LMFDB - Embedded newform 37.6.b.a.36.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,6,Mod(36,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("37.36");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 37 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	37.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.93420133308$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + \cdots + 178006118400 $ x^16 + 390x^14 + 60701x^12 + 4799932x^10 + 203487156x^8 + 4519465040x^6 + 48993644736x^4 + 211923220224*x^2 + 178006118400
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			36.1
Root			$-10.0606i$ of defining polynomial
Character	$\chi$	$=$	37.36
Dual form			37.6.b.a.36.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.0606i q^{2} -25.7580 q^{3} -69.2166 q^{4} -77.4051i q^{5} +259.142i q^{6} +168.370 q^{7} +374.424i q^{8} +420.475 q^{9} +O(q^{10})$	\(q-10.0606i q^{2} -25.7580 q^{3} -69.2166 q^{4} -77.4051i q^{5} +259.142i q^{6} +168.370 q^{7} +374.424i q^{8} +420.475 q^{9} -778.746 q^{10} -603.572 q^{11} +1782.88 q^{12} +268.415i q^{13} -1693.91i q^{14} +1993.80i q^{15} +1552.01 q^{16} -873.888i q^{17} -4230.25i q^{18} +706.696i q^{19} +5357.72i q^{20} -4336.87 q^{21} +6072.33i q^{22} +141.099i q^{23} -9644.40i q^{24} -2866.55 q^{25} +2700.43 q^{26} -4571.40 q^{27} -11654.0 q^{28} -6352.26i q^{29} +20058.9 q^{30} +5756.36i q^{31} -3632.68i q^{32} +15546.8 q^{33} -8791.88 q^{34} -13032.7i q^{35} -29103.9 q^{36} +(-3768.82 + 7425.63i) q^{37} +7109.82 q^{38} -6913.84i q^{39} +28982.3 q^{40} -4153.25 q^{41} +43631.8i q^{42} -8999.12i q^{43} +41777.2 q^{44} -32546.9i q^{45} +1419.55 q^{46} -22240.1 q^{47} -39976.7 q^{48} +11541.4 q^{49} +28839.4i q^{50} +22509.6i q^{51} -18578.8i q^{52} -13348.3 q^{53} +45991.3i q^{54} +46719.6i q^{55} +63041.6i q^{56} -18203.1i q^{57} -63907.9 q^{58} +7567.74i q^{59} -138004. i q^{60} -53741.6i q^{61} +57912.8 q^{62} +70795.3 q^{63} +13117.2 q^{64} +20776.7 q^{65} -156411. i q^{66} -1077.89 q^{67} +60487.6i q^{68} -3634.43i q^{69} -131117. q^{70} +41521.0 q^{71} +157436. i q^{72} -50604.9 q^{73} +(74706.6 + 37916.8i) q^{74} +73836.7 q^{75} -48915.1i q^{76} -101623. q^{77} -69557.7 q^{78} -9485.04i q^{79} -120134. i q^{80} +15574.8 q^{81} +41784.4i q^{82} +1476.60 q^{83} +300184. q^{84} -67643.4 q^{85} -90537.0 q^{86} +163622. i q^{87} -225992. i q^{88} +11185.2i q^{89} -327443. q^{90} +45193.0i q^{91} -9766.41i q^{92} -148272. i q^{93} +223750. i q^{94} +54701.9 q^{95} +93570.7i q^{96} -17542.0i q^{97} -116114. i q^{98} -253787. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9}+O(q^{10}) $ 16 * q - 18 * q^3 - 268 * q^4 + 190 * q^7 + 1394 * q^9	$ 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9} - 74 q^{10} - 1110 q^{11} + 1402 q^{12} + 2900 q^{16} - 7010 q^{21} - 12052 q^{25} + 4902 q^{26} + 4266 q^{27} - 16824 q^{28} + 19280 q^{30} - 2478 q^{33} + 20556 q^{34} - 51402 q^{36} - 11400 q^{37} + 12108 q^{38} + 16966 q^{40} + 3918 q^{41} + 125394 q^{44} + 17470 q^{46} + 3822 q^{47} - 78034 q^{48} - 32618 q^{49} - 24126 q^{53} - 164718 q^{58} - 81426 q^{62} + 219268 q^{63} + 158076 q^{64} + 98976 q^{65} + 23560 q^{67} - 222404 q^{70} - 50046 q^{71} - 196274 q^{73} + 141216 q^{74} + 214054 q^{75} - 239574 q^{77} - 90822 q^{78} + 317312 q^{81} - 215814 q^{83} + 438572 q^{84} - 346472 q^{85} + 197640 q^{86} - 857612 q^{90} - 132504 q^{95} - 574860 q^{99}+O(q^{100}) $ 16 * q - 18 * q^3 - 268 * q^4 + 190 * q^7 + 1394 * q^9 - 74 * q^10 - 1110 * q^11 + 1402 * q^12 + 2900 * q^16 - 7010 * q^21 - 12052 * q^25 + 4902 * q^26 + 4266 * q^27 - 16824 * q^28 + 19280 * q^30 - 2478 * q^33 + 20556 * q^34 - 51402 * q^36 - 11400 * q^37 + 12108 * q^38 + 16966 * q^40 + 3918 * q^41 + 125394 * q^44 + 17470 * q^46 + 3822 * q^47 - 78034 * q^48 - 32618 * q^49 - 24126 * q^53 - 164718 * q^58 - 81426 * q^62 + 219268 * q^63 + 158076 * q^64 + 98976 * q^65 + 23560 * q^67 - 222404 * q^70 - 50046 * q^71 - 196274 * q^73 + 141216 * q^74 + 214054 * q^75 - 239574 * q^77 - 90822 * q^78 + 317312 * q^81 - 215814 * q^83 + 438572 * q^84 - 346472 * q^85 + 197640 * q^86 - 857612 * q^90 - 132504 * q^95 - 574860 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$.

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

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 $

$2$		−	10.0606i		−	1.77849i	−0.457433	−	0.889244i	$-0.651231\pi$
$2$		−	10.0606i		−	1.77849i	0.457433	−	0.889244i	$-0.348769\pi$
$3$	−25.7580			−1.65238			−0.826188	−	0.563394i	$-0.809495\pi$
$3$	−25.7580			−1.65238			−0.826188	+	0.563394i	$0.809495\pi$
$4$	−69.2166			−2.16302
$5$		−	77.4051i		−	1.38466i	−0.721579	−	0.692332i	$-0.756583\pi$
$5$		−	77.4051i		−	1.38466i	0.721579	−	0.692332i	$-0.243417\pi$
$6$			259.142i			2.93873i
$7$	168.370			1.29873			0.649366	−	0.760476i	$-0.275034\pi$
$7$	168.370			1.29873			0.649366	+	0.760476i	$0.275034\pi$
$8$			374.424i			2.06842i
$9$	420.475			1.73035
$10$	−778.746			−2.46261
$11$	−603.572			−1.50400			−0.751999	−	0.659164i	$-0.770910\pi$
$11$	−603.572			−1.50400			−0.751999	+	0.659164i	$0.770910\pi$
$12$	1782.88			3.57412
$13$			268.415i			0.440503i	0.975443	+	0.220251i	$0.0706878\pi$
$13$			268.415i			0.440503i	−0.975443	+	0.220251i	$0.929312\pi$
$14$		−	1693.91i		−	2.30978i
$15$			1993.80i			2.28799i
$16$	1552.01			1.51564
$17$		−	873.888i		−	0.733388i	−0.930342	−	0.366694i	$-0.880490\pi$
$17$		−	873.888i		−	0.733388i	0.930342	−	0.366694i	$-0.119510\pi$
$18$		−	4230.25i		−	3.07741i
$19$			706.696i			0.449106i	0.974462	+	0.224553i	$0.0720921\pi$
$19$			706.696i			0.449106i	−0.974462	+	0.224553i	$0.927908\pi$
$20$			5357.72i			2.99506i
$21$	−4336.87			−2.14599
$22$			6072.33i			2.67484i
$23$			141.099i			0.0556167i	0.999613	+	0.0278083i	$0.00885281\pi$
$23$			141.099i			0.0556167i	−0.999613	+	0.0278083i	$0.991147\pi$
$24$		−	9644.40i		−	3.41781i
$25$	−2866.55			−0.917296
$26$	2700.43			0.783429
$27$	−4571.40			−1.20681
$28$	−11654.0			−2.80918
$29$		−	6352.26i		−	1.40260i	−0.712867	−	0.701300i	$-0.752604\pi$
$29$		−	6352.26i		−	1.40260i	0.712867	−	0.701300i	$-0.247396\pi$
$30$	20058.9			4.06916
$31$			5756.36i			1.07583i	0.842999	+	0.537915i	$0.180788\pi$
$31$			5756.36i			1.07583i	−0.842999	+	0.537915i	$0.819212\pi$
$32$		−	3632.68i		−	0.627123i
$33$	15546.8			2.48517
$34$	−8791.88			−1.30432
$35$		−	13032.7i		−	1.79831i
$36$	−29103.9			−3.74278
$37$	−3768.82	+	7425.63i	−0.452586	+	0.891721i
$37$	−3768.82	+	7425.63i	−0.452586	+	0.891721i
$38$	7109.82			0.798729
$39$		−	6913.84i		−	0.727876i
$40$	28982.3			2.86406
$41$	−4153.25			−0.385859			−0.192929	−	0.981213i	$-0.561799\pi$
$41$	−4153.25			−0.385859			−0.192929	+	0.981213i	$0.561799\pi$
$42$			43631.8i			3.81662i
$43$		−	8999.12i		−	0.742214i	−0.928590	−	0.371107i	$-0.878978\pi$
$43$		−	8999.12i		−	0.742214i	0.928590	−	0.371107i	$-0.121022\pi$
$44$	41777.2			3.25318
$45$		−	32546.9i		−	2.39595i
$46$	1419.55			0.0989136
$47$	−22240.1			−1.46856			−0.734280	−	0.678846i	$-0.762480\pi$
$47$	−22240.1			−1.46856			−0.734280	+	0.678846i	$0.762480\pi$
$48$	−39976.7			−2.50440
$49$	11541.4			0.686703
$50$			28839.4i			1.63140i
$51$			22509.6i			1.21183i
$52$		−	18578.8i		−	0.952816i
$53$	−13348.3			−0.652736			−0.326368	−	0.945243i	$-0.605825\pi$
$53$	−13348.3			−0.652736			−0.326368	+	0.945243i	$0.605825\pi$
$54$			45991.3i			2.14630i
$55$			46719.6i			2.08253i
$56$			63041.6i			2.68632i
$57$		−	18203.1i		−	0.742092i
$58$	−63907.9			−2.49451
$59$			7567.74i			0.283032i	0.989936	+	0.141516i	$0.0451977\pi$
$59$			7567.74i			0.283032i	−0.989936	+	0.141516i	$0.954802\pi$
$60$		−	138004.i		−	4.94896i
$61$		−	53741.6i		−	1.84921i	−0.380929	−	0.924604i	$-0.624396\pi$
$61$		−	53741.6i		−	1.84921i	0.380929	−	0.924604i	$-0.375604\pi$
$62$	57912.8			1.91335
$63$	70795.3			2.24726
$64$	13117.2			0.400305
$65$	20776.7			0.609949
$66$		−	156411.i		−	4.41985i
$67$	−1077.89			−0.0293351			−0.0146676	−	0.999892i	$-0.504669\pi$
$67$	−1077.89			−0.0293351			−0.0146676	+	0.999892i	$0.504669\pi$
$68$			60487.6i			1.58633i
$69$		−	3634.43i		−	0.0918997i
$70$	−131117.			−3.19827
$71$	41521.0			0.977512			0.488756	−	0.872421i	$-0.337451\pi$
$71$	41521.0			0.977512			0.488756	+	0.872421i	$0.337451\pi$
$72$			157436.i			3.57909i
$73$	−50604.9			−1.11144			−0.555719	−	0.831370i	$-0.687557\pi$
$73$	−50604.9			−1.11144			−0.555719	+	0.831370i	$0.687557\pi$
$74$	74706.6	+	37916.8i	1.58591	+	0.804919i
$75$	73836.7			1.51572
$76$		−	48915.1i		−	0.971425i
$77$	−101623.			−1.95329
$78$	−69557.7			−1.29452
$79$		−	9485.04i		−	0.170990i	−0.996339	−	0.0854952i	$-0.972753\pi$
$79$		−	9485.04i		−	0.170990i	0.996339	−	0.0854952i	$-0.0272472\pi$
$80$		−	120134.i		−	2.09865i
$81$	15574.8			0.263760
$82$			41784.4i			0.686246i
$83$	1476.60			0.0235271			0.0117636	−	0.999931i	$-0.496255\pi$
$83$	1476.60			0.0235271			0.0117636	+	0.999931i	$0.496255\pi$
$84$	300184.			4.64183
$85$	−67643.4			−1.01550
$86$	−90537.0			−1.32002
$87$			163622.i			2.31762i
$88$		−	225992.i		−	3.11090i
$89$			11185.2i			0.149682i	0.997195	+	0.0748409i	$0.0238449\pi$
$89$			11185.2i			0.149682i	−0.997195	+	0.0748409i	$0.976155\pi$
$90$	−327443.			−4.26118
$91$			45193.0i			0.572095i
$92$		−	9766.41i		−	0.120300i
$93$		−	148272.i		−	1.77768i
$94$			223750.i			2.61182i
$95$	54701.9			0.621861
$96$			93570.7i			1.03624i
$97$		−	17542.0i		−	0.189300i	−0.995511	−	0.0946500i	$-0.969827\pi$
$97$		−	17542.0i		−	0.189300i	0.995511	−	0.0946500i	$-0.0301732\pi$
$98$		−	116114.i		−	1.22129i
$99$	−253787.			−2.60244
$100$	198413.			1.98413
$101$	−21138.2			−0.206188			−0.103094	−	0.994672i	$-0.532874\pi$
$101$	−21138.2			−0.206188			−0.103094	+	0.994672i	$0.532874\pi$
$102$	226461.			2.15523
$103$			21654.8i			0.201123i	0.994931	+	0.100561i	$0.0320639\pi$
$103$			21654.8i			0.201123i	−0.994931	+	0.100561i	$0.967936\pi$
$104$	−100501.			−0.911143
$105$			335696.i			2.97148i
$106$			134293.i			1.16088i
$107$	−75852.3			−0.640485			−0.320243	−	0.947336i	$-0.603764\pi$
$107$	−75852.3			−0.640485			−0.320243	+	0.947336i	$0.603764\pi$
$108$	316417.			2.61036
$109$		−	224063.i		−	1.80636i	−0.429265	−	0.903179i	$-0.641227\pi$
$109$		−	224063.i		−	1.80636i	0.429265	−	0.903179i	$-0.358773\pi$
$110$	470029.			3.70376
$111$	97077.3	−	191269.i	0.747843	−	1.47346i
$112$	261312.			1.96840
$113$			84675.4i			0.623823i	0.950111	+	0.311911i	$0.100969\pi$
$113$			84675.4i			0.623823i	−0.950111	+	0.311911i	$0.899031\pi$
$114$	−183135.			−1.31980
$115$	10921.8			0.0770104
$116$			439682.i			3.03385i
$117$			112862.i			0.762224i
$118$	76136.3			0.503369
$119$		−	147136.i		−	0.952473i
$120$	−746526.			−4.73251
$121$	203248.			1.26201
$122$	−540675.			−3.28880
$123$	106979.			0.637584
$124$		−	398436.i		−	2.32704i
$125$		−	20005.2i		−	0.114517i
$126$		−	712247.i		−	3.99672i
$127$	256661.			1.41205			0.706026	−	0.708185i	$-0.250486\pi$
$127$	256661.			1.41205			0.706026	+	0.708185i	$0.250486\pi$
$128$		−	248213.i		−	1.33906i
$129$			231799.i			1.22642i
$130$		−	209027.i		−	1.08479i
$131$			79025.9i			0.402338i	0.979557	+	0.201169i	$0.0644741\pi$
$131$			79025.9i			0.402338i	−0.979557	+	0.201169i	$0.935526\pi$
$132$	−1.07610e6			−5.37548
$133$			118986.i			0.583268i
$134$			10844.3i			0.0521721i
$135$			353850.i			1.67103i
$136$	327204.			1.51695
$137$	55006.4			0.250387			0.125194	−	0.992132i	$-0.460045\pi$
$137$	55006.4			0.250387			0.125194	+	0.992132i	$0.460045\pi$
$138$	−36564.8			−0.163443
$139$	−305471.			−1.34101			−0.670507	−	0.741903i	$-0.733923\pi$
$139$	−305471.			−1.34101			−0.670507	+	0.741903i	$0.733923\pi$
$140$			902079.i			3.88978i
$141$	572860.			2.42662
$142$		−	417728.i		−	1.73849i
$143$		−	162008.i		−	0.662515i
$144$	652582.			2.62258
$145$	−491698.			−1.94213
$146$			509118.i			1.97668i
$147$	−297284.			−1.13469
$148$	260865.	−	513977.i	0.978953	−	1.92881i
$149$	304779.			1.12465			0.562327	−	0.826915i	$-0.309906\pi$
$149$	304779.			1.12465			0.562327	+	0.826915i	$0.309906\pi$
$150$		−	742845.i		−	2.69569i
$151$	16190.3			0.0577845			0.0288923	−	0.999583i	$-0.490802\pi$
$151$	16190.3			0.0577845			0.0288923	+	0.999583i	$0.490802\pi$
$152$	−264604.			−0.928938
$153$		−	367448.i		−	1.26902i
$154$			1.02240e6i			3.47390i
$155$	445572.			1.48967
$156$			478553.i			1.57441i
$157$	−312830.			−1.01288			−0.506441	−	0.862275i	$-0.669039\pi$
$157$	−312830.			−1.01288			−0.506441	+	0.862275i	$0.669039\pi$
$158$	−95425.7			−0.304104
$159$	343827.			1.07857
$160$	−281188.			−0.868355
$161$			23756.9i			0.0722311i
$162$		−	156692.i		−	0.469094i
$163$			436231.i			1.28602i	0.765858	+	0.643010i	$0.222315\pi$
$163$			436231.i			1.28602i	−0.765858	+	0.643010i	$0.777685\pi$
$164$	287474.			0.834621
$165$		−	1.20340e6i		−	3.44113i
$166$		−	14855.6i		−	0.0418427i
$167$		−	587366.i		−	1.62974i	−0.579646	−	0.814868i	$-0.696809\pi$
$167$		−	587366.i		−	1.62974i	0.579646	−	0.814868i	$-0.303191\pi$
$168$		−	1.62383e6i		−	4.43881i
$169$	299246.			0.805957
$170$			680537.i			1.80605i
$171$			297148.i			0.777110i
$172$			622889.i			1.60542i
$173$	−693665.			−1.76212			−0.881059	−	0.473007i	$-0.843169\pi$
$173$	−693665.			−1.76212			−0.881059	+	0.473007i	$0.843169\pi$
$174$	1.64614e6			4.12186
$175$	−482641.			−1.19132
$176$	−936750.			−2.27951
$177$		−	194930.i		−	0.467676i
$178$	112530.			0.266207
$179$		−	359738.i		−	0.839177i	−0.907714	−	0.419588i	$-0.862174\pi$
$179$		−	359738.i		−	0.839177i	0.907714	−	0.419588i	$-0.137826\pi$
$180$			2.25279e6i			5.18250i
$181$	259702.			0.589222			0.294611	−	0.955617i	$-0.404810\pi$
$181$	259702.			0.589222			0.294611	+	0.955617i	$0.404810\pi$
$182$	454671.			1.01746
$183$			1.38428e6i			3.05559i
$184$	−52830.9			−0.115038
$185$	574782.	+	291726.i	1.23473	+	0.626680i
$186$	−1.49172e6			−3.16158
$187$			527454.i			1.10301i
$188$	1.53938e6			3.17653
$189$	−769686.			−1.56733
$190$		−	550336.i		−	1.10597i
$191$		−	42814.5i		−	0.0849195i	−0.999098	−	0.0424598i	$-0.986481\pi$
$191$		−	42814.5i		−	0.0849195i	0.999098	−	0.0424598i	$-0.0135194\pi$
$192$	−337873.			−0.661455
$193$		−	923411.i		−	1.78444i	−0.451601	−	0.892220i	$-0.649147\pi$
$193$		−	923411.i		−	1.78444i	0.451601	−	0.892220i	$-0.350853\pi$
$194$	−176484.			−0.336668
$195$	−535166.			−1.00786
$196$	−798858.			−1.48535
$197$	−152054.			−0.279146			−0.139573	−	0.990212i	$-0.544573\pi$
$197$	−152054.			−0.279146			−0.139573	+	0.990212i	$0.544573\pi$
$198$			2.55326e6i			4.62841i
$199$		−	769973.i		−	1.37830i	−0.724620	−	0.689149i	$-0.757985\pi$
$199$		−	769973.i		−	1.37830i	0.724620	−	0.689149i	$-0.242015\pi$
$200$		−	1.07330e6i		−	1.89735i
$201$	27764.3			0.0484726
$202$			212664.i			0.366704i
$203$		−	1.06953e6i		−	1.82160i
$204$		−	1.55804e6i		−	2.62122i
$205$			321483.i			0.534285i
$206$	217861.			0.357694
$207$			59328.7i			0.0962363i
$208$			416583.i			0.667642i
$209$		−	426542.i		−	0.675454i
$210$	3.37732e6			5.28475
$211$	−330604.			−0.511213			−0.255607	−	0.966781i	$-0.582275\pi$
$211$	−330604.			−0.511213			−0.255607	+	0.966781i	$0.582275\pi$
$212$	923928.			1.41188
$213$	−1.06950e6			−1.61522
$214$			763123.i			1.13910i
$215$	−696578.			−1.02772
$216$		−	1.71164e6i		−	2.49619i
$217$			969198.i			1.39722i
$218$	−2.25422e6			−3.21259
$219$	1.30348e6			1.83651
$220$		−	3.23377e6i		−	4.50456i
$221$	234565.			0.323059
$222$	−1.92429e6	−	976661.i	−2.62053	−	1.33003i
$223$	−523244.			−0.704600			−0.352300	−	0.935887i	$-0.614600\pi$
$223$	−523244.			−0.704600			−0.352300	+	0.935887i	$0.614600\pi$
$224$		−	611635.i		−	0.814464i
$225$	−1.20531e6			−1.58724
$226$	851889.			1.10946
$227$			1.12179e6i			1.44492i	0.691410	+	0.722462i	$0.256990\pi$
$227$			1.12179e6i			1.44492i	−0.691410	+	0.722462i	$0.743010\pi$
$228$			1.25996e6i			1.60516i
$229$	−1.14774e6			−1.44629			−0.723146	−	0.690696i	$-0.757304\pi$
$229$	−1.14774e6			−1.44629			−0.723146	+	0.690696i	$0.757304\pi$
$230$		−	109880.i		−	0.136962i
$231$	2.61761e6			3.22757
$232$	2.37844e6			2.90116
$233$	989725.			1.19433			0.597165	−	0.802118i	$-0.296294\pi$
$233$	989725.			1.19433			0.597165	+	0.802118i	$0.296294\pi$
$234$	1.13546e6			1.35561
$235$			1.72150e6i			2.03346i
$236$		−	523813.i		−	0.612204i
$237$			244316.i			0.282540i
$238$	−1.48029e6			−1.69396
$239$		−	1.13545e6i		−	1.28580i	−0.765950	−	0.642900i	$-0.777731\pi$
$239$		−	1.13545e6i		−	1.28580i	0.765950	−	0.642900i	$-0.222269\pi$
$240$			3.09440e6i			3.46776i
$241$		−	157029.i		−	0.174156i	−0.996202	−	0.0870778i	$-0.972247\pi$
$241$		−	157029.i		−	0.174156i	0.996202	−	0.0870778i	$-0.0277529\pi$
$242$		−	2.04481e6i		−	2.24447i
$243$	709676.			0.770982
$244$			3.71981e6i			3.99987i
$245$		−	893365.i		−	0.950853i
$246$		−	1.07628e6i		−	1.13394i
$247$	−189688.			−0.197832
$248$	−2.15532e6			−2.22527
$249$	−38034.4			−0.0388757
$250$	−201266.			−0.203667
$251$		−	98057.8i		−	0.0982421i	−0.998793	−	0.0491211i	$-0.984358\pi$
$251$		−	98057.8i		−	0.0982421i	0.998793	−	0.0491211i	$-0.0156420\pi$
$252$	−4.90021e6			−4.86087
$253$		−	85163.5i		−	0.0836474i
$254$		−	2.58218e6i		−	2.51132i
$255$	1.74236e6			1.67798
$256$	−2.07744e6			−1.98120
$257$			1.81434e6i			1.71350i	0.515730	+	0.856751i	$0.327521\pi$
$257$			1.81434e6i			1.71350i	−0.515730	+	0.856751i	$0.672479\pi$
$258$	2.33205e6			2.18117
$259$	−634556.	+	1.25025e6i	−0.587788	+	1.15811i
$260$	−1.43809e6			−1.31933
$261$		−	2.67097e6i		−	2.42699i
$262$	795052.			0.715554
$263$	−797108.			−0.710604			−0.355302	−	0.934752i	$-0.615622\pi$
$263$	−797108.			−0.710604			−0.355302	+	0.934752i	$0.615622\pi$
$264$			5.82109e6i			5.14037i
$265$			1.03323e6i			0.903821i
$266$	1.19708e6			1.03733
$267$		−	288109.i		−	0.247331i
$268$	74608.0			0.0634524
$269$	1.28791e6			1.08519			0.542593	−	0.839996i	$-0.317443\pi$
$269$	1.28791e6			1.08519			0.542593	+	0.839996i	$0.317443\pi$
$270$	3.55996e6			2.97191
$271$	−1.27896e6			−1.05787			−0.528936	−	0.848662i	$-0.677409\pi$
$271$	−1.27896e6			−1.05787			−0.528936	+	0.848662i	$0.677409\pi$
$272$		−	1.35628e6i		−	1.11155i
$273$		−	1.16408e6i		−	0.945316i
$274$		−	553400.i		−	0.445310i
$275$	1.73017e6			1.37961
$276$			251563.i			0.198781i
$277$		−	1.40148e6i		−	1.09746i	−0.836000	−	0.548729i	$-0.815112\pi$
$277$		−	1.40148e6i		−	1.09746i	0.836000	−	0.548729i	$-0.184888\pi$
$278$			3.07324e6i			2.38498i
$279$			2.42041e6i			1.86156i
$280$	4.87975e6			3.71965
$281$		−	494703.i		−	0.373748i	−0.982384	−	0.186874i	$-0.940164\pi$
$281$		−	494703.i		−	0.373748i	0.982384	−	0.186874i	$-0.0598356\pi$
$282$		−	5.76335e6i		−	4.31571i
$283$		−	832541.i		−	0.617930i	−0.951073	−	0.308965i	$-0.900017\pi$
$283$		−	832541.i		−	0.617930i	0.951073	−	0.308965i	$-0.0999827\pi$
$284$	−2.87395e6			−2.11438
$285$	−1.40901e6			−1.02755
$286$	−1.62990e6			−1.17828
$287$	−699282.			−0.501127
$288$		−	1.52745e6i		−	1.08514i
$289$	656177.			0.462143
$290$			4.94680e6i			3.45405i
$291$			451848.i			0.312795i
$292$	3.50270e6			2.40406
$293$	951351.			0.647398			0.323699	−	0.946160i	$-0.395073\pi$
$293$	951351.			0.647398			0.323699	+	0.946160i	$0.395073\pi$
$294$			2.99087e6i			2.01804i
$295$	585781.			0.391905
$296$	−2.78033e6	−	1.41114e6i	−1.84445	−	0.936137i
$297$	2.75917e6			1.81504
$298$		−	3.06627e6i		−	2.00018i
$299$	−37873.2			−0.0244993
$300$	−5.11073e6			−3.27853
$301$		−	1.51518e6i		−	0.963936i
$302$		−	162885.i		−	0.102769i
$303$	544478.			0.340701
$304$			1.09680e6i			0.680681i
$305$	−4.15987e6			−2.56053
$306$	−3.69677e6			−2.25693
$307$	−458968.			−0.277931			−0.138965	−	0.990297i	$-0.544378\pi$
$307$	−458968.			−0.277931			−0.138965	+	0.990297i	$0.544378\pi$
$308$	7.03403e6			4.22501
$309$		−	557785.i		−	0.332331i
$310$		−	4.48274e6i		−	2.64935i
$311$			1.66612e6i			0.976799i	0.872620	+	0.488400i	$0.162419\pi$
$311$			1.66612e6i			0.976799i	−0.872620	+	0.488400i	$0.837581\pi$
$312$	2.58870e6			1.50555
$313$			1.75761e6i			1.01405i	0.861931	+	0.507026i	$0.169255\pi$
$313$			1.75761e6i			1.01405i	−0.861931	+	0.507026i	$0.830745\pi$
$314$			3.14727e6i			1.80140i
$315$		−	5.47992e6i		−	3.11170i
$316$			656523.i			0.369855i
$317$	1.17532e6			0.656912			0.328456	−	0.944519i	$-0.393472\pi$
$317$	1.17532e6			0.656912			0.328456	+	0.944519i	$0.393472\pi$
$318$		−	3.45912e6i		−	1.91822i
$319$			3.83405e6i			2.10951i
$320$		−	1.01534e6i		−	0.554289i
$321$	1.95380e6			1.05832
$322$	239009.			0.128462
$323$	617573.			0.329368
$324$	−1.07803e6			−0.570518
$325$		−	769426.i		−	0.404072i
$326$	4.38877e6			2.28717
$327$			5.77141e6i			2.98478i
$328$		−	1.55508e6i		−	0.798117i
$329$	−3.74456e6			−1.90727
$330$	−1.21070e7			−6.12001
$331$			65616.6i			0.0329188i	0.999865	+	0.0164594i	$0.00523942\pi$
$331$			65616.6i			0.0329188i	−0.999865	+	0.0164594i	$0.994761\pi$
$332$	−102206.			−0.0508896
$333$	−1.58470e6	+	3.12229e6i	−0.783132	+	1.54299i
$334$	−5.90928e6			−2.89847
$335$			83434.2i			0.0406193i
$336$	−6.73087e6			−3.25254
$337$	2.60182e6			1.24797			0.623983	−	0.781438i	$-0.285514\pi$
$337$	2.60182e6			1.24797			0.623983	+	0.781438i	$0.285514\pi$
$338$		−	3.01061e6i		−	1.43339i
$339$		−	2.18107e6i		−	1.03079i
$340$	4.68205e6			2.19654
$341$		−	3.47438e6i		−	1.61805i
$342$	2.98950e6			1.38208
$343$	−886566.			−0.406889
$344$	3.36948e6			1.53521
$345$	−281324.			−0.127250
$346$			6.97872e6i			3.13390i
$347$		−	908062.i		−	0.404848i	−0.979298	−	0.202424i	$-0.935118\pi$
$347$		−	908062.i		−	0.404848i	0.979298	−	0.202424i	$-0.0648819\pi$
$348$		−	1.13253e7i		−	5.01306i
$349$	3.30403e6			1.45205			0.726023	−	0.687670i	$-0.241367\pi$
$349$	3.30403e6			1.45205			0.726023	+	0.687670i	$0.241367\pi$
$350$			4.85568e6i			2.11875i
$351$		−	1.22703e6i		−	0.531604i
$352$			2.19259e6i			0.943192i
$353$			3.27707e6i			1.39974i	0.714269	+	0.699872i	$0.246759\pi$
$353$			3.27707e6i			1.39974i	−0.714269	+	0.699872i	$0.753241\pi$
$354$	−1.96112e6			−0.831756
$355$		−	3.21394e6i		−	1.35353i
$356$		−	774203.i		−	0.323765i
$357$			3.78994e6i			1.57384i
$358$	−3.61920e6			−1.49247
$359$	−2.67944e6			−1.09726			−0.548628	−	0.836067i	$-0.684849\pi$
$359$	−2.67944e6			−1.09726			−0.548628	+	0.836067i	$0.684849\pi$
$360$	1.21863e7			4.95583
$361$	1.97668e6			0.798304
$362$		−	2.61277e6i		−	1.04793i
$363$	−5.23527e6			−2.08532
$364$		−	3.12811e6i		−	1.23745i
$365$			3.91708e6i			1.53897i
$366$	1.39267e7			5.43433
$367$	1.54222e6			0.597695			0.298848	−	0.954301i	$-0.403398\pi$
$367$	1.54222e6			0.597695			0.298848	+	0.954301i	$0.403398\pi$
$368$			218988.i			0.0842946i
$369$	−1.74634e6			−0.667671
$370$	2.93495e6	−	5.78267e6i	1.11454	−	2.19596i
$371$	−2.24746e6			−0.847729
$372$			1.02629e7i			3.84515i
$373$	2.18057e6			0.811516			0.405758	−	0.913981i	$-0.367008\pi$
$373$	2.18057e6			0.811516			0.405758	+	0.913981i	$0.367008\pi$
$374$	5.30653e6			1.96170
$375$			515295.i			0.189225i
$376$		−	8.32721e6i		−	3.03760i
$377$	1.70504e6			0.617849
$378$			7.74354e6i			2.78747i
$379$	−562144.			−0.201025			−0.100512	−	0.994936i	$-0.532048\pi$
$379$	−562144.			−0.201025			−0.100512	+	0.994936i	$0.532048\pi$
$380$	−3.78628e6			−1.34510
$381$	−6.61108e6			−2.33324
$382$	−430742.			−0.151028
$383$		−	3.80268e6i		−	1.32462i	−0.749228	−	0.662312i	$-0.769575\pi$
$383$		−	3.80268e6i		−	1.32462i	0.749228	−	0.662312i	$-0.230425\pi$
$384$			6.39348e6i			2.21263i
$385$			7.86617e6i			2.70465i
$386$	−9.29012e6			−3.17361
$387$		−	3.78390e6i		−	1.28429i
$388$			1.21420e6i			0.409460i
$389$		−	2.54132e6i		−	0.851503i	−0.904840	−	0.425751i	$-0.860010\pi$
$389$		−	2.54132e6i		−	0.851503i	0.904840	−	0.425751i	$-0.139990\pi$
$390$			5.38412e6i			1.79248i
$391$	123305.			0.0407886
$392$			4.32138e6i			1.42039i
$393$		−	2.03555e6i		−	0.664814i
$394$			1.52976e6i			0.496459i
$395$	−734191.			−0.236764
$396$	1.75663e7			5.62914
$397$	3.65631e6			1.16430			0.582152	−	0.813080i	$-0.302211\pi$
$397$	3.65631e6			1.16430			0.582152	+	0.813080i	$0.302211\pi$
$398$	−7.74643e6			−2.45129
$399$		−	3.06485e6i		−	0.963778i
$400$	−4.44892e6			−1.39029
$401$		−	1.28914e6i		−	0.400350i	−0.979760	−	0.200175i	$-0.935849\pi$
$401$		−	1.28914e6i		−	0.400350i	0.979760	−	0.200175i	$-0.0641511\pi$
$402$		−	279327.i		−	0.0862080i
$403$	−1.54510e6			−0.473906
$404$	1.46311e6			0.445990
$405$		−	1.20557e6i		−	0.365219i
$406$	−1.07602e7			−3.23969
$407$	2.27476e6	−	4.48190e6i	0.680689	−	1.34115i
$408$	−8.42813e6			−2.50658
$409$		−	966332.i		−	0.285640i	−0.989749	−	0.142820i	$-0.954383\pi$
$409$		−	966332.i		−	0.285640i	0.989749	−	0.142820i	$-0.0456169\pi$
$410$	3.23433e6			0.950220
$411$	−1.41686e6			−0.413734
$412$		−	1.49887e6i		−	0.435033i
$413$			1.27418e6i			0.367583i
$414$	596885.			0.171155
$415$		−	114297.i		−	0.0325772i
$416$	975067.			0.276249
$417$	7.86833e6			2.21586
$418$	−4.29129e6			−1.20129
$419$	−1.23157e6			−0.342709			−0.171354	−	0.985209i	$-0.554814\pi$
$419$	−1.23157e6			−0.342709			−0.171354	+	0.985209i	$0.554814\pi$
$420$		−	2.32358e7i		−	6.42737i
$421$			3.22979e6i			0.888115i	0.895998	+	0.444057i	$0.146461\pi$
$421$			3.22979e6i			0.888115i	−0.895998	+	0.444057i	$0.853539\pi$
$422$			3.32609e6i			0.909187i
$423$	−9.35140e6			−2.54112
$424$		−	4.99793e6i		−	1.35013i
$425$			2.50505e6i			0.672734i
$426$			1.07598e7i			2.87265i
$427$		−	9.04846e6i		−	2.40162i
$428$	5.25024e6			1.38538
$429$			4.17300e6i			1.09473i
$430$			7.00802e6i			1.82778i
$431$		−	1.57396e6i		−	0.408131i	−0.978957	−	0.204065i	$-0.934584\pi$
$431$		−	1.57396e6i		−	0.408131i	0.978957	−	0.204065i	$-0.0654155\pi$
$432$	−7.09486e6			−1.82909
$433$	−1.33511e6			−0.342213			−0.171106	−	0.985253i	$-0.554734\pi$
$433$	−1.33511e6			−0.342213			−0.171106	+	0.985253i	$0.554734\pi$
$434$	9.75076e6			2.48493
$435$	1.26652e7			3.20913
$436$			1.55089e7i			3.90719i
$437$	−99714.2			−0.0249778
$438$		−	1.31139e7i		−	3.26622i
$439$		−	6.63631e6i		−	1.64348i	−0.569861	−	0.821741i	$-0.693003\pi$
$439$		−	6.63631e6i		−	1.64348i	0.569861	−	0.821741i	$-0.306997\pi$
$440$	−1.74929e7			−4.30755
$441$	4.85288e6			1.18824
$442$		−	2.35987e6i		−	0.574557i
$443$	5.21572e6			1.26271			0.631357	−	0.775493i	$-0.282498\pi$
$443$	5.21572e6			1.26271			0.631357	+	0.775493i	$0.282498\pi$
$444$	−6.71937e6	+	1.32390e7i	−1.61760	+	3.18712i
$445$	865792.			0.207259
$446$			5.26418e6i			1.25312i
$447$	−7.85049e6			−1.85835
$448$	2.20854e6			0.519889
$449$		−	6.96431e6i		−	1.63028i	−0.579264	−	0.815140i	$-0.696660\pi$
$449$		−	6.96431e6i		−	1.63028i	0.579264	−	0.815140i	$-0.303340\pi$
$450$			1.21262e7i			2.82289i
$451$	2.50679e6			0.580331
$452$		−	5.86095e6i		−	1.34934i
$453$	−417029.			−0.0954819
$454$	1.12859e7			2.56978
$455$	3.49817e6			0.792159
$456$	6.81566e6			1.53496
$457$			3.29043e6i			0.736990i	0.929630	+	0.368495i	$0.120127\pi$
$457$			3.29043e6i			0.736990i	−0.929630	+	0.368495i	$0.879873\pi$
$458$			1.15470e7i			2.57221i
$459$			3.99489e6i			0.885061i
$460$	−755970.			−0.166575
$461$		−	1.03373e6i		−	0.226546i	−0.993564	−	0.113273i	$-0.963866\pi$
$461$		−	1.03373e6i		−	0.226546i	0.993564	−	0.113273i	$-0.0361335\pi$
$462$		−	2.63349e7i		−	5.74020i
$463$			421953.i			0.0914770i	0.998953	+	0.0457385i	$0.0145641\pi$
$463$			421953.i			0.0914770i	−0.998953	+	0.0457385i	$0.985436\pi$
$464$		−	9.85879e6i		−	2.12583i
$465$	−1.14770e7			−2.46149
$466$		−	9.95727e6i		−	2.12410i
$467$		−	3.68933e6i		−	0.782808i	−0.920219	−	0.391404i	$-0.871990\pi$
$467$		−	3.68933e6i		−	0.782808i	0.920219	−	0.391404i	$-0.128010\pi$
$468$		−	7.81192e6i		−	1.64871i
$469$	−181484.			−0.0380984
$470$	1.73194e7			3.61649
$471$	8.05787e6			1.67366
$472$	−2.83354e6			−0.585429
$473$			5.43162e6i			1.11629i
$474$	2.45797e6			0.502495
$475$		−	2.02578e6i		−	0.411963i
$476$			1.01843e7i			2.06022i
$477$	−5.61264e6			−1.12946
$478$	−1.14234e7			−2.28678
$479$		−	320565.i		−	0.0638378i	−0.999490	−	0.0319189i	$-0.989838\pi$
$479$		−	320565.i		−	0.0638378i	0.999490	−	0.0319189i	$-0.0101618\pi$
$480$	7.24285e6			1.43485
$481$	−1.99315e6	−	1.01161e6i	−0.392805	−	0.199365i
$482$	−1.57981e6			−0.309734
$483$		−	611929.i		−	0.119353i
$484$	−1.40682e7			−2.72976
$485$	−1.35784e6			−0.262117
$486$		−	7.13980e6i		−	1.37118i
$487$		−	2.00859e6i		−	0.383768i	−0.981418	−	0.191884i	$-0.938540\pi$
$487$		−	2.00859e6i		−	0.383768i	0.981418	−	0.191884i	$-0.0614597\pi$
$488$	2.01221e7			3.82493
$489$		−	1.12365e7i		−	2.12499i
$490$	−8.98783e6			−1.69108
$491$	−7.62309e6			−1.42701			−0.713505	−	0.700650i	$-0.752894\pi$
$491$	−7.62309e6			−1.42701			−0.713505	+	0.700650i	$0.752894\pi$
$492$	−7.40476e6			−1.37911
$493$	−5.55117e6			−1.02865
$494$			1.90838e6i			0.351842i
$495$			1.96444e7i			3.60351i
$496$			8.93394e6i			1.63057i
$497$	6.99089e6			1.26953
$498$			382650.i			0.0691399i
$499$			2.42605e6i			0.436163i	0.975931	+	0.218082i	$0.0699799\pi$
$499$			2.42605e6i			0.436163i	−0.975931	+	0.218082i	$0.930020\pi$
$500$			1.38470e6i			0.247702i
$501$			1.51294e7i			2.69294i
$502$	−986525.			−0.174722
$503$			5.77617e6i			1.01793i	0.860786	+	0.508967i	$0.169973\pi$
$503$			5.77617e6i			1.01793i	−0.860786	+	0.508967i	$0.830027\pi$
$504$			2.65074e7i			4.64827i
$505$			1.63620e6i			0.285502i
$506$	−856800.			−0.148766
$507$	−7.70799e6			−1.33175
$508$	−1.77652e7			−3.05430
$509$	−6.24081e6			−1.06769			−0.533847	−	0.845581i	$-0.679254\pi$
$509$	−6.24081e6			−1.06769			−0.533847	+	0.845581i	$0.679254\pi$
$510$		−	1.75293e7i		−	2.98427i
$511$	−8.52034e6			−1.44346
$512$			1.29575e7i			2.18448i
$513$		−	3.23059e6i		−	0.541986i
$514$	1.82534e7			3.04744
$515$	1.67619e6			0.278488
$516$		−	1.60444e7i		−	2.65276i
$517$	1.34235e7			2.20871
$518$	1.25783e7	+	6.38405e6i	2.05968	+	1.04537i
$519$	1.78674e7			2.91168
$520$			7.77929e6i			1.26163i
$521$	522409.			0.0843173			0.0421586	−	0.999111i	$-0.486577\pi$
$521$	522409.			0.0843173			0.0421586	+	0.999111i	$0.486577\pi$
$522$	−2.68717e7			−4.31637
$523$		−	205708.i		−	0.0328849i	−0.999865	−	0.0164425i	$-0.994766\pi$
$523$		−	205708.i		−	0.0328849i	0.999865	−	0.0164425i	$-0.00523404\pi$
$524$		−	5.46991e6i		−	0.870266i
$525$	1.24319e7			1.96851
$526$			8.01942e6i			1.26380i
$527$	5.03042e6			0.789001
$528$	2.41288e7			3.76662
$529$	6.41643e6			0.996907
$530$	1.03950e7			1.60743
$531$			3.18204e6i			0.489745i
$532$		−	8.23583e6i		−	1.26162i
$533$		−	1.11480e6i		−	0.169972i
$534$	−2.89856e6			−0.439875
$535$			5.87135e6i			0.886857i
$536$		−	403588.i		−	0.0606772i
$537$			9.26613e6i			1.38664i
$538$		−	1.29572e7i		−	1.92999i
$539$	−6.96608e6			−1.03280
$540$		−	2.44923e7i		−	3.61447i
$541$			1.28578e6i			0.188875i	0.995531	+	0.0944374i	$0.0301052\pi$
$541$			1.28578e6i			0.188875i	−0.995531	+	0.0944374i	$0.969895\pi$
$542$			1.28671e7i			1.88141i
$543$	−6.68941e6			−0.973617
$544$	−3.17456e6			−0.459924
$545$	−1.73436e7			−2.50120
$546$	−1.17114e7			−1.68123
$547$			986186.i			0.140926i	0.997514	+	0.0704629i	$0.0224476\pi$
$547$			986186.i			0.140926i	−0.997514	+	0.0704629i	$0.977552\pi$
$548$	−3.80736e6			−0.541592
$549$		−	2.25970e7i		−	3.19978i
$550$		−	1.74066e7i		−	2.45362i
$551$	4.48912e6			0.629915
$552$	1.36082e6			0.190087
$553$		−	1.59700e6i		−	0.222070i
$554$	−1.40998e7			−1.95182
$555$	−1.48052e7	−	7.51428e6i	−2.04025	−	1.03551i
$556$	2.11437e7			2.90064
$557$			1.36157e7i			1.85952i	0.368163	+	0.929761i	$0.379987\pi$
$557$			1.36157e7i			1.85952i	−0.368163	+	0.929761i	$0.620013\pi$
$558$	2.43509e7			3.31077
$559$	2.41550e6			0.326947
$560$		−	2.02269e7i		−	2.72558i
$561$		−	1.35862e7i		−	1.82259i
$562$	−4.97703e6			−0.664706
$563$			6.75543e6i			0.898218i	0.893477	+	0.449109i	$0.148259\pi$
$563$			6.75543e6i			0.898218i	−0.893477	+	0.449109i	$0.851741\pi$
$564$	−3.96515e7			−5.24882
$565$	6.55431e6			0.863785
$566$	−8.37590e6			−1.09898
$567$	2.62232e6			0.342553
$568$			1.55464e7i			2.02190i
$569$		−	1.28082e7i		−	1.65846i	−0.558905	−	0.829232i	$-0.688778\pi$
$569$		−	1.28082e7i		−	1.65846i	0.558905	−	0.829232i	$-0.311222\pi$
$570$			1.41756e7i			1.82748i
$571$	414554.			0.0532097			0.0266049	−	0.999646i	$-0.491530\pi$
$571$	414554.			0.0532097			0.0266049	+	0.999646i	$0.491530\pi$
$572$			1.12136e7i			1.43303i
$573$			1.10282e6i			0.140319i
$574$			7.03523e6i			0.891249i
$575$		−	404468.i		−	0.0510170i
$576$	5.51545e6			0.692668
$577$		−	3.27944e6i		−	0.410072i	−0.978754	−	0.205036i	$-0.934269\pi$
$577$		−	3.27944e6i		−	0.410072i	0.978754	−	0.205036i	$-0.0657311\pi$
$578$		−	6.60156e6i		−	0.821915i
$579$			2.37852e7i			2.94857i
$580$	3.40337e7			4.20087
$581$	248616.			0.0305554
$582$	4.54588e6			0.556302
$583$	8.05669e6			0.981714
$584$		−	1.89477e7i		−	2.29892i
$585$	8.73608e6			1.05542
$586$		−	9.57120e6i		−	1.15139i
$587$		−	1.55344e7i		−	1.86080i	−0.366550	−	0.930398i	$-0.619461\pi$
$587$		−	1.55344e7i		−	1.86080i	0.366550	−	0.930398i	$-0.380539\pi$
$588$	2.05770e7			2.45436
$589$	−4.06800e6			−0.483162
$590$		−	5.89334e6i		−	0.696998i
$591$	3.91661e6			0.461255
$592$	−5.84925e6	+	1.15247e7i	−0.685956	+	1.35152i
$593$	−9.41131e6			−1.09904			−0.549520	−	0.835481i	$-0.685189\pi$
$593$	−9.41131e6			−1.09904			−0.549520	+	0.835481i	$0.685189\pi$
$594$		−	2.77590e7i		−	3.22803i
$595$	−1.13891e7			−1.31886
$596$	−2.10958e7			−2.43265
$597$			1.98330e7i			2.27747i
$598$			381029.i			0.0435717i
$599$	−1.54540e7			−1.75984			−0.879920	−	0.475122i	$-0.842404\pi$
$599$	−1.54540e7			−1.75984			−0.879920	+	0.475122i	$0.842404\pi$
$600$			2.76462e7i			3.13514i
$601$	−5.18219e6			−0.585231			−0.292616	−	0.956230i	$-0.594526\pi$
$601$	−5.18219e6			−0.585231			−0.292616	+	0.956230i	$0.594526\pi$
$602$	−1.52437e7			−1.71435
$603$	−453226.			−0.0507600
$604$	−1.12064e6			−0.124989
$605$		−	1.57324e7i		−	1.74746i
$606$		−	5.47780e6i		−	0.605933i
$607$			8.72049e6i			0.960659i	0.877088	+	0.480330i	$0.159483\pi$
$607$			8.72049e6i			0.960659i	−0.877088	+	0.480330i	$0.840517\pi$
$608$	2.56720e6			0.281644
$609$			2.75490e7i			3.00997i
$610$			4.18510e7i			4.55388i
$611$		−	5.96958e6i		−	0.646905i
$612$			2.54335e7i			2.74491i
$613$	4.99682e6			0.537084			0.268542	−	0.963268i	$-0.413458\pi$
$613$	4.99682e6			0.537084			0.268542	+	0.963268i	$0.413458\pi$
$614$			4.61752e6i			0.494296i
$615$		−	8.28076e6i		−	0.882841i
$616$		−	3.80502e7i		−	4.04022i
$617$	−1.00772e7			−1.06568			−0.532840	−	0.846216i	$-0.678875\pi$
$617$	−1.00772e7			−1.06568			−0.532840	+	0.846216i	$0.678875\pi$
$618$	−5.61168e6			−0.591046
$619$	1.33209e7			1.39735			0.698676	−	0.715438i	$-0.253773\pi$
$619$	1.33209e7			1.39735			0.698676	+	0.715438i	$0.253773\pi$
$620$	−3.08410e7			−3.22218
$621$		−	645021.i		−	0.0671189i
$622$	1.67622e7			1.73723
$623$			1.88325e6i			0.194396i
$624$		−	1.07304e7i		−	1.10320i
$625$	−1.05065e7			−1.07586
$626$	1.76826e7			1.80348
$627$			1.09869e7i			1.11610i
$628$	2.16530e7			2.19088
$629$	6.48917e6	+	3.29353e6i	0.653977	+	0.331921i
$630$	−5.51315e7			−5.53412
$631$		−	1.50505e7i		−	1.50479i	−0.658710	−	0.752397i	$-0.728898\pi$
$631$		−	1.50505e7i		−	1.50479i	0.658710	−	0.752397i	$-0.271102\pi$
$632$	3.55142e6			0.353679
$633$	8.51571e6			0.844717
$634$		−	1.18245e7i		−	1.16831i
$635$		−	1.98669e7i		−	1.95522i
$636$	−2.37985e7			−2.33296
$637$			3.09789e6i			0.302495i
$638$	3.85730e7			3.75173
$639$	1.74585e7			1.69144
$640$	−1.92130e7			−1.85415
$641$	1.20473e7			1.15810			0.579048	−	0.815293i	$-0.303424\pi$
$641$	1.20473e7			1.15810			0.579048	+	0.815293i	$0.303424\pi$
$642$		−	1.96565e7i		−	1.88222i
$643$			1.63347e6i			0.155806i	0.996961	+	0.0779029i	$0.0248224\pi$
$643$			1.63347e6i			0.155806i	−0.996961	+	0.0779029i	$0.975178\pi$
$644$		−	1.64437e6i		−	0.156237i
$645$	1.79425e7			1.69818
$646$		−	6.21319e6i		−	0.585778i
$647$		−	9.26473e6i		−	0.870106i	−0.900405	−	0.435053i	$-0.856730\pi$
$647$		−	9.26473e6i		−	0.870106i	0.900405	−	0.435053i	$-0.143270\pi$
$648$			5.83156e6i			0.545566i
$649$		−	4.56767e6i		−	0.425680i
$650$	−7.74092e6			−0.718636
$651$		−	2.49646e7i		−	2.30873i
$652$		−	3.01945e7i		−	2.78169i
$653$		−	3.31509e6i		−	0.304238i	−0.988362	−	0.152119i	$-0.951390\pi$
$653$		−	3.31509e6i		−	0.304238i	0.988362	−	0.152119i	$-0.0486096\pi$
$654$	5.80642e7			5.30840
$655$	6.11701e6			0.557104
$656$	−6.44589e6			−0.584822
$657$	−2.12781e7			−1.92318
$658$			3.76727e7i			3.39205i
$659$	245037.			0.0219796			0.0109898	−	0.999940i	$-0.496502\pi$
$659$	245037.			0.0219796			0.0109898	+	0.999940i	$0.496502\pi$
$660$			8.32955e7i			7.44323i
$661$		−	751834.i		−	0.0669296i	−0.999440	−	0.0334648i	$-0.989346\pi$
$661$		−	751834.i		−	0.0669296i	0.999440	−	0.0334648i	$-0.0106542\pi$
$662$	660145.			0.0585456
$663$	−6.04192e6			−0.533816
$664$			552875.i			0.0486639i
$665$	9.21015e6			0.807630
$666$	3.14123e7	+	1.59431e7i	2.74419	+	1.39279i
$667$	896300.			0.0780079
$668$			4.06555e7i			3.52515i
$669$	1.34777e7			1.16426
$670$	839402.			0.0722409
$671$			3.24369e7i			2.78121i
$672$			1.57545e7i			1.34580i
$673$	−1.60295e6			−0.136422			−0.0682108	−	0.997671i	$-0.521729\pi$
$673$	−1.60295e6			−0.136422			−0.0682108	+	0.997671i	$0.521729\pi$
$674$		−	2.61760e7i		−	2.21949i
$675$	1.31042e7			1.10701
$676$	−2.07128e7			−1.74330
$677$	1.08159e7			0.906964			0.453482	−	0.891266i	$-0.350182\pi$
$677$	1.08159e7			0.906964			0.453482	+	0.891266i	$0.350182\pi$
$678$	−2.19430e7			−1.83325
$679$		−	2.95355e6i		−	0.245850i
$680$		−	2.53273e7i		−	2.10047i
$681$		−	2.88950e7i		−	2.38756i
$682$	−3.49545e7			−2.87768
$683$			1.52237e7i			1.24873i	0.781132	+	0.624366i	$0.214642\pi$
$683$			1.52237e7i			1.24873i	−0.781132	+	0.624366i	$0.785358\pi$
$684$		−	2.05676e7i		−	1.68090i
$685$		−	4.25778e6i		−	0.346702i
$686$			8.91942e6i			0.723647i
$687$	2.95636e7			2.38982
$688$		−	1.39667e7i		−	1.12493i
$689$		−	3.58290e6i		−	0.287532i
$690$			2.83030e6i			0.226313i
$691$	3.49028e6			0.278077			0.139039	−	0.990287i	$-0.455599\pi$
$691$	3.49028e6			0.278077			0.139039	+	0.990287i	$0.455599\pi$
$692$	4.80132e7			3.81150
$693$	−4.27301e7			−3.37987
$694$	−9.13570e6			−0.720017
$695$			2.36450e7i			1.85685i
$696$	−6.12638e7			−4.79381
$697$			3.62948e6i			0.282984i
$698$		−	3.32407e7i		−	2.58245i
$699$	−2.54933e7			−1.97348
$700$	3.34068e7			2.57685
$701$		−	1.12118e7i		−	0.861746i	−0.902413	−	0.430873i	$-0.858206\pi$
$701$		−	1.12118e7i		−	0.861746i	0.902413	−	0.430873i	$-0.141794\pi$
$702$	−1.23447e7			−0.945452
$703$	−5.24766e6	−	2.66341e6i	−0.400477	−	0.203259i
$704$	−7.91718e6			−0.602058
$705$		−	4.43423e7i		−	3.36005i
$706$	3.29694e7			2.48943
$707$	−3.55903e6			−0.267783
$708$			1.34924e7i			1.01159i
$709$			1.73632e7i			1.29722i	0.761122	+	0.648609i	$0.224649\pi$
$709$			1.73632e7i			1.29722i	−0.761122	+	0.648609i	$0.775351\pi$
$710$	−3.23343e7			−2.40723
$711$		−	3.98822e6i		−	0.295873i
$712$	−4.18801e6			−0.309604
$713$	−812218.			−0.0598341
$714$	3.81293e7			2.79906
$715$	−1.25402e7			−0.917362
$716$			2.48998e7i			1.81516i
$717$			2.92469e7i			2.12463i
$718$			2.69569e7i			1.95146i
$719$	1.29054e7			0.930997			0.465498	−	0.885049i	$-0.345875\pi$
$719$	1.29054e7			0.930997			0.465498	+	0.885049i	$0.345875\pi$
$720$		−	5.05132e7i		−	3.63139i
$721$			3.64602e6i			0.261204i
$722$		−	1.98867e7i		−	1.41977i
$723$			4.04475e6i			0.287771i
$724$	−1.79757e7			−1.27450
$725$			1.82091e7i			1.28660i
$726$			5.26702e7i			3.70871i
$727$		−	5.33510e6i		−	0.374375i	−0.982324	−	0.187187i	$-0.940063\pi$
$727$		−	5.33510e6i		−	0.374375i	0.982324	−	0.187187i	$-0.0599372\pi$
$728$	−1.69213e7			−1.18333
$729$	−2.20645e7			−1.53771
$730$	3.94083e7			2.73704
$731$	−7.86422e6			−0.544330
$732$		−	9.58149e7i		−	6.60930i
$733$	−2.34611e7			−1.61283			−0.806415	−	0.591349i	$-0.798595\pi$
$733$	−2.34611e7			−1.61283			−0.806415	+	0.591349i	$0.798595\pi$
$734$		−	1.55157e7i		−	1.06299i
$735$			2.30113e7i			1.57117i
$736$	512569.			0.0348785
$737$	650584.			0.0441199
$738$			1.75693e7i			1.18744i
$739$	−1.21379e7			−0.817585			−0.408792	−	0.912627i	$-0.634050\pi$
$739$	−1.21379e7			−0.817585			−0.408792	+	0.912627i	$0.634050\pi$
$740$	−3.97844e7	−	2.01923e7i	−2.67075	−	1.35552i
$741$	4.88598e6			0.326893
$742$			2.26109e7i			1.50768i
$743$	−2.85192e7			−1.89524			−0.947622	−	0.319393i	$-0.896521\pi$
$743$	−2.85192e7			−1.89524			−0.947622	+	0.319393i	$0.896521\pi$
$744$	5.55167e7			3.67698
$745$		−	2.35914e7i		−	1.55727i
$746$		−	2.19379e7i		−	1.44327i
$747$	620875.			0.0407101
$748$		−	3.65086e7i		−	2.38584i
$749$	−1.27712e7			−0.831818
$750$	5.18420e6			0.336534
$751$	3.04518e7			1.97022			0.985108	−	0.171937i	$-0.0550026\pi$
$751$	3.04518e7			1.97022			0.985108	+	0.171937i	$0.0550026\pi$
$752$	−3.45169e7			−2.22580
$753$			2.52577e6i			0.162333i
$754$		−	1.71538e7i		−	1.09884i
$755$		−	1.25321e6i		−	0.0800122i
$756$	5.32751e7			3.39016
$757$			1.86061e7i			1.18009i	0.807369	+	0.590046i	$0.200891\pi$
$757$			1.86061e7i			1.18009i	−0.807369	+	0.590046i	$0.799109\pi$
$758$			5.65553e6i			0.357520i
$759$			2.19364e6i			0.138217i
$760$			2.04817e7i			1.28627i
$761$	1.70681e7			1.06837			0.534186	−	0.845367i	$-0.320618\pi$
$761$	1.70681e7			1.06837			0.534186	+	0.845367i	$0.320618\pi$
$762$			6.65118e7i			4.14965i
$763$		−	3.77254e7i		−	2.34597i
$764$			2.96348e6i			0.183683i
$765$	−2.84424e7			−1.75716
$766$	−3.82574e7			−2.35583
$767$	−2.03129e6			−0.124676
$768$	5.35107e7			3.27369
$769$			8.20099e6i			0.500092i	0.968234	+	0.250046i	$0.0804458\pi$
$769$			8.20099e6i			0.500092i	−0.968234	+	0.250046i	$0.919554\pi$
$770$	7.91387e7			4.81019
$771$		−	4.67337e7i		−	2.83135i
$772$			6.39154e7i			3.85978i
$773$	6.10300e6			0.367363			0.183681	−	0.982986i	$-0.441199\pi$
$773$	6.10300e6			0.367363			0.183681	+	0.982986i	$0.441199\pi$
$774$	−3.80685e7			−2.28409
$775$		−	1.65009e7i		−	0.986856i
$776$	6.56815e6			0.391551
$777$	1.63449e7	−	3.22040e7i	0.971247	−	1.91363i
$778$	−2.55674e7			−1.51439
$779$		−	2.93509e6i		−	0.173291i
$780$	3.70424e7			2.18003
$781$	−2.50609e7			−1.47018
$782$		−	1.24053e6i		−	0.0725420i
$783$			2.90387e7i			1.69267i
$784$	1.79124e7			1.04079
$785$			2.42146e7i			1.40250i
$786$	−2.04790e7			−1.18236
$787$	−2.58372e7			−1.48699			−0.743496	−	0.668740i	$-0.766834\pi$
$787$	−2.58372e7			−1.48699			−0.743496	+	0.668740i	$0.766834\pi$
$788$	1.05247e7			0.603799
$789$	2.05319e7			1.17419
$790$			7.38643e6i			0.421082i
$791$			1.42568e7i			0.810178i
$792$		−	9.50238e7i		−	5.38294i
$793$	1.44251e7			0.814581
$794$		−	3.67848e7i		−	2.07070i
$795$		−	2.66140e7i		−	1.49345i
$796$			5.32950e7i			2.98128i
$797$		−	9.74891e6i		−	0.543639i	−0.962348	−	0.271819i	$-0.912375\pi$
$797$		−	9.74891e6i		−	0.543639i	0.962348	−	0.271819i	$-0.0876253\pi$
$798$	−3.08344e7			−1.71407
$799$			1.94354e7i			1.07702i
$800$			1.04133e7i			0.575258i
$801$			4.70310e6i			0.259002i
$802$	−1.29696e7			−0.712018
$803$	3.05437e7			1.67160
$804$	−1.92175e6			−0.104847
$805$	1.83890e6			0.100016
$806$			1.55447e7i			0.842837i
$807$	−3.31739e7			−1.79314
$808$		−	7.91464e6i		−	0.426484i
$809$			1.05888e7i			0.568822i	0.958702	+	0.284411i	$0.0917980\pi$
$809$			1.05888e7i			0.568822i	−0.958702	+	0.284411i	$0.908202\pi$
$810$	−1.21288e7			−0.649538
$811$	2.21309e7			1.18153			0.590767	−	0.806842i	$-0.298825\pi$
$811$	2.21309e7			1.18153			0.590767	+	0.806842i	$0.298825\pi$
$812$			7.40293e7i			3.94016i
$813$	3.29434e7			1.74800
$814$	−4.50908e7	−	2.28855e7i	−2.38521	−	1.21060i
$815$	3.37665e7			1.78071
$816$			3.49352e7i			1.83670i
$817$	6.35964e6			0.333332
$818$	−9.72193e6			−0.508007
$819$			1.90025e7i			0.989924i
$820$		−	2.22520e7i		−	1.15567i
$821$	3.09814e7			1.60414			0.802071	−	0.597229i	$-0.203732\pi$
$821$	3.09814e7			1.60414			0.802071	+	0.597229i	$0.203732\pi$
$822$			1.42545e7i			0.735821i
$823$	1.92131e7			0.988776			0.494388	−	0.869241i	$-0.335392\pi$
$823$	1.92131e7			0.988776			0.494388	+	0.869241i	$0.335392\pi$
$824$	−8.10807e6			−0.416006
$825$	−4.45657e7			−2.27964
$826$	1.28191e7			0.653742
$827$		−	5.33826e6i		−	0.271416i	−0.990749	−	0.135708i	$-0.956669\pi$
$827$		−	5.33826e6i		−	0.271416i	0.990749	−	0.135708i	$-0.0433310\pi$
$828$		−	4.10653e6i		−	0.208161i
$829$			1.38953e7i			0.702235i	0.936331	+	0.351117i	$0.114198\pi$
$829$			1.38953e7i			0.702235i	−0.936331	+	0.351117i	$0.885802\pi$
$830$	−1.14990e6			−0.0579381
$831$			3.60994e7i			1.81342i
$832$			3.52085e6i			0.176336i
$833$		−	1.00859e7i		−	0.503619i
$834$		−	7.91605e7i		−	3.94088i
$835$	−4.54651e7			−2.25664
$836$			2.95238e7i			1.46102i
$837$		−	2.63146e7i		−	1.29833i
$838$			1.23904e7i			0.609504i
$839$	−1.11135e7			−0.545064			−0.272532	−	0.962147i	$-0.587861\pi$
$839$	−1.11135e7			−0.545064			−0.272532	+	0.962147i	$0.587861\pi$
$840$	−1.25693e8			−6.14626
$841$	−1.98401e7			−0.967285
$842$	3.24938e7			1.57950
$843$			1.27426e7i			0.617572i
$844$	2.28833e7			1.10576
$845$		−	2.31632e7i		−	1.11598i
$846$			9.40812e7i			4.51936i
$847$	3.42209e7			1.63901
$848$	−2.07168e7			−0.989310
$849$			2.14446e7i			1.02105i
$850$	2.52024e7			1.19645
$851$	−1.04775e6	−	531778.i	−0.0495945	−	0.0251713i
$852$	7.40271e7			3.49375
$853$		−	3.40951e7i		−	1.60443i	−0.597038	−	0.802213i	$-0.703656\pi$
$853$		−	3.40951e7i		−	1.60443i	0.597038	−	0.802213i	$-0.296344\pi$
$854$	−9.10334e7			−4.27126
$855$	2.30008e7			1.07604
$856$		−	2.84009e7i		−	1.32479i
$857$		−	1.49337e7i		−	0.694568i	−0.937760	−	0.347284i	$-0.887104\pi$
$857$		−	1.49337e7i		−	0.694568i	0.937760	−	0.347284i	$-0.112896\pi$
$858$	4.19831e7			1.94696
$859$			7.59887e6i			0.351371i	0.984446	+	0.175686i	$0.0562142\pi$
$859$			7.59887e6i			0.351371i	−0.984446	+	0.175686i	$0.943786\pi$
$860$	4.82148e7			2.22297
$861$	1.80121e7			0.828051
$862$	−1.58350e7			−0.725856
$863$	−7.69720e6			−0.351808			−0.175904	−	0.984407i	$-0.556285\pi$
$863$	−7.69720e6			−0.351808			−0.175904	+	0.984407i	$0.556285\pi$
$864$			1.66065e7i			0.756820i
$865$			5.36933e7i			2.43994i
$866$			1.34320e7i			0.608621i
$867$	−1.69018e7			−0.763634
$868$		−	6.70847e7i		−	3.02220i
$869$			5.72491e6i			0.257169i
$870$		−	1.27420e8i		−	5.70740i
$871$		−	289322.i		−	0.0129222i
$872$	8.38944e7			3.73630
$873$		−	7.37599e6i		−	0.327555i
$874$			1.00319e6i			0.0444227i
$875$		−	3.36828e6i		−	0.148726i
$876$	−9.02226e7			−3.97242
$877$	8.61090e6			0.378050			0.189025	−	0.981972i	$-0.439467\pi$
$877$	8.61090e6			0.378050			0.189025	+	0.981972i	$0.439467\pi$
$878$	−6.67655e7			−2.92291
$879$	−2.45049e7			−1.06975
$880$			7.25093e7i			3.15636i
$881$	2.80482e7			1.21749			0.608744	−	0.793367i	$-0.291674\pi$
$881$	2.80482e7			1.21749			0.608744	+	0.793367i	$0.291674\pi$
$882$		−	4.88231e7i		−	2.11326i
$883$			4.43487e7i			1.91416i	0.289816	+	0.957082i	$0.406406\pi$
$883$			4.43487e7i			1.91416i	−0.289816	+	0.957082i	$0.593594\pi$
$884$	−1.62358e7			−0.698783
$885$	−1.50886e7			−0.647574
$886$		−	5.24735e7i		−	2.24572i
$887$	−3.99164e7			−1.70350			−0.851749	−	0.523949i	$-0.824458\pi$
$887$	−3.99164e7			−1.70350			−0.851749	+	0.523949i	$0.824458\pi$
$888$	7.16158e7	+	3.63480e7i	3.04773	+	1.54685i
$889$	4.32140e7			1.83388
$890$		−	8.71043e6i		−	0.368608i
$891$	−9.40049e6			−0.396695
$892$	3.62172e7			1.52406
$893$		−	1.57170e7i		−	0.659539i
$894$			7.89811e7i			3.30506i
$895$	−2.78455e7			−1.16198
$896$		−	4.17917e7i		−	1.73908i
$897$	975537.			0.0404821
$898$	−7.00655e7			−2.89943
$899$	3.65660e7			1.50896
$900$	8.34277e7			3.43324
$901$			1.16650e7i			0.478709i
$902$		−	2.52199e7i		−	1.03211i
$903$			3.90280e7i			1.59279i
$904$	−3.17045e7			−1.29033
$905$		−	2.01023e7i		−	0.815875i
$906$			4.19558e6i			0.169813i
$907$		−	3.02380e6i		−	0.122049i	−0.998136	−	0.0610247i	$-0.980563\pi$
$907$		−	3.02380e6i		−	0.122049i	0.998136	−	0.0610247i	$-0.0194368\pi$
$908$		−	7.76462e7i		−	3.12540i
$909$	−8.88808e6			−0.356778
$910$		−	3.51939e7i		−	1.40885i
$911$			31968.4i			0.00127622i	1.00000		0.000638108i	$0.000203116\pi$
$911$			31968.4i			0.00127622i	−1.00000		0.000638108i	$0.999797\pi$
$912$		−	2.82514e7i		−	1.12474i
$913$	−891237.			−0.0353847
$914$	3.31038e7			1.31073
$915$	1.07150e8			4.23097
$916$	7.94429e7			3.12836
$917$			1.33056e7i			0.522529i
$918$	4.01912e7			1.57407
$919$			3.42133e7i			1.33631i	0.744024	+	0.668153i	$0.232915\pi$
$919$			3.42133e7i			1.33631i	−0.744024	+	0.668153i	$0.767085\pi$
$920$			4.08938e6i			0.159290i
$921$	1.18221e7			0.459246
$922$	−1.04000e7			−0.402910
$923$			1.11449e7i			0.430597i
$924$	−1.81183e8			−6.98130
$925$	1.08035e7	−	2.12859e7i	0.415156	−	0.817972i
$926$	4.24512e6			0.162691
$927$			9.10530e6i			0.348013i
$928$	−2.30758e7			−0.879602
$929$	743746.			0.0282739			0.0141369	−	0.999900i	$-0.495500\pi$
$929$	743746.			0.0282739			0.0141369	+	0.999900i	$0.495500\pi$
$930$			1.15467e8i			4.37773i
$931$			8.15627e6i			0.308402i
$932$	−6.85054e7			−2.58336
$933$		−	4.29159e7i		−	1.61404i
$934$	−3.71170e7			−1.39221
$935$	4.08277e7			1.52730
$936$	−4.22581e7			−1.57660
$937$	−2.62123e7			−0.975341			−0.487670	−	0.873028i	$-0.662153\pi$
$937$	−2.62123e7			−0.975341			−0.487670	+	0.873028i	$0.662153\pi$
$938$			1.82585e6i			0.0677576i
$939$		−	4.52724e7i		−	1.67560i
$940$		−	1.19156e8i		−	4.39842i
$941$	−8.93489e6			−0.328939			−0.164469	−	0.986382i	$-0.552591\pi$
$941$	−8.93489e6			−0.328939			−0.164469	+	0.986382i	$0.552591\pi$
$942$		−	8.10674e7i		−	2.97659i
$943$		−	586020.i		−	0.0214602i
$944$			1.17452e7i			0.428974i
$945$			5.95776e7i			2.17022i
$946$	5.46456e7			1.98530
$947$			4.14310e7i			1.50124i	0.660734	+	0.750620i	$0.270245\pi$
$947$			4.14310e7i			1.50124i	−0.660734	+	0.750620i	$0.729755\pi$
$948$		−	1.69107e7i		−	0.611141i
$949$		−	1.35831e7i		−	0.489592i
$950$	−2.03807e7			−0.732671
$951$	−3.02738e7			−1.08547
$952$	5.50913e7			1.97011
$953$	3.96432e6			0.141396			0.0706979	−	0.997498i	$-0.477477\pi$
$953$	3.96432e6			0.141396			0.0706979	+	0.997498i	$0.477477\pi$
$954$			5.64668e7i			2.00873i
$955$	−3.31406e6			−0.117585
$956$			7.85920e7i			2.78121i
$957$		−	9.87575e7i		−	3.48570i
$958$	−3.22509e6			−0.113535
$959$	9.26142e6			0.325185
$960$			2.61531e7i			0.915894i
$961$	−4.50658e6			−0.157412
$962$	−1.01774e7	+	2.00524e7i	−0.354569	+	0.698600i
$963$	−3.18940e7			−1.10826
$964$			1.08690e7i			0.376702i
$965$	−7.14768e7			−2.47085
$966$	−6.15641e6			−0.212268
$967$		−	5.27385e7i		−	1.81368i	−0.421472	−	0.906841i	$-0.638487\pi$
$967$		−	5.27385e7i		−	1.81368i	0.421472	−	0.906841i	$-0.361513\pi$
$968$			7.61009e7i			2.61037i
$969$	−1.59075e7			−0.544241
$970$			1.36608e7i			0.466172i
$971$	4.09048e7			1.39228			0.696140	−	0.717907i	$-0.254899\pi$
$971$	4.09048e7			1.39228			0.696140	+	0.717907i	$0.254899\pi$
$972$	−4.91214e7			−1.66765
$973$	−5.14321e7			−1.74162
$974$	−2.02077e7			−0.682526
$975$			1.98189e7i			0.667679i
$976$		−	8.34075e7i		−	2.80273i
$977$			2.65016e7i			0.888251i	0.895965	+	0.444126i	$0.146486\pi$
$977$			2.65016e7i			0.888251i	−0.895965	+	0.444126i	$0.853514\pi$
$978$	−1.13046e8			−3.77927
$979$		−	6.75108e6i		−	0.225121i
$980$			6.18357e7i			2.05672i
$981$		−	9.42128e7i		−	3.12563i
$982$			7.66932e7i			2.53792i
$983$	1.41878e7			0.468308			0.234154	−	0.972200i	$-0.424768\pi$
$983$	1.41878e7			0.468308			0.234154	+	0.972200i	$0.424768\pi$
$984$			4.00556e7i			1.31879i
$985$			1.17698e7i			0.386524i
$986$			5.58484e7i			1.82944i
$987$	9.64524e7			3.15152
$988$	1.31296e7			0.427915
$989$	1.26977e6			0.0412794
$990$	1.97635e8			6.40880
$991$		−	2.37057e7i		−	0.766775i	−0.923588	−	0.383388i	$-0.874757\pi$
$991$		−	2.37057e7i		−	0.766775i	0.923588	−	0.383388i	$-0.125243\pi$
$992$	2.09111e7			0.674678
$993$		−	1.69015e6i		−	0.0543942i
$994$		−	7.03329e7i		−	2.25784i
$995$	−5.95999e7			−1.90848
$996$	2.63261e6			0.0840888
$997$			5.24739e7i			1.67188i	0.548819	+	0.835941i	$0.315077\pi$
$997$			5.24739e7i			1.67188i	−0.548819	+	0.835941i	$0.684923\pi$
$998$	2.44077e7			0.775711
$999$	1.72288e7	−	3.39455e7i	0.546187	−	1.07614i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.b.a.36.1	✓	16
3.2	odd	2		333.6.c.d.73.16		16
4.3	odd	2		592.6.g.c.369.15		16
37.36	even	2	inner	37.6.b.a.36.16	yes	16
111.110	odd	2		333.6.c.d.73.1		16
148.147	odd	2		592.6.g.c.369.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.b.a.36.1	✓	16	1.1	even	1	trivial
37.6.b.a.36.16	yes	16	37.36	even	2	inner
333.6.c.d.73.1		16	111.110	odd	2
333.6.c.d.73.16		16	3.2	odd	2
592.6.g.c.369.15		16	4.3	odd	2
592.6.g.c.369.16		16	148.147	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-10.0606i q^{2} -25.7580 q^{3} -69.2166 q^{4} -77.4051i q^{5} +259.142i q^{6} +168.370 q^{7} +374.424i q^{8} +420.475 q^{9} +O(q^{10})\)	\(q-10.0606i q^{2} -25.7580 q^{3} -69.2166 q^{4} -77.4051i q^{5} +259.142i q^{6} +168.370 q^{7} +374.424i q^{8} +420.475 q^{9} -778.746 q^{10} -603.572 q^{11} +1782.88 q^{12} +268.415i q^{13} -1693.91i q^{14} +1993.80i q^{15} +1552.01 q^{16} -873.888i q^{17} -4230.25i q^{18} +706.696i q^{19} +5357.72i q^{20} -4336.87 q^{21} +6072.33i q^{22} +141.099i q^{23} -9644.40i q^{24} -2866.55 q^{25} +2700.43 q^{26} -4571.40 q^{27} -11654.0 q^{28} -6352.26i q^{29} +20058.9 q^{30} +5756.36i q^{31} -3632.68i q^{32} +15546.8 q^{33} -8791.88 q^{34} -13032.7i q^{35} -29103.9 q^{36} +(-3768.82 + 7425.63i) q^{37} +7109.82 q^{38} -6913.84i q^{39} +28982.3 q^{40} -4153.25 q^{41} +43631.8i q^{42} -8999.12i q^{43} +41777.2 q^{44} -32546.9i q^{45} +1419.55 q^{46} -22240.1 q^{47} -39976.7 q^{48} +11541.4 q^{49} +28839.4i q^{50} +22509.6i q^{51} -18578.8i q^{52} -13348.3 q^{53} +45991.3i q^{54} +46719.6i q^{55} +63041.6i q^{56} -18203.1i q^{57} -63907.9 q^{58} +7567.74i q^{59} -138004. i q^{60} -53741.6i q^{61} +57912.8 q^{62} +70795.3 q^{63} +13117.2 q^{64} +20776.7 q^{65} -156411. i q^{66} -1077.89 q^{67} +60487.6i q^{68} -3634.43i q^{69} -131117. q^{70} +41521.0 q^{71} +157436. i q^{72} -50604.9 q^{73} +(74706.6 + 37916.8i) q^{74} +73836.7 q^{75} -48915.1i q^{76} -101623. q^{77} -69557.7 q^{78} -9485.04i q^{79} -120134. i q^{80} +15574.8 q^{81} +41784.4i q^{82} +1476.60 q^{83} +300184. q^{84} -67643.4 q^{85} -90537.0 q^{86} +163622. i q^{87} -225992. i q^{88} +11185.2i q^{89} -327443. q^{90} +45193.0i q^{91} -9766.41i q^{92} -148272. i q^{93} +223750. i q^{94} +54701.9 q^{95} +93570.7i q^{96} -17542.0i q^{97} -116114. i q^{98} -253787. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9}+O(q^{10}) \) 16 * q - 18 * q^3 - 268 * q^4 + 190 * q^7 + 1394 * q^9	\( 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9} - 74 q^{10} - 1110 q^{11} + 1402 q^{12} + 2900 q^{16} - 7010 q^{21} - 12052 q^{25} + 4902 q^{26} + 4266 q^{27} - 16824 q^{28} + 19280 q^{30} - 2478 q^{33} + 20556 q^{34} - 51402 q^{36} - 11400 q^{37} + 12108 q^{38} + 16966 q^{40} + 3918 q^{41} + 125394 q^{44} + 17470 q^{46} + 3822 q^{47} - 78034 q^{48} - 32618 q^{49} - 24126 q^{53} - 164718 q^{58} - 81426 q^{62} + 219268 q^{63} + 158076 q^{64} + 98976 q^{65} + 23560 q^{67} - 222404 q^{70} - 50046 q^{71} - 196274 q^{73} + 141216 q^{74} + 214054 q^{75} - 239574 q^{77} - 90822 q^{78} + 317312 q^{81} - 215814 q^{83} + 438572 q^{84} - 346472 q^{85} + 197640 q^{86} - 857612 q^{90} - 132504 q^{95} - 574860 q^{99}+O(q^{100}) \) 16 * q - 18 * q^3 - 268 * q^4 + 190 * q^7 + 1394 * q^9 - 74 * q^10 - 1110 * q^11 + 1402 * q^12 + 2900 * q^16 - 7010 * q^21 - 12052 * q^25 + 4902 * q^26 + 4266 * q^27 - 16824 * q^28 + 19280 * q^30 - 2478 * q^33 + 20556 * q^34 - 51402 * q^36 - 11400 * q^37 + 12108 * q^38 + 16966 * q^40 + 3918 * q^41 + 125394 * q^44 + 17470 * q^46 + 3822 * q^47 - 78034 * q^48 - 32618 * q^49 - 24126 * q^53 - 164718 * q^58 - 81426 * q^62 + 219268 * q^63 + 158076 * q^64 + 98976 * q^65 + 23560 * q^67 - 222404 * q^70 - 50046 * q^71 - 196274 * q^73 + 141216 * q^74 + 214054 * q^75 - 239574 * q^77 - 90822 * q^78 + 317312 * q^81 - 215814 * q^83 + 438572 * q^84 - 346472 * q^85 + 197640 * q^86 - 857612 * q^90 - 132504 * q^95 - 574860 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more