LMFDB - Embedded newform 23.12.a.a.1.3

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			1.3
Root			$19.5149$ 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)

$f(q)$	$=$	$q-43.0298 q^{2} -312.230 q^{3} -196.437 q^{4} -1082.06 q^{5} +13435.2 q^{6} +38616.0 q^{7} +96577.7 q^{8} -79659.3 q^{9} +O(q^{10})$	\(q-43.0298 q^{2} -312.230 q^{3} -196.437 q^{4} -1082.06 q^{5} +13435.2 q^{6} +38616.0 q^{7} +96577.7 q^{8} -79659.3 q^{9} +46560.9 q^{10} +206509. q^{11} +61333.5 q^{12} +1.94280e6 q^{13} -1.66164e6 q^{14} +337852. q^{15} -3.75341e6 q^{16} +315485. q^{17} +3.42772e6 q^{18} -1.70155e7 q^{19} +212556. q^{20} -1.20571e7 q^{21} -8.88603e6 q^{22} +6.43634e6 q^{23} -3.01545e7 q^{24} -4.76573e7 q^{25} -8.35984e7 q^{26} +8.01827e7 q^{27} -7.58560e6 q^{28} -1.47468e8 q^{29} -1.45377e7 q^{30} +1.09615e7 q^{31} -3.62824e7 q^{32} -6.44783e7 q^{33} -1.35752e7 q^{34} -4.17849e7 q^{35} +1.56480e7 q^{36} +6.75584e8 q^{37} +7.32175e8 q^{38} -6.06602e8 q^{39} -1.04503e8 q^{40} +6.03098e8 q^{41} +5.18814e8 q^{42} -8.20201e8 q^{43} -4.05659e7 q^{44} +8.61962e7 q^{45} -2.76955e8 q^{46} +2.53136e8 q^{47} +1.17193e9 q^{48} -4.86128e8 q^{49} +2.05068e9 q^{50} -9.85039e7 q^{51} -3.81638e8 q^{52} -1.73866e9 q^{53} -3.45024e9 q^{54} -2.23455e8 q^{55} +3.72945e9 q^{56} +5.31276e9 q^{57} +6.34552e9 q^{58} -4.41902e9 q^{59} -6.63665e7 q^{60} -3.58492e9 q^{61} -4.71670e8 q^{62} -3.07613e9 q^{63} +9.24822e9 q^{64} -2.10223e9 q^{65} +2.77449e9 q^{66} -1.08482e10 q^{67} -6.19728e7 q^{68} -2.00962e9 q^{69} +1.79800e9 q^{70} -1.69770e10 q^{71} -7.69330e9 q^{72} -2.96417e10 q^{73} -2.90702e10 q^{74} +1.48800e10 q^{75} +3.34247e9 q^{76} +7.97455e9 q^{77} +2.61019e10 q^{78} -3.08415e10 q^{79} +4.06142e9 q^{80} -1.09241e10 q^{81} -2.59512e10 q^{82} +3.12203e10 q^{83} +2.36846e9 q^{84} -3.41374e8 q^{85} +3.52931e10 q^{86} +4.60440e10 q^{87} +1.99441e10 q^{88} +7.10416e10 q^{89} -3.70900e9 q^{90} +7.50233e10 q^{91} -1.26433e9 q^{92} -3.42250e9 q^{93} -1.08924e10 q^{94} +1.84118e10 q^{95} +1.13285e10 q^{96} -5.27501e10 q^{97} +2.09180e10 q^{98} -1.64503e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−43.0298	−0.950833	−0.475417	−	0.879761i	$-0.657703\pi$
$2$	−43.0298	−0.950833	−0.475417	+	0.879761i	$0.657703\pi$
$3$	−312.230	−0.741836	−0.370918	−	0.928666i	$-0.620957\pi$
$3$	−312.230	−0.741836	−0.370918	+	0.928666i	$0.620957\pi$
$4$	−196.437	−0.0959163
$5$	−1082.06	−0.154852	−0.0774260	−	0.996998i	$-0.524670\pi$
$5$	−1082.06	−0.154852	−0.0774260	+	0.996998i	$0.524670\pi$
$6$	13435.2	0.705363
$7$	38616.0	0.868417	0.434209	−	0.900812i	$-0.357028\pi$
$7$	38616.0	0.868417	0.434209	+	0.900812i	$0.357028\pi$
$8$	96577.7	1.04203
$9$	−79659.3	−0.449679
$10$	46560.9	0.147238
$11$	206509.	0.386615	0.193308	−	0.981138i	$-0.438078\pi$
$11$	206509.	0.386615	0.193308	+	0.981138i	$0.438078\pi$
$12$	61333.5	0.0711542
$13$	1.94280e6	1.45124	0.725622	−	0.688094i	$-0.241552\pi$
$13$	1.94280e6	1.45124	0.725622	+	0.688094i	$0.241552\pi$
$14$	−1.66164e6	−0.825720
$15$	337852.	0.114875
$16$	−3.75341e6	−0.894884
$17$	315485.	0.0538901	0.0269451	−	0.999637i	$-0.491422\pi$
$17$	315485.	0.0538901	0.0269451	+	0.999637i	$0.491422\pi$
$18$	3.42772e6	0.427570
$19$	−1.70155e7	−1.57652	−0.788262	−	0.615340i	$-0.789019\pi$
$19$	−1.70155e7	−1.57652	−0.788262	+	0.615340i	$0.789019\pi$
$20$	212556.	0.0148528
$21$	−1.20571e7	−0.644224
$22$	−8.88603e6	−0.367606
$23$	6.43634e6	0.208514
$23$	6.43634e6	0.208514
$24$	−3.01545e7	−0.773018
$25$	−4.76573e7	−0.976021
$26$	−8.35984e7	−1.37989
$27$	8.01827e7	1.07542
$28$	−7.58560e6	−0.0832954
$29$	−1.47468e8	−1.33509	−0.667543	−	0.744571i	$-0.732654\pi$
$29$	−1.47468e8	−1.33509	−0.667543	+	0.744571i	$0.732654\pi$
$30$	−1.45377e7	−0.109227
$31$	1.09615e7	0.0687669	0.0343834	−	0.999409i	$-0.489053\pi$
$31$	1.09615e7	0.0687669	0.0343834	+	0.999409i	$0.489053\pi$
$32$	−3.62824e7	−0.191148
$33$	−6.44783e7	−0.286805
$34$	−1.35752e7	−0.0512405
$35$	−4.17849e7	−0.134476
$36$	1.56480e7	0.0431315
$37$	6.75584e8	1.60166	0.800829	−	0.598893i	$-0.204392\pi$
$37$	6.75584e8	1.60166	0.800829	+	0.598893i	$0.204392\pi$
$38$	7.32175e8	1.49901
$39$	−6.06602e8	−1.07658
$40$	−1.04503e8	−0.161361
$41$	6.03098e8	0.812974	0.406487	−	0.913657i	$-0.366754\pi$
$41$	6.03098e8	0.812974	0.406487	+	0.913657i	$0.366754\pi$
$42$	5.18814e8	0.612549
$43$	−8.20201e8	−0.850832	−0.425416	−	0.904998i	$-0.639872\pi$
$43$	−8.20201e8	−0.850832	−0.425416	+	0.904998i	$0.639872\pi$
$44$	−4.05659e7	−0.0370827
$45$	8.61962e7	0.0696337
$46$	−2.76955e8	−0.198262
$47$	2.53136e8	0.160996	0.0804982	−	0.996755i	$-0.474349\pi$
$47$	2.53136e8	0.160996	0.0804982	+	0.996755i	$0.474349\pi$
$48$	1.17193e9	0.663857
$49$	−4.86128e8	−0.245851
$50$	2.05068e9	0.928033
$51$	−9.85039e7	−0.0399777
$52$	−3.81638e8	−0.139198
$53$	−1.73866e9	−0.571080	−0.285540	−	0.958367i	$-0.592173\pi$
$53$	−1.73866e9	−0.571080	−0.285540	+	0.958367i	$0.592173\pi$
$54$	−3.45024e9	−1.02255
$55$	−2.23455e8	−0.0598681
$56$	3.72945e9	0.904920
$57$	5.31276e9	1.16952
$58$	6.34552e9	1.26944
$59$	−4.41902e9	−0.804710	−0.402355	−	0.915484i	$-0.631808\pi$
$59$	−4.41902e9	−0.804710	−0.402355	+	0.915484i	$0.631808\pi$
$60$	−6.63665e7	−0.0110184
$61$	−3.58492e9	−0.543457	−0.271728	−	0.962374i	$-0.587595\pi$
$61$	−3.58492e9	−0.543457	−0.271728	+	0.962374i	$0.587595\pi$
$62$	−4.71670e8	−0.0653858
$63$	−3.07613e9	−0.390509
$64$	9.24822e9	1.07663
$65$	−2.10223e9	−0.224728
$66$	2.77449e9	0.272704
$67$	−1.08482e10	−0.981629	−0.490815	−	0.871264i	$-0.663301\pi$
$67$	−1.08482e10	−0.981629	−0.490815	+	0.871264i	$0.663301\pi$
$68$	−6.19728e7	−0.00516894
$69$	−2.00962e9	−0.154684
$70$	1.79800e9	0.127864
$71$	−1.69770e10	−1.11671	−0.558355	−	0.829602i	$-0.688567\pi$
$71$	−1.69770e10	−1.11671	−0.558355	+	0.829602i	$0.688567\pi$
$72$	−7.69330e9	−0.468580
$73$	−2.96417e10	−1.67351	−0.836753	−	0.547580i	$-0.815549\pi$
$73$	−2.96417e10	−1.67351	−0.836753	+	0.547580i	$0.815549\pi$
$74$	−2.90702e10	−1.52291
$75$	1.48800e10	0.724048
$76$	3.34247e9	0.151214
$77$	7.97455e9	0.335743
$78$	2.61019e10	1.02365
$79$	−3.08415e10	−1.12768	−0.563841	−	0.825883i	$-0.690677\pi$
$79$	−3.08415e10	−1.12768	−0.563841	+	0.825883i	$0.690677\pi$
$80$	4.06142e9	0.138575
$81$	−1.09241e10	−0.348110
$82$	−2.59512e10	−0.773003
$83$	3.12203e10	0.869978	0.434989	−	0.900436i	$-0.356752\pi$
$83$	3.12203e10	0.869978	0.434989	+	0.900436i	$0.356752\pi$
$84$	2.36846e9	0.0617916
$85$	−3.41374e8	−0.00834499
$86$	3.52931e10	0.808999
$87$	4.60440e10	0.990415
$88$	1.99441e10	0.402866
$89$	7.10416e10	1.34855	0.674276	−	0.738479i	$-0.264456\pi$
$89$	7.10416e10	1.34855	0.674276	+	0.738479i	$0.264456\pi$
$90$	−3.70900e9	−0.0662100
$91$	7.50233e10	1.26029
$92$	−1.26433e9	−0.0199999
$93$	−3.42250e9	−0.0510138
$94$	−1.08924e10	−0.153081
$95$	1.84118e10	0.244128
$96$	1.13285e10	0.141801
$97$	−5.27501e10	−0.623704	−0.311852	−	0.950131i	$-0.600949\pi$
$97$	−5.27501e10	−0.623704	−0.311852	+	0.950131i	$0.600949\pi$
$98$	2.09180e10	0.233763
$99$	−1.64503e10	−0.173853
$100$	9.36163e9	0.0936163
$101$	−1.11385e11	−1.05453	−0.527267	−	0.849700i	$-0.676783\pi$
$101$	−1.11385e11	−1.05453	−0.527267	+	0.849700i	$0.676783\pi$
$102$	4.23860e9	0.0380121
$103$	−1.88311e9	−0.0160055	−0.00800277	−	0.999968i	$-0.502547\pi$
$103$	−1.88311e9	−0.0160055	−0.00800277	+	0.999968i	$0.502547\pi$
$104$	1.87631e11	1.51224
$105$	1.30465e10	0.0997593
$106$	7.48142e10	0.543002
$107$	1.60864e10	0.110879	0.0554393	−	0.998462i	$-0.482344\pi$
$107$	1.60864e10	0.110879	0.0554393	+	0.998462i	$0.482344\pi$
$108$	−1.57508e10	−0.103151
$109$	−2.53355e11	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$109$	−2.53355e11	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$110$	9.61523e9	0.0569246
$111$	−2.10938e11	−1.18817
$112$	−1.44942e11	−0.777133
$113$	1.79788e11	0.917970	0.458985	−	0.888444i	$-0.348213\pi$
$113$	1.79788e11	0.917970	0.458985	+	0.888444i	$0.348213\pi$
$114$	−2.28607e11	−1.11202
$115$	−6.96452e9	−0.0322889
$116$	2.89681e10	0.128057
$117$	−1.54762e11	−0.652593
$118$	1.90149e11	0.765145
$119$	1.21828e10	0.0467991
$120$	3.26290e10	0.119703
$121$	−2.42666e11	−0.850529
$122$	1.54258e11	0.516737
$123$	−1.88305e11	−0.603094
$124$	−2.15323e9	−0.00659587
$125$	1.04403e11	0.305991
$126$	1.32365e11	0.371309
$127$	−1.01258e11	−0.271964	−0.135982	−	0.990711i	$-0.543419\pi$
$127$	−1.01258e11	−0.271964	−0.135982	+	0.990711i	$0.543419\pi$
$128$	−3.23643e11	−0.832551
$129$	2.56092e11	0.631178
$130$	9.04586e10	0.213679
$131$	1.87055e11	0.423622	0.211811	−	0.977311i	$-0.432064\pi$
$131$	1.87055e11	0.423622	0.211811	+	0.977311i	$0.432064\pi$
$132$	1.26659e10	0.0275093
$133$	−6.57072e11	−1.36908
$134$	4.66797e11	0.933366
$135$	−8.67626e10	−0.166532
$136$	3.04688e10	0.0561553
$137$	−4.86075e11	−0.860479	−0.430239	−	0.902715i	$-0.641571\pi$
$137$	−4.86075e11	−0.860479	−0.430239	+	0.902715i	$0.641571\pi$
$138$	8.64736e10	0.147078
$139$	−2.93762e11	−0.480191	−0.240095	−	0.970749i	$-0.577179\pi$
$139$	−2.93762e11	−0.480191	−0.240095	+	0.970749i	$0.577179\pi$
$140$	8.20809e9	0.0128985
$141$	−7.90368e10	−0.119433
$142$	7.30517e11	1.06180
$143$	4.01206e11	0.561073
$144$	2.98994e11	0.402410
$145$	1.59570e11	0.206741
$146$	1.27548e12	1.59123
$147$	1.51784e11	0.182381
$148$	−1.32709e11	−0.153625
$149$	3.30552e11	0.368735	0.184368	−	0.982857i	$-0.440976\pi$
$149$	3.30552e11	0.368735	0.184368	+	0.982857i	$0.440976\pi$
$150$	−6.40285e11	−0.688449
$151$	1.31138e12	1.35943	0.679714	−	0.733477i	$-0.262104\pi$
$151$	1.31138e12	1.35943	0.679714	+	0.733477i	$0.262104\pi$
$152$	−1.64332e12	−1.64279
$153$	−2.51313e10	−0.0242333
$154$	−3.43143e11	−0.319236
$155$	−1.18610e10	−0.0106487
$156$	1.19159e11	0.103262
$157$	−1.91767e12	−1.60445	−0.802225	−	0.597021i	$-0.796351\pi$
$157$	−1.91767e12	−1.60445	−0.802225	+	0.597021i	$0.796351\pi$
$158$	1.32710e12	1.07224
$159$	5.42862e11	0.423648
$160$	3.92597e10	0.0295997
$161$	2.48546e11	0.181078
$162$	4.70060e11	0.330995
$163$	−3.83256e11	−0.260890	−0.130445	−	0.991456i	$-0.541641\pi$
$163$	−3.83256e11	−0.260890	−0.130445	+	0.991456i	$0.541641\pi$
$164$	−1.18471e11	−0.0779775
$165$	6.97695e10	0.0444123
$166$	−1.34341e12	−0.827204
$167$	1.77486e12	1.05736	0.528682	−	0.848820i	$-0.322686\pi$
$167$	1.77486e12	1.05736	0.528682	+	0.848820i	$0.322686\pi$
$168$	−1.16445e12	−0.671303
$169$	1.98232e12	1.10611
$170$	1.46892e10	0.00793470
$171$	1.35544e12	0.708929
$172$	1.61118e11	0.0816087
$173$	2.60744e12	1.27927	0.639633	−	0.768680i	$-0.279086\pi$
$173$	2.60744e12	1.27927	0.639633	+	0.768680i	$0.279086\pi$
$174$	−1.98126e12	−0.941720
$175$	−1.84034e12	−0.847594
$176$	−7.75113e11	−0.345976
$177$	1.37975e12	0.596963
$178$	−3.05691e12	−1.28225
$179$	8.91823e11	0.362733	0.181367	−	0.983416i	$-0.441948\pi$
$179$	8.91823e11	0.362733	0.181367	+	0.983416i	$0.441948\pi$
$180$	−1.69321e10	−0.00667900
$181$	−2.22047e12	−0.849598	−0.424799	−	0.905288i	$-0.639655\pi$
$181$	−2.22047e12	−0.849598	−0.424799	+	0.905288i	$0.639655\pi$
$182$	−3.22824e12	−1.19832
$183$	1.11932e12	0.403156
$184$	6.21607e11	0.217279
$185$	−7.31023e11	−0.248020
$186$	1.47270e11	0.0485056
$187$	6.51504e10	0.0208347
$188$	−4.97253e10	−0.0154422
$189$	3.09634e12	0.933917
$190$	−7.92258e11	−0.232125
$191$	−3.78867e12	−1.07846	−0.539229	−	0.842159i	$-0.681284\pi$
$191$	−3.78867e12	−1.07846	−0.539229	+	0.842159i	$0.681284\pi$
$192$	−2.88757e12	−0.798686
$193$	−2.82578e12	−0.759580	−0.379790	−	0.925073i	$-0.624004\pi$
$193$	−2.82578e12	−0.759580	−0.379790	+	0.925073i	$0.624004\pi$
$194$	2.26983e12	0.593039
$195$	6.56380e11	0.166711
$196$	9.54933e10	0.0235811
$197$	−1.35534e12	−0.325451	−0.162725	−	0.986671i	$-0.552028\pi$
$197$	−1.35534e12	−0.325451	−0.162725	+	0.986671i	$0.552028\pi$
$198$	7.07855e11	0.165305
$199$	5.57108e11	0.126546	0.0632728	−	0.997996i	$-0.479846\pi$
$199$	5.57108e11	0.126546	0.0632728	+	0.997996i	$0.479846\pi$
$200$	−4.60263e12	−1.01705
$201$	3.38715e12	0.728208
$202$	4.79289e12	1.00269
$203$	−5.69464e12	−1.15941
$204$	1.93498e10	0.00383451
$205$	−6.52589e11	−0.125891
$206$	8.10297e10	0.0152186
$207$	−5.12714e11	−0.0937645
$208$	−7.29214e12	−1.29869
$209$	−3.51386e12	−0.609508
$210$	−5.61389e11	−0.0948544
$211$	8.47531e12	1.39509	0.697545	−	0.716541i	$-0.254276\pi$
$211$	8.47531e12	1.39509	0.697545	+	0.716541i	$0.254276\pi$
$212$	3.41536e11	0.0547759
$213$	5.30074e12	0.828416
$214$	−6.92194e11	−0.105427
$215$	8.87508e11	0.131753
$216$	7.74386e12	1.12063
$217$	4.23289e11	0.0597184
$218$	1.09018e13	1.49964
$219$	9.25504e12	1.24147
$220$	4.38948e10	0.00574233
$221$	6.12925e11	0.0782077
$222$	9.07661e12	1.12975
$223$	7.40652e12	0.899368	0.449684	−	0.893188i	$-0.351537\pi$
$223$	7.40652e12	0.899368	0.449684	+	0.893188i	$0.351537\pi$
$224$	−1.40108e12	−0.165997
$225$	3.79634e12	0.438896
$226$	−7.73623e12	−0.872836
$227$	1.62503e13	1.78945	0.894727	−	0.446614i	$-0.147370\pi$
$227$	1.62503e13	1.78945	0.894727	+	0.446614i	$0.147370\pi$
$228$	−1.04362e12	−0.112176
$229$	−1.45680e13	−1.52864	−0.764321	−	0.644836i	$-0.776926\pi$
$229$	−1.45680e13	−1.52864	−0.764321	+	0.644836i	$0.776926\pi$
$230$	2.99682e11	0.0307013
$231$	−2.48990e12	−0.249067
$232$	−1.42421e13	−1.39120
$233$	5.05225e11	0.0481978	0.0240989	−	0.999710i	$-0.492328\pi$
$233$	5.05225e11	0.0481978	0.0240989	+	0.999710i	$0.492328\pi$
$234$	6.65939e12	0.620508
$235$	−2.73909e11	−0.0249306
$236$	8.68056e11	0.0771848
$237$	9.62965e12	0.836555
$238$	−5.24222e11	−0.0444982
$239$	−1.33279e13	−1.10553	−0.552767	−	0.833336i	$-0.686428\pi$
$239$	−1.33279e13	−1.10553	−0.552767	+	0.833336i	$0.686428\pi$
$240$	−1.26810e12	−0.102800
$241$	−1.15327e13	−0.913774	−0.456887	−	0.889525i	$-0.651036\pi$
$241$	−1.15327e13	−0.913774	−0.456887	+	0.889525i	$0.651036\pi$
$242$	1.04419e13	0.808711
$243$	−1.07933e13	−0.817184
$244$	7.04209e11	0.0521264
$245$	5.26020e11	0.0380705
$246$	8.10274e12	0.573441
$247$	−3.30578e13	−2.28792
$248$	1.05863e12	0.0716574
$249$	−9.74794e12	−0.645381
$250$	−4.49244e12	−0.290946
$251$	−6.42998e12	−0.407384	−0.203692	−	0.979035i	$-0.565294\pi$
$251$	−6.42998e12	−0.407384	−0.203692	+	0.979035i	$0.565294\pi$
$252$	6.04264e11	0.0374562
$253$	1.32916e12	0.0806148
$254$	4.35713e12	0.258592
$255$	1.06587e11	0.00619062
$256$	−5.01407e12	−0.285017
$257$	−2.18026e13	−1.21304	−0.606520	−	0.795068i	$-0.707435\pi$
$257$	−2.18026e13	−1.21304	−0.606520	+	0.795068i	$0.707435\pi$
$258$	−1.10196e13	−0.600145
$259$	2.60884e13	1.39091
$260$	4.12955e11	0.0215551
$261$	1.17472e13	0.600360
$262$	−8.04895e12	−0.402793
$263$	−2.49009e13	−1.22028	−0.610139	−	0.792294i	$-0.708887\pi$
$263$	−2.49009e13	−1.22028	−0.610139	+	0.792294i	$0.708887\pi$
$264$	−6.22716e12	−0.298861
$265$	1.88134e12	0.0884329
$266$	2.82737e13	1.30177
$267$	−2.21813e13	−1.00040
$268$	2.13099e12	0.0941543
$269$	3.54642e13	1.53516	0.767578	−	0.640956i	$-0.221462\pi$
$269$	3.54642e13	1.53516	0.767578	+	0.640956i	$0.221462\pi$
$270$	3.73338e12	0.158344
$271$	4.02210e12	0.167156	0.0835781	−	0.996501i	$-0.473365\pi$
$271$	4.02210e12	0.167156	0.0835781	+	0.996501i	$0.473365\pi$
$272$	−1.18415e12	−0.0482254
$273$	−2.34246e13	−0.934925
$274$	2.09157e13	0.818172
$275$	−9.84165e12	−0.377344
$276$	3.94763e11	0.0148367
$277$	−5.86920e11	−0.0216242	−0.0108121	−	0.999942i	$-0.503442\pi$
$277$	−5.86920e11	−0.0216242	−0.0108121	+	0.999942i	$0.503442\pi$
$278$	1.26405e13	0.456581
$279$	−8.73183e11	−0.0309230
$280$	−4.03549e12	−0.140129
$281$	4.15886e13	1.41609	0.708044	−	0.706169i	$-0.249578\pi$
$281$	4.15886e13	1.41609	0.708044	+	0.706169i	$0.249578\pi$
$282$	3.40094e12	0.113561
$283$	1.50921e13	0.494225	0.247113	−	0.968987i	$-0.420518\pi$
$283$	1.50921e13	0.494225	0.247113	+	0.968987i	$0.420518\pi$
$284$	3.33491e12	0.107111
$285$	−5.74873e12	−0.181103
$286$	−1.72638e13	−0.533486
$287$	2.32893e13	0.706001
$288$	2.89023e12	0.0859554
$289$	−3.41724e13	−0.997096
$290$	−6.86625e12	−0.196576
$291$	1.64702e13	0.462686
$292$	5.82271e12	0.160517
$293$	−5.86560e13	−1.58687	−0.793434	−	0.608656i	$-0.791709\pi$
$293$	−5.86560e13	−1.58687	−0.793434	+	0.608656i	$0.791709\pi$
$294$	−6.53123e12	−0.173414
$295$	4.78164e12	0.124611
$296$	6.52463e13	1.66898
$297$	1.65584e13	0.415775
$298$	−1.42236e13	−0.350606
$299$	1.25045e13	0.302605
$300$	−2.92298e12	−0.0694480
$301$	−3.16729e13	−0.738877
$302$	−5.64285e13	−1.29259
$303$	3.47779e13	0.782292
$304$	6.38663e13	1.41081
$305$	3.87910e12	0.0841553
$306$	1.08139e12	0.0230418
$307$	−2.11135e13	−0.441875	−0.220937	−	0.975288i	$-0.570912\pi$
$307$	−2.11135e13	−0.441875	−0.220937	+	0.975288i	$0.570912\pi$
$308$	−1.56649e12	−0.0322033
$309$	5.87963e11	0.0118735
$310$	5.10375e11	0.0101251
$311$	4.69635e13	0.915331	0.457666	−	0.889124i	$-0.348686\pi$
$311$	4.69635e13	0.915331	0.457666	+	0.889124i	$0.348686\pi$
$312$	−5.85842e13	−1.12184
$313$	−1.36538e13	−0.256898	−0.128449	−	0.991716i	$-0.541000\pi$
$313$	−1.36538e13	−0.256898	−0.128449	+	0.991716i	$0.541000\pi$
$314$	8.25171e13	1.52557
$315$	3.32856e12	0.0604711
$316$	6.05840e12	0.108163
$317$	3.46611e13	0.608158	0.304079	−	0.952647i	$-0.401651\pi$
$317$	3.46611e13	0.608158	0.304079	+	0.952647i	$0.401651\pi$
$318$	−2.33592e13	−0.402819
$319$	−3.04535e13	−0.516164
$320$	−1.00071e13	−0.166719
$321$	−5.02266e12	−0.0822538
$322$	−1.06949e13	−0.172175
$323$	−5.36814e12	−0.0849591
$324$	2.14589e12	0.0333894
$325$	−9.25887e13	−1.41644
$326$	1.64914e13	0.248063
$327$	7.91050e13	1.17002
$328$	5.82458e13	0.847146
$329$	9.77513e12	0.139812
$330$	−3.00217e12	−0.0422287
$331$	8.21494e13	1.13645	0.568225	−	0.822873i	$-0.307630\pi$
$331$	8.21494e13	1.13645	0.568225	+	0.822873i	$0.307630\pi$
$332$	−6.13282e12	−0.0834451
$333$	−5.38165e13	−0.720232
$334$	−7.63721e13	−1.00538
$335$	1.17384e13	0.152007
$336$	4.52553e13	0.576505
$337$	4.85803e13	0.608829	0.304415	−	0.952540i	$-0.401539\pi$
$337$	4.85803e13	0.608829	0.304415	+	0.952540i	$0.401539\pi$
$338$	−8.52989e13	−1.05172
$339$	−5.61352e13	−0.680983
$340$	6.70583e10	0.000800421 0
$341$	2.26364e12	0.0265863
$342$	−5.83245e13	−0.674074
$343$	−9.51289e13	−1.08192
$344$	−7.92131e13	−0.886595
$345$	2.17453e12	0.0239531
$346$	−1.12198e14	−1.21637
$347$	1.83574e14	1.95884	0.979422	−	0.201822i	$-0.0646864\pi$
$347$	1.83574e14	1.95884	0.979422	+	0.201822i	$0.0646864\pi$
$348$	−9.04473e12	−0.0949970
$349$	−1.48395e14	−1.53419	−0.767096	−	0.641532i	$-0.778299\pi$
$349$	−1.48395e14	−1.53419	−0.767096	+	0.641532i	$0.778299\pi$
$350$	7.91892e13	0.805920
$351$	1.55779e14	1.56070
$352$	−7.49263e12	−0.0739009
$353$	−1.52171e14	−1.47765	−0.738823	−	0.673900i	$-0.764618\pi$
$353$	−1.52171e14	−1.47765	−0.738823	+	0.673900i	$0.764618\pi$
$354$	−5.93704e13	−0.567612
$355$	1.83702e13	0.172925
$356$	−1.39552e13	−0.129348
$357$	−3.80383e12	−0.0347173
$358$	−3.83750e13	−0.344899
$359$	−2.04271e14	−1.80795	−0.903975	−	0.427586i	$-0.859364\pi$
$359$	−2.04271e14	−1.80795	−0.903975	+	0.427586i	$0.859364\pi$
$360$	8.32463e12	0.0725606
$361$	1.73038e14	1.48543
$362$	9.55465e13	0.807826
$363$	7.57676e13	0.630953
$364$	−1.47373e13	−0.120882
$365$	3.20741e13	0.259146
$366$	−4.81641e13	−0.383334
$367$	2.16052e14	1.69393	0.846963	−	0.531652i	$-0.178429\pi$
$367$	2.16052e14	1.69393	0.846963	+	0.531652i	$0.178429\pi$
$368$	−2.41583e13	−0.186596
$369$	−4.80423e13	−0.365577
$370$	3.14558e13	0.235826
$371$	−6.71402e13	−0.495936
$372$	6.72305e11	0.00489305
$373$	−3.55193e13	−0.254721	−0.127361	−	0.991856i	$-0.540651\pi$
$373$	−3.55193e13	−0.254721	−0.127361	+	0.991856i	$0.540651\pi$
$374$	−2.80341e12	−0.0198104
$375$	−3.25978e13	−0.226995
$376$	2.44473e13	0.167764
$377$	−2.86502e14	−1.93753
$378$	−1.33235e14	−0.888000
$379$	1.49037e14	0.978990	0.489495	−	0.872006i	$-0.337181\pi$
$379$	1.49037e14	0.978990	0.489495	+	0.872006i	$0.337181\pi$
$380$	−3.61676e12	−0.0234158
$381$	3.16160e13	0.201753
$382$	1.63026e14	1.02543
$383$	−1.64077e12	−0.0101731	−0.00508657	−	0.999987i	$-0.501619\pi$
$383$	−1.64077e12	−0.0101731	−0.00508657	+	0.999987i	$0.501619\pi$
$384$	1.01051e14	0.617616
$385$	−8.62895e12	−0.0519905
$386$	1.21593e14	0.722234
$387$	6.53366e13	0.382601
$388$	1.03620e13	0.0598234
$389$	9.93469e13	0.565499	0.282749	−	0.959194i	$-0.408754\pi$
$389$	9.93469e13	0.565499	0.282749	+	0.959194i	$0.408754\pi$
$390$	−2.82439e13	−0.158515
$391$	2.03057e12	0.0112369
$392$	−4.69491e13	−0.256185
$393$	−5.84043e13	−0.314258
$394$	5.83202e13	0.309449
$395$	3.33724e13	0.174624
$396$	3.23145e12	0.0166753
$397$	3.52178e14	1.79231	0.896157	−	0.443738i	$-0.146348\pi$
$397$	3.52178e14	1.79231	0.896157	+	0.443738i	$0.146348\pi$
$398$	−2.39722e13	−0.120324
$399$	2.05158e14	1.01563
$400$	1.78877e14	0.873425
$401$	−3.82993e14	−1.84458	−0.922288	−	0.386503i	$-0.873683\pi$
$401$	−3.82993e14	−1.84458	−0.922288	+	0.386503i	$0.873683\pi$
$402$	−1.45748e14	−0.692405
$403$	2.12960e13	0.0997975
$404$	2.18802e13	0.101147
$405$	1.18205e13	0.0539055
$406$	2.45039e14	1.10241
$407$	1.39514e14	0.619225
$408$	−9.51327e12	−0.0416581
$409$	−7.78583e13	−0.336377	−0.168189	−	0.985755i	$-0.553792\pi$
$409$	−7.78583e13	−0.336377	−0.168189	+	0.985755i	$0.553792\pi$
$410$	2.80808e13	0.119701
$411$	1.51767e14	0.638334
$412$	3.69911e11	0.00153519
$413$	−1.70645e14	−0.698824
$414$	2.20620e13	0.0891544
$415$	−3.37823e13	−0.134718
$416$	−7.04895e13	−0.277403
$417$	9.17213e13	0.356223
$418$	1.51200e14	0.579540
$419$	2.09024e14	0.790713	0.395357	−	0.918528i	$-0.370621\pi$
$419$	2.09024e14	0.790713	0.395357	+	0.918528i	$0.370621\pi$
$420$	−2.56281e12	−0.00956854
$421$	−5.61516e13	−0.206924	−0.103462	−	0.994633i	$-0.532992\pi$
$421$	−5.61516e13	−0.206924	−0.103462	+	0.994633i	$0.532992\pi$
$422$	−3.64691e14	−1.32650
$423$	−2.01647e13	−0.0723967
$424$	−1.67916e14	−0.595085
$425$	−1.50351e13	−0.0525979
$426$	−2.28090e14	−0.787685
$427$	−1.38435e14	−0.471947
$428$	−3.15996e12	−0.0106351
$429$	−1.25269e14	−0.416224
$430$	−3.81893e13	−0.125275
$431$	1.19488e14	0.386989	0.193494	−	0.981101i	$-0.438018\pi$
$431$	1.19488e14	0.386989	0.193494	+	0.981101i	$0.438018\pi$
$432$	−3.00959e14	−0.962380
$433$	3.36496e14	1.06242	0.531210	−	0.847240i	$-0.321738\pi$
$433$	3.36496e14	1.06242	0.531210	+	0.847240i	$0.321738\pi$
$434$	−1.82140e13	−0.0567822
$435$	−4.98224e13	−0.153368
$436$	4.97682e13	0.151278
$437$	−1.09518e14	−0.328728
$438$	−3.98242e14	−1.18043
$439$	−7.83405e12	−0.0229315	−0.0114657	−	0.999934i	$-0.503650\pi$
$439$	−7.83405e12	−0.0229315	−0.0114657	+	0.999934i	$0.503650\pi$
$440$	−2.15808e13	−0.0623846
$441$	3.87246e13	0.110554
$442$	−2.63740e13	−0.0743625
$443$	−3.47294e14	−0.967112	−0.483556	−	0.875314i	$-0.660655\pi$
$443$	−3.47294e14	−0.967112	−0.483556	+	0.875314i	$0.660655\pi$
$444$	4.14359e13	0.113965
$445$	−7.68714e13	−0.208826
$446$	−3.18701e14	−0.855149
$447$	−1.03208e14	−0.273541
$448$	3.57129e14	0.934968
$449$	−4.82495e14	−1.24778	−0.623890	−	0.781512i	$-0.714449\pi$
$449$	−4.82495e14	−1.24778	−0.623890	+	0.781512i	$0.714449\pi$
$450$	−1.63356e14	−0.417317
$451$	1.24545e14	0.314308
$452$	−3.53169e13	−0.0880483
$453$	−4.09453e14	−1.00847
$454$	−6.99249e14	−1.70147
$455$	−8.11798e13	−0.195158
$456$	5.13094e14	1.21868
$457$	−1.41633e14	−0.332372	−0.166186	−	0.986094i	$-0.553145\pi$
$457$	−1.41633e14	−0.332372	−0.166186	+	0.986094i	$0.553145\pi$
$458$	6.26860e14	1.45348
$459$	2.52964e13	0.0579548
$460$	1.36809e12	0.00309703
$461$	9.47729e13	0.211997	0.105998	−	0.994366i	$-0.466196\pi$
$461$	9.47729e13	0.211997	0.105998	+	0.994366i	$0.466196\pi$
$462$	1.07140e14	0.236821
$463$	7.98720e14	1.74461	0.872306	−	0.488960i	$-0.162624\pi$
$463$	7.98720e14	1.74461	0.872306	+	0.488960i	$0.162624\pi$
$464$	5.53509e14	1.19475
$465$	3.70336e12	0.00789958
$466$	−2.17397e13	−0.0458281
$467$	−4.36216e14	−0.908780	−0.454390	−	0.890803i	$-0.650143\pi$
$467$	−4.36216e14	−0.908780	−0.454390	+	0.890803i	$0.650143\pi$
$468$	3.04010e13	0.0625944
$469$	−4.18916e14	−0.852464
$470$	1.17863e13	0.0237049
$471$	5.98755e14	1.19024
$472$	−4.26778e14	−0.838535
$473$	−1.69379e14	−0.328944
$474$	−4.14362e14	−0.795425
$475$	8.10913e14	1.53872
$476$	−2.39314e12	−0.00448880
$477$	1.38500e14	0.256803
$478$	5.73495e14	1.05118
$479$	−5.62976e14	−1.02010	−0.510052	−	0.860143i	$-0.670374\pi$
$479$	−5.62976e14	−1.02010	−0.510052	+	0.860143i	$0.670374\pi$
$480$	−1.22581e13	−0.0219581
$481$	1.31253e15	2.32440
$482$	4.96252e14	0.868847
$483$	−7.76036e13	−0.134330
$484$	4.76684e13	0.0815796
$485$	5.70788e13	0.0965818
$486$	4.64434e14	0.777005
$487$	−8.27744e14	−1.36926	−0.684631	−	0.728890i	$-0.740037\pi$
$487$	−8.27744e14	−1.36926	−0.684631	+	0.728890i	$0.740037\pi$
$488$	−3.46223e14	−0.566300
$489$	1.19664e14	0.193538
$490$	−2.26345e13	−0.0361987
$491$	7.86162e14	1.24326	0.621632	−	0.783309i	$-0.286470\pi$
$491$	7.86162e14	1.24326	0.621632	+	0.783309i	$0.286470\pi$
$492$	3.69901e13	0.0578465
$493$	−4.65240e13	−0.0719480
$494$	1.42247e15	2.17543
$495$	1.78003e13	0.0269214
$496$	−4.11429e13	−0.0615384
$497$	−6.55585e14	−0.969770
$498$	4.19452e14	0.613650
$499$	−2.77277e14	−0.401200	−0.200600	−	0.979673i	$-0.564289\pi$
$499$	−2.77277e14	−0.401200	−0.200600	+	0.979673i	$0.564289\pi$
$500$	−2.05086e13	−0.0293495
$501$	−5.54166e14	−0.784391
$502$	2.76681e14	0.387354
$503$	2.85278e13	0.0395043	0.0197522	−	0.999805i	$-0.493712\pi$
$503$	2.85278e13	0.0395043	0.0197522	+	0.999805i	$0.493712\pi$
$504$	−2.97085e14	−0.406923
$505$	1.20526e14	0.163297
$506$	−5.71935e13	−0.0766512
$507$	−6.18941e14	−0.820550
$508$	1.98909e13	0.0260858
$509$	−1.70310e14	−0.220950	−0.110475	−	0.993879i	$-0.535237\pi$
$509$	−1.70310e14	−0.220950	−0.110475	+	0.993879i	$0.535237\pi$
$510$	−4.58643e12	−0.00588625
$511$	−1.14465e15	−1.45330
$512$	8.78574e14	1.10355
$513$	−1.36435e15	−1.69543
$514$	9.38160e14	1.15340
$515$	2.03764e12	0.00247849
$516$	−5.03058e13	−0.0605403
$517$	5.22749e13	0.0622437
$518$	−1.12258e15	−1.32252
$519$	−8.14122e14	−0.949006
$520$	−2.03029e14	−0.234174
$521$	−5.15846e14	−0.588726	−0.294363	−	0.955694i	$-0.595107\pi$
$521$	−5.15846e14	−0.588726	−0.294363	+	0.955694i	$0.595107\pi$
$522$	−5.05480e14	−0.570842
$523$	1.10162e15	1.23104	0.615519	−	0.788122i	$-0.288946\pi$
$523$	1.10162e15	1.23104	0.615519	+	0.788122i	$0.288946\pi$
$524$	−3.67445e13	−0.0406322
$525$	5.74608e14	0.628776
$526$	1.07148e15	1.16028
$527$	3.45818e12	0.00370586
$528$	2.42014e14	0.256657
$529$	4.14265e13	0.0434783
$530$	−8.09535e13	−0.0840850
$531$	3.52016e14	0.361861
$532$	1.29073e14	0.131317
$533$	1.17170e15	1.17982
$534$	9.54459e14	0.951218
$535$	−1.74065e13	−0.0171698
$536$	−1.04770e15	−1.02289
$537$	−2.78454e14	−0.269089
$538$	−1.52602e15	−1.45968
$539$	−1.00390e14	−0.0950498
$540$	1.70433e13	0.0159731
$541$	−1.82633e15	−1.69431	−0.847156	−	0.531344i	$-0.821687\pi$
$541$	−1.82633e15	−1.69431	−0.847156	+	0.531344i	$0.821687\pi$
$542$	−1.73070e14	−0.158938
$543$	6.93299e14	0.630263
$544$	−1.14465e13	−0.0103010
$545$	2.74145e14	0.244231
$546$	1.00795e15	0.888958
$547$	−1.38097e15	−1.20574	−0.602870	−	0.797839i	$-0.705976\pi$
$547$	−1.38097e15	−1.20574	−0.602870	+	0.797839i	$0.705976\pi$
$548$	9.54829e13	0.0825339
$549$	2.85572e14	0.244381
$550$	4.23484e14	0.358792
$551$	2.50925e15	2.10479
$552$	−1.94084e14	−0.161185
$553$	−1.19098e15	−0.979299
$554$	2.52551e13	0.0205610
$555$	2.28248e14	0.183990
$556$	5.77055e13	0.0460581
$557$	−1.13770e15	−0.899130	−0.449565	−	0.893248i	$-0.648421\pi$
$557$	−1.13770e15	−0.899130	−0.449565	+	0.893248i	$0.648421\pi$
$558$	3.75729e13	0.0294026
$559$	−1.59349e15	−1.23476
$560$	1.56836e14	0.120341
$561$	−2.03419e13	−0.0154560
$562$	−1.78955e15	−1.34646
$563$	2.60488e15	1.94085	0.970424	−	0.241407i	$-0.0776089\pi$
$563$	2.60488e15	1.94085	0.970424	+	0.241407i	$0.0776089\pi$
$564$	1.55257e13	0.0114556
$565$	−1.94541e14	−0.142149
$566$	−6.49411e14	−0.469926
$567$	−4.21844e14	−0.302305
$568$	−1.63960e15	−1.16365
$569$	1.31768e15	0.926174	0.463087	−	0.886313i	$-0.346742\pi$
$569$	1.31768e15	0.926174	0.463087	+	0.886313i	$0.346742\pi$
$570$	2.47367e14	0.172199
$571$	6.32035e14	0.435755	0.217877	−	0.975976i	$-0.430087\pi$
$571$	6.32035e14	0.435755	0.217877	+	0.975976i	$0.430087\pi$
$572$	−7.88115e13	−0.0538160
$573$	1.18294e15	0.800039
$574$	−1.00213e15	−0.671289
$575$	−3.06739e14	−0.203514
$576$	−7.36706e14	−0.484140
$577$	−6.75315e14	−0.439581	−0.219791	−	0.975547i	$-0.570537\pi$
$577$	−6.75315e14	−0.439581	−0.219791	+	0.975547i	$0.570537\pi$
$578$	1.47043e15	0.948072
$579$	8.82294e14	0.563484
$580$	−3.13453e13	−0.0198298
$581$	1.20561e15	0.755504
$582$	−7.08708e14	−0.439938
$583$	−3.59048e14	−0.220788
$584$	−2.86273e15	−1.74385
$585$	1.67462e14	0.101055
$586$	2.52396e15	1.50885
$587$	−1.29464e15	−0.766726	−0.383363	−	0.923598i	$-0.625234\pi$
$587$	−1.29464e15	−0.766726	−0.383363	+	0.923598i	$0.625234\pi$
$588$	−2.98159e13	−0.0174933
$589$	−1.86515e14	−0.108413
$590$	−2.05753e14	−0.118484
$591$	4.23179e14	0.241431
$592$	−2.53575e15	−1.43330
$593$	4.59113e14	0.257110	0.128555	−	0.991702i	$-0.458966\pi$
$593$	4.59113e14	0.257110	0.128555	+	0.991702i	$0.458966\pi$
$594$	−7.12506e14	−0.395333
$595$	−1.31825e13	−0.00724694
$596$	−6.49325e13	−0.0353677
$597$	−1.73946e14	−0.0938762
$598$	−5.38068e14	−0.287727
$599$	−3.20772e15	−1.69961	−0.849803	−	0.527100i	$-0.823279\pi$
$599$	−3.20772e15	−1.69961	−0.849803	+	0.527100i	$0.823279\pi$
$600$	1.43708e15	0.754482
$601$	3.64695e15	1.89723	0.948615	−	0.316433i	$-0.102485\pi$
$601$	3.64695e15	1.89723	0.948615	+	0.316433i	$0.102485\pi$
$602$	1.36288e15	0.702549
$603$	8.64162e14	0.441418
$604$	−2.57604e14	−0.130391
$605$	2.62579e14	0.131706
$606$	−1.49649e15	−0.743829
$607$	−2.10487e14	−0.103678	−0.0518391	−	0.998655i	$-0.516508\pi$
$607$	−2.10487e14	−0.103678	−0.0518391	+	0.998655i	$0.516508\pi$
$608$	6.17363e14	0.301350
$609$	1.77804e15	0.860094
$610$	−1.66917e14	−0.0800177
$611$	4.91794e14	0.233645
$612$	4.93670e12	0.00232436
$613$	2.04699e15	0.955175	0.477588	−	0.878584i	$-0.341511\pi$
$613$	2.04699e15	0.955175	0.477588	+	0.878584i	$0.341511\pi$
$614$	9.08510e14	0.420149
$615$	2.03758e14	0.0933902
$616$	7.70164e14	0.349856
$617$	3.57753e15	1.61070	0.805350	−	0.592799i	$-0.201977\pi$
$617$	3.57753e15	1.61070	0.805350	+	0.592799i	$0.201977\pi$
$618$	−2.52999e13	−0.0112897
$619$	2.42301e15	1.07166	0.535830	−	0.844326i	$-0.319999\pi$
$619$	2.42301e15	1.07166	0.535830	+	0.844326i	$0.319999\pi$
$620$	2.32993e12	0.00102138
$621$	5.16083e14	0.224241
$622$	−2.02083e15	−0.870327
$623$	2.74335e15	1.17111
$624$	2.27683e15	0.963418
$625$	2.21404e15	0.928638
$626$	5.87521e14	0.244267
$627$	1.09713e15	0.452155
$628$	3.76701e14	0.153893
$629$	2.13136e14	0.0863136
$630$	−1.43227e14	−0.0574979
$631$	−4.53986e15	−1.80668	−0.903340	−	0.428926i	$-0.858892\pi$
$631$	−4.53986e15	−1.80668	−0.903340	+	0.428926i	$0.858892\pi$
$632$	−2.97860e15	−1.17508
$633$	−2.64625e15	−1.03493
$634$	−1.49146e15	−0.578257
$635$	1.09568e14	0.0421141
$636$	−1.06638e14	−0.0406348
$637$	−9.44451e14	−0.356790
$638$	1.31041e15	0.490786
$639$	1.35238e15	0.502161
$640$	3.50201e14	0.128922
$641$	−3.19239e15	−1.16519	−0.582596	−	0.812762i	$-0.697963\pi$
$641$	−3.19239e15	−1.16519	−0.582596	+	0.812762i	$0.697963\pi$
$642$	2.16124e14	0.0782096
$643$	2.03627e14	0.0730594	0.0365297	−	0.999333i	$-0.488370\pi$
$643$	2.03627e14	0.0730594	0.0365297	+	0.999333i	$0.488370\pi$
$644$	−4.88236e13	−0.0173683
$645$	−2.77107e14	−0.0977392
$646$	2.30990e14	0.0807819
$647$	−3.63748e15	−1.26133	−0.630663	−	0.776057i	$-0.717217\pi$
$647$	−3.63748e15	−1.26133	−0.630663	+	0.776057i	$0.717217\pi$
$648$	−1.05502e15	−0.362742
$649$	−9.12566e14	−0.311113
$650$	3.98407e15	1.34680
$651$	−1.32163e14	−0.0443013
$652$	7.52855e13	0.0250236
$653$	−3.38188e14	−0.111464	−0.0557321	−	0.998446i	$-0.517749\pi$
$653$	−3.38188e14	−0.111464	−0.0557321	+	0.998446i	$0.517749\pi$
$654$	−3.40387e15	−1.11249
$655$	−2.02405e14	−0.0655986
$656$	−2.26368e15	−0.727517
$657$	2.36124e15	0.752541
$658$	−4.20622e14	−0.132938
$659$	4.99044e15	1.56412	0.782059	−	0.623205i	$-0.214170\pi$
$659$	4.99044e15	1.56412	0.782059	+	0.623205i	$0.214170\pi$
$660$	−1.37053e13	−0.00425987
$661$	−1.45031e15	−0.447046	−0.223523	−	0.974699i	$-0.571756\pi$
$661$	−1.45031e15	−0.447046	−0.223523	+	0.974699i	$0.571756\pi$
$662$	−3.53487e15	−1.08057
$663$	−1.91374e14	−0.0580173
$664$	3.01519e15	0.906546
$665$	7.10992e14	0.212005
$666$	2.31571e15	0.684821
$667$	−9.49156e14	−0.278385
$668$	−3.48648e14	−0.101418
$669$	−2.31254e15	−0.667184
$670$	−5.05103e14	−0.144534
$671$	−7.40317e14	−0.210109
$672$	4.37460e14	0.123142
$673$	−2.44207e15	−0.681830	−0.340915	−	0.940094i	$-0.610737\pi$
$673$	−2.44207e15	−0.681830	−0.340915	+	0.940094i	$0.610737\pi$
$674$	−2.09040e15	−0.578895
$675$	−3.82129e15	−1.04964
$676$	−3.89400e14	−0.106094
$677$	−6.37123e15	−1.72181	−0.860906	−	0.508765i	$-0.830102\pi$
$677$	−6.37123e15	−1.72181	−0.860906	+	0.508765i	$0.830102\pi$
$678$	2.41548e15	0.647502
$679$	−2.03700e15	−0.541636
$680$	−3.29691e13	−0.00869576
$681$	−5.07385e15	−1.32748
$682$	−9.74040e13	−0.0252792
$683$	−6.28334e14	−0.161762	−0.0808810	−	0.996724i	$-0.525773\pi$
$683$	−6.28334e14	−0.161762	−0.0808810	+	0.996724i	$0.525773\pi$
$684$	−2.66259e14	−0.0679979
$685$	5.25963e14	0.133247
$686$	4.09338e15	1.02872
$687$	4.54858e15	1.13400
$688$	3.07855e15	0.761396
$689$	−3.37787e15	−0.828777
$690$	−9.35697e13	−0.0227754
$691$	2.43273e15	0.587441	0.293721	−	0.955891i	$-0.405106\pi$
$691$	2.43273e15	0.587441	0.293721	+	0.955891i	$0.405106\pi$
$692$	−5.12197e14	−0.122703
$693$	−6.35247e14	−0.150977
$694$	−7.89917e15	−1.86253
$695$	3.17868e14	0.0743584
$696$	4.44682e15	1.03205
$697$	1.90268e14	0.0438113
$698$	6.38541e15	1.45876
$699$	−1.57747e14	−0.0357549
$700$	3.61509e14	0.0812980
$701$	4.37786e15	0.976815	0.488408	−	0.872616i	$-0.337578\pi$
$701$	4.37786e15	0.976815	0.488408	+	0.872616i	$0.337578\pi$
$702$	−6.70314e15	−1.48397
$703$	−1.14954e16	−2.52505
$704$	1.90984e15	0.416243
$705$	8.55227e13	0.0184944
$706$	6.54787e15	1.40499
$707$	−4.30126e15	−0.915776
$708$	−2.71033e14	−0.0572585
$709$	1.69318e15	0.354934	0.177467	−	0.984127i	$-0.443210\pi$
$709$	1.69318e15	0.354934	0.177467	+	0.984127i	$0.443210\pi$
$710$	−7.90464e14	−0.164423
$711$	2.45681e15	0.507095
$712$	6.86103e15	1.40524
$713$	7.05518e13	0.0143389
$714$	1.63678e14	0.0330104
$715$	−4.34129e14	−0.0868832
$716$	−1.75187e14	−0.0347920
$717$	4.16136e15	0.820125
$718$	8.78972e15	1.71906
$719$	5.00841e15	0.972054	0.486027	−	0.873944i	$-0.338446\pi$
$719$	5.00841e15	0.972054	0.486027	+	0.873944i	$0.338446\pi$
$720$	−3.23530e14	−0.0623140
$721$	−7.27181e13	−0.0138995
$722$	−7.44578e15	−1.41239
$723$	3.60087e15	0.677871
$724$	4.36182e14	0.0814903
$725$	7.02793e15	1.30307
$726$	−3.26026e15	−0.599931
$727$	−1.57656e14	−0.0287919	−0.0143959	−	0.999896i	$-0.504583\pi$
$727$	−1.57656e14	−0.0287919	−0.0143959	+	0.999896i	$0.504583\pi$
$728$	7.24558e15	1.31326
$729$	5.30516e15	0.954327
$730$	−1.38014e15	−0.246404
$731$	−2.58761e14	−0.0458515
$732$	−2.19875e14	−0.0386692
$733$	−2.52921e15	−0.441481	−0.220741	−	0.975333i	$-0.570847\pi$
$733$	−2.52921e15	−0.441481	−0.220741	+	0.975333i	$0.570847\pi$
$734$	−9.29666e15	−1.61064
$735$	−1.64239e14	−0.0282421
$736$	−2.33526e14	−0.0398572
$737$	−2.24026e15	−0.379513
$738$	2.06725e15	0.347603
$739$	−1.03399e16	−1.72573	−0.862863	−	0.505437i	$-0.831331\pi$
$739$	−1.03399e16	−1.72573	−0.862863	+	0.505437i	$0.831331\pi$
$740$	1.43600e14	0.0237892
$741$	1.03216e16	1.69726
$742$	2.88903e15	0.471553
$743$	−4.98460e15	−0.807591	−0.403795	−	0.914849i	$-0.632309\pi$
$743$	−4.98460e15	−0.807591	−0.403795	+	0.914849i	$0.632309\pi$
$744$	−3.30537e14	−0.0531581
$745$	−3.57677e14	−0.0570994
$746$	1.52839e15	0.242197
$747$	−2.48699e15	−0.391211
$748$	−1.27979e13	−0.00199839
$749$	6.21193e14	0.0962889
$750$	1.40268e15	0.215834
$751$	8.80740e15	1.34533	0.672664	−	0.739948i	$-0.265150\pi$
$751$	8.80740e15	1.34533	0.672664	+	0.739948i	$0.265150\pi$
$752$	−9.50126e14	−0.144073
$753$	2.00763e15	0.302212
$754$	1.23281e16	1.84227
$755$	−1.41900e15	−0.210510
$756$	−6.08234e14	−0.0895779
$757$	7.02250e15	1.02675	0.513375	−	0.858164i	$-0.328395\pi$
$757$	7.02250e15	1.02675	0.513375	+	0.858164i	$0.328395\pi$
$758$	−6.41303e15	−0.930856
$759$	−4.15004e14	−0.0598030
$760$	1.77817e15	0.254389
$761$	1.00544e16	1.42805	0.714023	−	0.700122i	$-0.246871\pi$
$761$	1.00544e16	1.42805	0.714023	+	0.700122i	$0.246871\pi$
$762$	−1.36043e15	−0.191833
$763$	−9.78356e15	−1.36966
$764$	7.44233e14	0.103442
$765$	2.71936e13	0.00375257
$766$	7.06021e13	0.00967296
$767$	−8.58527e15	−1.16783
$768$	1.56555e15	0.211436
$769$	−5.29632e15	−0.710198	−0.355099	−	0.934829i	$-0.615553\pi$
$769$	−5.29632e15	−0.710198	−0.355099	+	0.934829i	$0.615553\pi$
$770$	3.71302e14	0.0494343
$771$	6.80742e15	0.899878
$772$	5.55087e14	0.0728561
$773$	−6.21737e14	−0.0810251	−0.0405126	−	0.999179i	$-0.512899\pi$
$773$	−6.21737e14	−0.0810251	−0.0405126	+	0.999179i	$0.512899\pi$
$774$	−2.81142e15	−0.363790
$775$	−5.22394e14	−0.0671179
$776$	−5.09448e15	−0.649921
$777$	−8.14558e15	−1.03183
$778$	−4.27488e15	−0.537695
$779$	−1.02620e16	−1.28167
$780$	−1.28937e14	−0.0159903
$781$	−3.50590e15	−0.431737
$782$	−8.73749e13	−0.0106844
$783$	−1.18244e16	−1.43578
$784$	1.82464e15	0.220008
$785$	2.07504e15	0.248452
$786$	2.51313e15	0.298807
$787$	−1.80142e15	−0.212694	−0.106347	−	0.994329i	$-0.533915\pi$
$787$	−1.80142e15	−0.212694	−0.106347	+	0.994329i	$0.533915\pi$
$788$	2.66239e14	0.0312160
$789$	7.77483e15	0.905247
$790$	−1.43601e15	−0.166038
$791$	6.94269e15	0.797181
$792$	−1.58874e15	−0.181160
$793$	−6.96478e15	−0.788688
$794$	−1.51541e16	−1.70419
$795$	−5.87410e14	−0.0656028
$796$	−1.09436e14	−0.0121378
$797$	1.19151e16	1.31244	0.656218	−	0.754571i	$-0.272155\pi$
$797$	1.19151e16	1.31244	0.656218	+	0.754571i	$0.272155\pi$
$798$	−8.82790e15	−0.965698
$799$	7.98607e13	0.00867612
$800$	1.72912e15	0.186565
$801$	−5.65912e15	−0.606415
$802$	1.64801e16	1.75388
$803$	−6.12127e15	−0.647003
$804$	−6.65360e14	−0.0698471
$805$	−2.68942e14	−0.0280402
$806$	−9.16361e14	−0.0948908
$807$	−1.10730e16	−1.13883
$808$	−1.07573e16	−1.09886
$809$	3.05658e15	0.310112	0.155056	−	0.987906i	$-0.450444\pi$
$809$	3.05658e15	0.310112	0.155056	+	0.987906i	$0.450444\pi$
$810$	−5.08634e14	−0.0512552
$811$	9.46409e15	0.947249	0.473624	−	0.880727i	$-0.342945\pi$
$811$	9.46409e15	0.947249	0.473624	+	0.880727i	$0.342945\pi$
$812$	1.11864e15	0.111207
$813$	−1.25582e15	−0.124002
$814$	−6.00326e15	−0.588780
$815$	4.14707e14	0.0403993
$816$	3.69726e14	0.0357754
$817$	1.39562e16	1.34136
$818$	3.35023e15	0.319839
$819$	−5.97630e15	−0.566724
$820$	1.28192e14	0.0120750
$821$	1.33311e16	1.24732	0.623660	−	0.781696i	$-0.285645\pi$
$821$	1.33311e16	1.24732	0.623660	+	0.781696i	$0.285645\pi$
$822$	−6.53052e15	−0.606949
$823$	1.02391e16	0.945285	0.472642	−	0.881254i	$-0.343300\pi$
$823$	1.02391e16	0.945285	0.472642	+	0.881254i	$0.343300\pi$
$824$	−1.81866e14	−0.0166783
$825$	3.07286e15	0.279928
$826$	7.34281e15	0.664465
$827$	−4.81227e15	−0.432583	−0.216292	−	0.976329i	$-0.569396\pi$
$827$	−4.81227e15	−0.432583	−0.216292	+	0.976329i	$0.569396\pi$
$828$	1.00716e14	0.00899355
$829$	−1.13763e16	−1.00914	−0.504570	−	0.863371i	$-0.668349\pi$
$829$	−1.13763e16	−1.00914	−0.504570	+	0.863371i	$0.668349\pi$
$830$	1.45365e15	0.128094
$831$	1.83254e14	0.0160416
$832$	1.79675e16	1.56246
$833$	−1.53366e14	−0.0132490
$834$	−3.94675e15	−0.338708
$835$	−1.92051e15	−0.163735
$836$	6.90250e14	0.0584617
$837$	8.78920e14	0.0739536
$838$	−8.99426e15	−0.751836
$839$	7.78815e15	0.646761	0.323380	−	0.946269i	$-0.395181\pi$
$839$	7.78815e15	0.646761	0.323380	+	0.946269i	$0.395181\pi$
$840$	1.26000e15	0.103953
$841$	9.54635e15	0.782455
$842$	2.41619e15	0.196750
$843$	−1.29852e16	−1.05050
$844$	−1.66486e15	−0.133812
$845$	−2.14499e15	−0.171283
$846$	8.67681e14	0.0688372
$847$	−9.37079e15	−0.738614
$848$	6.52591e15	0.511051
$849$	−4.71222e15	−0.366634
$850$	6.46959e14	0.0500118
$851$	4.34829e15	0.333969
$852$	−1.04126e15	−0.0794586
$853$	−6.19751e15	−0.469892	−0.234946	−	0.972008i	$-0.575491\pi$
$853$	−6.19751e15	−0.469892	−0.234946	+	0.972008i	$0.575491\pi$
$854$	5.95684e15	0.448743
$855$	−1.46667e15	−0.109779
$856$	1.55359e15	0.115539
$857$	1.88817e16	1.39524	0.697618	−	0.716470i	$-0.254243\pi$
$857$	1.88817e16	1.39524	0.697618	+	0.716470i	$0.254243\pi$
$858$	5.39028e15	0.395760
$859$	5.94279e15	0.433538	0.216769	−	0.976223i	$-0.430448\pi$
$859$	5.94279e15	0.433538	0.216769	+	0.976223i	$0.430448\pi$
$860$	−1.74339e14	−0.0126373
$861$	−7.27161e15	−0.523737
$862$	−5.14153e15	−0.367962
$863$	1.09810e16	0.780877	0.390439	−	0.920629i	$-0.372323\pi$
$863$	1.09810e16	0.780877	0.390439	+	0.920629i	$0.372323\pi$
$864$	−2.90922e15	−0.205566
$865$	−2.82141e15	−0.198097
$866$	−1.44793e16	−1.01018
$867$	1.06696e16	0.739682
$868$	−8.31494e13	−0.00572797
$869$	−6.36904e15	−0.435979
$870$	2.14385e15	0.145827
$871$	−2.10760e16	−1.42458
$872$	−2.44684e16	−1.64348
$873$	4.20203e15	0.280467
$874$	4.71253e15	0.312565
$875$	4.03163e15	0.265728
$876$	−1.81803e15	−0.119077
$877$	2.24358e16	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$877$	2.24358e16	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$878$	3.37098e14	0.0218040
$879$	1.83142e16	1.17720
$880$	8.38720e14	0.0535750
$881$	4.19537e15	0.266319	0.133160	−	0.991095i	$-0.457488\pi$
$881$	4.19537e15	0.266319	0.133160	+	0.991095i	$0.457488\pi$
$882$	−1.66631e15	−0.105118
$883$	−9.51760e15	−0.596683	−0.298341	−	0.954459i	$-0.596433\pi$
$883$	−9.51760e15	−0.596683	−0.298341	+	0.954459i	$0.596433\pi$
$884$	−1.20401e14	−0.00750139
$885$	−1.49297e15	−0.0924409
$886$	1.49440e16	0.919562
$887$	5.05010e15	0.308831	0.154415	−	0.988006i	$-0.450651\pi$
$887$	5.05010e15	0.308831	0.154415	+	0.988006i	$0.450651\pi$
$888$	−2.03719e16	−1.23811
$889$	−3.91020e15	−0.236178
$890$	3.30776e15	0.198559
$891$	−2.25591e15	−0.134585
$892$	−1.45491e15	−0.0862641
$893$	−4.30725e15	−0.253815
$894$	4.44103e15	0.260092
$895$	−9.65007e14	−0.0561699
$896$	−1.24978e16	−0.723002
$897$	−3.90430e15	−0.224483
$898$	2.07617e16	1.18643
$899$	−1.61647e15	−0.0918097
$900$	−7.45741e14	−0.0420973
$901$	−5.48521e14	−0.0307756
$902$	−5.35915e15	−0.298854
$903$	9.88924e15	0.548126
$904$	1.73635e16	0.956556
$905$	2.40269e15	0.131562
$906$	1.76187e16	0.958890
$907$	2.70042e16	1.46080	0.730400	−	0.683020i	$-0.239334\pi$
$907$	2.70042e16	1.46080	0.730400	+	0.683020i	$0.239334\pi$
$908$	−3.19216e15	−0.171638
$909$	8.87288e15	0.474202
$910$	3.49315e15	0.185562
$911$	5.90668e15	0.311883	0.155942	−	0.987766i	$-0.450159\pi$
$911$	5.90668e15	0.311883	0.155942	+	0.987766i	$0.450159\pi$
$912$	−1.99410e16	−1.04659
$913$	6.44728e15	0.336347
$914$	6.09443e15	0.316031
$915$	−1.21117e15	−0.0624295
$916$	2.86170e15	0.146622
$917$	7.22334e15	0.367880
$918$	−1.08850e15	−0.0551053
$919$	1.66963e16	0.840206	0.420103	−	0.907476i	$-0.361994\pi$
$919$	1.66963e16	0.840206	0.420103	+	0.907476i	$0.361994\pi$
$920$	−6.72617e14	−0.0336461
$921$	6.59227e15	0.327799
$922$	−4.07806e15	−0.201573
$923$	−3.29830e16	−1.62062
$924$	4.89107e14	0.0238895
$925$	−3.21965e16	−1.56325
$926$	−3.43687e16	−1.65884
$927$	1.50007e14	0.00719735
$928$	5.35049e15	0.255200
$929$	−1.01834e16	−0.482844	−0.241422	−	0.970420i	$-0.577614\pi$
$929$	−1.01834e16	−0.482844	−0.241422	+	0.970420i	$0.577614\pi$
$930$	−1.59355e14	−0.00751119
$931$	8.27172e15	0.387590
$932$	−9.92448e13	−0.00462296
$933$	−1.46634e16	−0.679026
$934$	1.87703e16	0.864098
$935$	−7.04967e13	−0.00322630
$936$	−1.49466e16	−0.680024
$937$	−1.48549e16	−0.671897	−0.335949	−	0.941880i	$-0.609057\pi$
$937$	−1.48549e16	−0.671897	−0.335949	+	0.941880i	$0.609057\pi$
$938$	1.80259e16	0.810551
$939$	4.26314e15	0.190576
$940$	5.38058e13	0.00239125
$941$	2.32953e16	1.02926	0.514631	−	0.857412i	$-0.327929\pi$
$941$	2.32953e16	1.02926	0.514631	+	0.857412i	$0.327929\pi$
$942$	−2.57643e16	−1.13172
$943$	3.88175e15	0.169517
$944$	1.65864e16	0.720122
$945$	−3.35043e15	−0.144619
$946$	7.28833e15	0.312771
$947$	1.50355e16	0.641495	0.320748	−	0.947165i	$-0.396066\pi$
$947$	1.50355e16	0.641495	0.320748	+	0.947165i	$0.396066\pi$
$948$	−1.89162e15	−0.0802393
$949$	−5.75880e16	−2.42866
$950$	−3.48934e16	−1.46307
$951$	−1.08223e16	−0.451154
$952$	1.17658e15	0.0487663
$953$	2.49426e16	1.02785	0.513926	−	0.857835i	$-0.328191\pi$
$953$	2.49426e16	1.02785	0.513926	+	0.857835i	$0.328191\pi$
$954$	−5.95964e15	−0.244177
$955$	4.09957e15	0.167001
$956$	2.61808e15	0.106039
$957$	9.50850e15	0.382910
$958$	2.42248e16	0.969950
$959$	−1.87703e16	−0.747255
$960$	3.12453e15	0.123678
$961$	−2.52883e16	−0.995271
$962$	−5.64778e16	−2.21011
$963$	−1.28143e15	−0.0498598
$964$	2.26545e15	0.0876459
$965$	3.05767e15	0.117622
$966$	3.33927e15	0.127725
$967$	−7.34030e15	−0.279170	−0.139585	−	0.990210i	$-0.544577\pi$
$967$	−7.34030e15	−0.279170	−0.139585	+	0.990210i	$0.544577\pi$
$968$	−2.34361e16	−0.886280
$969$	1.67610e15	0.0630257
$970$	−2.45609e15	−0.0918332
$971$	−5.08462e16	−1.89040	−0.945198	−	0.326497i	$-0.894132\pi$
$971$	−5.08462e16	−1.89040	−0.945198	+	0.326497i	$0.894132\pi$
$972$	2.12020e15	0.0783813
$973$	−1.13439e16	−0.417006
$974$	3.56176e16	1.30194
$975$	2.89090e16	1.05077
$976$	1.34557e16	0.486331
$977$	−1.67151e16	−0.600743	−0.300371	−	0.953822i	$-0.597111\pi$
$977$	−1.67151e16	−0.600743	−0.300371	+	0.953822i	$0.597111\pi$
$978$	−5.14913e15	−0.184022
$979$	1.46707e16	0.521371
$980$	−1.03330e14	−0.00365158
$981$	2.01821e16	0.709228
$982$	−3.38284e16	−1.18214
$983$	1.96972e16	0.684479	0.342239	−	0.939613i	$-0.388815\pi$
$983$	1.96972e16	0.684479	0.342239	+	0.939613i	$0.388815\pi$
$984$	−1.81861e16	−0.628444
$985$	1.46656e15	0.0503967
$986$	2.00192e15	0.0684105
$987$	−3.05209e15	−0.103718
$988$	6.49376e15	0.219449
$989$	−5.27910e15	−0.177411
$990$	−7.65942e14	−0.0255978
$991$	−1.33471e16	−0.443591	−0.221796	−	0.975093i	$-0.571192\pi$
$991$	−1.33471e16	−0.443591	−0.221796	+	0.975093i	$0.571192\pi$
$992$	−3.97708e14	−0.0131447
$993$	−2.56495e16	−0.843060
$994$	2.82097e16	0.922090
$995$	−6.02824e14	−0.0195958
$996$	1.91485e15	0.0619026
$997$	1.72159e16	0.553487	0.276743	−	0.960944i	$-0.410745\pi$
$997$	1.72159e16	0.553487	0.276743	+	0.960944i	$0.410745\pi$
$998$	1.19312e16	0.381474
$999$	5.41702e16	1.72246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.12.a.a.1.3	✓	8
3.2	odd	2		207.12.a.a.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.12.a.a.1.3	✓	8	1.1	even	1	trivial
207.12.a.a.1.6		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)

\(f(q)\)	\(=\)	\(q-43.0298 q^{2} -312.230 q^{3} -196.437 q^{4} -1082.06 q^{5} +13435.2 q^{6} +38616.0 q^{7} +96577.7 q^{8} -79659.3 q^{9} +O(q^{10})\)	\(q-43.0298 q^{2} -312.230 q^{3} -196.437 q^{4} -1082.06 q^{5} +13435.2 q^{6} +38616.0 q^{7} +96577.7 q^{8} -79659.3 q^{9} +46560.9 q^{10} +206509. q^{11} +61333.5 q^{12} +1.94280e6 q^{13} -1.66164e6 q^{14} +337852. q^{15} -3.75341e6 q^{16} +315485. q^{17} +3.42772e6 q^{18} -1.70155e7 q^{19} +212556. q^{20} -1.20571e7 q^{21} -8.88603e6 q^{22} +6.43634e6 q^{23} -3.01545e7 q^{24} -4.76573e7 q^{25} -8.35984e7 q^{26} +8.01827e7 q^{27} -7.58560e6 q^{28} -1.47468e8 q^{29} -1.45377e7 q^{30} +1.09615e7 q^{31} -3.62824e7 q^{32} -6.44783e7 q^{33} -1.35752e7 q^{34} -4.17849e7 q^{35} +1.56480e7 q^{36} +6.75584e8 q^{37} +7.32175e8 q^{38} -6.06602e8 q^{39} -1.04503e8 q^{40} +6.03098e8 q^{41} +5.18814e8 q^{42} -8.20201e8 q^{43} -4.05659e7 q^{44} +8.61962e7 q^{45} -2.76955e8 q^{46} +2.53136e8 q^{47} +1.17193e9 q^{48} -4.86128e8 q^{49} +2.05068e9 q^{50} -9.85039e7 q^{51} -3.81638e8 q^{52} -1.73866e9 q^{53} -3.45024e9 q^{54} -2.23455e8 q^{55} +3.72945e9 q^{56} +5.31276e9 q^{57} +6.34552e9 q^{58} -4.41902e9 q^{59} -6.63665e7 q^{60} -3.58492e9 q^{61} -4.71670e8 q^{62} -3.07613e9 q^{63} +9.24822e9 q^{64} -2.10223e9 q^{65} +2.77449e9 q^{66} -1.08482e10 q^{67} -6.19728e7 q^{68} -2.00962e9 q^{69} +1.79800e9 q^{70} -1.69770e10 q^{71} -7.69330e9 q^{72} -2.96417e10 q^{73} -2.90702e10 q^{74} +1.48800e10 q^{75} +3.34247e9 q^{76} +7.97455e9 q^{77} +2.61019e10 q^{78} -3.08415e10 q^{79} +4.06142e9 q^{80} -1.09241e10 q^{81} -2.59512e10 q^{82} +3.12203e10 q^{83} +2.36846e9 q^{84} -3.41374e8 q^{85} +3.52931e10 q^{86} +4.60440e10 q^{87} +1.99441e10 q^{88} +7.10416e10 q^{89} -3.70900e9 q^{90} +7.50233e10 q^{91} -1.26433e9 q^{92} -3.42250e9 q^{93} -1.08924e10 q^{94} +1.84118e10 q^{95} +1.13285e10 q^{96} -5.27501e10 q^{97} +2.09180e10 q^{98} -1.64503e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9}+O(q^{10}) \) 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

Representations

Groups

Database

Properties

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