	6.03857e15i		−	0.411656i	0.978588	−	0.205828i	$-0.0659887\pi$
$308$	−1.00019e15			−0.0667579
$309$	−1.04220e16			−0.681114
$310$		−	4.86406e14i		−	0.0311277i
$311$	−1.68566e16			−1.05640			−0.528200	−	0.849120i	$-0.677133\pi$
$311$	−1.68566e16			−1.05640			−0.528200	+	0.849120i	$0.677133\pi$
$312$	−3.19691e16	+	1.16493e15i	−1.96212	+	0.0714984i
$313$	2.02486e15			0.121718			0.0608592	−	0.998146i	$-0.480616\pi$
$313$	2.02486e15			0.121718			0.0608592	+	0.998146i	$0.480616\pi$
$314$		−	4.98253e14i		−	0.0293364i
$315$	2.57078e16			1.48268
$316$	9.14397e14			0.0516617
$317$			1.40459e16i			0.777438i	0.921356	+	0.388719i	$0.127082\pi$
$317$			1.40459e16i			0.777438i	−0.921356	+	0.388719i	$0.872918\pi$
$318$			1.03728e16i			0.562497i
$319$		−	4.17321e16i		−	2.21733i
$320$			1.85514e16i			0.965832i
$321$	4.89269e16			2.49612
$322$	−1.14511e16			−0.572512
$323$		−	2.31396e15i		−	0.113381i
$324$	4.49784e15			0.216004
$325$	−1.27144e14	−	3.48921e15i	−0.00598488	−	0.164242i
$326$	1.18945e16			0.548822
$327$		−	2.10905e16i		−	0.953953i
$328$	1.72578e16			0.765252
$329$	−1.06513e16			−0.463051
$330$		−	6.55423e16i		−	2.79371i
$331$			1.06454e16i			0.444917i	0.974942	+	0.222458i	$0.0714081\pi$
$331$			1.06454e16i			0.444917i	−0.974942	+	0.222458i	$0.928592\pi$
$332$		−	1.73361e15i		−	0.0710484i
$333$		−	1.14535e14i		−	0.00460311i
$334$	−4.31335e16			−1.70006
$335$	3.61803e16			1.39856
$336$		−	2.88046e16i		−	1.09209i
$337$	−3.10858e16			−1.15603			−0.578014	−	0.816027i	$-0.696172\pi$
$337$	−3.10858e16			−1.15603			−0.578014	+	0.816027i	$0.696172\pi$
$338$	1.92886e15	+	2.64316e16i	0.0703627	+	0.964195i
$339$	2.05305e16			0.734687
$340$			1.07725e15i			0.0378184i
$341$	−1.69582e15			−0.0584085
$342$	1.35014e16			0.456255
$343$			3.01000e16i			0.998057i
$344$		−	2.73443e16i		−	0.889686i
$345$			5.24900e16i			1.67592i
$346$			3.07984e16i			0.965013i
$347$	−4.50444e16			−1.38516			−0.692579	−	0.721342i	$-0.743526\pi$
$347$	−4.50444e16			−1.38516			−0.692579	+	0.721342i	$0.743526\pi$
$348$	−5.52147e15			−0.166644
$349$			1.59465e16i			0.472391i	0.971706	+	0.236195i	$0.0759005\pi$
$349$			1.59465e16i			0.472391i	−0.971706	+	0.236195i	$0.924099\pi$
$350$	3.36475e15			0.0978387
$351$	−3.97396e15	−	1.09057e17i	−0.113430	−	3.11284i
$352$	7.72554e15			0.216472
$353$			2.81276e16i			0.773744i	0.922133	+	0.386872i	$0.126444\pi$
$353$			2.81276e16i			0.773744i	−0.922133	+	0.386872i	$0.873556\pi$
$354$	6.31926e16			1.70665
$355$	2.62834e16			0.696943
$356$		−	1.01952e15i		−	0.0265443i
$357$		−	2.90550e16i		−	0.742810i
$358$		−	5.07391e16i		−	1.27381i
$359$			1.42870e16i			0.352231i	0.984369	+	0.176116i	$0.0563533\pi$
$359$			1.42870e16i			0.352231i	−0.984369	+	0.176116i	$0.943647\pi$
$360$	−1.02427e17			−2.47999
$361$	4.07030e16			0.967897
$362$			1.63446e16i			0.381741i
$363$	−1.45412e17			−3.33585
$364$	1.78532e15	−	6.50557e13i	0.0402306	−	0.00146597i
$365$	−2.91648e16			−0.645587
$366$			1.89546e16i			0.412180i
$367$	5.30985e16			1.13437			0.567183	−	0.823592i	$-0.308033\pi$
$367$	5.30985e16			1.13437			0.567183	+	0.823592i	$0.308033\pi$
$368$	4.26289e16			0.894733
$369$			9.49009e16i			1.95704i
$370$			7.62213e13i			0.00154442i
$371$		−	9.43970e15i		−	0.187944i
$372$			2.24370e14i			0.00438971i
$373$	8.49419e16			1.63311			0.816553	−	0.577270i	$-0.195882\pi$
$373$	8.49419e16			1.63311			0.816553	+	0.577270i	$0.195882\pi$
$374$	−5.36919e16			−1.01448
$375$		−	1.09268e17i		−	2.02903i
$376$	4.24378e16			0.774518
$377$	2.71439e15	+	7.44907e16i	0.0486916	+	1.33624i
$378$	1.05167e17			1.85431
$379$			7.54106e16i			1.30701i	0.756924	+	0.653503i	$0.226701\pi$
$379$			7.54106e16i			1.30701i	−0.756924	+	0.653503i	$0.773299\pi$
$380$	6.28498e14			0.0107081
$381$	−8.06331e16			−1.35052
$382$		−	6.19296e15i		−	0.101973i
$383$		−	3.22518e15i		−	0.0522110i	−0.999659	−	0.0261055i	$-0.991689\pi$
$383$		−	3.22518e15i		−	0.0522110i	0.999659	−	0.0261055i	$-0.00831058\pi$
$384$			1.06701e17i			1.69831i
$385$			5.96464e16i			0.933446i
$386$	−3.55689e16			−0.547335
$387$	1.50367e17			2.27526
$388$		−	4.67150e15i		−	0.0695104i
$389$	6.43418e16			0.941500			0.470750	−	0.882267i	$-0.343983\pi$
$389$	6.43418e16			0.941500			0.470750	+	0.882267i	$0.343983\pi$
$390$	4.26309e15	+	1.16992e17i	0.0613486	+	1.68358i
$391$	4.29994e16			0.608575
$392$		−	4.59358e16i		−	0.639429i
$393$	−2.64900e17			−3.62686
$394$	−1.25970e17			−1.69646
$395$		−	5.45301e16i		−	0.722362i
$396$			2.19139e16i			0.285562i
$397$		−	8.57754e16i		−	1.09957i	−0.835305	−	0.549787i	$-0.814709\pi$
$397$		−	8.57754e16i		−	1.09957i	0.835305	−	0.549787i	$-0.185291\pi$
$398$		−	1.21008e17i		−	1.52607i
$399$	−1.69515e16			−0.210322
$400$	−1.25259e16			−0.152904
$401$		−	8.64300e16i		−	1.03807i	−0.854753	−	0.519034i	$-0.826292\pi$
$401$		−	8.64300e16i		−	1.03807i	0.854753	−	0.519034i	$-0.173708\pi$
$402$	2.38588e17			2.81955
$403$	3.02700e15	−	1.10302e14i	0.0351989	−	0.00128263i
$404$	−5.01279e15			−0.0573589
$405$		−	2.68229e17i		−	3.02028i
$406$	−7.18336e16			−0.795992
$407$	2.65740e14			0.00289797
$408$			1.15763e17i			1.24245i
$409$			8.84315e16i			0.934126i	0.884224	+	0.467063i	$0.154688\pi$
$409$			8.84315e16i			0.934126i	−0.884224	+	0.467063i	$0.845312\pi$
$410$		−	6.31550e16i		−	0.656618i
$411$		−	2.63522e17i		−	2.69678i
$412$	−2.31891e15			−0.0233589
$413$	−5.75080e16			−0.570234
$414$			2.50891e17i			2.44896i
$415$	−1.03384e17			−0.993437
$416$	−1.37899e16	+	5.02495e14i	−0.130453	+	0.00475363i
$417$	−4.27478e16			−0.398134
$418$			3.13253e16i			0.287243i
$419$	7.71802e16			0.696811			0.348405	−	0.937344i	$-0.386723\pi$
$419$	7.71802e16			0.696811			0.348405	+	0.937344i	$0.386723\pi$
$420$	7.89167e15			0.0701534
$421$		−	1.56354e17i		−	1.36859i	−0.729204	−	0.684297i	$-0.760109\pi$
$421$		−	1.56354e17i		−	1.36859i	0.729204	−	0.684297i	$-0.239891\pi$
$422$		−	1.32359e17i		−	1.14083i
$423$			2.33367e17i			1.98074i
$424$			3.76105e16i			0.314363i
$425$	−1.26348e16			−0.104001
$426$	1.73324e17			1.40506
$427$		−	1.72495e16i		−	0.137719i
$428$	1.08863e16			0.0856047
$429$	4.07883e17	−	1.48630e16i	3.15910	−	0.115115i
$430$	−1.00067e17			−0.763387
$431$			1.47738e17i			1.11017i	0.831794	+	0.555085i	$0.187314\pi$
$431$			1.47738e17i			1.11017i	−0.831794	+	0.555085i	$0.812686\pi$
$432$	−3.91503e17			−2.89795
$433$	9.85012e16			0.718241			0.359120	−	0.933291i	$-0.383077\pi$
$433$	9.85012e16			0.718241			0.359120	+	0.933291i	$0.383077\pi$
$434$			2.91903e15i			0.0209679i
$435$			3.29273e17i			2.33011i
$436$		−	4.69269e15i		−	0.0327160i
$437$		−	2.50871e16i		−	0.172314i
$438$	−1.92325e17			−1.30153
$439$	2.62228e16			0.174848			0.0874239	−	0.996171i	$-0.472137\pi$
$439$	2.62228e16			0.174848			0.0874239	+	0.996171i	$0.472137\pi$
$440$		−	2.37648e17i		−	1.56132i
$441$	2.52602e17			1.63526
$442$	9.58387e16	−	3.49230e15i	0.611359	−	0.0222775i
$443$	−6.03593e16			−0.379420			−0.189710	−	0.981840i	$-0.560755\pi$
$443$	−6.03593e16			−0.379420			−0.189710	+	0.981840i	$0.560755\pi$
$444$		−	3.51595e13i		−	0.000217798i
$445$	−6.07992e16			−0.371157
$446$	2.16578e17			1.30298
$447$			3.78165e17i			2.24224i
$448$		−	1.11331e17i		−	0.650592i
$449$		−	1.69726e17i		−	0.977569i	−0.872405	−	0.488785i	$-0.837440\pi$
$449$		−	1.69726e17i		−	0.977569i	0.872405	−	0.488785i	$-0.162560\pi$
$450$		−	7.37207e16i		−	0.418512i
$451$	−2.20186e17			−1.23209
$452$	4.56809e15			0.0251962
$453$		−	1.88074e17i		−	1.02256i
$454$	−1.56242e17			−0.837401
$455$	−3.87960e15	−	1.06467e17i	−0.0204981	−	0.562526i
$456$	6.75397e16			0.351793
$457$			1.32615e17i			0.680986i	0.940247	+	0.340493i	$0.110594\pi$
$457$			1.32615e17i			0.680986i	−0.940247	+	0.340493i	$0.889406\pi$
$458$	−1.29678e17			−0.656510
$459$	−3.94906e17			−1.97111
$460$			1.16791e16i			0.0574758i
$461$			6.00255e15i			0.0291259i	0.999894	+	0.0145630i	$0.00463570\pi$
$461$			6.00255e15i			0.0291259i	−0.999894	+	0.0145630i	$0.995364\pi$
$462$			3.93334e17i			1.88186i
$463$			3.23963e17i			1.52834i	0.645016	+	0.764169i	$0.276851\pi$
$463$			3.23963e17i			1.52834i	−0.645016	+	0.764169i	$0.723149\pi$
$464$	2.67414e17			1.24399
$465$	1.33803e16			0.0613793
$466$			9.49045e16i			0.429317i
$467$	1.53186e17			0.683373			0.341687	−	0.939814i	$-0.389002\pi$
$467$	1.53186e17			0.683373			0.341687	+	0.939814i	$0.389002\pi$
$468$	−1.42535e15	−	3.91157e16i	−0.00627081	−	0.172089i
$469$	−2.17126e17			−0.942080
$470$		−	1.55302e17i		−	0.664569i
$471$	1.37062e16			0.0578471
$472$	2.29128e17			0.953797
$473$			3.48876e17i			1.43243i
$474$		−	3.59595e17i		−	1.45631i
$475$			7.37149e15i			0.0294474i
$476$		−	6.46481e15i		−	0.0254748i
$477$	−2.06821e17			−0.803943
$478$	2.40871e17			0.923642
$479$		−	2.82516e16i		−	0.106872i	−0.998571	−	0.0534358i	$-0.982983\pi$
$479$		−	2.82516e16i		−	0.106872i	0.998571	−	0.0534358i	$-0.0170173\pi$
$480$	−6.09559e16			−0.227482
$481$	−4.74341e14	+	1.72847e13i	−0.00174641	+	6.36382e-5i
$482$	2.05292e17			0.745705
$483$		−	3.15004e17i		−	1.12891i
$484$	−3.23545e16			−0.114404
$485$	−2.78585e17			−0.971933
$486$		−	8.94033e17i		−	3.07764i
$487$		−	1.42186e17i		−	0.482968i	−0.970405	−	0.241484i	$-0.922366\pi$
$487$		−	1.42186e17i		−	0.482968i	0.970405	−	0.241484i	$-0.0776341\pi$
$488$			6.87269e16i			0.230355i
$489$			3.27200e17i			1.08220i
$490$	−1.68103e17			−0.548656
$491$	1.85733e17			0.598218			0.299109	−	0.954219i	$-0.403311\pi$
$491$	1.85733e17			0.598218			0.299109	+	0.954219i	$0.403311\pi$
$492$			2.91322e16i			0.0925978i
$493$	2.69738e17			0.846131
$494$	−2.03751e15	−	5.59150e16i	−0.00630774	−	0.173102i
$495$	1.30683e18			3.99288
$496$		−	1.08666e16i		−	0.0327690i
$497$	−1.57732e17			−0.469466
$498$	−6.81758e17			−2.00281
$499$		−	4.33272e17i		−	1.25634i	−0.778076	−	0.628170i	$-0.783804\pi$
$499$		−	4.33272e17i		−	1.25634i	0.778076	−	0.628170i	$-0.216196\pi$
$500$		−	2.43123e16i		−	0.0695859i
$501$		−	1.18654e18i		−	3.35226i
$502$			4.98101e17i			1.38913i
$503$	−3.06362e17			−0.843419			−0.421709	−	0.906731i	$-0.638570\pi$
$503$	−3.06362e17			−0.843419			−0.421709	+	0.906731i	$0.638570\pi$
$504$	6.14688e17			1.67054
$505$			2.98938e17i			0.802024i
$506$	−5.82107e17			−1.54179
$507$	−7.27095e17	+	5.30602e16i	−1.90125	+	0.138745i
$508$	−1.79411e16			−0.0463163
$509$			2.56714e17i			0.654311i	0.944971	+	0.327155i	$0.106090\pi$
$509$			2.56714e17i			0.654311i	−0.944971	+	0.327155i	$0.893910\pi$
$510$	4.23638e17			1.06608
$511$	1.75024e17			0.434873
$512$			4.40095e17i			1.07967i
$513$			2.30399e17i			0.558108i
$514$			1.57785e17i			0.377404i
$515$			1.38288e17i			0.326617i
$516$	4.61590e16			0.107655
$517$	−5.41449e17			−1.24701
$518$		−	4.57421e14i		−	0.00104033i
$519$	−8.47218e17			−1.90286
$520$	1.54574e16	+	4.24196e17i	0.0342859	+	0.940904i
$521$	−2.71617e17			−0.594993			−0.297496	−	0.954723i	$-0.596152\pi$
$521$	−2.71617e17			−0.594993			−0.297496	+	0.954723i	$0.596152\pi$
$522$			1.57385e18i			3.40491i
$523$	7.18991e17			1.53625			0.768127	−	0.640298i	$-0.221189\pi$
$523$	7.18991e17			1.53625			0.768127	+	0.640298i	$0.221189\pi$
$524$	−5.89409e16			−0.124384
$525$			9.25594e16i			0.192923i
$526$		−	4.03468e17i		−	0.830618i
$527$		−	1.09611e16i		−	0.0222886i
$528$		−	1.46426e18i		−	2.94101i
$529$	−3.78523e16			−0.0750984
$530$	1.37636e17			0.269736
$531$			1.25998e18i			2.43922i
$532$	−3.77175e15			−0.00721302
$533$	3.93026e17	−	1.43216e16i	0.742497	−	0.0270561i
$534$	−4.00936e17			−0.748267
$535$		−	6.49207e17i		−	1.19697i
$536$	8.65091e17			1.57576
$537$	1.39576e18			2.51176
$538$		−	5.56612e17i		−	0.989619i
$539$			5.86079e17i			1.02951i
$540$		−	1.07261e17i		−	0.186158i
$541$		−	5.41301e16i		−	0.0928232i	−0.998922	−	0.0464116i	$-0.985221\pi$
$541$		−	5.41301e16i		−	0.0928232i	0.998922	−	0.0464116i	$-0.0147786\pi$
$542$	−8.57804e17			−1.45342
$543$	−4.49616e17			−0.752737
$544$			4.99346e16i			0.0826055i
$545$	−2.79849e17			−0.457453
$546$	−2.55837e16	−	7.02091e17i	−0.0413249	−	1.13407i
$547$	1.79153e17			0.285961			0.142980	−	0.989726i	$-0.454331\pi$
$547$	1.79153e17			0.285961			0.142980	+	0.989726i	$0.454331\pi$
$548$		−	5.86342e16i		−	0.0924864i
$549$	−3.77931e17			−0.589105
$550$	1.71044e17			0.263482
$551$		−	1.57373e17i		−	0.239577i
$552$			1.25506e18i			1.88826i
$553$			3.27247e17i			0.486589i
$554$		−	6.94529e16i		−	0.102065i
$555$	−2.09674e15			−0.00304537
$556$	−9.51148e15			−0.0136541
$557$		−	7.43876e16i		−	0.105546i	−0.998607	−	0.0527730i	$-0.983194\pi$
$557$		−	7.43876e16i		−	0.105546i	0.998607	−	0.0527730i	$-0.0168060\pi$
$558$	6.39550e16			0.0896916
$559$	−2.26921e16	−	6.22735e17i	−0.0314555	−	0.863231i
$560$	−3.82207e17			−0.523693
$561$		−	1.47698e18i		−	2.00040i
$562$	3.61382e17			0.483817
$563$	5.58736e17			0.739438			0.369719	−	0.929144i	$-0.379454\pi$
$563$	5.58736e17			0.739438			0.369719	+	0.929144i	$0.379454\pi$
$564$			7.16378e16i			0.0937191i
$565$		−	2.72418e17i		−	0.352307i
$566$			4.27179e17i			0.546139i
$567$			1.60970e18i			2.03448i
$568$	6.28451e17			0.785248
$569$	6.17454e17			0.762737			0.381368	−	0.924423i	$-0.375453\pi$
$569$	6.17454e17			0.762737			0.381368	+	0.924423i	$0.375453\pi$
$570$		−	2.47162e17i		−	0.301853i
$571$	4.66410e17			0.563162			0.281581	−	0.959537i	$-0.409141\pi$
$571$	4.66410e17			0.563162			0.281581	+	0.959537i	$0.409141\pi$
$572$	9.07549e16	−	3.30705e15i	0.108342	−	0.00394790i
$573$	1.70359e17			0.201076
$574$			3.79007e17i			0.442303i
$575$	−1.36982e17			−0.158060
$576$	−2.43923e18			−2.78295
$577$		−	1.40520e18i		−	1.58524i	−0.609713	−	0.792622i	$-0.708715\pi$
$577$		−	1.40520e18i		−	1.58524i	0.609713	−	0.792622i	$-0.291285\pi$
$578$			5.19619e17i			0.579635i
$579$		−	9.78450e17i		−	1.07926i
$580$			7.32640e16i			0.0799114i
$581$	6.20429e17			0.669187
$582$	−1.83711e18			−1.95945
$583$		−	4.79859e17i		−	0.506137i
$584$	−6.97346e17			−0.727386
$585$	−2.33267e18	+	8.50009e16i	−2.40625	+	0.0876820i
$586$	−6.31177e17			−0.643897
$587$			3.58281e17i			0.361473i	0.983532	+	0.180737i	$0.0578482\pi$
$587$			3.58281e17i			0.361473i	−0.983532	+	0.180737i	$0.942152\pi$
$588$	7.75427e16			0.0773728
$589$	−6.39499e15			−0.00631089
$590$		−	8.38499e17i		−	0.818398i
$591$		−	3.46526e18i		−	3.34516i
$592$			1.70283e15i			0.00162585i
$593$		−	2.27371e17i		−	0.214724i	−0.994220	−	0.107362i	$-0.965760\pi$
$593$		−	2.27371e17i		−	0.214724i	0.994220	−	0.107362i	$-0.0342403\pi$
$594$	5.34606e18			4.99370
$595$	−3.85529e17			−0.356202
$596$			8.41426e16i			0.0768980i
$597$	3.32875e18			3.00918
$598$	1.03905e18	−	3.78622e16i	0.929133	−	0.0338570i
$599$	−1.22701e18			−1.08536			−0.542681	−	0.839939i	$-0.682591\pi$
$599$	−1.22701e18			−1.08536			−0.542681	+	0.839939i	$0.682591\pi$
$600$		−	3.68783e17i		−	0.322692i
$601$	−1.49251e18			−1.29191			−0.645957	−	0.763374i	$-0.723541\pi$
$601$	−1.49251e18			−1.29191			−0.645957	+	0.763374i	$0.723541\pi$
$602$	6.00522e17			0.514224
$603$			4.75716e18i			4.02982i
$604$		−	4.18470e16i		−	0.0350690i
$605$			1.92946e18i			1.59965i
$606$			1.97132e18i			1.61691i
$607$	−1.66922e18			−1.35452			−0.677262	−	0.735742i	$-0.736834\pi$
$607$	−1.66922e18			−1.35452			−0.677262	+	0.735742i	$0.736834\pi$
$608$	2.91333e16			0.0233892
$609$		−	1.97604e18i		−	1.56958i
$610$	2.51507e17			0.197654
$611$	9.66474e17	−	3.52177e16i	0.751488	−	0.0273837i
$612$	−1.41642e17			−0.108970
$613$		−	9.85345e17i		−	0.750059i	−0.927013	−	0.375029i	$-0.877633\pi$
$613$		−	9.85345e17i		−	0.750059i	0.927013	−	0.375029i	$-0.122367\pi$
$614$	−5.28381e17			−0.397972
$615$	1.73730e18			1.29475
$616$			1.42618e18i			1.05172i
$617$			1.96101e18i			1.43096i	0.698636	+	0.715478i	$0.253791\pi$
$617$			1.96101e18i			1.43096i	−0.698636	+	0.715478i	$0.746209\pi$
$618$			9.11933e17i			0.658473i
$619$		−	1.98675e18i		−	1.41956i	−0.704424	−	0.709780i	$-0.748794\pi$
$619$		−	1.98675e18i		−	1.41956i	0.704424	−	0.709780i	$-0.251206\pi$
$620$	2.97715e15			0.00210501
$621$	−4.28143e18			−2.99566
$622$			1.47497e18i			1.02128i
$623$	3.64869e17			0.250014
$624$	9.52402e16	+	2.61367e18i	0.0645834	+	1.77235i
$625$	−1.20496e18			−0.808637
$626$		−	1.77177e17i		−	0.117672i
$627$	−8.61715e17			−0.566402
$628$	3.04967e15			0.00198388
$629$			1.71763e15i			0.00110586i
$630$		−	2.24946e18i		−	1.43339i
$631$			6.31756e17i			0.398436i	0.979955	+	0.199218i	$0.0638402\pi$
$631$			6.31756e17i			0.398436i	−0.979955	+	0.199218i	$0.936160\pi$
$632$		−	1.30384e18i		−	0.813888i
$633$	3.64100e18			2.24955
$634$	1.22903e18			0.751595
$635$			1.06992e18i			0.647620i
$636$	−6.34890e16			−0.0380388
$637$	−3.81205e16	−	1.04614e18i	−0.0226075	−	0.620415i
$638$	−3.65160e18			−2.14362
$639$			3.45587e18i			2.00818i
$640$	1.41581e18			0.814395
$641$	2.69186e18			1.53276			0.766381	−	0.642386i	$-0.222056\pi$
$641$	2.69186e18			1.53276			0.766381	+	0.642386i	$0.222056\pi$
$642$		−	4.28115e18i		−	2.41314i
$643$			1.54575e18i			0.862517i	0.902228	+	0.431259i	$0.141930\pi$
$643$			1.54575e18i			0.862517i	−0.902228	+	0.431259i	$0.858070\pi$
$644$		−	7.00891e16i		−	0.0387161i
$645$		−	2.75269e18i		−	1.50529i
$646$	−2.02474e17			−0.109612
$647$	−2.64371e18			−1.41689			−0.708444	−	0.705767i	$-0.750602\pi$
$647$	−2.64371e18			−1.41689			−0.708444	+	0.705767i	$0.750602\pi$
$648$		−	6.41349e18i		−	3.40296i
$649$	−2.92337e18			−1.53565
$650$	−3.05309e17	+	1.11253e16i	−0.158783	+	0.00578594i
$651$	−8.02981e16			−0.0413456
$652$			7.28029e16i			0.0371141i
$653$	3.47578e18			1.75435			0.877175	−	0.480170i	$-0.159425\pi$
$653$	3.47578e18			1.75435			0.877175	+	0.480170i	$0.159425\pi$
$654$	−1.84544e18			−0.922242
$655$			3.51495e18i			1.73920i
$656$		−	1.41092e18i		−	0.691239i
$657$		−	3.83473e18i		−	1.86020i
$658$			9.31999e17i			0.447659i
$659$	−6.64948e17			−0.316251			−0.158126	−	0.987419i	$-0.550545\pi$
$659$	−6.64948e17			−0.316251			−0.158126	+	0.987419i	$0.550545\pi$
$660$	4.01166e17			0.188925
$661$			1.05716e18i			0.492981i	0.969145	+	0.246491i	$0.0792774\pi$
$661$			1.05716e18i			0.492981i	−0.969145	+	0.246491i	$0.920723\pi$
$662$	9.31481e17			0.430127
$663$	9.60681e16	+	2.63638e18i	0.0439280	+	1.20551i
$664$	−2.47197e18			−1.11931
$665$			2.24929e17i			0.100856i
$666$	−1.00220e16			−0.00445010
$667$	2.92440e18			1.28594
$668$		−	2.64008e17i		−	0.114966i
$669$			5.95774e18i			2.56928i
$670$		−	3.16581e18i		−	1.35207i
$671$		−	8.76863e17i		−	0.370882i
$672$	3.65809e17			0.153234
$673$	9.84944e17			0.408614			0.204307	−	0.978907i	$-0.434506\pi$
$673$	9.84944e17			0.408614			0.204307	+	0.978907i	$0.434506\pi$
$674$			2.72004e18i			1.11760i
$675$	1.25804e18			0.511940
$676$	−1.61780e17	+	1.18060e16i	−0.0652037	+	0.00475828i
$677$	2.66054e18			1.06205			0.531023	−	0.847357i	$-0.321808\pi$
$677$	2.66054e18			1.06205			0.531023	+	0.847357i	$0.321808\pi$
$678$		−	1.79644e18i		−	0.710265i
$679$	1.67185e18			0.654701
$680$	1.53606e18			0.595799
$681$		−	4.29798e18i		−	1.65123i
$682$			1.48386e17i			0.0564670i
$683$		−	5.21508e17i		−	0.196574i	−0.995158	−	0.0982871i	$-0.968664\pi$
$683$		−	5.21508e17i		−	0.196574i	0.995158	−	0.0982871i	$-0.0313364\pi$
$684$			8.26379e16i			0.0308542i
$685$	−3.49666e18			−1.29320
$686$	2.63379e18			0.964881
$687$		−	3.56726e18i		−	1.29454i
$688$	−2.23556e18			−0.803639
$689$	3.12116e16	+	8.56537e17i	0.0111145	+	0.305015i
$690$	4.59293e18			1.62021
$691$		−	3.35433e18i		−	1.17219i	−0.810242	−	0.586096i	$-0.800664\pi$
$691$		−	3.35433e18i		−	1.17219i	0.810242	−	0.586096i	$-0.199336\pi$
$692$	−1.88508e17			−0.0652590
$693$	−7.84259e18			−2.68964
$694$			3.94144e18i			1.33911i
$695$			5.67217e17i			0.190919i
$696$			7.87310e18i			2.62534i
$697$		−	1.42319e18i		−	0.470163i
$698$	1.39534e18			0.456688
$699$	−2.61068e18			−0.846550
$700$			2.05947e16i			0.00661634i
$701$	−2.15474e18			−0.685848			−0.342924	−	0.939363i	$-0.611417\pi$
$701$	−2.15474e18			−0.685848			−0.342924	+	0.939363i	$0.611417\pi$
$702$	−9.54260e18	+	3.47726e17i	−3.00937	+	0.109659i
$703$	1.00212e15			0.000313118
$704$		−	5.65940e18i		−	1.75206i
$705$	4.27212e18			1.31043
$706$	2.46120e18			0.748024
$707$		−	1.79399e18i		−	0.540249i
$708$			3.86784e17i			0.115412i
$709$			6.64329e18i			1.96419i	0.188393	+	0.982094i	$0.439672\pi$
$709$			6.64329e18i			1.96419i	−0.188393	+	0.982094i	$0.560328\pi$
$710$		−	2.29982e18i		−	0.673776i
$711$	7.16988e18			2.08142
$712$	−1.45374e18			−0.418184
$713$		−	1.18836e17i		−	0.0338739i
$714$	−2.54234e18			−0.718118
$715$	−1.97216e17	−	5.41217e18i	−0.0552017	−	1.51489i
$716$	3.10560e17			0.0861411
$717$			6.62602e18i			1.82129i
$718$	1.25013e18			0.340523
$719$	−5.03748e18			−1.35980			−0.679901	−	0.733304i	$-0.737977\pi$
$719$	−5.03748e18			−1.35980			−0.679901	+	0.733304i	$0.737977\pi$
$720$			8.37404e18i			2.24013i
$721$		−	8.29899e17i		−	0.220012i
$722$		−	3.56155e18i		−	0.935723i
$723$			5.64729e18i			1.47042i
$724$	−1.00041e17			−0.0258152
$725$	−8.59295e17			−0.219758
$726$			1.27237e19i			3.22496i
$727$	2.44730e18			0.614770			0.307385	−	0.951585i	$-0.400546\pi$
$727$	2.44730e18			0.614770			0.307385	+	0.951585i	$0.400546\pi$
$728$	−9.27634e16	−	2.54569e18i	−0.0230953	−	0.633800i
$729$	1.12040e19			2.76469
$730$			2.55195e18i			0.624127i
$731$	−2.25499e18			−0.546614
$732$	−1.16016e17			−0.0278737
$733$			4.52485e18i			1.07753i	0.842457	+	0.538763i	$0.181108\pi$
$733$			4.52485e18i			1.07753i	−0.842457	+	0.538763i	$0.818892\pi$
$734$		−	4.64618e18i		−	1.09666i
$735$		−	4.62426e18i		−	1.08187i
$736$			5.41373e17i			0.125543i
$737$	−1.10374e19			−2.53704
$738$	8.30393e18			1.89198
$739$		−	5.57733e18i		−	1.25962i	−0.776751	−	0.629808i	$-0.783134\pi$
$739$		−	5.57733e18i		−	1.25962i	0.776751	−	0.629808i	$-0.216866\pi$
$740$	−4.66529e14			−0.000104441
$741$	1.53814e18	−	5.60488e16i	0.341332	−	0.0124379i
$742$	−8.25984e17			−0.181696
$743$		−	1.21489e18i		−	0.264916i	−0.991189	−	0.132458i	$-0.957713\pi$
$743$		−	1.21489e18i		−	0.264916i	0.991189	−	0.132458i	$-0.0422870\pi$
$744$	3.19931e17			0.0691563
$745$	5.01784e18			1.07523
$746$		−	7.43251e18i		−	1.57882i
$747$		−	1.35934e19i		−	2.86250i
$748$		−	3.28633e17i		−	0.0686041i
$749$			3.89603e18i			0.806289i
$750$	−9.56104e18			−1.96158
$751$	2.90122e18			0.590093			0.295047	−	0.955483i	$-0.404665\pi$
$751$	2.90122e18			0.590093			0.295047	+	0.955483i	$0.404665\pi$
$752$		−	3.46954e18i		−	0.699610i
$753$	−1.37020e19			−2.73916
$754$	6.51802e18	−	2.37512e17i	1.29182	−	0.0470730i
$755$	−2.49554e18			−0.490354
$756$			6.43697e17i			0.125398i
$757$	7.74004e18			1.49493			0.747463	−	0.664303i	$-0.231272\pi$
$757$	7.74004e18			1.49493			0.747463	+	0.664303i	$0.231272\pi$
$758$	6.59851e18			1.26356
$759$		−	1.60129e19i		−	3.04018i
$760$		−	8.96179e17i		−	0.168697i
$761$			3.77780e18i			0.705081i	0.935796	+	0.352541i	$0.114682\pi$
$761$			3.77780e18i			0.705081i	−0.935796	+	0.352541i	$0.885318\pi$
$762$			7.05548e18i			1.30563i
$763$	1.67943e18			0.308144
$764$	3.79053e16			0.00689594
$765$			8.44683e18i			1.52368i
$766$	−2.82207e17			−0.0504755
$767$	5.21814e18	−	1.90146e17i	0.925435	−	0.0337223i
$768$	−2.11683e18			−0.372253
$769$			5.74448e17i			0.100168i	0.998745	+	0.0500841i	$0.0159489\pi$
$769$			5.74448e17i			0.100168i	−0.998745	+	0.0500841i	$0.984051\pi$
$770$	5.21912e18			0.902418
$771$	−4.34045e18			−0.744185
$772$		−	2.17707e17i		−	0.0370135i
$773$			2.70309e18i			0.455715i	0.973694	+	0.227858i	$0.0731721\pi$
$773$			2.70309e18i			0.455715i	−0.973694	+	0.227858i	$0.926828\pi$
$774$		−	1.31573e19i		−	2.19963i
$775$			3.49182e16i			0.00578883i
$776$	−6.66112e18			−1.09508
$777$	1.25830e16			0.00205138
$778$		−	5.62997e18i		−	0.910204i
$779$	−8.30327e17			−0.133124
$780$	−7.16072e17	+	2.60932e16i	−0.113852	+	0.00414870i
$781$	−8.01818e18			−1.26428
$782$		−	3.76250e18i		−	0.588345i
$783$	−2.68577e19			−4.16502
$784$	−3.75552e18			−0.577585
$785$		−	1.81867e17i		−	0.0277397i
$786$			2.31791e19i			3.50630i
$787$			1.21559e19i			1.82369i	0.410539	+	0.911843i	$0.365341\pi$
$787$			1.21559e19i			1.82369i	−0.410539	+	0.911843i	$0.634659\pi$
$788$		−	7.71029e17i		−	0.114723i
$789$	1.10988e19			1.63786
$790$	−4.77144e18			−0.698350
$791$			1.63484e18i			0.237317i
$792$	3.12471e19			4.49880
$793$	5.70341e16	+	1.56518e18i	0.00814439	+	0.223506i
$794$	−7.50544e18			−1.06302
$795$			3.78617e18i			0.531880i
$796$	7.40654e17			0.103200
$797$	1.19259e19			1.64821			0.824103	−	0.566440i	$-0.191680\pi$
$797$	1.19259e19			1.64821			0.824103	+	0.566440i	$0.191680\pi$
$798$			1.48327e18i			0.203331i
$799$		−	3.49970e18i		−	0.475857i
$800$		−	1.59075e17i		−	0.0214544i
$801$		−	7.99417e18i		−	1.06945i
$802$	−7.56271e18			−1.00356
$803$	8.89719e18			1.17112
$804$			1.46033e18i			0.190672i
$805$	−4.17976e18			−0.541350
$806$	−9.65152e15	−	2.64865e17i	−0.00123999	−	0.0340289i
$807$	1.53116e19			1.95138
$808$			7.14777e18i			0.903643i
$809$	−2.02703e18			−0.254211			−0.127106	−	0.991889i	$-0.540569\pi$
$809$	−2.02703e18			−0.254211			−0.127106	+	0.991889i	$0.540569\pi$
$810$	−2.34703e19			−2.91988
$811$		−	5.40615e18i		−	0.667195i	−0.942716	−	0.333597i	$-0.891737\pi$
$811$		−	5.40615e18i		−	0.667195i	0.942716	−	0.333597i	$-0.108263\pi$
$812$		−	4.39673e17i		−	0.0538290i
$813$		−	2.35969e19i		−	2.86594i
$814$		−	2.32526e16i		−	0.00280164i
$815$	4.34160e18			0.518950
$816$	9.46435e18			1.12229
$817$			1.31562e18i			0.154770i
$818$	7.73785e18			0.903075
$819$	1.39988e19	−	5.10109e17i	1.62087	−	0.0590632i
$820$	3.86554e17			0.0444038
$821$			1.19996e19i			1.36753i	0.729704	+	0.683763i	$0.239658\pi$
$821$			1.19996e19i			1.36753i	−0.729704	+	0.683763i	$0.760342\pi$
$822$	−2.30584e19			−2.60713
$823$	−7.49361e17			−0.0840605			−0.0420303	−	0.999116i	$-0.513383\pi$
$823$	−7.49361e17			−0.0840605			−0.0420303	+	0.999116i	$0.513383\pi$
$824$			3.30655e18i			0.368001i
$825$			4.70517e18i			0.519547i
$826$			5.03201e18i			0.551279i
$827$			9.04543e18i			0.983204i	0.870820	+	0.491602i	$0.163588\pi$
$827$			9.04543e18i			0.983204i	−0.870820	+	0.491602i	$0.836412\pi$
$828$	−1.53563e18			−0.165611
$829$	5.57893e18			0.596962			0.298481	−	0.954416i	$-0.403520\pi$
$829$	5.57893e18			0.596962			0.298481	+	0.954416i	$0.403520\pi$
$830$			9.04620e18i			0.960414i
$831$	1.91055e18			0.201257
$832$	3.68107e17	+	1.01019e19i	0.0384744	+	1.05585i
$833$	−3.78817e18			−0.392859
$834$			3.74047e18i			0.384899i
$835$	−1.57441e19			−1.60752
$836$	−1.91734e17			−0.0194248
$837$			1.09139e18i			0.109714i
$838$		−	6.75335e18i		−	0.673648i
$839$		−	1.48570e19i		−	1.47054i	−0.677774	−	0.735271i	$-0.737055\pi$
$839$		−	1.48570e19i		−	1.47054i	0.677774	−	0.735271i	$-0.262945\pi$
$840$		−	1.12528e19i		−	1.10521i
$841$	8.08435e18			0.787900
$842$	−1.36811e19			−1.32310
$843$			9.94110e18i			0.954015i
$844$	8.10130e17			0.0771488
$845$	7.04052e17	+	9.64777e18i	0.0665328	+	0.911714i
$846$	2.04198e19			1.91489
$847$		−	1.15791e19i		−	1.07754i
$848$	3.07488e18			0.283959
$849$	−1.17511e19			−1.07690
$850$			1.10556e18i			0.100544i
$851$			1.86220e16i			0.00168067i
$852$			1.06087e18i			0.0950175i
$853$			1.30706e19i			1.16179i	0.813978	+	0.580895i	$0.197297\pi$
$853$			1.30706e19i			1.16179i	−0.813978	+	0.580895i	$0.802703\pi$
$854$	−1.50935e18			−0.133142
$855$	4.92812e18			0.431421
$856$		−	1.55229e19i		−	1.34863i
$857$	−1.17760e19			−1.01536			−0.507682	−	0.861544i	$-0.669498\pi$
$857$	−1.17760e19			−1.01536			−0.507682	+	0.861544i	$0.669498\pi$
$858$	−1.30053e18	−	3.56902e19i	−0.111289	−	3.05409i
$859$	−1.70251e19			−1.44588			−0.722942	−	0.690909i	$-0.757210\pi$
$859$	−1.70251e19			−1.44588			−0.722942	+	0.690909i	$0.757210\pi$
$860$		−	6.12480e17i		−	0.0516241i
$861$	−1.04259e19			−0.872156
$862$	1.29272e19			1.07327
$863$			1.24318e19i			1.02439i	0.858870	+	0.512194i	$0.171167\pi$
$863$			1.24318e19i			1.02439i	−0.858870	+	0.512194i	$0.828833\pi$
$864$		−	4.97196e18i		−	0.406620i
$865$			1.12417e19i			0.912487i
$866$		−	8.61895e18i		−	0.694366i
$867$	−1.42940e19			−1.14295
$868$	−1.78665e16			−0.00141795
$869$			1.66353e19i			1.31039i
$870$	2.88117e19			2.25265
$871$	1.97015e19	−	7.17909e17i	1.52891	−	0.0557123i
$872$	−6.69134e18			−0.515413
$873$		−	3.66297e19i		−	2.80053i
$874$	−2.19515e18			−0.166586
$875$	8.70096e18			0.655413
$876$		−	1.17717e18i		−	0.0880159i
$877$		−	2.36133e19i		−	1.75251i	−0.481850	−	0.876254i	$-0.660035\pi$
$877$		−	2.36133e19i		−	1.75251i	0.481850	−	0.876254i	$-0.339965\pi$
$878$		−	2.29452e18i		−	0.169036i
$879$		−	1.73628e19i		−	1.26967i
$880$	−1.94291e19			−1.41032
$881$	−5.00094e17			−0.0360336			−0.0180168	−	0.999838i	$-0.505735\pi$
$881$	−5.00094e17			−0.0360336			−0.0180168	+	0.999838i	$0.505735\pi$
$882$		−	2.21030e19i		−	1.58090i
$883$	−4.54937e18			−0.323003			−0.161501	−	0.986872i	$-0.551634\pi$
$883$	−4.54937e18			−0.323003			−0.161501	+	0.986872i	$0.551634\pi$
$884$	2.13754e16	+	5.86602e17i	0.00150652	+	0.0413431i
$885$	2.30659e19			1.61376
$886$			5.28150e18i			0.366807i
$887$	1.04938e19			0.723484			0.361742	−	0.932278i	$-0.382182\pi$
$887$	1.04938e19			0.723484			0.361742	+	0.932278i	$0.382182\pi$
$888$	−5.01342e16			−0.00343123
$889$		−	6.42079e18i		−	0.436242i
$890$			5.31999e18i			0.358819i
$891$			8.18275e19i			5.47891i
$892$			1.32561e18i			0.0881140i
$893$	−2.04182e18			−0.134736
$894$	3.30898e19			2.16771
$895$		−	1.85202e19i		−	1.20447i
$896$	−8.49658e18			−0.548583
$897$	1.04153e18	+	2.85827e19i	0.0667610	+	1.83211i
$898$	−1.48512e19			−0.945074
$899$		−	7.45465e17i		−	0.0470966i
$900$	4.51223e17			0.0283019
$901$	3.10160e18			0.193141
$902$			1.92665e19i			1.19113i
$903$			1.65195e19i			1.01397i
$904$		−	6.51368e18i		−	0.396946i
$905$			5.96593e18i			0.360963i
$906$	−1.64567e19			−0.988574
$907$	−8.50215e18			−0.507086			−0.253543	−	0.967324i	$-0.581596\pi$
$907$	−8.50215e18			−0.507086			−0.253543	+	0.967324i	$0.581596\pi$
$908$		−	9.56310e17i		−	0.0566293i
$909$	−3.93058e19			−2.31095
$910$	−9.31601e18	+	3.39469e17i	−0.543827	+	0.0198167i
$911$	−6.48478e18			−0.375859			−0.187930	−	0.982182i	$-0.560178\pi$
$911$	−6.48478e18			−0.375859			−0.187930	+	0.982182i	$0.560178\pi$
$912$		−	5.52177e18i		−	0.317769i
$913$	3.15390e19			1.80213
$914$	1.16040e19			0.658349
$915$			6.91860e18i			0.389745i
$916$		−	7.93724e17i		−	0.0443965i
$917$		−	2.10939e19i		−	1.17154i
$918$			3.45547e19i			1.90559i
$919$	−1.40012e19			−0.766682			−0.383341	−	0.923607i	$-0.625227\pi$
$919$	−1.40012e19			−0.766682			−0.383341	+	0.923607i	$0.625227\pi$
$920$	1.66534e19			0.905485
$921$		−	1.45350e19i		−	0.784742i
$922$	5.25229e17			0.0281577
$923$	1.43123e19	−	5.21529e17i	0.761899	−	0.0277631i
$924$	−2.40748e18			−0.127261
$925$		−	5.47180e15i		−	0.000287216i
$926$	2.83471e19			1.47753
$927$	−1.81828e19			−0.941116
$928$			3.39607e18i			0.174548i
$929$			3.60756e18i			0.184124i	0.995753	+	0.0920621i	$0.0293458\pi$
$929$			3.60756e18i			0.184124i	−0.995753	+	0.0920621i	$0.970654\pi$
$930$		−	1.17079e18i		−	0.0593390i
$931$			2.21012e18i			0.111236i
$932$	−5.80884e17			−0.0290326
$933$	−4.05743e19			−2.01382
$934$		−	1.34039e19i		−	0.660657i
$935$	−1.95980e19			−0.959261
$936$	−5.57754e19	+	2.03242e18i	−2.71113	+	0.0987916i
$937$	3.43026e19			1.65585			0.827923	−	0.560841i	$-0.189522\pi$
$937$	3.43026e19			1.65585			0.827923	+	0.560841i	$0.189522\pi$
$938$			1.89987e19i			0.910764i
$939$	4.87388e18			0.232032
$940$	9.50558e17			0.0449415
$941$		−	1.71617e19i		−	0.805802i	−0.915244	−	0.402901i	$-0.868002\pi$
$941$		−	1.71617e19i		−	0.805802i	0.915244	−	0.402901i	$-0.131998\pi$
$942$		−	1.19931e18i		−	0.0559242i
$943$		−	1.54297e19i		−	0.714547i
$944$		−	1.87326e19i		−	0.861549i
$945$	3.83869e19			1.75338
$946$	3.05270e19			1.38482
$947$		−	9.83424e18i		−	0.443063i	−0.975153	−	0.221532i	$-0.928894\pi$
$947$		−	9.83424e18i		−	0.443063i	0.975153	−	0.221532i	$-0.0711056\pi$
$948$	2.20098e18			0.0984830
$949$	−1.58813e19	+	5.78703e17i	−0.705757	+	0.0257173i
$950$	6.45013e17			0.0284685
$951$			3.38090e19i			1.48203i
$952$	−9.21821e18			−0.401335
$953$	−3.20088e19			−1.38409			−0.692047	−	0.721852i	$-0.743291\pi$
$953$	−3.20088e19			−1.38409			−0.692047	+	0.721852i	$0.743291\pi$
$954$			1.80971e19i			0.777219i
$955$		−	2.26049e18i		−	0.0964229i
$956$			1.47431e18i			0.0624613i
$957$		−	1.00450e20i		−	4.22691i
$958$	−2.47205e18			−0.103319
$959$	2.09842e19			0.871106
$960$			4.46537e19i			1.84117i
$961$	2.43873e19			0.998759
$962$	1.51242e15	+	4.15053e16i	6.15228e−5	+	0.00168836i
$963$	8.53609e19			3.44896
$964$			1.25654e18i			0.0504283i
$965$	−1.29830e19			−0.517544
$966$	−2.75631e19			−1.09138
$967$		−	7.50976e18i		−	0.295362i	−0.989035	−	0.147681i	$-0.952819\pi$
$967$		−	7.50976e18i		−	0.295362i	0.989035	−	0.147681i	$-0.0471808\pi$
$968$			4.61345e19i			1.80234i
$969$		−	5.56976e18i		−	0.216138i
$970$			2.43765e19i			0.939625i
$971$	−2.99621e19			−1.14722			−0.573611	−	0.819128i	$-0.694458\pi$
$971$	−2.99621e19			−1.14722			−0.573611	+	0.819128i	$0.694458\pi$
$972$	5.47212e18			0.208125
$973$		−	3.40399e18i		−	0.128604i
$974$	−1.24414e19			−0.466913
$975$	−3.06040e17	−	8.39862e18i	−0.0114090	−	0.313096i
$976$	5.61883e18			0.208076
$977$			3.69718e19i			1.36005i	0.733188	+	0.680026i	$0.238032\pi$
$977$			3.69718e19i			1.36005i	−0.733188	+	0.680026i	$0.761968\pi$
$978$	2.86304e19			1.04622
$979$	1.85478e19			0.673294
$980$		−	1.02891e18i		−	0.0371029i
$981$		−	3.67959e19i		−	1.31811i
$982$		−	1.62518e19i		−	0.578333i
$983$		−	4.64432e18i		−	0.164181i	−0.996625	−	0.0820907i	$-0.973840\pi$
$983$		−	4.64432e18i		−	0.164181i	0.996625	−	0.0820907i	$-0.0261597\pi$
$984$	4.15399e19			1.45880
$985$	−4.59803e19			−1.60412
$986$		−	2.36024e19i		−	0.818005i
$987$	−2.56379e19			−0.882717
$988$	3.42240e17	−	1.24710e16i	0.0117061	−	0.000426561i
$989$	−2.44477e19			−0.830735
$990$		−	1.14349e20i		−	3.86016i
$991$	1.34705e19			0.451758			0.225879	−	0.974155i	$-0.427475\pi$
$991$	1.34705e19			0.451758			0.225879	+	0.974155i	$0.427475\pi$
$992$	1.38002e17			0.00459791
$993$			2.56237e19i			0.848147i
$994$			1.38017e19i			0.453861i
$995$		−	4.41689e19i		−	1.44300i
$996$		−	4.17285e18i		−	0.135440i
$997$	2.44003e19			0.786823			0.393412	−	0.919362i	$-0.371295\pi$
$997$	2.44003e19			0.786823			0.393412	+	0.919362i	$0.371295\pi$
$998$	−3.79117e19			−1.21458
$999$		−	1.71024e17i		−	0.00544353i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.14.b.a.12.5	✓	14
3.2	odd	2		117.14.b.c.64.10		14
13.5	odd	4		169.14.a.d.1.5		14
13.8	odd	4		169.14.a.d.1.10		14
13.12	even	2	inner	13.14.b.a.12.10	yes	14
39.38	odd	2		117.14.b.c.64.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.14.b.a.12.5	✓	14	1.1	even	1	trivial
13.14.b.a.12.10	yes	14	13.12	even	2	inner
117.14.b.c.64.5		14	39.38	odd	2
117.14.b.c.64.10		14	3.2	odd	2
169.14.a.d.1.5		14	13.5	odd	4
169.14.a.d.1.10		14	13.8	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	13.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-87.5010i q^{2} +2407.03 q^{3} +535.569 q^{4} -31938.7i q^{5} -210617. i q^{6} +191671. i q^{7} -763671. i q^{8} +4.19945e6 q^{9} -2.79467e6 q^{10} +9.74341e6i q^{11} +1.28913e6 q^{12} +(-633744. - 1.73918e7i) q^{13} +1.67714e7 q^{14} -7.68772e7i q^{15} -6.24346e7 q^{16} -6.29773e7 q^{17} -3.67456e8i q^{18} +3.67427e7i q^{19} -1.71054e7i q^{20} +4.61357e8i q^{21} +8.52559e8 q^{22} -6.82777e8 q^{23} -1.83818e9i q^{24} +2.00624e8 q^{25} +(-1.52180e9 + 5.54533e7i) q^{26} +6.27061e9 q^{27} +1.02653e8i q^{28} -4.28310e9 q^{29} -6.72684e9 q^{30} +1.74048e8i q^{31} -7.92899e8i q^{32} +2.34526e10i q^{33} +5.51058e9i q^{34} +6.12171e9 q^{35} +2.24910e9 q^{36} -2.72739e7i q^{37} +3.21503e9 q^{38} +(-1.52544e9 - 4.18624e10i) q^{39} -2.43906e10 q^{40} +2.25984e10i q^{41} +4.03692e10 q^{42} +3.58063e10 q^{43} +5.21827e9i q^{44} -1.34125e11i q^{45} +5.97437e10i q^{46} +5.55708e10i q^{47} -1.50282e11 q^{48} +6.01513e10 q^{49} -1.75548e10i q^{50} -1.51588e11 q^{51} +(-3.39414e8 - 9.31449e9i) q^{52} -4.92496e10 q^{53} -5.48685e11i q^{54} +3.11192e11 q^{55} +1.46374e11 q^{56} +8.84407e10i q^{57} +3.74776e11i q^{58} +3.00035e11i q^{59} -4.11730e10i q^{60} -8.99954e10 q^{61} +1.52294e10 q^{62} +8.04912e11i q^{63} -5.80844e11 q^{64} +(-5.55470e11 + 2.02410e10i) q^{65} +2.05213e12 q^{66} +1.13281e12i q^{67} -3.37287e10 q^{68} -1.64346e12 q^{69} -5.35656e11i q^{70} +8.22933e11i q^{71} -3.20700e12i q^{72} -9.13149e11i q^{73} -2.38649e9 q^{74} +4.82908e11 q^{75} +1.96783e10i q^{76} -1.86753e12 q^{77} +(-3.66301e12 + 1.33477e11i) q^{78} +1.70734e12 q^{79} +1.99408e12i q^{80} +8.39824e12 q^{81} +1.97738e12 q^{82} -3.23695e12i q^{83} +2.47088e11i q^{84} +2.01141e12i q^{85} -3.13309e12i q^{86} -1.03095e13 q^{87} +7.44077e12 q^{88} -1.90362e12i q^{89} -1.17361e13 q^{90} +(3.33349e12 - 1.21470e11i) q^{91} -3.65674e11 q^{92} +4.18938e11i q^{93} +4.86250e12 q^{94} +1.17351e12 q^{95} -1.90853e12i q^{96} -8.72249e12i q^{97} -5.26330e12i q^{98} +4.09170e13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 1456 q^{3} - 49154 q^{4} + 5149262 q^{9} - 3968358 q^{10} - 21959110 q^{12} - 14362478 q^{13} + 72843942 q^{14} + 243447170 q^{16} - 178512492 q^{17} + 1330288056 q^{22} - 1641693744 q^{23} + 1601386654 q^{25}+ \cdots + 11940434574336 q^{95}+O(q^{100}) \) 14 * q + 1456 * q^3 - 49154 * q^4 + 5149262 * q^9 - 3968358 * q^10 - 21959110 * q^12 - 14362478 * q^13 + 72843942 * q^14 + 243447170 * q^16 - 178512492 * q^17 + 1330288056 * q^22 - 1641693744 * q^23 + 1601386654 * q^25 + 138020766 * q^26 + 6360372232 * q^27 - 10348326348 * q^29 - 11230427010 * q^30 + 20242816056 * q^35 + 33181294876 * q^36 - 72234754500 * q^38 + 3102574280 * q^39 - 50528567766 * q^40 + 90123482634 * q^42 - 42288191696 * q^43 + 256465114846 * q^48 + 202681235282 * q^49 - 454251099624 * q^51 + 605927200580 * q^52 - 541269433788 * q^53 + 294791560608 * q^55 - 1027260370698 * q^56 + 1693279412900 * q^61 + 1808450592168 * q^62 - 4815047793506 * q^64 + 534977124552 * q^65 + 1417411440480 * q^66 + 65016763122 * q^68 - 4207322259360 * q^69 + 7595100503238 * q^74 + 1662071406584 * q^75 - 1391939668824 * q^77 - 361798757502 * q^78 - 5060278211056 * q^79 + 6637938210830 * q^81 + 507818992992 * q^82 - 13033010125344 * q^87 + 2586979258080 * q^88 - 4680167629284 * q^90 + 3450737986176 * q^91 + 7241299383420 * q^92 + 11219792119974 * q^94 + 11940434574336 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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