LMFDB - Embedded newform 23.12.a.a.1.7

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			1.7
Root			$-33.5241$ 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)

$f(q)$	$=$	$q+63.0482 q^{2} -462.057 q^{3} +1927.07 q^{4} +5906.93 q^{5} -29131.8 q^{6} -36688.5 q^{7} -7624.20 q^{8} +36349.3 q^{9} +O(q^{10})$	\(q+63.0482 q^{2} -462.057 q^{3} +1927.07 q^{4} +5906.93 q^{5} -29131.8 q^{6} -36688.5 q^{7} -7624.20 q^{8} +36349.3 q^{9} +372421. q^{10} +19671.3 q^{11} -890417. q^{12} -1.25886e6 q^{13} -2.31314e6 q^{14} -2.72933e6 q^{15} -4.42734e6 q^{16} -6.08966e6 q^{17} +2.29176e6 q^{18} -6.45330e6 q^{19} +1.13831e7 q^{20} +1.69521e7 q^{21} +1.24024e6 q^{22} +6.43634e6 q^{23} +3.52281e6 q^{24} -1.39363e7 q^{25} -7.93688e7 q^{26} +6.50565e7 q^{27} -7.07014e7 q^{28} +4.85552e6 q^{29} -1.72080e8 q^{30} +4.74676e7 q^{31} -2.63521e8 q^{32} -9.08924e6 q^{33} -3.83942e8 q^{34} -2.16716e8 q^{35} +7.00478e7 q^{36} +3.61239e8 q^{37} -4.06869e8 q^{38} +5.81664e8 q^{39} -4.50356e7 q^{40} +7.61603e8 q^{41} +1.06880e9 q^{42} +8.63543e8 q^{43} +3.79080e7 q^{44} +2.14713e8 q^{45} +4.05800e8 q^{46} +2.26521e8 q^{47} +2.04568e9 q^{48} -6.31284e8 q^{49} -8.78661e8 q^{50} +2.81377e9 q^{51} -2.42591e9 q^{52} -4.77243e9 q^{53} +4.10169e9 q^{54} +1.16197e8 q^{55} +2.79720e8 q^{56} +2.98179e9 q^{57} +3.06132e8 q^{58} +4.62634e9 q^{59} -5.25963e9 q^{60} -8.02546e9 q^{61} +2.99274e9 q^{62} -1.33360e9 q^{63} -7.54735e9 q^{64} -7.43599e9 q^{65} -5.73060e8 q^{66} -5.86680e9 q^{67} -1.17352e10 q^{68} -2.97395e9 q^{69} -1.36636e10 q^{70} +5.53372e9 q^{71} -2.77135e8 q^{72} +1.09700e10 q^{73} +2.27755e10 q^{74} +6.43938e9 q^{75} -1.24360e10 q^{76} -7.21709e8 q^{77} +3.66729e10 q^{78} +3.09204e10 q^{79} -2.61520e10 q^{80} -3.64990e10 q^{81} +4.80177e10 q^{82} -6.79236e10 q^{83} +3.26680e10 q^{84} -3.59712e10 q^{85} +5.44448e10 q^{86} -2.24353e9 q^{87} -1.49978e8 q^{88} -5.23430e10 q^{89} +1.35373e10 q^{90} +4.61856e10 q^{91} +1.24033e10 q^{92} -2.19327e10 q^{93} +1.42817e10 q^{94} -3.81192e10 q^{95} +1.21762e11 q^{96} +5.03724e10 q^{97} -3.98013e10 q^{98} +7.15038e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	63.0482	1.39318	0.696591	−	0.717469i	$-0.254699\pi$
$2$	63.0482	1.39318	0.696591	+	0.717469i	$0.254699\pi$
$3$	−462.057	−1.09781	−0.548906	−	0.835884i	$-0.684956\pi$
$3$	−462.057	−1.09781	−0.548906	+	0.835884i	$0.684956\pi$
$4$	1927.07	0.940954
$5$	5906.93	0.845330	0.422665	−	0.906286i	$-0.361095\pi$
$5$	5906.93	0.845330	0.422665	+	0.906286i	$0.361095\pi$
$6$	−29131.8	−1.52945
$7$	−36688.5	−0.825069	−0.412534	−	0.910942i	$-0.635356\pi$
$7$	−36688.5	−0.825069	−0.412534	+	0.910942i	$0.635356\pi$
$8$	−7624.20	−0.0822620
$9$	36349.3	0.205193
$10$	372421.	1.17770
$11$	19671.3	0.0368276	0.0184138	−	0.999830i	$-0.494138\pi$
$11$	19671.3	0.0368276	0.0184138	+	0.999830i	$0.494138\pi$
$12$	−890417.	−1.03299
$13$	−1.25886e6	−0.940348	−0.470174	−	0.882574i	$-0.655809\pi$
$13$	−1.25886e6	−0.940348	−0.470174	+	0.882574i	$0.655809\pi$
$14$	−2.31314e6	−1.14947
$15$	−2.72933e6	−0.928015
$16$	−4.42734e6	−1.05556
$17$	−6.08966e6	−1.04022	−0.520108	−	0.854100i	$-0.674108\pi$
$17$	−6.08966e6	−1.04022	−0.520108	+	0.854100i	$0.674108\pi$
$18$	2.29176e6	0.285871
$19$	−6.45330e6	−0.597912	−0.298956	−	0.954267i	$-0.596638\pi$
$19$	−6.45330e6	−0.597912	−0.298956	+	0.954267i	$0.596638\pi$
$20$	1.13831e7	0.795417
$21$	1.69521e7	0.905771
$22$	1.24024e6	0.0513075
$23$	6.43634e6	0.208514
$23$	6.43634e6	0.208514
$24$	3.52281e6	0.0903083
$25$	−1.39363e7	−0.285416
$26$	−7.93688e7	−1.31008
$27$	6.50565e7	0.872549
$28$	−7.07014e7	−0.776352
$29$	4.85552e6	0.0439589	0.0219795	−	0.999758i	$-0.493003\pi$
$29$	4.85552e6	0.0439589	0.0219795	+	0.999758i	$0.493003\pi$
$30$	−1.72080e8	−1.29289
$31$	4.74676e7	0.297788	0.148894	−	0.988853i	$-0.452429\pi$
$31$	4.74676e7	0.297788	0.148894	+	0.988853i	$0.452429\pi$
$32$	−2.63521e8	−1.38832
$33$	−9.08924e6	−0.0404298
$34$	−3.83942e8	−1.44921
$35$	−2.16716e8	−0.697456
$36$	7.00478e7	0.193077
$37$	3.61239e8	0.856418	0.428209	−	0.903680i	$-0.359145\pi$
$37$	3.61239e8	0.856418	0.428209	+	0.903680i	$0.359145\pi$
$38$	−4.06869e8	−0.833000
$39$	5.81664e8	1.03233
$40$	−4.50356e7	−0.0695386
$41$	7.61603e8	1.02664	0.513319	−	0.858198i	$-0.328416\pi$
$41$	7.61603e8	1.02664	0.513319	+	0.858198i	$0.328416\pi$
$42$	1.06880e9	1.26190
$43$	8.63543e8	0.895792	0.447896	−	0.894086i	$-0.352173\pi$
$43$	8.63543e8	0.895792	0.447896	+	0.894086i	$0.352173\pi$
$44$	3.79080e7	0.0346530
$45$	2.14713e8	0.173456
$46$	4.05800e8	0.290498
$47$	2.26521e8	0.144069	0.0720344	−	0.997402i	$-0.477051\pi$
$47$	2.26521e8	0.144069	0.0720344	+	0.997402i	$0.477051\pi$
$48$	2.04568e9	1.15881
$49$	−6.31284e8	−0.319261
$50$	−8.78661e8	−0.397637
$51$	2.81377e9	1.14196
$52$	−2.42591e9	−0.884824
$53$	−4.77243e9	−1.56755	−0.783776	−	0.621044i	$-0.786709\pi$
$53$	−4.77243e9	−1.56755	−0.783776	+	0.621044i	$0.786709\pi$
$54$	4.10169e9	1.21562
$55$	1.16197e8	0.0311315
$56$	2.79720e8	0.0678718
$57$	2.98179e9	0.656396
$58$	3.06132e8	0.0612427
$59$	4.62634e9	0.842464	0.421232	−	0.906953i	$-0.361598\pi$
$59$	4.62634e9	0.842464	0.421232	+	0.906953i	$0.361598\pi$
$60$	−5.25963e9	−0.873219
$61$	−8.02546e9	−1.21662	−0.608311	−	0.793699i	$-0.708153\pi$
$61$	−8.02546e9	−1.21662	−0.608311	+	0.793699i	$0.708153\pi$
$62$	2.99274e9	0.414873
$63$	−1.33360e9	−0.169298
$64$	−7.54735e9	−0.878627
$65$	−7.43599e9	−0.794905
$66$	−5.73060e8	−0.0563260
$67$	−5.86680e9	−0.530872	−0.265436	−	0.964129i	$-0.585516\pi$
$67$	−5.86680e9	−0.530872	−0.265436	+	0.964129i	$0.585516\pi$
$68$	−1.17352e10	−0.978796
$69$	−2.97395e9	−0.228910
$70$	−1.36636e10	−0.971682
$71$	5.53372e9	0.363996	0.181998	−	0.983299i	$-0.441744\pi$
$71$	5.53372e9	0.363996	0.181998	+	0.983299i	$0.441744\pi$
$72$	−2.77135e8	−0.0168796
$73$	1.09700e10	0.619343	0.309672	−	0.950844i	$-0.399781\pi$
$73$	1.09700e10	0.619343	0.309672	+	0.950844i	$0.399781\pi$
$74$	2.27755e10	1.19315
$75$	6.43938e9	0.313334
$76$	−1.24360e10	−0.562608
$77$	−7.21709e8	−0.0303853
$78$	3.66729e10	1.43822
$79$	3.09204e10	1.13057	0.565284	−	0.824897i	$-0.308767\pi$
$79$	3.09204e10	1.13057	0.565284	+	0.824897i	$0.308767\pi$
$80$	−2.61520e10	−0.892297
$81$	−3.64990e10	−1.16309
$82$	4.80177e10	1.43029
$83$	−6.79236e10	−1.89274	−0.946371	−	0.323082i	$-0.895281\pi$
$83$	−6.79236e10	−1.89274	−0.946371	+	0.323082i	$0.895281\pi$
$84$	3.26680e10	0.852289
$85$	−3.59712e10	−0.879327
$86$	5.44448e10	1.24800
$87$	−2.24353e9	−0.0482587
$88$	−1.49978e8	−0.00302951
$89$	−5.23430e10	−0.993605	−0.496803	−	0.867864i	$-0.665493\pi$
$89$	−5.23430e10	−0.993605	−0.496803	+	0.867864i	$0.665493\pi$
$90$	1.35373e10	0.241656
$91$	4.61856e10	0.775852
$92$	1.24033e10	0.196202
$93$	−2.19327e10	−0.326916
$94$	1.42817e10	0.200714
$95$	−3.81192e10	−0.505433
$96$	1.21762e11	1.52412
$97$	5.03724e10	0.595591	0.297796	−	0.954630i	$-0.403749\pi$
$97$	5.03724e10	0.595591	0.297796	+	0.954630i	$0.403749\pi$
$98$	−3.98013e10	−0.444789
$99$	7.15038e8	0.00755676
$100$	−2.68564e10	−0.268564
$101$	−1.97452e11	−1.86936	−0.934682	−	0.355484i	$-0.884316\pi$
$101$	−1.97452e11	−1.86936	−0.934682	+	0.355484i	$0.884316\pi$
$102$	1.77403e11	1.59096
$103$	4.16982e10	0.354415	0.177208	−	0.984173i	$-0.443294\pi$
$103$	4.16982e10	0.354415	0.177208	+	0.984173i	$0.443294\pi$
$104$	9.59779e9	0.0773549
$105$	1.00135e11	0.765676
$106$	−3.00893e11	−2.18388
$107$	1.00950e10	0.0695818	0.0347909	−	0.999395i	$-0.488923\pi$
$107$	1.00950e10	0.0695818	0.0347909	+	0.999395i	$0.488923\pi$
$108$	1.25369e11	0.821029
$109$	2.01058e11	1.25163	0.625816	−	0.779971i	$-0.284766\pi$
$109$	2.01058e11	1.25163	0.625816	+	0.779971i	$0.284766\pi$
$110$	7.32600e9	0.0433718
$111$	−1.66913e11	−0.940187
$112$	1.62432e11	0.870910
$113$	−2.48207e11	−1.26731	−0.633655	−	0.773615i	$-0.718446\pi$
$113$	−2.48207e11	−1.26731	−0.633655	+	0.773615i	$0.718446\pi$
$114$	1.87997e11	0.914478
$115$	3.80190e10	0.176264
$116$	9.35695e9	0.0413633
$117$	−4.57587e10	−0.192953
$118$	2.91682e11	1.17370
$119$	2.23420e11	0.858250
$120$	2.08090e10	0.0763404
$121$	−2.84925e11	−0.998644
$122$	−5.05991e11	−1.69498
$123$	−3.51904e11	−1.12706
$124$	9.14735e10	0.280205
$125$	−3.70745e11	−1.08660
$126$	−8.40811e10	−0.235863
$127$	−1.61389e11	−0.433464	−0.216732	−	0.976231i	$-0.569540\pi$
$127$	−1.61389e11	−0.433464	−0.216732	+	0.976231i	$0.569540\pi$
$128$	6.38449e10	0.164237
$129$	−3.99006e11	−0.983412
$130$	−4.68826e11	−1.10745
$131$	−6.81653e8	−0.00154373	−0.000771865	−	1.00000i	$-0.500246\pi$
$131$	−6.81653e8	−0.00154373	−0.000771865		1.00000i	$0.500246\pi$
$132$	−1.75156e10	−0.0380425
$133$	2.36762e11	0.493319
$134$	−3.69891e11	−0.739600
$135$	3.84284e11	0.737593
$136$	4.64288e10	0.0855703
$137$	1.15713e11	0.204842	0.102421	−	0.994741i	$-0.467341\pi$
$137$	1.15713e11	0.204842	0.102421	+	0.994741i	$0.467341\pi$
$138$	−1.87502e11	−0.318913
$139$	6.89379e11	1.12688	0.563439	−	0.826158i	$-0.309478\pi$
$139$	6.89379e11	1.12688	0.563439	+	0.826158i	$0.309478\pi$
$140$	−4.17628e11	−0.656274
$141$	−1.04665e11	−0.158161
$142$	3.48891e11	0.507112
$143$	−2.47634e10	−0.0346307
$144$	−1.60931e11	−0.216593
$145$	2.86812e10	0.0371598
$146$	6.91639e11	0.862857
$147$	2.91689e11	0.350489
$148$	6.96135e11	0.805850
$149$	1.64263e12	1.83238	0.916188	−	0.400749i	$-0.131250\pi$
$149$	1.64263e12	1.83238	0.916188	+	0.400749i	$0.131250\pi$
$150$	4.05991e11	0.436531
$151$	1.66300e12	1.72393	0.861966	−	0.506966i	$-0.169233\pi$
$151$	1.66300e12	1.72393	0.861966	+	0.506966i	$0.169233\pi$
$152$	4.92013e10	0.0491855
$153$	−2.21355e11	−0.213445
$154$	−4.55024e10	−0.0423322
$155$	2.80387e11	0.251730
$156$	1.12091e12	0.971371
$157$	−6.06622e11	−0.507539	−0.253770	−	0.967265i	$-0.581671\pi$
$157$	−6.06622e11	−0.507539	−0.253770	+	0.967265i	$0.581671\pi$
$158$	1.94948e12	1.57509
$159$	2.20513e12	1.72088
$160$	−1.55660e12	−1.17359
$161$	−2.36140e11	−0.172039
$162$	−2.30119e12	−1.62039
$163$	−2.51013e12	−1.70870	−0.854348	−	0.519701i	$-0.826043\pi$
$163$	−2.51013e12	−1.70870	−0.854348	+	0.519701i	$0.826043\pi$
$164$	1.46766e12	0.966019
$165$	−5.36895e10	−0.0341765
$166$	−4.28246e12	−2.63693
$167$	−2.93039e12	−1.74576	−0.872880	−	0.487935i	$-0.837750\pi$
$167$	−2.93039e12	−1.74576	−0.872880	+	0.487935i	$0.837750\pi$
$168$	−1.29247e11	−0.0745106
$169$	−2.07434e11	−0.115745
$170$	−2.26792e12	−1.22506
$171$	−2.34573e11	−0.122687
$172$	1.66411e12	0.842899
$173$	2.55669e12	1.25436	0.627182	−	0.778873i	$-0.284208\pi$
$173$	2.55669e12	1.25436	0.627182	+	0.778873i	$0.284208\pi$
$174$	−1.41450e11	−0.0672331
$175$	5.11303e11	0.235488
$176$	−8.70914e10	−0.0388737
$177$	−2.13763e12	−0.924868
$178$	−3.30013e12	−1.38427
$179$	8.52846e11	0.346880	0.173440	−	0.984844i	$-0.444512\pi$
$179$	8.52846e11	0.346880	0.173440	+	0.984844i	$0.444512\pi$
$180$	4.13767e11	0.163214
$181$	−1.52665e12	−0.584126	−0.292063	−	0.956399i	$-0.594342\pi$
$181$	−1.52665e12	−0.584126	−0.292063	+	0.956399i	$0.594342\pi$
$182$	2.91192e12	1.08090
$183$	3.70822e12	1.33562
$184$	−4.90720e10	−0.0171528
$185$	2.13382e12	0.723956
$186$	−1.38282e12	−0.455453
$187$	−1.19791e11	−0.0383086
$188$	4.36522e11	0.135562
$189$	−2.38682e12	−0.719913
$190$	−2.40335e12	−0.704160
$191$	−7.77540e11	−0.221329	−0.110665	−	0.993858i	$-0.535298\pi$
$191$	−7.77540e11	−0.221329	−0.110665	+	0.993858i	$0.535298\pi$
$192$	3.48730e12	0.964568
$193$	−1.47440e12	−0.396324	−0.198162	−	0.980169i	$-0.563497\pi$
$193$	−1.47440e12	−0.396324	−0.198162	+	0.980169i	$0.563497\pi$
$194$	3.17589e12	0.829767
$195$	3.43585e12	0.872657
$196$	−1.21653e12	−0.300410
$197$	−6.59359e12	−1.58328	−0.791640	−	0.610988i	$-0.790772\pi$
$197$	−6.59359e12	−1.58328	−0.791640	+	0.610988i	$0.790772\pi$
$198$	4.50818e10	0.0105279
$199$	−2.44416e12	−0.555184	−0.277592	−	0.960699i	$-0.589536\pi$
$199$	−2.44416e12	−0.555184	−0.277592	+	0.960699i	$0.589536\pi$
$200$	1.06253e11	0.0234789
$201$	2.71079e12	0.582798
$202$	−1.24490e13	−2.60436
$203$	−1.78142e11	−0.0362691
$204$	5.42233e12	1.07453
$205$	4.49873e12	0.867848
$206$	2.62899e12	0.493765
$207$	2.33957e11	0.0427857
$208$	5.57340e12	0.992594
$209$	−1.26945e11	−0.0220196
$210$	6.31333e12	1.06673
$211$	5.92536e12	0.975352	0.487676	−	0.873025i	$-0.337845\pi$
$211$	5.92536e12	0.975352	0.487676	+	0.873025i	$0.337845\pi$
$212$	−9.19681e12	−1.47499
$213$	−2.55689e12	−0.399599
$214$	6.36471e11	0.0969400
$215$	5.10088e12	0.757241
$216$	−4.96004e11	−0.0717777
$217$	−1.74151e12	−0.245696
$218$	1.26764e13	1.74375
$219$	−5.06877e12	−0.679923
$220$	2.23920e11	0.0292933
$221$	7.66602e12	0.978166
$222$	−1.05236e13	−1.30985
$223$	−1.38412e13	−1.68072	−0.840362	−	0.542026i	$-0.817658\pi$
$223$	−1.38412e13	−1.68072	−0.840362	+	0.542026i	$0.817658\pi$
$224$	9.66819e12	1.14546
$225$	−5.06577e11	−0.0585654
$226$	−1.56490e13	−1.76559
$227$	1.53259e12	0.168766	0.0843829	−	0.996433i	$-0.473108\pi$
$227$	1.53259e12	0.168766	0.0843829	+	0.996433i	$0.473108\pi$
$228$	5.74613e12	0.617638
$229$	1.79068e11	0.0187898	0.00939490	−	0.999956i	$-0.497009\pi$
$229$	1.79068e11	0.0187898	0.00939490	+	0.999956i	$0.497009\pi$
$230$	2.39703e12	0.245567
$231$	3.33470e11	0.0333573
$232$	−3.70195e10	−0.00361615
$233$	1.78552e13	1.70337	0.851683	−	0.524057i	$-0.175582\pi$
$233$	1.78552e13	1.70337	0.851683	+	0.524057i	$0.175582\pi$
$234$	−2.88500e12	−0.268818
$235$	1.33804e12	0.121786
$236$	8.91529e12	0.792719
$237$	−1.42870e13	−1.24115
$238$	1.40862e13	1.19570
$239$	2.07089e12	0.171778	0.0858890	−	0.996305i	$-0.472627\pi$
$239$	2.07089e12	0.171778	0.0858890	+	0.996305i	$0.472627\pi$
$240$	1.20837e13	0.979575
$241$	3.69733e12	0.292951	0.146476	−	0.989214i	$-0.453207\pi$
$241$	3.69733e12	0.292951	0.146476	+	0.989214i	$0.453207\pi$
$242$	−1.79640e13	−1.39129
$243$	5.34002e12	0.404305
$244$	−1.54656e13	−1.14479
$245$	−3.72895e12	−0.269881
$246$	−2.21869e13	−1.57019
$247$	8.12380e12	0.562246
$248$	−3.61902e11	−0.0244967
$249$	3.13846e13	2.07788
$250$	−2.33748e13	−1.51383
$251$	−1.67093e13	−1.05865	−0.529325	−	0.848419i	$-0.677555\pi$
$251$	−1.67093e13	−1.05865	−0.529325	+	0.848419i	$0.677555\pi$
$252$	−2.56995e12	−0.159302
$253$	1.26611e11	0.00767908
$254$	−1.01753e13	−0.603894
$255$	1.66207e13	0.965336
$256$	1.94823e13	1.10744
$257$	1.26917e13	0.706133	0.353066	−	0.935598i	$-0.385139\pi$
$257$	1.26917e13	0.706133	0.353066	+	0.935598i	$0.385139\pi$
$258$	−2.51566e13	−1.37007
$259$	−1.32533e13	−0.706604
$260$	−1.43297e13	−0.747969
$261$	1.76495e11	0.00902006
$262$	−4.29770e10	−0.00215070
$263$	2.12935e13	1.04350	0.521749	−	0.853099i	$-0.325280\pi$
$263$	2.12935e13	1.04350	0.521749	+	0.853099i	$0.325280\pi$
$264$	6.92982e10	0.00332583
$265$	−2.81904e13	−1.32510
$266$	1.49274e13	0.687282
$267$	2.41854e13	1.09079
$268$	−1.13057e13	−0.499526
$269$	−1.94525e13	−0.842049	−0.421025	−	0.907049i	$-0.638329\pi$
$269$	−1.94525e13	−0.842049	−0.421025	+	0.907049i	$0.638329\pi$
$270$	2.42284e13	1.02760
$271$	−3.46906e13	−1.44172	−0.720859	−	0.693082i	$-0.756252\pi$
$271$	−3.46906e13	−1.44172	−0.720859	+	0.693082i	$0.756252\pi$
$272$	2.69610e13	1.09801
$273$	−2.13404e13	−0.851740
$274$	7.29548e12	0.285381
$275$	−2.74146e11	−0.0105112
$276$	−5.73103e12	−0.215394
$277$	−5.08687e13	−1.87418	−0.937092	−	0.349082i	$-0.886493\pi$
$277$	−5.08687e13	−1.87418	−0.937092	+	0.349082i	$0.886493\pi$
$278$	4.34641e13	1.56994
$279$	1.72541e12	0.0611041
$280$	1.65229e12	0.0573741
$281$	−3.49100e13	−1.18868	−0.594340	−	0.804214i	$-0.702587\pi$
$281$	−3.49100e13	−1.18868	−0.594340	+	0.804214i	$0.702587\pi$
$282$	−6.59896e12	−0.220346
$283$	−3.95659e13	−1.29567	−0.647837	−	0.761779i	$-0.724326\pi$
$283$	−3.95659e13	−1.29567	−0.647837	+	0.761779i	$0.724326\pi$
$284$	1.06639e13	0.342503
$285$	1.76132e13	0.554871
$286$	−1.56129e12	−0.0482469
$287$	−2.79420e13	−0.847047
$288$	−9.57882e12	−0.284874
$289$	2.81203e12	0.0820506
$290$	1.80830e12	0.0517703
$291$	−2.32749e13	−0.653848
$292$	2.11400e13	0.582773
$293$	3.35901e13	0.908740	0.454370	−	0.890813i	$-0.349864\pi$
$293$	3.35901e13	0.908740	0.454370	+	0.890813i	$0.349864\pi$
$294$	1.83905e13	0.488295
$295$	2.73274e13	0.712160
$296$	−2.75416e12	−0.0704507
$297$	1.27974e12	0.0321339
$298$	1.03565e14	2.55283
$299$	−8.10245e12	−0.196076
$300$	1.24092e13	0.294833
$301$	−3.16821e13	−0.739090
$302$	1.04849e14	2.40175
$303$	9.12340e13	2.05221
$304$	2.85710e13	0.631132
$305$	−4.74058e13	−1.02845
$306$	−1.39560e13	−0.297368
$307$	−2.03497e13	−0.425889	−0.212944	−	0.977064i	$-0.568305\pi$
$307$	−2.03497e13	−0.425889	−0.212944	+	0.977064i	$0.568305\pi$
$308$	−1.39079e12	−0.0285911
$309$	−1.92669e13	−0.389082
$310$	1.76779e13	0.350705
$311$	−2.59570e13	−0.505910	−0.252955	−	0.967478i	$-0.581402\pi$
$311$	−2.59570e13	−0.505910	−0.252955	+	0.967478i	$0.581402\pi$
$312$	−4.43472e12	−0.0849213
$313$	7.76402e13	1.46081	0.730404	−	0.683016i	$-0.239332\pi$
$313$	7.76402e13	1.46081	0.730404	+	0.683016i	$0.239332\pi$
$314$	−3.82464e13	−0.707094
$315$	−7.87748e12	−0.143113
$316$	5.95859e13	1.06381
$317$	−2.36641e13	−0.415206	−0.207603	−	0.978213i	$-0.566566\pi$
$317$	−2.36641e13	−0.415206	−0.207603	+	0.978213i	$0.566566\pi$
$318$	1.39029e14	2.39750
$319$	9.55143e10	0.00161890
$320$	−4.45816e13	−0.742730
$321$	−4.66446e12	−0.0763878
$322$	−1.48882e13	−0.239681
$323$	3.92984e13	0.621958
$324$	−7.03362e13	−1.09441
$325$	1.75439e13	0.268391
$326$	−1.58259e14	−2.38052
$327$	−9.29004e13	−1.37406
$328$	−5.80661e12	−0.0844533
$329$	−8.31070e12	−0.118867
$330$	−3.38503e12	−0.0476141
$331$	6.82678e12	0.0944413	0.0472207	−	0.998884i	$-0.484964\pi$
$331$	6.82678e12	0.0944413	0.0472207	+	0.998884i	$0.484964\pi$
$332$	−1.30894e14	−1.78098
$333$	1.31308e13	0.175731
$334$	−1.84756e14	−2.43216
$335$	−3.46547e13	−0.448762
$336$	−7.50529e13	−0.956096
$337$	1.17822e14	1.47660	0.738299	−	0.674474i	$-0.235629\pi$
$337$	1.17822e14	1.47660	0.738299	+	0.674474i	$0.235629\pi$
$338$	−1.30783e13	−0.161254
$339$	1.14686e14	1.39127
$340$	−6.93191e13	−0.827406
$341$	9.33748e11	0.0109668
$342$	−1.47894e13	−0.170926
$343$	9.57059e13	1.08848
$344$	−6.58382e12	−0.0736897
$345$	−1.75669e13	−0.193504
$346$	1.61194e14	1.74756
$347$	9.49373e13	1.01304	0.506518	−	0.862230i	$-0.330933\pi$
$347$	9.49373e13	1.01304	0.506518	+	0.862230i	$0.330933\pi$
$348$	−4.32344e12	−0.0454092
$349$	−1.56516e14	−1.61815	−0.809077	−	0.587703i	$-0.800032\pi$
$349$	−1.56516e14	−1.61815	−0.809077	+	0.587703i	$0.800032\pi$
$350$	3.22367e13	0.328078
$351$	−8.18970e13	−0.820500
$352$	−5.18380e12	−0.0511286
$353$	−1.73162e14	−1.68148	−0.840739	−	0.541441i	$-0.817879\pi$
$353$	−1.73162e14	−1.68148	−0.840739	+	0.541441i	$0.817879\pi$
$354$	−1.34774e14	−1.28851
$355$	3.26873e13	0.307697
$356$	−1.00869e14	−0.934937
$357$	−1.03233e14	−0.942198
$358$	5.37704e13	0.483266
$359$	1.34405e14	1.18959	0.594794	−	0.803878i	$-0.297234\pi$
$359$	1.34405e14	1.18959	0.594794	+	0.803878i	$0.297234\pi$
$360$	−1.63701e12	−0.0142688
$361$	−7.48451e13	−0.642501
$362$	−9.62524e13	−0.813794
$363$	1.31651e14	1.09632
$364$	8.90031e13	0.730041
$365$	6.47990e13	0.523550
$366$	2.33796e14	1.86077
$367$	6.31523e13	0.495138	0.247569	−	0.968870i	$-0.420368\pi$
$367$	6.31523e13	0.495138	0.247569	+	0.968870i	$0.420368\pi$
$368$	−2.84959e13	−0.220099
$369$	2.76837e13	0.210659
$370$	1.34533e14	1.00860
$371$	1.75093e14	1.29334
$372$	−4.22659e13	−0.307613
$373$	−9.36857e13	−0.671854	−0.335927	−	0.941888i	$-0.609050\pi$
$373$	−9.36857e13	−0.671854	−0.335927	+	0.941888i	$0.609050\pi$
$374$	−7.55263e12	−0.0533709
$375$	1.71305e14	1.19289
$376$	−1.72704e12	−0.0118514
$377$	−6.11242e12	−0.0413367
$378$	−1.50485e14	−1.00297
$379$	2.10198e14	1.38074	0.690371	−	0.723456i	$-0.257447\pi$
$379$	2.10198e14	1.38074	0.690371	+	0.723456i	$0.257447\pi$
$380$	−7.34585e13	−0.475589
$381$	7.45709e13	0.475863
$382$	−4.90225e13	−0.308352
$383$	1.14626e14	0.710709	0.355354	−	0.934732i	$-0.384360\pi$
$383$	1.14626e14	0.710709	0.355354	+	0.934732i	$0.384360\pi$
$384$	−2.95000e13	−0.180302
$385$	−4.26308e12	−0.0256856
$386$	−9.29582e13	−0.552151
$387$	3.13892e13	0.183810
$388$	9.70714e13	0.560424
$389$	1.75338e14	0.998052	0.499026	−	0.866587i	$-0.333691\pi$
$389$	1.75338e14	0.998052	0.499026	+	0.866587i	$0.333691\pi$
$390$	2.16624e14	1.21577
$391$	−3.91951e13	−0.216900
$392$	4.81303e12	0.0262631
$393$	3.14963e11	0.00169473
$394$	−4.15714e14	−2.20580
$395$	1.82645e14	0.955703
$396$	1.37793e12	0.00711056
$397$	−2.45922e14	−1.25155	−0.625777	−	0.780002i	$-0.715218\pi$
$397$	−2.45922e14	−1.25155	−0.625777	+	0.780002i	$0.715218\pi$
$398$	−1.54100e14	−0.773472
$399$	−1.09397e14	−0.541572
$400$	6.17009e13	0.301274
$401$	−2.28708e13	−0.110151	−0.0550755	−	0.998482i	$-0.517540\pi$
$401$	−2.28708e13	−0.110151	−0.0550755	+	0.998482i	$0.517540\pi$
$402$	1.70911e14	0.811943
$403$	−5.97550e13	−0.280025
$404$	−3.80505e14	−1.75899
$405$	−2.15597e14	−0.983194
$406$	−1.12315e13	−0.0505295
$407$	7.10604e12	0.0315398
$408$	−2.14527e13	−0.0939402
$409$	−7.66525e13	−0.331168	−0.165584	−	0.986196i	$-0.552951\pi$
$409$	−7.66525e13	−0.331168	−0.165584	+	0.986196i	$0.552951\pi$
$410$	2.83637e14	1.20907
$411$	−5.34658e13	−0.224878
$412$	8.03555e13	0.333488
$413$	−1.69733e14	−0.695091
$414$	1.47505e13	0.0596082
$415$	−4.01220e14	−1.59999
$416$	3.31736e14	1.30551
$417$	−3.18532e14	−1.23710
$418$	−8.00364e12	−0.0306773
$419$	4.66857e14	1.76606	0.883032	−	0.469313i	$-0.155498\pi$
$419$	4.66857e14	1.76606	0.883032	+	0.469313i	$0.155498\pi$
$420$	1.92968e14	0.720466
$421$	1.22687e14	0.452112	0.226056	−	0.974114i	$-0.427417\pi$
$421$	1.22687e14	0.452112	0.226056	+	0.974114i	$0.427417\pi$
$422$	3.73583e14	1.35884
$423$	8.23388e12	0.0295619
$424$	3.63859e13	0.128950
$425$	8.48676e13	0.296895
$426$	−1.61207e14	−0.556714
$427$	2.94442e14	1.00380
$428$	1.94538e13	0.0654732
$429$	1.14421e13	0.0380181
$430$	3.21601e14	1.05497
$431$	6.86465e13	0.222328	0.111164	−	0.993802i	$-0.464542\pi$
$431$	6.86465e13	0.222328	0.111164	+	0.993802i	$0.464542\pi$
$432$	−2.88027e14	−0.921028
$433$	−1.29355e14	−0.408413	−0.204207	−	0.978928i	$-0.565461\pi$
$433$	−1.29355e14	−0.408413	−0.204207	+	0.978928i	$0.565461\pi$
$434$	−1.09799e14	−0.342299
$435$	−1.32523e13	−0.0407945
$436$	3.87454e14	1.17773
$437$	−4.15357e13	−0.124673
$438$	−3.19577e14	−0.947256
$439$	−7.52221e13	−0.220186	−0.110093	−	0.993921i	$-0.535115\pi$
$439$	−7.52221e13	−0.220186	−0.110093	+	0.993921i	$0.535115\pi$
$440$	−8.85908e11	−0.00256094
$441$	−2.29467e13	−0.0655102
$442$	4.83329e14	1.36276
$443$	3.57633e14	0.995904	0.497952	−	0.867205i	$-0.334086\pi$
$443$	3.57633e14	0.995904	0.497952	+	0.867205i	$0.334086\pi$
$444$	−3.21654e14	−0.884672
$445$	−3.09186e14	−0.839925
$446$	−8.72661e14	−2.34155
$447$	−7.58987e14	−2.01161
$448$	2.76901e14	0.724928
$449$	2.80977e14	0.726634	0.363317	−	0.931666i	$-0.381644\pi$
$449$	2.80977e14	0.726634	0.363317	+	0.931666i	$0.381644\pi$
$450$	−3.19387e13	−0.0815923
$451$	1.49817e13	0.0378086
$452$	−4.78314e14	−1.19248
$453$	−7.68402e14	−1.89255
$454$	9.66272e13	0.235121
$455$	2.72815e14	0.655851
$456$	−2.27338e13	−0.0539964
$457$	2.77119e14	0.650319	0.325159	−	0.945659i	$-0.394582\pi$
$457$	2.77119e14	0.650319	0.325159	+	0.945659i	$0.394582\pi$
$458$	1.12899e13	0.0261776
$459$	−3.96172e14	−0.907640
$460$	7.32654e13	0.165856
$461$	−5.25735e14	−1.17601	−0.588006	−	0.808857i	$-0.700087\pi$
$461$	−5.25735e14	−1.17601	−0.588006	+	0.808857i	$0.700087\pi$
$462$	2.10247e13	0.0464728
$463$	1.86463e13	0.0407284	0.0203642	−	0.999793i	$-0.493517\pi$
$463$	1.86463e13	0.0407284	0.0203642	+	0.999793i	$0.493517\pi$
$464$	−2.14970e13	−0.0464013
$465$	−1.29555e14	−0.276352
$466$	1.12574e15	2.37310
$467$	−2.40505e14	−0.501051	−0.250525	−	0.968110i	$-0.580603\pi$
$467$	−2.40505e14	−0.501051	−0.250525	+	0.968110i	$0.580603\pi$
$468$	−8.81803e13	−0.181560
$469$	2.15244e14	0.438006
$470$	8.43611e13	0.169670
$471$	2.80294e14	0.557183
$472$	−3.52721e13	−0.0693028
$473$	1.69870e13	0.0329898
$474$	−9.00769e14	−1.72915
$475$	8.99355e13	0.170654
$476$	4.30547e14	0.807574
$477$	−1.73474e14	−0.321651
$478$	1.30566e14	0.239318
$479$	−6.85644e14	−1.24238	−0.621189	−	0.783661i	$-0.713350\pi$
$479$	−6.85644e14	−1.24238	−0.621189	+	0.783661i	$0.713350\pi$
$480$	7.19238e14	1.28838
$481$	−4.54750e14	−0.805331
$482$	2.33110e14	0.408134
$483$	1.09110e14	0.188866
$484$	−5.49071e14	−0.939678
$485$	2.97546e14	0.503472
$486$	3.36679e14	0.563269
$487$	1.06893e15	1.76824	0.884121	−	0.467258i	$-0.154758\pi$
$487$	1.06893e15	1.76824	0.884121	+	0.467258i	$0.154758\pi$
$488$	6.11877e13	0.100082
$489$	1.15982e15	1.87583
$490$	−2.35103e14	−0.375993
$491$	−8.10643e12	−0.0128198	−0.00640990	−	0.999979i	$-0.502040\pi$
$491$	−8.10643e12	−0.0128198	−0.00640990	+	0.999979i	$0.502040\pi$
$492$	−6.78144e14	−1.06051
$493$	−2.95685e13	−0.0457268
$494$	5.12191e14	0.783310
$495$	4.22368e12	0.00638796
$496$	−2.10155e14	−0.314333
$497$	−2.03024e14	−0.300322
$498$	1.97874e15	2.89486
$499$	1.05785e15	1.53063	0.765316	−	0.643655i	$-0.222583\pi$
$499$	1.05785e15	1.53063	0.765316	+	0.643655i	$0.222583\pi$
$500$	−7.14453e14	−1.02244
$501$	1.35401e15	1.91652
$502$	−1.05349e15	−1.47489
$503$	−1.18234e15	−1.63727	−0.818633	−	0.574318i	$-0.805267\pi$
$503$	−1.18234e15	−1.63727	−0.818633	+	0.574318i	$0.805267\pi$
$504$	1.01676e13	0.0139268
$505$	−1.16633e15	−1.58023
$506$	7.98260e12	0.0106983
$507$	9.58462e13	0.127067
$508$	−3.11008e14	−0.407870
$509$	1.07688e15	1.39708	0.698540	−	0.715571i	$-0.253834\pi$
$509$	1.07688e15	1.39708	0.698540	+	0.715571i	$0.253834\pi$
$510$	1.04791e15	1.34489
$511$	−4.02473e14	−0.511001
$512$	1.09757e15	1.37863
$513$	−4.19829e14	−0.521708
$514$	8.00186e14	0.983771
$515$	2.46308e14	0.299598
$516$	−7.68913e14	−0.925346
$517$	4.45595e12	0.00530570
$518$	−8.35598e14	−0.984427
$519$	−1.18133e15	−1.37706
$520$	5.66935e13	0.0653905
$521$	6.94779e14	0.792938	0.396469	−	0.918048i	$-0.370235\pi$
$521$	6.94779e14	0.792938	0.396469	+	0.918048i	$0.370235\pi$
$522$	1.11277e13	0.0125666
$523$	−8.98063e14	−1.00357	−0.501785	−	0.864992i	$-0.667323\pi$
$523$	−8.98063e14	−1.00357	−0.501785	+	0.864992i	$0.667323\pi$
$524$	−1.31360e12	−0.00145258
$525$	−2.36251e14	−0.258522
$526$	1.34252e15	1.45378
$527$	−2.89061e14	−0.309764
$528$	4.02412e13	0.0426760
$529$	4.14265e13	0.0434783
$530$	−1.77735e15	−1.84610
$531$	1.68164e14	0.172868
$532$	4.56257e14	0.464190
$533$	−9.58750e14	−0.965397
$534$	1.52485e15	1.51967
$535$	5.96304e13	0.0588196
$536$	4.47296e13	0.0436706
$537$	−3.94063e14	−0.380809
$538$	−1.22644e15	−1.17313
$539$	−1.24182e13	−0.0117576
$540$	7.40543e14	0.694041
$541$	9.36023e14	0.868364	0.434182	−	0.900825i	$-0.357038\pi$
$541$	9.36023e14	0.868364	0.434182	+	0.900825i	$0.357038\pi$
$542$	−2.18718e15	−2.00857
$543$	7.05398e14	0.641261
$544$	1.60475e15	1.44416
$545$	1.18764e15	1.05804
$546$	−1.34547e15	−1.18663
$547$	−7.75235e14	−0.676867	−0.338433	−	0.940990i	$-0.609897\pi$
$547$	−7.75235e14	−0.676867	−0.338433	+	0.940990i	$0.609897\pi$
$548$	2.22987e14	0.192746
$549$	−2.91720e14	−0.249642
$550$	−1.72844e13	−0.0146440
$551$	−3.13342e13	−0.0262836
$552$	2.26740e13	0.0188306
$553$	−1.13442e15	−0.932796
$554$	−3.20718e15	−2.61108
$555$	−9.85943e14	−0.794768
$556$	1.32848e15	1.06034
$557$	8.76481e14	0.692690	0.346345	−	0.938107i	$-0.387423\pi$
$557$	8.76481e14	0.692690	0.346345	+	0.938107i	$0.387423\pi$
$558$	1.08784e14	0.0851291
$559$	−1.08708e15	−0.842357
$560$	9.59475e14	0.736206
$561$	5.53504e13	0.0420557
$562$	−2.20101e15	−1.65605
$563$	9.21062e14	0.686266	0.343133	−	0.939287i	$-0.388512\pi$
$563$	9.21062e14	0.686266	0.343133	+	0.939287i	$0.388512\pi$
$564$	−2.01698e14	−0.148822
$565$	−1.46614e15	−1.07130
$566$	−2.49456e15	−1.80511
$567$	1.33909e15	0.959628
$568$	−4.21902e13	−0.0299430
$569$	−2.23293e15	−1.56949	−0.784744	−	0.619821i	$-0.787205\pi$
$569$	−2.23293e15	−1.56949	−0.784744	+	0.619821i	$0.787205\pi$
$570$	1.11048e15	0.773036
$571$	1.93035e15	1.33088	0.665438	−	0.746453i	$-0.268245\pi$
$571$	1.93035e15	1.33088	0.665438	+	0.746453i	$0.268245\pi$
$572$	−4.77208e13	−0.0325859
$573$	3.59268e14	0.242978
$574$	−1.76169e15	−1.18009
$575$	−8.96991e13	−0.0595134
$576$	−2.74341e14	−0.180288
$577$	−1.65643e15	−1.07822	−0.539110	−	0.842236i	$-0.681239\pi$
$577$	−1.65643e15	−1.07822	−0.539110	+	0.842236i	$0.681239\pi$
$578$	1.77293e14	0.114311
$579$	6.81256e14	0.435089
$580$	5.52708e13	0.0349657
$581$	2.49201e15	1.56164
$582$	−1.46744e15	−0.910929
$583$	−9.38797e13	−0.0577291
$584$	−8.36376e13	−0.0509484
$585$	−2.70293e14	−0.163109
$586$	2.11780e15	1.26604
$587$	2.72188e15	1.61198	0.805989	−	0.591930i	$-0.201634\pi$
$587$	2.72188e15	1.61198	0.805989	+	0.591930i	$0.201634\pi$
$588$	5.62106e14	0.329794
$589$	−3.06323e14	−0.178051
$590$	1.72295e15	0.992168
$591$	3.04661e15	1.73814
$592$	−1.59933e15	−0.904000
$593$	4.98659e14	0.279256	0.139628	−	0.990204i	$-0.455409\pi$
$593$	4.98659e14	0.279256	0.139628	+	0.990204i	$0.455409\pi$
$594$	8.06856e13	0.0447683
$595$	1.31973e15	0.725505
$596$	3.16546e15	1.72418
$597$	1.12934e15	0.609488
$598$	−5.10845e14	−0.273170
$599$	1.48148e15	0.784961	0.392481	−	0.919760i	$-0.371617\pi$
$599$	1.48148e15	0.784961	0.392481	+	0.919760i	$0.371617\pi$
$600$	−4.90951e13	−0.0257755
$601$	5.77634e14	0.300499	0.150249	−	0.988648i	$-0.451992\pi$
$601$	5.77634e14	0.300499	0.150249	+	0.988648i	$0.451992\pi$
$602$	−1.99750e15	−1.02969
$603$	−2.13254e14	−0.108931
$604$	3.20473e15	1.62214
$605$	−1.68303e15	−0.844184
$606$	5.75214e15	2.85910
$607$	−2.32972e15	−1.14754	−0.573768	−	0.819018i	$-0.694519\pi$
$607$	−2.32972e15	−1.14754	−0.573768	+	0.819018i	$0.694519\pi$
$608$	1.70058e15	0.830096
$609$	8.23115e13	0.0398167
$610$	−2.98885e15	−1.43281
$611$	−2.85158e14	−0.135475
$612$	−4.26567e14	−0.200842
$613$	5.63875e14	0.263118	0.131559	−	0.991308i	$-0.458002\pi$
$613$	5.63875e14	0.263118	0.131559	+	0.991308i	$0.458002\pi$
$614$	−1.28301e15	−0.593340
$615$	−2.07867e15	−0.952735
$616$	5.50245e12	0.00249955
$617$	−8.95548e14	−0.403200	−0.201600	−	0.979468i	$-0.564614\pi$
$617$	−8.95548e14	−0.403200	−0.201600	+	0.979468i	$0.564614\pi$
$618$	−1.21474e15	−0.542061
$619$	2.34383e15	1.03664	0.518319	−	0.855187i	$-0.326558\pi$
$619$	2.34383e15	1.03664	0.518319	+	0.855187i	$0.326558\pi$
$620$	5.40327e14	0.236866
$621$	4.18726e14	0.181939
$622$	−1.63654e15	−0.704824
$623$	1.92039e15	0.819793
$624$	−2.57522e15	−1.08968
$625$	−1.50948e15	−0.633121
$626$	4.89508e15	2.03517
$627$	5.86557e13	0.0241734
$628$	−1.16900e15	−0.477571
$629$	−2.19982e15	−0.890860
$630$	−4.96661e14	−0.199382
$631$	−3.87543e15	−1.54226	−0.771132	−	0.636676i	$-0.780309\pi$
$631$	−3.87543e15	−1.54226	−0.771132	+	0.636676i	$0.780309\pi$
$632$	−2.35744e14	−0.0930028
$633$	−2.73785e15	−1.07075
$634$	−1.49198e15	−0.578457
$635$	−9.53313e14	−0.366421
$636$	4.24945e15	1.61927
$637$	7.94697e14	0.300217
$638$	6.02200e12	0.00225542
$639$	2.01147e14	0.0746894
$640$	3.77127e14	0.138835
$641$	2.94714e15	1.07567	0.537837	−	0.843049i	$-0.319241\pi$
$641$	2.94714e15	1.07567	0.537837	+	0.843049i	$0.319241\pi$
$642$	−2.94086e14	−0.106422
$643$	−1.19917e15	−0.430248	−0.215124	−	0.976587i	$-0.569016\pi$
$643$	−1.19917e15	−0.430248	−0.215124	+	0.976587i	$0.569016\pi$
$644$	−4.55058e14	−0.161881
$645$	−2.35690e15	−0.831308
$646$	2.47769e15	0.866500
$647$	7.96020e14	0.276026	0.138013	−	0.990430i	$-0.455928\pi$
$647$	7.96020e14	0.276026	0.138013	+	0.990430i	$0.455928\pi$
$648$	2.78275e14	0.0956780
$649$	9.10060e13	0.0310259
$650$	1.10611e15	0.373917
$651$	8.04677e14	0.269728
$652$	−4.83721e15	−1.60780
$653$	−6.62244e14	−0.218271	−0.109135	−	0.994027i	$-0.534808\pi$
$653$	−6.62244e14	−0.218271	−0.109135	+	0.994027i	$0.534808\pi$
$654$	−5.85720e15	−1.91431
$655$	−4.02648e12	−0.00130496
$656$	−3.37187e15	−1.08368
$657$	3.98753e14	0.127085
$658$	−5.23974e14	−0.165603
$659$	−3.15279e15	−0.988154	−0.494077	−	0.869418i	$-0.664494\pi$
$659$	−3.15279e15	−0.988154	−0.494077	+	0.869418i	$0.664494\pi$
$660$	−1.03464e14	−0.0321585
$661$	5.86059e15	1.80648	0.903240	−	0.429136i	$-0.141182\pi$
$661$	5.86059e15	1.80648	0.903240	+	0.429136i	$0.141182\pi$
$662$	4.30416e14	0.131574
$663$	−3.54214e15	−1.07384
$664$	5.17863e14	0.155701
$665$	1.39853e15	0.417017
$666$	8.27874e14	0.244825
$667$	3.12518e13	0.00916607
$668$	−5.64707e15	−1.64268
$669$	6.39541e15	1.84512
$670$	−2.18492e15	−0.625207
$671$	−1.57871e14	−0.0448052
$672$	−4.46725e15	−1.25750
$673$	−9.60039e13	−0.0268044	−0.0134022	−	0.999910i	$-0.504266\pi$
$673$	−9.60039e13	−0.0268044	−0.0134022	+	0.999910i	$0.504266\pi$
$674$	7.42847e15	2.05717
$675$	−9.06650e14	−0.249040
$676$	−3.99740e14	−0.108911
$677$	−8.02272e14	−0.216812	−0.108406	−	0.994107i	$-0.534575\pi$
$677$	−8.02272e14	−0.216812	−0.108406	+	0.994107i	$0.534575\pi$
$678$	7.23073e15	1.93829
$679$	−1.84809e15	−0.491404
$680$	2.74251e14	0.0723352
$681$	−7.08145e14	−0.185273
$682$	5.88711e13	0.0152788
$683$	1.74903e14	0.0450280	0.0225140	−	0.999747i	$-0.492833\pi$
$683$	1.74903e14	0.0450280	0.0225140	+	0.999747i	$0.492833\pi$
$684$	−4.52040e14	−0.115443
$685$	6.83507e14	0.173159
$686$	6.03408e15	1.51645
$687$	−8.27394e13	−0.0206277
$688$	−3.82320e15	−0.945562
$689$	6.00781e15	1.47404
$690$	−1.10756e15	−0.269587
$691$	4.17910e15	1.00915	0.504573	−	0.863369i	$-0.331650\pi$
$691$	4.17910e15	1.00915	0.504573	+	0.863369i	$0.331650\pi$
$692$	4.92692e15	1.18030
$693$	−2.62336e13	−0.00623484
$694$	5.98562e15	1.41134
$695$	4.07211e15	0.952584
$696$	1.71051e13	0.00396985
$697$	−4.63790e15	−1.06793
$698$	−9.86807e15	−2.25438
$699$	−8.25013e15	−1.86998
$700$	9.85319e14	0.221584
$701$	3.79984e15	0.847844	0.423922	−	0.905699i	$-0.360653\pi$
$701$	3.79984e15	0.847844	0.423922	+	0.905699i	$0.360653\pi$
$702$	−5.16346e15	−1.14311
$703$	−2.33119e15	−0.512063
$704$	−1.48466e14	−0.0323577
$705$	−6.18251e14	−0.133698
$706$	−1.09175e16	−2.34260
$707$	7.24421e15	1.54235
$708$	−4.11937e15	−0.870258
$709$	−1.69921e15	−0.356198	−0.178099	−	0.984013i	$-0.556995\pi$
$709$	−1.69921e15	−0.356198	−0.178099	+	0.984013i	$0.556995\pi$
$710$	2.06087e15	0.428677
$711$	1.12394e15	0.231985
$712$	3.99074e14	0.0817360
$713$	3.05518e14	0.0620932
$714$	−6.50864e15	−1.31265
$715$	−1.46275e14	−0.0292744
$716$	1.64350e15	0.326398
$717$	−9.56866e14	−0.188580
$718$	8.47400e15	1.65731
$719$	−3.16482e15	−0.614242	−0.307121	−	0.951670i	$-0.599366\pi$
$719$	−3.16482e15	−0.614242	−0.307121	+	0.951670i	$0.599366\pi$
$720$	−9.50606e14	−0.183093
$721$	−1.52984e15	−0.292417
$722$	−4.71885e15	−0.895121
$723$	−1.70838e15	−0.321605
$724$	−2.94196e15	−0.549636
$725$	−6.76682e13	−0.0125466
$726$	8.30038e15	1.52738
$727$	−3.18866e15	−0.582331	−0.291165	−	0.956673i	$-0.594043\pi$
$727$	−3.18866e15	−0.582331	−0.291165	+	0.956673i	$0.594043\pi$
$728$	−3.52128e14	−0.0638232
$729$	3.99829e15	0.719238
$730$	4.08546e15	0.729400
$731$	−5.25868e15	−0.931818
$732$	7.14601e15	1.25676
$733$	−1.58589e15	−0.276822	−0.138411	−	0.990375i	$-0.544199\pi$
$733$	−1.58589e15	−0.276822	−0.138411	+	0.990375i	$0.544199\pi$
$734$	3.98164e15	0.689817
$735$	1.72298e15	0.296279
$736$	−1.69611e15	−0.289486
$737$	−1.15407e14	−0.0195507
$738$	1.74541e15	0.293486
$739$	−5.85781e14	−0.0977668	−0.0488834	−	0.998804i	$-0.515566\pi$
$739$	−5.85781e14	−0.0977668	−0.0488834	+	0.998804i	$0.515566\pi$
$740$	4.11202e15	0.681209
$741$	−3.75366e15	−0.617240
$742$	1.10393e16	1.80185
$743$	−4.98356e15	−0.807423	−0.403711	−	0.914886i	$-0.632280\pi$
$743$	−4.98356e15	−0.807423	−0.403711	+	0.914886i	$0.632280\pi$
$744$	1.67219e14	0.0268928
$745$	9.70288e15	1.54896
$746$	−5.90672e15	−0.936014
$747$	−2.46898e15	−0.388377
$748$	−2.30847e14	−0.0360467
$749$	−3.70370e14	−0.0574098
$750$	1.08005e16	1.66191
$751$	−9.70268e15	−1.48208	−0.741041	−	0.671460i	$-0.765667\pi$
$751$	−9.70268e15	−1.48208	−0.741041	+	0.671460i	$0.765667\pi$
$752$	−1.00288e15	−0.152073
$753$	7.72063e15	1.16220
$754$	−3.85377e14	−0.0575895
$755$	9.82324e15	1.45729
$756$	−4.59958e15	−0.677405
$757$	5.14522e15	0.752275	0.376137	−	0.926564i	$-0.377252\pi$
$757$	5.14522e15	0.752275	0.376137	+	0.926564i	$0.377252\pi$
$758$	1.32526e16	1.92362
$759$	−5.85015e13	−0.00843019
$760$	2.90628e14	0.0415780
$761$	1.05285e15	0.149538	0.0747691	−	0.997201i	$-0.476178\pi$
$761$	1.05285e15	0.149538	0.0747691	+	0.997201i	$0.476178\pi$
$762$	4.70156e15	0.662963
$763$	−7.37652e15	−1.03268
$764$	−1.49838e15	−0.208261
$765$	−1.30753e15	−0.180432
$766$	7.22699e15	0.990146
$767$	−5.82391e15	−0.792209
$768$	−9.00192e15	−1.21576
$769$	−3.03268e15	−0.406661	−0.203330	−	0.979110i	$-0.565177\pi$
$769$	−3.03268e15	−0.406661	−0.203330	+	0.979110i	$0.565177\pi$
$770$	−2.68780e14	−0.0357847
$771$	−5.86427e15	−0.775201
$772$	−2.84128e15	−0.372922
$773$	−2.20133e15	−0.286878	−0.143439	−	0.989659i	$-0.545816\pi$
$773$	−2.20133e15	−0.286878	−0.143439	+	0.989659i	$0.545816\pi$
$774$	1.97903e15	0.256081
$775$	−6.61524e14	−0.0849937
$776$	−3.84050e14	−0.0489946
$777$	6.12378e15	0.775719
$778$	1.10547e16	1.39047
$779$	−4.91485e15	−0.613839
$780$	6.62113e15	0.821130
$781$	1.08855e14	0.0134051
$782$	−2.47118e15	−0.302181
$783$	3.15883e14	0.0383563
$784$	2.79491e15	0.336999
$785$	−3.58327e15	−0.429039
$786$	1.98578e13	0.00236106
$787$	−1.46805e16	−1.73332	−0.866662	−	0.498896i	$-0.833739\pi$
$787$	−1.46805e16	−1.73332	−0.866662	+	0.498896i	$0.833739\pi$
$788$	−1.27063e16	−1.48979
$789$	−9.83882e15	−1.14556
$790$	1.15154e16	1.33147
$791$	9.10634e15	1.04562
$792$	−5.45159e12	−0.000621634 0
$793$	1.01029e16	1.14405
$794$	−1.55049e16	−1.74364
$795$	1.30255e16	1.45471
$796$	−4.71007e15	−0.522403
$797$	−5.96819e15	−0.657389	−0.328694	−	0.944436i	$-0.606609\pi$
$797$	−5.96819e15	−0.657389	−0.328694	+	0.944436i	$0.606609\pi$
$798$	−6.89730e15	−0.754507
$799$	−1.37943e15	−0.149863
$800$	3.67252e15	0.396250
$801$	−1.90263e15	−0.203881
$802$	−1.44197e15	−0.153460
$803$	2.15794e14	0.0228089
$804$	5.22389e15	0.548386
$805$	−1.39486e15	−0.145430
$806$	−3.76744e15	−0.390125
$807$	8.98815e15	0.924413
$808$	1.50541e15	0.153778
$809$	−1.92359e16	−1.95162	−0.975811	−	0.218616i	$-0.929846\pi$
$809$	−1.92359e16	−1.95162	−0.975811	+	0.218616i	$0.929846\pi$
$810$	−1.35930e16	−1.36977
$811$	−6.78672e15	−0.679274	−0.339637	−	0.940557i	$-0.610304\pi$
$811$	−6.78672e15	−0.679274	−0.339637	+	0.940557i	$0.610304\pi$
$812$	−3.43292e14	−0.0341276
$813$	1.60290e16	1.58274
$814$	4.48023e14	0.0439406
$815$	−1.48272e16	−1.44441
$816$	−1.24575e16	−1.20541
$817$	−5.57270e15	−0.535605
$818$	−4.83280e15	−0.461377
$819$	1.67882e15	0.159199
$820$	8.66938e15	0.816605
$821$	−2.43919e15	−0.228222	−0.114111	−	0.993468i	$-0.536402\pi$
$821$	−2.43919e15	−0.228222	−0.114111	+	0.993468i	$0.536402\pi$
$822$	−3.37092e15	−0.313295
$823$	−1.56725e16	−1.44690	−0.723452	−	0.690375i	$-0.757445\pi$
$823$	−1.56725e16	−1.44690	−0.723452	+	0.690375i	$0.757445\pi$
$824$	−3.17915e14	−0.0291549
$825$	1.26671e14	0.0115393
$826$	−1.07014e16	−0.968387
$827$	−2.07202e16	−1.86257	−0.931287	−	0.364286i	$-0.881313\pi$
$827$	−2.07202e16	−1.86257	−0.931287	+	0.364286i	$0.881313\pi$
$828$	4.50852e14	0.0402594
$829$	2.05111e16	1.81944	0.909721	−	0.415219i	$-0.136295\pi$
$829$	2.05111e16	1.81944	0.909721	+	0.415219i	$0.136295\pi$
$830$	−2.52962e16	−2.22908
$831$	2.35042e16	2.05750
$832$	9.50105e15	0.826215
$833$	3.84430e15	0.332101
$834$	−2.00829e16	−1.72351
$835$	−1.73096e16	−1.47574
$836$	−2.44632e14	−0.0207195
$837$	3.08807e15	0.259835
$838$	2.94345e16	2.46045
$839$	7.83218e15	0.650417	0.325208	−	0.945642i	$-0.394566\pi$
$839$	7.83218e15	0.650417	0.325208	+	0.945642i	$0.394566\pi$
$840$	−7.63450e14	−0.0629861
$841$	−1.21769e16	−0.998068
$842$	7.73517e15	0.629873
$843$	1.61304e16	1.30495
$844$	1.14186e16	0.917761
$845$	−1.22530e15	−0.0978429
$846$	5.19131e14	0.0411851
$847$	1.04534e16	0.823950
$848$	2.11291e16	1.65464
$849$	1.82817e16	1.42241
$850$	5.35075e15	0.413628
$851$	2.32506e15	0.178575
$852$	−4.92732e15	−0.376004
$853$	6.10283e15	0.462713	0.231356	−	0.972869i	$-0.425684\pi$
$853$	6.10283e15	0.462713	0.231356	+	0.972869i	$0.425684\pi$
$854$	1.85640e16	1.39847
$855$	−1.38561e15	−0.103711
$856$	−7.69663e13	−0.00572394
$857$	−3.99346e15	−0.295090	−0.147545	−	0.989055i	$-0.547137\pi$
$857$	−3.99346e15	−0.295090	−0.147545	+	0.989055i	$0.547137\pi$
$858$	7.21402e14	0.0529660
$859$	9.49836e15	0.692925	0.346463	−	0.938064i	$-0.387383\pi$
$859$	9.49836e15	0.692925	0.346463	+	0.938064i	$0.387383\pi$
$860$	9.82978e15	0.712528
$861$	1.29108e16	0.929899
$862$	4.32804e15	0.309743
$863$	−1.43014e16	−1.01700	−0.508498	−	0.861063i	$-0.669799\pi$
$863$	−1.43014e16	−1.01700	−0.508498	+	0.861063i	$0.669799\pi$
$864$	−1.71438e16	−1.21138
$865$	1.51022e16	1.06035
$866$	−8.15559e15	−0.568993
$867$	−1.29932e15	−0.0900762
$868$	−3.35602e15	−0.231189
$869$	6.08244e14	0.0416360
$870$	−8.35536e14	−0.0568341
$871$	7.38547e15	0.499204
$872$	−1.53291e15	−0.102962
$873$	1.83100e15	0.122211
$874$	−2.61875e15	−0.173692
$875$	1.36021e16	0.896521
$876$	−9.76789e15	−0.639776
$877$	1.45300e16	0.945734	0.472867	−	0.881134i	$-0.343219\pi$
$877$	1.45300e16	0.945734	0.472867	+	0.881134i	$0.343219\pi$
$878$	−4.74262e15	−0.306760
$879$	−1.55205e16	−0.997626
$880$	−5.14443e14	−0.0328611
$881$	1.27659e15	0.0810374	0.0405187	−	0.999179i	$-0.487099\pi$
$881$	1.27659e15	0.0810374	0.0405187	+	0.999179i	$0.487099\pi$
$882$	−1.44675e15	−0.0912675
$883$	1.51964e16	0.952703	0.476351	−	0.879255i	$-0.341959\pi$
$883$	1.51964e16	0.952703	0.476351	+	0.879255i	$0.341959\pi$
$884$	1.47730e16	0.920409
$885$	−1.26268e16	−0.781819
$886$	2.25481e16	1.38747
$887$	−1.09944e16	−0.672344	−0.336172	−	0.941801i	$-0.609132\pi$
$887$	−1.09944e16	−0.672344	−0.336172	+	0.941801i	$0.609132\pi$
$888$	1.27258e15	0.0773416
$889$	5.92111e15	0.357638
$890$	−1.94936e16	−1.17017
$891$	−7.17981e14	−0.0428337
$892$	−2.66730e16	−1.58148
$893$	−1.46181e15	−0.0861404
$894$	−4.78527e16	−2.80253
$895$	5.03770e15	0.293228
$896$	−2.34237e15	−0.135507
$897$	3.74379e15	0.215255
$898$	1.77151e16	1.01233
$899$	2.30480e14	0.0130905
$900$	−9.76211e14	−0.0551074
$901$	2.90624e16	1.63059
$902$	9.44569e14	0.0526742
$903$	1.46389e16	0.811383
$904$	1.89238e15	0.104252
$905$	−9.01780e15	−0.493780
$906$	−4.84464e16	−2.63667
$907$	6.77580e15	0.366539	0.183270	−	0.983063i	$-0.441332\pi$
$907$	6.77580e15	0.366539	0.183270	+	0.983063i	$0.441332\pi$
$908$	2.95342e15	0.158801
$909$	−7.17725e15	−0.383581
$910$	1.72005e16	0.913720
$911$	5.46049e15	0.288324	0.144162	−	0.989554i	$-0.453951\pi$
$911$	5.46049e15	0.288324	0.144162	+	0.989554i	$0.453951\pi$
$912$	−1.32014e16	−0.692865
$913$	−1.33614e15	−0.0697051
$914$	1.74718e16	0.906012
$915$	2.19042e16	1.12904
$916$	3.45077e14	0.0176803
$917$	2.50088e13	0.00127368
$918$	−2.49779e16	−1.26451
$919$	−1.11516e16	−0.561180	−0.280590	−	0.959828i	$-0.590530\pi$
$919$	−1.11516e16	−0.561180	−0.280590	+	0.959828i	$0.590530\pi$
$920$	−2.89865e14	−0.0144998
$921$	9.40270e15	0.467546
$922$	−3.31466e16	−1.63840
$923$	−6.96618e15	−0.342283
$924$	6.42622e14	0.0313877
$925$	−5.03436e15	−0.244436
$926$	1.17561e15	0.0567420
$927$	1.51570e15	0.0727235
$928$	−1.27953e15	−0.0610292
$929$	3.13210e16	1.48508	0.742540	−	0.669802i	$-0.233621\pi$
$929$	3.13210e16	1.48508	0.742540	+	0.669802i	$0.233621\pi$
$930$	−8.16820e15	−0.385008
$931$	4.07387e15	0.190890
$932$	3.44084e16	1.60279
$933$	1.19936e16	0.555394
$934$	−1.51634e16	−0.698054
$935$	−7.07599e14	−0.0323835
$936$	3.48873e14	0.0158727
$937$	−1.48520e16	−0.671764	−0.335882	−	0.941904i	$-0.609034\pi$
$937$	−1.48520e16	−0.671764	−0.335882	+	0.941904i	$0.609034\pi$
$938$	1.35707e16	0.610221
$939$	−3.58742e16	−1.60369
$940$	2.57850e15	0.114595
$941$	−1.91985e16	−0.848252	−0.424126	−	0.905603i	$-0.639419\pi$
$941$	−1.91985e16	−0.848252	−0.424126	+	0.905603i	$0.639419\pi$
$942$	1.76720e16	0.776257
$943$	4.90194e15	0.214069
$944$	−2.04824e16	−0.889271
$945$	−1.40988e16	−0.608565
$946$	1.07100e15	0.0459608
$947$	5.67931e14	0.0242310	0.0121155	−	0.999927i	$-0.496143\pi$
$947$	5.67931e14	0.0242310	0.0121155	+	0.999927i	$0.496143\pi$
$948$	−2.75321e16	−1.16787
$949$	−1.38097e16	−0.582398
$950$	5.67027e15	0.237752
$951$	1.09341e16	0.455818
$952$	−1.70340e15	−0.0706014
$953$	−4.93265e15	−0.203268	−0.101634	−	0.994822i	$-0.532407\pi$
$953$	−4.93265e15	−0.203268	−0.101634	+	0.994822i	$0.532407\pi$
$954$	−1.09372e16	−0.448118
$955$	−4.59287e15	−0.187097
$956$	3.99075e15	0.161635
$957$	−4.41330e13	−0.00177725
$958$	−4.32286e16	−1.73086
$959$	−4.24532e15	−0.169008
$960$	2.05992e16	0.815379
$961$	−2.31553e16	−0.911322
$962$	−2.86711e16	−1.12197
$963$	3.66946e14	0.0142777
$964$	7.12504e15	0.275653
$965$	−8.70917e15	−0.335024
$966$	6.87918e15	0.263125
$967$	−3.45711e16	−1.31482	−0.657412	−	0.753531i	$-0.728349\pi$
$967$	−3.45711e16	−1.31482	−0.657412	+	0.753531i	$0.728349\pi$
$968$	2.17232e15	0.0821505
$969$	−1.81581e16	−0.682793
$970$	1.87598e16	0.701427
$971$	2.36533e16	0.879398	0.439699	−	0.898145i	$-0.355085\pi$
$971$	2.36533e16	0.879398	0.439699	+	0.898145i	$0.355085\pi$
$972$	1.02906e16	0.380432
$973$	−2.52923e16	−0.929752
$974$	6.73944e16	2.46348
$975$	−8.10627e15	−0.294643
$976$	3.55314e16	1.28422
$977$	−1.29933e16	−0.466981	−0.233491	−	0.972359i	$-0.575015\pi$
$977$	−1.29933e16	−0.466981	−0.233491	+	0.972359i	$0.575015\pi$
$978$	7.31247e16	2.61337
$979$	−1.02965e15	−0.0365920
$980$	−7.18595e15	−0.253946
$981$	7.30834e15	0.256826
$982$	−5.11096e14	−0.0178603
$983$	3.29275e16	1.14423	0.572116	−	0.820173i	$-0.306123\pi$
$983$	3.29275e16	1.14423	0.572116	+	0.820173i	$0.306123\pi$
$984$	2.68298e15	0.0927139
$985$	−3.89478e16	−1.33839
$986$	−1.86424e15	−0.0637057
$987$	3.84001e15	0.130493
$988$	1.56552e16	0.529047
$989$	5.55806e15	0.186786
$990$	2.66295e14	0.00889958
$991$	4.04527e16	1.34444	0.672221	−	0.740350i	$-0.265340\pi$
$991$	4.04527e16	1.34444	0.672221	+	0.740350i	$0.265340\pi$
$992$	−1.25087e16	−0.413427
$993$	−3.15436e15	−0.103679
$994$	−1.28003e16	−0.418402
$995$	−1.44375e16	−0.469314
$996$	6.04804e16	1.95519
$997$	−6.00152e16	−1.92947	−0.964735	−	0.263223i	$-0.915214\pi$
$997$	−6.00152e16	−1.92947	−0.964735	+	0.263223i	$0.915214\pi$
$998$	6.66954e16	2.13245
$999$	2.35010e16	0.747267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.12.a.a.1.7	✓	8
3.2	odd	2		207.12.a.a.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.12.a.a.1.7	✓	8	1.1	even	1	trivial
207.12.a.a.1.2		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)

\(f(q)\)	\(=\)	\(q+63.0482 q^{2} -462.057 q^{3} +1927.07 q^{4} +5906.93 q^{5} -29131.8 q^{6} -36688.5 q^{7} -7624.20 q^{8} +36349.3 q^{9} +O(q^{10})\)	\(q+63.0482 q^{2} -462.057 q^{3} +1927.07 q^{4} +5906.93 q^{5} -29131.8 q^{6} -36688.5 q^{7} -7624.20 q^{8} +36349.3 q^{9} +372421. q^{10} +19671.3 q^{11} -890417. q^{12} -1.25886e6 q^{13} -2.31314e6 q^{14} -2.72933e6 q^{15} -4.42734e6 q^{16} -6.08966e6 q^{17} +2.29176e6 q^{18} -6.45330e6 q^{19} +1.13831e7 q^{20} +1.69521e7 q^{21} +1.24024e6 q^{22} +6.43634e6 q^{23} +3.52281e6 q^{24} -1.39363e7 q^{25} -7.93688e7 q^{26} +6.50565e7 q^{27} -7.07014e7 q^{28} +4.85552e6 q^{29} -1.72080e8 q^{30} +4.74676e7 q^{31} -2.63521e8 q^{32} -9.08924e6 q^{33} -3.83942e8 q^{34} -2.16716e8 q^{35} +7.00478e7 q^{36} +3.61239e8 q^{37} -4.06869e8 q^{38} +5.81664e8 q^{39} -4.50356e7 q^{40} +7.61603e8 q^{41} +1.06880e9 q^{42} +8.63543e8 q^{43} +3.79080e7 q^{44} +2.14713e8 q^{45} +4.05800e8 q^{46} +2.26521e8 q^{47} +2.04568e9 q^{48} -6.31284e8 q^{49} -8.78661e8 q^{50} +2.81377e9 q^{51} -2.42591e9 q^{52} -4.77243e9 q^{53} +4.10169e9 q^{54} +1.16197e8 q^{55} +2.79720e8 q^{56} +2.98179e9 q^{57} +3.06132e8 q^{58} +4.62634e9 q^{59} -5.25963e9 q^{60} -8.02546e9 q^{61} +2.99274e9 q^{62} -1.33360e9 q^{63} -7.54735e9 q^{64} -7.43599e9 q^{65} -5.73060e8 q^{66} -5.86680e9 q^{67} -1.17352e10 q^{68} -2.97395e9 q^{69} -1.36636e10 q^{70} +5.53372e9 q^{71} -2.77135e8 q^{72} +1.09700e10 q^{73} +2.27755e10 q^{74} +6.43938e9 q^{75} -1.24360e10 q^{76} -7.21709e8 q^{77} +3.66729e10 q^{78} +3.09204e10 q^{79} -2.61520e10 q^{80} -3.64990e10 q^{81} +4.80177e10 q^{82} -6.79236e10 q^{83} +3.26680e10 q^{84} -3.59712e10 q^{85} +5.44448e10 q^{86} -2.24353e9 q^{87} -1.49978e8 q^{88} -5.23430e10 q^{89} +1.35373e10 q^{90} +4.61856e10 q^{91} +1.24033e10 q^{92} -2.19327e10 q^{93} +1.42817e10 q^{94} -3.81192e10 q^{95} +1.21762e11 q^{96} +5.03724e10 q^{97} -3.98013e10 q^{98} +7.15038e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9}+O(q^{10}) \) 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

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Database

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