LMFDB - Embedded newform 21.7.d.a.13.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,7,Mod(13,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("21.13");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	21.d (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:	$4.83113575602$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 212x^{6} - 1123x^{5} + 44168x^{4} - 138697x^{3} + 660109x^{2} + 680340x + 1040400 $ x^8 - x^7 + 212x^6 - 1123x^5 + 44168x^4 - 138697x^3 + 660109x^2 + 680340x + 1040400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{7}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			13.4
Root			$-0.564972 + 0.978561i$ of defining polynomial
Character	$\chi$	$=$	21.13
Dual form			21.7.d.a.13.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.129945 q^{2} +15.5885i q^{3} -63.9831 q^{4} -126.447i q^{5} -2.02564i q^{6} +(-213.284 - 268.624i) q^{7} +16.6307 q^{8} -243.000 q^{9} +O(q^{10})$	\(q-0.129945 q^{2} +15.5885i q^{3} -63.9831 q^{4} -126.447i q^{5} -2.02564i q^{6} +(-213.284 - 268.624i) q^{7} +16.6307 q^{8} -243.000 q^{9} +16.4311i q^{10} -1712.19 q^{11} -997.398i q^{12} -309.447i q^{13} +(27.7152 + 34.9063i) q^{14} +1971.11 q^{15} +4092.76 q^{16} +5991.87i q^{17} +31.5766 q^{18} -3030.11i q^{19} +8090.45i q^{20} +(4187.43 - 3324.77i) q^{21} +222.491 q^{22} +409.222 q^{23} +259.248i q^{24} -363.744 q^{25} +40.2111i q^{26} -3788.00i q^{27} +(13646.6 + 17187.4i) q^{28} -33121.1 q^{29} -256.135 q^{30} -52443.3i q^{31} -1596.20 q^{32} -26690.4i q^{33} -778.612i q^{34} +(-33966.6 + 26969.1i) q^{35} +15547.9 q^{36} -54089.1 q^{37} +393.747i q^{38} +4823.81 q^{39} -2102.90i q^{40} +59391.1i q^{41} +(-544.135 + 432.037i) q^{42} +86604.1 q^{43} +109551. q^{44} +30726.5i q^{45} -53.1764 q^{46} -180225. i q^{47} +63799.8i q^{48} +(-26668.5 + 114587. i) q^{49} +47.2666 q^{50} -93403.9 q^{51} +19799.4i q^{52} +144578. q^{53} +492.231i q^{54} +216501. i q^{55} +(-3547.08 - 4467.42i) q^{56} +47234.7 q^{57} +4303.92 q^{58} -74972.5i q^{59} -126118. q^{60} -398720. i q^{61} +6814.74i q^{62} +(51828.1 + 65275.6i) q^{63} -261729. q^{64} -39128.6 q^{65} +3468.28i q^{66} -126500. q^{67} -383378. i q^{68} +6379.15i q^{69} +(4413.78 - 3504.49i) q^{70} -238387. q^{71} -4041.27 q^{72} +125014. i q^{73} +7028.61 q^{74} -5670.20i q^{75} +193876. i q^{76} +(365184. + 459936. i) q^{77} -626.829 q^{78} +176687. q^{79} -517515. i q^{80} +59049.0 q^{81} -7717.57i q^{82} +395381. i q^{83} +(-267925. + 212729. i) q^{84} +757651. q^{85} -11253.8 q^{86} -516307. i q^{87} -28475.0 q^{88} +795472. i q^{89} -3992.75i q^{90} +(-83125.0 + 66000.3i) q^{91} -26183.3 q^{92} +817510. q^{93} +23419.3i q^{94} -383147. q^{95} -24882.3i q^{96} -451838. i q^{97} +(3465.44 - 14889.9i) q^{98} +416063. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 10 q^{2} + 346 q^{4} + 28 q^{7} - 1462 q^{8} - 1944 q^{9}+O(q^{10}) $ 8 * q + 10 * q^2 + 346 * q^4 + 28 * q^7 - 1462 * q^8 - 1944 * q^9	$ 8 q + 10 q^{2} + 346 q^{4} + 28 q^{7} - 1462 q^{8} - 1944 q^{9} - 6848 q^{11} + 12082 q^{14} - 4536 q^{15} + 28466 q^{16} - 2430 q^{18} + 6804 q^{21} - 7764 q^{22} - 24320 q^{23} - 77056 q^{25} - 30142 q^{28} + 60496 q^{29} + 89424 q^{30} + 5082 q^{32} + 103656 q^{35} - 84078 q^{36} - 39112 q^{37} + 108864 q^{39} - 267624 q^{42} - 29272 q^{43} - 577884 q^{44} + 564972 q^{46} - 94864 q^{49} + 240154 q^{50} + 103032 q^{51} + 232288 q^{53} + 225722 q^{56} + 180792 q^{57} + 987684 q^{58} - 375192 q^{60} - 6804 q^{63} - 690734 q^{64} - 836304 q^{65} - 2163848 q^{67} - 366744 q^{70} - 506288 q^{71} + 355266 q^{72} + 512324 q^{74} + 1536304 q^{77} + 1272024 q^{78} - 93272 q^{79} + 472392 q^{81} - 653184 q^{84} + 3740760 q^{85} + 3846452 q^{86} - 2077548 q^{88} + 1890336 q^{91} - 9701580 q^{92} - 2153952 q^{93} + 4154832 q^{95} - 507542 q^{98} + 1664064 q^{99}+O(q^{100}) $ 8 * q + 10 * q^2 + 346 * q^4 + 28 * q^7 - 1462 * q^8 - 1944 * q^9 - 6848 * q^11 + 12082 * q^14 - 4536 * q^15 + 28466 * q^16 - 2430 * q^18 + 6804 * q^21 - 7764 * q^22 - 24320 * q^23 - 77056 * q^25 - 30142 * q^28 + 60496 * q^29 + 89424 * q^30 + 5082 * q^32 + 103656 * q^35 - 84078 * q^36 - 39112 * q^37 + 108864 * q^39 - 267624 * q^42 - 29272 * q^43 - 577884 * q^44 + 564972 * q^46 - 94864 * q^49 + 240154 * q^50 + 103032 * q^51 + 232288 * q^53 + 225722 * q^56 + 180792 * q^57 + 987684 * q^58 - 375192 * q^60 - 6804 * q^63 - 690734 * q^64 - 836304 * q^65 - 2163848 * q^67 - 366744 * q^70 - 506288 * q^71 + 355266 * q^72 + 512324 * q^74 + 1536304 * q^77 + 1272024 * q^78 - 93272 * q^79 + 472392 * q^81 - 653184 * q^84 + 3740760 * q^85 + 3846452 * q^86 - 2077548 * q^88 + 1890336 * q^91 - 9701580 * q^92 - 2153952 * q^93 + 4154832 * q^95 - 507542 * q^98 + 1664064 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$.

$n$	$8$	$10$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	−0.129945			−0.0162431			−0.00812155	−	0.999967i	$-0.502585\pi$
$2$	−0.129945			−0.0162431			−0.00812155	+	0.999967i	$0.502585\pi$
$3$			15.5885i			0.577350i
$3$			15.5885i			0.577350i
$4$	−63.9831			−0.999736
$5$		−	126.447i		−	1.01157i	−0.862659	−	0.505786i	$-0.831202\pi$
$5$		−	126.447i		−	1.01157i	0.862659	−	0.505786i	$-0.168798\pi$
$6$		−	2.02564i		−	0.00937796i
$7$	−213.284	−	268.624i	−0.621820	−	0.783160i
$7$	−213.284	−	268.624i	−0.621820	−	0.783160i
$8$	16.6307			0.0324819
$9$	−243.000			−0.333333
$10$			16.4311i			0.0164311i
$11$	−1712.19			−1.28640			−0.643198	−	0.765700i	$-0.722393\pi$
$11$	−1712.19			−1.28640			−0.643198	+	0.765700i	$0.722393\pi$
$12$		−	997.398i		−	0.577198i
$13$		−	309.447i		−	0.140850i	−0.997517	−	0.0704250i	$-0.977564\pi$
$13$		−	309.447i		−	0.140850i	0.997517	−	0.0704250i	$-0.0224355\pi$
$14$	27.7152	+	34.9063i	0.0101003	+	0.0127210i
$15$	1971.11			0.584032
$16$	4092.76			0.999209
$17$			5991.87i			1.21959i	0.792558	+	0.609797i	$0.208749\pi$
$17$			5991.87i			1.21959i	−0.792558	+	0.609797i	$0.791251\pi$
$18$	31.5766			0.00541437
$19$		−	3030.11i		−	0.441771i	−0.975300	−	0.220886i	$-0.929105\pi$
$19$		−	3030.11i		−	0.441771i	0.975300	−	0.220886i	$-0.0708948\pi$
$20$			8090.45i			1.01131i
$21$	4187.43	−	3324.77i	0.452158	−	0.359008i
$22$	222.491			0.0208951
$23$	409.222			0.0336338			0.0168169	−	0.999859i	$-0.494647\pi$
$23$	409.222			0.0336338			0.0168169	+	0.999859i	$0.494647\pi$
$24$			259.248i			0.0187535i
$25$	−363.744			−0.0232796
$26$			40.2111i			0.00228784i
$27$		−	3788.00i		−	0.192450i
$28$	13646.6	+	17187.4i	0.621656	+	0.782953i
$29$	−33121.1			−1.35804			−0.679018	−	0.734122i	$-0.737594\pi$
$29$	−33121.1			−1.35804			−0.679018	+	0.734122i	$0.737594\pi$
$30$	−256.135			−0.00948649
$31$		−	52443.3i		−	1.76037i	−0.474627	−	0.880187i	$-0.657417\pi$
$31$		−	52443.3i		−	1.76037i	0.474627	−	0.880187i	$-0.342583\pi$
$32$	−1596.20			−0.0487122
$33$		−	26690.4i		−	0.742701i
$34$		−	778.612i		−	0.0198100i
$35$	−33966.6	+	26969.1i	−0.792223	+	0.629017i
$36$	15547.9			0.333245
$37$	−54089.1			−1.06784			−0.533918	−	0.845536i	$-0.679281\pi$
$37$	−54089.1			−1.06784			−0.533918	+	0.845536i	$0.679281\pi$
$38$			393.747i			0.00717574i
$39$	4823.81			0.0813198
$40$		−	2102.90i		−	0.0328578i
$41$			59391.1i			0.861727i	0.902417	+	0.430863i	$0.141791\pi$
$41$			59391.1i			0.861727i	−0.902417	+	0.430863i	$0.858209\pi$
$42$	−544.135	+	432.037i	−0.00734444	+	0.00583141i
$43$	86604.1			1.08926			0.544632	−	0.838675i	$-0.316669\pi$
$43$	86604.1			1.08926			0.544632	+	0.838675i	$0.316669\pi$
$44$	109551.			1.28606
$45$			30726.5i			0.337191i
$46$	−53.1764			−0.000546318
$47$		−	180225.i		−	1.73589i	−0.496662	−	0.867944i	$-0.665441\pi$
$47$		−	180225.i		−	1.73589i	0.496662	−	0.867944i	$-0.334559\pi$
$48$			63799.8i			0.576893i
$49$	−26668.5	+	114587.i	−0.226679	+	0.973970i
$50$	47.2666			0.000378133
$51$	−93403.9			−0.704133
$52$			19799.4i			0.140813i
$53$	144578.			0.971127			0.485563	−	0.874202i	$-0.338615\pi$
$53$	144578.			0.971127			0.485563	+	0.874202i	$0.338615\pi$
$54$			492.231i			0.00312599i
$55$			216501.i			1.30128i
$56$	−3547.08	−	4467.42i	−0.0201979	−	0.0254385i
$57$	47234.7			0.255057
$58$	4303.92			0.0220587
$59$		−	74972.5i		−	0.365044i	−0.983202	−	0.182522i	$-0.941574\pi$
$59$		−	74972.5i		−	0.365044i	0.983202	−	0.182522i	$-0.0584261\pi$
$60$	−126118.			−0.583878
$61$		−	398720.i		−	1.75662i	−0.478090	−	0.878311i	$-0.658671\pi$
$61$		−	398720.i		−	1.75662i	0.478090	−	0.878311i	$-0.341329\pi$
$62$			6814.74i			0.0285940i
$63$	51828.1	+	65275.6i	0.207273	+	0.261053i
$64$	−261729.			−0.998417
$65$	−39128.6			−0.142480
$66$			3468.28i			0.0120638i
$67$	−126500.			−0.420596			−0.210298	−	0.977637i	$-0.567443\pi$
$67$	−126500.			−0.420596			−0.210298	+	0.977637i	$0.567443\pi$
$68$		−	383378.i		−	1.21927i
$69$			6379.15i			0.0194185i
$70$	4413.78	−	3504.49i	0.0128682	−	0.0102172i
$71$	−238387.			−0.666050			−0.333025	−	0.942918i	$-0.608069\pi$
$71$	−238387.			−0.666050			−0.333025	+	0.942918i	$0.608069\pi$
$72$	−4041.27			−0.0108273
$73$			125014.i			0.321359i	0.987007	+	0.160679i	$0.0513685\pi$
$73$			125014.i			0.321359i	−0.987007	+	0.160679i	$0.948632\pi$
$74$	7028.61			0.0173450
$75$		−	5670.20i		−	0.0134405i
$76$			193876.i			0.441655i
$77$	365184.	+	459936.i	0.799907	+	1.00745i
$78$	−626.829			−0.00132089
$79$	176687.			0.358363			0.179182	−	0.983816i	$-0.442655\pi$
$79$	176687.			0.358363			0.179182	+	0.983816i	$0.442655\pi$
$80$		−	517515.i		−	1.01077i
$81$	59049.0			0.111111
$82$		−	7717.57i		−	0.0139971i
$83$			395381.i			0.691483i	0.938330	+	0.345741i	$0.112373\pi$
$83$			395381.i			0.691483i	−0.938330	+	0.345741i	$0.887627\pi$
$84$	−267925.	+	212729.i	−0.452038	+	0.358913i
$85$	757651.			1.23371
$86$	−11253.8			−0.0176930
$87$		−	516307.i		−	0.784062i
$88$	−28475.0			−0.0417846
$89$			795472.i			1.12838i	0.825646	+	0.564189i	$0.190811\pi$
$89$			795472.i			1.12838i	−0.825646	+	0.564189i	$0.809189\pi$
$90$		−	3992.75i		−	0.00547703i
$91$	−83125.0	+	66000.3i	−0.110308	+	0.0875834i
$92$	−26183.3			−0.0336249
$93$	817510.			1.01635
$94$			23419.3i			0.0281962i
$95$	−383147.			−0.446884
$96$		−	24882.3i		−	0.0281240i
$97$		−	451838.i		−	0.495071i	−0.968879	−	0.247535i	$-0.920379\pi$
$97$		−	451838.i		−	0.495071i	0.968879	−	0.247535i	$-0.0796207\pi$
$98$	3465.44	−	14889.9i	0.00368197	−	0.0158203i
$99$	416063.			0.428798
$100$	23273.4			0.0232734
$101$			1.16635e6i			1.13205i	0.824389	+	0.566024i	$0.191519\pi$
$101$			1.16635e6i			1.13205i	−0.824389	+	0.566024i	$0.808481\pi$
$102$	12137.4			0.0114373
$103$		−	858846.i		−	0.785965i	−0.919546	−	0.392983i	$-0.871443\pi$
$103$		−	858846.i		−	0.785965i	0.919546	−	0.392983i	$-0.128557\pi$
$104$		−	5146.34i		−	0.00457508i
$105$	−420406.	−	529486.i	−0.363163	−	0.457390i
$106$	−18787.2			−0.0157741
$107$	−2.42759e6			−1.98164			−0.990818	−	0.135206i	$-0.956830\pi$
$107$	−2.42759e6			−1.98164			−0.990818	+	0.135206i	$0.956830\pi$
$108$			242368.i			0.192399i
$109$	6326.41			0.00488515			0.00244258	−	0.999997i	$-0.499223\pi$
$109$	6326.41			0.00488515			0.00244258	+	0.999997i	$0.499223\pi$
$110$		−	28133.2i		−	0.0211369i
$111$		−	843166.i		−	0.616516i
$112$	−872921.	−	1.09941e6i	−0.621328	−	0.782540i
$113$	−1.67712e6			−1.16233			−0.581165	−	0.813786i	$-0.697403\pi$
$113$	−1.67712e6			−1.16233			−0.581165	+	0.813786i	$0.697403\pi$
$114$	−6137.91			−0.00414291
$115$		−	51744.8i		−	0.0340230i
$116$	2.11919e6			1.35768
$117$			75195.7i			0.0469500i
$118$			9742.29i			0.00592946i
$119$	1.60956e6	−	1.27797e6i	0.955137	−	0.758368i
$120$	32781.0			0.0189705
$121$	1.16004e6			0.654812
$122$			51811.6i			0.0285330i
$123$	−925815.			−0.497518
$124$			3.35549e6i			1.75991i
$125$		−	1.92973e6i		−	0.988024i
$126$	−6734.80	−	8482.23i	−0.00336677	−	0.00424032i
$127$	−452109.			−0.220715			−0.110358	−	0.993892i	$-0.535200\pi$
$127$	−452109.			−0.220715			−0.110358	+	0.993892i	$0.535200\pi$
$128$	136167.			0.0649296
$129$			1.35002e6i			0.628887i
$130$	5084.56			0.00231432
$131$			1.49608e6i			0.665488i	0.943017	+	0.332744i	$0.107974\pi$
$131$			1.49608e6i			0.665488i	−0.943017	+	0.332744i	$0.892026\pi$
$132$			1.70774e6i			0.742505i
$133$	−813960.	+	646275.i	−0.345978	+	0.274702i
$134$	16438.0			0.00683179
$135$	−478979.			−0.194677
$136$			99649.2i			0.0396148i
$137$	1.87487e6			0.729137			0.364569	−	0.931177i	$-0.381216\pi$
$137$	1.87487e6			0.729137			0.364569	+	0.931177i	$0.381216\pi$
$138$		−	828.937i		−	0.000315417i
$139$		−	3.06381e6i		−	1.14082i	−0.821360	−	0.570411i	$-0.806784\pi$
$139$		−	3.06381e6i		−	1.14082i	0.821360	−	0.570411i	$-0.193216\pi$
$140$	2.17329e6	−	1.72557e6i	0.792014	−	0.628851i
$141$	2.80943e6			1.00222
$142$	30977.1			0.0108187
$143$			529833.i			0.181189i
$144$	−994540.			−0.333070
$145$			4.18805e6i			1.37375i
$146$		−	16244.9i		−	0.00521986i
$147$	−1.78623e6	−	415721.i	−0.562322	−	0.130873i
$148$	3.46079e6			1.06756
$149$	3.88149e6			1.17338			0.586691	−	0.809811i	$-0.300430\pi$
$149$	3.88149e6			1.17338			0.586691	+	0.809811i	$0.300430\pi$
$150$			736.813i			0.000218315i
$151$	−3.68952e6			−1.07162			−0.535808	−	0.844340i	$-0.679993\pi$
$151$	−3.68952e6			−1.07162			−0.535808	+	0.844340i	$0.679993\pi$
$152$		−	50393.0i		−	0.0143496i
$153$		−	1.45602e6i		−	0.406531i
$154$	−47453.8	−	59766.3i	−0.0129930	−	0.0163642i
$155$	−6.63128e6			−1.78075
$156$	−308642.			−0.0812983
$157$		−	1.87601e6i		−	0.484771i	−0.970180	−	0.242385i	$-0.922070\pi$
$157$		−	1.87601e6i		−	0.484771i	0.970180	−	0.242385i	$-0.0779298\pi$
$158$	−22959.6			−0.00582093
$159$			2.25375e6i			0.560680i
$160$			201834.i			0.0492759i
$161$	−87280.8	−	109927.i	−0.0209142	−	0.0263406i
$162$	−7673.11			−0.00180479
$163$	−3.26117e6			−0.753028			−0.376514	−	0.926411i	$-0.622877\pi$
$163$	−3.26117e6			−0.753028			−0.376514	+	0.926411i	$0.622877\pi$
$164$		−	3.80003e6i		−	0.861499i
$165$	−3.37491e6			−0.751296
$166$		−	51377.7i		−	0.0112318i
$167$		−	6.64462e6i		−	1.42666i	−0.700827	−	0.713331i	$-0.747186\pi$
$167$		−	6.64462e6i		−	1.42666i	0.700827	−	0.713331i	$-0.252814\pi$
$168$	69640.1	−	55293.5i	0.0146870	−	0.0116613i
$169$	4.73105e6			0.980161
$170$	−98452.9			−0.0200393
$171$			736317.i			0.147257i
$172$	−5.54120e6			−1.08898
$173$			2.18353e6i			0.421717i	0.977517	+	0.210859i	$0.0676260\pi$
$173$			2.18353e6i			0.421717i	−0.977517	+	0.210859i	$0.932374\pi$
$174$			67091.5i			0.0127356i
$175$	77580.8	+	97710.2i	0.0144757	+	0.0182316i
$176$	−7.00759e6			−1.28538
$177$	1.16870e6			0.210758
$178$		−	103367.i		−	0.0183284i
$179$	1.07228e7			1.86960			0.934800	−	0.355175i	$-0.115579\pi$
$179$	1.07228e7			1.86960			0.934800	+	0.355175i	$0.115579\pi$
$180$		−	1.96598e6i		−	0.337102i
$181$			3.15432e6i			0.531950i	0.963980	+	0.265975i	$0.0856938\pi$
$181$			3.15432e6i			0.531950i	−0.963980	+	0.265975i	$0.914306\pi$
$182$	10801.7	−	8576.40i	0.00179175	−	0.00142263i
$183$	6.21543e6			1.01419
$184$	6805.68			0.00109249
$185$			6.83939e6i			1.08019i
$186$	−106231.			−0.0165087
$187$		−	1.02592e7i		−	1.56888i
$188$			1.15314e7i			1.73543i
$189$	−1.01755e6	+	807920.i	−0.150719	+	0.119669i
$190$	49788.0			0.00725878
$191$	−2.52981e6			−0.363068			−0.181534	−	0.983385i	$-0.558106\pi$
$191$	−2.52981e6			−0.363068			−0.181534	+	0.983385i	$0.558106\pi$
$192$		−	4.07995e6i		−	0.576437i
$193$	8.41552e6			1.17060			0.585300	−	0.810817i	$-0.300977\pi$
$193$	8.41552e6			1.17060			0.585300	+	0.810817i	$0.300977\pi$
$194$			58714.0i			0.00804149i
$195$		−	609954.i		−	0.0822609i
$196$	1.70634e6	−	7.33160e6i	0.226619	−	0.973713i
$197$	−1.18968e7			−1.55608			−0.778040	−	0.628214i	$-0.783786\pi$
$197$	−1.18968e7			−1.55608			−0.778040	+	0.628214i	$0.783786\pi$
$198$	−54065.2			−0.00696502
$199$			1.39371e7i			1.76853i	0.466986	+	0.884265i	$0.345340\pi$
$199$			1.39371e7i			1.76853i	−0.466986	+	0.884265i	$0.654660\pi$
$200$	−6049.33			−0.000756166
$201$		−	1.97194e6i		−	0.242831i
$202$		−	151561.i		−	0.0183880i
$203$	7.06422e6	+	8.89712e6i	0.844454	+	1.06356i
$204$	5.97628e6			0.703947
$205$	7.50980e6			0.871699
$206$			111603.i			0.0127665i
$207$	−99441.1			−0.0112113
$208$		−	1.26649e6i		−	0.140739i
$209$			5.18813e6i			0.568292i
$210$	54629.7	+	68804.0i	0.00589889	+	0.00742944i
$211$	6.73451e6			0.716900			0.358450	−	0.933549i	$-0.383305\pi$
$211$	6.73451e6			0.716900			0.358450	+	0.933549i	$0.383305\pi$
$212$	−9.25058e6			−0.970870
$213$		−	3.71608e6i		−	0.384544i
$214$	315453.			0.0321879
$215$		−	1.09508e7i		−	1.10187i
$216$		−	62997.2i		−	0.00625115i
$217$	−1.40875e7	+	1.11853e7i	−1.37865	+	1.09464i
$218$	−822.085			−7.93501e−5
$219$	−1.94878e6			−0.185537
$220$		−	1.38524e7i		−	1.30094i
$221$	1.85417e6			0.171780
$222$			109565.i			0.0100141i
$223$			110295.i			0.00994586i	0.999988	+	0.00497293i	$0.00158294\pi$
$223$			110295.i			0.00994586i	−0.999988	+	0.00497293i	$0.998417\pi$
$224$	340445.	+	428778.i	0.0302902	+	0.0381494i
$225$	88389.7			0.00775986
$226$	217934.			0.0188799
$227$		−	1.45179e7i		−	1.24115i	−0.784146	−	0.620577i	$-0.786898\pi$
$227$		−	1.45179e7i		−	1.24115i	0.784146	−	0.620577i	$-0.213102\pi$
$228$	−3.02222e6			−0.254989
$229$		−	1.47979e7i		−	1.23224i	−0.787654	−	0.616118i	$-0.788704\pi$
$229$		−	1.47979e7i		−	1.23224i	0.787654	−	0.616118i	$-0.211296\pi$
$230$			6723.97i			0.000552640i
$231$	−7.16969e6	+	5.69265e6i	−0.581653	+	0.461826i
$232$	−550829.			−0.0441116
$233$	−2.43183e6			−0.192249			−0.0961247	−	0.995369i	$-0.530645\pi$
$233$	−2.43183e6			−0.192249			−0.0961247	+	0.995369i	$0.530645\pi$
$234$		−	9771.30i		−	0.000762614i
$235$	−2.27889e7			−1.75598
$236$			4.79697e6i			0.364948i
$237$			2.75428e6i			0.206901i
$238$	−209154.	+	166066.i	−0.0155144	+	0.0123183i
$239$	1.31161e6			0.0960750			0.0480375	−	0.998846i	$-0.484703\pi$
$239$	1.31161e6			0.0960750			0.0480375	+	0.998846i	$0.484703\pi$
$240$	8.06727e6			0.583570
$241$		−	5.31020e6i		−	0.379367i	−0.981845	−	0.189684i	$-0.939254\pi$
$241$		−	5.31020e6i		−	0.379367i	0.981845	−	0.189684i	$-0.0607462\pi$
$242$	−150741.			−0.0106362
$243$			920483.i			0.0641500i
$244$			2.55113e7i			1.75616i
$245$	1.44891e7	+	3.37215e6i	0.985241	+	0.229302i
$246$	120305.			0.00808124
$247$	−937659.			−0.0622235
$248$		−	872172.i		−	0.0571804i
$249$	−6.16338e6			−0.399228
$250$			250759.i			0.0160486i
$251$		−	1.11880e7i		−	0.707511i	−0.935338	−	0.353756i	$-0.884904\pi$
$251$		−	1.11880e7i		−	0.707511i	0.935338	−	0.353756i	$-0.115096\pi$
$252$	−3.31612e6	−	4.17654e6i	−0.207219	−	0.260984i
$253$	−700667.			−0.0432664
$254$	58749.2			0.00358510
$255$			1.18106e7i			0.712282i
$256$	1.67330e7			0.997363
$257$		−	2.38951e7i		−	1.40770i	−0.710350	−	0.703848i	$-0.751464\pi$
$257$		−	2.38951e7i		−	1.40770i	0.710350	−	0.703848i	$-0.248536\pi$
$258$		−	175429.i		−	0.0102151i
$259$	1.15364e7	+	1.45296e7i	0.664003	+	0.836287i
$260$	2.50357e6			0.142442
$261$	8.04843e6			0.452679
$262$		−	194408.i		−	0.0108096i
$263$	1.18819e7			0.653156			0.326578	−	0.945170i	$-0.394104\pi$
$263$	1.18819e7			0.653156			0.326578	+	0.945170i	$0.394104\pi$
$264$		−	443882.i		−	0.0241244i
$265$		−	1.82814e7i		−	0.982365i
$266$	105770.	−	83980.1i	0.00561975	−	0.00446202i
$267$	−1.24002e7			−0.651469
$268$	8.09385e6			0.420485
$269$			6.86914e6i			0.352895i	0.984310	+	0.176448i	$0.0564606\pi$
$269$			6.86914e6i			0.352895i	−0.984310	+	0.176448i	$0.943539\pi$
$270$	62240.9			0.00316216
$271$			2.60681e7i			1.30979i	0.755720	+	0.654895i	$0.227287\pi$
$271$			2.60681e7i			1.30979i	−0.755720	+	0.654895i	$0.772713\pi$
$272$			2.45233e7i			1.21863i
$273$	−1.02884e6	−	1.29579e6i	−0.0505663	−	0.0636864i
$274$	−243630.			−0.0118435
$275$	622799.			0.0299467
$276$		−	408158.i		−	0.0194134i
$277$	−1.38886e7			−0.653462			−0.326731	−	0.945117i	$-0.605947\pi$
$277$	−1.38886e7			−0.653462			−0.326731	+	0.945117i	$0.605947\pi$
$278$			398127.i			0.0185305i
$279$			1.27437e7i			0.586791i
$280$	−564890.	+	448516.i	−0.0257329	+	0.0204317i
$281$	175938.			0.00792942			0.00396471	−	0.999992i	$-0.498738\pi$
$281$	175938.			0.00792942			0.00396471	+	0.999992i	$0.498738\pi$
$282$	−365071.			−0.0162791
$283$			7.28129e6i			0.321254i	0.987015	+	0.160627i	$0.0513517\pi$
$283$			7.28129e6i			0.321254i	−0.987015	+	0.160627i	$0.948648\pi$
$284$	1.52527e7			0.665875
$285$		−	5.97267e6i		−	0.258008i
$286$		−	68849.1i		−	0.00294307i
$287$	1.59539e7	−	1.26672e7i	0.674870	−	0.535839i
$288$	387877.			0.0162374
$289$	−1.17649e7			−0.487410
$290$		−	544216.i		−	0.0223140i
$291$	7.04345e6			0.285829
$292$		−	7.99878e6i		−	0.321274i
$293$		−	3.01379e7i		−	1.19815i	−0.800693	−	0.599074i	$-0.795535\pi$
$293$		−	3.01379e7i		−	1.19815i	0.800693	−	0.599074i	$-0.204465\pi$
$294$	232111.	+	54020.8i	0.00913385	+	0.00212579i
$295$	−9.48001e6			−0.369269
$296$	−899543.			−0.0346854
$297$			6.48577e6i			0.247567i
$298$	−504380.			−0.0190594
$299$		−	126633.i		−	0.00473732i
$300$			362797.i			0.0134369i
$301$	−1.84713e7	−	2.32639e7i	−0.677326	−	0.853068i
$302$	479435.			0.0174064
$303$	−1.81816e7			−0.653588
$304$		−	1.24015e7i		−	0.441422i
$305$	−5.04168e7			−1.77695
$306$			189203.i			0.00660333i
$307$		−	2.03724e7i		−	0.704088i	−0.935983	−	0.352044i	$-0.885487\pi$
$307$		−	2.03724e7i		−	0.704088i	0.935983	−	0.352044i	$-0.114513\pi$
$308$	−2.33656e7	−	2.94281e7i	−0.799696	−	1.00719i
$309$	1.33881e7			0.453777
$310$	861701.			0.0289249
$311$			3.95263e7i			1.31403i	0.753878	+	0.657014i	$0.228181\pi$
$311$			3.95263e7i			1.31403i	−0.753878	+	0.657014i	$0.771819\pi$
$312$	80223.5			0.00264142
$313$			1.96234e7i			0.639944i	0.947427	+	0.319972i	$0.103674\pi$
$313$			1.96234e7i			0.639944i	−0.947427	+	0.319972i	$0.896326\pi$
$314$			243778.i			0.00787418i
$315$	8.25388e6	−	6.55349e6i	0.264074	−	0.209672i
$316$	−1.13050e7			−0.358269
$317$	−4.23585e6			−0.132973			−0.0664864	−	0.997787i	$-0.521179\pi$
$317$	−4.23585e6			−0.132973			−0.0664864	+	0.997787i	$0.521179\pi$
$318$		−	292864.i		−	0.00910719i
$319$	5.67097e7			1.74697
$320$			3.30948e7i			1.00997i
$321$		−	3.78424e7i		−	1.14410i
$322$	11341.7	+	14284.4i	0.000339711	+	0.000427854i
$323$	1.81560e7			0.538782
$324$	−3.77814e6			−0.111082
$325$			112559.i			0.00327893i
$326$	423773.			0.0122315
$327$			98619.0i			0.00282044i
$328$			987718.i			0.0279906i
$329$	−4.84128e7	+	3.84392e7i	−1.35948	+	1.07941i
$330$	438553.			0.0122034
$331$	1.00966e7			0.278414			0.139207	−	0.990263i	$-0.455545\pi$
$331$	1.00966e7			0.278414			0.139207	+	0.990263i	$0.455545\pi$
$332$		−	2.52977e7i		−	0.691300i
$333$	1.31437e7			0.355946
$334$			863435.i			0.0231734i
$335$			1.59955e7i			0.425464i
$336$	1.71381e7	−	1.36075e7i	0.451800	−	0.358724i
$337$	3.34900e7			0.875035			0.437517	−	0.899210i	$-0.355858\pi$
$337$	3.34900e7			0.875035			0.437517	+	0.899210i	$0.355858\pi$
$338$	−614776.			−0.0159209
$339$		−	2.61438e7i		−	0.671072i
$340$	−4.84769e7			−1.23338
$341$			8.97930e7i			2.26454i
$342$		−	95680.6i		−	0.00239191i
$343$	3.64687e7	−	1.72757e7i	0.903727	−	0.428108i
$344$	1.44029e6			0.0353814
$345$	806621.			0.0196432
$346$		−	283739.i		−	0.00685000i
$347$	5.89372e6			0.141059			0.0705296	−	0.997510i	$-0.477531\pi$
$347$	5.89372e6			0.141059			0.0705296	+	0.997510i	$0.477531\pi$
$348$			3.30349e7i			0.783855i
$349$		−	3.16743e7i		−	0.745128i	−0.928007	−	0.372564i	$-0.878479\pi$
$349$		−	3.16743e7i		−	0.745128i	0.928007	−	0.372564i	$-0.121521\pi$
$350$	−10081.2	−	12696.9i	−0.000235131	−	0.000296138i
$351$	−1.17219e6			−0.0271066
$352$	2.73300e6			0.0626631
$353$		−	3.77824e6i		−	0.0858945i	−0.999077	−	0.0429473i	$-0.986325\pi$
$353$		−	3.77824e6i		−	0.0858945i	0.999077	−	0.0429473i	$-0.0136747\pi$
$354$	−151867.			−0.00342337
$355$			3.01432e7i			0.673758i
$356$		−	5.08967e7i		−	1.12808i
$357$	1.99216e7	+	2.50905e7i	0.437844	+	0.551449i
$358$	−1.39337e6			−0.0303681
$359$	6.52661e7			1.41060			0.705300	−	0.708909i	$-0.250812\pi$
$359$	6.52661e7			1.41060			0.705300	+	0.708909i	$0.250812\pi$
$360$			511005.i			0.0109526i
$361$	3.78643e7			0.804838
$362$		−	409888.i		−	0.00864051i
$363$			1.80832e7i			0.378056i
$364$	5.31859e6	−	4.22291e6i	0.110279	−	0.0875603i
$365$	1.58076e7			0.325078
$366$	−807663.			−0.0164735
$367$			5.40939e7i			1.09433i	0.837024	+	0.547167i	$0.184294\pi$
$367$			5.40939e7i			1.09433i	−0.837024	+	0.547167i	$0.815706\pi$
$368$	1.67485e6			0.0336072
$369$		−	1.44320e7i		−	0.287242i
$370$		−	888743.i		−	0.0175457i
$371$	−3.08363e7	−	3.88372e7i	−0.603866	−	0.760547i
$372$	−5.23069e7			−1.01608
$373$	−8.24014e7			−1.58784			−0.793922	−	0.608019i	$-0.791964\pi$
$373$	−8.24014e7			−1.58784			−0.793922	+	0.608019i	$0.791964\pi$
$374$			1.33313e6i			0.0254835i
$375$	3.00816e7			0.570436
$376$		−	2.99728e6i		−	0.0563850i
$377$			1.02492e7i			0.191279i
$378$	132225.	−	104985.i	0.00244815	−	0.00194380i
$379$	−7.77855e7			−1.42883			−0.714415	−	0.699722i	$-0.753307\pi$
$379$	−7.77855e7			−1.42883			−0.714415	+	0.699722i	$0.753307\pi$
$380$	2.45149e7			0.446766
$381$		−	7.04768e6i		−	0.127430i
$382$	328736.			0.00589736
$383$		−	4.27625e7i		−	0.761145i	−0.924751	−	0.380572i	$-0.875727\pi$
$383$		−	4.27625e7i		−	0.761145i	0.924751	−	0.380572i	$-0.124273\pi$
$384$			2.12264e6i			0.0374871i
$385$	5.81573e7	−	4.61763e7i	1.01911	−	0.809164i
$386$	−1.09355e6			−0.0190142
$387$	−2.10448e7			−0.363088
$388$			2.89100e7i			0.494940i
$389$	−9.61157e7			−1.63285			−0.816423	−	0.577454i	$-0.804046\pi$
$389$	−9.61157e7			−1.63285			−0.816423	+	0.577454i	$0.804046\pi$
$390$			79260.4i			0.00133617i
$391$			2.45201e6i			0.0410196i
$392$	−443518.	+	1.90566e6i	−0.00736297	+	0.0316364i
$393$	−2.33215e7			−0.384220
$394$	1.54593e6			0.0252756
$395$		−	2.23415e7i		−	0.362511i
$396$	−2.66210e7			−0.428685
$397$		−	3.49802e7i		−	0.559050i	−0.960138	−	0.279525i	$-0.909823\pi$
$397$		−	3.49802e7i		−	0.559050i	0.960138	−	0.279525i	$-0.0901770\pi$
$398$		−	1.81105e6i		−	0.0287264i
$399$	−1.00744e7	−	1.26884e7i	−0.158599	−	0.199750i
$400$	−1.48871e6			−0.0232612
$401$	1.38216e7			0.214350			0.107175	−	0.994240i	$-0.465819\pi$
$401$	1.38216e7			0.214350			0.107175	+	0.994240i	$0.465819\pi$
$402$			256243.i			0.00394434i
$403$	−1.62284e7			−0.247949
$404$		−	7.46267e7i		−	1.13175i
$405$		−	7.46655e6i		−	0.112397i
$406$	−917959.	−	1.15614e6i	−0.0137166	−	0.0172755i
$407$	9.26110e7			1.37366
$408$	−1.55338e6			−0.0228716
$409$		−	2.17072e7i		−	0.317273i	−0.987337	−	0.158637i	$-0.949290\pi$
$409$		−	2.17072e7i		−	0.317273i	0.987337	−	0.158637i	$-0.0507098\pi$
$410$	−975860.			−0.0141591
$411$			2.92263e7i			0.420967i
$412$			5.49516e7i			0.785758i
$413$	−2.01394e7	+	1.59905e7i	−0.285888	+	0.226992i
$414$	12921.9			0.000182106
$415$	4.99946e7			0.699485
$416$			493940.i			0.00686111i
$417$	4.77601e7			0.658653
$418$		−	674171.i		−	0.00923084i
$419$		−	1.45726e7i		−	0.198105i	−0.995082	−	0.0990526i	$-0.968419\pi$
$419$		−	1.45726e7i		−	0.198105i	0.995082	−	0.0990526i	$-0.0315812\pi$
$420$	2.68989e7	+	3.38782e7i	0.363067	+	0.457270i
$421$	−3.30617e7			−0.443076			−0.221538	−	0.975152i	$-0.571108\pi$
$421$	−3.30617e7			−0.443076			−0.221538	+	0.975152i	$0.571108\pi$
$422$	−875114.			−0.0116447
$423$			4.37947e7i			0.578630i
$424$	2.40445e6			0.0315441
$425$		−	2.17950e6i		−	0.0283916i
$426$			482886.i			0.00624620i
$427$	−1.07106e8	+	8.50407e7i	−1.37572	+	1.09230i
$428$	1.55325e8			1.98111
$429$	−8.25928e6			−0.104609
$430$			1.42300e6i			0.0178978i
$431$	−6.69583e7			−0.836320			−0.418160	−	0.908373i	$-0.637325\pi$
$431$	−6.69583e7			−0.836320			−0.418160	+	0.908373i	$0.637325\pi$
$432$		−	1.55033e7i		−	0.192298i
$433$			5.25595e7i			0.647422i	0.946156	+	0.323711i	$0.104931\pi$
$433$			5.25595e7i			0.647422i	−0.946156	+	0.323711i	$0.895069\pi$
$434$	1.83060e6	−	1.45348e6i	0.0223936	−	0.0177803i
$435$	−6.52853e7			−0.793136
$436$	−404784.			−0.00488386
$437$		−	1.23999e6i		−	0.0148584i
$438$	253233.			0.00301369
$439$		−	5.71226e7i		−	0.675172i	−0.941295	−	0.337586i	$-0.890390\pi$
$439$		−	5.71226e7i		−	0.675172i	0.941295	−	0.337586i	$-0.109610\pi$
$440$			3.60057e6i			0.0422682i
$441$	6.48045e6	−	2.78445e7i	0.0755596	−	0.324657i
$442$	−240940.			−0.00279024
$443$	−1.79007e7			−0.205901			−0.102950	−	0.994686i	$-0.532828\pi$
$443$	−1.79007e7			−0.205901			−0.102950	+	0.994686i	$0.532828\pi$
$444$			5.39484e7i			0.616353i
$445$	1.00585e8			1.14144
$446$		−	14332.3i		−	0.000161552i
$447$			6.05064e7i			0.677453i
$448$	5.58227e7	+	7.03067e7i	0.620836	+	0.781920i
$449$	9.24309e7			1.02112			0.510562	−	0.859841i	$-0.329437\pi$
$449$	9.24309e7			1.02112			0.510562	+	0.859841i	$0.329437\pi$
$450$	−11485.8			−0.000126044
$451$		−	1.01689e8i		−	1.10852i
$452$	1.07308e8			1.16202
$453$		−	5.75140e7i		−	0.618698i
$454$			1.88652e6i			0.0201602i
$455$	8.34551e6	+	1.05109e7i	0.0885970	+	0.111585i
$456$	785549.			0.00828474
$457$	7.68645e7			0.805336			0.402668	−	0.915346i	$-0.368083\pi$
$457$	7.68645e7			0.805336			0.402668	+	0.915346i	$0.368083\pi$
$458$			1.92291e6i			0.0200153i
$459$	2.26972e7			0.234711
$460$			3.31079e6i			0.0340141i
$461$			1.48359e8i			1.51429i	0.653245	+	0.757146i	$0.273407\pi$
$461$			1.48359e8i			1.51429i	−0.653245	+	0.757146i	$0.726593\pi$
$462$	931664.	−	739731.i	0.00944786	−	0.00750150i
$463$	−9.55853e7			−0.963048			−0.481524	−	0.876433i	$-0.659917\pi$
$463$	−9.55853e7			−0.963048			−0.481524	+	0.876433i	$0.659917\pi$
$464$	−1.35557e8			−1.35696
$465$		−	1.03371e8i		−	1.02811i
$466$	316003.			0.00312273
$467$		−	1.17781e8i		−	1.15645i	−0.815879	−	0.578223i	$-0.803746\pi$
$467$		−	1.17781e8i		−	1.15645i	0.815879	−	0.578223i	$-0.196254\pi$
$468$		−	4.81126e6i		−	0.0469376i
$469$	2.69804e7	+	3.39809e7i	0.261535	+	0.329394i
$470$	2.96130e6			0.0285225
$471$	2.92441e7			0.279882
$472$		−	1.24685e6i		−	0.0118573i
$473$	−1.48283e8			−1.40122
$474$		−	357904.i		−	0.00336072i
$475$			1.10218e6i			0.0102843i
$476$	−1.02985e8	+	8.17686e7i	−0.954885	+	0.758168i
$477$	−3.51326e7			−0.323709
$478$	−170437.			−0.00156056
$479$		−	6.53373e7i		−	0.594504i	−0.954799	−	0.297252i	$-0.903930\pi$
$479$		−	6.53373e7i		−	0.594504i	0.954799	−	0.297252i	$-0.0960701\pi$
$480$	−3.14628e6			−0.0284495
$481$			1.67377e7i			0.150405i
$482$			690033.i			0.00616210i
$483$	1.71359e6	−	1.36057e6i	0.0152078	−	0.0120748i
$484$	−7.42230e7			−0.654640
$485$	−5.71334e7			−0.500800
$486$		−	119612.i		−	0.00104200i
$487$	4.16652e7			0.360734			0.180367	−	0.983599i	$-0.442271\pi$
$487$	4.16652e7			0.360734			0.180367	+	0.983599i	$0.442271\pi$
$488$		−	6.63101e6i		−	0.0570585i
$489$		−	5.08367e7i		−	0.434761i
$490$	−1.88278e6	−	438193.i	−0.0160034	−	0.00372458i
$491$	−1.21529e8			−1.02668			−0.513340	−	0.858186i	$-0.671592\pi$
$491$	−1.21529e8			−1.02668			−0.513340	+	0.858186i	$0.671592\pi$
$492$	5.92365e7			0.497387
$493$		−	1.98457e8i		−	1.65625i
$494$	121844.			0.00101070
$495$		−	5.26097e7i		−	0.433761i
$496$		−	2.14638e8i		−	1.75898i
$497$	5.08442e7	+	6.40364e7i	0.414164	+	0.521624i
$498$	800899.			0.00648470
$499$	−1.61827e8			−1.30241			−0.651207	−	0.758900i	$-0.725737\pi$
$499$	−1.61827e8			−1.30241			−0.651207	+	0.758900i	$0.725737\pi$
$500$			1.23470e8i			0.987763i
$501$	1.03579e8			0.823683
$502$			1.45383e6i			0.0114922i
$503$			9.31731e7i			0.732127i	0.930590	+	0.366064i	$0.119295\pi$
$503$			9.31731e7i			0.732127i	−0.930590	+	0.366064i	$0.880705\pi$
$504$	861940.	+	1.08558e6i	0.00673264	+	0.00847952i
$505$	1.47481e8			1.14515
$506$	91048.1			0.000702780
$507$			7.37498e7i			0.565896i
$508$	2.89273e7			0.220657
$509$			3.50371e7i			0.265689i	0.991137	+	0.132845i	$0.0424112\pi$
$509$			3.50371e7i			0.265689i	−0.991137	+	0.132845i	$0.957589\pi$
$510$		−	1.53473e6i		−	0.0115697i
$511$	3.35817e7	−	2.66635e7i	0.251675	−	0.199827i
$512$	−1.08891e7			−0.0811299
$513$	−1.14780e7			−0.0850189
$514$			3.10504e6i			0.0228654i
$515$	−1.08598e8			−0.795061
$516$		−	8.63787e7i		−	0.628721i
$517$			3.08580e8i			2.23304i
$518$	−1.49909e6	−	1.88805e6i	−0.0107855	−	0.0135839i
$519$	−3.40379e7			−0.243479
$520$	−650738.			−0.00462803
$521$			7.99145e7i			0.565083i	0.959255	+	0.282542i	$0.0911775\pi$
$521$			7.99145e7i			0.565083i	−0.959255	+	0.282542i	$0.908822\pi$
$522$	−1.04585e6			−0.00735291
$523$		−	2.05964e8i		−	1.43975i	−0.694104	−	0.719874i	$-0.744199\pi$
$523$		−	2.05964e8i		−	1.43975i	0.694104	−	0.719874i	$-0.255801\pi$
$524$		−	9.57237e7i		−	0.665312i
$525$	−1.52315e6	+	1.20937e6i	−0.0105260	+	0.00835756i
$526$	−1.54399e6			−0.0106093
$527$	3.14233e8			2.14694
$528$		−	1.09237e8i		−	0.742113i
$529$	−1.47868e8			−0.998869
$530$			2.37558e6i			0.0159567i
$531$			1.82183e7i			0.121681i
$532$	5.20797e7	−	4.13507e7i	0.345886	−	0.274630i
$533$	1.83784e7			0.121374
$534$	1.61134e6			0.0105819
$535$			3.06960e8i			2.00457i
$536$	−2.10379e6			−0.0136618
$537$			1.67152e8i			1.07941i
$538$		−	892610.i		−	0.00573211i
$539$	4.56616e7	−	1.96194e8i	0.291598	−	1.25291i
$540$	3.06466e7			0.194626
$541$	1.83229e8			1.15718			0.578591	−	0.815618i	$-0.303603\pi$
$541$	1.83229e8			1.15718			0.578591	+	0.815618i	$0.303603\pi$
$542$		−	3.38741e6i		−	0.0212750i
$543$	−4.91710e7			−0.307121
$544$		−	9.56422e6i		−	0.0594091i
$545$		−	799954.i		−	0.00494169i
$546$	133693.	+	168381.i	0.000821354	+	0.00103446i
$547$	1.16700e8			0.713030			0.356515	−	0.934290i	$-0.383965\pi$
$547$	1.16700e8			0.713030			0.356515	+	0.934290i	$0.383965\pi$
$548$	−1.19960e8			−0.728945
$549$			9.68889e7i			0.585541i
$550$	−80929.5			−0.000486428
$551$			1.00361e8i			0.599941i
$552$			106090.i			0.000630750i
$553$	−3.76846e7	−	4.74624e7i	−0.222838	−	0.280656i
$554$	1.80476e6			0.0106143
$555$	−1.06616e8			−0.623651
$556$			1.96032e8i			1.14052i
$557$	2.15387e8			1.24639			0.623194	−	0.782067i	$-0.285835\pi$
$557$	2.15387e8			1.24639			0.623194	+	0.782067i	$0.285835\pi$
$558$		−	1.65598e6i		−	0.00953132i
$559$		−	2.67994e7i		−	0.153423i
$560$	−1.39017e8	+	1.10378e8i	−0.791596	+	0.628519i
$561$	1.59925e8			0.905793
$562$	−22862.3			−0.000128798
$563$		−	2.02663e8i		−	1.13566i	−0.823144	−	0.567832i	$-0.807782\pi$
$563$		−	2.02663e8i		−	1.13566i	0.823144	−	0.567832i	$-0.192218\pi$
$564$	−1.79756e8			−1.00195
$565$			2.12067e8i			1.17578i
$566$		−	946166.i		−	0.00521817i
$567$	−1.25942e7	−	1.58620e7i	−0.0690912	−	0.0870178i
$568$	−3.96455e6			−0.0216346
$569$	1.01282e8			0.549786			0.274893	−	0.961475i	$-0.411358\pi$
$569$	1.01282e8			0.549786			0.274893	+	0.961475i	$0.411358\pi$
$570$			776118.i			0.00419086i
$571$	1.03141e7			0.0554015			0.0277007	−	0.999616i	$-0.491181\pi$
$571$	1.03141e7			0.0554015			0.0277007	+	0.999616i	$0.491181\pi$
$572$		−	3.39004e7i		−	0.181141i
$573$		−	3.94359e7i		−	0.209618i
$574$	−2.07312e6	+	1.64604e6i	−0.0109620	+	0.00870370i
$575$	−148852.			−0.000782981
$576$	6.36002e7			0.332806
$577$		−	1.87163e8i		−	0.974298i	−0.873319	−	0.487149i	$-0.838037\pi$
$577$		−	1.87163e8i		−	0.974298i	0.873319	−	0.487149i	$-0.161963\pi$
$578$	1.52879e6			0.00791705
$579$			1.31185e8i			0.675847i
$580$		−	2.67965e8i		−	1.37339i
$581$	1.06209e8	−	8.43286e7i	0.541542	−	0.429978i
$582$	−915261.			−0.00464276
$583$	−2.47546e8			−1.24925
$584$			2.07908e6i			0.0104384i
$585$	9.50824e6			0.0474933
$586$			3.91627e6i			0.0194617i
$587$		−	2.89983e8i		−	1.43370i	−0.697226	−	0.716851i	$-0.745583\pi$
$587$		−	2.89983e8i		−	1.43370i	0.697226	−	0.716851i	$-0.254417\pi$
$588$	1.14288e8	+	2.65991e7i	0.562173	+	0.130839i
$589$	−1.58909e8			−0.777683
$590$	1.23188e6			0.00599808
$591$		−	1.85453e8i		−	0.898404i
$592$	−2.21374e8			−1.06699
$593$			2.00338e8i			0.960725i	0.877070	+	0.480362i	$0.159495\pi$
$593$			2.00338e8i			0.960725i	−0.877070	+	0.480362i	$0.840505\pi$
$594$		−	842793.i		−	0.00402126i
$595$	−1.61595e8	−	2.03523e8i	−0.767145	−	0.966191i
$596$	−2.48350e8			−1.17307
$597$	−2.17257e8			−1.02106
$598$			16455.3i			7.69488e-5i
$599$	9.02892e7			0.420103			0.210051	−	0.977690i	$-0.432637\pi$
$599$	9.02892e7			0.420103			0.210051	+	0.977690i	$0.432637\pi$
$600$		−	94299.7i		−	0.000436573i
$601$			1.39581e7i			0.0642989i	0.999483	+	0.0321495i	$0.0102353\pi$
$601$			1.39581e7i			0.0642989i	−0.999483	+	0.0321495i	$0.989765\pi$
$602$	2.40025e6	+	3.02303e6i	0.0110019	+	0.0138565i
$603$	3.07394e7			0.140199
$604$	2.36067e8			1.07133
$605$		−	1.46683e8i		−	0.662391i
$606$	2.36261e6			0.0106163
$607$			2.43169e8i			1.08728i	0.839318	+	0.543640i	$0.182954\pi$
$607$			2.43169e8i			1.08728i	−0.839318	+	0.543640i	$0.817046\pi$
$608$			4.83666e6i			0.0215196i
$609$	−1.38692e8	+	1.10120e8i	−0.614046	+	0.487546i
$610$	6.55140e6			0.0288632
$611$	−5.57702e7			−0.244500
$612$			9.31609e7i			0.406424i
$613$	1.49233e8			0.647862			0.323931	−	0.946081i	$-0.394995\pi$
$613$	1.49233e8			0.647862			0.323931	+	0.946081i	$0.394995\pi$
$614$			2.64729e6i			0.0114366i
$615$			1.17066e8i			0.503276i
$616$	6.07328e6	+	7.64907e6i	0.0259825	+	0.0327240i
$617$	−3.21313e8			−1.36796			−0.683978	−	0.729502i	$-0.739752\pi$
$617$	−3.21313e8			−1.36796			−0.683978	+	0.729502i	$0.739752\pi$
$618$	−1.73971e6			−0.00737075
$619$			2.09399e8i			0.882880i	0.897291	+	0.441440i	$0.145532\pi$
$619$			2.09399e8i			0.882880i	−0.897291	+	0.441440i	$0.854468\pi$
$620$	4.24290e8			1.78028
$621$		−	1.55013e6i		−	0.00647283i
$622$		−	5.13624e6i		−	0.0213439i
$623$	2.13683e8	−	1.69662e8i	0.883700	−	0.701649i
$624$	1.97427e7			0.0812554
$625$	−2.49692e8			−1.02274
$626$		−	2.54996e6i		−	0.0103947i
$627$	−8.08749e7			−0.328104
$628$			1.20033e8i			0.484643i
$629$		−	3.24095e8i		−	1.30233i
$630$	−1.07255e6	+	851592.i	−0.00428939	+	0.00340573i
$631$	2.34959e8			0.935200			0.467600	−	0.883940i	$-0.345119\pi$
$631$	2.34959e8			0.935200			0.467600	+	0.883940i	$0.345119\pi$
$632$	2.93844e6			0.0116403
$633$			1.04981e8i			0.413902i
$634$	550427.			0.00215989
$635$			5.71676e7i			0.223269i
$636$		−	1.44202e8i		−	0.560532i
$637$	3.54585e7	+	8.25251e6i	0.137184	+	0.0319277i
$638$	−7.36914e6			−0.0283762
$639$	5.79280e7			0.222017
$640$		−	1.72179e7i		−	0.0656810i
$641$	2.91254e7			0.110586			0.0552928	−	0.998470i	$-0.482391\pi$
$641$	2.91254e7			0.110586			0.0552928	+	0.998470i	$0.482391\pi$
$642$			4.91742e6i			0.0185837i
$643$			2.76839e7i			0.104135i	0.998644	+	0.0520673i	$0.0165810\pi$
$643$			2.76839e7i			0.104135i	−0.998644	+	0.0520673i	$0.983419\pi$
$644$	5.58450e6	+	7.03347e6i	0.0209087	+	0.0263337i
$645$	1.70706e8			0.636165
$646$	−2.35928e6			−0.00875149
$647$			1.41555e8i			0.522651i	0.965251	+	0.261325i	$0.0841596\pi$
$647$			1.41555e8i			0.522651i	−0.965251	+	0.261325i	$0.915840\pi$
$648$	982029.			0.00360910
$649$			1.28367e8i			0.469591i
$650$		−	14626.5i		−	5.32600e-5i
$651$	−1.74362e8	−	2.19603e8i	−0.631989	−	0.795967i
$652$	2.08660e8			0.752829
$653$	1.96392e7			0.0705319			0.0352659	−	0.999378i	$-0.488772\pi$
$653$	1.96392e7			0.0705319			0.0352659	+	0.999378i	$0.488772\pi$
$654$		−	12815.0i		−	4.58128e-5i
$655$	1.89174e8			0.673189
$656$			2.43073e8i			0.861045i
$657$		−	3.03784e7i		−	0.107120i
$658$	6.29099e6	−	4.99498e6i	0.0220822	−	0.0175330i
$659$	−9.27456e7			−0.324069			−0.162034	−	0.986785i	$-0.551806\pi$
$659$	−9.27456e7			−0.324069			−0.162034	+	0.986785i	$0.551806\pi$
$660$	2.15938e8			0.751098
$661$		−	3.38156e8i		−	1.17088i	−0.810715	−	0.585441i	$-0.800922\pi$
$661$		−	3.38156e8i		−	1.17088i	0.810715	−	0.585441i	$-0.199078\pi$
$662$	−1.31200e6			−0.00452230
$663$			2.89036e7i			0.0991771i
$664$			6.57548e6i			0.0224607i
$665$	8.17193e7	+	1.02922e8i	0.277881	+	0.349981i
$666$	−1.70795e6			−0.00578166
$667$	−1.35539e7			−0.0456759
$668$			4.25144e8i			1.42629i
$669$	−1.71933e6			−0.00574224
$670$		−	2.07853e6i		−	0.00691085i
$671$			6.82685e8i			2.25971i
$672$	−6.68398e6	+	5.30701e6i	−0.0220256	+	0.0174881i
$673$	2.17685e8			0.714140			0.357070	−	0.934078i	$-0.383776\pi$
$673$	2.17685e8			0.714140			0.357070	+	0.934078i	$0.383776\pi$
$674$	−4.35185e6			−0.0142133
$675$			1.37786e6i			0.00448016i
$676$	−3.02707e8			−0.979903
$677$		−	5.26139e7i		−	0.169564i	−0.996400	−	0.0847821i	$-0.972981\pi$
$677$		−	5.26139e7i		−	0.169564i	0.996400	−	0.0847821i	$-0.0270194\pi$
$678$			3.39725e6i			0.0109003i
$679$	−1.21374e8	+	9.63700e7i	−0.387720	+	0.307845i
$680$	1.26003e7			0.0400732
$681$	2.26311e8			0.716581
$682$		−	1.16681e7i		−	0.0367831i
$683$	−4.41207e8			−1.38478			−0.692389	−	0.721525i	$-0.743442\pi$
$683$	−4.41207e8			−1.38478			−0.692389	+	0.721525i	$0.743442\pi$
$684$		−	4.71118e7i		−	0.147218i
$685$		−	2.37071e8i		−	0.737575i
$686$	−4.73892e6	+	2.24489e6i	−0.0146793	+	0.00695381i
$687$	2.30677e8			0.711432
$688$	3.54450e8			1.08840
$689$		−	4.47394e7i		−	0.136783i
$690$	−104816.			−0.000319067
$691$		−	5.12612e8i		−	1.55365i	−0.629714	−	0.776827i	$-0.716828\pi$
$691$		−	5.12612e8i		−	1.55365i	0.629714	−	0.776827i	$-0.283172\pi$
$692$		−	1.39709e8i		−	0.421606i
$693$	−8.87397e7	−	1.11764e8i	−0.266636	−	0.335818i
$694$	−765859.			−0.00229124
$695$	−3.87409e8			−1.15402
$696$		−	8.58658e6i		−	0.0254679i
$697$	−3.55863e8			−1.05096
$698$			4.11591e6i			0.0121032i
$699$		−	3.79084e7i		−	0.110995i
$700$	−4.96386e6	−	6.25180e6i	−0.0144719	−	0.0182268i
$701$	−5.97226e8			−1.73374			−0.866871	−	0.498532i	$-0.833873\pi$
$701$	−5.97226e8			−1.73374			−0.866871	+	0.498532i	$0.833873\pi$
$702$	152319.			0.000440295
$703$			1.63896e8i			0.471740i
$704$	4.48130e8			1.28436
$705$		−	3.55243e8i		−	1.01381i
$706$			490963.i			0.00139519i
$707$	3.13310e8	−	2.48764e8i	0.886575	−	0.703931i
$708$	−7.47774e7			−0.210703
$709$	9.44291e7			0.264952			0.132476	−	0.991186i	$-0.457707\pi$
$709$	9.44291e7			0.264952			0.132476	+	0.991186i	$0.457707\pi$
$710$		−	3.91695e6i		−	0.0109439i
$711$	−4.29350e7			−0.119454
$712$			1.32293e7i			0.0366519i
$713$		−	2.14610e7i		−	0.0592081i
$714$	−2.58871e6	−	3.26039e6i	−0.00711195	−	0.00895724i
$715$	6.69956e7			0.183286
$716$	−6.86077e8			−1.86911
$717$			2.04459e7i			0.0554689i
$718$	−8.48099e6			−0.0229125
$719$		−	3.95754e8i		−	1.06473i	−0.846516	−	0.532363i	$-0.821304\pi$
$719$		−	3.95754e8i		−	1.06473i	0.846516	−	0.532363i	$-0.178696\pi$
$720$			1.25756e8i			0.336924i
$721$	−2.30706e8	+	1.83178e8i	−0.615536	+	0.488729i
$722$	−4.92027e6			−0.0130731
$723$	8.27778e7			0.219028
$724$		−	2.01823e8i		−	0.531809i
$725$	1.20476e7			0.0316145
$726$		−	2.34982e6i		−	0.00614081i
$727$		−	1.30205e8i		−	0.338862i	−0.985542	−	0.169431i	$-0.945807\pi$
$727$		−	1.30205e8i		−	0.338862i	0.985542	−	0.169431i	$-0.0541930\pi$
$728$	−1.38243e6	+	1.09763e6i	−0.00358302	+	0.00284488i
$729$	−1.43489e7			−0.0370370
$730$	−2.05412e6			−0.00528027
$731$			5.18920e8i			1.32846i
$732$	−3.97682e8			−1.01392
$733$			1.88929e8i			0.479719i	0.970808	+	0.239860i	$0.0771014\pi$
$733$			1.88929e8i			0.479719i	−0.970808	+	0.239860i	$0.922899\pi$
$734$		−	7.02922e6i		−	0.0177754i
$735$	−5.25665e7	+	2.25862e8i	−0.132388	+	0.568829i
$736$	−653201.			−0.00163838
$737$	2.16592e8			0.541053
$738$			1.87537e6i			0.00466571i
$739$	1.25077e8			0.309917			0.154959	−	0.987921i	$-0.450476\pi$
$739$	1.25077e8			0.309917			0.154959	+	0.987921i	$0.450476\pi$
$740$		−	4.37605e8i		−	1.07991i
$741$		−	1.46167e7i		−	0.0359247i
$742$	4.00702e6	+	5.04670e6i	0.00980867	+	0.0123537i
$743$	7.56594e7			0.184457			0.0922287	−	0.995738i	$-0.470601\pi$
$743$	7.56594e7			0.184457			0.0922287	+	0.995738i	$0.470601\pi$
$744$	1.35958e7			0.0330131
$745$		−	4.90801e8i		−	1.18696i
$746$	1.07076e7			0.0257915
$747$		−	9.60776e7i		−	0.230494i
$748$			6.56417e8i			1.56847i
$749$	5.17767e8	+	6.52108e8i	1.23222	+	1.55194i
$750$	−3.90895e6			−0.00926565
$751$	6.97668e8			1.64713			0.823567	−	0.567219i	$-0.191981\pi$
$751$	6.97668e8			1.64713			0.823567	+	0.567219i	$0.191981\pi$
$752$		−	7.37618e8i		−	1.73451i
$753$	1.74404e8			0.408482
$754$		−	1.33184e6i		−	0.00310697i
$755$			4.66528e8i			1.08402i
$756$	6.51057e7	−	5.16933e7i	0.150679	−	0.119638i
$757$	−7.06421e8			−1.62846			−0.814228	−	0.580545i	$-0.802840\pi$
$757$	−7.06421e8			−1.62846			−0.814228	+	0.580545i	$0.802840\pi$
$758$	1.01078e7			0.0232087
$759$		−	1.09223e7i		−	0.0249798i
$760$	−6.37202e6			−0.0145156
$761$			1.05701e8i			0.239842i	0.992783	+	0.119921i	$0.0382642\pi$
$761$			1.05701e8i			0.239842i	−0.992783	+	0.119921i	$0.961736\pi$
$762$			915810.i			0.00206986i
$763$	−1.34933e6	−	1.69943e6i	−0.00303769	−	0.00382586i
$764$	1.61865e8			0.362972
$765$	−1.84109e8			−0.411236
$766$			5.55677e6i			0.0123634i
$767$	−2.32000e7			−0.0514165
$768$			2.60841e8i			0.575828i
$769$			1.09507e8i			0.240803i	0.992725	+	0.120401i	$0.0384181\pi$
$769$			1.09507e8i			0.240803i	−0.992725	+	0.120401i	$0.961582\pi$
$770$	−7.55724e6	+	6.00037e6i	−0.0165535	+	0.0131433i
$771$	3.72487e8			0.812734
$772$	−5.38451e8			−1.17029
$773$		−	6.65641e8i		−	1.44112i	−0.693390	−	0.720562i	$-0.743884\pi$
$773$		−	6.65641e8i		−	1.44112i	0.693390	−	0.720562i	$-0.256116\pi$
$774$	2.73466e6			0.00589768
$775$			1.90759e7i			0.0409808i
$776$		−	7.51440e6i		−	0.0160809i
$777$	−2.26495e8	+	1.79834e8i	−0.482831	+	0.383362i
$778$	1.24897e7			0.0265225
$779$	1.79961e8			0.380686
$780$			3.90268e7i			0.0822392i
$781$	4.08164e8			0.856804
$782$		−	318626.i		−	0.000666286i
$783$			1.25463e8i			0.261354i
$784$	−1.09148e8	+	4.68975e8i	−0.226499	+	0.973199i
$785$	−2.37215e8			−0.490381
$786$	3.03051e6			0.00624092
$787$			5.04559e8i			1.03511i	0.855649	+	0.517556i	$0.173158\pi$
$787$			5.04559e8i			1.03511i	−0.855649	+	0.517556i	$0.826842\pi$
$788$	7.61195e8			1.55567
$789$			1.85220e8i			0.377100i
$790$			2.90316e6i			0.00588830i
$791$	3.57704e8	+	4.50515e8i	0.722761	+	0.910291i
$792$	6.91943e6			0.0139282
$793$	−1.23383e8			−0.247420
$794$			4.54550e6i			0.00908071i
$795$	2.84980e8			0.567169
$796$		−	8.91737e8i		−	1.76806i
$797$		−	2.81935e8i		−	0.556896i	−0.960451	−	0.278448i	$-0.910180\pi$
$797$		−	2.81935e8i		−	0.556896i	0.960451	−	0.278448i	$-0.0898199\pi$
$798$	1.30912e6	+	1.64879e6i	0.00257615	+	0.00324456i
$799$	1.07988e9			2.11708
$800$	580608.			0.00113400
$801$		−	1.93300e8i		−	0.376126i
$802$	−1.79604e6			−0.00348172
$803$		−	2.14048e8i		−	0.413394i
$804$			1.26171e8i			0.242767i
$805$	−1.38999e7	+	1.10364e7i	−0.0266455	+	0.0211562i
$806$	2.10880e6			0.00402746
$807$	−1.07079e8			−0.203744
$808$			1.93973e7i			0.0367711i
$809$	−1.57957e8			−0.298327			−0.149163	−	0.988813i	$-0.547658\pi$
$809$	−1.57957e8			−0.298327			−0.149163	+	0.988813i	$0.547658\pi$
$810$			970239.i			0.00182568i
$811$			4.25633e8i			0.797944i	0.916963	+	0.398972i	$0.130633\pi$
$811$			4.25633e8i			0.797944i	−0.916963	+	0.398972i	$0.869367\pi$
$812$	−4.51991e8	−	5.69266e8i	−0.844231	−	1.06328i
$813$	−4.06361e8			−0.756207
$814$	−1.20343e7			−0.0223125
$815$			4.12364e8i			0.761743i
$816$	−3.82280e8			−0.703576
$817$		−	2.62420e8i		−	0.481205i
$818$			2.82074e6i			0.00515351i
$819$	2.01994e7	−	1.60381e7i	0.0367694	−	0.0291945i
$820$	−4.80500e8			−0.871469
$821$	4.74360e8			0.857192			0.428596	−	0.903496i	$-0.359008\pi$
$821$	4.74360e8			0.857192			0.428596	+	0.903496i	$0.359008\pi$
$822$		−	3.79781e6i		−	0.00683782i
$823$	−5.27780e8			−0.946790			−0.473395	−	0.880850i	$-0.656972\pi$
$823$	−5.27780e8			−0.946790			−0.473395	+	0.880850i	$0.656972\pi$
$824$		−	1.42832e7i		−	0.0255297i
$825$			9.70847e6i			0.0172898i
$826$	2.61701e6	−	2.07788e6i	0.00464371	−	0.00368706i
$827$	−1.80928e8			−0.319881			−0.159941	−	0.987127i	$-0.551130\pi$
$827$	−1.80928e8			−0.319881			−0.159941	+	0.987127i	$0.551130\pi$
$828$	6.36255e6			0.0112083
$829$			6.30932e8i			1.10744i	0.832704	+	0.553718i	$0.186792\pi$
$829$			6.30932e8i			1.10744i	−0.832704	+	0.553718i	$0.813208\pi$
$830$	−6.49654e6			−0.0113618
$831$		−	2.16502e8i		−	0.377277i
$832$			8.09914e7i			0.140627i
$833$	−6.86587e8	−	1.59794e8i	−1.18785	−	0.276456i
$834$	−6.20618e6			−0.0106986
$835$	−8.40190e8			−1.44317
$836$		−	3.31953e8i		−	0.568142i
$837$	−1.98655e8			−0.338784
$838$			1.89364e6i			0.00321784i
$839$		−	5.09418e8i		−	0.862559i	−0.902218	−	0.431280i	$-0.858062\pi$
$839$		−	5.09418e8i		−	0.862559i	0.902218	−	0.431280i	$-0.141938\pi$
$840$	−6.99167e6	−	8.80576e6i	−0.0117962	−	0.0148569i
$841$	5.02186e8			0.844261
$842$	4.29620e6			0.00719694
$843$			2.74261e6i			0.00457805i
$844$	−4.30895e8			−0.716710
$845$		−	5.98225e8i		−	0.991505i
$846$		−	5.69090e6i		−	0.00939874i
$847$	−2.47418e8	−	3.11614e8i	−0.407176	−	0.512823i
$848$	5.91725e8			0.970358
$849$	−1.13504e8			−0.185476
$850$			283215.i			0.000461169i
$851$	−2.21345e7			−0.0359154
$852$			2.37766e8i			0.384443i
$853$		−	5.01892e7i		−	0.0808655i	−0.999182	−	0.0404328i	$-0.987126\pi$
$853$		−	5.01892e7i		−	0.0808655i	0.999182	−	0.0404328i	$-0.0128737\pi$
$854$	1.39178e7	−	1.10506e7i	0.0223459	−	0.0177424i
$855$	9.31047e7			0.148961
$856$	−4.03726e7			−0.0643673
$857$		−	3.59511e8i		−	0.571176i	−0.958353	−	0.285588i	$-0.907811\pi$
$857$		−	3.59511e8i		−	0.571176i	0.958353	−	0.285588i	$-0.0921889\pi$
$858$	1.07325e6			0.00169918
$859$		−	5.39220e8i		−	0.850719i	−0.905024	−	0.425360i	$-0.860148\pi$
$859$		−	5.39220e8i		−	0.850719i	0.905024	−	0.425360i	$-0.139852\pi$
$860$			7.00666e8i			1.10158i
$861$	1.97462e8	+	2.48696e8i	0.309367	+	0.389636i
$862$	8.70088e6			0.0135844
$863$	4.55454e8			0.708618			0.354309	−	0.935128i	$-0.384716\pi$
$863$	4.55454e8			0.708618			0.354309	+	0.935128i	$0.384716\pi$
$864$			6.04640e6i			0.00937466i
$865$	2.76100e8			0.426598
$866$		−	6.82984e6i		−	0.0105161i
$867$		−	1.83396e8i		−	0.281406i
$868$	9.01364e8	−	7.15673e8i	1.37829	−	1.09435i
$869$	−3.02522e8			−0.460997
$870$	8.48349e6			0.0128830
$871$			3.91450e7i			0.0592410i
$872$	105213.			0.000158679
$873$			1.09797e8i			0.165024i
$874$			161130.i			0.000241347i
$875$	−5.18373e8	+	4.11582e8i	−0.773781	+	0.614373i
$876$	1.24689e8			0.185488
$877$	1.03677e8			0.153704			0.0768519	−	0.997043i	$-0.475513\pi$
$877$	1.03677e8			0.153704			0.0768519	+	0.997043i	$0.475513\pi$
$878$			7.42279e6i			0.0109669i
$879$	4.69804e8			0.691751
$880$			8.86086e8i			1.30025i
$881$		−	7.15577e6i		−	0.0104648i	−0.999986	−	0.00523238i	$-0.998334\pi$
$881$		−	7.15577e6i		−	0.0104648i	0.999986	−	0.00523238i	$-0.00166552\pi$
$882$	−842102.	+	3.61825e6i	−0.00122732	+	0.00527343i
$883$	−6.93530e8			−1.00736			−0.503678	−	0.863892i	$-0.668020\pi$
$883$	−6.93530e8			−1.00736			−0.503678	+	0.863892i	$0.668020\pi$
$884$	−1.18635e8			−0.171734
$885$		−	1.47779e8i		−	0.213198i
$886$	2.32610e6			0.00334447
$887$			4.47709e8i			0.641541i	0.947157	+	0.320771i	$0.103942\pi$
$887$			4.47709e8i			0.641541i	−0.947157	+	0.320771i	$0.896058\pi$
$888$		−	1.40225e7i		−	0.0200256i
$889$	9.64278e7	+	1.21447e8i	0.137245	+	0.172855i
$890$	−1.30705e7			−0.0185405
$891$	−1.01103e8			−0.142933
$892$		−	7.05703e6i		−	0.00994323i
$893$	−5.46102e8			−0.766866
$894$		−	7.86250e6i		−	0.0110039i
$895$		−	1.35586e9i		−	1.89124i
$896$	−2.90423e7	−	3.65778e7i	−0.0403745	−	0.0508502i
$897$	1.97401e6			0.00273509
$898$	−1.20109e7			−0.0165862
$899$			1.73698e9i			2.39065i
$900$	−5.65545e6			−0.00775781
$901$			8.66294e8i			1.18438i
$902$			1.32140e7i			0.0180058i
$903$	3.62649e8	−	2.87939e8i	0.492519	−	0.391055i
$904$	−2.78918e7			−0.0377547
$905$	3.98853e8			0.538106
$906$			7.47365e6i			0.0100496i
$907$	9.55816e8			1.28101			0.640505	−	0.767954i	$-0.278725\pi$
$907$	9.55816e8			1.28101			0.640505	+	0.767954i	$0.278725\pi$
$908$			9.28899e8i			1.24083i
$909$		−	2.83423e8i		−	0.377349i
$910$	−1.08446e6	−	1.36583e6i	−0.00143909	−	0.00181248i
$911$	−4.54844e8			−0.601600			−0.300800	−	0.953687i	$-0.597254\pi$
$911$	−4.54844e8			−0.601600			−0.300800	+	0.953687i	$0.597254\pi$
$912$	1.93320e8			0.254855
$913$		−	6.76968e8i		−	0.889520i
$914$	−9.98814e6			−0.0130812
$915$		−	7.85920e8i		−	1.02592i
$916$			9.46816e8i			1.23191i
$917$	4.01882e8	−	3.19090e8i	0.521183	−	0.413814i
$918$	−2.94938e6			−0.00381244
$919$	6.39790e8			0.824311			0.412156	−	0.911114i	$-0.364776\pi$
$919$	6.39790e8			0.824311			0.412156	+	0.911114i	$0.364776\pi$
$920$		−	860555.i		−	0.00110513i
$921$	3.17574e8			0.406505
$922$		−	1.92784e7i		−	0.0245968i
$923$			7.37682e7i			0.0938132i
$924$	4.58739e8	−	3.64234e8i	0.581500	−	0.461705i
$925$	1.96746e7			0.0248588
$926$	1.24208e7			0.0156429
$927$			2.08699e8i			0.261988i
$928$	5.28680e7			0.0661529
$929$			1.33414e9i			1.66400i	0.554774	+	0.832001i	$0.312805\pi$
$929$			1.33414e9i			1.66400i	−0.554774	+	0.832001i	$0.687195\pi$
$930$			1.34326e7i			0.0166998i
$931$	3.47210e8	+	8.08086e7i	0.430272	+	0.100140i
$932$	1.55596e8			0.192199
$933$	−6.16153e8			−0.758654
$934$			1.53051e7i			0.0187843i
$935$	−1.29724e9			−1.58704
$936$			1.25056e6i			0.00152503i
$937$		−	8.60795e8i		−	1.04636i	−0.852222	−	0.523180i	$-0.824746\pi$
$937$		−	8.60795e8i		−	1.04636i	0.852222	−	0.523180i	$-0.175254\pi$
$938$	−3.50597e6	−	4.41564e6i	−0.00424815	−	0.00535038i
$939$	−3.05899e8			−0.369472
$940$	1.45810e9			1.75551
$941$		−	5.91787e8i		−	0.710225i	−0.934823	−	0.355113i	$-0.884442\pi$
$941$		−	5.91787e8i		−	0.710225i	0.934823	−	0.355113i	$-0.115558\pi$
$942$	−3.80012e6			−0.00454616
$943$			2.43042e7i			0.0289831i
$944$		−	3.06844e8i		−	0.364756i
$945$	1.02159e8	+	1.28665e8i	0.121054	+	0.152463i
$946$	1.92686e7			0.0227602
$947$	−5.14296e8			−0.605568			−0.302784	−	0.953059i	$-0.597916\pi$
$947$	−5.14296e8			−0.605568			−0.302784	+	0.953059i	$0.597916\pi$
$948$		−	1.76227e8i		−	0.206847i
$949$	3.86853e7			0.0452634
$950$		−	143223.i		−	0.000167048i
$951$		−	6.60304e7i		−	0.0767719i
$952$	2.67682e7	−	2.12536e7i	0.0310247	−	0.0246333i
$953$	1.59021e9			1.83729			0.918643	−	0.395088i	$-0.129286\pi$
$953$	1.59021e9			1.83729			0.918643	+	0.395088i	$0.129286\pi$
$954$	4.56530e6			0.00525804
$955$			3.19886e8i			0.367270i
$956$	−8.39208e7			−0.0960496
$957$			8.84017e8i			1.00861i
$958$			8.49024e6i			0.00965659i
$959$	−3.99880e8	−	5.03634e8i	−0.453392	−	0.571031i
$960$	−5.15896e8			−0.583108
$961$	−1.86280e9			−2.09892
$962$		−	2.17498e6i		−	0.00244304i
$963$	5.89904e8			0.660545
$964$			3.39763e8i			0.379267i
$965$		−	1.06411e9i		−	1.18415i
$966$	−222672.	+	176799.i	−0.000247022	+	0.000196132i
$967$	−1.42749e9			−1.57868			−0.789340	−	0.613956i	$-0.789577\pi$
$967$	−1.42749e9			−1.57868			−0.789340	+	0.613956i	$0.789577\pi$
$968$	1.92923e7			0.0212696
$969$			2.83024e8i			0.311066i
$970$	7.42419e6			0.00813455
$971$		−	2.95070e8i		−	0.322306i	−0.986929	−	0.161153i	$-0.948479\pi$
$971$		−	2.95070e8i		−	0.322306i	0.986929	−	0.161153i	$-0.0515212\pi$
$972$		−	5.88954e7i		−	0.0641331i
$973$	−8.23013e8	+	6.53463e8i	−0.893445	+	0.709386i
$974$	−5.41418e6			−0.00585944
$975$	−1.75463e6			−0.00189309
$976$		−	1.63186e9i		−	1.75523i
$977$	−7.35775e8			−0.788971			−0.394486	−	0.918902i	$-0.629077\pi$
$977$	−7.35775e8			−0.788971			−0.394486	+	0.918902i	$0.629077\pi$
$978$			6.60596e6i			0.00706187i
$979$		−	1.36200e9i		−	1.45154i
$980$	−9.27056e8	−	2.15760e8i	−0.984981	−	0.229242i
$981$	−1.53732e6			−0.00162838
$982$	1.57920e7			0.0166765
$983$		−	2.41971e8i		−	0.254744i	−0.991855	−	0.127372i	$-0.959346\pi$
$983$		−	2.41971e8i		−	0.254744i	0.991855	−	0.127372i	$-0.0406541\pi$
$984$	−1.53970e7			−0.0161604
$985$			1.50431e9i			1.57409i
$986$			2.57885e7i			0.0269027i
$987$	−5.99208e8	−	7.54680e8i	−0.623198	−	0.784895i
$988$	5.99944e7			0.0622071
$989$	3.54403e7			0.0366361
$990$			6.83636e6i			0.00704562i
$991$	−3.12806e8			−0.321407			−0.160703	−	0.987003i	$-0.551376\pi$
$991$	−3.12806e8			−0.321407			−0.160703	+	0.987003i	$0.551376\pi$
$992$			8.37101e7i			0.0857517i
$993$			1.57390e8i			0.160742i
$994$	−6.60694e6	−	8.32120e6i	−0.00672731	−	0.00847279i
$995$	1.76229e9			1.78900
$996$	3.94352e8			0.399122
$997$			1.39722e8i			0.140987i	0.997512	+	0.0704933i	$0.0224573\pi$
$997$			1.39722e8i			0.140987i	−0.997512	+	0.0704933i	$0.977543\pi$
$998$	2.10286e7			0.0211553
$999$			2.04889e8i			0.205505i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.7.d.a.13.4	yes	8
3.2	odd	2		63.7.d.f.55.6		8
4.3	odd	2		336.7.f.a.97.1		8
7.2	even	3		147.7.f.b.31.3		8
7.3	odd	6		147.7.f.b.19.3		8
7.4	even	3		147.7.f.c.19.3		8
7.5	odd	6		147.7.f.c.31.3		8
7.6	odd	2	inner	21.7.d.a.13.3	✓	8
21.20	even	2		63.7.d.f.55.5		8
28.27	even	2		336.7.f.a.97.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.7.d.a.13.3	✓	8	7.6	odd	2	inner
21.7.d.a.13.4	yes	8	1.1	even	1	trivial
63.7.d.f.55.5		8	21.20	even	2
63.7.d.f.55.6		8	3.2	odd	2
147.7.f.b.19.3		8	7.3	odd	6
147.7.f.b.31.3		8	7.2	even	3
147.7.f.c.19.3		8	7.4	even	3
147.7.f.c.31.3		8	7.5	odd	6
336.7.f.a.97.1		8	4.3	odd	2
336.7.f.a.97.8		8	28.27	even	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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	21.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.129945 q^{2} +15.5885i q^{3} -63.9831 q^{4} -126.447i q^{5} -2.02564i q^{6} +(-213.284 - 268.624i) q^{7} +16.6307 q^{8} -243.000 q^{9} +O(q^{10})\)	\(q-0.129945 q^{2} +15.5885i q^{3} -63.9831 q^{4} -126.447i q^{5} -2.02564i q^{6} +(-213.284 - 268.624i) q^{7} +16.6307 q^{8} -243.000 q^{9} +16.4311i q^{10} -1712.19 q^{11} -997.398i q^{12} -309.447i q^{13} +(27.7152 + 34.9063i) q^{14} +1971.11 q^{15} +4092.76 q^{16} +5991.87i q^{17} +31.5766 q^{18} -3030.11i q^{19} +8090.45i q^{20} +(4187.43 - 3324.77i) q^{21} +222.491 q^{22} +409.222 q^{23} +259.248i q^{24} -363.744 q^{25} +40.2111i q^{26} -3788.00i q^{27} +(13646.6 + 17187.4i) q^{28} -33121.1 q^{29} -256.135 q^{30} -52443.3i q^{31} -1596.20 q^{32} -26690.4i q^{33} -778.612i q^{34} +(-33966.6 + 26969.1i) q^{35} +15547.9 q^{36} -54089.1 q^{37} +393.747i q^{38} +4823.81 q^{39} -2102.90i q^{40} +59391.1i q^{41} +(-544.135 + 432.037i) q^{42} +86604.1 q^{43} +109551. q^{44} +30726.5i q^{45} -53.1764 q^{46} -180225. i q^{47} +63799.8i q^{48} +(-26668.5 + 114587. i) q^{49} +47.2666 q^{50} -93403.9 q^{51} +19799.4i q^{52} +144578. q^{53} +492.231i q^{54} +216501. i q^{55} +(-3547.08 - 4467.42i) q^{56} +47234.7 q^{57} +4303.92 q^{58} -74972.5i q^{59} -126118. q^{60} -398720. i q^{61} +6814.74i q^{62} +(51828.1 + 65275.6i) q^{63} -261729. q^{64} -39128.6 q^{65} +3468.28i q^{66} -126500. q^{67} -383378. i q^{68} +6379.15i q^{69} +(4413.78 - 3504.49i) q^{70} -238387. q^{71} -4041.27 q^{72} +125014. i q^{73} +7028.61 q^{74} -5670.20i q^{75} +193876. i q^{76} +(365184. + 459936. i) q^{77} -626.829 q^{78} +176687. q^{79} -517515. i q^{80} +59049.0 q^{81} -7717.57i q^{82} +395381. i q^{83} +(-267925. + 212729. i) q^{84} +757651. q^{85} -11253.8 q^{86} -516307. i q^{87} -28475.0 q^{88} +795472. i q^{89} -3992.75i q^{90} +(-83125.0 + 66000.3i) q^{91} -26183.3 q^{92} +817510. q^{93} +23419.3i q^{94} -383147. q^{95} -24882.3i q^{96} -451838. i q^{97} +(3465.44 - 14889.9i) q^{98} +416063. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{2} + 346 q^{4} + 28 q^{7} - 1462 q^{8} - 1944 q^{9}+O(q^{10}) \) 8 * q + 10 * q^2 + 346 * q^4 + 28 * q^7 - 1462 * q^8 - 1944 * q^9	\( 8 q + 10 q^{2} + 346 q^{4} + 28 q^{7} - 1462 q^{8} - 1944 q^{9} - 6848 q^{11} + 12082 q^{14} - 4536 q^{15} + 28466 q^{16} - 2430 q^{18} + 6804 q^{21} - 7764 q^{22} - 24320 q^{23} - 77056 q^{25} - 30142 q^{28} + 60496 q^{29} + 89424 q^{30} + 5082 q^{32} + 103656 q^{35} - 84078 q^{36} - 39112 q^{37} + 108864 q^{39} - 267624 q^{42} - 29272 q^{43} - 577884 q^{44} + 564972 q^{46} - 94864 q^{49} + 240154 q^{50} + 103032 q^{51} + 232288 q^{53} + 225722 q^{56} + 180792 q^{57} + 987684 q^{58} - 375192 q^{60} - 6804 q^{63} - 690734 q^{64} - 836304 q^{65} - 2163848 q^{67} - 366744 q^{70} - 506288 q^{71} + 355266 q^{72} + 512324 q^{74} + 1536304 q^{77} + 1272024 q^{78} - 93272 q^{79} + 472392 q^{81} - 653184 q^{84} + 3740760 q^{85} + 3846452 q^{86} - 2077548 q^{88} + 1890336 q^{91} - 9701580 q^{92} - 2153952 q^{93} + 4154832 q^{95} - 507542 q^{98} + 1664064 q^{99}+O(q^{100}) \) 8 * q + 10 * q^2 + 346 * q^4 + 28 * q^7 - 1462 * q^8 - 1944 * q^9 - 6848 * q^11 + 12082 * q^14 - 4536 * q^15 + 28466 * q^16 - 2430 * q^18 + 6804 * q^21 - 7764 * q^22 - 24320 * q^23 - 77056 * q^25 - 30142 * q^28 + 60496 * q^29 + 89424 * q^30 + 5082 * q^32 + 103656 * q^35 - 84078 * q^36 - 39112 * q^37 + 108864 * q^39 - 267624 * q^42 - 29272 * q^43 - 577884 * q^44 + 564972 * q^46 - 94864 * q^49 + 240154 * q^50 + 103032 * q^51 + 232288 * q^53 + 225722 * q^56 + 180792 * q^57 + 987684 * q^58 - 375192 * q^60 - 6804 * q^63 - 690734 * q^64 - 836304 * q^65 - 2163848 * q^67 - 366744 * q^70 - 506288 * q^71 + 355266 * q^72 + 512324 * q^74 + 1536304 * q^77 + 1272024 * q^78 - 93272 * q^79 + 472392 * q^81 - 653184 * q^84 + 3740760 * q^85 + 3846452 * q^86 - 2077548 * q^88 + 1890336 * q^91 - 9701580 * q^92 - 2153952 * q^93 + 4154832 * q^95 - 507542 * q^98 + 1664064 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more