LMFDB - Embedded newform 22.20.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [22,20,Mod(1,22)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(22, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("22.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 22 = 2 \cdot 11 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	22.a (trivial)

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:	yes
Analytic conductor:	$50.3396732424$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{51151}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 51151 $ x^2 - 51151
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-226.166$ of defining polynomial
Character	$\chi$	$=$	22.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+512.000 q^{2} +345.029 q^{3} +262144. q^{4} +2.58768e6 q^{5} +176655. q^{6} -6.04760e6 q^{7} +1.34218e8 q^{8} -1.16214e9 q^{9} +O(q^{10})$	\(q+512.000 q^{2} +345.029 q^{3} +262144. q^{4} +2.58768e6 q^{5} +176655. q^{6} -6.04760e6 q^{7} +1.34218e8 q^{8} -1.16214e9 q^{9} +1.32489e9 q^{10} -2.35795e9 q^{11} +9.04472e7 q^{12} -4.78264e10 q^{13} -3.09637e9 q^{14} +8.92825e8 q^{15} +6.87195e10 q^{16} -6.28794e10 q^{17} -5.95017e11 q^{18} -3.64145e11 q^{19} +6.78346e11 q^{20} -2.08660e9 q^{21} -1.20727e12 q^{22} +9.37990e12 q^{23} +4.63090e10 q^{24} -1.23774e13 q^{25} -2.44871e13 q^{26} -8.01986e11 q^{27} -1.58534e12 q^{28} -3.35449e13 q^{29} +4.57127e11 q^{30} -1.29644e14 q^{31} +3.51844e13 q^{32} -8.13560e11 q^{33} -3.21942e13 q^{34} -1.56493e13 q^{35} -3.04649e14 q^{36} -7.83280e14 q^{37} -1.86442e14 q^{38} -1.65015e13 q^{39} +3.47313e14 q^{40} -7.69712e14 q^{41} -1.06834e12 q^{42} +1.44991e15 q^{43} -6.18122e14 q^{44} -3.00726e15 q^{45} +4.80251e15 q^{46} +3.06293e15 q^{47} +2.37102e13 q^{48} -1.13623e16 q^{49} -6.33722e15 q^{50} -2.16952e13 q^{51} -1.25374e16 q^{52} +2.07671e16 q^{53} -4.10617e14 q^{54} -6.10162e15 q^{55} -8.11695e14 q^{56} -1.25641e14 q^{57} -1.71750e16 q^{58} -9.94239e16 q^{59} +2.34049e14 q^{60} -1.23775e17 q^{61} -6.63779e16 q^{62} +7.02817e15 q^{63} +1.80144e16 q^{64} -1.23760e17 q^{65} -4.16543e14 q^{66} -2.06797e16 q^{67} -1.64834e16 q^{68} +3.23633e15 q^{69} -8.01243e15 q^{70} -5.58506e17 q^{71} -1.55980e17 q^{72} -6.99092e17 q^{73} -4.01039e17 q^{74} -4.27055e15 q^{75} -9.54585e16 q^{76} +1.42599e16 q^{77} -8.44876e15 q^{78} +8.88603e16 q^{79} +1.77824e17 q^{80} +1.35044e18 q^{81} -3.94092e17 q^{82} +1.08872e18 q^{83} -5.46989e14 q^{84} -1.62712e17 q^{85} +7.42353e17 q^{86} -1.15740e16 q^{87} -3.16478e17 q^{88} +5.47613e18 q^{89} -1.53972e18 q^{90} +2.89235e17 q^{91} +2.45888e18 q^{92} -4.47310e16 q^{93} +1.56822e18 q^{94} -9.42293e17 q^{95} +1.21396e16 q^{96} +4.21172e18 q^{97} -5.81751e18 q^{98} +2.74027e18 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1024 q^{2} + 16974 q^{3} + 524288 q^{4} - 2948510 q^{5} + 8690688 q^{6} + 84929956 q^{7} + 268435456 q^{8} - 2047881204 q^{9}+O(q^{10}) $ 2 * q + 1024 * q^2 + 16974 * q^3 + 524288 * q^4 - 2948510 * q^5 + 8690688 * q^6 + 84929956 * q^7 + 268435456 * q^8 - 2047881204 * q^9	$ 2 q + 1024 q^{2} + 16974 q^{3} + 524288 q^{4} - 2948510 q^{5} + 8690688 q^{6} + 84929956 q^{7} + 268435456 q^{8} - 2047881204 q^{9} - 1509637120 q^{10} - 4715895382 q^{11} + 4449632256 q^{12} + 12931357856 q^{13} + 43484137472 q^{14} - 91168385490 q^{15} + 137438953472 q^{16} - 459787304876 q^{17} - 1048515176448 q^{18} - 1766628927912 q^{19} - 772934205440 q^{20} + 1510776580572 q^{21} - 2414538435584 q^{22} - 3266600096146 q^{23} + 2278211715072 q^{24} - 801420243000 q^{25} + 6620855222272 q^{26} - 34858123423398 q^{27} + 22263878385664 q^{28} - 97113536421264 q^{29} - 46678213370880 q^{30} - 154254203154130 q^{31} + 70368744177664 q^{32} - 40023804107034 q^{33} - 235411100096512 q^{34} - 519318683122780 q^{35} - 536839770341376 q^{36} - 571392764844202 q^{37} - 904514011090944 q^{38} + 993838018166208 q^{39} - 395742313185280 q^{40} - 184921680043224 q^{41} + 773517609252864 q^{42} + 21\!\cdots\!04 q^{43}+ \cdots + 48\!\cdots\!64 q^{99}+O(q^{100}) $ 2 * q + 1024 * q^2 + 16974 * q^3 + 524288 * q^4 - 2948510 * q^5 + 8690688 * q^6 + 84929956 * q^7 + 268435456 * q^8 - 2047881204 * q^9 - 1509637120 * q^10 - 4715895382 * q^11 + 4449632256 * q^12 + 12931357856 * q^13 + 43484137472 * q^14 - 91168385490 * q^15 + 137438953472 * q^16 - 459787304876 * q^17 - 1048515176448 * q^18 - 1766628927912 * q^19 - 772934205440 * q^20 + 1510776580572 * q^21 - 2414538435584 * q^22 - 3266600096146 * q^23 + 2278211715072 * q^24 - 801420243000 * q^25 + 6620855222272 * q^26 - 34858123423398 * q^27 + 22263878385664 * q^28 - 97113536421264 * q^29 - 46678213370880 * q^30 - 154254203154130 * q^31 + 70368744177664 * q^32 - 40023804107034 * q^33 - 235411100096512 * q^34 - 519318683122780 * q^35 - 536839770341376 * q^36 - 571392764844202 * q^37 - 904514011090944 * q^38 + 993838018166208 * q^39 - 395742313185280 * q^40 - 184921680043224 * q^41 + 773517609252864 * q^42 + 2116703700483204 * q^43 - 1236243679019008 * q^44 + 1896364379272140 * q^45 - 1672499249226752 * q^46 - 7456201298413488 * q^47 + 1166444398116864 * q^48 - 14484301062165318 * q^49 - 410327164416000 * q^50 - 6621865965055188 * q^51 + 3389877873803264 * q^52 - 14846639027241356 * q^53 - 17847359192779776 * q^54 + 6952432346390410 * q^55 + 11399105733459968 * q^56 - 23447497932939960 * q^57 - 49722130647687168 * q^58 + 27278259292488522 * q^59 - 23899245245890560 * q^60 - 221405767994019480 * q^61 - 78978152014914560 * q^62 - 73554176903617512 * q^63 + 36028797018963968 * q^64 - 460126589237118240 * q^65 - 20492187702801408 * q^66 - 211824895464439422 * q^67 - 120530483249414144 * q^68 - 207061904815621470 * q^69 - 265891165758863360 * q^70 - 752788827699476886 * q^71 - 274861962414784512 * q^72 - 1171002362670305344 * q^73 - 292553095600231424 * q^74 + 188225715573790200 * q^75 - 463111173678563328 * q^76 - 200260393646931596 * q^77 + 508845065301098496 * q^78 + 1558374634535268220 * q^79 - 202620064350863360 * q^80 + 1813578176100871890 * q^81 - 94679900182130688 * q^82 + 974129749376372 * q^83 + 396041015937466368 * q^84 + 2034647452871174740 * q^85 + 1083752294647400448 * q^86 - 1068654199999982256 * q^87 - 632956763657732096 * q^88 + 3903208985507920622 * q^89 + 970938562187335680 * q^90 + 5816830265117419168 * q^91 - 856319615604097024 * q^92 - 453968381403456030 * q^93 - 3817575064787705856 * q^94 + 6822127176303479320 * q^95 + 597219531835834368 * q^96 + 17667658647655093842 * q^97 - 7415962143828642816 * q^98 + 4828796756414099964 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	512.000	0.707107
$2$	512.000	0.707107
$3$	345.029	0.0101205	0.00506027	−	0.999987i	$-0.498389\pi$
$3$	345.029	0.0101205	0.00506027	+	0.999987i	$0.498389\pi$
$4$	262144.	0.500000
$5$	2.58768e6	0.592511	0.296255	−	0.955109i	$-0.404262\pi$
$5$	2.58768e6	0.592511	0.296255	+	0.955109i	$0.404262\pi$
$6$	176655.	0.00715630
$7$	−6.04760e6	−0.0566437	−0.0283219	−	0.999599i	$-0.509016\pi$
$7$	−6.04760e6	−0.0566437	−0.0283219	+	0.999599i	$0.509016\pi$
$8$	1.34218e8	0.353553
$9$	−1.16214e9	−0.999898
$10$	1.32489e9	0.418968
$11$	−2.35795e9	−0.301511
$11$	−2.35795e9	−0.301511
$12$	9.04472e7	0.00506027
$13$	−4.78264e10	−1.25085	−0.625426	−	0.780283i	$-0.715075\pi$
$13$	−4.78264e10	−1.25085	−0.625426	+	0.780283i	$0.715075\pi$
$14$	−3.09637e9	−0.0400532
$15$	8.92825e8	0.00599652
$16$	6.87195e10	0.250000
$17$	−6.28794e10	−0.128601	−0.0643004	−	0.997931i	$-0.520482\pi$
$17$	−6.28794e10	−0.128601	−0.0643004	+	0.997931i	$0.520482\pi$
$18$	−5.95017e11	−0.707034
$19$	−3.64145e11	−0.258890	−0.129445	−	0.991587i	$-0.541320\pi$
$19$	−3.64145e11	−0.258890	−0.129445	+	0.991587i	$0.541320\pi$
$20$	6.78346e11	0.296255
$21$	−2.08660e9	−0.000573265 0
$22$	−1.20727e12	−0.213201
$23$	9.37990e12	1.08588	0.542942	−	0.839770i	$-0.317310\pi$
$23$	9.37990e12	1.08588	0.542942	+	0.839770i	$0.317310\pi$
$24$	4.63090e10	0.00357815
$25$	−1.23774e13	−0.648931
$26$	−2.44871e13	−0.884486
$27$	−8.01986e11	−0.0202400
$28$	−1.58534e12	−0.0283219
$29$	−3.35449e13	−0.429384	−0.214692	−	0.976682i	$-0.568875\pi$
$29$	−3.35449e13	−0.429384	−0.214692	+	0.976682i	$0.568875\pi$
$30$	4.57127e11	0.00424018
$31$	−1.29644e14	−0.880678	−0.440339	−	0.897831i	$-0.645142\pi$
$31$	−1.29644e14	−0.880678	−0.440339	+	0.897831i	$0.645142\pi$
$32$	3.51844e13	0.176777
$33$	−8.13560e11	−0.00305146
$34$	−3.21942e13	−0.0909344
$35$	−1.56493e13	−0.0335620
$36$	−3.04649e14	−0.499949
$37$	−7.83280e14	−0.990833	−0.495417	−	0.868656i	$-0.664985\pi$
$37$	−7.83280e14	−0.990833	−0.495417	+	0.868656i	$0.664985\pi$
$38$	−1.86442e14	−0.183063
$39$	−1.65015e13	−0.0126593
$40$	3.47313e14	0.209484
$41$	−7.69712e14	−0.367182	−0.183591	−	0.983003i	$-0.558772\pi$
$41$	−7.69712e14	−0.367182	−0.183591	+	0.983003i	$0.558772\pi$
$42$	−1.06834e12	−0.000405359 0
$43$	1.44991e15	0.439937	0.219968	−	0.975507i	$-0.429405\pi$
$43$	1.44991e15	0.439937	0.219968	+	0.975507i	$0.429405\pi$
$44$	−6.18122e14	−0.150756
$45$	−3.00726e15	−0.592450
$46$	4.80251e15	0.767836
$47$	3.06293e15	0.399215	0.199607	−	0.979876i	$-0.436033\pi$
$47$	3.06293e15	0.399215	0.199607	+	0.979876i	$0.436033\pi$
$48$	2.37102e13	0.00253013
$49$	−1.13623e16	−0.996791
$50$	−6.33722e15	−0.458864
$51$	−2.16952e13	−0.00130151
$52$	−1.25374e16	−0.625426
$53$	2.07671e16	0.864482	0.432241	−	0.901758i	$-0.357723\pi$
$53$	2.07671e16	0.864482	0.432241	+	0.901758i	$0.357723\pi$
$54$	−4.10617e14	−0.0143119
$55$	−6.10162e15	−0.178649
$56$	−8.11695e14	−0.0200266
$57$	−1.25641e14	−0.00262011
$58$	−1.71750e16	−0.303621
$59$	−9.94239e16	−1.49416	−0.747079	−	0.664735i	$-0.768544\pi$
$59$	−9.94239e16	−1.49416	−0.747079	+	0.664735i	$0.768544\pi$
$60$	2.34049e14	0.00299826
$61$	−1.23775e17	−1.35519	−0.677596	−	0.735434i	$-0.736978\pi$
$61$	−1.23775e17	−1.35519	−0.677596	+	0.735434i	$0.736978\pi$
$62$	−6.63779e16	−0.622734
$63$	7.02817e15	0.0566379
$64$	1.80144e16	0.125000
$65$	−1.23760e17	−0.741144
$66$	−4.16543e14	−0.00215770
$67$	−2.06797e16	−0.0928611	−0.0464306	−	0.998922i	$-0.514785\pi$
$67$	−2.06797e16	−0.0928611	−0.0464306	+	0.998922i	$0.514785\pi$
$68$	−1.64834e16	−0.0643004
$69$	3.23633e15	0.0109897
$70$	−8.01243e15	−0.0237319
$71$	−5.58506e17	−1.44569	−0.722843	−	0.691013i	$-0.757165\pi$
$71$	−5.58506e17	−1.44569	−0.722843	+	0.691013i	$0.757165\pi$
$72$	−1.55980e17	−0.353517
$73$	−6.99092e17	−1.38985	−0.694924	−	0.719083i	$-0.744562\pi$
$73$	−6.99092e17	−1.38985	−0.694924	+	0.719083i	$0.744562\pi$
$74$	−4.01039e17	−0.700625
$75$	−4.27055e15	−0.00656753
$76$	−9.54585e16	−0.129445
$77$	1.42599e16	0.0170787
$78$	−8.44876e15	−0.00895147
$79$	8.88603e16	0.0834162	0.0417081	−	0.999130i	$-0.486720\pi$
$79$	8.88603e16	0.0834162	0.0417081	+	0.999130i	$0.486720\pi$
$80$	1.77824e17	0.148128
$81$	1.35044e18	0.999693
$82$	−3.94092e17	−0.259637
$83$	1.08872e18	0.639258	0.319629	−	0.947543i	$-0.396442\pi$
$83$	1.08872e18	0.639258	0.319629	+	0.947543i	$0.396442\pi$
$84$	−5.46989e14	−0.000286632 0
$85$	−1.62712e17	−0.0761973
$86$	7.42353e17	0.311082
$87$	−1.15740e16	−0.00434560
$88$	−3.16478e17	−0.106600
$89$	5.47613e18	1.65679	0.828397	−	0.560141i	$-0.189253\pi$
$89$	5.47613e18	1.65679	0.828397	+	0.560141i	$0.189253\pi$
$90$	−1.53972e18	−0.418925
$91$	2.89235e17	0.0708529
$92$	2.45888e18	0.542942
$93$	−4.47310e16	−0.00891293
$94$	1.56822e18	0.282288
$95$	−9.42293e17	−0.153395
$96$	1.21396e16	0.00178907
$97$	4.21172e18	0.562508	0.281254	−	0.959633i	$-0.409250\pi$
$97$	4.21172e18	0.562508	0.281254	+	0.959633i	$0.409250\pi$
$98$	−5.81751e18	−0.704838
$99$	2.74027e18	0.301480
$100$	−3.24466e18	−0.324466
$101$	1.32232e19	1.20305	0.601527	−	0.798852i	$-0.294559\pi$
$101$	1.32232e19	1.20305	0.601527	+	0.798852i	$0.294559\pi$
$102$	−1.11079e16	−0.000920305 0
$103$	−5.86597e18	−0.442982	−0.221491	−	0.975162i	$-0.571092\pi$
$103$	−5.86597e18	−0.442982	−0.221491	+	0.975162i	$0.571092\pi$
$104$	−6.41915e18	−0.442243
$105$	−5.39945e15	−0.000339665 0
$106$	1.06328e19	0.611281
$107$	1.47950e19	0.777980	0.388990	−	0.921242i	$-0.372824\pi$
$107$	1.47950e19	0.777980	0.388990	+	0.921242i	$0.372824\pi$
$108$	−2.10236e17	−0.0101200
$109$	1.43992e19	0.635020	0.317510	−	0.948255i	$-0.397153\pi$
$109$	1.43992e19	0.635020	0.317510	+	0.948255i	$0.397153\pi$
$110$	−3.12403e18	−0.126324
$111$	−2.70254e17	−0.0100278
$112$	−4.15588e17	−0.0141609
$113$	−2.32029e18	−0.0726603	−0.0363302	−	0.999340i	$-0.511567\pi$
$113$	−2.32029e18	−0.0726603	−0.0363302	+	0.999340i	$0.511567\pi$
$114$	−6.43280e16	−0.00185270
$115$	2.42722e19	0.643398
$116$	−8.79361e18	−0.214692
$117$	5.55811e19	1.25072
$118$	−5.09050e19	−1.05653
$119$	3.80269e17	0.00728442
$120$	1.19833e17	0.00212009
$121$	5.55992e18	0.0909091
$122$	−6.33730e19	−0.958266
$123$	−2.65573e17	−0.00371608
$124$	−3.39855e19	−0.440339
$125$	−8.13849e19	−0.977009
$126$	3.59842e18	0.0400491
$127$	1.03267e20	1.06617	0.533085	−	0.846062i	$-0.321033\pi$
$127$	1.03267e20	1.06617	0.533085	+	0.846062i	$0.321033\pi$
$128$	9.22337e18	0.0883883
$129$	5.00260e17	0.00445239
$130$	−6.33650e19	−0.524068
$131$	3.22631e19	0.248101	0.124051	−	0.992276i	$-0.460411\pi$
$131$	3.22631e19	0.248101	0.124051	+	0.992276i	$0.460411\pi$
$132$	−2.13270e17	−0.00152573
$133$	2.20221e18	0.0146645
$134$	−1.05880e19	−0.0656627
$135$	−2.07529e18	−0.0119924
$136$	−8.43953e18	−0.0454672
$137$	−1.41009e20	−0.708602	−0.354301	−	0.935132i	$-0.615281\pi$
$137$	−1.41009e20	−0.708602	−0.354301	+	0.935132i	$0.615281\pi$
$138$	1.65700e18	0.00777091
$139$	−2.29103e20	−1.00321	−0.501604	−	0.865097i	$-0.667257\pi$
$139$	−2.29103e20	−1.00321	−0.501604	+	0.865097i	$0.667257\pi$
$140$	−4.10236e18	−0.0167810
$141$	1.05680e18	0.00404027
$142$	−2.85955e20	−1.02225
$143$	1.12772e20	0.377146
$144$	−7.98618e19	−0.249974
$145$	−8.68037e19	−0.254415
$146$	−3.57935e20	−0.982771
$147$	−3.92033e18	−0.0100881
$148$	−2.05332e20	−0.495417
$149$	4.49255e20	1.01677	0.508386	−	0.861129i	$-0.330242\pi$
$149$	4.49255e20	1.01677	0.508386	+	0.861129i	$0.330242\pi$
$150$	−2.18652e18	−0.00464394
$151$	−3.52850e20	−0.703572	−0.351786	−	0.936080i	$-0.614426\pi$
$151$	−3.52850e20	−0.703572	−0.351786	+	0.936080i	$0.614426\pi$
$152$	−4.88748e19	−0.0915315
$153$	7.30748e19	0.128588
$154$	7.30108e18	0.0120765
$155$	−3.35478e20	−0.521811
$156$	−4.32577e18	−0.00632965
$157$	7.25681e20	0.999307	0.499653	−	0.866225i	$-0.333461\pi$
$157$	7.25681e20	0.999307	0.499653	+	0.866225i	$0.333461\pi$
$158$	4.54965e19	0.0589842
$159$	7.16526e18	0.00874901
$160$	9.10460e19	0.104742
$161$	−5.67259e19	−0.0615085
$162$	6.91424e20	0.706890
$163$	1.51331e21	1.45930	0.729652	−	0.683819i	$-0.239682\pi$
$163$	1.51331e21	1.45930	0.729652	+	0.683819i	$0.239682\pi$
$164$	−2.01775e20	−0.183591
$165$	−2.10524e18	−0.00180802
$166$	5.57427e20	0.452024
$167$	1.49425e21	1.14450	0.572252	−	0.820078i	$-0.306070\pi$
$167$	1.49425e21	1.14450	0.572252	+	0.820078i	$0.306070\pi$
$168$	−2.80058e17	−0.000202680 0
$169$	8.25447e20	0.564632
$170$	−8.33085e19	−0.0538796
$171$	4.23189e20	0.258864
$172$	3.80085e20	0.219968
$173$	−4.78814e20	−0.262258	−0.131129	−	0.991365i	$-0.541860\pi$
$173$	−4.78814e20	−0.262258	−0.131129	+	0.991365i	$0.541860\pi$
$174$	−5.92587e18	−0.00307280
$175$	7.48534e19	0.0367579
$176$	−1.62037e20	−0.0753778
$177$	−3.43041e19	−0.0151217
$178$	2.80378e21	1.17153
$179$	1.18993e21	0.471430	0.235715	−	0.971822i	$-0.424257\pi$
$179$	1.18993e21	0.471430	0.235715	+	0.971822i	$0.424257\pi$
$180$	−7.88334e20	−0.296225
$181$	−2.04299e21	−0.728315	−0.364157	−	0.931337i	$-0.618643\pi$
$181$	−2.04299e21	−0.728315	−0.364157	+	0.931337i	$0.618643\pi$
$182$	1.48088e20	0.0501006
$183$	−4.27061e19	−0.0137153
$184$	1.25895e21	0.383918
$185$	−2.02688e21	−0.587079
$186$	−2.29023e19	−0.00630240
$187$	1.48266e20	0.0387746
$188$	8.02928e20	0.199607
$189$	4.85009e18	0.00114647
$190$	−4.82454e20	−0.108467
$191$	−2.27547e21	−0.486693	−0.243346	−	0.969939i	$-0.578245\pi$
$191$	−2.27547e21	−0.486693	−0.243346	+	0.969939i	$0.578245\pi$
$192$	6.21549e18	0.00126507
$193$	−2.10049e21	−0.406936	−0.203468	−	0.979082i	$-0.565221\pi$
$193$	−2.10049e21	−0.406936	−0.203468	+	0.979082i	$0.565221\pi$
$194$	2.15640e21	0.397753
$195$	−4.27006e19	−0.00750077
$196$	−2.97856e21	−0.498396
$197$	−3.99494e21	−0.636914	−0.318457	−	0.947937i	$-0.603165\pi$
$197$	−3.99494e21	−0.636914	−0.318457	+	0.947937i	$0.603165\pi$
$198$	1.40302e21	0.213179
$199$	−7.82617e21	−1.13356	−0.566781	−	0.823869i	$-0.691811\pi$
$199$	−7.82617e21	−1.13356	−0.566781	+	0.823869i	$0.691811\pi$
$200$	−1.66126e21	−0.229432
$201$	−7.13510e18	−0.000939804 0
$202$	6.77030e21	0.850688
$203$	2.02866e20	0.0243219
$204$	−5.68726e18	−0.000650754 0
$205$	−1.99177e21	−0.217559
$206$	−3.00338e21	−0.313236
$207$	−1.09008e22	−1.08577
$208$	−3.28661e21	−0.312713
$209$	8.58636e20	0.0780583
$210$	−2.76452e18	−0.000240180 0
$211$	−1.75150e22	−1.45454	−0.727272	−	0.686349i	$-0.759212\pi$
$211$	−1.75150e22	−1.45454	−0.727272	+	0.686349i	$0.759212\pi$
$212$	5.44398e21	0.432241
$213$	−1.92701e20	−0.0146311
$214$	7.57503e21	0.550115
$215$	3.75191e21	0.260667
$216$	−1.07641e20	−0.00715593
$217$	7.84037e20	0.0498849
$218$	7.37240e21	0.449027
$219$	−2.41207e20	−0.0140660
$220$	−1.59950e21	−0.0893243
$221$	3.00730e21	0.160861
$222$	−1.38370e20	−0.00709070
$223$	−1.22754e21	−0.0602753	−0.0301376	−	0.999546i	$-0.509595\pi$
$223$	−1.22754e21	−0.0602753	−0.0301376	+	0.999546i	$0.509595\pi$
$224$	−2.12781e20	−0.0100133
$225$	1.43843e22	0.648865
$226$	−1.18799e21	−0.0513786
$227$	−2.07995e22	−0.862594	−0.431297	−	0.902210i	$-0.641944\pi$
$227$	−2.07995e22	−0.862594	−0.431297	+	0.902210i	$0.641944\pi$
$228$	−3.29359e19	−0.00131005
$229$	−3.42595e22	−1.30721	−0.653603	−	0.756837i	$-0.726744\pi$
$229$	−3.42595e22	−1.30721	−0.653603	+	0.756837i	$0.726744\pi$
$230$	1.24274e22	0.454951
$231$	4.92008e18	0.000172846 0
$232$	−4.50233e21	−0.151810
$233$	3.66760e22	1.18714	0.593568	−	0.804784i	$-0.297719\pi$
$233$	3.66760e22	1.18714	0.593568	+	0.804784i	$0.297719\pi$
$234$	2.84575e22	0.884396
$235$	7.92588e21	0.236539
$236$	−2.60634e22	−0.747079
$237$	3.06594e19	0.000844217 0
$238$	1.94698e20	0.00515086
$239$	4.00822e22	1.01899	0.509497	−	0.860472i	$-0.329831\pi$
$239$	4.00822e22	1.01899	0.509497	+	0.860472i	$0.329831\pi$
$240$	6.13545e19	0.00149913
$241$	2.05296e22	0.482191	0.241096	−	0.970501i	$-0.422493\pi$
$241$	2.05296e22	0.482191	0.241096	+	0.970501i	$0.422493\pi$
$242$	2.84668e21	0.0642824
$243$	1.39806e21	0.0303575
$244$	−3.24470e22	−0.677596
$245$	−2.94021e22	−0.590610
$246$	−1.35973e20	−0.00262766
$247$	1.74158e22	0.323834
$248$	−1.74006e22	−0.311367
$249$	3.75641e20	0.00646963
$250$	−4.16691e22	−0.690850
$251$	1.00140e23	1.59848	0.799242	−	0.601009i	$-0.205235\pi$
$251$	1.00140e23	1.59848	0.799242	+	0.601009i	$0.205235\pi$
$252$	1.84239e21	0.0283190
$253$	−2.21173e22	−0.327406
$254$	5.28728e22	0.753896
$255$	−5.61403e19	−0.000771157 0
$256$	4.72237e21	0.0625000
$257$	6.11474e22	0.779854	0.389927	−	0.920846i	$-0.372500\pi$
$257$	6.11474e22	0.779854	0.389927	+	0.920846i	$0.372500\pi$
$258$	2.56133e20	0.00314832
$259$	4.73696e21	0.0561245
$260$	−3.24429e22	−0.370572
$261$	3.89840e22	0.429340
$262$	1.65187e22	0.175434
$263$	−4.53968e22	−0.464993	−0.232496	−	0.972597i	$-0.574689\pi$
$263$	−4.53968e22	−0.464993	−0.232496	+	0.972597i	$0.574689\pi$
$264$	−1.09194e20	−0.00107885
$265$	5.37388e22	0.512215
$266$	1.12753e21	0.0103694
$267$	1.88942e21	0.0167676
$268$	−5.42106e21	−0.0464306
$269$	2.08928e22	0.172723	0.0863616	−	0.996264i	$-0.472476\pi$
$269$	2.08928e22	0.172723	0.0863616	+	0.996264i	$0.472476\pi$
$270$	−1.06255e21	−0.00847993
$271$	3.76342e22	0.289984	0.144992	−	0.989433i	$-0.453684\pi$
$271$	3.76342e22	0.289984	0.144992	+	0.989433i	$0.453684\pi$
$272$	−4.32104e21	−0.0321502
$273$	9.97944e19	0.000717069 0
$274$	−7.21967e22	−0.501057
$275$	2.91852e22	0.195660
$276$	8.48386e20	0.00549486
$277$	−1.65712e23	−1.03704	−0.518520	−	0.855066i	$-0.673517\pi$
$277$	−1.65712e23	−1.03704	−0.518520	+	0.855066i	$0.673517\pi$
$278$	−1.17301e23	−0.709375
$279$	1.50665e23	0.880588
$280$	−2.10041e21	−0.0118660
$281$	3.01997e23	1.64927	0.824636	−	0.565664i	$-0.191380\pi$
$281$	3.01997e23	1.64927	0.824636	+	0.565664i	$0.191380\pi$
$282$	5.41080e20	0.00285690
$283$	−5.67345e22	−0.289652	−0.144826	−	0.989457i	$-0.546262\pi$
$283$	−5.67345e22	−0.289652	−0.144826	+	0.989457i	$0.546262\pi$
$284$	−1.46409e23	−0.722843
$285$	−3.25118e20	−0.00155244
$286$	5.77394e22	0.266683
$287$	4.65491e21	0.0207986
$288$	−4.08893e22	−0.176759
$289$	−2.35119e23	−0.983462
$290$	−4.44435e22	−0.179898
$291$	1.45317e21	0.00569288
$292$	−1.83263e23	−0.694924
$293$	−1.36033e23	−0.499346	−0.249673	−	0.968330i	$-0.580323\pi$
$293$	−1.36033e23	−0.499346	−0.249673	+	0.968330i	$0.580323\pi$
$294$	−2.00721e21	−0.00713334
$295$	−2.57278e23	−0.885305
$296$	−1.05130e23	−0.350312
$297$	1.89104e21	0.00610260
$298$	2.30019e23	0.718966
$299$	−4.48607e23	−1.35828
$300$	−1.11950e21	−0.00328376
$301$	−8.76847e21	−0.0249197
$302$	−1.80659e23	−0.497501
$303$	4.56240e21	0.0121755
$304$	−2.50239e22	−0.0647226
$305$	−3.20292e23	−0.802966
$306$	3.74143e22	0.0909251
$307$	8.68387e22	0.204597	0.102298	−	0.994754i	$-0.467380\pi$
$307$	8.68387e22	0.204597	0.102298	+	0.994754i	$0.467380\pi$
$308$	3.73815e21	0.00853936
$309$	−2.02393e21	−0.00448321
$310$	−1.71765e23	−0.368976
$311$	−1.18626e23	−0.247147	−0.123573	−	0.992335i	$-0.539435\pi$
$311$	−1.18626e23	−0.247147	−0.123573	+	0.992335i	$0.539435\pi$
$312$	−2.21479e21	−0.00447574
$313$	2.27586e23	0.446143	0.223072	−	0.974802i	$-0.428392\pi$
$313$	2.27586e23	0.446143	0.223072	+	0.974802i	$0.428392\pi$
$314$	3.71549e23	0.706617
$315$	1.81867e22	0.0335586
$316$	2.32942e22	0.0417081
$317$	2.40139e23	0.417253	0.208627	−	0.977995i	$-0.433101\pi$
$317$	2.40139e23	0.417253	0.208627	+	0.977995i	$0.433101\pi$
$318$	3.66861e21	0.00618649
$319$	7.90972e22	0.129464
$320$	4.66156e22	0.0740638
$321$	5.10469e21	0.00787358
$322$	−2.90436e22	−0.0434931
$323$	2.28972e22	0.0332935
$324$	3.54009e23	0.499846
$325$	5.91966e23	0.811717
$326$	7.74815e23	1.03188
$327$	4.96814e21	0.00642674
$328$	−1.03309e23	−0.129818
$329$	−1.85233e22	−0.0226130
$330$	−1.07788e21	−0.00127846
$331$	−2.68216e23	−0.309114	−0.154557	−	0.987984i	$-0.549395\pi$
$331$	−2.68216e23	−0.309114	−0.154557	+	0.987984i	$0.549395\pi$
$332$	2.85403e23	0.319629
$333$	9.10283e23	0.990732
$334$	7.65057e23	0.809287
$335$	−5.35126e22	−0.0550212
$336$	−1.43390e20	−0.000143316 0
$337$	−1.18483e24	−1.15125	−0.575627	−	0.817712i	$-0.695242\pi$
$337$	−1.18483e24	−1.15125	−0.575627	+	0.817712i	$0.695242\pi$
$338$	4.22629e23	0.399255
$339$	−8.00567e20	−0.000735361 0
$340$	−4.26540e22	−0.0380986
$341$	3.05694e23	0.265535
$342$	2.16673e23	0.183044
$343$	1.37651e23	0.113106
$344$	1.94603e23	0.155541
$345$	8.37461e21	0.00651153
$346$	−2.45153e23	−0.185445
$347$	6.45525e23	0.475098	0.237549	−	0.971376i	$-0.423656\pi$
$347$	6.45525e23	0.475098	0.237549	+	0.971376i	$0.423656\pi$
$348$	−3.03405e21	−0.00217280
$349$	1.60464e24	1.11825	0.559123	−	0.829085i	$-0.311138\pi$
$349$	1.60464e24	1.11825	0.559123	+	0.829085i	$0.311138\pi$
$350$	3.83250e22	0.0259917
$351$	3.83561e22	0.0253173
$352$	−8.29629e22	−0.0533002
$353$	1.24051e24	0.775786	0.387893	−	0.921704i	$-0.373203\pi$
$353$	1.24051e24	0.775786	0.387893	+	0.921704i	$0.373203\pi$
$354$	−1.75637e22	−0.0106926
$355$	−1.44524e24	−0.856584
$356$	1.43553e24	0.828397
$357$	1.31204e20	7.37222e−5 0
$358$	6.09244e23	0.333352
$359$	−3.22264e24	−1.71717	−0.858587	−	0.512669i	$-0.828657\pi$
$359$	−3.22264e24	−1.71717	−0.858587	+	0.512669i	$0.828657\pi$
$360$	−4.03627e23	−0.209463
$361$	−1.84582e24	−0.932976
$362$	−1.04601e24	−0.514996
$363$	1.91833e21	0.000920048 0
$364$	7.58213e22	0.0354265
$365$	−1.80903e24	−0.823500
$366$	−2.18655e22	−0.00969816
$367$	3.76155e24	1.62570	0.812848	−	0.582476i	$-0.197916\pi$
$367$	3.76155e24	1.62570	0.812848	+	0.582476i	$0.197916\pi$
$368$	6.44582e23	0.271471
$369$	8.94515e23	0.367144
$370$	−1.03776e24	−0.415128
$371$	−1.25591e23	−0.0489674
$372$	−1.17260e22	−0.00445647
$373$	−4.10641e24	−1.52135	−0.760675	−	0.649133i	$-0.775132\pi$
$373$	−4.10641e24	−1.52135	−0.760675	+	0.649133i	$0.775132\pi$
$374$	7.59123e22	0.0274178
$375$	−2.80801e22	−0.00988785
$376$	4.11099e23	0.141144
$377$	1.60434e24	0.537096
$378$	2.48325e21	0.000810677 0
$379$	−2.67952e24	−0.853068	−0.426534	−	0.904471i	$-0.640266\pi$
$379$	−2.67952e24	−0.853068	−0.426534	+	0.904471i	$0.640266\pi$
$380$	−2.47017e23	−0.0766976
$381$	3.56301e22	0.0107902
$382$	−1.16504e24	−0.344144
$383$	−6.24638e24	−1.79987	−0.899933	−	0.436029i	$-0.856385\pi$
$383$	−6.24638e24	−1.79987	−0.899933	+	0.436029i	$0.856385\pi$
$384$	3.18233e21	0.000894537 0
$385$	3.69002e22	0.0101193
$386$	−1.07545e24	−0.287747
$387$	−1.68500e24	−0.439892
$388$	1.10408e24	0.281254
$389$	4.20327e23	0.104488	0.0522440	−	0.998634i	$-0.483363\pi$
$389$	4.20327e23	0.104488	0.0522440	+	0.998634i	$0.483363\pi$
$390$	−2.18627e22	−0.00530384
$391$	−5.89802e23	−0.139646
$392$	−1.52503e24	−0.352419
$393$	1.11317e22	0.00251092
$394$	−2.04541e24	−0.450366
$395$	2.29942e23	0.0494250
$396$	7.18346e23	0.150740
$397$	1.14770e24	0.235136	0.117568	−	0.993065i	$-0.462490\pi$
$397$	1.14770e24	0.235136	0.117568	+	0.993065i	$0.462490\pi$
$398$	−4.00700e24	−0.801549
$399$	7.59824e20	0.000148413 0
$400$	−8.50567e23	−0.162233
$401$	6.63970e24	1.23673	0.618367	−	0.785889i	$-0.287794\pi$
$401$	6.63970e24	1.23673	0.618367	+	0.785889i	$0.287794\pi$
$402$	−3.65317e21	−0.000664542 0
$403$	6.20042e24	1.10160
$404$	3.46640e24	0.601527
$405$	3.49450e24	0.592329
$406$	1.03868e23	0.0171982
$407$	1.84693e24	0.298747
$408$	−2.91188e21	−0.000460152 0
$409$	4.66547e24	0.720317	0.360159	−	0.932891i	$-0.382723\pi$
$409$	4.66547e24	0.720317	0.360159	+	0.932891i	$0.382723\pi$
$410$	−1.01979e24	−0.153838
$411$	−4.86522e22	−0.00717142
$412$	−1.53773e24	−0.221491
$413$	6.01276e23	0.0846347
$414$	−5.58120e24	−0.767758
$415$	2.81728e24	0.378767
$416$	−1.68274e24	−0.221122
$417$	−7.90473e22	−0.0101530
$418$	4.39622e23	0.0551956
$419$	−1.56577e23	−0.0192174	−0.00960868	−	0.999954i	$-0.503059\pi$
$419$	−1.56577e23	−0.0192174	−0.00960868	+	0.999954i	$0.503059\pi$
$420$	−1.41543e21	−0.000169833 0
$421$	−3.41897e22	−0.00401066	−0.00200533	−	0.999998i	$-0.500638\pi$
$421$	−3.41897e22	−0.00401066	−0.00200533	+	0.999998i	$0.500638\pi$
$422$	−8.96768e24	−1.02852
$423$	−3.55956e24	−0.399174
$424$	2.78732e24	0.305640
$425$	7.78282e23	0.0834530
$426$	−9.86628e22	−0.0103458
$427$	7.48544e23	0.0767631
$428$	3.87842e24	0.388990
$429$	3.89097e22	0.00381692
$430$	1.92098e24	0.184320
$431$	−6.02019e24	−0.565036	−0.282518	−	0.959262i	$-0.591170\pi$
$431$	−6.02019e24	−0.565036	−0.282518	+	0.959262i	$0.591170\pi$
$432$	−5.51121e22	−0.00506001
$433$	7.62433e24	0.684805	0.342402	−	0.939553i	$-0.388759\pi$
$433$	7.62433e24	0.684805	0.342402	+	0.939553i	$0.388759\pi$
$434$	4.01427e23	0.0352739
$435$	−2.99498e22	−0.00257481
$436$	3.77467e24	0.317510
$437$	−3.41565e24	−0.281125
$438$	−1.23498e23	−0.00994617
$439$	3.40148e24	0.268074	0.134037	−	0.990976i	$-0.457206\pi$
$439$	3.40148e24	0.268074	0.134037	+	0.990976i	$0.457206\pi$
$440$	−8.18946e23	−0.0631619
$441$	1.32046e25	0.996689
$442$	1.53974e24	0.113746
$443$	−5.08549e24	−0.367703	−0.183851	−	0.982954i	$-0.558857\pi$
$443$	−5.08549e24	−0.367703	−0.183851	+	0.982954i	$0.558857\pi$
$444$	−7.08455e22	−0.00501388
$445$	1.41705e25	0.981668
$446$	−6.28500e23	−0.0426211
$447$	1.55006e23	0.0102903
$448$	−1.08944e23	−0.00708046
$449$	3.21734e23	0.0204718	0.0102359	−	0.999948i	$-0.496742\pi$
$449$	3.21734e23	0.0204718	0.0102359	+	0.999948i	$0.496742\pi$
$450$	7.36475e24	0.458817
$451$	1.81494e24	0.110710
$452$	−6.08250e23	−0.0363302
$453$	−1.21743e23	−0.00712053
$454$	−1.06493e25	−0.609946
$455$	7.48449e23	0.0419811
$456$	−1.68632e22	−0.000926348 0
$457$	2.83959e25	1.52775	0.763875	−	0.645364i	$-0.223294\pi$
$457$	2.83959e25	1.52775	0.763875	+	0.645364i	$0.223294\pi$
$458$	−1.75409e25	−0.924335
$459$	5.04284e22	0.00260288
$460$	6.36281e24	0.321699
$461$	−9.77800e24	−0.484274	−0.242137	−	0.970242i	$-0.577848\pi$
$461$	−9.77800e24	−0.484274	−0.242137	+	0.970242i	$0.577848\pi$
$462$	2.51908e21	0.000122220 0
$463$	1.39305e25	0.662138	0.331069	−	0.943607i	$-0.392591\pi$
$463$	1.39305e25	0.662138	0.331069	+	0.943607i	$0.392591\pi$
$464$	−2.30519e24	−0.107346
$465$	−1.15750e23	−0.00528101
$466$	1.87781e25	0.839431
$467$	−3.64842e25	−1.59807	−0.799033	−	0.601287i	$-0.794655\pi$
$467$	−3.64842e25	−1.59807	−0.799033	+	0.601287i	$0.794655\pi$
$468$	1.45703e25	0.625362
$469$	1.25063e23	0.00526000
$470$	4.05805e24	0.167258
$471$	2.50381e23	0.0101135
$472$	−1.33444e25	−0.528265
$473$	−3.41881e24	−0.132646
$474$	1.56976e22	0.000596951 0
$475$	4.50717e24	0.168002
$476$	9.96853e22	0.00364221
$477$	−2.41344e25	−0.864393
$478$	2.05221e25	0.720538
$479$	−5.15289e25	−1.77363	−0.886815	−	0.462124i	$-0.847087\pi$
$479$	−5.15289e25	−1.77363	−0.886815	+	0.462124i	$0.847087\pi$
$480$	3.14135e22	0.00106005
$481$	3.74615e25	1.23939
$482$	1.05112e25	0.340961
$483$	−1.95721e22	−0.000622499 0
$484$	1.45750e24	0.0454545
$485$	1.08986e25	0.333292
$486$	7.15805e23	0.0214660
$487$	−6.08197e25	−1.78862	−0.894312	−	0.447444i	$-0.852334\pi$
$487$	−6.08197e25	−1.78862	−0.894312	+	0.447444i	$0.852334\pi$
$488$	−1.66129e25	−0.479133
$489$	5.22135e23	0.0147689
$490$	−1.50539e25	−0.417624
$491$	−4.37774e25	−1.19118	−0.595588	−	0.803290i	$-0.703081\pi$
$491$	−4.37774e25	−1.19118	−0.595588	+	0.803290i	$0.703081\pi$
$492$	−6.96183e22	−0.00185804
$493$	2.10928e24	0.0552191
$494$	8.91688e24	0.228985
$495$	7.09096e24	0.178630
$496$	−8.90909e24	−0.220170
$497$	3.37762e24	0.0818890
$498$	1.92328e23	0.00457472
$499$	−5.91618e25	−1.38066	−0.690329	−	0.723496i	$-0.742534\pi$
$499$	−5.91618e25	−1.38066	−0.690329	+	0.723496i	$0.742534\pi$
$500$	−2.13346e25	−0.488505
$501$	5.15560e23	0.0115830
$502$	5.12718e25	1.13030
$503$	−2.71878e25	−0.588136	−0.294068	−	0.955785i	$-0.595009\pi$
$503$	−2.71878e25	−0.588136	−0.294068	+	0.955785i	$0.595009\pi$
$504$	9.43305e23	0.0200245
$505$	3.42176e25	0.712822
$506$	−1.13241e25	−0.231511
$507$	2.84803e23	0.00571438
$508$	2.70708e25	0.533085
$509$	2.92714e23	0.00565749	0.00282875	−	0.999996i	$-0.499100\pi$
$509$	2.92714e23	0.00565749	0.00282875	+	0.999996i	$0.499100\pi$
$510$	−2.87438e22	−0.000545290 0
$511$	4.22783e24	0.0787262
$512$	2.41785e24	0.0441942
$513$	2.92040e23	0.00523995
$514$	3.13075e25	0.551440
$515$	−1.51793e25	−0.262472
$516$	1.31140e23	0.00222620
$517$	−7.22222e24	−0.120368
$518$	2.42533e24	0.0396860
$519$	−1.65205e23	−0.00265419
$520$	−1.66107e25	−0.262034
$521$	−9.81725e25	−1.52066	−0.760329	−	0.649538i	$-0.774962\pi$
$521$	−9.81725e25	−1.52066	−0.760329	+	0.649538i	$0.774962\pi$
$522$	1.99598e25	0.303589
$523$	−1.03966e26	−1.55284	−0.776421	−	0.630215i	$-0.782967\pi$
$523$	−1.03966e26	−1.55284	−0.776421	+	0.630215i	$0.782967\pi$
$524$	8.45758e24	0.124051
$525$	2.58266e22	0.000372009 0
$526$	−2.32432e25	−0.328799
$527$	8.15195e24	0.113256
$528$	−5.59074e22	−0.000762864 0
$529$	1.33670e25	0.179145
$530$	2.75142e25	0.362190
$531$	1.15545e26	1.49401
$532$	5.77295e23	0.00733225
$533$	3.68126e25	0.459291
$534$	9.67384e23	0.0118565
$535$	3.82847e25	0.460962
$536$	−2.77558e24	−0.0328314
$537$	4.10560e23	0.00477113
$538$	1.06971e25	0.122134
$539$	2.67918e25	0.300544
$540$	−5.44024e23	−0.00599622
$541$	1.62615e26	1.76111	0.880556	−	0.473942i	$-0.157169\pi$
$541$	1.62615e26	1.76111	0.880556	+	0.473942i	$0.157169\pi$
$542$	1.92687e25	0.205050
$543$	−7.04889e23	−0.00737093
$544$	−2.21237e24	−0.0227336
$545$	3.72606e25	0.376256
$546$	5.10948e22	0.000507045 0
$547$	1.44943e26	1.41357	0.706787	−	0.707427i	$-0.250144\pi$
$547$	1.44943e26	1.41357	0.706787	+	0.707427i	$0.250144\pi$
$548$	−3.69647e25	−0.354301
$549$	1.43845e26	1.35505
$550$	1.49428e25	0.138353
$551$	1.22152e25	0.111163
$552$	4.34373e23	0.00388546
$553$	−5.37392e23	−0.00472500
$554$	−8.48444e25	−0.733298
$555$	−6.99332e23	−0.00594155
$556$	−6.00581e25	−0.501604
$557$	−2.66891e25	−0.219134	−0.109567	−	0.993979i	$-0.534946\pi$
$557$	−2.66891e25	−0.219134	−0.109567	+	0.993979i	$0.534946\pi$
$558$	7.71406e25	0.622670
$559$	−6.93440e25	−0.550296
$560$	−1.07541e24	−0.00839050
$561$	5.11561e22	0.000392419 0
$562$	1.54622e26	1.16621
$563$	−2.78210e25	−0.206321	−0.103160	−	0.994665i	$-0.532895\pi$
$563$	−2.78210e25	−0.206321	−0.103160	+	0.994665i	$0.532895\pi$
$564$	2.77033e23	0.00202013
$565$	−6.00418e24	−0.0430520
$566$	−2.90481e25	−0.204815
$567$	−8.16690e24	−0.0566263
$568$	−7.49614e25	−0.511127
$569$	1.85493e26	1.24383	0.621917	−	0.783084i	$-0.286354\pi$
$569$	1.85493e26	1.24383	0.621917	+	0.783084i	$0.286354\pi$
$570$	−1.66461e23	−0.00109774
$571$	−9.98871e25	−0.647838	−0.323919	−	0.946085i	$-0.605001\pi$
$571$	−9.98871e25	−0.647838	−0.323919	+	0.946085i	$0.605001\pi$
$572$	2.95626e25	0.188573
$573$	−7.85104e23	−0.00492559
$574$	2.38331e24	0.0147068
$575$	−1.16099e26	−0.704664
$576$	−2.09353e25	−0.124987
$577$	1.29720e26	0.761794	0.380897	−	0.924617i	$-0.375615\pi$
$577$	1.29720e26	0.761794	0.380897	+	0.924617i	$0.375615\pi$
$578$	−1.20381e26	−0.695413
$579$	−7.24729e23	−0.00411841
$580$	−2.27551e25	−0.127207
$581$	−6.58417e24	−0.0362100
$582$	7.44021e23	0.00402547
$583$	−4.89678e25	−0.260651
$584$	−9.38306e25	−0.491386
$585$	1.43826e26	0.741068
$586$	−6.96489e25	−0.353091
$587$	−2.96599e26	−1.47947	−0.739737	−	0.672896i	$-0.765050\pi$
$587$	−2.96599e26	−1.47947	−0.739737	+	0.672896i	$0.765050\pi$
$588$	−1.02769e24	−0.00504403
$589$	4.72094e25	0.227999
$590$	−1.31726e26	−0.626005
$591$	−1.37837e24	−0.00644591
$592$	−5.38266e25	−0.247708
$593$	−3.27043e25	−0.148110	−0.0740551	−	0.997254i	$-0.523594\pi$
$593$	−3.27043e25	−0.148110	−0.0740551	+	0.997254i	$0.523594\pi$
$594$	9.68213e23	0.00431519
$595$	9.84017e23	0.00431610
$596$	1.17769e26	0.508386
$597$	−2.70025e24	−0.0114722
$598$	−2.29687e26	−0.960450
$599$	−3.62773e26	−1.49307	−0.746534	−	0.665347i	$-0.768284\pi$
$599$	−3.62773e26	−1.49307	−0.746534	+	0.665347i	$0.768284\pi$
$600$	−5.73184e23	−0.00232197
$601$	−2.61429e26	−1.04243	−0.521215	−	0.853425i	$-0.674521\pi$
$601$	−2.61429e26	−1.04243	−0.521215	+	0.853425i	$0.674521\pi$
$602$	−4.48946e24	−0.0176209
$603$	2.40328e25	0.0928516
$604$	−9.24974e25	−0.351786
$605$	1.43873e25	0.0538646
$606$	2.33595e24	0.00860941
$607$	5.02506e26	1.82326	0.911631	−	0.411010i	$-0.134824\pi$
$607$	5.02506e26	1.82326	0.911631	+	0.411010i	$0.134824\pi$
$608$	−1.28122e25	−0.0457658
$609$	6.99948e22	0.000246151 0
$610$	−1.63989e26	−0.567783
$611$	−1.46489e26	−0.499359
$612$	1.91561e25	0.0642938
$613$	3.87214e26	1.27961	0.639804	−	0.768538i	$-0.279016\pi$
$613$	3.87214e26	1.27961	0.639804	+	0.768538i	$0.279016\pi$
$614$	4.44614e25	0.144672
$615$	−6.87218e23	−0.00220182
$616$	1.91393e24	0.00603824
$617$	1.12801e23	0.000350433 0	0.000175216	−	1.00000i	$-0.499944\pi$
$617$	1.12801e23	0.000350433 0	0.000175216		1.00000i	$0.499944\pi$
$618$	−1.03625e24	−0.00317011
$619$	−3.71693e26	−1.11976	−0.559878	−	0.828575i	$-0.689152\pi$
$619$	−3.71693e26	−1.11976	−0.559878	+	0.828575i	$0.689152\pi$
$620$	−8.79437e25	−0.260906
$621$	−7.52255e24	−0.0219783
$622$	−6.07364e25	−0.174759
$623$	−3.31174e25	−0.0938470
$624$	−1.13397e24	−0.00316482
$625$	2.54813e25	0.0700426
$626$	1.16524e26	0.315471
$627$	2.96254e23	0.000789992 0
$628$	1.90233e26	0.499653
$629$	4.92521e25	0.127422
$630$	9.31159e24	0.0237295
$631$	−5.71909e26	−1.43565	−0.717823	−	0.696225i	$-0.754862\pi$
$631$	−5.71909e26	−1.43565	−0.717823	+	0.696225i	$0.754862\pi$
$632$	1.19266e25	0.0294921
$633$	−6.04318e24	−0.0147208
$634$	1.22951e26	0.295043
$635$	2.67223e26	0.631717
$636$	1.87833e24	0.00437451
$637$	5.43419e26	1.24684
$638$	4.04978e25	0.0915450
$639$	6.49064e26	1.44554
$640$	2.38672e25	0.0523710
$641$	−7.23640e26	−1.56448	−0.782242	−	0.622975i	$-0.785924\pi$
$641$	−7.23640e26	−1.56448	−0.782242	+	0.622975i	$0.785924\pi$
$642$	2.61360e24	0.00556746
$643$	5.02561e26	1.05484	0.527418	−	0.849606i	$-0.323160\pi$
$643$	5.02561e26	1.05484	0.527418	+	0.849606i	$0.323160\pi$
$644$	−1.48703e25	−0.0307543
$645$	1.29452e24	0.00263809
$646$	1.17234e25	0.0235420
$647$	−5.70800e26	−1.12952	−0.564760	−	0.825255i	$-0.691031\pi$
$647$	−5.70800e26	−1.12952	−0.564760	+	0.825255i	$0.691031\pi$
$648$	1.81253e26	0.353445
$649$	2.34436e26	0.450506
$650$	3.03086e26	0.573971
$651$	2.70515e23	0.000504862 0
$652$	3.96705e26	0.729652
$653$	−8.84073e26	−1.60256	−0.801278	−	0.598293i	$-0.795846\pi$
$653$	−8.84073e26	−1.60256	−0.801278	+	0.598293i	$0.795846\pi$
$654$	2.54369e24	0.00454439
$655$	8.34867e25	0.147003
$656$	−5.28942e25	−0.0917955
$657$	8.12445e26	1.38971
$658$	−9.48395e24	−0.0159898
$659$	−3.91234e26	−0.650168	−0.325084	−	0.945685i	$-0.605393\pi$
$659$	−3.91234e26	−0.650168	−0.325084	+	0.945685i	$0.605393\pi$
$660$	−5.51875e23	−0.000904010 0
$661$	2.71921e26	0.439065	0.219533	−	0.975605i	$-0.429547\pi$
$661$	2.71921e26	0.439065	0.219533	+	0.975605i	$0.429547\pi$
$662$	−1.37327e26	−0.218577
$663$	1.03760e24	0.00162799
$664$	1.46126e26	0.226012
$665$	5.69861e24	0.00868888
$666$	4.66065e26	0.700553
$667$	−3.14648e26	−0.466262
$668$	3.91709e26	0.572252
$669$	−4.23536e23	−0.000610018 0
$670$	−2.73984e25	−0.0389059
$671$	2.91856e26	0.408606
$672$	−7.34156e22	−0.000101340 0
$673$	−4.51297e26	−0.614213	−0.307106	−	0.951675i	$-0.599361\pi$
$673$	−4.51297e26	−0.614213	−0.307106	+	0.951675i	$0.599361\pi$
$674$	−6.06632e26	−0.814059
$675$	9.92649e24	0.0131344
$676$	2.16386e26	0.282316
$677$	1.48745e26	0.191359	0.0956796	−	0.995412i	$-0.469498\pi$
$677$	1.48745e26	0.191359	0.0956796	+	0.995412i	$0.469498\pi$
$678$	−4.09890e23	−0.000519979 0
$679$	−2.54708e25	−0.0318625
$680$	−2.18388e25	−0.0269398
$681$	−7.17641e24	−0.00872991
$682$	1.56516e26	0.187761
$683$	1.48608e27	1.75810	0.879052	−	0.476727i	$-0.158177\pi$
$683$	1.48608e27	1.75810	0.879052	+	0.476727i	$0.158177\pi$
$684$	1.10936e26	0.129432
$685$	−3.64887e26	−0.419854
$686$	7.04772e25	0.0799778
$687$	−1.18205e25	−0.0132296
$688$	9.96370e25	0.109984
$689$	−9.93218e26	−1.08134
$690$	4.28780e24	0.00460435
$691$	−7.59042e26	−0.803941	−0.401970	−	0.915653i	$-0.631675\pi$
$691$	−7.59042e26	−0.803941	−0.401970	+	0.915653i	$0.631675\pi$
$692$	−1.25518e26	−0.131129
$693$	−1.65721e25	−0.0170770
$694$	3.30509e26	0.335945
$695$	−5.92847e26	−0.594411
$696$	−1.55343e24	−0.00153640
$697$	4.83990e25	0.0472199
$698$	8.21578e26	0.790719
$699$	1.26543e25	0.0120144
$700$	1.96224e25	0.0183789
$701$	1.54787e27	1.43026	0.715128	−	0.698993i	$-0.246368\pi$
$701$	1.54787e27	1.43026	0.715128	+	0.698993i	$0.246368\pi$
$702$	1.96383e25	0.0179020
$703$	2.85228e26	0.256517
$704$	−4.24770e25	−0.0376889
$705$	2.73466e24	0.00239390
$706$	6.35143e26	0.548564
$707$	−7.99689e25	−0.0681454
$708$	−8.99261e24	−0.00756084
$709$	−6.05830e26	−0.502587	−0.251294	−	0.967911i	$-0.580856\pi$
$709$	−6.05830e26	−0.502587	−0.251294	+	0.967911i	$0.580856\pi$
$710$	−7.39962e26	−0.605696
$711$	−1.03268e26	−0.0834077
$712$	7.34993e26	0.585765
$713$	−1.21605e27	−0.956315
$714$	6.71764e22	5.21295e−5 0
$715$	2.91819e26	0.223463
$716$	3.11933e26	0.235715
$717$	1.38295e25	0.0103128
$718$	−1.64999e27	−1.21422
$719$	−1.47676e27	−1.07247	−0.536236	−	0.844068i	$-0.680154\pi$
$719$	−1.47676e27	−1.07247	−0.536236	+	0.844068i	$0.680154\pi$
$720$	−2.06657e26	−0.148113
$721$	3.54750e25	0.0250921
$722$	−9.45059e26	−0.659714
$723$	7.08332e24	0.00488003
$724$	−5.35557e26	−0.364157
$725$	4.15198e26	0.278641
$726$	9.82186e23	0.000650572 0
$727$	3.01692e27	1.97236	0.986181	−	0.165669i	$-0.0529783\pi$
$727$	3.01692e27	1.97236	0.986181	+	0.165669i	$0.0529783\pi$
$728$	3.88205e25	0.0250503
$729$	−1.56908e27	−0.999386
$730$	−9.26224e26	−0.582302
$731$	−9.11693e25	−0.0565762
$732$	−1.11951e25	−0.00685763
$733$	−2.82053e27	−1.70546	−0.852732	−	0.522349i	$-0.825056\pi$
$733$	−2.82053e27	−1.70546	−0.852732	+	0.522349i	$0.825056\pi$
$734$	1.92591e27	1.14954
$735$	−1.01446e25	−0.00597728
$736$	3.30026e26	0.191959
$737$	4.87617e25	0.0279987
$738$	4.57991e26	0.259610
$739$	1.19184e27	0.666951	0.333476	−	0.942759i	$-0.391779\pi$
$739$	1.19184e27	0.666951	0.333476	+	0.942759i	$0.391779\pi$
$740$	−5.31335e26	−0.293540
$741$	6.00894e24	0.00327737
$742$	−6.43028e25	−0.0346252
$743$	−1.82741e27	−0.971501	−0.485751	−	0.874097i	$-0.661454\pi$
$743$	−1.82741e27	−0.971501	−0.485751	+	0.874097i	$0.661454\pi$
$744$	−6.00369e24	−0.00315120
$745$	1.16253e27	0.602448
$746$	−2.10248e27	−1.07576
$747$	−1.26525e27	−0.639193
$748$	3.88671e25	0.0193873
$749$	−8.94742e25	−0.0440677
$750$	−1.43770e25	−0.00699177
$751$	1.93421e27	0.928804	0.464402	−	0.885624i	$-0.346269\pi$
$751$	1.93421e27	0.928804	0.464402	+	0.885624i	$0.346269\pi$
$752$	2.10483e26	0.0998037
$753$	3.45513e25	0.0161775
$754$	8.21420e26	0.379785
$755$	−9.13064e26	−0.416874
$756$	1.27142e24	0.000573235 0
$757$	−1.24431e27	−0.554008	−0.277004	−	0.960869i	$-0.589342\pi$
$757$	−1.24431e27	−0.554008	−0.277004	+	0.960869i	$0.589342\pi$
$758$	−1.37191e27	−0.603210
$759$	−7.63111e24	−0.00331353
$760$	−1.26472e26	−0.0542334
$761$	1.62980e26	0.0690208	0.0345104	−	0.999404i	$-0.489013\pi$
$761$	1.62980e26	0.0690208	0.0345104	+	0.999404i	$0.489013\pi$
$762$	1.82426e25	0.00762983
$763$	−8.70807e25	−0.0359699
$764$	−5.96502e26	−0.243346
$765$	1.89094e26	0.0761895
$766$	−3.19815e27	−1.27270
$767$	4.75509e27	1.86897
$768$	1.62935e24	0.000632533 0
$769$	−3.63280e26	−0.139297	−0.0696483	−	0.997572i	$-0.522188\pi$
$769$	−3.63280e26	−0.139297	−0.0696483	+	0.997572i	$0.522188\pi$
$770$	1.88929e25	0.00715544
$771$	2.10976e25	0.00789254
$772$	−5.50630e26	−0.203468
$773$	−2.05704e27	−0.750824	−0.375412	−	0.926858i	$-0.622499\pi$
$773$	−2.05704e27	−0.750824	−0.375412	+	0.926858i	$0.622499\pi$
$774$	−8.62720e26	−0.311050
$775$	1.60466e27	0.571500
$776$	5.65288e26	0.198877
$777$	1.63439e24	0.000568010 0
$778$	2.15207e26	0.0738841
$779$	2.80287e26	0.0950599
$780$	−1.11937e25	−0.00375038
$781$	1.31693e27	0.435891
$782$	−3.01979e26	−0.0987443
$783$	2.69026e25	0.00869075
$784$	−7.80813e26	−0.249198
$785$	1.87783e27	0.592100
$786$	5.69943e24	0.00177549
$787$	2.83950e27	0.873939	0.436970	−	0.899476i	$-0.356052\pi$
$787$	2.83950e27	0.873939	0.436970	+	0.899476i	$0.356052\pi$
$788$	−1.04725e27	−0.318457
$789$	−1.56632e25	−0.00470597
$790$	1.17731e26	0.0349488
$791$	1.40322e25	0.00411575
$792$	3.67793e26	0.106589
$793$	5.91974e27	1.69515
$794$	5.87623e26	0.166266
$795$	1.85414e25	0.00518388
$796$	−2.05158e27	−0.566781
$797$	1.26285e26	0.0344744	0.0172372	−	0.999851i	$-0.494513\pi$
$797$	1.26285e26	0.0344744	0.0172372	+	0.999851i	$0.494513\pi$
$798$	3.89030e23	0.000104944 0
$799$	−1.92595e26	−0.0513393
$800$	−4.35490e26	−0.114716
$801$	−6.36404e27	−1.65662
$802$	3.39952e27	0.874504
$803$	1.64842e27	0.419055
$804$	−1.87042e24	−0.000469902 0
$805$	−1.46789e26	−0.0364445
$806$	3.17462e27	0.778948
$807$	7.20862e24	0.00174805
$808$	1.77479e27	0.425344
$809$	4.54094e27	1.07556	0.537780	−	0.843085i	$-0.319263\pi$
$809$	4.54094e27	1.07556	0.537780	+	0.843085i	$0.319263\pi$
$810$	1.78919e27	0.418840
$811$	7.10547e27	1.64397	0.821986	−	0.569507i	$-0.192866\pi$
$811$	7.10547e27	1.64397	0.821986	+	0.569507i	$0.192866\pi$
$812$	5.31802e25	0.0121610
$813$	1.29849e25	0.00293479
$814$	9.45630e26	0.211246
$815$	3.91597e27	0.864653
$816$	−1.49088e24	−0.000325377 0
$817$	−5.27978e26	−0.113895
$818$	2.38872e27	0.509341
$819$	−3.36132e26	−0.0708457
$820$	−5.22131e26	−0.108780
$821$	7.96541e25	0.0164039	0.00820197	−	0.999966i	$-0.497389\pi$
$821$	7.96541e25	0.0164039	0.00820197	+	0.999966i	$0.497389\pi$
$822$	−2.49099e25	−0.00507096
$823$	−2.76793e27	−0.557002	−0.278501	−	0.960436i	$-0.589838\pi$
$823$	−2.76793e27	−0.557002	−0.278501	+	0.960436i	$0.589838\pi$
$824$	−7.87317e26	−0.156618
$825$	1.00697e25	0.00198018
$826$	3.07853e26	0.0598458
$827$	5.70167e27	1.09572	0.547860	−	0.836570i	$-0.315443\pi$
$827$	5.70167e27	1.09572	0.547860	+	0.836570i	$0.315443\pi$
$828$	−2.85757e27	−0.542887
$829$	−9.94200e27	−1.86726	−0.933632	−	0.358233i	$-0.883379\pi$
$829$	−9.94200e27	−1.86726	−0.933632	+	0.358233i	$0.883379\pi$
$830$	1.44245e27	0.267829
$831$	−5.71753e25	−0.0104954
$832$	−8.61564e26	−0.156357
$833$	7.14456e26	0.128188
$834$	−4.04722e25	−0.00717925
$835$	3.86665e27	0.678131
$836$	2.25086e26	0.0390292
$837$	1.03973e26	0.0178250
$838$	−8.01673e25	−0.0135887
$839$	−6.37699e27	−1.06875	−0.534376	−	0.845247i	$-0.679453\pi$
$839$	−6.37699e27	−1.06875	−0.534376	+	0.845247i	$0.679453\pi$
$840$	−7.24702e23	−0.000120090 0
$841$	−4.97800e27	−0.815629
$842$	−1.75051e25	−0.00283596
$843$	1.04198e26	0.0166915
$844$	−4.59145e27	−0.727272
$845$	2.13600e27	0.334551
$846$	−1.82249e27	−0.282259
$847$	−3.36242e25	−0.00514943
$848$	1.42711e27	0.216120
$849$	−1.95750e25	−0.00293143
$850$	3.98480e26	0.0590102
$851$	−7.34708e27	−1.07593
$852$	−5.05153e25	−0.00731555
$853$	−9.25535e27	−1.32549	−0.662746	−	0.748845i	$-0.730609\pi$
$853$	−9.25535e27	−1.32549	−0.662746	+	0.748845i	$0.730609\pi$
$854$	3.83255e26	0.0542797
$855$	1.09508e27	0.153380
$856$	1.98575e27	0.275058
$857$	−8.46592e27	−1.15973	−0.579864	−	0.814713i	$-0.696894\pi$
$857$	−8.46592e27	−1.15973	−0.579864	+	0.814713i	$0.696894\pi$
$858$	1.99217e25	0.00269897
$859$	−5.24101e27	−0.702231	−0.351116	−	0.936332i	$-0.614198\pi$
$859$	−5.24101e27	−0.702231	−0.351116	+	0.936332i	$0.614198\pi$
$860$	9.83540e26	0.130334
$861$	1.60608e24	0.000210492 0
$862$	−3.08234e27	−0.399541
$863$	−3.67642e27	−0.471327	−0.235663	−	0.971835i	$-0.575726\pi$
$863$	−3.67642e27	−0.471327	−0.235663	+	0.971835i	$0.575726\pi$
$864$	−2.82174e25	−0.00357797
$865$	−1.23902e27	−0.155391
$866$	3.90366e27	0.484230
$867$	−8.11227e25	−0.00995316
$868$	2.05531e26	0.0249424
$869$	−2.09528e26	−0.0251509
$870$	−1.53343e25	−0.00182067
$871$	9.89037e26	0.116156
$872$	1.93263e27	0.224513
$873$	−4.89462e27	−0.562450
$874$	−1.74881e27	−0.198785
$875$	4.92183e26	0.0553414
$876$	−6.32310e25	−0.00703300
$877$	3.01762e25	0.00332023	0.00166012	−	0.999999i	$-0.499472\pi$
$877$	3.01762e25	0.00332023	0.00166012	+	0.999999i	$0.499472\pi$
$878$	1.74156e27	0.189557
$879$	−4.69353e25	−0.00505365
$880$	−4.19300e26	−0.0446622
$881$	−3.12357e27	−0.329139	−0.164570	−	0.986365i	$-0.552624\pi$
$881$	−3.12357e27	−0.329139	−0.164570	+	0.986365i	$0.552624\pi$
$882$	6.76077e27	0.704766
$883$	−6.56027e27	−0.676542	−0.338271	−	0.941049i	$-0.609842\pi$
$883$	−6.56027e27	−0.676542	−0.338271	+	0.941049i	$0.609842\pi$
$884$	7.88344e26	0.0804303
$885$	−8.87682e25	−0.00895976
$886$	−2.60377e27	−0.260005
$887$	2.29870e27	0.227095	0.113548	−	0.993533i	$-0.463779\pi$
$887$	2.29870e27	0.227095	0.113548	+	0.993533i	$0.463779\pi$
$888$	−3.62729e25	−0.00354535
$889$	−6.24518e26	−0.0603918
$890$	7.25529e27	0.694144
$891$	−3.18426e27	−0.301419
$892$	−3.21792e26	−0.0301376
$893$	−1.11535e27	−0.103353
$894$	7.93630e25	0.00727632
$895$	3.07916e27	0.279328
$896$	−5.57793e25	−0.00500664
$897$	−1.54782e26	−0.0137465
$898$	1.64728e26	0.0144758
$899$	4.34891e27	0.378149
$900$	3.77075e27	0.324432
$901$	−1.30582e27	−0.111173
$902$	9.29249e26	0.0782835
$903$	−3.02537e24	−0.000252200 0
$904$	−3.11424e26	−0.0256893
$905$	−5.28660e27	−0.431534
$906$	−6.23326e25	−0.00503497
$907$	−3.65191e26	−0.0291912	−0.0145956	−	0.999893i	$-0.504646\pi$
$907$	−3.65191e26	−0.0291912	−0.0145956	+	0.999893i	$0.504646\pi$
$908$	−5.45245e27	−0.431297
$909$	−1.53673e28	−1.20293
$910$	3.83206e26	0.0296851
$911$	1.82592e28	1.39977	0.699885	−	0.714256i	$-0.253235\pi$
$911$	1.82592e28	1.39977	0.699885	+	0.714256i	$0.253235\pi$
$912$	−8.63396e24	−0.000655027 0
$913$	−2.56716e27	−0.192744
$914$	1.45387e28	1.08028
$915$	−1.10510e26	−0.00812644
$916$	−8.98093e27	−0.653603
$917$	−1.95114e26	−0.0140534
$918$	2.58193e25	0.00184052
$919$	−2.35496e27	−0.166145	−0.0830725	−	0.996544i	$-0.526473\pi$
$919$	−2.35496e27	−0.166145	−0.0830725	+	0.996544i	$0.526473\pi$
$920$	3.25776e27	0.227476
$921$	2.99618e25	0.00207063
$922$	−5.00634e27	−0.342434
$923$	2.67114e28	1.80834
$924$	1.28977e24	8.64229e−5 0
$925$	9.69495e27	0.642982
$926$	7.13244e27	0.468202
$927$	6.81709e27	0.442937
$928$	−1.18026e27	−0.0759051
$929$	9.74001e27	0.620026	0.310013	−	0.950732i	$-0.399667\pi$
$929$	9.74001e27	0.620026	0.310013	+	0.950732i	$0.399667\pi$
$930$	−5.92639e25	−0.00373424
$931$	4.13754e27	0.258060
$932$	9.61438e27	0.593568
$933$	−4.09293e25	−0.00250126
$934$	−1.86799e28	−1.13000
$935$	3.83666e26	0.0229743
$936$	7.45997e27	0.442198
$937$	−6.10388e27	−0.358163	−0.179081	−	0.983834i	$-0.557313\pi$
$937$	−6.10388e27	−0.358163	−0.179081	+	0.983834i	$0.557313\pi$
$938$	6.40321e25	0.00371938
$939$	7.85237e25	0.00451521
$940$	2.07772e27	0.118270
$941$	1.51509e28	0.853762	0.426881	−	0.904308i	$-0.359612\pi$
$941$	1.51509e28	0.853762	0.426881	+	0.904308i	$0.359612\pi$
$942$	1.28195e26	0.00715134
$943$	−7.21982e27	−0.398717
$944$	−6.83236e27	−0.373540
$945$	1.25505e25	0.000679296 0
$946$	−1.75043e27	−0.0937948
$947$	−1.44086e28	−0.764361	−0.382180	−	0.924088i	$-0.624827\pi$
$947$	−1.44086e28	−0.764361	−0.382180	+	0.924088i	$0.624827\pi$
$948$	8.03717e24	0.000422108 0
$949$	3.34351e28	1.73850
$950$	2.30767e27	0.118795
$951$	8.28548e25	0.00422282
$952$	5.10389e25	0.00257543
$953$	2.59956e28	1.29872	0.649362	−	0.760479i	$-0.275036\pi$
$953$	2.59956e28	1.29872	0.649362	+	0.760479i	$0.275036\pi$
$954$	−1.23568e28	−0.611218
$955$	−5.88821e27	−0.288371
$956$	1.05073e28	0.509497
$957$	2.72908e25	0.00131025
$958$	−2.63828e28	−1.25415
$959$	8.52767e26	0.0401378
$960$	1.60837e25	0.000749565 0
$961$	−4.86302e27	−0.224406
$962$	1.91803e28	0.876378
$963$	−1.71939e28	−0.777901
$964$	5.38172e27	0.241096
$965$	−5.43540e27	−0.241114
$966$	−1.00209e25	−0.000440173 0
$967$	1.49073e28	0.648407	0.324203	−	0.945987i	$-0.394904\pi$
$967$	1.49073e28	0.648407	0.324203	+	0.945987i	$0.394904\pi$
$968$	7.46239e26	0.0321412
$969$	7.90020e24	0.000336948 0
$970$	5.58009e27	0.235673
$971$	1.73196e28	0.724361	0.362181	−	0.932108i	$-0.382032\pi$
$971$	1.73196e28	0.724361	0.362181	+	0.932108i	$0.382032\pi$
$972$	3.66492e26	0.0151787
$973$	1.38553e27	0.0568254
$974$	−3.11397e28	−1.26475
$975$	2.04245e26	0.00821501
$976$	−8.50578e27	−0.338798
$977$	1.77264e28	0.699235	0.349617	−	0.936893i	$-0.386312\pi$
$977$	1.77264e28	0.699235	0.349617	+	0.936893i	$0.386312\pi$
$978$	2.67333e26	0.0104432
$979$	−1.29124e28	−0.499542
$980$	−7.70758e27	−0.295305
$981$	−1.67339e28	−0.634955
$982$	−2.24140e28	−0.842289
$983$	−1.23485e28	−0.459573	−0.229787	−	0.973241i	$-0.573803\pi$
$983$	−1.23485e28	−0.459573	−0.229787	+	0.973241i	$0.573803\pi$
$984$	−3.56446e25	−0.00131383
$985$	−1.03376e28	−0.377378
$986$	1.07995e27	0.0390458
$987$	−6.39109e24	−0.000228856 0
$988$	4.56544e27	0.161917
$989$	1.36000e28	0.477721
$990$	3.63057e27	0.126311
$991$	−3.39030e28	−1.16826	−0.584129	−	0.811661i	$-0.698564\pi$
$991$	−3.39030e28	−1.16826	−0.584129	+	0.811661i	$0.698564\pi$
$992$	−4.56145e27	−0.155683
$993$	−9.25422e25	−0.00312840
$994$	1.72934e27	0.0579043
$995$	−2.02517e28	−0.671648
$996$	9.84721e25	0.00323482
$997$	3.89859e28	1.26854	0.634269	−	0.773112i	$-0.281301\pi$
$997$	3.89859e28	1.26854	0.634269	+	0.773112i	$0.281301\pi$
$998$	−3.02908e28	−0.976273
$999$	6.28180e26	0.0200545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.20.a.b.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.20.a.b.1.1	✓	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	22.a (trivial)

\(f(q)\)	\(=\)	\(q+512.000 q^{2} +345.029 q^{3} +262144. q^{4} +2.58768e6 q^{5} +176655. q^{6} -6.04760e6 q^{7} +1.34218e8 q^{8} -1.16214e9 q^{9} +O(q^{10})\)	\(q+512.000 q^{2} +345.029 q^{3} +262144. q^{4} +2.58768e6 q^{5} +176655. q^{6} -6.04760e6 q^{7} +1.34218e8 q^{8} -1.16214e9 q^{9} +1.32489e9 q^{10} -2.35795e9 q^{11} +9.04472e7 q^{12} -4.78264e10 q^{13} -3.09637e9 q^{14} +8.92825e8 q^{15} +6.87195e10 q^{16} -6.28794e10 q^{17} -5.95017e11 q^{18} -3.64145e11 q^{19} +6.78346e11 q^{20} -2.08660e9 q^{21} -1.20727e12 q^{22} +9.37990e12 q^{23} +4.63090e10 q^{24} -1.23774e13 q^{25} -2.44871e13 q^{26} -8.01986e11 q^{27} -1.58534e12 q^{28} -3.35449e13 q^{29} +4.57127e11 q^{30} -1.29644e14 q^{31} +3.51844e13 q^{32} -8.13560e11 q^{33} -3.21942e13 q^{34} -1.56493e13 q^{35} -3.04649e14 q^{36} -7.83280e14 q^{37} -1.86442e14 q^{38} -1.65015e13 q^{39} +3.47313e14 q^{40} -7.69712e14 q^{41} -1.06834e12 q^{42} +1.44991e15 q^{43} -6.18122e14 q^{44} -3.00726e15 q^{45} +4.80251e15 q^{46} +3.06293e15 q^{47} +2.37102e13 q^{48} -1.13623e16 q^{49} -6.33722e15 q^{50} -2.16952e13 q^{51} -1.25374e16 q^{52} +2.07671e16 q^{53} -4.10617e14 q^{54} -6.10162e15 q^{55} -8.11695e14 q^{56} -1.25641e14 q^{57} -1.71750e16 q^{58} -9.94239e16 q^{59} +2.34049e14 q^{60} -1.23775e17 q^{61} -6.63779e16 q^{62} +7.02817e15 q^{63} +1.80144e16 q^{64} -1.23760e17 q^{65} -4.16543e14 q^{66} -2.06797e16 q^{67} -1.64834e16 q^{68} +3.23633e15 q^{69} -8.01243e15 q^{70} -5.58506e17 q^{71} -1.55980e17 q^{72} -6.99092e17 q^{73} -4.01039e17 q^{74} -4.27055e15 q^{75} -9.54585e16 q^{76} +1.42599e16 q^{77} -8.44876e15 q^{78} +8.88603e16 q^{79} +1.77824e17 q^{80} +1.35044e18 q^{81} -3.94092e17 q^{82} +1.08872e18 q^{83} -5.46989e14 q^{84} -1.62712e17 q^{85} +7.42353e17 q^{86} -1.15740e16 q^{87} -3.16478e17 q^{88} +5.47613e18 q^{89} -1.53972e18 q^{90} +2.89235e17 q^{91} +2.45888e18 q^{92} -4.47310e16 q^{93} +1.56822e18 q^{94} -9.42293e17 q^{95} +1.21396e16 q^{96} +4.21172e18 q^{97} -5.81751e18 q^{98} +2.74027e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1024 q^{2} + 16974 q^{3} + 524288 q^{4} - 2948510 q^{5} + 8690688 q^{6} + 84929956 q^{7} + 268435456 q^{8} - 2047881204 q^{9}+O(q^{10}) \) 2 * q + 1024 * q^2 + 16974 * q^3 + 524288 * q^4 - 2948510 * q^5 + 8690688 * q^6 + 84929956 * q^7 + 268435456 * q^8 - 2047881204 * q^9	\( 2 q + 1024 q^{2} + 16974 q^{3} + 524288 q^{4} - 2948510 q^{5} + 8690688 q^{6} + 84929956 q^{7} + 268435456 q^{8} - 2047881204 q^{9} - 1509637120 q^{10} - 4715895382 q^{11} + 4449632256 q^{12} + 12931357856 q^{13} + 43484137472 q^{14} - 91168385490 q^{15} + 137438953472 q^{16} - 459787304876 q^{17} - 1048515176448 q^{18} - 1766628927912 q^{19} - 772934205440 q^{20} + 1510776580572 q^{21} - 2414538435584 q^{22} - 3266600096146 q^{23} + 2278211715072 q^{24} - 801420243000 q^{25} + 6620855222272 q^{26} - 34858123423398 q^{27} + 22263878385664 q^{28} - 97113536421264 q^{29} - 46678213370880 q^{30} - 154254203154130 q^{31} + 70368744177664 q^{32} - 40023804107034 q^{33} - 235411100096512 q^{34} - 519318683122780 q^{35} - 536839770341376 q^{36} - 571392764844202 q^{37} - 904514011090944 q^{38} + 993838018166208 q^{39} - 395742313185280 q^{40} - 184921680043224 q^{41} + 773517609252864 q^{42} + 21\!\cdots\!04 q^{43}+ \cdots + 48\!\cdots\!64 q^{99}+O(q^{100}) \) 2 * q + 1024 * q^2 + 16974 * q^3 + 524288 * q^4 - 2948510 * q^5 + 8690688 * q^6 + 84929956 * q^7 + 268435456 * q^8 - 2047881204 * q^9 - 1509637120 * q^10 - 4715895382 * q^11 + 4449632256 * q^12 + 12931357856 * q^13 + 43484137472 * q^14 - 91168385490 * q^15 + 137438953472 * q^16 - 459787304876 * q^17 - 1048515176448 * q^18 - 1766628927912 * q^19 - 772934205440 * q^20 + 1510776580572 * q^21 - 2414538435584 * q^22 - 3266600096146 * q^23 + 2278211715072 * q^24 - 801420243000 * q^25 + 6620855222272 * q^26 - 34858123423398 * q^27 + 22263878385664 * q^28 - 97113536421264 * q^29 - 46678213370880 * q^30 - 154254203154130 * q^31 + 70368744177664 * q^32 - 40023804107034 * q^33 - 235411100096512 * q^34 - 519318683122780 * q^35 - 536839770341376 * q^36 - 571392764844202 * q^37 - 904514011090944 * q^38 + 993838018166208 * q^39 - 395742313185280 * q^40 - 184921680043224 * q^41 + 773517609252864 * q^42 + 2116703700483204 * q^43 - 1236243679019008 * q^44 + 1896364379272140 * q^45 - 1672499249226752 * q^46 - 7456201298413488 * q^47 + 1166444398116864 * q^48 - 14484301062165318 * q^49 - 410327164416000 * q^50 - 6621865965055188 * q^51 + 3389877873803264 * q^52 - 14846639027241356 * q^53 - 17847359192779776 * q^54 + 6952432346390410 * q^55 + 11399105733459968 * q^56 - 23447497932939960 * q^57 - 49722130647687168 * q^58 + 27278259292488522 * q^59 - 23899245245890560 * q^60 - 221405767994019480 * q^61 - 78978152014914560 * q^62 - 73554176903617512 * q^63 + 36028797018963968 * q^64 - 460126589237118240 * q^65 - 20492187702801408 * q^66 - 211824895464439422 * q^67 - 120530483249414144 * q^68 - 207061904815621470 * q^69 - 265891165758863360 * q^70 - 752788827699476886 * q^71 - 274861962414784512 * q^72 - 1171002362670305344 * q^73 - 292553095600231424 * q^74 + 188225715573790200 * q^75 - 463111173678563328 * q^76 - 200260393646931596 * q^77 + 508845065301098496 * q^78 + 1558374634535268220 * q^79 - 202620064350863360 * q^80 + 1813578176100871890 * q^81 - 94679900182130688 * q^82 + 974129749376372 * q^83 + 396041015937466368 * q^84 + 2034647452871174740 * q^85 + 1083752294647400448 * q^86 - 1068654199999982256 * q^87 - 632956763657732096 * q^88 + 3903208985507920622 * q^89 + 970938562187335680 * q^90 + 5816830265117419168 * q^91 - 856319615604097024 * q^92 - 453968381403456030 * q^93 - 3817575064787705856 * q^94 + 6822127176303479320 * q^95 + 597219531835834368 * q^96 + 17667658647655093842 * q^97 - 7415962143828642816 * q^98 + 4828796756414099964 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

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