LMFDB - Embedded newform 23.12.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,12,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	23.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:	$17.6718931529$
Analytic rank:	$1$
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} - 2672x^{6} - 1234x^{5} + 2202967x^{4} + 2386582x^{3} - 543567396x^{2} - 1204011928x + 23305583840 $ x^8 - 2672x^6 - 1234x^5 + 2202967x^4 + 2386582x^3 - 543567396x^2 - 1204011928x + 23305583840
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$35.1157$ of defining polynomial
Character	$\chi$	$=$	23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-74.2314 q^{2} +504.234 q^{3} +3462.31 q^{4} -3053.92 q^{5} -37430.0 q^{6} -27639.7 q^{7} -104986. q^{8} +77105.4 q^{9} +O(q^{10})$	\(q-74.2314 q^{2} +504.234 q^{3} +3462.31 q^{4} -3053.92 q^{5} -37430.0 q^{6} -27639.7 q^{7} -104986. q^{8} +77105.4 q^{9} +226697. q^{10} +290643. q^{11} +1.74581e6 q^{12} +601170. q^{13} +2.05173e6 q^{14} -1.53989e6 q^{15} +702457. q^{16} -1.10350e6 q^{17} -5.72364e6 q^{18} -5.48307e6 q^{19} -1.05736e7 q^{20} -1.39369e7 q^{21} -2.15749e7 q^{22} +6.43634e6 q^{23} -5.29376e7 q^{24} -3.95017e7 q^{25} -4.46257e7 q^{26} -5.04444e7 q^{27} -9.56970e7 q^{28} -3.06394e7 q^{29} +1.14308e8 q^{30} -2.75631e7 q^{31} +1.62867e8 q^{32} +1.46552e8 q^{33} +8.19141e7 q^{34} +8.44094e7 q^{35} +2.66963e8 q^{36} -5.57307e8 q^{37} +4.07016e8 q^{38} +3.03131e8 q^{39} +3.20619e8 q^{40} -8.61514e8 q^{41} +1.03455e9 q^{42} -9.46660e8 q^{43} +1.00630e9 q^{44} -2.35474e8 q^{45} -4.77779e8 q^{46} -2.89426e9 q^{47} +3.54203e8 q^{48} -1.21337e9 q^{49} +2.93227e9 q^{50} -5.56421e8 q^{51} +2.08144e9 q^{52} +3.88609e9 q^{53} +3.74456e9 q^{54} -8.87601e8 q^{55} +2.90178e9 q^{56} -2.76475e9 q^{57} +2.27440e9 q^{58} +8.66045e9 q^{59} -5.33158e9 q^{60} -6.39897e9 q^{61} +2.04605e9 q^{62} -2.13117e9 q^{63} -1.35285e10 q^{64} -1.83593e9 q^{65} -1.08788e10 q^{66} -3.58690e9 q^{67} -3.82064e9 q^{68} +3.24543e9 q^{69} -6.26583e9 q^{70} +9.32716e9 q^{71} -8.09499e9 q^{72} +1.42089e10 q^{73} +4.13697e10 q^{74} -1.99181e10 q^{75} -1.89841e10 q^{76} -8.03328e9 q^{77} -2.25018e10 q^{78} -2.30781e10 q^{79} -2.14525e9 q^{80} -3.90948e10 q^{81} +6.39514e10 q^{82} -2.53549e10 q^{83} -4.82537e10 q^{84} +3.36999e9 q^{85} +7.02719e10 q^{86} -1.54494e10 q^{87} -3.05134e10 q^{88} +7.03025e10 q^{89} +1.74796e10 q^{90} -1.66162e10 q^{91} +2.22846e10 q^{92} -1.38983e10 q^{93} +2.14845e11 q^{94} +1.67449e10 q^{95} +8.21231e10 q^{96} +1.14492e11 q^{97} +9.00706e10 q^{98} +2.24101e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9}+O(q^{10}) $ 8 * q - 32 * q^2 - 992 * q^3 + 5120 * q^4 - 11466 * q^5 + 34618 * q^6 - 54118 * q^7 - 155568 * q^8 + 648814 * q^9	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9} + 517892 q^{10} - 291462 q^{11} - 884188 q^{12} - 2211306 q^{13} - 939584 q^{14} - 2205330 q^{15} - 8561344 q^{16} - 5775330 q^{17} - 51349034 q^{18} - 21015588 q^{19} - 65503576 q^{20} - 36171230 q^{21} - 83047784 q^{22} + 51490744 q^{23} - 129286728 q^{24} - 36491644 q^{25} - 119299562 q^{26} - 394617320 q^{27} - 392796032 q^{28} - 322285430 q^{29} - 646885140 q^{30} - 415184840 q^{31} + 31831744 q^{32} - 549306602 q^{33} - 28224252 q^{34} + 603721008 q^{35} + 690703676 q^{36} + 176642018 q^{37} + 554685496 q^{38} + 2251149264 q^{39} + 1337904816 q^{40} + 357962218 q^{41} + 340644280 q^{42} + 2500461376 q^{43} + 5064743472 q^{44} + 385017072 q^{45} - 205962976 q^{46} + 261795200 q^{47} + 4421752784 q^{48} + 2656605924 q^{49} + 1642758328 q^{50} + 6771514570 q^{51} + 3841657212 q^{52} + 3542935060 q^{53} + 18173306686 q^{54} - 10100187604 q^{55} + 7995463104 q^{56} - 14761628752 q^{57} - 9113565454 q^{58} + 930905396 q^{59} + 19344914040 q^{60} - 25338655048 q^{61} + 4385691666 q^{62} - 25499316044 q^{63} - 34067008768 q^{64} - 25954746658 q^{65} + 13172584012 q^{66} - 3123467482 q^{67} - 37358480280 q^{68} - 6384852256 q^{69} - 35719175696 q^{70} - 52612263236 q^{71} - 9100886376 q^{72} - 67014176274 q^{73} + 10171443276 q^{74} - 87540153860 q^{75} + 17955918576 q^{76} - 44516617816 q^{77} - 25596104778 q^{78} - 27683357604 q^{79} + 74357773216 q^{80} + 55141240264 q^{81} + 73615849126 q^{82} - 12253964262 q^{83} + 168565479344 q^{84} + 58779027600 q^{85} + 90522557252 q^{86} - 129275944888 q^{87} + 33736356800 q^{88} + 10662817760 q^{89} + 450294422856 q^{90} - 28336741418 q^{91} + 32954076160 q^{92} + 164368292014 q^{93} + 285145948346 q^{94} - 64104297380 q^{95} + 208023008864 q^{96} - 124519454530 q^{97} + 215615498272 q^{98} + 186256571332 q^{99}+O(q^{100}) \) 8 * q - 32 * q^2 - 992 * q^3 + 5120 * q^4 - 11466 * q^5 + 34618 * q^6 - 54118 * q^7 - 155568 * q^8 + 648814 * q^9 + 517892 * q^10 - 291462 * q^11 - 884188 * q^12 - 2211306 * q^13 - 939584 * q^14 - 2205330 * q^15 - 8561344 * q^16 - 5775330 * q^17 - 51349034 * q^18 - 21015588 * q^19 - 65503576 * q^20 - 36171230 * q^21 - 83047784 * q^22 + 51490744 * q^23 - 129286728 * q^24 - 36491644 * q^25 - 119299562 * q^26 - 394617320 * q^27 - 392796032 * q^28 - 322285430 * q^29 - 646885140 * q^30 - 415184840 * q^31 + 31831744 * q^32 - 549306602 * q^33 - 28224252 * q^34 + 603721008 * q^35 + 690703676 * q^36 + 176642018 * q^37 + 554685496 * q^38 + 2251149264 * q^39 + 1337904816 * q^40 + 357962218 * q^41 + 340644280 * q^42 + 2500461376 * q^43 + 5064743472 * q^44 + 385017072 * q^45 - 205962976 * q^46 + 261795200 * q^47 + 4421752784 * q^48 + 2656605924 * q^49 + 1642758328 * q^50 + 6771514570 * q^51 + 3841657212 * q^52 + 3542935060 * q^53 + 18173306686 * q^54 - 10100187604 * q^55 + 7995463104 * q^56 - 14761628752 * q^57 - 9113565454 * q^58 + 930905396 * q^59 + 19344914040 * q^60 - 25338655048 * q^61 + 4385691666 * q^62 - 25499316044 * q^63 - 34067008768 * q^64 - 25954746658 * q^65 + 13172584012 * q^66 - 3123467482 * q^67 - 37358480280 * q^68 - 6384852256 * q^69 - 35719175696 * q^70 - 52612263236 * q^71 - 9100886376 * q^72 - 67014176274 * q^73 + 10171443276 * q^74 - 87540153860 * q^75 + 17955918576 * q^76 - 44516617816 * q^77 - 25596104778 * q^78 - 27683357604 * q^79 + 74357773216 * q^80 + 55141240264 * q^81 + 73615849126 * q^82 - 12253964262 * q^83 + 168565479344 * q^84 + 58779027600 * q^85 + 90522557252 * q^86 - 129275944888 * q^87 + 33736356800 * q^88 + 10662817760 * q^89 + 450294422856 * q^90 - 28336741418 * q^91 + 32954076160 * q^92 + 164368292014 * q^93 + 285145948346 * q^94 - 64104297380 * q^95 + 208023008864 * q^96 - 124519454530 * q^97 + 215615498272 * q^98 + 186256571332 * 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 $

$2$	−74.2314	−1.64030	−0.820149	−	0.572150i	$-0.806110\pi$
$2$	−74.2314	−1.64030	−0.820149	+	0.572150i	$0.806110\pi$
$3$	504.234	1.19802	0.599012	−	0.800740i	$-0.295560\pi$
$3$	504.234	1.19802	0.599012	+	0.800740i	$0.295560\pi$
$4$	3462.31	1.69058
$5$	−3053.92	−0.437041	−0.218521	−	0.975832i	$-0.570123\pi$
$5$	−3053.92	−0.437041	−0.218521	+	0.975832i	$0.570123\pi$
$6$	−37430.0	−1.96512
$7$	−27639.7	−0.621575	−0.310788	−	0.950479i	$-0.600593\pi$
$7$	−27639.7	−0.621575	−0.310788	+	0.950479i	$0.600593\pi$
$8$	−104986.	−1.13276
$9$	77105.4	0.435262
$10$	226697.	0.716878
$11$	290643.	0.544127	0.272063	−	0.962279i	$-0.412294\pi$
$11$	290643.	0.544127	0.272063	+	0.962279i	$0.412294\pi$
$12$	1.74581e6	2.02536
$13$	601170.	0.449065	0.224533	−	0.974467i	$-0.427915\pi$
$13$	601170.	0.449065	0.224533	+	0.974467i	$0.427915\pi$
$14$	2.05173e6	1.01957
$15$	−1.53989e6	−0.523586
$16$	702457.	0.167479
$17$	−1.10350e6	−0.188496	−0.0942479	−	0.995549i	$-0.530045\pi$
$17$	−1.10350e6	−0.188496	−0.0942479	+	0.995549i	$0.530045\pi$
$18$	−5.72364e6	−0.713960
$19$	−5.48307e6	−0.508018	−0.254009	−	0.967202i	$-0.581749\pi$
$19$	−5.48307e6	−0.508018	−0.254009	+	0.967202i	$0.581749\pi$
$20$	−1.05736e7	−0.738853
$21$	−1.39369e7	−0.744662
$22$	−2.15749e7	−0.892531
$23$	6.43634e6	0.208514
$23$	6.43634e6	0.208514
$24$	−5.29376e7	−1.35707
$25$	−3.95017e7	−0.808995
$26$	−4.46257e7	−0.736601
$27$	−5.04444e7	−0.676570
$28$	−9.56970e7	−1.05082
$29$	−3.06394e7	−0.277390	−0.138695	−	0.990335i	$-0.544291\pi$
$29$	−3.06394e7	−0.277390	−0.138695	+	0.990335i	$0.544291\pi$
$30$	1.14308e8	0.858838
$31$	−2.75631e7	−0.172918	−0.0864588	−	0.996255i	$-0.527555\pi$
$31$	−2.75631e7	−0.172918	−0.0864588	+	0.996255i	$0.527555\pi$
$32$	1.62867e8	0.858041
$33$	1.46552e8	0.651877
$34$	8.19141e7	0.309189
$35$	8.44094e7	0.271654
$36$	2.66963e8	0.735845
$37$	−5.57307e8	−1.32125	−0.660625	−	0.750716i	$-0.729709\pi$
$37$	−5.57307e8	−1.32125	−0.660625	+	0.750716i	$0.729709\pi$
$38$	4.07016e8	0.833301
$39$	3.03131e8	0.537991
$40$	3.20619e8	0.495061
$41$	−8.61514e8	−1.16132	−0.580659	−	0.814147i	$-0.697205\pi$
$41$	−8.61514e8	−1.16132	−0.580659	+	0.814147i	$0.697205\pi$
$42$	1.03455e9	1.22147
$43$	−9.46660e8	−0.982013	−0.491007	−	0.871156i	$-0.663371\pi$
$43$	−9.46660e8	−0.982013	−0.491007	+	0.871156i	$0.663371\pi$
$44$	1.00630e9	0.919890
$45$	−2.35474e8	−0.190228
$46$	−4.77779e8	−0.342026
$47$	−2.89426e9	−1.84077	−0.920385	−	0.391013i	$-0.872125\pi$
$47$	−2.89426e9	−1.84077	−0.920385	+	0.391013i	$0.872125\pi$
$48$	3.54203e8	0.200644
$49$	−1.21337e9	−0.613644
$50$	2.93227e9	1.32699
$51$	−5.56421e8	−0.225823
$52$	2.08144e9	0.759180
$53$	3.88609e9	1.27642	0.638212	−	0.769861i	$-0.279674\pi$
$53$	3.88609e9	1.27642	0.638212	+	0.769861i	$0.279674\pi$
$54$	3.74456e9	1.10978
$55$	−8.87601e8	−0.237806
$56$	2.90178e9	0.704093
$57$	−2.76475e9	−0.608618
$58$	2.27440e9	0.455003
$59$	8.66045e9	1.57708	0.788541	−	0.614982i	$-0.210837\pi$
$59$	8.66045e9	1.57708	0.788541	+	0.614982i	$0.210837\pi$
$60$	−5.33158e9	−0.885164
$61$	−6.39897e9	−0.970055	−0.485027	−	0.874499i	$-0.661190\pi$
$61$	−6.39897e9	−0.970055	−0.485027	+	0.874499i	$0.661190\pi$
$62$	2.04605e9	0.283637
$63$	−2.13117e9	−0.270548
$64$	−1.35285e10	−1.57492
$65$	−1.83593e9	−0.196260
$66$	−1.08788e10	−1.06927
$67$	−3.58690e9	−0.324570	−0.162285	−	0.986744i	$-0.551886\pi$
$67$	−3.58690e9	−0.324570	−0.162285	+	0.986744i	$0.551886\pi$
$68$	−3.82064e9	−0.318667
$69$	3.24543e9	0.249805
$70$	−6.26583e9	−0.445594
$71$	9.32716e9	0.613520	0.306760	−	0.951787i	$-0.400755\pi$
$71$	9.32716e9	0.613520	0.306760	+	0.951787i	$0.400755\pi$
$72$	−8.09499e9	−0.493046
$73$	1.42089e10	0.802204	0.401102	−	0.916033i	$-0.368627\pi$
$73$	1.42089e10	0.802204	0.401102	+	0.916033i	$0.368627\pi$
$74$	4.13697e10	2.16725
$75$	−1.99181e10	−0.969195
$76$	−1.89841e10	−0.858845
$77$	−8.03328e9	−0.338216
$78$	−2.25018e10	−0.882466
$79$	−2.30781e10	−0.843824	−0.421912	−	0.906637i	$-0.638641\pi$
$79$	−2.30781e10	−0.843824	−0.421912	+	0.906637i	$0.638641\pi$
$80$	−2.14525e9	−0.0731952
$81$	−3.90948e10	−1.24581
$82$	6.39514e10	1.90491
$83$	−2.53549e10	−0.706534	−0.353267	−	0.935523i	$-0.614929\pi$
$83$	−2.53549e10	−0.706534	−0.353267	+	0.935523i	$0.614929\pi$
$84$	−4.82537e10	−1.25891
$85$	3.36999e9	0.0823805
$86$	7.02719e10	1.61079
$87$	−1.54494e10	−0.332320
$88$	−3.05134e10	−0.616363
$89$	7.03025e10	1.33452	0.667261	−	0.744824i	$-0.267466\pi$
$89$	7.03025e10	1.33452	0.667261	+	0.744824i	$0.267466\pi$
$90$	1.74796e10	0.312030
$91$	−1.66162e10	−0.279128
$92$	2.22846e10	0.352510
$93$	−1.38983e10	−0.207160
$94$	2.14845e11	3.01941
$95$	1.67449e10	0.222025
$96$	8.21231e10	1.02795
$97$	1.14492e11	1.35373	0.676864	−	0.736108i	$-0.263339\pi$
$97$	1.14492e11	1.35373	0.676864	+	0.736108i	$0.263339\pi$
$98$	9.00706e10	1.00656
$99$	2.24101e10	0.236838
$100$	−1.36767e11	−1.36767
$101$	2.56430e9	0.0242774	0.0121387	−	0.999926i	$-0.496136\pi$
$101$	2.56430e9	0.0242774	0.0121387	+	0.999926i	$0.496136\pi$
$102$	4.13039e10	0.370416
$103$	−2.63698e9	−0.0224131	−0.0112066	−	0.999937i	$-0.503567\pi$
$103$	−2.63698e9	−0.0224131	−0.0112066	+	0.999937i	$0.503567\pi$
$104$	−6.31145e10	−0.508681
$105$	4.25621e10	0.325448
$106$	−2.88470e11	−2.09372
$107$	−1.55795e11	−1.07385	−0.536926	−	0.843630i	$-0.680414\pi$
$107$	−1.55795e11	−1.07385	−0.536926	+	0.843630i	$0.680414\pi$
$108$	−1.74654e11	−1.14379
$109$	1.08496e11	0.675408	0.337704	−	0.941252i	$-0.390350\pi$
$109$	1.08496e11	0.675408	0.337704	+	0.941252i	$0.390350\pi$
$110$	6.58879e10	0.390073
$111$	−2.81014e11	−1.58289
$112$	−1.94157e10	−0.104101
$113$	−1.37528e11	−0.702196	−0.351098	−	0.936339i	$-0.614192\pi$
$113$	−1.37528e11	−0.702196	−0.351098	+	0.936339i	$0.614192\pi$
$114$	2.05232e11	0.998315
$115$	−1.96561e10	−0.0911294
$116$	−1.06083e11	−0.468950
$117$	4.63535e10	0.195461
$118$	−6.42878e11	−2.58689
$119$	3.05003e10	0.117164
$120$	1.61667e11	0.593096
$121$	−2.00838e11	−0.703926
$122$	4.75005e11	1.59118
$123$	−4.34405e11	−1.39129
$124$	−9.54320e10	−0.292331
$125$	2.69752e11	0.790606
$126$	1.58200e11	0.443780
$127$	2.05402e11	0.551677	0.275838	−	0.961204i	$-0.411045\pi$
$127$	2.05402e11	0.551677	0.275838	+	0.961204i	$0.411045\pi$
$128$	6.70687e11	1.72530
$129$	−4.77338e11	−1.17648
$130$	1.36283e11	0.321925
$131$	−3.90531e10	−0.0884429	−0.0442215	−	0.999022i	$-0.514081\pi$
$131$	−3.90531e10	−0.0884429	−0.0442215	+	0.999022i	$0.514081\pi$
$132$	5.07409e11	1.10205
$133$	1.51550e11	0.315772
$134$	2.66261e11	0.532391
$135$	1.54053e11	0.295689
$136$	1.15852e11	0.213520
$137$	9.19783e11	1.62825	0.814127	−	0.580686i	$-0.197216\pi$
$137$	9.19783e11	1.62825	0.814127	+	0.580686i	$0.197216\pi$
$138$	−2.40913e11	−0.409755
$139$	−4.78750e11	−0.782578	−0.391289	−	0.920268i	$-0.627971\pi$
$139$	−4.78750e11	−0.782578	−0.391289	+	0.920268i	$0.627971\pi$
$140$	2.92251e11	0.459253
$141$	−1.45939e12	−2.20529
$142$	−6.92369e11	−1.00636
$143$	1.74726e11	0.244348
$144$	5.41633e10	0.0728972
$145$	9.35702e10	0.121231
$146$	−1.05475e12	−1.31585
$147$	−6.11825e11	−0.735160
$148$	−1.92957e12	−2.23368
$149$	1.07931e12	1.20398	0.601991	−	0.798503i	$-0.294374\pi$
$149$	1.07931e12	1.20398	0.601991	+	0.798503i	$0.294374\pi$
$150$	1.47855e12	1.58977
$151$	−3.51191e11	−0.364058	−0.182029	−	0.983293i	$-0.558266\pi$
$151$	−3.51191e11	−0.364058	−0.182029	+	0.983293i	$0.558266\pi$
$152$	5.75646e11	0.575461
$153$	−8.50855e10	−0.0820451
$154$	5.96322e11	0.554775
$155$	8.41756e10	0.0755722
$156$	1.04953e12	0.909516
$157$	−2.19660e11	−0.183782	−0.0918909	−	0.995769i	$-0.529291\pi$
$157$	−2.19660e11	−0.183782	−0.0918909	+	0.995769i	$0.529291\pi$
$158$	1.71312e12	1.38412
$159$	1.95950e12	1.52919
$160$	−4.97382e11	−0.374999
$161$	−1.77898e11	−0.129607
$162$	2.90206e12	2.04350
$163$	−3.99291e11	−0.271805	−0.135903	−	0.990722i	$-0.543393\pi$
$163$	−3.99291e11	−0.271805	−0.135903	+	0.990722i	$0.543393\pi$
$164$	−2.98282e12	−1.96330
$165$	−4.47559e11	−0.284897
$166$	1.88213e12	1.15893
$167$	−2.26685e12	−1.35046	−0.675232	−	0.737606i	$-0.735956\pi$
$167$	−2.26685e12	−1.35046	−0.675232	+	0.737606i	$0.735956\pi$
$168$	1.46318e12	0.843521
$169$	−1.43075e12	−0.798341
$170$	−2.50159e11	−0.135129
$171$	−4.22774e11	−0.221121
$172$	−3.27763e12	−1.66017
$173$	−1.87280e11	−0.0918835	−0.0459417	−	0.998944i	$-0.514629\pi$
$173$	−1.87280e11	−0.0918835	−0.0459417	+	0.998944i	$0.514629\pi$
$174$	1.14683e12	0.545104
$175$	1.09181e12	0.502851
$176$	2.04164e11	0.0911298
$177$	4.36690e12	1.88938
$178$	−5.21866e12	−2.18902
$179$	2.46461e12	1.00244	0.501218	−	0.865321i	$-0.332885\pi$
$179$	2.46461e12	1.00244	0.501218	+	0.865321i	$0.332885\pi$
$180$	−8.15282e11	−0.321595
$181$	4.22119e12	1.61511	0.807556	−	0.589791i	$-0.200790\pi$
$181$	4.22119e12	1.61511	0.807556	+	0.589791i	$0.200790\pi$
$182$	1.23344e12	0.457853
$183$	−3.22658e12	−1.16215
$184$	−6.75726e11	−0.236196
$185$	1.70197e12	0.577441
$186$	1.03169e12	0.339804
$187$	−3.20723e11	−0.102566
$188$	−1.00208e13	−3.11197
$189$	1.39427e12	0.420539
$190$	−1.24300e12	−0.364187
$191$	5.43923e12	1.54830	0.774148	−	0.633005i	$-0.218179\pi$
$191$	5.43923e12	1.54830	0.774148	+	0.633005i	$0.218179\pi$
$192$	−6.82152e12	−1.88679
$193$	−1.80014e12	−0.483884	−0.241942	−	0.970291i	$-0.577784\pi$
$193$	−1.80014e12	−0.483884	−0.241942	+	0.970291i	$0.577784\pi$
$194$	−8.49892e12	−2.22052
$195$	−9.25737e11	−0.235124
$196$	−4.20108e12	−1.03741
$197$	−5.34581e11	−0.128366	−0.0641829	−	0.997938i	$-0.520444\pi$
$197$	−5.34581e11	−0.128366	−0.0641829	+	0.997938i	$0.520444\pi$
$198$	−1.66354e12	−0.388485
$199$	5.12267e12	1.16360	0.581801	−	0.813331i	$-0.302348\pi$
$199$	5.12267e12	1.16360	0.581801	+	0.813331i	$0.302348\pi$
$200$	4.14712e12	0.916394
$201$	−1.80864e12	−0.388842
$202$	−1.90352e11	−0.0398222
$203$	8.46863e11	0.172419
$204$	−1.92650e12	−0.381771
$205$	2.63099e12	0.507544
$206$	1.95747e11	0.0367642
$207$	4.96277e11	0.0907584
$208$	4.22297e11	0.0752089
$209$	−1.59362e12	−0.276426
$210$	−3.15945e12	−0.533832
$211$	5.76989e12	0.949761	0.474880	−	0.880050i	$-0.342491\pi$
$211$	5.76989e12	0.949761	0.474880	+	0.880050i	$0.342491\pi$
$212$	1.34548e13	2.15790
$213$	4.70308e12	0.735012
$214$	1.15649e13	1.76144
$215$	2.89102e12	0.429180
$216$	5.29596e12	0.766388
$217$	7.61836e11	0.107481
$218$	−8.05378e12	−1.10787
$219$	7.16462e12	0.961060
$220$	−3.07315e12	−0.402030
$221$	−6.63389e11	−0.0846469
$222$	2.08600e13	2.59641
$223$	−2.05978e12	−0.250117	−0.125059	−	0.992149i	$-0.539912\pi$
$223$	−2.05978e12	−0.250117	−0.125059	+	0.992149i	$0.539912\pi$
$224$	−4.50159e12	−0.533337
$225$	−3.04579e12	−0.352125
$226$	1.02089e13	1.15181
$227$	−1.40077e13	−1.54250	−0.771252	−	0.636530i	$-0.780369\pi$
$227$	−1.40077e13	−1.54250	−0.771252	+	0.636530i	$0.780369\pi$
$228$	−9.57243e12	−1.02892
$229$	−2.54414e12	−0.266960	−0.133480	−	0.991052i	$-0.542615\pi$
$229$	−2.54414e12	−0.266960	−0.133480	+	0.991052i	$0.542615\pi$
$230$	1.45910e12	0.149479
$231$	−4.05066e12	−0.405191
$232$	3.21671e12	0.314215
$233$	−1.56918e13	−1.49698	−0.748489	−	0.663147i	$-0.769221\pi$
$233$	−1.56918e13	−1.49698	−0.748489	+	0.663147i	$0.769221\pi$
$234$	−3.44089e12	−0.320614
$235$	8.83884e12	0.804493
$236$	2.99851e13	2.66618
$237$	−1.16368e13	−1.01092
$238$	−2.26408e12	−0.192184
$239$	5.82457e12	0.483143	0.241571	−	0.970383i	$-0.422337\pi$
$239$	5.82457e12	0.483143	0.241571	+	0.970383i	$0.422337\pi$
$240$	−1.08171e12	−0.0876897
$241$	−2.01618e13	−1.59748	−0.798741	−	0.601675i	$-0.794500\pi$
$241$	−2.01618e13	−1.59748	−0.798741	+	0.601675i	$0.794500\pi$
$242$	1.49085e13	1.15465
$243$	−1.07769e13	−0.815940
$244$	−2.21552e13	−1.63995
$245$	3.70555e12	0.268188
$246$	3.22465e13	2.28213
$247$	−3.29626e12	−0.228133
$248$	2.89374e12	0.195874
$249$	−1.27848e13	−0.846445
$250$	−2.00241e13	−1.29683
$251$	−3.07508e13	−1.94828	−0.974140	−	0.225946i	$-0.927453\pi$
$251$	−3.07508e13	−1.94828	−0.974140	+	0.225946i	$0.927453\pi$
$252$	−7.37876e12	−0.457383
$253$	1.87068e12	0.113458
$254$	−1.52473e13	−0.904915
$255$	1.69926e12	0.0986938
$256$	−2.20797e13	−1.25509
$257$	1.30106e13	0.723880	0.361940	−	0.932201i	$-0.382115\pi$
$257$	1.30106e13	0.723880	0.361940	+	0.932201i	$0.382115\pi$
$258$	3.54335e13	1.92977
$259$	1.54038e13	0.821257
$260$	−6.35654e12	−0.331793
$261$	−2.36246e12	−0.120737
$262$	2.89897e12	0.145073
$263$	9.83668e12	0.482050	0.241025	−	0.970519i	$-0.422516\pi$
$263$	9.83668e12	0.482050	0.241025	+	0.970519i	$0.422516\pi$
$264$	−1.53859e13	−0.738418
$265$	−1.18678e13	−0.557850
$266$	−1.12498e13	−0.517960
$267$	3.54490e13	1.59879
$268$	−1.24189e13	−0.548710
$269$	−1.68229e13	−0.728222	−0.364111	−	0.931355i	$-0.618627\pi$
$269$	−1.68229e13	−0.728222	−0.364111	+	0.931355i	$0.618627\pi$
$270$	−1.14356e13	−0.485018
$271$	3.00319e13	1.24810	0.624052	−	0.781382i	$-0.285485\pi$
$271$	3.00319e13	1.24810	0.624052	+	0.781382i	$0.285485\pi$
$272$	−7.75159e11	−0.0315691
$273$	−8.37844e12	−0.334402
$274$	−6.82768e13	−2.67082
$275$	−1.14809e13	−0.440196
$276$	1.12367e13	0.422316
$277$	−3.09039e13	−1.13861	−0.569305	−	0.822127i	$-0.692788\pi$
$277$	−3.09039e13	−1.13861	−0.569305	+	0.822127i	$0.692788\pi$
$278$	3.55383e13	1.28366
$279$	−2.12527e12	−0.0752645
$280$	−8.86180e12	−0.307718
$281$	3.78610e13	1.28916	0.644580	−	0.764537i	$-0.277032\pi$
$281$	3.78610e13	1.28916	0.644580	+	0.764537i	$0.277032\pi$
$282$	1.08332e14	3.61733
$283$	−2.74113e13	−0.897643	−0.448822	−	0.893621i	$-0.648156\pi$
$283$	−2.74113e13	−0.897643	−0.448822	+	0.893621i	$0.648156\pi$
$284$	3.22935e13	1.03720
$285$	8.44334e12	0.265991
$286$	−1.29702e13	−0.400804
$287$	2.38120e13	0.721846
$288$	1.25579e13	0.373473
$289$	−3.30542e13	−0.964469
$290$	−6.94585e12	−0.198855
$291$	5.77309e13	1.62180
$292$	4.91956e13	1.35619
$293$	−4.90592e13	−1.32724	−0.663619	−	0.748071i	$-0.730980\pi$
$293$	−4.90592e13	−1.32724	−0.663619	+	0.748071i	$0.730980\pi$
$294$	4.54167e13	1.20588
$295$	−2.64483e13	−0.689250
$296$	5.85095e13	1.49666
$297$	−1.46613e13	−0.368140
$298$	−8.01184e13	−1.97489
$299$	3.86934e12	0.0936365
$300$	−6.89626e13	−1.63850
$301$	2.61654e13	0.610395
$302$	2.60694e13	0.597164
$303$	1.29301e12	0.0290849
$304$	−3.85163e12	−0.0850823
$305$	1.95420e13	0.423954
$306$	6.31602e12	0.134578
$307$	1.01713e13	0.212870	0.106435	−	0.994320i	$-0.466056\pi$
$307$	1.01713e13	0.212870	0.106435	+	0.994320i	$0.466056\pi$
$308$	−2.78137e13	−0.571781
$309$	−1.32966e12	−0.0268515
$310$	−6.24848e12	−0.123961
$311$	6.35579e13	1.23876	0.619380	−	0.785091i	$-0.287384\pi$
$311$	6.35579e13	1.23876	0.619380	+	0.785091i	$0.287384\pi$
$312$	−3.18245e13	−0.609412
$313$	−9.34616e12	−0.175849	−0.0879244	−	0.996127i	$-0.528023\pi$
$313$	−9.34616e12	−0.175849	−0.0879244	+	0.996127i	$0.528023\pi$
$314$	1.63057e13	0.301457
$315$	6.50842e12	0.118241
$316$	−7.99036e13	−1.42655
$317$	−3.53537e12	−0.0620310	−0.0310155	−	0.999519i	$-0.509874\pi$
$317$	−3.53537e12	−0.0620310	−0.0310155	+	0.999519i	$0.509874\pi$
$318$	−1.45456e14	−2.50832
$319$	−8.90512e12	−0.150935
$320$	4.13149e13	0.688306
$321$	−7.85575e13	−1.28650
$322$	1.32057e13	0.212595
$323$	6.05055e12	0.0957593
$324$	−1.35358e14	−2.10614
$325$	−2.37473e13	−0.363291
$326$	2.96400e13	0.445842
$327$	5.47072e13	0.809155
$328$	9.04469e13	1.31549
$329$	7.99965e13	1.14418
$330$	3.32229e13	0.467317
$331$	7.49637e13	1.03704	0.518522	−	0.855064i	$-0.326482\pi$
$331$	7.49637e13	1.03704	0.518522	+	0.855064i	$0.326482\pi$
$332$	−8.77866e13	−1.19445
$333$	−4.29714e13	−0.575091
$334$	1.68272e14	2.21516
$335$	1.09541e13	0.141850
$336$	−9.79007e12	−0.124715
$337$	9.53662e13	1.19517	0.597585	−	0.801805i	$-0.296127\pi$
$337$	9.53662e13	1.19517	0.597585	+	0.801805i	$0.296127\pi$
$338$	1.06207e14	1.30952
$339$	−6.93461e13	−0.841248
$340$	1.16679e13	0.139271
$341$	−8.01103e12	−0.0940891
$342$	3.13832e13	0.362705
$343$	8.81900e13	1.00300
$344$	9.93860e13	1.11238
$345$	−9.91127e12	−0.109175
$346$	1.39020e13	0.150716
$347$	−1.68126e14	−1.79400	−0.896999	−	0.442032i	$-0.854258\pi$
$347$	−1.68126e14	−1.79400	−0.896999	+	0.442032i	$0.854258\pi$
$348$	−5.34907e13	−0.561813
$349$	1.11235e14	1.15001	0.575006	−	0.818149i	$-0.305000\pi$
$349$	1.11235e14	1.15001	0.575006	+	0.818149i	$0.305000\pi$
$350$	−8.10469e13	−0.824826
$351$	−3.03257e13	−0.303824
$352$	4.73361e13	0.466883
$353$	−1.59352e14	−1.54738	−0.773689	−	0.633565i	$-0.781591\pi$
$353$	−1.59352e14	−1.54738	−0.773689	+	0.633565i	$0.781591\pi$
$354$	−3.24161e14	−3.09915
$355$	−2.84844e13	−0.268134
$356$	2.43409e14	2.25612
$357$	1.53793e13	0.140366
$358$	−1.82952e14	−1.64430
$359$	−6.13035e13	−0.542582	−0.271291	−	0.962497i	$-0.587451\pi$
$359$	−6.13035e13	−0.542582	−0.271291	+	0.962497i	$0.587451\pi$
$360$	2.47214e13	0.215482
$361$	−8.64262e13	−0.741918
$362$	−3.13345e14	−2.64927
$363$	−1.01270e14	−0.843320
$364$	−5.75302e13	−0.471888
$365$	−4.33929e13	−0.350596
$366$	2.39514e14	1.90627
$367$	−6.18852e13	−0.485203	−0.242602	−	0.970126i	$-0.578001\pi$
$367$	−6.18852e13	−0.485203	−0.242602	+	0.970126i	$0.578001\pi$
$368$	4.52126e12	0.0349218
$369$	−6.64274e13	−0.505478
$370$	−1.26340e14	−0.947176
$371$	−1.07410e14	−0.793394
$372$	−4.81201e13	−0.350220
$373$	6.92035e13	0.496283	0.248142	−	0.968724i	$-0.420180\pi$
$373$	6.92035e13	0.496283	0.248142	+	0.968724i	$0.420180\pi$
$374$	2.38078e13	0.168238
$375$	1.36018e14	0.947165
$376$	3.03857e14	2.08514
$377$	−1.84195e13	−0.124566
$378$	−1.03499e14	−0.689809
$379$	2.74009e14	1.79990	0.899951	−	0.435991i	$-0.143602\pi$
$379$	2.74009e14	1.79990	0.899951	+	0.435991i	$0.143602\pi$
$380$	5.79758e13	0.375351
$381$	1.03571e14	0.660922
$382$	−4.03762e14	−2.53967
$383$	1.73098e14	1.07325	0.536624	−	0.843822i	$-0.319700\pi$
$383$	1.73098e14	1.07325	0.536624	+	0.843822i	$0.319700\pi$
$384$	3.38183e14	2.06695
$385$	2.45330e13	0.147814
$386$	1.33627e14	0.793715
$387$	−7.29926e13	−0.427433
$388$	3.96407e14	2.28858
$389$	6.62868e13	0.377315	0.188658	−	0.982043i	$-0.439586\pi$
$389$	6.62868e13	0.377315	0.188658	+	0.982043i	$0.439586\pi$
$390$	6.87188e13	0.385674
$391$	−7.10248e12	−0.0393041
$392$	1.27387e14	0.695109
$393$	−1.96919e13	−0.105957
$394$	3.96827e13	0.210558
$395$	7.04788e13	0.368786
$396$	7.75908e13	0.400393
$397$	−1.72730e14	−0.879063	−0.439531	−	0.898227i	$-0.644855\pi$
$397$	−1.72730e14	−0.879063	−0.439531	+	0.898227i	$0.644855\pi$
$398$	−3.80263e14	−1.90865
$399$	7.64169e13	0.378302
$400$	−2.77483e13	−0.135490
$401$	1.57209e14	0.757151	0.378575	−	0.925570i	$-0.376414\pi$
$401$	1.57209e14	0.757151	0.378575	+	0.925570i	$0.376414\pi$
$402$	1.34258e14	0.637817
$403$	−1.65701e13	−0.0776513
$404$	8.87841e12	0.0410428
$405$	1.19392e14	0.544470
$406$	−6.28638e13	−0.282818
$407$	−1.61978e14	−0.718928
$408$	5.84164e13	0.255802
$409$	3.57851e14	1.54605	0.773026	−	0.634375i	$-0.218742\pi$
$409$	3.57851e14	1.54605	0.773026	+	0.634375i	$0.218742\pi$
$410$	−1.95302e14	−0.832523
$411$	4.63786e14	1.95069
$412$	−9.13005e12	−0.0378912
$413$	−2.39372e14	−0.980276
$414$	−3.68393e13	−0.148871
$415$	7.74320e13	0.308785
$416$	9.79108e13	0.385316
$417$	−2.41402e14	−0.937548
$418$	1.18296e14	0.453422
$419$	7.11258e13	0.269060	0.134530	−	0.990909i	$-0.457047\pi$
$419$	7.11258e13	0.269060	0.134530	+	0.990909i	$0.457047\pi$
$420$	1.47363e14	0.550196
$421$	−1.99739e13	−0.0736058	−0.0368029	−	0.999323i	$-0.511717\pi$
$421$	−1.99739e13	−0.0736058	−0.0368029	+	0.999323i	$0.511717\pi$
$422$	−4.28307e14	−1.55789
$423$	−2.23163e14	−0.801218
$424$	−4.07984e14	−1.44588
$425$	4.35900e13	0.152492
$426$	−3.49116e14	−1.20564
$427$	1.76866e14	0.602962
$428$	−5.39412e14	−1.81543
$429$	8.81029e13	0.292735
$430$	−2.14605e14	−0.703984
$431$	−7.80000e13	−0.252621	−0.126311	−	0.991991i	$-0.540314\pi$
$431$	−7.80000e13	−0.252621	−0.126311	+	0.991991i	$0.540314\pi$
$432$	−3.54351e13	−0.113311
$433$	−1.88457e14	−0.595015	−0.297508	−	0.954719i	$-0.596155\pi$
$433$	−1.88457e14	−0.595015	−0.297508	+	0.954719i	$0.596155\pi$
$434$	−5.65522e13	−0.176302
$435$	4.71813e13	0.145238
$436$	3.75645e14	1.14183
$437$	−3.52909e13	−0.105929
$438$	−5.31840e14	−1.57643
$439$	6.46769e14	1.89319	0.946595	−	0.322425i	$-0.104498\pi$
$439$	6.46769e14	1.89319	0.946595	+	0.322425i	$0.104498\pi$
$440$	9.31856e13	0.269376
$441$	−9.35577e13	−0.267096
$442$	4.92443e13	0.138846
$443$	−4.12743e14	−1.14937	−0.574685	−	0.818375i	$-0.694875\pi$
$443$	−4.12743e14	−1.14937	−0.574685	+	0.818375i	$0.694875\pi$
$444$	−9.72955e14	−2.67600
$445$	−2.14698e14	−0.583242
$446$	1.52900e14	0.410267
$447$	5.44223e14	1.44240
$448$	3.73923e14	0.978933
$449$	−4.82320e14	−1.24733	−0.623663	−	0.781693i	$-0.714356\pi$
$449$	−4.82320e14	−1.24733	−0.623663	+	0.781693i	$0.714356\pi$
$450$	2.26094e14	0.577590
$451$	−2.50393e14	−0.631904
$452$	−4.76163e14	−1.18712
$453$	−1.77083e14	−0.436150
$454$	1.03982e15	2.53017
$455$	5.07444e13	0.121990
$456$	2.90260e14	0.689416
$457$	−5.29906e14	−1.24354	−0.621770	−	0.783200i	$-0.713586\pi$
$457$	−5.29906e14	−1.24354	−0.621770	+	0.783200i	$0.713586\pi$
$458$	1.88855e14	0.437893
$459$	5.56652e13	0.127531
$460$	−6.80554e13	−0.154062
$461$	−5.09200e14	−1.13902	−0.569512	−	0.821983i	$-0.692868\pi$
$461$	−5.09200e14	−1.13902	−0.569512	+	0.821983i	$0.692868\pi$
$462$	3.00686e14	0.664634
$463$	3.17151e14	0.692742	0.346371	−	0.938098i	$-0.387414\pi$
$463$	3.17151e14	0.692742	0.346371	+	0.938098i	$0.387414\pi$
$464$	−2.15229e13	−0.0464570
$465$	4.24442e13	0.0905373
$466$	1.16483e15	2.45549
$467$	1.65726e14	0.345261	0.172631	−	0.984987i	$-0.444773\pi$
$467$	1.65726e14	0.345261	0.172631	+	0.984987i	$0.444773\pi$
$468$	1.60490e14	0.330442
$469$	9.91407e13	0.201744
$470$	−6.56120e14	−1.31961
$471$	−1.10760e14	−0.220175
$472$	−9.09226e14	−1.78645
$473$	−2.75140e14	−0.534340
$474$	8.63816e14	1.65821
$475$	2.16591e14	0.410984
$476$	1.05601e14	0.198076
$477$	2.99638e14	0.555579
$478$	−4.32366e14	−0.792498
$479$	6.60343e14	1.19653	0.598266	−	0.801298i	$-0.295857\pi$
$479$	6.60343e14	1.19653	0.598266	+	0.801298i	$0.295857\pi$
$480$	−2.50797e14	−0.449258
$481$	−3.35037e14	−0.593328
$482$	1.49664e15	2.62035
$483$	−8.97025e13	−0.155273
$484$	−6.95364e14	−1.19004
$485$	−3.49650e14	−0.591635
$486$	7.99983e14	1.33839
$487$	9.43941e14	1.56148	0.780738	−	0.624858i	$-0.214843\pi$
$487$	9.43941e14	1.56148	0.780738	+	0.624858i	$0.214843\pi$
$488$	6.71803e14	1.09884
$489$	−2.01336e14	−0.325629
$490$	−2.75068e14	−0.439908
$491$	−2.00478e12	−0.00317043	−0.00158522	−	0.999999i	$-0.500505\pi$
$491$	−2.00478e12	−0.00317043	−0.00158522	+	0.999999i	$0.500505\pi$
$492$	−1.50404e15	−2.35208
$493$	3.38104e13	0.0522869
$494$	2.44686e14	0.374206
$495$	−6.84388e13	−0.103508
$496$	−1.93619e13	−0.0289601
$497$	−2.57800e14	−0.381349
$498$	9.49037e14	1.38842
$499$	−7.11052e14	−1.02884	−0.514420	−	0.857538i	$-0.671993\pi$
$499$	−7.11052e14	−1.02884	−0.514420	+	0.857538i	$0.671993\pi$
$500$	9.33965e14	1.33658
$501$	−1.14303e15	−1.61789
$502$	2.28268e15	3.19576
$503$	−6.28377e14	−0.870154	−0.435077	−	0.900393i	$-0.643279\pi$
$503$	−6.28377e14	−0.870154	−0.435077	+	0.900393i	$0.643279\pi$
$504$	2.23743e14	0.306465
$505$	−7.83118e12	−0.0106102
$506$	−1.38863e14	−0.186105
$507$	−7.21436e14	−0.956431
$508$	7.11166e14	0.932654
$509$	1.01555e15	1.31751	0.658757	−	0.752356i	$-0.271083\pi$
$509$	1.01555e15	1.31751	0.658757	+	0.752356i	$0.271083\pi$
$510$	−1.26139e14	−0.161887
$511$	−3.92730e14	−0.498630
$512$	2.65443e14	0.333416
$513$	2.76590e14	0.343710
$514$	−9.65798e14	−1.18738
$515$	8.05314e12	0.00979547
$516$	−1.65269e15	−1.98893
$517$	−8.41197e14	−1.00161
$518$	−1.14345e15	−1.34711
$519$	−9.44329e13	−0.110079
$520$	1.92747e14	0.222315
$521$	−5.04518e14	−0.575797	−0.287898	−	0.957661i	$-0.592957\pi$
$521$	−5.04518e14	−0.575797	−0.287898	+	0.957661i	$0.592957\pi$
$522$	1.75369e14	0.198045
$523$	3.57291e13	0.0399267	0.0199633	−	0.999801i	$-0.493645\pi$
$523$	3.57291e13	0.0399267	0.0199633	+	0.999801i	$0.493645\pi$
$524$	−1.35214e14	−0.149520
$525$	5.50530e14	0.602428
$526$	−7.30191e14	−0.790705
$527$	3.04158e13	0.0325942
$528$	1.02947e14	0.109176
$529$	4.14265e13	0.0434783
$530$	8.80963e14	0.915041
$531$	6.67767e14	0.686444
$532$	5.24714e14	0.533837
$533$	−5.17917e14	−0.521507
$534$	−2.63143e15	−2.62249
$535$	4.75787e14	0.469317
$536$	3.76574e14	0.367658
$537$	1.24274e15	1.20094
$538$	1.24879e15	1.19450
$539$	−3.52659e14	−0.333900
$540$	5.33379e14	0.499886
$541$	−1.85232e15	−1.71843	−0.859214	−	0.511616i	$-0.829047\pi$
$541$	−1.85232e15	−1.71843	−0.859214	+	0.511616i	$0.829047\pi$
$542$	−2.22931e15	−2.04726
$543$	2.12847e15	1.93494
$544$	−1.79723e14	−0.161737
$545$	−3.31337e14	−0.295181
$546$	6.21944e14	0.548519
$547$	−1.27987e15	−1.11747	−0.558736	−	0.829346i	$-0.688713\pi$
$547$	−1.27987e15	−1.11747	−0.558736	+	0.829346i	$0.688713\pi$
$548$	3.18457e15	2.75269
$549$	−4.93395e14	−0.422228
$550$	8.52243e14	0.722053
$551$	1.67998e14	0.140919
$552$	−3.40724e14	−0.282969
$553$	6.37872e14	0.524500
$554$	2.29404e15	1.86766
$555$	8.58193e14	0.691789
$556$	−1.65758e15	−1.32301
$557$	−5.99234e13	−0.0473579	−0.0236790	−	0.999720i	$-0.507538\pi$
$557$	−5.99234e13	−0.0473579	−0.0236790	+	0.999720i	$0.507538\pi$
$558$	1.57762e14	0.123456
$559$	−5.69104e14	−0.440988
$560$	5.92940e13	0.0454964
$561$	−1.61720e14	−0.122876
$562$	−2.81047e15	−2.11461
$563$	5.44298e14	0.405546	0.202773	−	0.979226i	$-0.435005\pi$
$563$	5.44298e14	0.405546	0.202773	+	0.979226i	$0.435005\pi$
$564$	−5.05284e15	−3.72821
$565$	4.19998e14	0.306889
$566$	2.03478e15	1.47240
$567$	1.08057e15	0.774364
$568$	−9.79221e14	−0.694968
$569$	−2.09809e15	−1.47471	−0.737356	−	0.675504i	$-0.763926\pi$
$569$	−2.09809e15	−1.47471	−0.737356	+	0.675504i	$0.763926\pi$
$570$	−6.26761e14	−0.436305
$571$	5.27163e14	0.363452	0.181726	−	0.983349i	$-0.441832\pi$
$571$	5.27163e14	0.363452	0.181726	+	0.983349i	$0.441832\pi$
$572$	6.04955e14	0.413090
$573$	2.74265e15	1.85490
$574$	−1.76760e15	−1.18404
$575$	−2.54246e14	−0.168687
$576$	−1.04312e15	−0.685504
$577$	−1.41077e15	−0.918313	−0.459156	−	0.888355i	$-0.651848\pi$
$577$	−1.41077e15	−0.918313	−0.459156	+	0.888355i	$0.651848\pi$
$578$	2.45366e15	1.58202
$579$	−9.07694e14	−0.579705
$580$	3.23969e14	0.204951
$581$	7.00803e14	0.439164
$582$	−4.28545e15	−2.66023
$583$	1.12946e15	0.694537
$584$	−1.49174e15	−0.908701
$585$	−1.41560e14	−0.0854246
$586$	3.64174e15	2.17707
$587$	−4.47356e14	−0.264938	−0.132469	−	0.991187i	$-0.542290\pi$
$587$	−4.47356e14	−0.264938	−0.132469	+	0.991187i	$0.542290\pi$
$588$	−2.11833e15	−1.24285
$589$	1.51131e14	0.0878453
$590$	1.96330e15	1.13058
$591$	−2.69554e14	−0.153785
$592$	−3.91485e14	−0.221282
$593$	1.56949e15	0.878936	0.439468	−	0.898258i	$-0.355167\pi$
$593$	1.56949e15	0.878936	0.439468	+	0.898258i	$0.355167\pi$
$594$	1.08833e15	0.603859
$595$	−9.31454e13	−0.0512057
$596$	3.73689e15	2.03543
$597$	2.58303e15	1.39402
$598$	−2.87227e14	−0.153592
$599$	2.37114e15	1.25635	0.628173	−	0.778073i	$-0.283803\pi$
$599$	2.37114e15	1.25635	0.628173	+	0.778073i	$0.283803\pi$
$600$	2.09112e15	1.09786
$601$	5.12944e14	0.266846	0.133423	−	0.991059i	$-0.457403\pi$
$601$	5.12944e14	0.266846	0.133423	+	0.991059i	$0.457403\pi$
$602$	−1.94229e15	−1.00123
$603$	−2.76569e14	−0.141273
$604$	−1.21593e15	−0.615469
$605$	6.13344e14	0.307645
$606$	−9.59820e13	−0.0477079
$607$	−5.52151e14	−0.271969	−0.135985	−	0.990711i	$-0.543420\pi$
$607$	−5.52151e14	−0.271969	−0.135985	+	0.990711i	$0.543420\pi$
$608$	−8.93011e14	−0.435900
$609$	4.27017e14	0.206562
$610$	−1.45063e15	−0.695411
$611$	−1.73994e15	−0.826625
$612$	−2.94592e14	−0.138704
$613$	−1.93574e15	−0.903263	−0.451631	−	0.892205i	$-0.649158\pi$
$613$	−1.93574e15	−0.903263	−0.451631	+	0.892205i	$0.649158\pi$
$614$	−7.55029e14	−0.349171
$615$	1.32664e15	0.608050
$616$	8.43382e14	0.383116
$617$	2.44575e15	1.10114	0.550571	−	0.834788i	$-0.314410\pi$
$617$	2.44575e15	1.10114	0.550571	+	0.834788i	$0.314410\pi$
$618$	9.87024e13	0.0440445
$619$	−3.88710e15	−1.71920	−0.859602	−	0.510963i	$-0.829289\pi$
$619$	−3.88710e15	−1.71920	−0.859602	+	0.510963i	$0.829289\pi$
$620$	2.91442e14	0.127761
$621$	−3.24678e14	−0.141075
$622$	−4.71799e15	−2.03194
$623$	−1.94314e15	−0.829507
$624$	2.12937e14	0.0901021
$625$	1.10499e15	0.463467
$626$	6.93779e14	0.288444
$627$	−8.03557e14	−0.331165
$628$	−7.60530e14	−0.310698
$629$	6.14986e14	0.249050
$630$	−4.83129e14	−0.193950
$631$	3.21076e15	1.27775	0.638875	−	0.769310i	$-0.279400\pi$
$631$	3.21076e15	1.27775	0.638875	+	0.769310i	$0.279400\pi$
$632$	2.42288e15	0.955846
$633$	2.90938e15	1.13784
$634$	2.62436e14	0.101749
$635$	−6.27282e14	−0.241106
$636$	6.78438e15	2.58521
$637$	−7.29445e14	−0.275566
$638$	6.61040e14	0.247579
$639$	7.19175e14	0.267042
$640$	−2.04822e15	−0.754028
$641$	2.96215e15	1.08115	0.540577	−	0.841294i	$-0.318206\pi$
$641$	2.96215e15	1.08115	0.540577	+	0.841294i	$0.318206\pi$
$642$	5.83143e15	2.11024
$643$	3.70290e15	1.32856	0.664280	−	0.747484i	$-0.268738\pi$
$643$	3.70290e15	1.32856	0.664280	+	0.747484i	$0.268738\pi$
$644$	−6.15939e14	−0.219112
$645$	1.45775e15	0.514169
$646$	−4.49141e14	−0.157074
$647$	−4.18033e15	−1.44956	−0.724781	−	0.688979i	$-0.758059\pi$
$647$	−4.18033e15	−1.44956	−0.724781	+	0.688979i	$0.758059\pi$
$648$	4.10441e15	1.41120
$649$	2.51710e15	0.858133
$650$	1.76279e15	0.595906
$651$	3.84144e14	0.128765
$652$	−1.38247e15	−0.459509
$653$	−2.50787e15	−0.826575	−0.413287	−	0.910601i	$-0.635619\pi$
$653$	−2.50787e15	−0.826575	−0.413287	+	0.910601i	$0.635619\pi$
$654$	−4.06099e15	−1.32726
$655$	1.19265e14	0.0386532
$656$	−6.05177e14	−0.194496
$657$	1.09558e15	0.349169
$658$	−5.93825e15	−1.87679
$659$	−5.16044e15	−1.61740	−0.808699	−	0.588223i	$-0.799828\pi$
$659$	−5.16044e15	−1.61740	−0.808699	+	0.588223i	$0.799828\pi$
$660$	−1.54959e15	−0.481642
$661$	3.36177e15	1.03624	0.518120	−	0.855308i	$-0.326632\pi$
$661$	3.36177e15	1.03624	0.518120	+	0.855308i	$0.326632\pi$
$662$	−5.56467e15	−1.70106
$663$	−3.34504e14	−0.101409
$664$	2.66191e15	0.800331
$665$	−4.62823e14	−0.138005
$666$	3.18983e15	0.943320
$667$	−1.97206e14	−0.0578398
$668$	−7.84854e15	−2.28307
$669$	−1.03861e15	−0.299646
$670$	−8.13139e14	−0.232677
$671$	−1.85982e15	−0.527833
$672$	−2.26986e15	−0.638951
$673$	1.84933e15	0.516336	0.258168	−	0.966100i	$-0.416881\pi$
$673$	1.84933e15	0.516336	0.258168	+	0.966100i	$0.416881\pi$
$674$	−7.07917e15	−1.96044
$675$	1.99264e15	0.547341
$676$	−4.95371e15	−1.34966
$677$	3.19086e14	0.0862322	0.0431161	−	0.999070i	$-0.486271\pi$
$677$	3.19086e14	0.0862322	0.0431161	+	0.999070i	$0.486271\pi$
$678$	5.14766e15	1.37990
$679$	−3.16453e15	−0.841444
$680$	−3.53802e14	−0.0933170
$681$	−7.06319e15	−1.84796
$682$	5.94671e14	0.154334
$683$	1.73514e14	0.0446705	0.0223353	−	0.999751i	$-0.492890\pi$
$683$	1.73514e14	0.0446705	0.0223353	+	0.999751i	$0.492890\pi$
$684$	−1.46377e15	−0.373823
$685$	−2.80894e15	−0.711615
$686$	−6.54647e15	−1.64522
$687$	−1.28284e15	−0.319824
$688$	−6.64988e14	−0.164467
$689$	2.33620e15	0.573197
$690$	7.35728e14	0.179080
$691$	−4.50087e15	−1.08685	−0.543423	−	0.839459i	$-0.682872\pi$
$691$	−4.50087e15	−1.08685	−0.543423	+	0.839459i	$0.682872\pi$
$692$	−6.48420e14	−0.155336
$693$	−6.19409e14	−0.147213
$694$	1.24802e16	2.94269
$695$	1.46207e15	0.342019
$696$	1.62197e15	0.376438
$697$	9.50677e14	0.218903
$698$	−8.25715e15	−1.88636
$699$	−7.91235e15	−1.79342
$700$	3.78020e15	0.850110
$701$	−1.93212e14	−0.0431108	−0.0215554	−	0.999768i	$-0.506862\pi$
$701$	−1.93212e14	−0.0431108	−0.0215554	+	0.999768i	$0.506862\pi$
$702$	2.25112e15	0.498362
$703$	3.05576e15	0.671219
$704$	−3.93196e15	−0.856957
$705$	4.45685e15	0.963802
$706$	1.18289e16	2.53816
$707$	−7.08765e13	−0.0150902
$708$	1.51195e16	3.19415
$709$	−4.30822e15	−0.903117	−0.451558	−	0.892242i	$-0.649132\pi$
$709$	−4.30822e15	−0.903117	−0.451558	+	0.892242i	$0.649132\pi$
$710$	2.11444e15	0.439819
$711$	−1.77945e15	−0.367285
$712$	−7.38078e15	−1.51169
$713$	−1.77406e14	−0.0360558
$714$	−1.14163e15	−0.230242
$715$	−5.33599e14	−0.106790
$716$	8.53324e15	1.69470
$717$	2.93695e15	0.578817
$718$	4.55064e15	0.889997
$719$	5.32440e15	1.03338	0.516692	−	0.856171i	$-0.327163\pi$
$719$	5.32440e15	1.03338	0.516692	+	0.856171i	$0.327163\pi$
$720$	−1.65410e14	−0.0318591
$721$	7.28854e13	0.0139315
$722$	6.41554e15	1.21697
$723$	−1.01663e16	−1.91382
$724$	1.46151e16	2.73047
$725$	1.21031e15	0.224407
$726$	7.51739e15	1.38330
$727$	−7.84961e15	−1.43354	−0.716769	−	0.697311i	$-0.754380\pi$
$727$	−7.84961e15	−1.43354	−0.716769	+	0.697311i	$0.754380\pi$
$728$	1.74446e15	0.316184
$729$	1.49146e15	0.268293
$730$	3.22111e15	0.575083
$731$	1.04463e15	0.185105
$732$	−1.11714e16	−1.96471
$733$	7.89487e15	1.37808	0.689038	−	0.724726i	$-0.258033\pi$
$733$	7.89487e15	1.37808	0.689038	+	0.724726i	$0.258033\pi$
$734$	4.59383e15	0.795878
$735$	1.86847e15	0.321296
$736$	1.04827e15	0.178914
$737$	−1.04251e15	−0.176607
$738$	4.93100e15	0.829134
$739$	−3.18982e15	−0.532380	−0.266190	−	0.963921i	$-0.585765\pi$
$739$	−3.18982e15	−0.532380	−0.266190	+	0.963921i	$0.585765\pi$
$740$	5.89275e15	0.976210
$741$	−1.66209e15	−0.273309
$742$	7.97321e15	1.30140
$743$	7.65391e15	1.24007	0.620033	−	0.784575i	$-0.287119\pi$
$743$	7.65391e15	1.24007	0.620033	+	0.784575i	$0.287119\pi$
$744$	1.45913e15	0.234661
$745$	−3.29611e15	−0.526190
$746$	−5.13708e15	−0.814053
$747$	−1.95500e15	−0.307528
$748$	−1.11044e15	−0.173395
$749$	4.30614e15	0.667479
$750$	−1.00968e16	−1.55363
$751$	−8.66782e15	−1.32401	−0.662004	−	0.749501i	$-0.730294\pi$
$751$	−8.66782e15	−1.32401	−0.662004	+	0.749501i	$0.730294\pi$
$752$	−2.03310e15	−0.308290
$753$	−1.55056e16	−2.33409
$754$	1.36731e15	0.204326
$755$	1.07251e15	0.159108
$756$	4.82738e15	0.710955
$757$	−7.33013e15	−1.07173	−0.535864	−	0.844305i	$-0.680014\pi$
$757$	−7.33013e15	−1.07173	−0.535864	+	0.844305i	$0.680014\pi$
$758$	−2.03401e16	−2.95238
$759$	9.43261e14	0.135926
$760$	−1.75798e15	−0.251500
$761$	−3.81883e15	−0.542393	−0.271197	−	0.962524i	$-0.587419\pi$
$761$	−3.81883e15	−0.542393	−0.271197	+	0.962524i	$0.587419\pi$
$762$	−7.68822e15	−1.08411
$763$	−2.99878e15	−0.419817
$764$	1.88323e16	2.61752
$765$	2.59844e14	0.0358571
$766$	−1.28493e16	−1.76045
$767$	5.20641e15	0.708212
$768$	−1.11334e16	−1.50362
$769$	1.17634e15	0.157738	0.0788691	−	0.996885i	$-0.474869\pi$
$769$	1.17634e15	0.157738	0.0788691	+	0.996885i	$0.474869\pi$
$770$	−1.82112e15	−0.242460
$771$	6.56041e15	0.867225
$772$	−6.23264e15	−0.818045
$773$	5.13291e15	0.668924	0.334462	−	0.942409i	$-0.391445\pi$
$773$	5.13291e15	0.668924	0.334462	+	0.942409i	$0.391445\pi$
$774$	5.41834e15	0.701118
$775$	1.08879e15	0.139889
$776$	−1.20201e16	−1.53344
$777$	7.76713e15	0.983886
$778$	−4.92057e15	−0.618910
$779$	4.72374e15	0.589970
$780$	−3.20519e15	−0.397496
$781$	2.71088e15	0.333833
$782$	5.27227e14	0.0644704
$783$	1.54559e15	0.187674
$784$	−8.52344e14	−0.102772
$785$	6.70824e14	0.0803203
$786$	1.46176e15	0.173801
$787$	2.88740e15	0.340915	0.170457	−	0.985365i	$-0.445475\pi$
$787$	2.88740e15	0.340915	0.170457	+	0.985365i	$0.445475\pi$
$788$	−1.85088e15	−0.217012
$789$	4.95999e15	0.577507
$790$	−5.23174e15	−0.604919
$791$	3.80122e15	0.436468
$792$	−2.35275e15	−0.268280
$793$	−3.84687e15	−0.435618
$794$	1.28220e16	1.44193
$795$	−5.98415e15	−0.668318
$796$	1.77362e16	1.96716
$797$	9.74329e15	1.07321	0.536605	−	0.843834i	$-0.319707\pi$
$797$	9.74329e15	1.07321	0.536605	+	0.843834i	$0.319707\pi$
$798$	−5.67254e15	−0.620528
$799$	3.19381e15	0.346977
$800$	−6.43352e15	−0.694150
$801$	5.42071e15	0.580867
$802$	−1.16698e16	−1.24195
$803$	4.12972e15	0.436501
$804$	−6.26206e15	−0.657368
$805$	5.43288e14	0.0566438
$806$	1.23003e15	0.127371
$807$	−8.48270e15	−0.872428
$808$	−2.69216e14	−0.0275004
$809$	−5.06254e15	−0.513632	−0.256816	−	0.966460i	$-0.582673\pi$
$809$	−5.06254e15	−0.513632	−0.256816	+	0.966460i	$0.582673\pi$
$810$	−8.86267e15	−0.893094
$811$	5.12491e15	0.512946	0.256473	−	0.966551i	$-0.417440\pi$
$811$	5.12491e15	0.512946	0.256473	+	0.966551i	$0.417440\pi$
$812$	2.93210e15	0.291488
$813$	1.51431e16	1.49526
$814$	1.20238e16	1.17926
$815$	1.21940e15	0.118790
$816$	−3.90862e14	−0.0378205
$817$	5.19060e15	0.498880
$818$	−2.65638e16	−2.53599
$819$	−1.28120e15	−0.121494
$820$	9.10931e15	0.858043
$821$	1.52178e16	1.42385	0.711923	−	0.702257i	$-0.247824\pi$
$821$	1.52178e16	1.42385	0.711923	+	0.702257i	$0.247824\pi$
$822$	−3.44275e16	−3.19971
$823$	−8.69843e15	−0.803048	−0.401524	−	0.915848i	$-0.631519\pi$
$823$	−8.69843e15	−0.803048	−0.401524	+	0.915848i	$0.631519\pi$
$824$	2.76846e14	0.0253886
$825$	−5.78906e15	−0.527365
$826$	1.77689e16	1.60794
$827$	1.13855e16	1.02346	0.511730	−	0.859146i	$-0.329005\pi$
$827$	1.13855e16	1.02346	0.511730	+	0.859146i	$0.329005\pi$
$828$	1.71826e15	0.153434
$829$	−2.06524e16	−1.83198	−0.915990	−	0.401201i	$-0.868593\pi$
$829$	−2.06524e16	−1.83198	−0.915990	+	0.401201i	$0.868593\pi$
$830$	−5.74789e15	−0.506499
$831$	−1.55828e16	−1.36408
$832$	−8.13292e15	−0.707242
$833$	1.33895e15	0.115669
$834$	1.79197e16	1.53786
$835$	6.92279e15	0.590208
$836$	−5.51759e15	−0.467321
$837$	1.39041e15	0.116991
$838$	−5.27977e15	−0.441339
$839$	−1.05575e16	−0.876741	−0.438370	−	0.898794i	$-0.644444\pi$
$839$	−1.05575e16	−0.876741	−0.438370	+	0.898794i	$0.644444\pi$
$840$	−4.46843e15	−0.368654
$841$	−1.12617e16	−0.923055
$842$	1.48269e15	0.120736
$843$	1.90908e16	1.54444
$844$	1.99771e16	1.60565
$845$	4.36941e15	0.348908
$846$	1.65657e16	1.31424
$847$	5.55111e15	0.437543
$848$	2.72981e15	0.213774
$849$	−1.38217e16	−1.07540
$850$	−3.23575e15	−0.250133
$851$	−3.58702e15	−0.275500
$852$	1.62835e16	1.24260
$853$	−2.39013e16	−1.81218	−0.906090	−	0.423085i	$-0.860947\pi$
$853$	−2.39013e16	−1.81218	−0.906090	+	0.423085i	$0.860947\pi$
$854$	−1.31290e16	−0.989038
$855$	1.29112e15	0.0966391
$856$	1.63563e16	1.21641
$857$	1.43142e16	1.05773	0.528863	−	0.848707i	$-0.322619\pi$
$857$	1.43142e16	1.05773	0.528863	+	0.848707i	$0.322619\pi$
$858$	−6.54000e15	−0.480173
$859$	1.37286e16	1.00153	0.500763	−	0.865584i	$-0.333053\pi$
$859$	1.37286e16	1.00153	0.500763	+	0.865584i	$0.333053\pi$
$860$	1.00096e16	0.725564
$861$	1.20068e16	0.864789
$862$	5.79005e15	0.414374
$863$	3.96945e15	0.282274	0.141137	−	0.989990i	$-0.454924\pi$
$863$	3.96945e15	0.282274	0.141137	+	0.989990i	$0.454924\pi$
$864$	−8.21573e15	−0.580524
$865$	5.71937e14	0.0401569
$866$	1.39894e16	0.976003
$867$	−1.66671e16	−1.15546
$868$	2.63771e15	0.181706
$869$	−6.70750e15	−0.459147
$870$	−3.50234e15	−0.238233
$871$	−2.15634e15	−0.145753
$872$	−1.13905e16	−0.765073
$873$	8.82796e15	0.589226
$874$	2.61970e15	0.173755
$875$	−7.45587e15	−0.491421
$876$	2.48061e16	1.62475
$877$	5.30799e15	0.345487	0.172744	−	0.984967i	$-0.444737\pi$
$877$	5.30799e15	0.345487	0.172744	+	0.984967i	$0.444737\pi$
$878$	−4.80106e16	−3.10540
$879$	−2.47374e16	−1.59006
$880$	−6.23502e14	−0.0398275
$881$	2.51764e16	1.59818	0.799089	−	0.601212i	$-0.205315\pi$
$881$	2.51764e16	1.59818	0.799089	+	0.601212i	$0.205315\pi$
$882$	6.94493e15	0.438117
$883$	−4.83599e15	−0.303180	−0.151590	−	0.988443i	$-0.548439\pi$
$883$	−4.83599e15	−0.303180	−0.151590	+	0.988443i	$0.548439\pi$
$884$	−2.29686e15	−0.143102
$885$	−1.33362e16	−0.825739
$886$	3.06385e16	1.88531
$887$	−1.19150e16	−0.728639	−0.364320	−	0.931274i	$-0.618698\pi$
$887$	−1.19150e16	−0.728639	−0.364320	+	0.931274i	$0.618698\pi$
$888$	2.95025e16	1.79303
$889$	−5.67725e15	−0.342909
$890$	1.59374e16	0.956691
$891$	−1.13626e16	−0.677878
$892$	−7.13158e15	−0.422843
$893$	1.58694e16	0.935144
$894$	−4.03985e16	−2.36597
$895$	−7.52673e15	−0.438106
$896$	−1.85376e16	−1.07240
$897$	1.95105e15	0.112179
$898$	3.58033e16	2.04599
$899$	8.44517e14	0.0479656
$900$	−1.05455e16	−0.595295
$901$	−4.28828e15	−0.240601
$902$	1.85870e16	1.03651
$903$	1.31935e16	0.731268
$904$	1.44385e16	0.795417
$905$	−1.28912e16	−0.705871
$906$	1.31451e16	0.715417
$907$	1.79423e16	0.970596	0.485298	−	0.874349i	$-0.338711\pi$
$907$	1.79423e16	0.970596	0.485298	+	0.874349i	$0.338711\pi$
$908$	−4.84991e16	−2.60772
$909$	1.97722e14	0.0105670
$910$	−3.76683e15	−0.200101
$911$	2.23381e16	1.17949	0.589746	−	0.807589i	$-0.299228\pi$
$911$	2.23381e16	1.17949	0.589746	+	0.807589i	$0.299228\pi$
$912$	−1.94212e15	−0.101931
$913$	−7.36924e15	−0.384444
$914$	3.93357e16	2.03978
$915$	9.85373e15	0.507907
$916$	−8.80858e15	−0.451316
$917$	1.07941e15	0.0549740
$918$	−4.13211e15	−0.209188
$919$	−2.83253e16	−1.42541	−0.712704	−	0.701465i	$-0.752530\pi$
$919$	−2.83253e16	−1.42541	−0.712704	+	0.701465i	$0.752530\pi$
$920$	2.06361e15	0.103227
$921$	5.12871e15	0.255024
$922$	3.77986e16	1.86834
$923$	5.60722e15	0.275510
$924$	−1.40246e16	−0.685007
$925$	2.20146e16	1.06889
$926$	−2.35426e16	−1.13630
$927$	−2.03326e14	−0.00975559
$928$	−4.99014e15	−0.238012
$929$	1.57937e16	0.748853	0.374426	−	0.927257i	$-0.377840\pi$
$929$	1.57937e16	0.748853	0.374426	+	0.927257i	$0.377840\pi$
$930$	−3.15070e15	−0.148508
$931$	6.65302e15	0.311742
$932$	−5.43299e16	−2.53076
$933$	3.20481e16	1.48407
$934$	−1.23021e16	−0.566332
$935$	9.79464e14	0.0448254
$936$	−4.86647e15	−0.221410
$937$	−1.23668e16	−0.559357	−0.279678	−	0.960094i	$-0.590228\pi$
$937$	−1.23668e16	−0.559357	−0.279678	+	0.960094i	$0.590228\pi$
$938$	−7.35936e15	−0.330921
$939$	−4.71266e15	−0.210671
$940$	3.06028e16	1.36006
$941$	−2.21299e16	−0.977768	−0.488884	−	0.872349i	$-0.662596\pi$
$941$	−2.21299e16	−0.977768	−0.488884	+	0.872349i	$0.662596\pi$
$942$	8.22188e15	0.361153
$943$	−5.54500e15	−0.242151
$944$	6.08360e15	0.264128
$945$	−4.25798e15	−0.183793
$946$	2.04240e16	0.876477
$947$	2.28513e16	0.974958	0.487479	−	0.873135i	$-0.337917\pi$
$947$	2.28513e16	0.974958	0.487479	+	0.873135i	$0.337917\pi$
$948$	−4.02901e16	−1.70904
$949$	8.54197e15	0.360242
$950$	−1.60778e16	−0.674136
$951$	−1.78265e15	−0.0743146
$952$	−3.20210e15	−0.132719
$953$	−2.60293e16	−1.07264	−0.536318	−	0.844016i	$-0.680185\pi$
$953$	−2.60293e16	−1.07264	−0.536318	+	0.844016i	$0.680185\pi$
$954$	−2.22426e16	−0.911316
$955$	−1.66110e16	−0.676669
$956$	2.01664e16	0.816791
$957$	−4.49027e15	−0.180824
$958$	−4.90182e16	−1.96267
$959$	−2.54225e16	−1.01208
$960$	2.08324e16	0.824607
$961$	−2.46488e16	−0.970099
$962$	2.48703e16	0.973234
$963$	−1.20127e16	−0.467407
$964$	−6.98064e16	−2.70067
$965$	5.49749e15	0.211478
$966$	6.65875e15	0.254694
$967$	4.03376e15	0.153414	0.0767068	−	0.997054i	$-0.475559\pi$
$967$	4.03376e15	0.153414	0.0767068	+	0.997054i	$0.475559\pi$
$968$	2.10852e16	0.797376
$969$	3.05089e15	0.114722
$970$	2.59550e16	0.970458
$971$	1.87535e16	0.697231	0.348615	−	0.937266i	$-0.386652\pi$
$971$	1.87535e16	0.697231	0.348615	+	0.937266i	$0.386652\pi$
$972$	−3.73128e16	−1.37941
$973$	1.32325e16	0.486431
$974$	−7.00701e16	−2.56129
$975$	−1.19742e16	−0.435232
$976$	−4.49501e15	−0.162464
$977$	2.24370e15	0.0806390	0.0403195	−	0.999187i	$-0.487162\pi$
$977$	2.24370e15	0.0806390	0.0403195	+	0.999187i	$0.487162\pi$
$978$	1.49455e16	0.534130
$979$	2.04329e16	0.726150
$980$	1.28297e16	0.453393
$981$	8.36559e15	0.293980
$982$	1.48818e14	0.00520046
$983$	7.37829e15	0.256396	0.128198	−	0.991749i	$-0.459081\pi$
$983$	7.37829e15	0.256396	0.128198	+	0.991749i	$0.459081\pi$
$984$	4.56064e16	1.57599
$985$	1.63257e15	0.0561011
$986$	−2.50980e15	−0.0857661
$987$	4.03370e16	1.37075
$988$	−1.14127e16	−0.385677
$989$	−6.09303e15	−0.204764
$990$	5.08031e15	0.169784
$991$	−3.15435e16	−1.04835	−0.524174	−	0.851611i	$-0.675626\pi$
$991$	−3.15435e16	−1.04835	−0.524174	+	0.851611i	$0.675626\pi$
$992$	−4.48912e15	−0.148370
$993$	3.77993e16	1.24240
$994$	1.91369e16	0.625526
$995$	−1.56442e16	−0.508542
$996$	−4.42650e16	−1.43098
$997$	6.64518e15	0.213640	0.106820	−	0.994278i	$-0.465933\pi$
$997$	6.64518e15	0.213640	0.106820	+	0.994278i	$0.465933\pi$
$998$	5.27824e16	1.68761
$999$	2.81130e16	0.893918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.12.a.a.1.1	✓	8
3.2	odd	2		207.12.a.a.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.12.a.a.1.1	✓	8	1.1	even	1	trivial
207.12.a.a.1.8		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	23.a (trivial)

\(f(q)\)	\(=\)	\(q-74.2314 q^{2} +504.234 q^{3} +3462.31 q^{4} -3053.92 q^{5} -37430.0 q^{6} -27639.7 q^{7} -104986. q^{8} +77105.4 q^{9} +O(q^{10})\)	\(q-74.2314 q^{2} +504.234 q^{3} +3462.31 q^{4} -3053.92 q^{5} -37430.0 q^{6} -27639.7 q^{7} -104986. q^{8} +77105.4 q^{9} +226697. q^{10} +290643. q^{11} +1.74581e6 q^{12} +601170. q^{13} +2.05173e6 q^{14} -1.53989e6 q^{15} +702457. q^{16} -1.10350e6 q^{17} -5.72364e6 q^{18} -5.48307e6 q^{19} -1.05736e7 q^{20} -1.39369e7 q^{21} -2.15749e7 q^{22} +6.43634e6 q^{23} -5.29376e7 q^{24} -3.95017e7 q^{25} -4.46257e7 q^{26} -5.04444e7 q^{27} -9.56970e7 q^{28} -3.06394e7 q^{29} +1.14308e8 q^{30} -2.75631e7 q^{31} +1.62867e8 q^{32} +1.46552e8 q^{33} +8.19141e7 q^{34} +8.44094e7 q^{35} +2.66963e8 q^{36} -5.57307e8 q^{37} +4.07016e8 q^{38} +3.03131e8 q^{39} +3.20619e8 q^{40} -8.61514e8 q^{41} +1.03455e9 q^{42} -9.46660e8 q^{43} +1.00630e9 q^{44} -2.35474e8 q^{45} -4.77779e8 q^{46} -2.89426e9 q^{47} +3.54203e8 q^{48} -1.21337e9 q^{49} +2.93227e9 q^{50} -5.56421e8 q^{51} +2.08144e9 q^{52} +3.88609e9 q^{53} +3.74456e9 q^{54} -8.87601e8 q^{55} +2.90178e9 q^{56} -2.76475e9 q^{57} +2.27440e9 q^{58} +8.66045e9 q^{59} -5.33158e9 q^{60} -6.39897e9 q^{61} +2.04605e9 q^{62} -2.13117e9 q^{63} -1.35285e10 q^{64} -1.83593e9 q^{65} -1.08788e10 q^{66} -3.58690e9 q^{67} -3.82064e9 q^{68} +3.24543e9 q^{69} -6.26583e9 q^{70} +9.32716e9 q^{71} -8.09499e9 q^{72} +1.42089e10 q^{73} +4.13697e10 q^{74} -1.99181e10 q^{75} -1.89841e10 q^{76} -8.03328e9 q^{77} -2.25018e10 q^{78} -2.30781e10 q^{79} -2.14525e9 q^{80} -3.90948e10 q^{81} +6.39514e10 q^{82} -2.53549e10 q^{83} -4.82537e10 q^{84} +3.36999e9 q^{85} +7.02719e10 q^{86} -1.54494e10 q^{87} -3.05134e10 q^{88} +7.03025e10 q^{89} +1.74796e10 q^{90} -1.66162e10 q^{91} +2.22846e10 q^{92} -1.38983e10 q^{93} +2.14845e11 q^{94} +1.67449e10 q^{95} +8.21231e10 q^{96} +1.14492e11 q^{97} +9.00706e10 q^{98} +2.24101e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9}+O(q^{10}) \) 8 * q - 32 * q^2 - 992 * q^3 + 5120 * q^4 - 11466 * q^5 + 34618 * q^6 - 54118 * q^7 - 155568 * q^8 + 648814 * q^9	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9} + 517892 q^{10} - 291462 q^{11} - 884188 q^{12} - 2211306 q^{13} - 939584 q^{14} - 2205330 q^{15} - 8561344 q^{16} - 5775330 q^{17} - 51349034 q^{18} - 21015588 q^{19} - 65503576 q^{20} - 36171230 q^{21} - 83047784 q^{22} + 51490744 q^{23} - 129286728 q^{24} - 36491644 q^{25} - 119299562 q^{26} - 394617320 q^{27} - 392796032 q^{28} - 322285430 q^{29} - 646885140 q^{30} - 415184840 q^{31} + 31831744 q^{32} - 549306602 q^{33} - 28224252 q^{34} + 603721008 q^{35} + 690703676 q^{36} + 176642018 q^{37} + 554685496 q^{38} + 2251149264 q^{39} + 1337904816 q^{40} + 357962218 q^{41} + 340644280 q^{42} + 2500461376 q^{43} + 5064743472 q^{44} + 385017072 q^{45} - 205962976 q^{46} + 261795200 q^{47} + 4421752784 q^{48} + 2656605924 q^{49} + 1642758328 q^{50} + 6771514570 q^{51} + 3841657212 q^{52} + 3542935060 q^{53} + 18173306686 q^{54} - 10100187604 q^{55} + 7995463104 q^{56} - 14761628752 q^{57} - 9113565454 q^{58} + 930905396 q^{59} + 19344914040 q^{60} - 25338655048 q^{61} + 4385691666 q^{62} - 25499316044 q^{63} - 34067008768 q^{64} - 25954746658 q^{65} + 13172584012 q^{66} - 3123467482 q^{67} - 37358480280 q^{68} - 6384852256 q^{69} - 35719175696 q^{70} - 52612263236 q^{71} - 9100886376 q^{72} - 67014176274 q^{73} + 10171443276 q^{74} - 87540153860 q^{75} + 17955918576 q^{76} - 44516617816 q^{77} - 25596104778 q^{78} - 27683357604 q^{79} + 74357773216 q^{80} + 55141240264 q^{81} + 73615849126 q^{82} - 12253964262 q^{83} + 168565479344 q^{84} + 58779027600 q^{85} + 90522557252 q^{86} - 129275944888 q^{87} + 33736356800 q^{88} + 10662817760 q^{89} + 450294422856 q^{90} - 28336741418 q^{91} + 32954076160 q^{92} + 164368292014 q^{93} + 285145948346 q^{94} - 64104297380 q^{95} + 208023008864 q^{96} - 124519454530 q^{97} + 215615498272 q^{98} + 186256571332 q^{99}+O(q^{100}) \) 8 * q - 32 * q^2 - 992 * q^3 + 5120 * q^4 - 11466 * q^5 + 34618 * q^6 - 54118 * q^7 - 155568 * q^8 + 648814 * q^9 + 517892 * q^10 - 291462 * q^11 - 884188 * q^12 - 2211306 * q^13 - 939584 * q^14 - 2205330 * q^15 - 8561344 * q^16 - 5775330 * q^17 - 51349034 * q^18 - 21015588 * q^19 - 65503576 * q^20 - 36171230 * q^21 - 83047784 * q^22 + 51490744 * q^23 - 129286728 * q^24 - 36491644 * q^25 - 119299562 * q^26 - 394617320 * q^27 - 392796032 * q^28 - 322285430 * q^29 - 646885140 * q^30 - 415184840 * q^31 + 31831744 * q^32 - 549306602 * q^33 - 28224252 * q^34 + 603721008 * q^35 + 690703676 * q^36 + 176642018 * q^37 + 554685496 * q^38 + 2251149264 * q^39 + 1337904816 * q^40 + 357962218 * q^41 + 340644280 * q^42 + 2500461376 * q^43 + 5064743472 * q^44 + 385017072 * q^45 - 205962976 * q^46 + 261795200 * q^47 + 4421752784 * q^48 + 2656605924 * q^49 + 1642758328 * q^50 + 6771514570 * q^51 + 3841657212 * q^52 + 3542935060 * q^53 + 18173306686 * q^54 - 10100187604 * q^55 + 7995463104 * q^56 - 14761628752 * q^57 - 9113565454 * q^58 + 930905396 * q^59 + 19344914040 * q^60 - 25338655048 * q^61 + 4385691666 * q^62 - 25499316044 * q^63 - 34067008768 * q^64 - 25954746658 * q^65 + 13172584012 * q^66 - 3123467482 * q^67 - 37358480280 * q^68 - 6384852256 * q^69 - 35719175696 * q^70 - 52612263236 * q^71 - 9100886376 * q^72 - 67014176274 * q^73 + 10171443276 * q^74 - 87540153860 * q^75 + 17955918576 * q^76 - 44516617816 * q^77 - 25596104778 * q^78 - 27683357604 * q^79 + 74357773216 * q^80 + 55141240264 * q^81 + 73615849126 * q^82 - 12253964262 * q^83 + 168565479344 * q^84 + 58779027600 * q^85 + 90522557252 * q^86 - 129275944888 * q^87 + 33736356800 * q^88 + 10662817760 * q^89 + 450294422856 * q^90 - 28336741418 * q^91 + 32954076160 * q^92 + 164368292014 * q^93 + 285145948346 * q^94 - 64104297380 * q^95 + 208023008864 * q^96 - 124519454530 * q^97 + 215615498272 * q^98 + 186256571332 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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