LMFDB - Embedded newform 13.14.b.a.12.7

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			12.7
Root			$-10.5212i$ of defining polynomial
Character	$\chi$	$=$	13.12
Dual form			13.14.b.a.12.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-10.5212i q^{2} -1591.09 q^{3} +8081.30 q^{4} -9424.40i q^{5} +16740.2i q^{6} +484840. i q^{7} -171215. i q^{8} +937259. q^{9} -99156.0 q^{10} -8.93854e6i q^{11} -1.28581e7 q^{12} +(4.96535e6 - 1.66799e7i) q^{13} +5.10109e6 q^{14} +1.49951e7i q^{15} +6.44007e7 q^{16} -1.57696e8 q^{17} -9.86109e6i q^{18} -3.07483e8i q^{19} -7.61615e7i q^{20} -7.71426e8i q^{21} -9.40441e7 q^{22} -6.52271e8 q^{23} +2.72419e8i q^{24} +1.13188e9 q^{25} +(-1.75493e8 - 5.22414e7i) q^{26} +1.04545e9 q^{27} +3.91814e9i q^{28} -1.77749e9 q^{29} +1.57767e8 q^{30} -5.73027e9i q^{31} -2.08016e9i q^{32} +1.42221e10i q^{33} +1.65915e9i q^{34} +4.56932e9 q^{35} +7.57428e9 q^{36} +4.96227e9i q^{37} -3.23509e9 q^{38} +(-7.90034e9 + 2.65394e10i) q^{39} -1.61360e9 q^{40} -1.33251e10i q^{41} -8.11632e9 q^{42} +3.47041e10 q^{43} -7.22350e10i q^{44} -8.83311e9i q^{45} +6.86267e9i q^{46} -5.71534e10i q^{47} -1.02468e11 q^{48} -1.38180e11 q^{49} -1.19088e10i q^{50} +2.50909e11 q^{51} +(4.01265e10 - 1.34796e11i) q^{52} -8.98677e10 q^{53} -1.09994e10i q^{54} -8.42404e10 q^{55} +8.30116e10 q^{56} +4.89235e11i q^{57} +1.87013e10i q^{58} -2.02674e11i q^{59} +1.21180e11i q^{60} +9.24532e10 q^{61} -6.02893e10 q^{62} +4.54420e11i q^{63} +5.05684e11 q^{64} +(-1.57198e11 - 4.67955e10i) q^{65} +1.49633e11 q^{66} -2.80350e11i q^{67} -1.27439e12 q^{68} +1.03782e12 q^{69} -4.80748e10i q^{70} +1.45170e12i q^{71} -1.60473e11i q^{72} +6.04461e11i q^{73} +5.22090e10 q^{74} -1.80093e12 q^{75} -2.48487e12i q^{76} +4.33376e12 q^{77} +(2.79226e11 + 8.31211e10i) q^{78} -2.70387e12 q^{79} -6.06938e11i q^{80} -3.15770e12 q^{81} -1.40196e11 q^{82} -3.96858e12i q^{83} -6.23413e12i q^{84} +1.48619e12i q^{85} -3.65129e11i q^{86} +2.82815e12 q^{87} -1.53041e12 q^{88} +5.41762e12i q^{89} -9.29349e10 q^{90} +(8.08710e12 + 2.40740e12i) q^{91} -5.27120e12 q^{92} +9.11740e12i q^{93} -6.01322e11 q^{94} -2.89785e12 q^{95} +3.30974e12i q^{96} +4.83182e12i q^{97} +1.45382e12i q^{98} -8.37773e12i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	10.5212i		−	0.116244i	−0.998309	−	0.0581220i	$-0.981489\pi$
$2$		−	10.5212i		−	0.116244i	0.998309	−	0.0581220i	$-0.0185112\pi$
$3$	−1591.09			−1.26011			−0.630054	−	0.776551i	$-0.716967\pi$
$3$	−1591.09			−1.26011			−0.630054	+	0.776551i	$0.716967\pi$
$4$	8081.30			0.986487
$5$		−	9424.40i		−	0.269742i	−0.990863	−	0.134871i	$-0.956938\pi$
$5$		−	9424.40i		−	0.269742i	0.990863	−	0.134871i	$-0.0430620\pi$
$6$			16740.2i			0.146480i
$7$			484840.i			1.55762i	0.627261	+	0.778809i	$0.284176\pi$
$7$			484840.i			1.55762i	−0.627261	+	0.778809i	$0.715824\pi$
$8$		−	171215.i		−	0.230917i
$9$	937259.			0.587873
$10$	−99156.0			−0.0313559
$11$		−	8.93854e6i		−	1.52130i	−0.649164	−	0.760649i	$-0.724881\pi$
$11$		−	8.93854e6i		−	1.52130i	0.649164	−	0.760649i	$-0.275119\pi$
$12$	−1.28581e7			−1.24308
$13$	4.96535e6	−	1.66799e7i	0.285311	−	0.958435i
$13$	4.96535e6	−	1.66799e7i	0.285311	−	0.958435i
$14$	5.10109e6			0.181064
$15$			1.49951e7i			0.339904i
$16$	6.44007e7			0.959645
$17$	−1.57696e8			−1.58453			−0.792267	−	0.610174i	$-0.791099\pi$
$17$	−1.57696e8			−1.58453			−0.792267	+	0.610174i	$0.791099\pi$
$18$		−	9.86109e6i		−	0.0683366i
$19$		−	3.07483e8i		−	1.49942i	−0.661767	−	0.749709i	$-0.730193\pi$
$19$		−	3.07483e8i		−	1.49942i	0.661767	−	0.749709i	$-0.269807\pi$
$20$		−	7.61615e7i		−	0.266097i
$21$		−	7.71426e8i		−	1.96277i
$22$	−9.40441e7			−0.176842
$23$	−6.52271e8			−0.918749			−0.459374	−	0.888243i	$-0.651926\pi$
$23$	−6.52271e8			−0.918749			−0.459374	+	0.888243i	$0.651926\pi$
$24$			2.72419e8i			0.290980i
$25$	1.13188e9			0.927239
$26$	−1.75493e8	−	5.22414e7i	−0.111412	−	0.0331656i
$27$	1.04545e9			0.519325
$28$			3.91814e9i			1.53657i
$29$	−1.77749e9			−0.554906			−0.277453	−	0.960739i	$-0.589490\pi$
$29$	−1.77749e9			−0.554906			−0.277453	+	0.960739i	$0.589490\pi$
$30$	1.57767e8			0.0395118
$31$		−	5.73027e9i		−	1.15964i	−0.814744	−	0.579821i	$-0.803122\pi$
$31$		−	5.73027e9i		−	1.15964i	0.814744	−	0.579821i	$-0.196878\pi$
$32$		−	2.08016e9i		−	0.342470i
$33$			1.42221e10i			1.91700i
$34$			1.65915e9i			0.184192i
$35$	4.56932e9			0.420155
$36$	7.57428e9			0.579929
$37$			4.96227e9i			0.317957i	0.987282	+	0.158979i	$0.0508201\pi$
$37$			4.96227e9i			0.317957i	−0.987282	+	0.158979i	$0.949180\pi$
$38$	−3.23509e9			−0.174298
$39$	−7.90034e9	+	2.65394e10i	−0.359522	+	1.20773i
$40$	−1.61360e9			−0.0622881
$41$		−	1.33251e10i		−	0.438101i	−0.975714	−	0.219050i	$-0.929704\pi$
$41$		−	1.33251e10i		−	0.438101i	0.975714	−	0.219050i	$-0.0702959\pi$
$42$	−8.11632e9			−0.228160
$43$	3.47041e10			0.837213			0.418606	−	0.908168i	$-0.362519\pi$
$43$	3.47041e10			0.837213			0.418606	+	0.908168i	$0.362519\pi$
$44$		−	7.22350e10i		−	1.50074i
$45$		−	8.83311e9i		−	0.158574i
$46$			6.86267e9i			0.106799i
$47$		−	5.71534e10i		−	0.773403i	−0.922205	−	0.386701i	$-0.873614\pi$
$47$		−	5.71534e10i		−	0.773403i	0.922205	−	0.386701i	$-0.126386\pi$
$48$	−1.02468e11			−1.20926
$49$	−1.38180e11			−1.42617
$50$		−	1.19088e10i		−	0.107786i
$51$	2.50909e11			1.99668
$52$	4.01265e10	−	1.34796e11i	0.281455	−	0.945484i
$53$	−8.98677e10			−0.556943			−0.278471	−	0.960445i	$-0.589828\pi$
$53$	−8.98677e10			−0.556943			−0.278471	+	0.960445i	$0.589828\pi$
$54$		−	1.09994e10i		−	0.0603683i
$55$	−8.42404e10			−0.410358
$56$	8.30116e10			0.359680
$57$			4.89235e11i			1.88943i
$58$			1.87013e10i			0.0645044i
$59$		−	2.02674e11i		−	0.625548i	−0.949828	−	0.312774i	$-0.898742\pi$
$59$		−	2.02674e11i		−	0.625548i	0.949828	−	0.312774i	$-0.101258\pi$
$60$			1.21180e11i			0.335311i
$61$	9.24532e10			0.229762			0.114881	−	0.993379i	$-0.463351\pi$
$61$	9.24532e10			0.229762			0.114881	+	0.993379i	$0.463351\pi$
$62$	−6.02893e10			−0.134801
$63$			4.54420e11i			0.915681i
$64$	5.05684e11			0.919835
$65$	−1.57198e11	−	4.67955e10i	−0.258530	−	0.0769604i
$66$	1.49633e11			0.222839
$67$		−	2.80350e11i		−	0.378629i	−0.981917	−	0.189314i	$-0.939373\pi$
$67$		−	2.80350e11i		−	0.378629i	0.981917	−	0.189314i	$-0.0606266\pi$
$68$	−1.27439e12			−1.56312
$69$	1.03782e12			1.15772
$70$		−	4.80748e10i		−	0.0488405i
$71$			1.45170e12i			1.34492i	0.740132	+	0.672461i	$0.234763\pi$
$71$			1.45170e12i			1.34492i	−0.740132	+	0.672461i	$0.765237\pi$
$72$		−	1.60473e11i		−	0.135750i
$73$			6.04461e11i			0.467487i	0.972298	+	0.233744i	$0.0750976\pi$
$73$			6.04461e11i			0.467487i	−0.972298	+	0.233744i	$0.924902\pi$
$74$	5.22090e10			0.0369606
$75$	−1.80093e12			−1.16842
$76$		−	2.48487e12i		−	1.47916i
$77$	4.33376e12			2.36960
$78$	2.79226e11	+	8.31211e10i	0.140391	+	0.0417923i
$79$	−2.70387e12			−1.25144			−0.625719	−	0.780048i	$-0.715194\pi$
$79$	−2.70387e12			−1.25144			−0.625719	+	0.780048i	$0.715194\pi$
$80$		−	6.06938e11i		−	0.258857i
$81$	−3.15770e12			−1.24228
$82$	−1.40196e11			−0.0509266
$83$		−	3.96858e12i		−	1.33238i	−0.745783	−	0.666189i	$-0.767925\pi$
$83$		−	3.96858e12i		−	1.33238i	0.745783	−	0.666189i	$-0.232075\pi$
$84$		−	6.23413e12i		−	1.93624i
$85$			1.48619e12i			0.427416i
$86$		−	3.65129e11i		−	0.0973209i
$87$	2.82815e12			0.699242
$88$	−1.53041e12			−0.351293
$89$			5.41762e12i			1.15551i	0.816211	+	0.577755i	$0.196071\pi$
$89$			5.41762e12i			1.15551i	−0.816211	+	0.577755i	$0.803929\pi$
$90$	−9.29349e10			−0.0184333
$91$	8.08710e12	+	2.40740e12i	1.49288	+	0.444405i
$92$	−5.27120e12			−0.906334
$93$			9.11740e12i			1.46127i
$94$	−6.01322e11			−0.0899034
$95$	−2.89785e12			−0.404457
$96$			3.30974e12i			0.431549i
$97$			4.83182e12i			0.588971i	0.955656	+	0.294486i	$0.0951483\pi$
$97$			4.83182e12i			0.588971i	−0.955656	+	0.294486i	$0.904852\pi$
$98$			1.45382e12i			0.165784i
$99$		−	8.37773e12i		−	0.894329i
$100$	9.14710e12			0.914710
$101$	3.49579e12			0.327685			0.163842	−	0.986487i	$-0.447611\pi$
$101$	3.49579e12			0.327685			0.163842	+	0.986487i	$0.447611\pi$
$102$		−	2.63986e12i		−	0.232102i
$103$	1.10612e12			0.0912764			0.0456382	−	0.998958i	$-0.485468\pi$
$103$	1.10612e12			0.0912764			0.0456382	+	0.998958i	$0.485468\pi$
$104$	−2.85585e12	−	8.50141e11i	−0.221319	−	0.0658831i
$105$	−7.27023e12			−0.529441
$106$			9.45516e11i			0.0647412i
$107$	−1.27698e13			−0.822601			−0.411300	−	0.911500i	$-0.634925\pi$
$107$	−1.27698e13			−0.822601			−0.411300	+	0.911500i	$0.634925\pi$
$108$	8.44860e12			0.512307
$109$			2.66370e12i			0.152130i	0.997103	+	0.0760648i	$0.0242356\pi$
$109$			2.66370e12i			0.152130i	−0.997103	+	0.0760648i	$0.975764\pi$
$110$			8.86310e11i			0.0477016i
$111$		−	7.89544e12i		−	0.400661i
$112$			3.12240e13i			1.49476i
$113$	2.11333e13			0.954898			0.477449	−	0.878659i	$-0.341562\pi$
$113$	2.11333e13			0.954898			0.477449	+	0.878659i	$0.341562\pi$
$114$	5.14734e12			0.219635
$115$			6.14726e12i			0.247825i
$116$	−1.43644e13			−0.547408
$117$	4.65382e12	−	1.56334e13i	0.167726	−	0.563438i
$118$	−2.13238e12			−0.0727162
$119$		−	7.64571e13i		−	2.46810i
$120$	2.56738e12			0.0784897
$121$	−4.53747e13			−1.31434
$122$		−	9.72719e11i		−	0.0267084i
$123$			2.12014e13i			0.552055i
$124$		−	4.63080e13i		−	1.14397i
$125$		−	2.21717e13i		−	0.519858i
$126$	4.78105e12			0.106442
$127$	−1.62029e13			−0.342665			−0.171332	−	0.985213i	$-0.554807\pi$
$127$	−1.62029e13			−0.342665			−0.171332	+	0.985213i	$0.554807\pi$
$128$		−	2.23611e13i		−	0.449395i
$129$	−5.52175e13			−1.05498
$130$	−4.92344e11	+	1.65392e12i	−0.00894617	+	0.0300526i
$131$	3.39907e13			0.587620			0.293810	−	0.955864i	$-0.405077\pi$
$131$	3.39907e13			0.587620			0.293810	+	0.955864i	$0.405077\pi$
$132$			1.14933e14i			1.89110i
$133$	1.49080e14			2.33552
$134$	−2.94961e12			−0.0440133
$135$		−	9.85275e12i		−	0.140084i
$136$			2.69998e13i			0.365896i
$137$			1.05808e14i			1.36720i	0.729855	+	0.683602i	$0.239587\pi$
$137$			1.05808e14i			1.36720i	−0.729855	+	0.683602i	$0.760413\pi$
$138$		−	1.09192e13i		−	0.134578i
$139$	−1.00309e14			−1.17963			−0.589813	−	0.807540i	$-0.700799\pi$
$139$	−1.00309e14			−1.17963			−0.589813	+	0.807540i	$0.700799\pi$
$140$	3.69261e13			0.414478
$141$			9.09364e13i			0.974571i
$142$	1.52736e13			0.156339
$143$	−1.49094e14	−	4.43830e13i	−1.45806	−	0.434042i
$144$	6.03601e13			0.564149
$145$			1.67518e13i			0.149682i
$146$	6.35965e12			0.0543425
$147$	2.19858e14			1.79713
$148$			4.01016e13i			0.313661i
$149$		−	2.26562e14i		−	1.69619i	−0.529840	−	0.848097i	$-0.677748\pi$
$149$		−	2.26562e14i		−	1.69619i	0.529840	−	0.848097i	$-0.322252\pi$
$150$			1.89480e13i			0.135822i
$151$			3.54725e13i			0.243524i	0.992559	+	0.121762i	$0.0388544\pi$
$151$			3.54725e13i			0.243524i	−0.992559	+	0.121762i	$0.961146\pi$
$152$	−5.26456e13			−0.346241
$153$	−1.47802e14			−0.931505
$154$		−	4.55963e13i		−	0.275451i
$155$	−5.40044e13			−0.312804
$156$	−6.38451e13	+	2.14473e14i	−0.354664	+	1.19141i
$157$	2.09005e13			0.111381			0.0556903	−	0.998448i	$-0.482264\pi$
$157$	2.09005e13			0.111381			0.0556903	+	0.998448i	$0.482264\pi$
$158$			2.84479e13i			0.145472i
$159$	1.42988e14			0.701808
$160$	−1.96043e13			−0.0923786
$161$		−	3.16247e14i		−	1.43106i
$162$			3.32228e13i			0.144407i
$163$			3.97849e11i			0.00166149i	1.00000		0.000830746i	$0.000264435\pi$
$163$			3.97849e11i			0.00166149i	−1.00000		0.000830746i	$0.999736\pi$
$164$		−	1.07684e14i		−	0.432181i
$165$	1.34034e14			0.517095
$166$	−4.17542e13			−0.154881
$167$		−	8.60410e13i		−	0.306936i	−0.988154	−	0.153468i	$-0.950956\pi$
$167$		−	8.60410e13i		−	0.306936i	0.988154	−	0.153468i	$-0.0490442\pi$
$168$	−1.32079e14			−0.453236
$169$	−2.53566e14	−	1.65644e14i	−0.837196	−	0.546904i
$170$	1.56365e13			0.0496845
$171$		−	2.88191e14i		−	0.881468i
$172$	2.80454e14			0.825900
$173$	−1.16395e14			−0.330092			−0.165046	−	0.986286i	$-0.552777\pi$
$173$	−1.16395e14			−0.330092			−0.165046	+	0.986286i	$0.552777\pi$
$174$		−	2.97555e13i		−	0.0812826i
$175$			5.48782e14i			1.44428i
$176$		−	5.75648e14i		−	1.45990i
$177$			3.22474e14i			0.788258i
$178$	5.69998e13			0.134321
$179$	−3.69710e14			−0.840071			−0.420036	−	0.907508i	$-0.637982\pi$
$179$	−3.69710e14			−0.840071			−0.420036	+	0.907508i	$0.637982\pi$
$180$		−	7.13830e13i		−	0.156431i
$181$	4.63653e14			0.980126			0.490063	−	0.871687i	$-0.336974\pi$
$181$	4.63653e14			0.980126			0.490063	+	0.871687i	$0.336974\pi$
$182$	2.53287e13	−	8.50859e13i	0.0516594	−	0.173538i
$183$	−1.47102e14			−0.289525
$184$			1.11678e14i			0.212155i
$185$	4.67664e13			0.0857665
$186$	9.59260e13			0.169864
$187$			1.40957e15i			2.41055i
$188$		−	4.61874e14i		−	0.762952i
$189$			5.06876e14i			0.808909i
$190$			3.04888e13i			0.0470156i
$191$	1.07831e15			1.60705			0.803524	−	0.595273i	$-0.202956\pi$
$191$	1.07831e15			1.60705			0.803524	+	0.595273i	$0.202956\pi$
$192$	−8.04592e14			−1.15909
$193$			3.07847e14i			0.428759i	0.976750	+	0.214379i	$0.0687729\pi$
$193$			3.07847e14i			0.428759i	−0.976750	+	0.214379i	$0.931227\pi$
$194$	5.08365e13			0.0684643
$195$	2.50118e14	+	7.44560e13i	0.325776	+	0.0969784i
$196$	−1.11668e15			−1.40690
$197$			2.82605e14i			0.344468i	0.985056	+	0.172234i	$0.0550985\pi$
$197$			2.82605e14i			0.344468i	−0.985056	+	0.172234i	$0.944901\pi$
$198$	−8.81437e13			−0.103960
$199$	8.39524e14			0.958270			0.479135	−	0.877741i	$-0.340950\pi$
$199$	8.39524e14			0.958270			0.479135	+	0.877741i	$0.340950\pi$
$200$		−	1.93795e14i		−	0.214115i
$201$			4.46063e14i			0.477113i
$202$		−	3.67799e13i		−	0.0380914i
$203$		−	8.61796e14i		−	0.864331i
$204$	2.02767e15			1.96970
$205$	−1.25581e14			−0.118174
$206$		−	1.16377e13i		−	0.0106103i
$207$	−6.11347e14			−0.540108
$208$	3.19772e14	−	1.07420e15i	0.273797	−	0.919757i
$209$	−2.74845e15			−2.28106
$210$			7.64915e13i			0.0615443i
$211$	−5.03465e14			−0.392765			−0.196383	−	0.980527i	$-0.562919\pi$
$211$	−5.03465e14			−0.392765			−0.196383	+	0.980527i	$0.562919\pi$
$212$	−7.26248e14			−0.549417
$213$		−	2.30979e15i		−	1.69475i
$214$			1.34353e14i			0.0956223i
$215$		−	3.27065e14i		−	0.225832i
$216$		−	1.78996e14i		−	0.119921i
$217$	2.77826e15			1.80628
$218$	2.80253e13			0.0176841
$219$		−	9.61755e14i		−	0.589084i
$220$	−6.80772e14			−0.404813
$221$	−7.83014e14	+	2.63035e15i	−0.452085	+	1.51867i
$222$	−8.30694e13			−0.0465744
$223$		−	1.00251e15i		−	0.545891i	−0.962029	−	0.272946i	$-0.912002\pi$
$223$		−	1.00251e15i		−	0.545891i	0.962029	−	0.272946i	$-0.0879979\pi$
$224$	1.00855e15			0.533437
$225$	1.06087e15			0.545099
$226$		−	2.22347e14i		−	0.111001i
$227$		−	3.07498e15i		−	1.49168i	−0.666127	−	0.745838i	$-0.732049\pi$
$227$		−	3.07498e15i		−	1.49168i	0.666127	−	0.745838i	$-0.267951\pi$
$228$			3.95366e15i			1.86390i
$229$			1.07094e15i			0.490722i	0.969432	+	0.245361i	$0.0789066\pi$
$229$			1.07094e15i			0.490722i	−0.969432	+	0.245361i	$0.921093\pi$
$230$	6.46766e13			0.0288082
$231$	−6.89542e15			−2.98595
$232$			3.04332e14i			0.128137i
$233$	2.93496e15			1.20168			0.600839	−	0.799370i	$-0.294833\pi$
$233$	2.93496e15			1.20168			0.600839	+	0.799370i	$0.294833\pi$
$234$	−1.64482e14	−	4.89638e13i	−0.0654962	−	0.0194972i
$235$	−5.38636e14			−0.208619
$236$		−	1.63787e15i		−	0.617095i
$237$	4.30211e15			1.57695
$238$	−8.04420e14			−0.286901
$239$			1.17148e15i			0.406581i	0.979119	+	0.203290i	$0.0651635\pi$
$239$			1.17148e15i			0.406581i	−0.979119	+	0.203290i	$0.934836\pi$
$240$			9.65696e14i			0.326187i
$241$		−	3.11024e15i		−	1.02255i	−0.859418	−	0.511273i	$-0.829174\pi$
$241$		−	3.11024e15i		−	1.02255i	0.859418	−	0.511273i	$-0.170826\pi$
$242$			4.77397e14i			0.152785i
$243$	3.35742e15			1.04608
$244$	7.47143e14			0.226657
$245$			1.30227e15i			0.384699i
$246$	2.23064e14			0.0641730
$247$	−5.12880e15	−	1.52676e15i	−1.43710	−	0.427800i
$248$	−9.81106e14			−0.267781
$249$			6.31438e15i			1.67894i
$250$	−2.33273e14			−0.0604303
$251$	4.91205e15			1.23989			0.619946	−	0.784645i	$-0.287155\pi$
$251$	4.91205e15			1.23989			0.619946	+	0.784645i	$0.287155\pi$
$252$			3.67231e15i			0.903308i
$253$			5.83035e15i			1.39769i
$254$			1.70474e14i			0.0398327i
$255$		−	2.36466e15i		−	0.538590i
$256$	3.90730e15			0.867595
$257$	−5.04425e15			−1.09202			−0.546011	−	0.837778i	$-0.683854\pi$
$257$	−5.04425e15			−1.09202			−0.546011	+	0.837778i	$0.683854\pi$
$258$			5.80954e14i			0.122635i
$259$	−2.40590e15			−0.495256
$260$	−1.27037e15	−	3.78168e14i	−0.255037	−	0.0759204i
$261$	−1.66597e15			−0.326214
$262$		−	3.57623e14i		−	0.0683072i
$263$	8.92638e15			1.66327			0.831635	−	0.555322i	$-0.187405\pi$
$263$	8.92638e15			1.66327			0.831635	+	0.555322i	$0.187405\pi$
$264$	2.43502e15			0.442668
$265$			8.46949e14i			0.150231i
$266$		−	1.56850e15i		−	0.271490i
$267$		−	8.61994e15i		−	1.45607i
$268$		−	2.26559e15i		−	0.373513i
$269$	−3.73207e15			−0.600565			−0.300283	−	0.953850i	$-0.597081\pi$
$269$	−3.73207e15			−0.600565			−0.300283	+	0.953850i	$0.597081\pi$
$270$	−1.03663e14			−0.0162839
$271$		−	8.73516e15i		−	1.33959i	−0.742548	−	0.669793i	$-0.766383\pi$
$271$		−	8.73516e15i		−	1.33959i	0.742548	−	0.669793i	$-0.233617\pi$
$272$	−1.01557e16			−1.52059
$273$	−1.28673e16	−	3.83040e15i	−1.88118	−	0.559999i
$274$	1.11322e15			0.158929
$275$		−	1.01174e16i		−	1.41061i
$276$	8.38697e15			1.14208
$277$	6.78204e15			0.902073			0.451037	−	0.892505i	$-0.351054\pi$
$277$	6.78204e15			0.902073			0.451037	+	0.892505i	$0.351054\pi$
$278$			1.05537e15i			0.137124i
$279$		−	5.37075e15i		−	0.681722i
$280$		−	7.82335e14i		−	0.0970210i
$281$			1.06581e16i			1.29149i	0.763554	+	0.645744i	$0.223453\pi$
$281$			1.06581e16i			1.29149i	−0.763554	+	0.645744i	$0.776547\pi$
$282$	9.56760e14			0.113288
$283$	−1.38154e16			−1.59864			−0.799320	−	0.600906i	$-0.794807\pi$
$283$	−1.38154e16			−1.59864			−0.799320	+	0.600906i	$0.794807\pi$
$284$			1.17316e16i			1.32675i
$285$	4.61075e15			0.509659
$286$	−4.66962e14	+	1.56865e15i	−0.0504548	+	0.169491i
$287$	6.46052e15			0.682394
$288$		−	1.94965e15i		−	0.201329i
$289$	1.49633e16			1.51075
$290$	1.76249e14			0.0173996
$291$		−	7.68788e15i		−	0.742168i
$292$			4.88483e15i			0.461170i
$293$			1.55600e16i			1.43671i	0.695676	+	0.718356i	$0.255105\pi$
$293$			1.55600e16i			1.43671i	−0.695676	+	0.718356i	$0.744895\pi$
$294$		−	2.31317e15i		−	0.208906i
$295$	−1.91008e15			−0.168737
$296$	8.49613e14			0.0734218
$297$		−	9.34480e15i		−	0.790047i
$298$	−2.38370e15			−0.197172
$299$	−3.23875e15	+	1.08798e16i	−0.262129	+	0.880561i
$300$	−1.45539e16			−1.15263
$301$			1.68259e16i			1.30406i
$302$	3.73213e14			0.0283082
$303$	−5.56213e15			−0.412918
$304$		−	1.98021e16i		−	1.43891i
$305$		−	8.71317e14i		−	0.0619765i
$306$			1.55505e15i			0.108282i
$307$		−	6.49159e15i		−	0.442539i	−0.975213	−	0.221269i	$-0.928980\pi$
$307$		−	6.49159e15i		−	0.442539i	0.975213	−	0.221269i	$-0.0710200\pi$
$308$	3.50224e16			2.33758
$309$	−1.75993e15			−0.115018
$310$			5.68190e14i			0.0363616i
$311$	−3.81669e15			−0.239191			−0.119595	−	0.992823i	$-0.538160\pi$
$311$	−3.81669e15			−0.239191			−0.119595	+	0.992823i	$0.538160\pi$
$312$	4.54393e15	+	1.35265e15i	0.278886	+	0.0830199i
$313$	−2.89419e15			−0.173976			−0.0869879	−	0.996209i	$-0.527724\pi$
$313$	−2.89419e15			−0.173976			−0.0869879	+	0.996209i	$0.527724\pi$
$314$		−	2.19899e14i		−	0.0129473i
$315$	4.28264e15			0.246998
$316$	−2.18508e16			−1.23453
$317$			2.55802e16i			1.41586i	0.706285	+	0.707928i	$0.250370\pi$
$317$			2.55802e16i			1.41586i	−0.706285	+	0.707928i	$0.749630\pi$
$318$		−	1.50441e15i		−	0.0815809i
$319$			1.58881e16i			0.844177i
$320$		−	4.76577e15i		−	0.248118i
$321$	2.03179e16			1.03657
$322$	−3.32729e15			−0.166352
$323$			4.84888e16i			2.37588i
$324$	−2.55184e16			−1.22549
$325$	5.62020e15	−	1.88798e16i	0.264551	−	0.888699i
$326$	4.18584e12			0.000193138
$327$		−	4.23820e15i		−	0.191700i
$328$	−2.28145e15			−0.101165
$329$	2.77102e16			1.20467
$330$		−	1.41020e15i		−	0.0601092i
$331$		−	3.12279e16i		−	1.30515i	−0.757724	−	0.652576i	$-0.773688\pi$
$331$		−	3.12279e16i		−	1.30515i	0.757724	−	0.652576i	$-0.226312\pi$
$332$		−	3.20713e16i		−	1.31437i
$333$			4.65093e15i			0.186918i
$334$	−9.05254e14			−0.0356795
$335$	−2.64213e15			−0.102132
$336$		−	4.96803e16i		−	1.88356i
$337$	2.78440e16			1.03547			0.517734	−	0.855541i	$-0.326776\pi$
$337$	2.78440e16			1.03547			0.517734	+	0.855541i	$0.326776\pi$
$338$	−1.74277e15	+	2.66781e15i	−0.0635742	+	0.0973189i
$339$	−3.36250e16			−1.20327
$340$			1.20103e16i			0.421640i
$341$	−5.12202e16			−1.76416
$342$	−3.03212e15			−0.102465
$343$		−	2.00197e16i		−	0.663813i
$344$		−	5.94185e15i		−	0.193327i
$345$		−	9.78087e15i		−	0.312287i
$346$			1.22461e15i			0.0383712i
$347$	−1.94815e15			−0.0599075			−0.0299538	−	0.999551i	$-0.509536\pi$
$347$	−1.94815e15			−0.0599075			−0.0299538	+	0.999551i	$0.509536\pi$
$348$	2.28551e16			0.689793
$349$		−	1.33538e16i		−	0.395585i	−0.980244	−	0.197792i	$-0.936623\pi$
$349$		−	1.33538e16i		−	0.395585i	0.980244	−	0.197792i	$-0.0633772\pi$
$350$	5.77384e15			0.167889
$351$	5.19103e15	−	1.74381e16i	0.148169	−	0.497739i
$352$	−1.85936e16			−0.520998
$353$		−	4.75413e16i		−	1.30778i	−0.756589	−	0.653891i	$-0.773136\pi$
$353$		−	4.75413e16i		−	1.30778i	0.756589	−	0.653891i	$-0.226864\pi$
$354$	3.39281e15			0.0916302
$355$	1.36814e16			0.362782
$356$			4.37814e16i			1.13990i
$357$			1.21650e17i			3.11007i
$358$			3.88979e15i			0.0976531i
$359$		−	7.29791e15i		−	0.179922i	−0.995945	−	0.0899610i	$-0.971326\pi$
$359$		−	7.29791e15i		−	0.179922i	0.995945	−	0.0899610i	$-0.0286742\pi$
$360$	−1.51236e15			−0.0366175
$361$	−5.24929e16			−1.24826
$362$		−	4.87818e15i		−	0.113934i
$363$	7.21955e16			1.65622
$364$	6.53543e16	+	1.94549e16i	1.47270	+	0.438400i
$365$	5.69668e15			0.126101
$366$			1.54769e15i			0.0336555i
$367$	−1.64056e16			−0.350480			−0.175240	−	0.984526i	$-0.556070\pi$
$367$	−1.64056e16			−0.350480			−0.175240	+	0.984526i	$0.556070\pi$
$368$	−4.20067e16			−0.881673
$369$		−	1.24890e16i		−	0.257548i
$370$		−	4.92039e14i		−	0.00996983i
$371$		−	4.35714e16i		−	0.867504i
$372$			7.36805e16i			1.44153i
$373$	−2.44029e16			−0.469174			−0.234587	−	0.972095i	$-0.575374\pi$
$373$	−2.44029e16			−0.469174			−0.234587	+	0.972095i	$0.575374\pi$
$374$	1.48303e16			0.280211
$375$			3.52773e16i			0.655077i
$376$	−9.78549e15			−0.178592
$377$	−8.82585e15	+	2.96484e16i	−0.158321	+	0.531841i
$378$	5.33294e15			0.0940308
$379$			3.31354e16i			0.574298i	0.957886	+	0.287149i	$0.0927075\pi$
$379$			3.31354e16i			0.574298i	−0.957886	+	0.287149i	$0.907292\pi$
$380$	−2.34184e16			−0.398991
$381$	2.57804e16			0.431794
$382$		−	1.13452e16i		−	0.186809i
$383$		−	1.10336e17i		−	1.78617i	−0.449885	−	0.893087i	$-0.648535\pi$
$383$		−	1.10336e17i		−	1.78617i	0.449885	−	0.893087i	$-0.351465\pi$
$384$			3.55786e16i			0.566286i
$385$		−	4.08431e16i		−	0.639181i
$386$	3.23892e15			0.0498406
$387$	3.25267e16			0.492175
$388$			3.90474e16i			0.581013i
$389$	3.38919e16			0.495934			0.247967	−	0.968768i	$-0.420238\pi$
$389$	3.38919e16			0.495934			0.247967	+	0.968768i	$0.420238\pi$
$390$	7.83367e14	−	2.63154e15i	0.0112731	−	0.0378695i
$391$	1.02860e17			1.45579
$392$			2.36585e16i			0.329328i
$393$	−5.40824e16			−0.740464
$394$	2.97334e15			0.0400423
$395$			2.54824e16i			0.337566i
$396$		−	6.77030e16i		−	0.882245i
$397$			9.19933e16i			1.17928i	0.807665	+	0.589641i	$0.200731\pi$
$397$			9.19933e16i			1.17928i	−0.807665	+	0.589641i	$0.799269\pi$
$398$		−	8.83280e15i		−	0.111393i
$399$	−2.37200e17			−2.94301
$400$	7.28941e16			0.889820
$401$		−	9.31194e16i		−	1.11841i	−0.829029	−	0.559206i	$-0.811106\pi$
$401$		−	9.31194e16i		−	1.11841i	0.829029	−	0.559206i	$-0.188894\pi$
$402$	4.69312e15			0.0554615
$403$	−9.55805e16	−	2.84528e16i	−1.11144	−	0.330858i
$404$	2.82505e16			0.323257
$405$			2.97595e16i			0.335095i
$406$	−9.06713e15			−0.100473
$407$	4.43554e16			0.483708
$408$		−	4.29592e16i		−	0.461069i
$409$			3.59137e16i			0.379366i	0.981845	+	0.189683i	$0.0607460\pi$
$409$			3.59137e16i			0.379366i	−0.981845	+	0.189683i	$0.939254\pi$
$410$			1.32126e15i			0.0137370i
$411$		−	1.68350e17i		−	1.72282i
$412$	8.93886e15			0.0900430
$413$	9.82645e16			0.974365
$414$			6.43210e15i			0.0627842i
$415$	−3.74015e16			−0.359398
$416$	−3.46970e16	−	1.03287e16i	−0.328235	−	0.0977104i
$417$	1.59601e17			1.48646
$418$			2.89170e16i			0.265160i
$419$	−4.75608e16			−0.429396			−0.214698	−	0.976681i	$-0.568877\pi$
$419$	−4.75608e16			−0.429396			−0.214698	+	0.976681i	$0.568877\pi$
$420$	−5.87529e16			−0.522287
$421$		−	9.27249e16i		−	0.811638i	−0.913953	−	0.405819i	$-0.866986\pi$
$421$		−	9.27249e16i		−	0.811638i	0.913953	−	0.405819i	$-0.133014\pi$
$422$			5.29705e15i			0.0456566i
$423$		−	5.35675e16i		−	0.454663i
$424$			1.53867e16i			0.128608i
$425$	−1.78493e17			−1.46924
$426$	−2.43018e16			−0.197004
$427$			4.48250e16i			0.357881i
$428$	−1.03197e17			−0.811485
$429$	2.37223e17	+	7.06175e16i	1.83732	+	0.546940i
$430$	−3.44112e15			−0.0262515
$431$		−	1.37175e17i		−	1.03080i	−0.856951	−	0.515398i	$-0.827644\pi$
$431$		−	1.37175e17i		−	1.03080i	0.856951	−	0.515398i	$-0.172356\pi$
$432$	6.73277e16			0.498367
$433$	−1.71417e17			−1.24992			−0.624960	−	0.780657i	$-0.714885\pi$
$433$	−1.71417e17			−1.24992			−0.624960	+	0.780657i	$0.714885\pi$
$434$		−	2.92306e16i		−	0.209969i
$435$		−	2.66536e16i		−	0.188615i
$436$			2.15262e16i			0.150074i
$437$			2.00562e17i			1.37759i
$438$	−1.01188e16			−0.0684775
$439$	2.77445e17			1.84994			0.924969	−	0.380043i	$-0.124091\pi$
$439$	2.77445e17			1.84994			0.924969	+	0.380043i	$0.124091\pi$
$440$			1.44232e16i			0.0947587i
$441$	−1.29511e17			−0.838408
$442$	2.76745e16	+	8.23825e15i	0.176537	+	0.0525521i
$443$	6.85915e16			0.431168			0.215584	−	0.976485i	$-0.430835\pi$
$443$	6.85915e16			0.431168			0.215584	+	0.976485i	$0.430835\pi$
$444$		−	6.38054e16i		−	0.395247i
$445$	5.10578e16			0.311690
$446$	−1.05476e16			−0.0634565
$447$			3.60481e17i			2.13739i
$448$			2.45176e17i			1.43275i
$449$		−	1.23361e16i		−	0.0710523i	−0.999369	−	0.0355262i	$-0.988689\pi$
$449$		−	1.23361e16i		−	0.0710523i	0.999369	−	0.0355262i	$-0.0113107\pi$
$450$		−	1.11616e16i		−	0.0633644i
$451$	−1.19107e17			−0.666482
$452$	1.70784e17			0.941995
$453$		−	5.64401e16i		−	0.306866i
$454$	−3.23525e16			−0.173398
$455$	2.26883e16	−	7.62160e16i	0.119875	−	0.402691i
$456$	8.37642e16			0.436302
$457$		−	5.59570e16i		−	0.287342i	−0.989626	−	0.143671i	$-0.954109\pi$
$457$		−	5.59570e16i		−	0.287342i	0.989626	−	0.143671i	$-0.0458907\pi$
$458$	1.12676e16			0.0570435
$459$	−1.64863e17			−0.822888
$460$			4.96779e16i			0.244477i
$461$			8.94329e16i			0.433951i	0.976177	+	0.216976i	$0.0696192\pi$
$461$			8.94329e16i			0.433951i	−0.976177	+	0.216976i	$0.930381\pi$
$462$			7.25481e16i			0.347099i
$463$			3.48225e17i			1.64280i	0.570355	+	0.821399i	$0.306806\pi$
$463$			3.48225e17i			1.64280i	−0.570355	+	0.821399i	$0.693194\pi$
$464$	−1.14471e17			−0.532513
$465$	8.59260e16			0.394167
$466$		−	3.08793e16i		−	0.139688i
$467$	−1.04214e17			−0.464906			−0.232453	−	0.972608i	$-0.574675\pi$
$467$	−1.04214e17			−0.464906			−0.232453	+	0.972608i	$0.574675\pi$
$468$	3.76089e16	−	1.26338e17i	0.165460	−	0.555824i
$469$	1.35925e17			0.589759
$470$			5.66710e15i			0.0242507i
$471$	−3.32547e16			−0.140352
$472$	−3.47008e16			−0.144450
$473$		−	3.10204e17i		−	1.27365i
$474$		−	4.52634e16i		−	0.183311i
$475$		−	3.48035e17i		−	1.39032i
$476$		−	6.17873e17i		−	2.43475i
$477$	−8.42293e16			−0.327411
$478$	1.23253e16			0.0472625
$479$		−	2.52054e16i		−	0.0953481i	−0.998863	−	0.0476741i	$-0.984819\pi$
$479$		−	2.52054e16i		−	0.0953481i	0.998863	−	0.0476741i	$-0.0151809\pi$
$480$	3.11923e16			0.116407
$481$	8.27703e16	+	2.46394e16i	0.304741	+	0.0907167i
$482$	−3.27235e16			−0.118865
$483$			5.03178e17i			1.80329i
$484$	−3.66687e17			−1.29658
$485$	4.55370e16			0.158870
$486$		−	3.53241e16i		−	0.121600i
$487$		−	2.44127e17i		−	0.829235i	−0.909996	−	0.414618i	$-0.863915\pi$
$487$		−	2.44127e17i		−	0.829235i	0.909996	−	0.414618i	$-0.136085\pi$
$488$		−	1.58293e16i		−	0.0530560i
$489$		−	6.33015e14i		−	0.00209366i
$490$	1.37014e16			0.0447189
$491$	1.89093e17			0.609039			0.304520	−	0.952506i	$-0.401504\pi$
$491$	1.89093e17			0.609039			0.304520	+	0.952506i	$0.401504\pi$
$492$			1.71335e17i			0.544595i
$493$	2.80302e17			0.879268
$494$	−1.60634e16	+	5.39611e16i	−0.0497292	+	0.167054i
$495$	−7.89551e16			−0.241238
$496$		−	3.69033e17i		−	1.11284i
$497$	−7.03841e17			−2.09488
$498$	6.64349e16			0.195167
$499$			3.67156e17i			1.06462i	0.846548	+	0.532312i	$0.178677\pi$
$499$			3.67156e17i			1.06462i	−0.846548	+	0.532312i	$0.821323\pi$
$500$		−	1.79176e17i		−	0.512833i
$501$			1.36899e17i			0.386773i
$502$		−	5.16807e16i		−	0.144130i
$503$	5.79880e17			1.59642			0.798208	−	0.602382i	$-0.205781\pi$
$503$	5.79880e17			1.59642			0.798208	+	0.602382i	$0.205781\pi$
$504$	7.78034e16			0.211446
$505$		−	3.29457e16i		−	0.0883904i
$506$	6.13422e16			0.162473
$507$	4.03447e17	+	2.63555e17i	1.05496	+	0.689158i
$508$	−1.30941e17			−0.338034
$509$			1.01334e17i			0.258280i	0.991626	+	0.129140i	$0.0412216\pi$
$509$			1.01334e17i			0.258280i	−0.991626	+	0.129140i	$0.958778\pi$
$510$	−2.48791e16			−0.0626078
$511$	−2.93067e17			−0.728166
$512$		−	2.24292e17i		−	0.550248i
$513$		−	3.21458e17i		−	0.778685i
$514$			5.30715e16i			0.126941i
$515$		−	1.04245e16i		−	0.0246211i
$516$	−4.46230e17			−1.04072
$517$	−5.10867e17			−1.17658
$518$			2.53130e16i			0.0575705i
$519$	1.85195e17			0.415951
$520$	−8.01207e15	+	2.69147e16i	−0.0177715	+	0.0596991i
$521$	−3.30472e17			−0.723917			−0.361959	−	0.932194i	$-0.617892\pi$
$521$	−3.30472e17			−0.723917			−0.361959	+	0.932194i	$0.617892\pi$
$522$			1.75280e16i			0.0379204i
$523$	3.43602e17			0.734166			0.367083	−	0.930188i	$-0.380356\pi$
$523$	3.43602e17			0.734166			0.367083	+	0.930188i	$0.380356\pi$
$524$	2.74689e17			0.579679
$525$		−	8.73164e17i		−	1.81995i
$526$		−	9.39162e16i		−	0.193345i
$527$			9.03638e17i			1.83749i
$528$			9.15910e17i			1.83964i
$529$	−7.85794e16			−0.155900
$530$	8.91092e15			0.0174634
$531$		−	1.89958e17i		−	0.367743i
$532$	1.20476e18			2.30396
$533$	−2.22261e17	−	6.61636e16i	−0.419891	−	0.124995i
$534$	−9.06921e16			−0.169259
$535$			1.20348e17i			0.221890i
$536$	−4.80000e16			−0.0874319
$537$	5.88243e17			1.05858
$538$			3.92659e16i			0.0698121i
$539$			1.23513e18i			2.16963i
$540$		−	7.96231e16i		−	0.138191i
$541$			1.83938e16i			0.0315420i	0.999876	+	0.0157710i	$0.00502027\pi$
$541$			1.83938e16i			0.0315420i	−0.999876	+	0.0157710i	$0.994980\pi$
$542$	−9.19043e16			−0.155719
$543$	−7.37715e17			−1.23506
$544$			3.28033e17i			0.542655i
$545$	2.51038e16			0.0410357
$546$	−4.03004e16	+	1.35380e17i	−0.0650964	+	0.218676i
$547$	2.41353e17			0.385243			0.192621	−	0.981273i	$-0.438301\pi$
$547$	2.41353e17			0.385243			0.192621	+	0.981273i	$0.438301\pi$
$548$			8.55063e17i			1.34873i
$549$	8.66527e16			0.135071
$550$	−1.06447e17			−0.163974
$551$			5.46547e17i			0.832037i
$552$		−	1.77691e17i		−	0.267338i
$553$		−	1.31094e18i		−	1.94926i
$554$		−	7.13552e16i		−	0.104861i
$555$	−7.44098e16			−0.108075
$556$	−8.10629e17			−1.16369
$557$		−	1.57109e17i		−	0.222916i	−0.993769	−	0.111458i	$-0.964448\pi$
$557$		−	1.57109e17i		−	0.222916i	0.993769	−	0.111458i	$-0.0355521\pi$
$558$	−5.65067e16			−0.0792460
$559$	1.72318e17	−	5.78862e17i	0.238866	−	0.802414i
$560$	2.94267e17			0.403200
$561$		−	2.24276e18i		−	3.03755i
$562$	1.12136e17			0.150128
$563$	7.27362e17			0.962601			0.481300	−	0.876556i	$-0.340165\pi$
$563$	7.27362e17			0.962601			0.481300	+	0.876556i	$0.340165\pi$
$564$			7.34885e17i			0.961402i
$565$		−	1.99169e17i		−	0.257576i
$566$			1.45354e17i			0.185832i
$567$		−	1.53098e18i		−	1.93499i
$568$	2.48552e17			0.310566
$569$	1.04594e18			1.29205			0.646024	−	0.763318i	$-0.276431\pi$
$569$	1.04594e18			1.29205			0.646024	+	0.763318i	$0.276431\pi$
$570$		−	4.85106e16i		−	0.0592448i
$571$	−1.18728e18			−1.43356			−0.716782	−	0.697297i	$-0.754386\pi$
$571$	−1.18728e18			−1.43356			−0.716782	+	0.697297i	$0.754386\pi$
$572$	−1.20488e18	−	3.58672e17i	−1.43836	−	0.428177i
$573$	−1.71570e18			−2.02505
$574$		−	6.79724e16i		−	0.0793241i
$575$	−7.38294e17			−0.851900
$576$	4.73957e17			0.540746
$577$			2.53500e17i			0.285980i	0.989724	+	0.142990i	$0.0456716\pi$
$577$			2.53500e17i			0.285980i	−0.989724	+	0.142990i	$0.954328\pi$
$578$		−	1.57432e17i		−	0.175615i
$579$		−	4.89814e17i		−	0.540282i
$580$			1.35376e17i			0.147659i
$581$	1.92412e18			2.07533
$582$	−8.08857e16			−0.0862725
$583$			8.03286e17i			0.847275i
$584$	1.03493e17			0.107951
$585$	−1.47336e17	−	4.38595e16i	−0.151983	−	0.0452429i
$586$	1.63710e17			0.167009
$587$			1.71807e18i			1.73338i	0.498851	+	0.866688i	$0.333756\pi$
$587$			1.71807e18i			1.73338i	−0.498851	+	0.866688i	$0.666244\pi$
$588$	1.77674e18			1.77285
$589$	−1.76196e18			−1.73879
$590$			2.00964e16i			0.0196146i
$591$		−	4.49651e17i		−	0.434067i
$592$			3.19573e17i			0.305126i
$593$			1.35319e18i			1.27792i	0.769239	+	0.638962i	$0.220636\pi$
$593$			1.35319e18i			1.27792i	−0.769239	+	0.638962i	$0.779364\pi$
$594$	−9.83185e16			−0.0918382
$595$	−7.20562e17			−0.665750
$596$		−	1.83091e18i		−	1.67327i
$597$	−1.33576e18			−1.20752
$598$	1.14469e17	+	3.40756e16i	0.102360	+	0.0304709i
$599$	1.59111e18			1.40743			0.703714	−	0.710484i	$-0.251524\pi$
$599$	1.59111e18			1.40743			0.703714	+	0.710484i	$0.251524\pi$
$600$			3.08346e17i			0.269809i
$601$	1.95821e18			1.69502			0.847510	−	0.530779i	$-0.178100\pi$
$601$	1.95821e18			1.69502			0.847510	+	0.530779i	$0.178100\pi$
$602$	1.77029e17			0.151589
$603$		−	2.62760e17i		−	0.222586i
$604$			2.86664e17i			0.240233i
$605$			4.27630e17i			0.354534i
$606$			5.85203e16i			0.0479992i
$607$	1.19586e18			0.970407			0.485203	−	0.874401i	$-0.338746\pi$
$607$	1.19586e18			0.970407			0.485203	+	0.874401i	$0.338746\pi$
$608$	−6.39615e17			−0.513506
$609$			1.37120e18i			1.08915i
$610$	−9.16729e15			−0.00720439
$611$	−9.53315e17	−	2.83787e17i	−0.741256	−	0.220660i
$612$	−1.19443e18			−0.918918
$613$		−	1.29101e18i		−	0.982736i	−0.870952	−	0.491368i	$-0.836497\pi$
$613$		−	1.29101e18i		−	0.982736i	0.870952	−	0.491368i	$-0.163503\pi$
$614$	−6.82993e16			−0.0514425
$615$	1.99811e17			0.148912
$616$		−	7.42003e17i		−	0.547181i
$617$			1.31171e18i			0.957157i	0.878045	+	0.478579i	$0.158848\pi$
$617$			1.31171e18i			0.957157i	−0.878045	+	0.478579i	$0.841152\pi$
$618$			1.85166e16i			0.0133702i
$619$		−	1.04438e18i		−	0.746227i	−0.927786	−	0.373114i	$-0.878290\pi$
$619$		−	1.04438e18i		−	0.746227i	0.927786	−	0.373114i	$-0.121710\pi$
$620$	−4.36426e17			−0.308578
$621$	−6.81917e17			−0.477129
$622$			4.01562e16i			0.0278045i
$623$	−2.62668e18			−1.79984
$624$	−5.08787e17	+	1.70915e18i	−0.345014	+	1.15899i
$625$	1.17274e18			0.787012
$626$			3.04503e16i			0.0202236i
$627$	4.37304e18			2.87438
$628$	1.68904e17			0.109876
$629$		−	7.82528e17i		−	0.503814i
$630$		−	4.50585e16i		−	0.0287120i
$631$			2.69060e18i			1.69691i	0.529270	+	0.848454i	$0.322466\pi$
$631$			2.69060e18i			1.69691i	−0.529270	+	0.848454i	$0.677534\pi$
$632$			4.62942e17i			0.288979i
$633$	8.01060e17			0.494927
$634$	2.69135e17			0.164585
$635$			1.52703e17i			0.0924311i
$636$	1.15553e18			0.692325
$637$	−6.86114e17	+	2.30484e18i	−0.406902	+	1.36689i
$638$	1.67162e17			0.0981304
$639$			1.36062e18i			0.790644i
$640$	−2.10740e17			−0.121221
$641$	2.11037e17			0.120166			0.0600830	−	0.998193i	$-0.480863\pi$
$641$	2.11037e17			0.120166			0.0600830	+	0.998193i	$0.480863\pi$
$642$		−	2.13769e17i		−	0.120494i
$643$		−	2.90287e18i		−	1.61978i	−0.586582	−	0.809890i	$-0.699527\pi$
$643$		−	2.90287e18i		−	1.61978i	0.586582	−	0.809890i	$-0.300473\pi$
$644$		−	2.55569e18i		−	1.41172i
$645$			5.20392e17i			0.284572i
$646$	5.10160e17			0.276182
$647$	−1.40372e18			−0.752318			−0.376159	−	0.926555i	$-0.622755\pi$
$647$	−1.40372e18			−0.752318			−0.376159	+	0.926555i	$0.622755\pi$
$648$			5.40645e17i			0.286863i
$649$	−1.81161e18			−0.951644
$650$	−1.98638e17	−	5.91312e16i	−0.103306	−	0.0307525i
$651$	−4.42048e18			−2.27611
$652$			3.21514e15i			0.00163904i
$653$	−2.10572e17			−0.106283			−0.0531415	−	0.998587i	$-0.516923\pi$
$653$	−2.10572e17			−0.106283			−0.0531415	+	0.998587i	$0.516923\pi$
$654$	−4.45910e16			−0.0222839
$655$		−	3.20342e17i		−	0.158506i
$656$		−	8.58143e17i		−	0.420421i
$657$			5.66537e17i			0.274823i
$658$		−	2.91545e17i		−	0.140035i
$659$	3.91215e18			1.86063			0.930315	−	0.366762i	$-0.119534\pi$
$659$	3.91215e18			1.86063			0.930315	+	0.366762i	$0.119534\pi$
$660$	1.08317e18			0.510108
$661$		−	3.04004e18i		−	1.41765i	−0.705382	−	0.708827i	$-0.749225\pi$
$661$		−	3.04004e18i		−	1.41765i	0.705382	−	0.708827i	$-0.250775\pi$
$662$	−3.28555e17			−0.151716
$663$	1.24585e18	−	4.18514e18i	0.569676	−	1.91369i
$664$	−6.79479e17			−0.307669
$665$		−	1.40499e18i		−	0.629989i
$666$	4.89334e16			0.0217281
$667$	1.15940e18			0.509819
$668$		−	6.95323e17i		−	0.302789i
$669$			1.59508e18i			0.687882i
$670$			2.77984e16i			0.0118722i
$671$		−	8.26397e17i		−	0.349536i
$672$	−1.60469e18			−0.672189
$673$	−2.30373e18			−0.955727			−0.477863	−	0.878434i	$-0.658589\pi$
$673$	−2.30373e18			−0.955727			−0.477863	+	0.878434i	$0.658589\pi$
$674$		−	2.92952e17i		−	0.120367i
$675$	1.18333e18			0.481538
$676$	−2.04914e18	−	1.33862e18i	−0.825883	−	0.539514i
$677$	1.02215e18			0.408028			0.204014	−	0.978968i	$-0.434601\pi$
$677$	1.02215e18			0.408028			0.204014	+	0.978968i	$0.434601\pi$
$678$			3.53776e17i			0.139873i
$679$	−2.34266e18			−0.917392
$680$	2.54457e17			0.0986976
$681$			4.89259e18i			1.87967i
$682$			5.38898e17i			0.205073i
$683$		−	4.36365e17i		−	0.164481i	−0.996613	−	0.0822404i	$-0.973792\pi$
$683$		−	4.36365e17i		−	0.164481i	0.996613	−	0.0822404i	$-0.0262075\pi$
$684$		−	2.32896e18i		−	0.869557i
$685$	9.97173e17			0.368792
$686$	−2.10631e17			−0.0771643
$687$		−	1.70397e18i		−	0.618363i
$688$	2.23497e18			0.803427
$689$	−4.46225e17	+	1.49899e18i	−0.158902	+	0.533793i
$690$	−1.02907e17			−0.0363014
$691$			3.72116e18i			1.30038i	0.759770	+	0.650192i	$0.225312\pi$
$691$			3.72116e18i			1.30038i	−0.759770	+	0.650192i	$0.774688\pi$
$692$	−9.40623e17			−0.325631
$693$	4.06185e18			1.39302
$694$			2.04969e16i			0.00696388i
$695$			9.45354e17i			0.318195i
$696$		−	4.84221e17i		−	0.161467i
$697$			2.10130e18i			0.694186i
$698$	−1.40498e17			−0.0459843
$699$	−4.66980e18			−1.51425
$700$			4.43487e18i			1.42477i
$701$	4.34458e18			1.38287			0.691435	−	0.722439i	$-0.256979\pi$
$701$	4.34458e18			1.38287			0.691435	+	0.722439i	$0.256979\pi$
$702$	−1.83469e17	−	5.46158e16i	−0.0578591	−	0.0172237i
$703$	1.52581e18			0.476751
$704$		−	4.52008e18i		−	1.39934i
$705$	8.57021e17			0.262883
$706$	−5.00191e17			−0.152022
$707$			1.69490e18i			0.510408i
$708$			2.60601e18i			0.777607i
$709$		−	2.53785e18i		−	0.750352i	−0.926954	−	0.375176i	$-0.877582\pi$
$709$		−	2.53785e18i		−	0.750352i	0.926954	−	0.375176i	$-0.122418\pi$
$710$		−	1.43945e17i		−	0.0421712i
$711$	−2.53423e18			−0.735687
$712$	9.27576e17			0.266827
$713$			3.73769e18i			1.06542i
$714$	1.27991e18			0.361527
$715$	−4.18283e17	+	1.40512e18i	−0.117080	+	0.393301i
$716$	−2.98774e18			−0.828719
$717$		−	1.86393e18i		−	0.512336i
$718$	−7.67827e16			−0.0209148
$719$	−1.35407e18			−0.365515			−0.182757	−	0.983158i	$-0.558502\pi$
$719$	−1.35407e18			−0.365515			−0.182757	+	0.983158i	$0.558502\pi$
$720$		−	5.68858e17i		−	0.152175i
$721$			5.36289e17i			0.142174i
$722$			5.52289e17i			0.145102i
$723$			4.94869e18i			1.28852i
$724$	3.74692e18			0.966882
$725$	−2.01191e18			−0.514531
$726$		−	7.59583e17i		−	0.192525i
$727$	7.18735e18			1.80549			0.902745	−	0.430175i	$-0.141548\pi$
$727$	7.18735e18			1.80549			0.902745	+	0.430175i	$0.141548\pi$
$728$	4.12182e17	−	1.38463e18i	0.102621	−	0.344730i
$729$	−3.07574e17			−0.0758963
$730$		−	5.99359e16i		−	0.0146585i
$731$	−5.47269e18			−1.32659
$732$	−1.18877e18			−0.285613
$733$		−	3.36376e18i		−	0.801031i	−0.916290	−	0.400515i	$-0.868831\pi$
$733$		−	3.36376e18i		−	0.801031i	0.916290	−	0.400515i	$-0.131169\pi$
$734$			1.72607e17i			0.0407412i
$735$		−	2.07203e18i		−	0.484762i
$736$			1.35683e18i			0.314644i
$737$	−2.50592e18			−0.576007
$738$	−1.31400e17			−0.0299383
$739$		−	3.62815e18i		−	0.819401i	−0.912220	−	0.409700i	$-0.865633\pi$
$739$		−	3.62815e18i		−	0.819401i	0.912220	−	0.409700i	$-0.134367\pi$
$740$	3.77933e17			0.0846076
$741$	8.16041e18	+	2.42922e18i	1.81090	+	0.539075i
$742$	−4.58423e17			−0.100842
$743$		−	3.41321e18i		−	0.744278i	−0.928177	−	0.372139i	$-0.878624\pi$
$743$		−	3.41321e18i		−	0.744278i	0.928177	−	0.372139i	$-0.121376\pi$
$744$	1.56103e18			0.337433
$745$	−2.13521e18			−0.457535
$746$			2.56748e17i			0.0545387i
$747$		−	3.71959e18i		−	0.783269i
$748$			1.13911e19i			2.37797i
$749$		−	6.19130e18i		−	1.28130i
$750$	3.71160e17			0.0761487
$751$	2.14113e18			0.435495			0.217748	−	0.976005i	$-0.430129\pi$
$751$	2.14113e18			0.435495			0.217748	+	0.976005i	$0.430129\pi$
$752$		−	3.68071e18i		−	0.742192i
$753$	−7.81554e18			−1.56240
$754$	3.11936e17	+	9.28585e16i	0.0618233	+	0.0184038i
$755$	3.34307e17			0.0656886
$756$			4.09622e18i			0.797979i
$757$	−7.70654e18			−1.48846			−0.744229	−	0.667925i	$-0.767183\pi$
$757$	−7.70654e18			−1.48846			−0.744229	+	0.667925i	$0.767183\pi$
$758$	3.48624e17			0.0667587
$759$		−	9.27663e18i		−	1.76124i
$760$			4.96154e17i			0.0933959i
$761$		−	1.20918e18i		−	0.225679i	−0.993613	−	0.112840i	$-0.964005\pi$
$761$		−	1.20918e18i		−	0.225679i	0.993613	−	0.112840i	$-0.0359946\pi$
$762$		−	2.71241e17i		−	0.0501935i
$763$	−1.29147e18			−0.236960
$764$	8.71419e18			1.58533
$765$			1.39294e18i			0.251266i
$766$	−1.16086e18			−0.207632
$767$	−3.38060e18	−	1.00635e18i	−0.599547	−	0.178476i
$768$	−6.21689e18			−1.09326
$769$			8.72057e18i			1.52063i	0.649554	+	0.760315i	$0.274955\pi$
$769$			8.72057e18i			1.52063i	−0.649554	+	0.760315i	$0.725045\pi$
$770$	−4.29718e17			−0.0743009
$771$	8.02588e18			1.37607
$772$			2.48781e18i			0.422965i
$773$		−	1.00121e19i		−	1.68794i	−0.536392	−	0.843969i	$-0.680213\pi$
$773$		−	1.00121e19i		−	1.68794i	0.536392	−	0.843969i	$-0.319787\pi$
$774$		−	3.42220e17i		−	0.0572123i
$775$		−	6.48600e18i		−	1.07527i
$776$	8.27278e17			0.136004
$777$	3.82802e18			0.624076
$778$		−	3.56584e17i		−	0.0576493i
$779$	−4.09723e18			−0.656897
$780$	2.02128e18	+	6.01702e17i	0.321374	+	0.0956679i
$781$	1.29761e19			2.04603
$782$		−	1.08221e18i		−	0.169227i
$783$	−1.85827e18			−0.288176
$784$	−8.89891e18			−1.36862
$785$		−	1.96975e17i		−	0.0300441i
$786$			5.69011e17i			0.0860745i
$787$		−	1.12189e19i		−	1.68312i	−0.540166	−	0.841558i	$-0.681639\pi$
$787$		−	1.12189e19i		−	1.68312i	0.540166	−	0.841558i	$-0.318361\pi$
$788$			2.28381e18i			0.339813i
$789$	−1.42027e19			−2.09590
$790$	2.68105e17			0.0392400
$791$			1.02462e19i			1.48737i
$792$	−1.43439e18			−0.206516
$793$	4.59063e17	−	1.54211e18i	0.0655536	−	0.220212i
$794$	9.67880e17			0.137084
$795$		−	1.34758e18i		−	0.189307i
$796$	6.78445e18			0.945322
$797$	−4.70000e18			−0.649559			−0.324780	−	0.945790i	$-0.605290\pi$
$797$	−4.70000e18			−0.649559			−0.324780	+	0.945790i	$0.605290\pi$
$798$			2.49563e18i			0.342107i
$799$			9.01283e18i			1.22548i
$800$		−	2.35450e18i		−	0.317552i
$801$			5.07771e18i			0.679292i
$802$	−9.79727e17			−0.130009
$803$	5.40300e18			0.711187
$804$			3.60477e18i			0.470666i
$805$	−2.98044e18			−0.386017
$806$	−2.99357e17	+	1.00562e18i	−0.0384603	+	0.129198i
$807$	5.93808e18			0.756777
$808$		−	5.98530e17i		−	0.0756680i
$809$	5.07622e18			0.636612			0.318306	−	0.947988i	$-0.396886\pi$
$809$	5.07622e18			0.636612			0.318306	+	0.947988i	$0.396886\pi$
$810$	3.13105e17			0.0389527
$811$			9.93674e18i			1.22633i	0.789954	+	0.613166i	$0.210105\pi$
$811$			9.93674e18i			1.22633i	−0.789954	+	0.613166i	$0.789895\pi$
$812$		−	6.96444e18i		−	0.852652i
$813$			1.38985e19i			1.68802i
$814$		−	4.66672e17i		−	0.0562281i
$815$	3.74949e15			0.000448175
$816$	1.61587e19			1.91611
$817$		−	1.06709e19i		−	1.25533i
$818$	3.77855e17			0.0440990
$819$	7.57971e18	+	2.25636e18i	0.877621	+	0.261254i
$820$	−1.01486e18			−0.116577
$821$			5.68777e18i			0.648204i	0.946022	+	0.324102i	$0.105062\pi$
$821$			5.68777e18i			0.648204i	−0.946022	+	0.324102i	$0.894938\pi$
$822$	−1.77124e18			−0.200268
$823$	−1.08701e19			−1.21936			−0.609681	−	0.792647i	$-0.708703\pi$
$823$	−1.08701e19			−1.21936			−0.609681	+	0.792647i	$0.708703\pi$
$824$		−	1.89383e17i		−	0.0210773i
$825$			1.60977e19i			1.77752i
$826$		−	1.03386e18i		−	0.113264i
$827$			1.05437e19i			1.14605i	0.819536	+	0.573027i	$0.194231\pi$
$827$			1.05437e19i			1.14605i	−0.819536	+	0.573027i	$0.805769\pi$
$828$	−4.94048e18			−0.532809
$829$	8.40690e17			0.0899562			0.0449781	−	0.998988i	$-0.485678\pi$
$829$	8.40690e17			0.0899562			0.0449781	+	0.998988i	$0.485678\pi$
$830$			3.93508e17i			0.0417779i
$831$	−1.07909e19			−1.13671
$832$	2.51090e18	−	8.43479e18i	0.262439	−	0.881602i
$833$	2.17904e19			2.25982
$834$		−	1.67920e18i		−	0.172791i
$835$	−8.10885e17			−0.0827937
$836$	−2.22111e19			−2.25024
$837$		−	5.99071e18i		−	0.602231i
$838$			5.00396e17i			0.0499147i
$839$		−	2.41432e18i		−	0.238969i	−0.992836	−	0.119484i	$-0.961876\pi$
$839$		−	2.41432e18i		−	0.238969i	0.992836	−	0.119484i	$-0.0381242\pi$
$840$			1.24477e18i			0.122257i
$841$	−7.10117e18			−0.692079
$842$	−9.75577e17			−0.0943480
$843$		−	1.69581e19i		−	1.62741i
$844$	−4.06865e18			−0.387458
$845$	−1.56109e18	+	2.38971e18i	−0.147523	+	0.225827i
$846$	−5.63594e17			−0.0528517
$847$		−	2.19995e19i		−	2.04725i
$848$	−5.78754e18			−0.534467
$849$	2.19816e19			2.01446
$850$			1.87796e18i			0.170790i
$851$		−	3.23674e18i		−	0.292123i
$852$		−	1.86661e19i		−	1.67185i
$853$			1.72105e18i			0.152977i	0.997070	+	0.0764883i	$0.0243708\pi$
$853$			1.72105e18i			0.152977i	−0.997070	+	0.0764883i	$0.975629\pi$
$854$	4.71613e17			0.0416015
$855$	−2.71603e18			−0.237769
$856$			2.18637e18i			0.189953i
$857$	7.99164e18			0.689066			0.344533	−	0.938774i	$-0.388037\pi$
$857$	7.99164e18			0.689066			0.344533	+	0.938774i	$0.388037\pi$
$858$	7.42981e17	−	2.49587e18i	0.0635785	−	0.213577i
$859$	−1.85761e18			−0.157760			−0.0788802	−	0.996884i	$-0.525134\pi$
$859$	−1.85761e18			−0.157760			−0.0788802	+	0.996884i	$0.525134\pi$
$860$		−	2.64312e18i		−	0.222780i
$861$	−1.02793e19			−0.859890
$862$	−1.44325e18			−0.119824
$863$			7.68131e18i			0.632944i	0.948602	+	0.316472i	$0.102498\pi$
$863$			7.68131e18i			0.632944i	−0.948602	+	0.316472i	$0.897502\pi$
$864$		−	2.17471e18i		−	0.177853i
$865$			1.09695e18i			0.0890397i
$866$			1.80351e18i			0.145296i
$867$	−2.38081e19			−1.90371
$868$	2.24520e19			1.78187
$869$			2.41686e19i			1.90381i
$870$	−2.80428e17			−0.0219253
$871$	−4.67622e18	−	1.39203e18i	−0.362891	−	0.108027i
$872$	4.56065e17			0.0351293
$873$			4.52866e18i			0.346240i
$874$	2.11016e18			0.160136
$875$	1.07497e19			0.809739
$876$		−	7.77223e18i		−	0.581124i
$877$		−	8.04195e17i		−	0.0596849i	−0.999555	−	0.0298424i	$-0.990499\pi$
$877$		−	8.04195e17i		−	0.0596849i	0.999555	−	0.0298424i	$-0.00950055\pi$
$878$		−	2.91905e18i		−	0.215044i
$879$		−	2.47574e19i		−	1.81041i
$880$	−5.42514e18			−0.393798
$881$	−5.69712e18			−0.410499			−0.205249	−	0.978710i	$-0.565801\pi$
$881$	−5.69712e18			−0.410499			−0.205249	+	0.978710i	$0.565801\pi$
$882$			1.36261e18i			0.0974598i
$883$	1.22609e19			0.870521			0.435261	−	0.900304i	$-0.356656\pi$
$883$	1.22609e19			0.870521			0.435261	+	0.900304i	$0.356656\pi$
$884$	−6.32778e18	+	2.12567e19i	−0.445976	+	1.49815i
$885$	3.03913e18			0.212627
$886$		−	7.21665e17i		−	0.0501206i
$887$	−5.52404e18			−0.380849			−0.190425	−	0.981702i	$-0.560986\pi$
$887$	−5.52404e18			−0.380849			−0.190425	+	0.981702i	$0.560986\pi$
$888$	−1.35181e18			−0.0925194
$889$		−	7.85582e18i		−	0.533740i
$890$		−	5.37189e17i		−	0.0362320i
$891$			2.82253e19i			1.88987i
$892$		−	8.10157e18i		−	0.538515i
$893$	−1.75737e19			−1.15965
$894$	3.79269e18			0.248458
$895$			3.48429e18i			0.226603i
$896$	1.08415e19			0.699986
$897$	5.15316e18	−	1.73108e19i	0.330311	−	1.10960i
$898$	−1.29791e17			−0.00825940
$899$			1.01855e19i			0.643492i
$900$	8.57320e18			0.537733
$901$	1.41717e19			0.882495
$902$			1.25314e18i			0.0774744i
$903$		−	2.67716e19i		−	1.64325i
$904$		−	3.61833e18i		−	0.220502i
$905$		−	4.36965e18i		−	0.264381i
$906$	−5.93817e17			−0.0356713
$907$	−2.25630e19			−1.34570			−0.672851	−	0.739778i	$-0.734931\pi$
$907$	−2.25630e19			−1.34570			−0.672851	+	0.739778i	$0.734931\pi$
$908$		−	2.48499e19i		−	1.47152i
$909$	3.27646e18			0.192637
$910$	−8.01884e17	−	2.38708e17i	−0.0468104	−	0.0139347i
$911$	2.45982e19			1.42572			0.712858	−	0.701308i	$-0.247400\pi$
$911$	2.45982e19			1.42572			0.712858	+	0.701308i	$0.247400\pi$
$912$			3.15071e19i			1.81318i
$913$	−3.54733e19			−2.02694
$914$	−5.88734e17			−0.0334017
$915$			1.38635e18i			0.0780971i
$916$			8.65462e18i			0.484091i
$917$			1.64800e19i			0.915287i
$918$			1.73456e18i			0.0956557i
$919$	2.18624e19			1.19715			0.598574	−	0.801068i	$-0.295734\pi$
$919$	2.18624e19			1.19715			0.598574	+	0.801068i	$0.295734\pi$
$920$	1.05250e18			0.0572271
$921$			1.03287e19i			0.557647i
$922$	9.40941e17			0.0504442
$923$	2.42143e19	+	7.20820e18i	1.28902	+	0.383721i
$924$	−5.57240e19			−2.94560
$925$			5.61671e18i			0.294822i
$926$	3.66375e18			0.190965
$927$	1.03672e18			0.0536589
$928$			3.69746e18i			0.190039i
$929$			1.92132e19i			0.980613i	0.871550	+	0.490306i	$0.163115\pi$
$929$			1.92132e19i			0.980613i	−0.871550	+	0.490306i	$0.836885\pi$
$930$		−	9.04045e17i		−	0.0458196i
$931$			4.24882e19i			2.13843i
$932$	2.37183e19			1.18544
$933$	6.07272e18			0.301406
$934$			1.09645e18i			0.0540425i
$935$	1.32843e19			0.650226
$936$	−2.67667e18	−	7.96802e17i	−0.130107	−	0.0387309i
$937$	5.79468e18			0.279719			0.139859	−	0.990171i	$-0.455335\pi$
$937$	5.79468e18			0.279719			0.139859	+	0.990171i	$0.455335\pi$
$938$		−	1.43009e18i		−	0.0685559i
$939$	4.60493e18			0.219228
$940$	−4.35288e18			−0.205800
$941$		−	7.73986e18i		−	0.363413i	−0.983353	−	0.181707i	$-0.941838\pi$
$941$		−	7.73986e18i		−	0.363413i	0.983353	−	0.181707i	$-0.0581621\pi$
$942$			3.49880e17i			0.0163150i
$943$			8.69154e18i			0.402505i
$944$		−	1.30524e19i		−	0.600304i
$945$	4.77700e18			0.218197
$946$	−3.26372e18			−0.148054
$947$			5.11547e18i			0.230468i	0.993338	+	0.115234i	$0.0367618\pi$
$947$			5.11547e18i			0.230468i	−0.993338	+	0.115234i	$0.963238\pi$
$948$	3.47667e19			1.55564
$949$	1.00824e19	+	3.00136e18i	0.448056	+	0.133379i
$950$	−3.66175e18			−0.161616
$951$		−	4.07006e19i		−	1.78413i
$952$	−1.30906e19			−0.569926
$953$	−4.16248e19			−1.79990			−0.899950	−	0.435993i	$-0.856397\pi$
$953$	−4.16248e19			−1.79990			−0.899950	+	0.435993i	$0.856397\pi$
$954$			8.86193e17i			0.0380596i
$955$		−	1.01625e19i		−	0.433488i
$956$			9.46705e18i			0.401087i
$957$		−	2.52795e19i		−	1.06375i
$958$	−2.65191e17			−0.0110836
$959$	−5.12997e19			−2.12958
$960$			7.58280e18i			0.312656i
$961$	−8.41842e18			−0.344769
$962$	2.59236e17	−	8.70843e17i	0.0105453	−	0.0354243i
$963$	−1.19686e19			−0.483585
$964$		−	2.51348e19i		−	1.00873i
$965$	2.90128e18			0.115654
$966$	5.29404e18			0.209621
$967$		−	1.03305e19i		−	0.406301i	−0.979147	−	0.203151i	$-0.934882\pi$
$967$		−	1.03305e19i		−	0.406301i	0.979147	−	0.203151i	$-0.0651181\pi$
$968$			7.76882e18i			0.303505i
$969$		−	7.71502e19i		−	2.99387i
$970$		−	4.79104e17i		−	0.0184677i
$971$	1.84127e19			0.705007			0.352503	−	0.935811i	$-0.385330\pi$
$971$	1.84127e19			0.705007			0.352503	+	0.935811i	$0.385330\pi$
$972$	2.71323e19			1.03195
$973$		−	4.86338e19i		−	1.83741i
$974$	−2.56851e18			−0.0963935
$975$	−8.94227e18	+	3.00395e19i	−0.333363	+	1.11986i
$976$	5.95405e18			0.220490
$977$		−	2.54146e18i		−	0.0934908i	−0.998907	−	0.0467454i	$-0.985115\pi$
$977$		−	2.54146e18i		−	0.0934908i	0.998907	−	0.0467454i	$-0.0148849\pi$
$978$	−6.66007e15			−0.000243375
$979$	4.84256e19			1.75787
$980$			1.05240e19i			0.379501i
$981$			2.49658e18i			0.0894328i
$982$		−	1.98948e18i		−	0.0707971i
$983$			2.09604e19i			0.740970i	0.928838	+	0.370485i	$0.120809\pi$
$983$			2.09604e19i			0.740970i	−0.928838	+	0.370485i	$0.879191\pi$
$984$	3.63000e18			0.127479
$985$	2.66338e18			0.0929175
$986$		−	2.94911e18i		−	0.102210i
$987$	−4.40896e19			−1.51801
$988$	−4.14474e19	−	1.23382e19i	−1.41768	−	0.422020i
$989$	−2.26365e19			−0.769188
$990$			8.30702e17i			0.0280425i
$991$	−4.12487e18			−0.138335			−0.0691674	−	0.997605i	$-0.522034\pi$
$991$	−4.12487e18			−0.138335			−0.0691674	+	0.997605i	$0.522034\pi$
$992$	−1.19199e19			−0.397142
$993$			4.96866e19i			1.64463i
$994$			7.40525e18i			0.243516i
$995$		−	7.91201e18i		−	0.258486i
$996$			5.10284e19i			1.65625i
$997$	4.59640e19			1.48217			0.741086	−	0.671410i	$-0.234311\pi$
$997$	4.59640e19			1.48217			0.741086	+	0.671410i	$0.234311\pi$
$998$	3.86292e18			0.123756
$999$			5.18780e18i			0.165123i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.14.b.a.12.7	✓	14
3.2	odd	2		117.14.b.c.64.8		14
13.5	odd	4		169.14.a.d.1.7		14
13.8	odd	4		169.14.a.d.1.8		14
13.12	even	2	inner	13.14.b.a.12.8	yes	14
39.38	odd	2		117.14.b.c.64.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.14.b.a.12.7	✓	14	1.1	even	1	trivial
13.14.b.a.12.8	yes	14	13.12	even	2	inner
117.14.b.c.64.7		14	39.38	odd	2
117.14.b.c.64.8		14	3.2	odd	2
169.14.a.d.1.7		14	13.5	odd	4
169.14.a.d.1.8		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)

\(f(q)\)	\(=\)	\(q-10.5212i q^{2} -1591.09 q^{3} +8081.30 q^{4} -9424.40i q^{5} +16740.2i q^{6} +484840. i q^{7} -171215. i q^{8} +937259. q^{9} -99156.0 q^{10} -8.93854e6i q^{11} -1.28581e7 q^{12} +(4.96535e6 - 1.66799e7i) q^{13} +5.10109e6 q^{14} +1.49951e7i q^{15} +6.44007e7 q^{16} -1.57696e8 q^{17} -9.86109e6i q^{18} -3.07483e8i q^{19} -7.61615e7i q^{20} -7.71426e8i q^{21} -9.40441e7 q^{22} -6.52271e8 q^{23} +2.72419e8i q^{24} +1.13188e9 q^{25} +(-1.75493e8 - 5.22414e7i) q^{26} +1.04545e9 q^{27} +3.91814e9i q^{28} -1.77749e9 q^{29} +1.57767e8 q^{30} -5.73027e9i q^{31} -2.08016e9i q^{32} +1.42221e10i q^{33} +1.65915e9i q^{34} +4.56932e9 q^{35} +7.57428e9 q^{36} +4.96227e9i q^{37} -3.23509e9 q^{38} +(-7.90034e9 + 2.65394e10i) q^{39} -1.61360e9 q^{40} -1.33251e10i q^{41} -8.11632e9 q^{42} +3.47041e10 q^{43} -7.22350e10i q^{44} -8.83311e9i q^{45} +6.86267e9i q^{46} -5.71534e10i q^{47} -1.02468e11 q^{48} -1.38180e11 q^{49} -1.19088e10i q^{50} +2.50909e11 q^{51} +(4.01265e10 - 1.34796e11i) q^{52} -8.98677e10 q^{53} -1.09994e10i q^{54} -8.42404e10 q^{55} +8.30116e10 q^{56} +4.89235e11i q^{57} +1.87013e10i q^{58} -2.02674e11i q^{59} +1.21180e11i q^{60} +9.24532e10 q^{61} -6.02893e10 q^{62} +4.54420e11i q^{63} +5.05684e11 q^{64} +(-1.57198e11 - 4.67955e10i) q^{65} +1.49633e11 q^{66} -2.80350e11i q^{67} -1.27439e12 q^{68} +1.03782e12 q^{69} -4.80748e10i q^{70} +1.45170e12i q^{71} -1.60473e11i q^{72} +6.04461e11i q^{73} +5.22090e10 q^{74} -1.80093e12 q^{75} -2.48487e12i q^{76} +4.33376e12 q^{77} +(2.79226e11 + 8.31211e10i) q^{78} -2.70387e12 q^{79} -6.06938e11i q^{80} -3.15770e12 q^{81} -1.40196e11 q^{82} -3.96858e12i q^{83} -6.23413e12i q^{84} +1.48619e12i q^{85} -3.65129e11i q^{86} +2.82815e12 q^{87} -1.53041e12 q^{88} +5.41762e12i q^{89} -9.29349e10 q^{90} +(8.08710e12 + 2.40740e12i) q^{91} -5.27120e12 q^{92} +9.11740e12i q^{93} -6.01322e11 q^{94} -2.89785e12 q^{95} +3.30974e12i q^{96} +4.83182e12i q^{97} +1.45382e12i q^{98} -8.37773e12i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 1456 q^{3} - 49154 q^{4} + 5149262 q^{9} - 3968358 q^{10} - 21959110 q^{12} - 14362478 q^{13} + 72843942 q^{14} + 243447170 q^{16} - 178512492 q^{17} + 1330288056 q^{22} - 1641693744 q^{23} + 1601386654 q^{25}+ \cdots + 11940434574336 q^{95}+O(q^{100}) \) 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

Related objects

Downloads

Learn more