LMFDB - Embedded newform 252.6.b.d.55.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,6,Mod(55,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("252.55");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	252.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:	$40.4167225929$
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} - 4 x^{15} - 674 x^{14} + 3404 x^{13} + 173721 x^{12} - 919512 x^{11} - 21981508 x^{10} + \cdots + 224266997486896 $ x^16 - 4x^15 - 674x^14 + 3404x^13 + 173721x^12 - 919512x^11 - 21981508x^10 + 94817024x^9 + 1652635052x^8 - 4373672096x^7 - 73339451096x^6 + 61523648112x^5 + 1688776741892x^4 + 257949039296x^3 - 14815785541168x^2 + 39188323305792*x + 224266997486896
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{46}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			55.16
Root			$-16.2859 - 1.80141i$ of defining polynomial
Character	$\chi$	$=$	252.55
Dual form			252.6.b.d.55.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.36116 + 3.60282i) q^{2} +(6.03935 + 31.4249i) q^{4} +57.4402i q^{5} +(15.0189 + 128.769i) q^{7} +(-86.8799 + 158.808i) q^{8} +O(q^{10})$	\(q+(4.36116 + 3.60282i) q^{2} +(6.03935 + 31.4249i) q^{4} +57.4402i q^{5} +(15.0189 + 128.769i) q^{7} +(-86.8799 + 158.808i) q^{8} +(-206.947 + 250.506i) q^{10} +158.969i q^{11} +470.875i q^{13} +(-398.432 + 615.692i) q^{14} +(-951.052 + 379.572i) q^{16} -1264.55i q^{17} +789.647 q^{19} +(-1805.05 + 346.901i) q^{20} +(-572.737 + 693.288i) q^{22} -97.3529i q^{23} -174.374 q^{25} +(-1696.48 + 2053.56i) q^{26} +(-3955.85 + 1249.65i) q^{28} -3573.18 q^{29} +7667.40 q^{31} +(-5515.22 - 1771.10i) q^{32} +(4555.95 - 5514.90i) q^{34} +(-7396.51 + 862.689i) q^{35} -6348.19 q^{37} +(3443.77 + 2844.96i) q^{38} +(-9121.94 - 4990.40i) q^{40} -16135.9i q^{41} +12188.5i q^{43} +(-4995.59 + 960.069i) q^{44} +(350.745 - 424.571i) q^{46} +12836.8 q^{47} +(-16355.9 + 3867.94i) q^{49} +(-760.474 - 628.240i) q^{50} +(-14797.2 + 2843.78i) q^{52} -11994.1 q^{53} -9131.20 q^{55} +(-21754.3 - 8802.31i) q^{56} +(-15583.2 - 12873.5i) q^{58} -13155.9 q^{59} +2131.54i q^{61} +(33438.7 + 27624.3i) q^{62} +(-17671.8 - 27594.4i) q^{64} -27047.1 q^{65} -13483.5i q^{67} +(39738.4 - 7637.07i) q^{68} +(-35365.4 - 22886.0i) q^{70} -7742.28i q^{71} +58398.9i q^{73} +(-27685.5 - 22871.4i) q^{74} +(4768.95 + 24814.6i) q^{76} +(-20470.3 + 2387.54i) q^{77} +45786.8i q^{79} +(-21802.7 - 54628.6i) q^{80} +(58134.7 - 70371.0i) q^{82} -52783.3 q^{83} +72636.0 q^{85} +(-43912.9 + 53155.8i) q^{86} +(-25245.5 - 13811.2i) q^{88} +96346.5i q^{89} +(-60634.0 + 7072.02i) q^{91} +(3059.31 - 587.949i) q^{92} +(55983.2 + 46248.6i) q^{94} +45357.4i q^{95} -118900. i q^{97} +(-85266.0 - 42058.6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{2} - 48 q^{4} - 608 q^{8}+O(q^{10}) $ 16 * q - 8 * q^2 - 48 * q^4 - 608 * q^8	$ 16 q - 8 q^{2} - 48 q^{4} - 608 q^{8} - 1176 q^{14} - 3008 q^{16} + 1552 q^{22} - 9776 q^{25} - 14672 q^{28} - 26592 q^{29} - 11648 q^{32} - 26272 q^{37} + 12256 q^{44} + 20208 q^{46} + 8848 q^{49} - 5992 q^{50} + 41888 q^{53} - 38304 q^{56} - 144400 q^{58} + 45312 q^{64} - 66688 q^{65} + 79296 q^{70} - 348464 q^{74} - 320992 q^{77} + 78080 q^{85} + 78448 q^{86} - 66112 q^{88} - 446944 q^{92} - 224840 q^{98}+O(q^{100}) $ 16 * q - 8 * q^2 - 48 * q^4 - 608 * q^8 - 1176 * q^14 - 3008 * q^16 + 1552 * q^22 - 9776 * q^25 - 14672 * q^28 - 26592 * q^29 - 11648 * q^32 - 26272 * q^37 + 12256 * q^44 + 20208 * q^46 + 8848 * q^49 - 5992 * q^50 + 41888 * q^53 - 38304 * q^56 - 144400 * q^58 + 45312 * q^64 - 66688 * q^65 + 79296 * q^70 - 348464 * q^74 - 320992 * q^77 + 78080 * q^85 + 78448 * q^86 - 66112 * q^88 - 446944 * q^92 - 224840 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	4.36116	+	3.60282i	0.770951	+	0.636895i
$2$	4.36116	+	3.60282i	0.770951	+	0.636895i
$3$		0			0
$3$		0			0
$4$	6.03935	+	31.4249i	0.188730	+	0.982029i
$5$			57.4402i			1.02752i	0.857934	+	0.513761i	$0.171748\pi$
$5$			57.4402i			1.02752i	−0.857934	+	0.513761i	$0.828252\pi$
$6$		0			0
$7$	15.0189	+	128.769i	0.115849	+	0.993267i
$7$	15.0189	+	128.769i	0.115849	+	0.993267i
$8$	−86.8799	+	158.808i	−0.479948	+	0.877297i
$9$		0			0
$10$	−206.947	+	250.506i	−0.654423	+	0.792168i
$11$			158.969i			0.396123i	0.980190	+	0.198062i	$0.0634646\pi$
$11$			158.969i			0.396123i	−0.980190	+	0.198062i	$0.936535\pi$
$12$		0			0
$13$			470.875i			0.772764i	0.922339	+	0.386382i	$0.126275\pi$
$13$			470.875i			0.772764i	−0.922339	+	0.386382i	$0.873725\pi$
$14$	−398.432	+	615.692i	−0.543293	+	0.839543i
$15$		0			0
$16$	−951.052	+	379.572i	−0.928762	+	0.370676i
$17$		−	1264.55i		−	1.06124i	−0.847610	−	0.530621i	$-0.821959\pi$
$17$		−	1264.55i		−	1.06124i	0.847610	−	0.530621i	$-0.178041\pi$
$18$		0			0
$19$	789.647			0.501821			0.250910	−	0.968010i	$-0.419270\pi$
$19$	789.647			0.501821			0.250910	+	0.968010i	$0.419270\pi$
$20$	−1805.05	+	346.901i	−1.00906	+	0.193924i
$21$		0			0
$22$	−572.737	+	693.288i	−0.252289	+	0.305392i
$23$		−	97.3529i		−	0.0383733i	−0.999816	−	0.0191867i	$-0.993892\pi$
$23$		−	97.3529i		−	0.0383733i	0.999816	−	0.0191867i	$-0.00610768\pi$
$24$		0			0
$25$	−174.374			−0.0557998
$26$	−1696.48	+	2053.56i	−0.492170	+	0.595763i
$27$		0			0
$28$	−3955.85	+	1249.65i	−0.953553	+	0.301226i
$29$	−3573.18			−0.788970			−0.394485	−	0.918902i	$-0.629077\pi$
$29$	−3573.18			−0.788970			−0.394485	+	0.918902i	$0.629077\pi$
$30$		0			0
$31$	7667.40			1.43299			0.716496	−	0.697591i	$-0.245745\pi$
$31$	7667.40			1.43299			0.716496	+	0.697591i	$0.245745\pi$
$32$	−5515.22	−	1771.10i	−0.952112	−	0.305751i
$33$		0			0
$34$	4555.95	−	5514.90i	0.675899	−	0.818164i
$35$	−7396.51	+	862.689i	−1.02060	+	0.119038i
$36$		0			0
$37$	−6348.19			−0.762335			−0.381168	−	0.924506i	$-0.624478\pi$
$37$	−6348.19			−0.762335			−0.381168	+	0.924506i	$0.624478\pi$
$38$	3443.77	+	2844.96i	0.386879	+	0.319607i
$39$		0			0
$40$	−9121.94	−	4990.40i	−0.901441	−	0.493157i
$41$		−	16135.9i		−	1.49911i	−0.661944	−	0.749554i	$-0.730268\pi$
$41$		−	16135.9i		−	1.49911i	0.661944	−	0.749554i	$-0.269732\pi$
$42$		0			0
$43$			12188.5i			1.00526i	0.864502	+	0.502630i	$0.167634\pi$
$43$			12188.5i			1.00526i	−0.864502	+	0.502630i	$0.832366\pi$
$44$	−4995.59	+	960.069i	−0.389005	+	0.0747603i
$45$		0			0
$46$	350.745	−	424.571i	0.0244398	−	0.0295839i
$47$	12836.8			0.847639			0.423820	−	0.905747i	$-0.360689\pi$
$47$	12836.8			0.847639			0.423820	+	0.905747i	$0.360689\pi$
$48$		0			0
$49$	−16355.9	+	3867.94i	−0.973158	+	0.230138i
$50$	−760.474	−	628.240i	−0.0430189	−	0.0355386i
$51$		0			0
$52$	−14797.2	+	2843.78i	−0.758877	+	0.145844i
$53$	−11994.1			−0.586516			−0.293258	−	0.956033i	$-0.594739\pi$
$53$	−11994.1			−0.586516			−0.293258	+	0.956033i	$0.594739\pi$
$54$		0			0
$55$	−9131.20			−0.407025
$56$	−21754.3	−	8802.31i	−0.926992	−	0.375082i
$57$		0			0
$58$	−15583.2	−	12873.5i	−0.608257	−	0.502491i
$59$	−13155.9			−0.492028			−0.246014	−	0.969266i	$-0.579121\pi$
$59$	−13155.9			−0.492028			−0.246014	+	0.969266i	$0.579121\pi$
$60$		0			0
$61$			2131.54i			0.0733447i	0.999327	+	0.0366724i	$0.0116758\pi$
$61$			2131.54i			0.0733447i	−0.999327	+	0.0366724i	$0.988324\pi$
$62$	33438.7	+	27624.3i	1.10477	+	0.912666i
$63$		0			0
$64$	−17671.8	−	27594.4i	−0.539300	−	0.842114i
$65$	−27047.1			−0.794032
$66$		0			0
$67$		−	13483.5i		−	0.366956i	−0.983024	−	0.183478i	$-0.941264\pi$
$67$		−	13483.5i		−	0.366956i	0.983024	−	0.183478i	$-0.0587357\pi$
$68$	39738.4	−	7637.07i	1.04217	−	0.200288i
$69$		0			0
$70$	−35365.4	−	22886.0i	−0.862649	−	0.558245i
$71$		−	7742.28i		−	0.182273i	−0.995838	−	0.0911366i	$-0.970950\pi$
$71$		−	7742.28i		−	0.182273i	0.995838	−	0.0911366i	$-0.0290500\pi$
$72$		0			0
$73$			58398.9i			1.28262i	0.767283	+	0.641309i	$0.221608\pi$
$73$			58398.9i			1.28262i	−0.767283	+	0.641309i	$0.778392\pi$
$74$	−27685.5	−	22871.4i	−0.587723	−	0.485527i
$75$		0			0
$76$	4768.95	+	24814.6i	0.0947085	+	0.492803i
$77$	−20470.3	+	2387.54i	−0.393456	+	0.0458906i
$78$		0			0
$79$			45786.8i			0.825415i	0.910864	+	0.412708i	$0.135417\pi$
$79$			45786.8i			0.825415i	−0.910864	+	0.412708i	$0.864583\pi$
$80$	−21802.7	−	54628.6i	−0.380878	−	0.954323i
$81$		0			0
$82$	58134.7	−	70371.0i	0.954774	−	1.15574i
$83$	−52783.3			−0.841010			−0.420505	−	0.907290i	$-0.638147\pi$
$83$	−52783.3			−0.841010			−0.420505	+	0.907290i	$0.638147\pi$
$84$		0			0
$85$	72636.0			1.09045
$86$	−43912.9	+	53155.8i	−0.640245	+	0.775006i
$87$		0			0
$88$	−25245.5	−	13811.2i	−0.347518	−	0.190119i
$89$			96346.5i			1.28932i	0.764469	+	0.644660i	$0.223001\pi$
$89$			96346.5i			1.28932i	−0.764469	+	0.644660i	$0.776999\pi$
$90$		0			0
$91$	−60634.0	+	7072.02i	−0.767561	+	0.0895241i
$92$	3059.31	−	587.949i	0.0376837	−	0.00724219i
$93$		0			0
$94$	55983.2	+	46248.6i	0.653488	+	0.539857i
$95$			45357.4i			0.515632i
$96$		0			0
$97$		−	118900.i		−	1.28308i	−0.767089	−	0.641541i	$-0.778296\pi$
$97$		−	118900.i		−	1.28308i	0.767089	−	0.641541i	$-0.221704\pi$
$98$	−85266.0	−	42058.6i	−0.896831	−	0.442374i
$99$		0			0
$100$	−1053.11	−	5479.70i	−0.0105311	−	0.0547970i
$101$		−	145117.i		−	1.41551i	−0.706457	−	0.707756i	$-0.749708\pi$
$101$		−	145117.i		−	1.41551i	0.706457	−	0.707756i	$-0.250292\pi$
$102$		0			0
$103$	115967.			1.07707			0.538533	−	0.842605i	$-0.318979\pi$
$103$	115967.			1.07707			0.538533	+	0.842605i	$0.318979\pi$
$104$	−74778.5	−	40909.5i	−0.677944	−	0.370887i
$105$		0			0
$106$	−52308.4	−	43212.8i	−0.452175	−	0.373549i
$107$			3375.47i			0.0285020i	0.999898	+	0.0142510i	$0.00453638\pi$
$107$			3375.47i			0.0285020i	−0.999898	+	0.0142510i	$0.995464\pi$
$108$		0			0
$109$	64089.9			0.516682			0.258341	−	0.966054i	$-0.416824\pi$
$109$	64089.9			0.516682			0.258341	+	0.966054i	$0.416824\pi$
$110$	−39822.6	−	32898.1i	−0.313796	−	0.259232i
$111$		0			0
$112$	−63160.9	−	116765.i	−0.475777	−	0.879566i
$113$	99376.8			0.732131			0.366065	−	0.930589i	$-0.380705\pi$
$113$	99376.8			0.732131			0.366065	+	0.930589i	$0.380705\pi$
$114$		0			0
$115$	5591.97			0.0394294
$116$	−21579.7	−	112287.i	−0.148902	−	0.774791i
$117$		0			0
$118$	−57374.9	−	47398.3i	−0.379330	−	0.313370i
$119$	162835.	−	18992.2i	1.05410	−	0.122944i
$120$		0			0
$121$	135780.			0.843086
$122$	−7679.56	+	9295.98i	−0.0467129	+	0.0565452i
$123$		0			0
$124$	46306.1	+	240948.i	0.270448	+	1.40724i
$125$			169484.i			0.970186i
$126$		0			0
$127$			107605.i			0.592000i	0.955188	+	0.296000i	$0.0956528\pi$
$127$			107605.i			0.592000i	−0.955188	+	0.296000i	$0.904347\pi$
$128$	22348.3	−	184012.i	0.120564	−	0.992706i
$129$		0			0
$130$	−117957.	−	97446.0i	−0.612159	−	0.505715i
$131$	89421.7			0.455265			0.227633	−	0.973747i	$-0.426901\pi$
$131$	89421.7			0.455265			0.227633	+	0.973747i	$0.426901\pi$
$132$		0			0
$133$	11859.6	+	101682.i	0.0581356	+	0.498442i
$134$	48578.5	−	58803.5i	0.233713	−	0.282905i
$135$		0			0
$136$	200820.	+	109864.i	0.931024	+	0.509340i
$137$	−175103.			−0.797064			−0.398532	−	0.917154i	$-0.630480\pi$
$137$	−175103.			−0.797064			−0.398532	+	0.917154i	$0.630480\pi$
$138$		0			0
$139$	−309736.			−1.35974			−0.679868	−	0.733334i	$-0.737963\pi$
$139$	−309736.			−1.35974			−0.679868	+	0.733334i	$0.737963\pi$
$140$	−71780.1	−	227225.i	−0.309516	−	0.979796i
$141$		0			0
$142$	27894.0	−	33765.3i	0.116089	−	0.140524i
$143$	−74854.4			−0.306110
$144$		0			0
$145$		−	205244.i		−	0.810683i
$146$	−210401.	+	254687.i	−0.816893	+	0.988835i
$147$		0			0
$148$	−38339.0	−	199492.i	−0.143875	−	0.748635i
$149$	406369.			1.49953			0.749765	−	0.661704i	$-0.230166\pi$
$149$	406369.			1.49953			0.749765	+	0.661704i	$0.230166\pi$
$150$		0			0
$151$			397636.i			1.41920i	0.704605	+	0.709599i	$0.251124\pi$
$151$			397636.i			1.41920i	−0.704605	+	0.709599i	$0.748876\pi$
$152$	−68604.4	+	125402.i	−0.240848	+	0.440246i
$153$		0			0
$154$	−97875.8	−	63338.2i	−0.332563	−	0.215211i
$155$			440417.i			1.47243i
$156$		0			0
$157$		−	35153.4i		−	0.113820i	−0.998379	−	0.0569099i	$-0.981875\pi$
$157$		−	35153.4i		−	0.113820i	0.998379	−	0.0569099i	$-0.0181248\pi$
$158$	−164962.	+	199683.i	−0.525703	+	0.636354i
$159$		0			0
$160$	101732.	−	316795.i	0.314165	−	0.978315i
$161$	12536.0	−	1462.13i	0.0381150	−	0.00444552i
$162$		0			0
$163$			585666.i			1.72656i	0.504729	+	0.863278i	$0.331593\pi$
$163$			585666.i			1.72656i	−0.504729	+	0.863278i	$0.668407\pi$
$164$	507069.	−	97450.2i	1.47217	−	0.282926i
$165$		0			0
$166$	−230196.	−	190169.i	−0.648377	−	0.535635i
$167$	67790.9			0.188096			0.0940482	−	0.995568i	$-0.470019\pi$
$167$	67790.9			0.188096			0.0940482	+	0.995568i	$0.470019\pi$
$168$		0			0
$169$	149570.			0.402836
$170$	316777.	+	261695.i	0.840681	+	0.694501i
$171$		0			0
$172$	−383022.	+	73610.5i	−0.987194	+	0.189722i
$173$			262400.i			0.666574i	0.942825	+	0.333287i	$0.108158\pi$
$173$			262400.i			0.666574i	−0.942825	+	0.333287i	$0.891842\pi$
$174$		0			0
$175$	−2618.91	−	22454.0i	−0.00646437	−	0.0554241i
$176$	−60340.2	−	151188.i	−0.146834	−	0.367904i
$177$		0			0
$178$	−347119.	+	420182.i	−0.821162	+	0.994003i
$179$			258955.i			0.604076i	0.953296	+	0.302038i	$0.0976669\pi$
$179$			258955.i			0.604076i	−0.953296	+	0.302038i	$0.902333\pi$
$180$		0			0
$181$			108699.i			0.246619i	0.992368	+	0.123310i	$0.0393509\pi$
$181$			108699.i			0.246619i	−0.992368	+	0.123310i	$0.960649\pi$
$182$	−289914.	−	187611.i	−0.648769	−	0.419837i
$183$		0			0
$184$	15460.4	+	8458.01i	0.0336648	+	0.0184172i
$185$		−	364641.i		−	0.783315i
$186$		0			0
$187$	201024.			0.420382
$188$	77525.8	+	403395.i	0.159975	+	0.832407i
$189$		0			0
$190$	−163415.	+	197811.i	−0.328403	+	0.397527i
$191$			420780.i			0.834588i	0.908772	+	0.417294i	$0.137021\pi$
$191$			420780.i			0.834588i	−0.908772	+	0.417294i	$0.862979\pi$
$192$		0			0
$193$	254156.			0.491141			0.245571	−	0.969379i	$-0.421025\pi$
$193$	254156.			0.491141			0.245571	+	0.969379i	$0.421025\pi$
$194$	428377.	−	518543.i	0.817188	−	0.989192i
$195$		0			0
$196$	−220328.	−	490622.i	−0.409666	−	0.912235i
$197$	800528.			1.46964			0.734820	−	0.678262i	$-0.237267\pi$
$197$	800528.			1.46964			0.734820	+	0.678262i	$0.237267\pi$
$198$		0			0
$199$	−529483.			−0.947806			−0.473903	−	0.880577i	$-0.657155\pi$
$199$	−529483.			−0.947806			−0.473903	+	0.880577i	$0.657155\pi$
$200$	15149.6	−	27692.0i	0.0267810	−	0.0489530i
$201$		0			0
$202$	522829.	−	632876.i	0.901532	−	1.09129i
$203$	−53665.3	−	460115.i	−0.0914015	−	0.783657i
$204$		0			0
$205$	926847.			1.54036
$206$	505751.	+	417809.i	0.830364	+	0.685977i
$207$		0			0
$208$	−178731.	−	447827.i	−0.286445	−	0.717714i
$209$			125529.i			0.198783i
$210$		0			0
$211$		−	410954.i		−	0.635459i	−0.948181	−	0.317730i	$-0.897080\pi$
$211$		−	410954.i		−	0.635459i	0.948181	−	0.317730i	$-0.102920\pi$
$212$	−72436.9	−	376915.i	−0.110693	−	0.575976i
$213$		0			0
$214$	−12161.2	+	14720.9i	−0.0181528	+	0.0219736i
$215$	−700108.			−1.03293
$216$		0			0
$217$	115156.	+	987323.i	0.166011	+	1.42334i
$218$	279506.	+	230905.i	0.398337	+	0.329072i
$219$		0			0
$220$	−55146.6	−	286947.i	−0.0768178	−	0.399711i
$221$	595445.			0.820089
$222$		0			0
$223$	1.11136e6			1.49655			0.748275	−	0.663389i	$-0.230883\pi$
$223$	1.11136e6			1.49655			0.748275	+	0.663389i	$0.230883\pi$
$224$	145230.	−	736789.i	0.193391	−	0.981122i
$225$		0			0
$226$	433398.	+	358037.i	0.564437	+	0.466290i
$227$	−969097.			−1.24825			−0.624126	−	0.781323i	$-0.714545\pi$
$227$	−969097.			−1.24825			−0.624126	+	0.781323i	$0.714545\pi$
$228$		0			0
$229$			772460.i			0.973391i	0.873572	+	0.486696i	$0.161798\pi$
$229$			772460.i			0.973391i	−0.873572	+	0.486696i	$0.838202\pi$
$230$	24387.5	+	20146.9i	0.0303981	+	0.0251124i
$231$		0			0
$232$	310438.	−	567449.i	0.378664	−	0.692161i
$233$	558911.			0.674455			0.337227	−	0.941423i	$-0.390511\pi$
$233$	558911.			0.674455			0.337227	+	0.941423i	$0.390511\pi$
$234$		0			0
$235$			737347.i			0.870968i
$236$	−79453.0	−	413423.i	−0.0928604	−	0.483186i
$237$		0			0
$238$	778573.	+	503837.i	0.890958	+	0.576564i
$239$		−	1.00644e6i		−	1.13971i	−0.821745	−	0.569856i	$-0.806999\pi$
$239$		−	1.00644e6i		−	1.13971i	0.821745	−	0.569856i	$-0.193001\pi$
$240$		0			0
$241$		−	535358.i		−	0.593747i	−0.954917	−	0.296874i	$-0.904056\pi$
$241$		−	535358.i		−	0.593747i	0.954917	−	0.296874i	$-0.0959440\pi$
$242$	592157.	+	489191.i	0.649978	+	0.536957i
$243$		0			0
$244$	−66983.5	+	12873.1i	−0.0720267	+	0.0138423i
$245$	−222175.	−	939484.i	−0.236472	−	0.999940i
$246$		0			0
$247$			371825.i			0.387789i
$248$	−666143.	+	1.21764e6i	−0.687762	+	1.25716i
$249$		0			0
$250$	−610622.	+	739148.i	−0.617906	+	0.747965i
$251$	496876.			0.497810			0.248905	−	0.968528i	$-0.419929\pi$
$251$	496876.			0.497810			0.248905	+	0.968528i	$0.419929\pi$
$252$		0			0
$253$	15476.1			0.0152006
$254$	−387680.	+	469280.i	−0.377042	+	0.456402i
$255$		0			0
$256$	760426.	−	721987.i	0.725198	−	0.688540i
$257$			1.31818e6i			1.24493i	0.782649	+	0.622463i	$0.213868\pi$
$257$			1.31818e6i			1.24493i	−0.782649	+	0.622463i	$0.786132\pi$
$258$		0			0
$259$	−95342.9	−	817450.i	−0.0883159	−	0.757202i
$260$	−163347.	−	849954.i	−0.149857	−	0.779762i
$261$		0			0
$262$	389982.	+	322170.i	0.350987	+	0.289956i
$263$			1.15800e6i			1.03233i	0.856489	+	0.516165i	$0.172641\pi$
$263$			1.15800e6i			1.03233i	−0.856489	+	0.516165i	$0.827359\pi$
$264$		0			0
$265$		−	688946.i		−	0.602658i
$266$	−314620.	+	486179.i	−0.272636	+	0.421300i
$267$		0			0
$268$	423717.	−	81431.4i	0.360362	−	0.0692556i
$269$		−	960355.i		−	0.809192i	−0.914496	−	0.404596i	$-0.867412\pi$
$269$		−	960355.i		−	0.809192i	0.914496	−	0.404596i	$-0.132588\pi$
$270$		0			0
$271$	1.06159e6			0.878080			0.439040	−	0.898468i	$-0.355319\pi$
$271$	1.06159e6			0.878080			0.439040	+	0.898468i	$0.355319\pi$
$272$	479989.	+	1.20265e6i	0.393377	+	0.985640i
$273$		0			0
$274$	−763653.	−	630866.i	−0.614497	−	0.507646i
$275$		−	27720.1i		−	0.0221036i
$276$		0			0
$277$	−214106.			−0.167660			−0.0838301	−	0.996480i	$-0.526715\pi$
$277$	−214106.			−0.167660			−0.0838301	+	0.996480i	$0.526715\pi$
$278$	−1.35081e6	−	1.11592e6i	−1.04829	−	0.866009i
$279$		0			0
$280$	505606.	−	1.24957e6i	0.385405	−	0.952503i
$281$	−2.36986e6			−1.79043			−0.895213	−	0.445638i	$-0.852977\pi$
$281$	−2.36986e6			−1.79043			−0.895213	+	0.445638i	$0.852977\pi$
$282$		0			0
$283$	24158.6			0.0179310			0.00896551	−	0.999960i	$-0.497146\pi$
$283$	24158.6			0.0179310			0.00896551	+	0.999960i	$0.497146\pi$
$284$	243301.	−	46758.3i	0.178998	−	0.0344004i
$285$		0			0
$286$	−326452.	−	269687.i	−0.235996	−	0.194960i
$287$	2.07780e6	−	242343.i	1.48901	−	0.173670i
$288$		0			0
$289$	−179232.			−0.126233
$290$	739459.	−	895102.i	0.516320	−	0.624997i
$291$		0			0
$292$	−1.83518e6	+	352691.i	−1.25957	+	0.242068i
$293$		−	1.15738e6i		−	0.787602i	−0.919196	−	0.393801i	$-0.871160\pi$
$293$		−	1.15738e6i		−	0.787602i	0.919196	−	0.393801i	$-0.128840\pi$
$294$		0			0
$295$		−	755677.i		−	0.505570i
$296$	551530.	−	1.00814e6i	0.365881	−	0.668794i
$297$		0			0
$298$	1.77224e6	+	1.46408e6i	1.15606	+	0.955043i
$299$	45841.0			0.0296535
$300$		0			0
$301$	−1.56950e6	+	183058.i	−0.998491	+	0.116459i
$302$	−1.43261e6	+	1.73415e6i	−0.903881	+	1.09413i
$303$		0			0
$304$	−750995.	+	299728.i	−0.466072	+	0.186013i
$305$	−122436.			−0.0753633
$306$		0			0
$307$	−3.29883e6			−1.99762			−0.998812	−	0.0487344i	$-0.984481\pi$
$307$	−3.29883e6			−1.99762			−0.998812	+	0.0487344i	$0.984481\pi$
$308$	−198655.	−	628857.i	−0.119323	−	0.377725i
$309$		0			0
$310$	−1.58674e6	+	1.92073e6i	−0.937783	+	1.13517i
$311$	761941.			0.446704			0.223352	−	0.974738i	$-0.428300\pi$
$311$	761941.			0.446704			0.223352	+	0.974738i	$0.428300\pi$
$312$		0			0
$313$		−	3.40261e6i		−	1.96314i	−0.191103	−	0.981570i	$-0.561206\pi$
$313$		−	3.40261e6i		−	1.96314i	0.191103	−	0.981570i	$-0.438794\pi$
$314$	126651.	−	153309.i	0.0724913	−	0.0877495i
$315$		0			0
$316$	−1.43885e6	+	276523.i	−0.810582	+	0.155780i
$317$	−2.02634e6			−1.13257			−0.566284	−	0.824210i	$-0.691620\pi$
$317$	−2.02634e6			−1.13257			−0.566284	+	0.824210i	$0.691620\pi$
$318$		0			0
$319$		−	568025.i		−	0.312529i
$320$	1.58503e6	−	1.01507e6i	0.865290	−	0.554142i
$321$		0			0
$322$	59939.4	+	38788.5i	0.0322161	+	0.0208479i
$323$		−	998548.i		−	0.532553i
$324$		0			0
$325$		−	82108.5i		−	0.0431201i
$326$	−2.11005e6	+	2.55418e6i	−1.09963	+	1.33109i
$327$		0			0
$328$	2.56250e6	+	1.40188e6i	1.31516	+	0.719494i
$329$	192794.	+	1.65298e6i	0.0981984	+	0.841932i
$330$		0			0
$331$		−	370514.i		−	0.185881i	−0.995672	−	0.0929404i	$-0.970373\pi$
$331$		−	370514.i		−	0.185881i	0.995672	−	0.0929404i	$-0.0296266\pi$
$332$	−318777.	−	1.65871e6i	−0.158724	−	0.825896i
$333$		0			0
$334$	295647.	+	244239.i	0.145013	+	0.119798i
$335$	774492.			0.377055
$336$		0			0
$337$	−2.99209e6			−1.43516			−0.717580	−	0.696477i	$-0.754750\pi$
$337$	−2.99209e6			−1.43516			−0.717580	+	0.696477i	$0.754750\pi$
$338$	652298.	+	538874.i	0.310566	+	0.256564i
$339$		0			0
$340$	438675.	+	2.28258e6i	0.205800	+	1.07085i
$341$			1.21888e6i			0.567642i
$342$		0			0
$343$	−743717.	−	2.04803e6i	−0.341328	−	0.939944i
$344$	−1.93562e6	−	1.05893e6i	−0.881911	−	0.482472i
$345$		0			0
$346$	−945380.	+	1.14437e6i	−0.424538	+	0.513896i
$347$			1.26324e6i			0.563198i	0.959532	+	0.281599i	$0.0908647\pi$
$347$			1.26324e6i			0.563198i	−0.959532	+	0.281599i	$0.909135\pi$
$348$		0			0
$349$			2.44388e6i			1.07403i	0.843572	+	0.537016i	$0.180448\pi$
$349$			2.44388e6i			1.07403i	−0.843572	+	0.537016i	$0.819552\pi$
$350$	69476.3	−	107361.i	0.0303156	−	0.0468464i
$351$		0			0
$352$	281550.	−	876748.i	0.121115	−	0.377154i
$353$			2.27918e6i			0.973515i	0.873537	+	0.486757i	$0.161820\pi$
$353$			2.27918e6i			0.973515i	−0.873537	+	0.486757i	$0.838180\pi$
$354$		0			0
$355$	444718.			0.187290
$356$	−3.02768e6	+	581871.i	−1.26615	+	0.243333i
$357$		0			0
$358$	−932968.	+	1.12934e6i	−0.384733	+	0.465712i
$359$		−	3.23332e6i		−	1.32408i	−0.749471	−	0.662038i	$-0.769692\pi$
$359$		−	3.23332e6i		−	1.32408i	0.749471	−	0.662038i	$-0.230308\pi$
$360$		0			0
$361$	−1.85256e6			−0.748176
$362$	−391621.	+	474051.i	−0.157071	+	0.190131i
$363$		0			0
$364$	−588428.	−	1.86271e6i	−0.232777	−	0.736871i
$365$	−3.35444e6			−1.31792
$366$		0			0
$367$	1.98513e6			0.769351			0.384675	−	0.923052i	$-0.374313\pi$
$367$	1.98513e6			0.769351			0.384675	+	0.923052i	$0.374313\pi$
$368$	36952.5	+	92587.7i	0.0142241	+	0.0356397i
$369$		0			0
$370$	1.31374e6	−	1.59026e6i	0.498890	−	0.603898i
$371$	−180139.	−	1.54447e6i	−0.0679474	−	0.582567i
$372$		0			0
$373$	−2.04828e6			−0.762286			−0.381143	−	0.924516i	$-0.624469\pi$
$373$	−2.04828e6			−0.762286			−0.381143	+	0.924516i	$0.624469\pi$
$374$	876698.	+	724255.i	0.324094	+	0.267739i
$375$		0			0
$376$	−1.11526e6	+	2.03858e6i	−0.406823	+	0.743632i
$377$		−	1.68252e6i		−	0.609687i
$378$		0			0
$379$		−	4.24315e6i		−	1.51737i	−0.651459	−	0.758684i	$-0.725843\pi$
$379$		−	4.24315e6i		−	1.51737i	0.651459	−	0.758684i	$-0.274157\pi$
$380$	−1.42535e6	+	273930.i	−0.506365	+	0.0973150i
$381$		0			0
$382$	−1.51600e6	+	1.83509e6i	−0.531545	+	0.643426i
$383$	−3.35304e6			−1.16800			−0.583998	−	0.811755i	$-0.698512\pi$
$383$	−3.35304e6			−1.16800			−0.583998	+	0.811755i	$0.698512\pi$
$384$		0			0
$385$	−137141.	−	1.17582e6i	−0.0471536	−	0.404285i
$386$	1.10841e6	+	915677.i	0.378646	+	0.312805i
$387$		0			0
$388$	3.73644e6	−	718082.i	1.26002	−	0.242156i
$389$	−2.22696e6			−0.746170			−0.373085	−	0.927797i	$-0.621700\pi$
$389$	−2.22696e6			−0.746170			−0.373085	+	0.927797i	$0.621700\pi$
$390$		0			0
$391$	−123108.			−0.0407233
$392$	806737.	−	2.93348e6i	0.265165	−	0.964203i
$393$		0			0
$394$	3.49123e6	+	2.88416e6i	1.13302	+	0.936006i
$395$	−2.63000e6			−0.848132
$396$		0			0
$397$		−	51809.1i		−	0.0164980i	−0.999966	−	0.00824898i	$-0.997374\pi$
$397$		−	51809.1i		−	0.0164980i	0.999966	−	0.00824898i	$-0.00262576\pi$
$398$	−2.30916e6	−	1.90763e6i	−0.730711	−	0.603653i
$399$		0			0
$400$	165839.	−	66187.7i	0.0518248	−	0.0206837i
$401$	26813.0			0.00832693			0.00416346	−	0.999991i	$-0.498675\pi$
$401$	26813.0			0.00832693			0.00416346	+	0.999991i	$0.498675\pi$
$402$		0			0
$403$			3.61038e6i			1.10737i
$404$	4.56028e6	−	876410.i	1.39007	−	0.267149i
$405$		0			0
$406$	1.42367e6	−	2.19998e6i	0.428641	−	0.662374i
$407$		−	1.00917e6i		−	0.301979i
$408$		0			0
$409$		−	1.46358e6i		−	0.432623i	−0.976324	−	0.216311i	$-0.930597\pi$
$409$		−	1.46358e6i		−	0.432623i	0.976324	−	0.216311i	$-0.0694026\pi$
$410$	4.04213e6	+	3.33927e6i	1.18755	+	0.981050i
$411$		0			0
$412$	700367.	+	3.64426e6i	0.203274	+	1.05771i
$413$	−197587.	−	1.69407e6i	−0.0570011	−	0.488715i
$414$		0			0
$415$		−	3.03188e6i		−	0.864156i
$416$	833965.	−	2.59698e6i	0.236273	−	0.735758i
$417$		0			0
$418$	−452260.	+	547453.i	−0.126604	+	0.153252i
$419$	2.55441e6			0.710813			0.355407	−	0.934712i	$-0.384342\pi$
$419$	2.55441e6			0.710813			0.355407	+	0.934712i	$0.384342\pi$
$420$		0			0
$421$	719641.			0.197884			0.0989420	−	0.995093i	$-0.468454\pi$
$421$	719641.			0.197884			0.0989420	+	0.995093i	$0.468454\pi$
$422$	1.48060e6	−	1.79224e6i	0.404721	−	0.489908i
$423$		0			0
$424$	1.04205e6	−	1.90476e6i	0.281497	−	0.514549i
$425$			220505.i			0.0592171i
$426$		0			0
$427$	−274476.	+	32013.4i	−0.0728509	+	0.00849693i
$428$	−106074.	+	20385.6i	−0.0279898	+	0.00537917i
$429$		0			0
$430$	−3.05328e6	−	2.52237e6i	−0.796335	−	0.657865i
$431$			4.59239e6i			1.19082i	0.803422	+	0.595409i	$0.203010\pi$
$431$			4.59239e6i			1.19082i	−0.803422	+	0.595409i	$0.796990\pi$
$432$		0			0
$433$			4.14912e6i			1.06350i	0.846902	+	0.531749i	$0.178465\pi$
$433$			4.14912e6i			1.06350i	−0.846902	+	0.531749i	$0.821535\pi$
$434$	−3.05493e6	+	4.72075e6i	−0.778534	+	1.20306i
$435$		0			0
$436$	387062.	+	2.01402e6i	0.0975133	+	0.507397i
$437$		−	76874.4i		−	0.0192565i
$438$		0			0
$439$	2.31118e6			0.572365			0.286183	−	0.958175i	$-0.407614\pi$
$439$	2.31118e6			0.572365			0.286183	+	0.958175i	$0.407614\pi$
$440$	793318.	−	1.45011e6i	0.195351	−	0.357082i
$441$		0			0
$442$	2.59683e6	+	2.14528e6i	0.632248	+	0.522310i
$443$		−	2.53712e6i		−	0.614231i	−0.951672	−	0.307115i	$-0.900636\pi$
$443$		−	2.53712e6i		−	0.614231i	0.951672	−	0.307115i	$-0.0993637\pi$
$444$		0			0
$445$	−5.53416e6			−1.32480
$446$	4.84680e6	+	4.00402e6i	1.15377	+	0.953145i
$447$		0			0
$448$	3.28789e6	−	2.69001e6i	0.773966	−	0.633227i
$449$	7.26004e6			1.69951			0.849754	−	0.527180i	$-0.176751\pi$
$449$	7.26004e6			1.69951			0.849754	+	0.527180i	$0.176751\pi$
$450$		0			0
$451$	2.56510e6			0.593831
$452$	600171.	+	3.12291e6i	0.138175	+	0.718974i
$453$		0			0
$454$	−4.22638e6	−	3.49148e6i	−0.962341	−	0.795006i
$455$	−406218.	−	3.48283e6i	−0.0919880	−	0.788685i
$456$		0			0
$457$	4.37440e6			0.979779			0.489889	−	0.871785i	$-0.337037\pi$
$457$	4.37440e6			0.979779			0.489889	+	0.871785i	$0.337037\pi$
$458$	−2.78304e6	+	3.36882e6i	−0.619948	+	0.750437i
$459$		0			0
$460$	33771.9	+	175727.i	0.00744150	+	0.0387208i
$461$		−	6.25764e6i		−	1.37138i	−0.727893	−	0.685691i	$-0.759500\pi$
$461$		−	6.25764e6i		−	1.37138i	0.727893	−	0.685691i	$-0.240500\pi$
$462$		0			0
$463$		−	517004.i		−	0.112083i	−0.998428	−	0.0560417i	$-0.982152\pi$
$463$		−	517004.i		−	0.112083i	0.998428	−	0.0560417i	$-0.0178480\pi$
$464$	3.39828e6	−	1.35628e6i	0.732765	−	0.292452i
$465$		0			0
$466$	2.43750e6	+	2.01366e6i	0.519971	+	0.429557i
$467$	−536211.			−0.113774			−0.0568870	−	0.998381i	$-0.518117\pi$
$467$	−536211.			−0.113774			−0.0568870	+	0.998381i	$0.518117\pi$
$468$		0			0
$469$	1.73625e6	−	202507.i	0.364486	−	0.0425116i
$470$	−2.65653e6	+	3.21568e6i	−0.554715	+	0.671473i
$471$		0			0
$472$	1.14298e6	−	2.08926e6i	0.236148	−	0.431655i
$473$	−1.93759e6			−0.398207
$474$		0			0
$475$	−137694.			−0.0280015
$476$	1.58024e6	+	5.00237e6i	0.319674	+	1.01195i
$477$		0			0
$478$	3.62604e6	−	4.38926e6i	0.725876	−	0.878661i
$479$	3.92296e6			0.781222			0.390611	−	0.920556i	$-0.372264\pi$
$479$	3.92296e6			0.781222			0.390611	+	0.920556i	$0.372264\pi$
$480$		0			0
$481$		−	2.98920e6i		−	0.589105i
$482$	1.92880e6	−	2.33478e6i	0.378155	−	0.457750i
$483$		0			0
$484$	820023.	+	4.26687e6i	0.159115	+	0.827935i
$485$	6.82966e6			1.31839
$486$		0			0
$487$			149201.i			0.0285069i	0.999898	+	0.0142535i	$0.00453717\pi$
$487$			149201.i			0.0285069i	−0.999898	+	0.0142535i	$0.995463\pi$
$488$	−338505.	−	185188.i	−0.0643451	−	0.0352017i
$489$		0			0
$490$	2.41585e6	−	4.89769e6i	0.454549	−	0.921513i
$491$			6.23790e6i			1.16771i	0.811858	+	0.583854i	$0.198456\pi$
$491$			6.23790e6i			1.16771i	−0.811858	+	0.583854i	$0.801544\pi$
$492$		0			0
$493$			4.51847e6i			0.837287i
$494$	−1.33962e6	+	1.62158e6i	−0.246981	+	0.298966i
$495$		0			0
$496$	−7.29210e6	+	2.91033e6i	−1.33091	+	0.531176i
$497$	996965.	−	116281.i	0.181046	−	0.0211162i
$498$		0			0
$499$		−	5.58426e6i		−	1.00396i	−0.864880	−	0.501978i	$-0.832606\pi$
$499$		−	5.58426e6i		−	1.00396i	0.864880	−	0.501978i	$-0.167394\pi$
$500$	−5.32604e6	+	1.02358e6i	−0.952751	+	0.183103i
$501$		0			0
$502$	2.16695e6	+	1.79016e6i	0.383787	+	0.317053i
$503$	−582506.			−0.102655			−0.0513275	−	0.998682i	$-0.516345\pi$
$503$	−582506.			−0.102655			−0.0513275	+	0.998682i	$0.516345\pi$
$504$		0			0
$505$	8.33552e6			1.45447
$506$	67493.6	+	55757.6i	0.0117189	+	0.00968117i
$507$		0			0
$508$	−3.38147e6	+	649862.i	−0.581361	+	0.111728i
$509$		−	3.94492e6i		−	0.674906i	−0.941342	−	0.337453i	$-0.890435\pi$
$509$		−	3.94492e6i		−	0.674906i	0.941342	−	0.337453i	$-0.109565\pi$
$510$		0			0
$511$	−7.51996e6	+	877087.i	−1.27398	+	0.148590i
$512$	5.91752e6	−	409018.i	0.997620	−	0.0689553i
$513$		0			0
$514$	−4.74919e6	+	5.74881e6i	−0.792887	+	0.959777i
$515$			6.66118e6i			1.10671i
$516$		0			0
$517$			2.04065e6i			0.335770i
$518$	2.52932e6	−	3.90853e6i	0.414171	−	0.640013i
$519$		0			0
$520$	2.34985e6	−	4.29529e6i	0.381094	−	0.696601i
$521$			8.44596e6i			1.36318i	0.731732	+	0.681592i	$0.238712\pi$
$521$			8.44596e6i			1.36318i	−0.731732	+	0.681592i	$0.761288\pi$
$522$		0			0
$523$	9.92373e6			1.58643			0.793215	−	0.608942i	$-0.208406\pi$
$523$	9.92373e6			1.58643			0.793215	+	0.608942i	$0.208406\pi$
$524$	540049.	+	2.81007e6i	0.0859221	+	0.447084i
$525$		0			0
$526$	−4.17206e6	+	5.05021e6i	−0.657485	+	0.795875i
$527$		−	9.69582e6i		−	1.52075i
$528$		0			0
$529$	6.42687e6			0.998527
$530$	2.48215e6	−	3.00460e6i	0.383830	−	0.464619i
$531$		0			0
$532$	−3.12372e6	+	986781.i	−0.478513	+	0.151162i
$533$	7.59797e6			1.15846
$534$		0			0
$535$	−193888.			−0.0292864
$536$	2.14128e6	+	1.17144e6i	0.321930	+	0.176120i
$537$		0			0
$538$	3.45999e6	−	4.18826e6i	0.515370	−	0.623847i
$539$	−614882.	−	2.60007e6i	−0.0911632	−	0.385491i
$540$		0			0
$541$	321014.			0.0471553			0.0235777	−	0.999722i	$-0.492494\pi$
$541$	321014.			0.0471553			0.0235777	+	0.999722i	$0.492494\pi$
$542$	4.62976e6	+	3.82472e6i	0.676956	+	0.559245i
$543$		0			0
$544$	−2.23964e6	+	6.97428e6i	−0.324475	+	1.01042i
$545$			3.68134e6i			0.530902i
$546$		0			0
$547$			1.51583e6i			0.216612i	0.994118	+	0.108306i	$0.0345426\pi$
$547$			1.51583e6i			0.216612i	−0.994118	+	0.108306i	$0.965457\pi$
$548$	−1.05751e6	−	5.50261e6i	−0.150430	−	0.782740i
$549$		0			0
$550$	99870.6	−	120892.i	0.0140777	−	0.0170408i
$551$	−2.82155e6			−0.395922
$552$		0			0
$553$	−5.89591e6	+	687667.i	−0.819857	+	0.0956237i
$554$	−933751.	−	771387.i	−0.129258	−	0.106782i
$555$		0			0
$556$	−1.87061e6	−	9.73344e6i	−0.256623	−	1.33530i
$557$	−3.61423e6			−0.493603			−0.246802	−	0.969066i	$-0.579380\pi$
$557$	−3.61423e6			−0.493603			−0.246802	+	0.969066i	$0.579380\pi$
$558$		0			0
$559$	−5.73925e6			−0.776829
$560$	6.70702e6	−	3.62797e6i	0.903773	−	0.488871i
$561$		0			0
$562$	−1.03353e7	−	8.53818e6i	−1.38033	−	1.14031i
$563$	2.03635e6			0.270758			0.135379	−	0.990794i	$-0.456775\pi$
$563$	2.03635e6			0.270758			0.135379	+	0.990794i	$0.456775\pi$
$564$		0			0
$565$			5.70822e6i			0.752280i
$566$	105359.	+	87039.0i	0.0138239	+	0.0114202i
$567$		0			0
$568$	1.22953e6	+	672648.i	0.159908	+	0.0874817i
$569$	2.48345e6			0.321569			0.160784	−	0.986990i	$-0.448598\pi$
$569$	2.48345e6			0.321569			0.160784	+	0.986990i	$0.448598\pi$
$570$		0			0
$571$			1.10072e7i			1.41283i	0.707800	+	0.706413i	$0.249688\pi$
$571$			1.10072e7i			1.41283i	−0.707800	+	0.706413i	$0.750312\pi$
$572$	−452072.	−	2.35230e6i	−0.0577721	−	0.300609i
$573$		0			0
$574$	9.93472e6	+	6.42904e6i	1.25857	+	0.814454i
$575$			16975.9i			0.00214122i
$576$		0			0
$577$		−	1.27799e7i		−	1.59804i	−0.601305	−	0.799020i	$-0.705352\pi$
$577$		−	1.27799e7i		−	1.59804i	0.601305	−	0.799020i	$-0.294648\pi$
$578$	−781659.	−	645741.i	−0.0973190	−	0.0803968i
$579$		0			0
$580$	6.44979e6	−	1.23954e6i	0.796114	−	0.153000i
$581$	−792747.	−	6.79685e6i	−0.0974304	−	0.835347i
$582$		0			0
$583$		−	1.90670e6i		−	0.232333i
$584$	−9.27419e6	−	5.07369e6i	−1.12524	−	0.615590i
$585$		0			0
$586$	4.16983e6	−	5.04751e6i	0.501620	−	0.607202i
$587$	4.85191e6			0.581189			0.290595	−	0.956846i	$-0.406147\pi$
$587$	4.85191e6			0.581189			0.290595	+	0.956846i	$0.406147\pi$
$588$		0			0
$589$	6.05454e6			0.719106
$590$	2.72257e6	−	3.29562e6i	0.321995	−	0.389769i
$591$		0			0
$592$	6.03747e6	−	2.40960e6i	0.708028	−	0.282579i
$593$			548014.i			0.0639963i	0.999488	+	0.0319982i	$0.0101871\pi$
$593$			548014.i			0.0639963i	−0.999488	+	0.0319982i	$0.989813\pi$
$594$		0			0
$595$	1.09091e6	+	9.35326e6i	0.126328	+	1.08311i
$596$	2.45421e6	+	1.27701e7i	0.283006	+	1.47258i
$597$		0			0
$598$	199920.	+	165157.i	0.0228614	+	0.0188862i
$599$			7.36280e6i			0.838447i	0.907883	+	0.419224i	$0.137698\pi$
$599$			7.36280e6i			0.838447i	−0.907883	+	0.419224i	$0.862302\pi$
$600$		0			0
$601$			1.02485e6i			0.115738i	0.998324	+	0.0578690i	$0.0184306\pi$
$601$			1.02485e6i			0.115738i	−0.998324	+	0.0578690i	$0.981569\pi$
$602$	−7.50434e6	−	4.85628e6i	−0.843959	−	0.546150i
$603$		0			0
$604$	−1.24957e7	+	2.40146e6i	−1.39369	+	0.267845i
$605$			7.79922e6i			0.866289i
$606$		0			0
$607$	−1.10484e7			−1.21710			−0.608551	−	0.793515i	$-0.708249\pi$
$607$	−1.10484e7			−1.21710			−0.608551	+	0.793515i	$0.708249\pi$
$608$	−4.35507e6	−	1.39854e6i	−0.477790	−	0.153432i
$609$		0			0
$610$	−533963.	−	441115.i	−0.0581014	−	0.0479985i
$611$			6.04451e6i			0.655025i
$612$		0			0
$613$	−4.22758e6			−0.454403			−0.227201	−	0.973848i	$-0.572958\pi$
$613$	−4.22758e6			−0.454403			−0.227201	+	0.973848i	$0.572958\pi$
$614$	−1.43867e7	−	1.18851e7i	−1.54007	−	1.27228i
$615$		0			0
$616$	1.39929e6	−	3.45826e6i	0.148579	−	0.367203i
$617$	−3.30914e6			−0.349947			−0.174974	−	0.984573i	$-0.555984\pi$
$617$	−3.30914e6			−0.349947			−0.174974	+	0.984573i	$0.555984\pi$
$618$		0			0
$619$	−2.47693e6			−0.259828			−0.129914	−	0.991525i	$-0.541470\pi$
$619$	−2.47693e6			−0.259828			−0.129914	+	0.991525i	$0.541470\pi$
$620$	−1.38401e7	+	2.65983e6i	−1.44597	+	0.277891i
$621$		0			0
$622$	3.32294e6	+	2.74514e6i	0.344387	+	0.284504i
$623$	−1.24064e7	+	1.44702e6i	−1.28064	+	0.149367i
$624$		0			0
$625$	−1.02801e7			−1.05269
$626$	1.22590e7	−	1.48393e7i	1.25031	−	1.51348i
$627$		0			0
$628$	1.10469e6	−	212304.i	0.111774	−	0.0214812i
$629$			8.02762e6i			0.809021i
$630$		0			0
$631$			1.31268e7i			1.31245i	0.754564	+	0.656227i	$0.227849\pi$
$631$			1.31268e7i			1.31245i	−0.754564	+	0.656227i	$0.772151\pi$
$632$	−7.27129e6	−	3.97795e6i	−0.724134	−	0.396156i
$633$		0			0
$634$	−8.83719e6	−	7.30055e6i	−0.873155	−	0.721327i
$635$	−6.18083e6			−0.608292
$636$		0			0
$637$	−1.82131e6	−	7.70156e6i	−0.177843	−	0.752022i
$638$	2.04649e6	−	2.47725e6i	0.199048	−	0.240945i
$639$		0			0
$640$	1.05697e7	+	1.28369e6i	1.02003	+	0.123882i
$641$	−5.80499e6			−0.558029			−0.279014	−	0.960287i	$-0.590008\pi$
$641$	−5.80499e6			−0.558029			−0.279014	+	0.960287i	$0.590008\pi$
$642$		0			0
$643$	6.94518e6			0.662455			0.331227	−	0.943551i	$-0.392537\pi$
$643$	6.94518e6			0.662455			0.331227	+	0.943551i	$0.392537\pi$
$644$	121657.	+	385114.i	0.0115591	+	0.0365910i
$645$		0			0
$646$	3.59759e6	−	4.35482e6i	0.339180	−	0.410572i
$647$	2.14392e6			0.201348			0.100674	−	0.994919i	$-0.467900\pi$
$647$	2.14392e6			0.201348			0.100674	+	0.994919i	$0.467900\pi$
$648$		0			0
$649$		−	2.09138e6i		−	0.194904i
$650$	295822.	−	358088.i	0.0274630	−	0.0332435i
$651$		0			0
$652$	−1.84045e7	+	3.53704e6i	−1.69553	+	0.325852i
$653$	−8.38727e6			−0.769728			−0.384864	−	0.922973i	$-0.625752\pi$
$653$	−8.38727e6			−0.769728			−0.384864	+	0.922973i	$0.625752\pi$
$654$		0			0
$655$			5.13640e6i			0.467795i
$656$	6.12473e6	+	1.53461e7i	0.555683	+	1.39231i
$657$		0			0
$658$	−5.11458e6	+	7.90349e6i	−0.460516	+	0.711630i
$659$			4.89205e6i			0.438811i	0.975634	+	0.219406i	$0.0704118\pi$
$659$			4.89205e6i			0.438811i	−0.975634	+	0.219406i	$0.929588\pi$
$660$		0			0
$661$			7.44510e6i			0.662776i	0.943495	+	0.331388i	$0.107517\pi$
$661$			7.44510e6i			0.662776i	−0.943495	+	0.331388i	$0.892483\pi$
$662$	1.33490e6	−	1.61587e6i	0.118386	−	0.143305i
$663$		0			0
$664$	4.58581e6	−	8.38239e6i	0.403641	−	0.737816i
$665$	−5.84063e6	+	681219.i	−0.512160	+	0.0597355i
$666$		0			0
$667$			347860.i			0.0302754i
$668$	409413.	+	2.13033e6i	0.0354994	+	0.184716i
$669$		0			0
$670$	3.37768e6	+	2.79036e6i	0.290691	+	0.240145i
$671$	−338849.			−0.0290536
$672$		0			0
$673$	2.31902e7			1.97364			0.986818	−	0.161835i	$-0.0517413\pi$
$673$	2.31902e7			1.97364			0.986818	+	0.161835i	$0.0517413\pi$
$674$	−1.30490e7	−	1.07800e7i	−1.10644	−	0.914046i
$675$		0			0
$676$	903306.	+	4.70023e6i	0.0760271	+	0.395596i
$677$			3.31996e6i			0.278395i	0.990265	+	0.139197i	$0.0444523\pi$
$677$			3.31996e6i			0.278395i	−0.990265	+	0.139197i	$0.955548\pi$
$678$		0			0
$679$	1.53107e7	−	1.78575e6i	1.27444	−	0.148644i
$680$	−6.31061e6	+	1.15352e7i	−0.523358	+	0.956646i
$681$		0			0
$682$	−4.39140e6	+	5.31572e6i	−0.361528	+	0.437624i
$683$			2.32567e7i			1.90764i	0.300384	+	0.953818i	$0.402885\pi$
$683$			2.32567e7i			1.90764i	−0.300384	+	0.953818i	$0.597115\pi$
$684$		0			0
$685$		−	1.00580e7i		−	0.819000i
$686$	4.13524e6	−	1.16113e7i	0.335498	−	0.942041i
$687$		0			0
$688$	−4.62641e6	−	1.15919e7i	−0.372626	−	0.933647i
$689$		−	5.64774e6i		−	0.453238i
$690$		0			0
$691$	−2.00025e7			−1.59363			−0.796817	−	0.604221i	$-0.793485\pi$
$691$	−2.00025e7			−1.59363			−0.796817	+	0.604221i	$0.793485\pi$
$692$	−8.24590e6	+	1.58473e6i	−0.654595	+	0.125802i
$693$		0			0
$694$	−4.55122e6	+	5.50917e6i	−0.358698	+	0.434198i
$695$		−	1.77913e7i		−	1.39716i
$696$		0			0
$697$	−2.04046e7			−1.59091
$698$	−8.80487e6	+	1.06582e7i	−0.684045	+	0.828025i
$699$		0			0
$700$	689799.	−	217907.i	0.0532081	−	0.0168084i
$701$	1.08996e7			0.837752			0.418876	−	0.908043i	$-0.362424\pi$
$701$	1.08996e7			0.837752			0.418876	+	0.908043i	$0.362424\pi$
$702$		0			0
$703$	−5.01283e6			−0.382556
$704$	4.38665e6	−	2.80926e6i	0.333581	−	0.213629i
$705$		0			0
$706$	−8.21149e6	+	9.93988e6i	−0.620027	+	0.750532i
$707$	1.86865e7	−	2.17949e6i	1.40598	−	0.163986i
$708$		0			0
$709$	2.33993e7			1.74819			0.874094	−	0.485758i	$-0.161456\pi$
$709$	2.33993e7			1.74819			0.874094	+	0.485758i	$0.161456\pi$
$710$	1.93948e6	+	1.60224e6i	0.144391	+	0.119284i
$711$		0			0
$712$	−1.53006e7	−	8.37057e6i	−1.13112	−	0.618807i
$713$		−	746444.i		−	0.0549887i
$714$		0			0
$715$		−	4.29965e6i		−	0.314534i
$716$	−8.13763e6	+	1.56392e6i	−0.593220	+	0.114007i
$717$		0			0
$718$	1.16491e7	−	1.41010e7i	0.843297	−	1.02080i
$719$	1.75505e7			1.26610			0.633049	−	0.774111i	$-0.281803\pi$
$719$	1.75505e7			1.26610			0.633049	+	0.774111i	$0.281803\pi$
$720$		0			0
$721$	1.74170e6	+	1.49330e7i	0.124777	+	1.06981i
$722$	−8.07929e6	−	6.67443e6i	−0.576807	−	0.476509i
$723$		0			0
$724$	−3.41584e6	+	656469.i	−0.242187	+	0.0465444i
$725$	623072.			0.0440244
$726$		0			0
$727$	5.05539e6			0.354747			0.177373	−	0.984144i	$-0.443240\pi$
$727$	5.05539e6			0.354747			0.177373	+	0.984144i	$0.443240\pi$
$728$	4.14478e6	−	1.02436e7i	0.289850	−	0.716346i
$729$		0			0
$730$	−1.46292e7	−	1.20855e7i	−1.01605	−	0.839375i
$731$	1.54130e7			1.06682
$732$		0			0
$733$		−	6.21029e6i		−	0.426925i	−0.976951	−	0.213463i	$-0.931526\pi$
$733$		−	6.21029e6i		−	0.426925i	0.976951	−	0.213463i	$-0.0684742\pi$
$734$	8.65747e6	+	7.15208e6i	0.593131	+	0.489995i
$735$		0			0
$736$	−172422.	+	536923.i	−0.0117327	+	0.0365357i
$737$	2.14345e6			0.145360
$738$		0			0
$739$			1.45763e7i			0.981828i	0.871208	+	0.490914i	$0.163337\pi$
$739$			1.45763e7i			0.981828i	−0.871208	+	0.490914i	$0.836663\pi$
$740$	1.14588e7	−	2.20220e6i	0.769239	−	0.147835i
$741$		0			0
$742$	4.77885e6	−	7.38470e6i	0.318650	−	0.492406i
$743$		−	1.79332e7i		−	1.19175i	−0.803076	−	0.595876i	$-0.796805\pi$
$743$		−	1.79332e7i		−	1.19175i	0.803076	−	0.595876i	$-0.203195\pi$
$744$		0			0
$745$			2.33419e7i			1.54080i
$746$	−8.93288e6	−	7.37960e6i	−0.587685	−	0.485496i
$747$		0			0
$748$	1.21406e6	+	6.31717e6i	0.0793387	+	0.412828i
$749$	−434656.	+	50695.9i	−0.0283101	+	0.00330193i
$750$		0			0
$751$		−	2.38059e7i		−	1.54023i	−0.637907	−	0.770114i	$-0.720199\pi$
$751$		−	2.38059e7i		−	1.54023i	0.637907	−	0.770114i	$-0.279801\pi$
$752$	−1.22084e7	+	4.87248e6i	−0.787255	+	0.314200i
$753$		0			0
$754$	6.06183e6	−	7.33774e6i	0.388307	−	0.470039i
$755$	−2.28403e7			−1.45826
$756$		0			0
$757$	1.46912e7			0.931786			0.465893	−	0.884841i	$-0.345733\pi$
$757$	1.46912e7			0.931786			0.465893	+	0.884841i	$0.345733\pi$
$758$	1.52873e7	−	1.85051e7i	0.966403	−	1.16982i
$759$		0			0
$760$	−7.20311e6	−	3.94065e6i	−0.452362	−	0.247476i
$761$		−	2.10069e7i		−	1.31492i	−0.753489	−	0.657461i	$-0.771630\pi$
$761$		−	2.10069e7i		−	1.31492i	0.753489	−	0.657461i	$-0.228370\pi$
$762$		0			0
$763$	962561.	+	8.25279e6i	0.0598573	+	0.513203i
$764$	−1.32230e7	+	2.54124e6i	−0.819589	+	0.157512i
$765$		0			0
$766$	−1.46231e7	−	1.20804e7i	−0.900467	−	0.743890i
$767$		−	6.19477e6i		−	0.380222i
$768$		0			0
$769$		−	2.23601e7i		−	1.36351i	−0.731582	−	0.681753i	$-0.761218\pi$
$769$		−	2.23601e7i		−	1.36351i	0.731582	−	0.681753i	$-0.238782\pi$
$770$	3.63816e6	−	5.62201e6i	0.221134	−	0.341715i
$771$		0			0
$772$	1.53494e6	+	7.98682e6i	0.0926930	+	0.482315i
$773$		−	2.98328e7i		−	1.79575i	−0.440254	−	0.897873i	$-0.645111\pi$
$773$		−	2.98328e7i		−	1.79575i	0.440254	−	0.897873i	$-0.354889\pi$
$774$		0			0
$775$	−1.33700e6			−0.0799607
$776$	1.88823e7	+	1.03301e7i	1.12564	+	0.615812i
$777$		0			0
$778$	−9.71210e6	−	8.02332e6i	−0.575260	−	0.475232i
$779$		−	1.27416e7i		−	0.752283i
$780$		0			0
$781$	1.23078e6			0.0722027
$782$	−536892.	−	443535.i	−0.0313957	−	0.0259365i
$783$		0			0
$784$	1.40871e7	−	9.88685e6i	0.818525	−	0.574470i
$785$	2.01922e6			0.116952
$786$		0			0
$787$	−239187.			−0.0137658			−0.00688288	−	0.999976i	$-0.502191\pi$
$787$	−239187.			−0.0137658			−0.00688288	+	0.999976i	$0.502191\pi$
$788$	4.83467e6	+	2.51565e7i	0.277365	+	1.44323i
$789$		0			0
$790$	−1.14698e7	−	9.47543e6i	−0.653868	−	0.540171i
$791$	1.49253e6	+	1.27966e7i	0.0848168	+	0.727201i
$792$		0			0
$793$	−1.00369e6			−0.0566782
$794$	186659.	−	225948.i	0.0105075	−	0.0127191i
$795$		0			0
$796$	−3.19773e6	−	1.66390e7i	−0.178879	−	0.930773i
$797$			2.64867e7i			1.47701i	0.674250	+	0.738503i	$0.264467\pi$
$797$			2.64867e7i			1.47701i	−0.674250	+	0.738503i	$0.735533\pi$
$798$		0			0
$799$		−	1.62328e7i		−	0.899550i
$800$	961713.	+	308834.i	0.0531277	+	0.0170608i
$801$		0			0
$802$	116936.	+	96602.5i	0.00641965	+	0.00530338i
$803$	−9.28360e6			−0.508075
$804$		0			0
$805$	83985.3	+	720072.i	0.00456787	+	0.0391639i
$806$	−1.30076e7	+	1.57454e7i	−0.705275	+	0.853724i
$807$		0			0
$808$	2.30456e7	+	1.26077e7i	1.24182	+	0.679372i
$809$	−1.19183e7			−0.640238			−0.320119	−	0.947377i	$-0.603723\pi$
$809$	−1.19183e7			−0.640238			−0.320119	+	0.947377i	$0.603723\pi$
$810$		0			0
$811$	2.57548e7			1.37501			0.687507	−	0.726178i	$-0.258705\pi$
$811$	2.57548e7			1.37501			0.687507	+	0.726178i	$0.258705\pi$
$812$	1.41350e7	−	4.46522e6i	0.752324	−	0.237658i
$813$		0			0
$814$	3.63584e6	−	4.40113e6i	0.192329	−	0.232811i
$815$	−3.36407e7			−1.77407
$816$		0			0
$817$			9.62459e6i			0.504460i
$818$	5.27303e6	−	6.38291e6i	0.275535	−	0.333531i
$819$		0			0
$820$	5.59756e6	+	2.91261e7i	0.290713	+	1.51268i
$821$	7.54910e6			0.390874			0.195437	−	0.980716i	$-0.437387\pi$
$821$	7.54910e6			0.390874			0.195437	+	0.980716i	$0.437387\pi$
$822$		0			0
$823$			1.51474e7i			0.779540i	0.920912	+	0.389770i	$0.127445\pi$
$823$			1.51474e7i			0.779540i	−0.920912	+	0.389770i	$0.872555\pi$
$824$	−1.00752e7	+	1.84165e7i	−0.516935	+	0.944906i
$825$		0			0
$826$	5.24172e6	−	8.09997e6i	0.267315	−	0.413079i
$827$			2.77159e7i			1.40917i	0.709618	+	0.704587i	$0.248868\pi$
$827$			2.77159e7i			1.40917i	−0.709618	+	0.704587i	$0.751132\pi$
$828$		0			0
$829$			7.60207e6i			0.384189i	0.981376	+	0.192095i	$0.0615281\pi$
$829$			7.60207e6i			0.384189i	−0.981376	+	0.192095i	$0.938472\pi$
$830$	1.09233e7	−	1.32225e7i	0.550376	−	0.666221i
$831$		0			0
$832$	1.29935e7	−	8.32119e6i	0.650755	−	0.416752i
$833$	4.89120e6	+	2.06828e7i	0.244232	+	1.03276i
$834$		0			0
$835$			3.89392e6i			0.193273i
$836$	−3.94475e6	+	758115.i	−0.195211	+	0.0375163i
$837$		0			0
$838$	1.11402e7	+	9.20308e6i	0.548002	+	0.452713i
$839$	−1.38922e6			−0.0681343			−0.0340671	−	0.999420i	$-0.510846\pi$
$839$	−1.38922e6			−0.0681343			−0.0340671	+	0.999420i	$0.510846\pi$
$840$		0			0
$841$	−7.74351e6			−0.377527
$842$	3.13847e6	+	2.59274e6i	0.152559	+	0.126031i
$843$		0			0
$844$	1.29142e7	−	2.48190e6i	0.624039	−	0.119930i
$845$			8.59133e6i			0.413922i
$846$		0			0
$847$	2.03927e6	+	1.74842e7i	0.0976709	+	0.837410i
$848$	1.14071e7	−	4.55265e6i	0.544734	−	0.217407i
$849$		0			0
$850$	−794441.	+	961658.i	−0.0377150	+	0.0456534i
$851$			618015.i			0.0292533i
$852$		0			0
$853$			5.36484e6i			0.252455i	0.992001	+	0.126228i	$0.0402870\pi$
$853$			5.36484e6i			0.252455i	−0.992001	+	0.126228i	$0.959713\pi$
$854$	−1.31237e6	−	849273.i	−0.0615761	−	0.0398477i
$855$		0			0
$856$	−536051.	−	293260.i	−0.0250047	−	0.0136795i
$857$		−	2.62171e7i		−	1.21936i	−0.792648	−	0.609680i	$-0.791298\pi$
$857$		−	2.62171e7i		−	1.21936i	0.792648	−	0.609680i	$-0.208702\pi$
$858$		0			0
$859$	7.99661e6			0.369763			0.184881	−	0.982761i	$-0.440810\pi$
$859$	7.99661e6			0.369763			0.184881	+	0.982761i	$0.440810\pi$
$860$	−4.22820e6	−	2.20009e7i	−0.194944	−	1.01436i
$861$		0			0
$862$	−1.65456e7	+	2.00281e7i	−0.758426	+	0.918062i
$863$			1.54584e6i			0.0706539i	0.999376	+	0.0353270i	$0.0112473\pi$
$863$			1.54584e6i			0.0706539i	−0.999376	+	0.0353270i	$0.988753\pi$
$864$		0			0
$865$	−1.50723e7			−0.684919
$866$	−1.49485e7	+	1.80950e7i	−0.677336	+	0.819904i
$867$		0			0
$868$	−3.03311e7	+	9.58156e6i	−1.36643	+	0.431655i
$869$	−7.27868e6			−0.326966
$870$		0			0
$871$	6.34902e6			0.283571
$872$	−5.56812e6	+	1.01780e7i	−0.247981	+	0.453284i
$873$		0			0
$874$	276965.	−	335261.i	0.0122644	−	0.0148458i
$875$	−2.18243e7	+	2.54547e6i	−0.963653	+	0.112395i
$876$		0			0
$877$	−2.38529e7			−1.04723			−0.523616	−	0.851954i	$-0.675417\pi$
$877$	−2.38529e7			−1.04723			−0.523616	+	0.851954i	$0.675417\pi$
$878$	1.00794e7	+	8.32678e6i	0.441265	+	0.364537i
$879$		0			0
$880$	8.68425e6	−	3.46595e6i	0.378030	−	0.150875i
$881$		−	1.44171e7i		−	0.625806i	−0.949785	−	0.312903i	$-0.898699\pi$
$881$		−	1.44171e7i		−	0.625806i	0.949785	−	0.312903i	$-0.101301\pi$
$882$		0			0
$883$		−	1.90137e7i		−	0.820663i	−0.911936	−	0.410331i	$-0.865413\pi$
$883$		−	1.90137e7i		−	0.820663i	0.911936	−	0.410331i	$-0.134587\pi$
$884$	3.59610e6	+	1.87118e7i	0.154775	+	0.805351i
$885$		0			0
$886$	9.14079e6	−	1.10648e7i	0.391200	−	0.473542i
$887$	3.15674e7			1.34719			0.673597	−	0.739099i	$-0.264748\pi$
$887$	3.15674e7			1.34719			0.673597	+	0.739099i	$0.264748\pi$
$888$		0			0
$889$	−1.38561e7	+	1.61610e6i	−0.588014	+	0.0685827i
$890$	−2.41353e7	−	1.99386e7i	−1.02136	−	0.843761i
$891$		0			0
$892$	6.71187e6	+	3.49243e7i	0.282443	+	1.46965i
$893$	1.01365e7			0.425363
$894$		0			0
$895$	−1.48744e7			−0.620700
$896$	2.40306e7	+	114109.i	0.999989	+	0.00474844i
$897$		0			0
$898$	3.16621e7	+	2.61566e7i	1.31024	+	1.08241i
$899$	−2.73970e7			−1.13059
$900$		0			0
$901$			1.51672e7i			0.622435i
$902$	1.11868e7	+	9.24160e6i	0.457815	+	0.378208i
$903$		0			0
$904$	−8.63384e6	+	1.57818e7i	−0.351385	+	0.642296i
$905$	−6.24366e6			−0.253407
$906$		0			0
$907$			2.02021e7i			0.815416i	0.913112	+	0.407708i	$0.133672\pi$
$907$			2.02021e7i			0.815416i	−0.913112	+	0.407708i	$0.866328\pi$
$908$	−5.85272e6	−	3.04538e7i	−0.235582	−	1.22582i
$909$		0			0
$910$	1.07764e7	−	1.66527e7i	0.431391	−	0.666624i
$911$		−	3.73893e7i		−	1.49263i	−0.665594	−	0.746314i	$-0.731822\pi$
$911$		−	3.73893e7i		−	1.49263i	0.665594	−	0.746314i	$-0.268178\pi$
$912$		0			0
$913$		−	8.39090e6i		−	0.333144i
$914$	1.90774e7	+	1.57602e7i	0.755361	+	0.624016i
$915$		0			0
$916$	−2.42745e7	+	4.66516e6i	−0.955898	+	0.183708i
$917$	1.34302e6	+	1.15147e7i	0.0527421	+	0.452200i
$918$		0			0
$919$		−	2.91420e7i		−	1.13823i	−0.822258	−	0.569116i	$-0.807286\pi$
$919$		−	2.91420e7i		−	1.13823i	0.822258	−	0.569116i	$-0.192714\pi$
$920$	−485830.	+	888048.i	−0.0189241	+	0.0345913i
$921$		0			0
$922$	2.25452e7	−	2.72906e7i	0.873426	−	1.05727i
$923$	3.64564e6			0.140854
$924$		0			0
$925$	1.10696e6			0.0425382
$926$	1.86267e6	−	2.25473e6i	0.0713853	−	0.0864107i
$927$		0			0
$928$	1.97069e7	+	6.32846e6i	0.751187	+	0.241228i
$929$			1.92953e7i			0.733522i	0.930315	+	0.366761i	$0.119533\pi$
$929$			1.92953e7i			0.733522i	−0.930315	+	0.366761i	$0.880467\pi$
$930$		0			0
$931$	−1.29154e7	+	3.05430e6i	−0.488351	+	0.115488i
$932$	3.37546e6	+	1.75637e7i	0.127290	+	0.662334i
$933$		0			0
$934$	−2.33850e6	−	1.93187e6i	−0.0877142	−	0.0724621i
$935$			1.15469e7i			0.431952i
$936$		0			0
$937$		−	1.21648e7i		−	0.452641i	−0.974053	−	0.226321i	$-0.927330\pi$
$937$		−	1.21648e7i		−	0.452641i	0.974053	−	0.226321i	$-0.0726697\pi$
$938$	8.30165e6	+	5.37224e6i	0.308076	+	0.199365i
$939$		0			0
$940$	−2.31711e7	+	4.45310e6i	−0.855315	+	0.164377i
$941$			1.53178e7i			0.563925i	0.959425	+	0.281963i	$0.0909854\pi$
$941$			1.53178e7i			0.563925i	−0.959425	+	0.281963i	$0.909015\pi$
$942$		0			0
$943$	−1.57087e6			−0.0575257
$944$	1.25119e7	−	4.99361e6i	0.456977	−	0.182383i
$945$		0			0
$946$	−8.45013e6	−	6.98079e6i	−0.306998	−	0.253616i
$947$			2.79730e7i			1.01359i	0.862065	+	0.506797i	$0.169171\pi$
$947$			2.79730e7i			1.01359i	−0.862065	+	0.506797i	$0.830829\pi$
$948$		0			0
$949$	−2.74985e7			−0.991161
$950$	−600506.	−	496088.i	−0.0215878	−	0.0178340i
$951$		0			0
$952$	−1.11310e7	+	2.75095e7i	−0.398053	+	0.983761i
$953$	3.07038e7			1.09512			0.547558	−	0.836768i	$-0.315558\pi$
$953$	3.07038e7			1.09512			0.547558	+	0.836768i	$0.315558\pi$
$954$		0			0
$955$	−2.41697e7			−0.857556
$956$	3.16274e7	−	6.07827e6i	1.11923	−	0.215097i
$957$		0			0
$958$	1.71086e7	+	1.41337e7i	0.602284	+	0.497557i
$959$	−2.62986e6	−	2.25479e7i	−0.0923392	−	0.791697i
$960$		0			0
$961$	3.01599e7			1.05347
$962$	1.07696e7	−	1.30364e7i	0.375198	−	0.454171i
$963$		0			0
$964$	1.68236e7	−	3.23321e6i	0.583077	−	0.112058i
$965$			1.45987e7i			0.504658i
$966$		0			0
$967$		−	4.85110e7i		−	1.66830i	−0.551537	−	0.834150i	$-0.685959\pi$
$967$		−	4.85110e7i		−	1.66830i	0.551537	−	0.834150i	$-0.314041\pi$
$968$	−1.17965e7	+	2.15629e7i	−0.404638	+	0.739637i
$969$		0			0
$970$	2.97852e7	+	2.46061e7i	1.01642	+	0.839678i
$971$	3.83854e7			1.30653			0.653263	−	0.757131i	$-0.273400\pi$
$971$	3.83854e7			1.30653			0.653263	+	0.757131i	$0.273400\pi$
$972$		0			0
$973$	−4.65190e6	−	3.98844e7i	−0.157524	−	1.35058i
$974$	−537545.	+	650690.i	−0.0181559	+	0.0219774i
$975$		0			0
$976$	−809074.	−	2.02721e6i	−0.0271872	−	0.0681198i
$977$	−1.00019e7			−0.335233			−0.167616	−	0.985852i	$-0.553607\pi$
$977$	−1.00019e7			−0.335233			−0.167616	+	0.985852i	$0.553607\pi$
$978$		0			0
$979$	−1.53161e7			−0.510730
$980$	2.81814e7	−	1.26557e7i	0.937341	−	0.420941i
$981$		0			0
$982$	−2.24740e7	+	2.72044e7i	−0.743708	+	0.900246i
$983$	5.82198e7			1.92171			0.960853	−	0.277059i	$-0.0893596\pi$
$983$	5.82198e7			1.92171			0.960853	+	0.277059i	$0.0893596\pi$
$984$		0			0
$985$			4.59825e7i			1.51009i
$986$	−1.62793e7	+	1.97058e7i	−0.533264	+	0.645507i
$987$		0			0
$988$	−1.16846e7	+	2.24558e6i	−0.380820	+	0.0731874i
$989$	1.18658e6			0.0385752
$990$		0			0
$991$		−	3.63684e7i		−	1.17636i	−0.808731	−	0.588179i	$-0.799845\pi$
$991$		−	3.63684e7i		−	1.17636i	0.808731	−	0.588179i	$-0.200155\pi$
$992$	−4.22874e7	−	1.35797e7i	−1.36437	−	0.438139i
$993$		0			0
$994$	4.76686e6	+	3.08477e6i	0.153026	+	0.0990277i
$995$		−	3.04136e7i		−	0.973891i
$996$		0			0
$997$		−	5.28050e7i		−	1.68243i	−0.540700	−	0.841215i	$-0.681841\pi$
$997$		−	5.28050e7i		−	1.68243i	0.540700	−	0.841215i	$-0.318159\pi$
$998$	2.01191e7	−	2.43538e7i	0.639414	−	0.774000i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.6.b.d.55.16		16
3.2	odd	2		28.6.d.b.27.1	✓	16
4.3	odd	2	inner	252.6.b.d.55.14		16
7.6	odd	2	inner	252.6.b.d.55.15		16
12.11	even	2		28.6.d.b.27.4	yes	16
21.20	even	2		28.6.d.b.27.2	yes	16
24.5	odd	2		448.6.f.d.447.16		16
24.11	even	2		448.6.f.d.447.2		16
28.27	even	2	inner	252.6.b.d.55.13		16
84.83	odd	2		28.6.d.b.27.3	yes	16
168.83	odd	2		448.6.f.d.447.15		16
168.125	even	2		448.6.f.d.447.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.d.b.27.1	✓	16	3.2	odd	2
28.6.d.b.27.2	yes	16	21.20	even	2
28.6.d.b.27.3	yes	16	84.83	odd	2
28.6.d.b.27.4	yes	16	12.11	even	2
252.6.b.d.55.13		16	28.27	even	2	inner
252.6.b.d.55.14		16	4.3	odd	2	inner
252.6.b.d.55.15		16	7.6	odd	2	inner
252.6.b.d.55.16		16	1.1	even	1	trivial
448.6.f.d.447.1		16	168.125	even	2
448.6.f.d.447.2		16	24.11	even	2
448.6.f.d.447.15		16	168.83	odd	2
448.6.f.d.447.16		16	24.5	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(4.36116 + 3.60282i) q^{2} +(6.03935 + 31.4249i) q^{4} +57.4402i q^{5} +(15.0189 + 128.769i) q^{7} +(-86.8799 + 158.808i) q^{8} +O(q^{10})\)	\(q+(4.36116 + 3.60282i) q^{2} +(6.03935 + 31.4249i) q^{4} +57.4402i q^{5} +(15.0189 + 128.769i) q^{7} +(-86.8799 + 158.808i) q^{8} +(-206.947 + 250.506i) q^{10} +158.969i q^{11} +470.875i q^{13} +(-398.432 + 615.692i) q^{14} +(-951.052 + 379.572i) q^{16} -1264.55i q^{17} +789.647 q^{19} +(-1805.05 + 346.901i) q^{20} +(-572.737 + 693.288i) q^{22} -97.3529i q^{23} -174.374 q^{25} +(-1696.48 + 2053.56i) q^{26} +(-3955.85 + 1249.65i) q^{28} -3573.18 q^{29} +7667.40 q^{31} +(-5515.22 - 1771.10i) q^{32} +(4555.95 - 5514.90i) q^{34} +(-7396.51 + 862.689i) q^{35} -6348.19 q^{37} +(3443.77 + 2844.96i) q^{38} +(-9121.94 - 4990.40i) q^{40} -16135.9i q^{41} +12188.5i q^{43} +(-4995.59 + 960.069i) q^{44} +(350.745 - 424.571i) q^{46} +12836.8 q^{47} +(-16355.9 + 3867.94i) q^{49} +(-760.474 - 628.240i) q^{50} +(-14797.2 + 2843.78i) q^{52} -11994.1 q^{53} -9131.20 q^{55} +(-21754.3 - 8802.31i) q^{56} +(-15583.2 - 12873.5i) q^{58} -13155.9 q^{59} +2131.54i q^{61} +(33438.7 + 27624.3i) q^{62} +(-17671.8 - 27594.4i) q^{64} -27047.1 q^{65} -13483.5i q^{67} +(39738.4 - 7637.07i) q^{68} +(-35365.4 - 22886.0i) q^{70} -7742.28i q^{71} +58398.9i q^{73} +(-27685.5 - 22871.4i) q^{74} +(4768.95 + 24814.6i) q^{76} +(-20470.3 + 2387.54i) q^{77} +45786.8i q^{79} +(-21802.7 - 54628.6i) q^{80} +(58134.7 - 70371.0i) q^{82} -52783.3 q^{83} +72636.0 q^{85} +(-43912.9 + 53155.8i) q^{86} +(-25245.5 - 13811.2i) q^{88} +96346.5i q^{89} +(-60634.0 + 7072.02i) q^{91} +(3059.31 - 587.949i) q^{92} +(55983.2 + 46248.6i) q^{94} +45357.4i q^{95} -118900. i q^{97} +(-85266.0 - 42058.6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{2} - 48 q^{4} - 608 q^{8}+O(q^{10}) \) 16 * q - 8 * q^2 - 48 * q^4 - 608 * q^8	\( 16 q - 8 q^{2} - 48 q^{4} - 608 q^{8} - 1176 q^{14} - 3008 q^{16} + 1552 q^{22} - 9776 q^{25} - 14672 q^{28} - 26592 q^{29} - 11648 q^{32} - 26272 q^{37} + 12256 q^{44} + 20208 q^{46} + 8848 q^{49} - 5992 q^{50} + 41888 q^{53} - 38304 q^{56} - 144400 q^{58} + 45312 q^{64} - 66688 q^{65} + 79296 q^{70} - 348464 q^{74} - 320992 q^{77} + 78080 q^{85} + 78448 q^{86} - 66112 q^{88} - 446944 q^{92} - 224840 q^{98}+O(q^{100}) \) 16 * q - 8 * q^2 - 48 * q^4 - 608 * q^8 - 1176 * q^14 - 3008 * q^16 + 1552 * q^22 - 9776 * q^25 - 14672 * q^28 - 26592 * q^29 - 11648 * q^32 - 26272 * q^37 + 12256 * q^44 + 20208 * q^46 + 8848 * q^49 - 5992 * q^50 + 41888 * q^53 - 38304 * q^56 - 144400 * q^58 + 45312 * q^64 - 66688 * q^65 + 79296 * q^70 - 348464 * q^74 - 320992 * q^77 + 78080 * q^85 + 78448 * q^86 - 66112 * q^88 - 446944 * q^92 - 224840 * q^98

Introduction

L-functions

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