LMFDB - Embedded newform 23.15.b.b.22.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,15,Mod(22,23)]

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([1]))

N = Newforms(chi, 15, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	23.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.5956626749$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.3
Character	$\chi$	$=$	23.22
Dual form			23.15.b.b.22.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-181.785 q^{2} +3188.39 q^{3} +16661.8 q^{4} -67250.0i q^{5} -579603. q^{6} +311198. i q^{7} -50505.8 q^{8} +5.38289e6 q^{9} +O(q^{10})$	\(q-181.785 q^{2} +3188.39 q^{3} +16661.8 q^{4} -67250.0i q^{5} -579603. q^{6} +311198. i q^{7} -50505.8 q^{8} +5.38289e6 q^{9} +1.22251e7i q^{10} +3.75331e7i q^{11} +5.31245e7 q^{12} +2.15516e6 q^{13} -5.65712e7i q^{14} -2.14420e8i q^{15} -2.63806e8 q^{16} -3.10672e8i q^{17} -9.78530e8 q^{18} +4.08118e8i q^{19} -1.12051e9i q^{20} +9.92223e8i q^{21} -6.82296e9i q^{22} +(-2.02724e9 + 2.73553e9i) q^{23} -1.61032e8 q^{24} +1.58095e9 q^{25} -3.91776e8 q^{26} +1.91279e9 q^{27} +5.18513e9i q^{28} +4.76662e9 q^{29} +3.89783e10i q^{30} +3.01377e10 q^{31} +4.87835e10 q^{32} +1.19670e11i q^{33} +5.64756e10i q^{34} +2.09281e10 q^{35} +8.96889e10 q^{36} +1.00808e10i q^{37} -7.41898e10i q^{38} +6.87151e9 q^{39} +3.39651e9i q^{40} -3.94420e10 q^{41} -1.80371e11i q^{42} +3.58465e11i q^{43} +6.25371e11i q^{44} -3.62000e11i q^{45} +(3.68523e11 - 4.97279e11i) q^{46} -2.29667e11 q^{47} -8.41119e11 q^{48} +5.81379e11 q^{49} -2.87393e11 q^{50} -9.90546e11i q^{51} +3.59089e10 q^{52} +1.68482e12i q^{53} -3.47717e11 q^{54} +2.52410e12 q^{55} -1.57173e10i q^{56} +1.30124e12i q^{57} -8.66501e11 q^{58} +2.91842e12 q^{59} -3.57262e12i q^{60} +5.28265e12i q^{61} -5.47859e12 q^{62} +1.67515e12i q^{63} -4.54592e12 q^{64} -1.44935e11i q^{65} -2.17543e13i q^{66} -2.07055e12i q^{67} -5.17637e12i q^{68} +(-6.46365e12 + 8.72195e12i) q^{69} -3.80442e12 q^{70} -4.52726e12 q^{71} -2.71867e11 q^{72} +2.15813e13 q^{73} -1.83254e12i q^{74} +5.04070e12 q^{75} +6.80000e12i q^{76} -1.16802e13 q^{77} -1.24914e12 q^{78} -8.95682e12i q^{79} +1.77410e13i q^{80} -1.96475e13 q^{81} +7.16997e12 q^{82} -1.16910e13i q^{83} +1.65323e13i q^{84} -2.08927e13 q^{85} -6.51637e13i q^{86} +1.51979e13 q^{87} -1.89564e12i q^{88} -1.38713e13i q^{89} +6.58061e13i q^{90} +6.70683e11i q^{91} +(-3.37776e13 + 4.55789e13i) q^{92} +9.60909e13 q^{93} +4.17500e13 q^{94} +2.74460e13 q^{95} +1.55541e14 q^{96} +1.14504e14i q^{97} -1.05686e14 q^{98} +2.02037e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10}) $ 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9} - 61408584 q^{12} - 75796032 q^{13} + 61604880 q^{16} + 2078422368 q^{18} + 11061310696 q^{23} - 31030771920 q^{24} - 46421538840 q^{25} - 2664699128 q^{26} + 53616275568 q^{27} + 34752122960 q^{29} - 66504561216 q^{31} - 119385530304 q^{32} + 268767587760 q^{35} - 826304812272 q^{36} - 531040317600 q^{39} - 586388906608 q^{41} - 219014904864 q^{46} + 552854704544 q^{47} + 1314198459696 q^{48} - 2682585958584 q^{49} + 1292374320920 q^{50} - 614830645416 q^{52} - 756511540560 q^{54} - 4886166096240 q^{55} - 4069251029352 q^{58} + 17167207160144 q^{59} + 7479493046896 q^{62} + 18607147469856 q^{64} - 1866422538000 q^{69} + 10676769015360 q^{70} + 15000981757712 q^{71} + 5949703969824 q^{72} + 17535136179168 q^{73} - 79533751619520 q^{75} + 53823905395344 q^{77} + 6433358950512 q^{78} + 149774494087944 q^{81} - 27510435600840 q^{82} - 170228194057680 q^{85} - 286575506293872 q^{87} + 133024510658856 q^{92} + 66347882278032 q^{93} - 205878296932464 q^{94} - 211992774994560 q^{95} + 494703018436320 q^{96} - 537018388090408 q^{98}+O(q^{100}) \) 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9 - 61408584 * q^12 - 75796032 * q^13 + 61604880 * q^16 + 2078422368 * q^18 + 11061310696 * q^23 - 31030771920 * q^24 - 46421538840 * q^25 - 2664699128 * q^26 + 53616275568 * q^27 + 34752122960 * q^29 - 66504561216 * q^31 - 119385530304 * q^32 + 268767587760 * q^35 - 826304812272 * q^36 - 531040317600 * q^39 - 586388906608 * q^41 - 219014904864 * q^46 + 552854704544 * q^47 + 1314198459696 * q^48 - 2682585958584 * q^49 + 1292374320920 * q^50 - 614830645416 * q^52 - 756511540560 * q^54 - 4886166096240 * q^55 - 4069251029352 * q^58 + 17167207160144 * q^59 + 7479493046896 * q^62 + 18607147469856 * q^64 - 1866422538000 * q^69 + 10676769015360 * q^70 + 15000981757712 * q^71 + 5949703969824 * q^72 + 17535136179168 * q^73 - 79533751619520 * q^75 + 53823905395344 * q^77 + 6433358950512 * q^78 + 149774494087944 * q^81 - 27510435600840 * q^82 - 170228194057680 * q^85 - 286575506293872 * q^87 + 133024510658856 * q^92 + 66347882278032 * q^93 - 205878296932464 * q^94 - 211992774994560 * q^95 + 494703018436320 * q^96 - 537018388090408 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$	−181.785			−1.42020			−0.710098	−	0.704103i	$-0.751349\pi$
$2$	−181.785			−1.42020			−0.710098	+	0.704103i	$0.751349\pi$
$3$	3188.39			1.45789			0.728943	−	0.684575i	$-0.240012\pi$
$3$	3188.39			1.45789			0.728943	+	0.684575i	$0.240012\pi$
$4$	16661.8			1.01696
$5$		−	67250.0i		−	0.860800i	−0.902638	−	0.430400i	$-0.858372\pi$
$5$		−	67250.0i		−	0.860800i	0.902638	−	0.430400i	$-0.141628\pi$
$6$	−579603.			−2.07048
$7$			311198.i			0.377877i	0.981989	+	0.188939i	$0.0605048\pi$
$7$			311198.i			0.377877i	−0.981989	+	0.188939i	$0.939495\pi$
$8$	−50505.8			−0.0240830
$9$	5.38289e6			1.12543
$10$			1.22251e7i			1.22251i
$11$			3.75331e7i			1.92604i	0.269425	+	0.963021i	$0.413166\pi$
$11$			3.75331e7i			1.92604i	−0.269425	+	0.963021i	$0.586834\pi$
$12$	5.31245e7			1.48261
$13$	2.15516e6			0.0343460			0.0171730	−	0.999853i	$-0.494533\pi$
$13$	2.15516e6			0.0343460			0.0171730	+	0.999853i	$0.494533\pi$
$14$		−	5.65712e7i		−	0.536660i
$15$		−	2.14420e8i		−	1.25495i
$16$	−2.63806e8			−0.982755
$17$		−	3.10672e8i		−	0.757112i	−0.925578	−	0.378556i	$-0.876421\pi$
$17$		−	3.10672e8i		−	0.757112i	0.925578	−	0.378556i	$-0.123579\pi$
$18$	−9.78530e8			−1.59833
$19$			4.08118e8i			0.456574i	0.973594	+	0.228287i	$0.0733124\pi$
$19$			4.08118e8i			0.456574i	−0.973594	+	0.228287i	$0.926688\pi$
$20$		−	1.12051e9i		−	0.875397i
$21$			9.92223e8i			0.550902i
$22$		−	6.82296e9i		−	2.73536i
$23$	−2.02724e9	+	2.73553e9i	−0.595403	+	0.803427i
$23$	−2.02724e9	+	2.73553e9i	−0.595403	+	0.803427i
$24$	−1.61032e8			−0.0351103
$25$	1.58095e9			0.259023
$26$	−3.91776e8			−0.0487781
$27$	1.91279e9			0.182861
$28$			5.18513e9i			0.384285i
$29$	4.76662e9			0.276328			0.138164	−	0.990409i	$-0.455880\pi$
$29$	4.76662e9			0.276328			0.138164	+	0.990409i	$0.455880\pi$
$30$			3.89783e10i			1.78227i
$31$	3.01377e10			1.09541			0.547707	−	0.836670i	$-0.315501\pi$
$31$	3.01377e10			1.09541			0.547707	+	0.836670i	$0.315501\pi$
$32$	4.87835e10			1.41979
$33$			1.19670e11i			2.80795i
$34$			5.64756e10i			1.07525i
$35$	2.09281e10			0.325277
$36$	8.96889e10			1.14451
$37$			1.00808e10i			0.106190i	0.998589	+	0.0530950i	$0.0169086\pi$
$37$			1.00808e10i			0.106190i	−0.998589	+	0.0530950i	$0.983091\pi$
$38$		−	7.41898e10i		−	0.648424i
$39$	6.87151e9			0.0500726
$40$			3.39651e9i			0.0207307i
$41$	−3.94420e10			−0.202522			−0.101261	−	0.994860i	$-0.532288\pi$
$41$	−3.94420e10			−0.202522			−0.101261	+	0.994860i	$0.532288\pi$
$42$		−	1.80371e11i		−	0.782389i
$43$			3.58465e11i			1.31877i	0.751807	+	0.659383i	$0.229183\pi$
$43$			3.58465e11i			1.31877i	−0.751807	+	0.659383i	$0.770817\pi$
$44$			6.25371e11i			1.95870i
$45$		−	3.62000e11i		−	0.968770i
$46$	3.68523e11	−	4.97279e11i	0.845589	−	1.14102i
$47$	−2.29667e11			−0.453329			−0.226664	−	0.973973i	$-0.572782\pi$
$47$	−2.29667e11			−0.453329			−0.226664	+	0.973973i	$0.572782\pi$
$48$	−8.41119e11			−1.43274
$49$	5.81379e11			0.857209
$50$	−2.87393e11			−0.367864
$51$		−	9.90546e11i		−	1.10378i
$52$	3.59089e10			0.0349284
$53$			1.68482e12i			1.43424i	0.696950	+	0.717120i	$0.254540\pi$
$53$			1.68482e12i			1.43424i	−0.696950	+	0.717120i	$0.745460\pi$
$54$	−3.47717e11			−0.259699
$55$	2.52410e12			1.65794
$56$		−	1.57173e10i		−	0.00910043i
$57$			1.30124e12i			0.665632i
$58$	−8.66501e11			−0.392440
$59$	2.91842e12			1.17269			0.586346	−	0.810061i	$-0.300566\pi$
$59$	2.91842e12			1.17269			0.586346	+	0.810061i	$0.300566\pi$
$60$		−	3.57262e12i		−	1.27623i
$61$			5.28265e12i			1.68090i	0.541886	+	0.840452i	$0.317710\pi$
$61$			5.28265e12i			1.68090i	−0.541886	+	0.840452i	$0.682290\pi$
$62$	−5.47859e12			−1.55570
$63$			1.67515e12i			0.425274i
$64$	−4.54592e12			−1.03362
$65$		−	1.44935e11i		−	0.0295651i
$66$		−	2.17543e13i		−	3.98784i
$67$		−	2.07055e12i		−	0.341635i	−0.985303	−	0.170817i	$-0.945359\pi$
$67$		−	2.07055e12i		−	0.341635i	0.985303	−	0.170817i	$-0.0546408\pi$
$68$		−	5.17637e12i		−	0.769951i
$69$	−6.46365e12	+	8.72195e12i	−0.868029	+	1.17130i
$70$	−3.80442e12			−0.461957
$71$	−4.52726e12			−0.497768			−0.248884	−	0.968533i	$-0.580064\pi$
$71$	−4.52726e12			−0.497768			−0.248884	+	0.968533i	$0.580064\pi$
$72$	−2.71867e11			−0.0271037
$73$	2.15813e13			1.95352			0.976761	−	0.214330i	$-0.0687568\pi$
$73$	2.15813e13			1.95352			0.976761	+	0.214330i	$0.0687568\pi$
$74$		−	1.83254e12i		−	0.150811i
$75$	5.04070e12			0.377626
$76$			6.80000e12i			0.464316i
$77$	−1.16802e13			−0.727808
$78$	−1.24914e12			−0.0711129
$79$		−	8.95682e12i		−	0.466406i	−0.972428	−	0.233203i	$-0.925079\pi$
$79$		−	8.95682e12i		−	0.466406i	0.972428	−	0.233203i	$-0.0749207\pi$
$80$			1.77410e13i			0.845956i
$81$	−1.96475e13			−0.858838
$82$	7.16997e12			0.287621
$83$		−	1.16910e13i		−	0.430829i	−0.976523	−	0.215415i	$-0.930890\pi$
$83$		−	1.16910e13i		−	0.430829i	0.976523	−	0.215415i	$-0.0691103\pi$
$84$			1.65323e13i			0.560244i
$85$	−2.08927e13			−0.651722
$86$		−	6.51637e13i		−	1.87291i
$87$	1.51979e13			0.402854
$88$		−	1.89564e12i		−	0.0463849i
$89$		−	1.38713e13i		−	0.313608i	−0.987630	−	0.156804i	$-0.949881\pi$
$89$		−	1.38713e13i		−	0.313608i	0.987630	−	0.156804i	$-0.0501192\pi$
$90$			6.58061e13i			1.37584i
$91$			6.70683e11i			0.0129786i
$92$	−3.37776e13	+	4.55789e13i	−0.605500	+	0.817051i
$93$	9.60909e13			1.59699
$94$	4.17500e13			0.643816
$95$	2.74460e13			0.393019
$96$	1.55541e14			2.06989
$97$			1.14504e14i			1.41716i	0.705628	+	0.708582i	$0.250665\pi$
$97$			1.14504e14i			1.41716i	−0.705628	+	0.708582i	$0.749335\pi$
$98$	−1.05686e14			−1.21740
$99$			2.02037e14i			2.16762i
$100$	2.63416e13			0.263416
$101$	2.38748e13			0.222684			0.111342	−	0.993782i	$-0.464485\pi$
$101$	2.38748e13			0.222684			0.111342	+	0.993782i	$0.464485\pi$
$102$			1.80067e14i			1.56759i
$103$			8.10131e13i			0.658711i	0.944206	+	0.329355i	$0.106831\pi$
$103$			8.10131e13i			0.658711i	−0.944206	+	0.329355i	$0.893169\pi$
$104$	−1.08848e11			−0.000827156
$105$	6.67270e13			0.474216
$106$		−	3.06275e14i		−	2.03690i
$107$		−	2.41968e14i		−	1.50686i	−0.657530	−	0.753428i	$-0.728399\pi$
$107$		−	2.41968e14i		−	1.50686i	0.657530	−	0.753428i	$-0.271601\pi$
$108$	3.18706e13			0.185962
$109$		−	2.94959e13i		−	0.161353i	−0.996740	−	0.0806763i	$-0.974292\pi$
$109$		−	2.94959e13i		−	0.161353i	0.996740	−	0.0806763i	$-0.0257080\pi$
$110$	−4.58844e14			−2.35460
$111$			3.21416e13i			0.154813i
$112$		−	8.20961e13i		−	0.371361i
$113$		−	2.72019e14i		−	1.15625i	−0.815950	−	0.578123i	$-0.803785\pi$
$113$		−	2.72019e14i		−	1.15625i	0.815950	−	0.578123i	$-0.196215\pi$
$114$		−	2.36546e14i		−	0.945328i
$115$	1.83964e14	+	1.36332e14i	0.691590	+	0.512523i
$116$	7.94207e13			0.281014
$117$	1.16010e13			0.0386540
$118$	−5.30526e14			−1.66545
$119$	9.66807e13			0.286096
$120$			1.08294e13i			0.0302229i
$121$	−1.02899e15			−2.70964
$122$		−	9.60307e14i		−	2.38721i
$123$	−1.25757e14			−0.295254
$124$	5.02149e14			1.11399
$125$		−	5.16781e14i		−	1.08377i
$126$		−	3.04517e14i		−	0.603973i
$127$	5.70502e13			0.107061			0.0535305	−	0.998566i	$-0.482953\pi$
$127$	5.70502e13			0.107061			0.0535305	+	0.998566i	$0.482953\pi$
$128$	2.71112e13			0.0481591
$129$			1.14293e15i			1.92261i
$130$			2.63470e13i			0.0419882i
$131$	7.08256e14			1.06977			0.534886	−	0.844924i	$-0.320355\pi$
$131$	7.08256e14			1.06977			0.534886	+	0.844924i	$0.320355\pi$
$132$			1.99393e15i			2.85557i
$133$	−1.27006e14			−0.172529
$134$			3.76395e14i			0.485188i
$135$		−	1.28635e14i		−	0.157407i
$136$			1.56907e13i			0.0182335i
$137$			1.13878e15i			1.25718i	0.777739	+	0.628588i	$0.216367\pi$
$137$			1.13878e15i			1.25718i	−0.777739	+	0.628588i	$0.783633\pi$
$138$	1.17500e15	−	1.58552e15i	1.23277	−	1.66348i
$139$	1.83021e15			1.82557			0.912785	−	0.408440i	$-0.133927\pi$
$139$	1.83021e15			1.82557			0.912785	+	0.408440i	$0.133927\pi$
$140$	3.48700e14			0.330793
$141$	−7.32269e14			−0.660902
$142$	8.22988e14			0.706928
$143$			8.08900e13i			0.0661519i
$144$	−1.42004e15			−1.10602
$145$		−	3.20555e14i		−	0.237863i
$146$	−3.92317e15			−2.77439
$147$	1.85366e15			1.24971
$148$			1.67965e14i			0.107991i
$149$		−	1.26166e15i		−	0.773820i	−0.922117	−	0.386910i	$-0.873542\pi$
$149$		−	1.26166e15i		−	0.773820i	0.922117	−	0.386910i	$-0.126458\pi$
$150$	−9.16324e14			−0.536303
$151$	−1.92174e15			−1.07363			−0.536816	−	0.843699i	$-0.680373\pi$
$151$	−1.92174e15			−1.07363			−0.536816	+	0.843699i	$0.680373\pi$
$152$		−	2.06123e13i		−	0.0109957i
$153$		−	1.67232e15i		−	0.852076i
$154$	2.12330e15			1.03363
$155$		−	2.02676e15i		−	0.942933i
$156$	1.14492e14			0.0509217
$157$			2.79926e15i			1.19055i	0.803524	+	0.595273i	$0.202956\pi$
$157$			2.79926e15i			1.19055i	−0.803524	+	0.595273i	$0.797044\pi$
$158$			1.62822e15i			0.662388i
$159$			5.37186e15i			2.09096i
$160$		−	3.28069e15i		−	1.22215i
$161$	−8.51292e14	−	6.30875e14i	−0.303597	−	0.224989i
$162$	3.57162e15			1.21972
$163$	−3.41387e13			−0.0111669			−0.00558347	−	0.999984i	$-0.501777\pi$
$163$	−3.41387e13			−0.0111669			−0.00558347	+	0.999984i	$0.501777\pi$
$164$	−6.57176e14			−0.205956
$165$	8.04784e15			2.41708
$166$			2.12525e15i			0.611862i
$167$	−2.36773e15			−0.653606			−0.326803	−	0.945092i	$-0.605971\pi$
$167$	−2.36773e15			−0.653606			−0.326803	+	0.945092i	$0.605971\pi$
$168$		−	5.01130e13i		−	0.0132674i
$169$	−3.93273e15			−0.998820
$170$	3.79798e15			0.925573
$171$			2.19686e15i			0.513841i
$172$			5.97269e15i			1.34113i
$173$	−5.80481e15			−1.25160			−0.625799	−	0.779984i	$-0.715227\pi$
$173$	−5.80481e15			−1.25160			−0.625799	+	0.779984i	$0.715227\pi$
$174$	−2.76275e15			−0.572132
$175$			4.91990e14i			0.0978790i
$176$		−	9.90147e15i		−	1.89283i
$177$	9.30508e15			1.70965
$178$			2.52160e15i			0.445385i
$179$	−7.01559e14			−0.119150			−0.0595748	−	0.998224i	$-0.518974\pi$
$179$	−7.01559e14			−0.119150			−0.0595748	+	0.998224i	$0.518974\pi$
$180$		−	6.03158e15i		−	0.985198i
$181$		−	7.05657e15i		−	1.10878i	−0.832258	−	0.554388i	$-0.812952\pi$
$181$		−	7.05657e15i		−	1.10878i	0.832258	−	0.554388i	$-0.187048\pi$
$182$		−	1.21920e14i		−	0.0184321i
$183$			1.68432e16i			2.45056i
$184$	1.02387e14	−	1.38160e14i	0.0143391	−	0.0193490i
$185$	6.77935e14			0.0914083
$186$	−1.74679e16			−2.26804
$187$	1.16605e16			1.45823
$188$	−3.82667e15			−0.461016
$189$			5.95258e14i			0.0690992i
$190$	−4.98927e15			−0.558164
$191$		−	6.25751e15i		−	0.674789i	−0.941363	−	0.337395i	$-0.890454\pi$
$191$		−	6.25751e15i		−	0.674789i	0.941363	−	0.337395i	$-0.109546\pi$
$192$	−1.44942e16			−1.50690
$193$	1.25896e16			1.26215			0.631077	−	0.775720i	$-0.282613\pi$
$193$	1.25896e16			1.26215			0.631077	+	0.775720i	$0.282613\pi$
$194$		−	2.08152e16i		−	2.01265i
$195$		−	4.62109e14i		−	0.0431025i
$196$	9.68683e15			0.871745
$197$	−1.48336e16			−1.28820			−0.644099	−	0.764942i	$-0.722767\pi$
$197$	−1.48336e16			−1.28820			−0.644099	+	0.764942i	$0.722767\pi$
$198$		−	3.67273e16i		−	3.07845i
$199$			2.07457e16i			1.67863i	0.543643	+	0.839316i	$0.317044\pi$
$199$			2.07457e16i			1.67863i	−0.543643	+	0.839316i	$0.682956\pi$
$200$	−7.98472e13			−0.00623806
$201$		−	6.60173e15i		−	0.498064i
$202$	−4.34008e15			−0.316255
$203$			1.48336e15i			0.104418i
$204$		−	1.65043e16i		−	1.12250i
$205$			2.65247e15i			0.174331i
$206$		−	1.47270e16i		−	0.935499i
$207$	−1.09124e16	+	1.47251e16i	−0.670084	+	0.904200i
$208$	−5.68545e14			−0.0337537
$209$	−1.53180e16			−0.879380
$210$	−1.21300e16			−0.673481
$211$	7.84404e15			0.421272			0.210636	−	0.977565i	$-0.432447\pi$
$211$	7.84404e15			0.421272			0.210636	+	0.977565i	$0.432447\pi$
$212$			2.80722e16i			1.45856i
$213$	−1.44347e16			−0.725688
$214$			4.39862e16i			2.14003i
$215$	2.41068e16			1.13519
$216$	−9.66071e13			−0.00440385
$217$			9.37881e15i			0.413932i
$218$			5.36191e15i			0.229152i
$219$	6.88098e16			2.84801
$220$	4.20562e16			1.68605
$221$		−	6.69549e14i		−	0.0260038i
$222$		−	5.84286e15i		−	0.219864i
$223$	1.98133e16			0.722477			0.361238	−	0.932474i	$-0.382354\pi$
$223$	1.98133e16			0.722477			0.361238	+	0.932474i	$0.382354\pi$
$224$			1.51814e16i			0.536506i
$225$	8.51009e15			0.291512
$226$			4.94490e16i			1.64210i
$227$		−	4.21960e15i		−	0.135860i	−0.997690	−	0.0679298i	$-0.978361\pi$
$227$		−	4.21960e15i		−	0.135860i	0.997690	−	0.0679298i	$-0.0216394\pi$
$228$			2.16811e16i			0.676919i
$229$		−	3.33158e16i		−	1.00879i	−0.863472	−	0.504396i	$-0.831715\pi$
$229$		−	3.33158e16i		−	1.00879i	0.863472	−	0.504396i	$-0.168285\pi$
$230$	−3.34420e16	−	2.47832e16i	−0.982194	−	0.727883i
$231$	−3.72412e16			−1.06106
$232$	−2.40742e14			−0.00665481
$233$	−3.73751e16			−1.00252			−0.501258	−	0.865298i	$-0.667129\pi$
$233$	−3.73751e16			−1.00252			−0.501258	+	0.865298i	$0.667129\pi$
$234$	−2.10889e15			−0.0548963
$235$			1.54451e16i			0.390226i
$236$	4.86263e16			1.19258
$237$		−	2.85579e16i		−	0.679967i
$238$	−1.75751e16			−0.406312
$239$	−5.42487e16			−1.21788			−0.608939	−	0.793217i	$-0.708405\pi$
$239$	−5.42487e16			−1.21788			−0.608939	+	0.793217i	$0.708405\pi$
$240$			5.65652e16i			1.23331i
$241$		−	5.36056e16i		−	1.13525i	−0.823287	−	0.567625i	$-0.807862\pi$
$241$		−	5.36056e16i		−	1.13525i	0.823287	−	0.567625i	$-0.192138\pi$
$242$	1.87054e17			3.84822
$243$	−7.17927e16			−1.43495
$244$			8.80186e16i			1.70941i
$245$		−	3.90977e16i		−	0.737885i
$246$	2.28607e16			0.419318
$247$			8.79561e14i			0.0156815i
$248$	−1.52213e15			−0.0263809
$249$		−	3.72755e16i		−	0.628099i
$250$			9.39430e16i			1.53916i
$251$		−	1.32445e16i		−	0.211017i	−0.994418	−	0.105509i	$-0.966353\pi$
$251$		−	1.32445e16i		−	0.211017i	0.994418	−	0.105509i	$-0.0336471\pi$
$252$			2.79110e16i			0.432486i
$253$	−1.02673e17	−	7.60888e16i	−1.54744	−	1.14677i
$254$	−1.03709e16			−0.152048
$255$	−6.66142e16			−0.950136
$256$	6.95520e16			0.965227
$257$	−5.99781e16			−0.809955			−0.404978	−	0.914327i	$-0.632721\pi$
$257$	−5.99781e16			−0.809955			−0.404978	+	0.914327i	$0.632721\pi$
$258$		−	2.07768e17i		−	2.73048i
$259$	−3.13713e15			−0.0401268
$260$		−	2.41488e15i		−	0.0300664i
$261$	2.56582e16			0.310987
$262$	−1.28750e17			−1.51929
$263$		−	8.21385e16i		−	0.943750i	−0.881666	−	0.471875i	$-0.843577\pi$
$263$		−	8.21385e16i		−	0.943750i	0.881666	−	0.471875i	$-0.156423\pi$
$264$		−	6.04405e15i		−	0.0676239i
$265$	1.13304e17			1.23459
$266$	2.30878e16			0.245025
$267$		−	4.42272e16i		−	0.457205i
$268$		−	3.44991e16i		−	0.347428i
$269$	−5.26625e15			−0.0516697			−0.0258348	−	0.999666i	$-0.508224\pi$
$269$	−5.26625e15			−0.0516697			−0.0258348	+	0.999666i	$0.508224\pi$
$270$			2.33840e16i			0.223549i
$271$	−1.55951e16			−0.145279			−0.0726395	−	0.997358i	$-0.523142\pi$
$271$	−1.55951e16			−0.145279			−0.0726395	+	0.997358i	$0.523142\pi$
$272$			8.19573e16i			0.744055i
$273$			2.13840e15i			0.0189213i
$274$		−	2.07013e17i		−	1.78544i
$275$			5.93381e16i			0.498890i
$276$	−1.07696e17	+	1.45324e17i	−0.882749	+	1.19117i
$277$	−9.44828e16			−0.755082			−0.377541	−	0.925993i	$-0.623230\pi$
$277$	−9.44828e16			−0.755082			−0.377541	+	0.925993i	$0.623230\pi$
$278$	−3.32706e17			−2.59267
$279$	1.62228e17			1.23281
$280$	−1.05699e15			−0.00783365
$281$			2.09339e17i			1.51323i	0.653860	+	0.756616i	$0.273149\pi$
$281$			2.09339e17i			1.51323i	−0.653860	+	0.756616i	$0.726851\pi$
$282$	1.33116e17			0.938610
$283$		−	2.81129e17i		−	1.93375i	−0.255248	−	0.966876i	$-0.582157\pi$
$283$		−	2.81129e17i		−	1.93375i	0.255248	−	0.966876i	$-0.417843\pi$
$284$	−7.54324e16			−0.506209
$285$	8.75086e16			0.572976
$286$		−	1.47046e16i		−	0.0939487i
$287$		−	1.22743e16i		−	0.0765284i
$288$	2.62597e17			1.59787
$289$	7.18606e16			0.426782
$290$			5.82722e16i			0.337812i
$291$			3.65085e17i			2.06606i
$292$	3.59585e17			1.98665
$293$		−	2.42458e16i		−	0.130786i	−0.997860	−	0.0653932i	$-0.979170\pi$
$293$		−	2.42458e16i		−	0.130786i	0.997860	−	0.0653932i	$-0.0208302\pi$
$294$	−3.36969e17			−1.77484
$295$		−	1.96264e17i		−	1.00945i
$296$		−	5.09139e14i		−	0.00255737i
$297$			7.17931e16i			0.352199i
$298$			2.29352e17i			1.09898i
$299$	−4.36904e15	+	5.89551e15i	−0.0204497	+	0.0275945i
$300$	8.39873e16			0.384030
$301$	−1.11554e17			−0.498332
$302$	3.49343e17			1.52477
$303$	7.61222e16			0.324648
$304$		−	1.07664e17i		−	0.448700i
$305$	3.55258e17			1.44692
$306$			3.04002e17i			1.21011i
$307$	−4.20462e17			−1.63591			−0.817953	−	0.575286i	$-0.804891\pi$
$307$	−4.20462e17			−1.63591			−0.817953	+	0.575286i	$0.804891\pi$
$308$	−1.94614e17			−0.740150
$309$			2.58302e17i			0.960325i
$310$			3.68435e17i			1.33915i
$311$	1.11017e17			0.394516			0.197258	−	0.980352i	$-0.436796\pi$
$311$	1.11017e17			0.394516			0.197258	+	0.980352i	$0.436796\pi$
$312$	−3.47051e14			−0.00120590
$313$			4.44281e17i			1.50955i	0.655985	+	0.754774i	$0.272254\pi$
$313$			4.44281e17i			1.50955i	−0.655985	+	0.754774i	$0.727746\pi$
$314$		−	5.08864e17i		−	1.69081i
$315$	1.12654e17			0.366076
$316$		−	1.49237e17i		−	0.474315i
$317$	−2.78837e17			−0.866834			−0.433417	−	0.901194i	$-0.642692\pi$
$317$	−2.78837e17			−0.866834			−0.433417	+	0.901194i	$0.642692\pi$
$318$		−	9.76525e17i		−	2.96957i
$319$			1.78906e17i			0.532219i
$320$			3.05713e17i			0.889743i
$321$		−	7.71490e17i		−	2.19682i
$322$	1.54752e17	+	1.14684e17i	0.431167	+	0.319529i
$323$	1.26791e17			0.345677
$324$	−3.27363e17			−0.873402
$325$	3.40721e15			0.00889641
$326$	6.20591e15			0.0158592
$327$		−	9.40445e16i		−	0.235234i
$328$	1.99205e15			0.00487734
$329$		−	7.14720e16i		−	0.171303i
$330$	−1.46298e18			−3.43273
$331$	2.32813e17			0.534825			0.267413	−	0.963582i	$-0.413831\pi$
$331$	2.32813e17			0.534825			0.267413	+	0.963582i	$0.413831\pi$
$332$		−	1.94793e17i		−	0.438135i
$333$			5.42639e16i			0.119509i
$334$	4.30417e17			0.928249
$335$	−1.39244e17			−0.294079
$336$		−	2.61755e17i		−	0.541402i
$337$			8.38946e17i			1.69951i	0.527176	+	0.849756i	$0.323251\pi$
$337$			8.38946e17i			1.69951i	−0.527176	+	0.849756i	$0.676749\pi$
$338$	7.14912e17			1.41852
$339$		−	8.67304e17i		−	1.68567i
$340$	−3.48111e17			−0.662774
$341$			1.13116e18i			2.10981i
$342$		−	3.99356e17i		−	0.729755i
$343$			3.91986e17i			0.701797i
$344$		−	1.81046e16i		−	0.0317599i
$345$	5.86551e17	+	4.34681e17i	1.00826	+	0.747200i
$346$	1.05523e18			1.77752
$347$	1.14116e18			1.88382			0.941911	−	0.335861i	$-0.109027\pi$
$347$	1.14116e18			1.88382			0.941911	+	0.335861i	$0.109027\pi$
$348$	2.53224e17			0.409686
$349$	4.65052e17			0.737435			0.368717	−	0.929542i	$-0.379797\pi$
$349$	4.65052e17			0.737435			0.368717	+	0.929542i	$0.379797\pi$
$350$		−	8.94364e16i		−	0.139007i
$351$	4.12238e15			0.00628056
$352$			1.83100e18i			2.73457i
$353$	−1.11174e18			−1.62772			−0.813860	−	0.581062i	$-0.802638\pi$
$353$	−1.11174e18			−1.62772			−0.813860	+	0.581062i	$0.802638\pi$
$354$	−1.69153e18			−2.42804
$355$			3.04458e17i			0.428479i
$356$		−	2.31122e17i		−	0.318926i
$357$	3.08256e17			0.417094
$358$	1.27533e17			0.169216
$359$		−	1.22544e18i		−	1.59452i	−0.603635	−	0.797261i	$-0.706282\pi$
$359$		−	1.22544e18i		−	1.59452i	0.603635	−	0.797261i	$-0.293718\pi$
$360$			1.82831e16i			0.0233309i
$361$	6.32446e17			0.791541
$362$			1.28278e18i			1.57468i
$363$	−3.28081e18			−3.95035
$364$			1.11748e16i			0.0131987i
$365$		−	1.45135e18i		−	1.68159i
$366$		−	3.06184e18i		−	3.48028i
$367$			4.74002e17i			0.528589i	0.964442	+	0.264295i	$0.0851391\pi$
$367$			4.74002e17i			0.528589i	−0.964442	+	0.264295i	$0.914861\pi$
$368$	5.34799e17	−	7.21650e17i	0.585135	−	0.789572i
$369$	−2.12312e17			−0.227924
$370$	−1.23238e17			−0.129818
$371$	−5.24313e17			−0.541967
$372$	1.60105e18			1.62407
$373$		−	1.51919e18i		−	1.51234i	−0.654374	−	0.756171i	$-0.727068\pi$
$373$		−	1.51919e18i		−	1.51234i	0.654374	−	0.756171i	$-0.272932\pi$
$374$	−2.11971e18			−2.07097
$375$		−	1.64770e18i		−	1.58001i
$376$	1.15995e16			0.0109175
$377$	1.02728e16			0.00949076
$378$		−	1.08209e17i		−	0.0981344i
$379$		−	1.56124e18i		−	1.38994i	−0.719039	−	0.694970i	$-0.755418\pi$
$379$		−	1.56124e18i		−	1.38994i	0.719039	−	0.694970i	$-0.244582\pi$
$380$	4.57300e17			0.399683
$381$	1.81899e17			0.156083
$382$			1.13752e18i			0.958333i
$383$		−	1.03658e18i		−	0.857457i	−0.903433	−	0.428728i	$-0.858962\pi$
$383$		−	1.03658e18i		−	0.857457i	0.903433	−	0.428728i	$-0.141038\pi$
$384$	8.64411e16			0.0702105
$385$			7.85497e17i			0.626497i
$386$	−2.28861e18			−1.79251
$387$			1.92958e18i			1.48418i
$388$			1.90785e18i			1.44120i
$389$			2.48507e18i			1.84370i	0.387541	+	0.921852i	$0.373325\pi$
$389$			2.48507e18i			1.84370i	−0.387541	+	0.921852i	$0.626675\pi$
$390$			8.40045e16i			0.0612140i
$391$	8.49853e17	+	6.29808e17i	0.608284	+	0.450787i
$392$	−2.93630e16			−0.0206442
$393$	2.25820e18			1.55960
$394$	2.69652e18			1.82949
$395$	−6.02346e17			−0.401483
$396$			3.36630e18i			2.20438i
$397$	2.16452e18			1.39260			0.696302	−	0.717749i	$-0.254828\pi$
$397$	2.16452e18			1.39260			0.696302	+	0.717749i	$0.254828\pi$
$398$		−	3.77125e18i		−	2.38399i
$399$	−4.04944e17			−0.251527
$400$	−4.17065e17			−0.254556
$401$			1.44467e17i			0.0866477i	0.999061	+	0.0433238i	$0.0137947\pi$
$401$			1.44467e17i			0.0866477i	−0.999061	+	0.0433238i	$0.986205\pi$
$402$			1.20010e18i			0.707349i
$403$	6.49517e16			0.0376231
$404$	3.97797e17			0.226460
$405$			1.32129e18i			0.739288i
$406$		−	2.69654e17i		−	0.148294i
$407$	−3.78364e17			−0.204526
$408$			5.00283e16i			0.0265824i
$409$	−2.35055e18			−1.22774			−0.613871	−	0.789406i	$-0.710388\pi$
$409$	−2.35055e18			−1.22774			−0.613871	+	0.789406i	$0.710388\pi$
$410$		−	4.82180e17i		−	0.247584i
$411$			3.63088e18i			1.83282i
$412$			1.34983e18i			0.669881i
$413$			9.08208e17i			0.443134i
$414$	1.98372e18	−	2.67680e18i	0.951651	−	1.28414i
$415$	−7.86220e17			−0.370858
$416$	1.05136e17			0.0487641
$417$	5.83545e18			2.66147
$418$	2.78458e18			1.24889
$419$			1.57400e18i			0.694235i	0.937822	+	0.347118i	$0.112840\pi$
$419$			1.57400e18i			0.694235i	−0.937822	+	0.347118i	$0.887160\pi$
$420$	1.11179e18			0.482258
$421$			4.07118e18i			1.73678i	0.495880	+	0.868391i	$0.334846\pi$
$421$			4.07118e18i			1.73678i	−0.495880	+	0.868391i	$0.665154\pi$
$422$	−1.42593e18			−0.598289
$423$	−1.23627e18			−0.510190
$424$		−	8.50930e16i		−	0.0345408i
$425$		−	4.91158e17i		−	0.196109i
$426$	2.62401e18			1.03062
$427$	−1.64395e18			−0.635176
$428$		−	4.03163e18i		−	1.53241i
$429$			2.57909e17i			0.0964419i
$430$	−4.38226e18			−1.61220
$431$		−	2.40149e18i		−	0.869242i	−0.900614	−	0.434621i	$-0.856882\pi$
$431$		−	2.40149e18i		−	0.869242i	0.900614	−	0.434621i	$-0.143118\pi$
$432$	−5.04607e17			−0.179708
$433$		−	1.75344e18i		−	0.614433i	−0.951640	−	0.307217i	$-0.900602\pi$
$433$		−	1.75344e18i		−	0.614433i	0.951640	−	0.307217i	$-0.0993976\pi$
$434$		−	1.70493e18i		−	0.587865i
$435$		−	1.02206e18i		−	0.346777i
$436$		−	4.91455e17i		−	0.164089i
$437$	−1.11642e18	−	8.27355e17i	−0.366824	−	0.271845i
$438$	−1.25086e19			−4.04474
$439$	1.82287e18			0.580102			0.290051	−	0.957011i	$-0.406328\pi$
$439$	1.82287e18			0.580102			0.290051	+	0.957011i	$0.406328\pi$
$440$	−1.27482e17			−0.0399282
$441$	3.12950e18			0.964728
$442$			1.21714e17i			0.0369305i
$443$	2.95789e18			0.883398			0.441699	−	0.897163i	$-0.354376\pi$
$443$	2.95789e18			0.883398			0.441699	+	0.897163i	$0.354376\pi$
$444$			5.35538e17i			0.157438i
$445$	−9.32846e17			−0.269954
$446$	−3.60177e18			−1.02606
$447$		−	4.02268e18i		−	1.12814i
$448$		−	1.41468e18i		−	0.390583i
$449$	1.58509e18			0.430852			0.215426	−	0.976520i	$-0.430886\pi$
$449$	1.58509e18			0.430852			0.215426	+	0.976520i	$0.430886\pi$
$450$	−1.54701e18			−0.414005
$451$		−	1.48038e18i		−	0.390066i
$452$		−	4.53234e18i		−	1.17585i
$453$	−6.12726e18			−1.56523
$454$			7.67060e17i			0.192947i
$455$	4.51034e16			0.0111720
$456$		−	6.57202e16i		−	0.0160304i
$457$		−	3.55314e18i		−	0.853491i	−0.904372	−	0.426745i	$-0.859660\pi$
$457$		−	3.55314e18i		−	0.853491i	0.904372	−	0.426745i	$-0.140340\pi$
$458$			6.05632e18i			1.43268i
$459$		−	5.94252e17i		−	0.138446i
$460$	3.06518e18	+	2.27154e18i	0.703318	+	0.521214i
$461$	−4.85201e18			−1.09652			−0.548258	−	0.836309i	$-0.684709\pi$
$461$	−4.85201e18			−1.09652			−0.548258	+	0.836309i	$0.684709\pi$
$462$	6.76990e18			1.50691
$463$	3.47574e18			0.762044			0.381022	−	0.924566i	$-0.375572\pi$
$463$	3.47574e18			0.762044			0.381022	+	0.924566i	$0.375572\pi$
$464$	−1.25746e18			−0.271563
$465$		−	6.46212e18i		−	1.37469i
$466$	6.79424e18			1.42377
$467$		−	4.22676e18i		−	0.872548i	−0.899814	−	0.436274i	$-0.856298\pi$
$467$		−	4.22676e18i		−	0.872548i	0.899814	−	0.436274i	$-0.143702\pi$
$468$	1.93294e17			0.0393095
$469$	6.44351e17			0.129096
$470$		−	2.80769e18i		−	0.554197i
$471$			8.92515e18i			1.73568i
$472$	−1.47397e17			−0.0282420
$473$	−1.34543e19			−2.54000
$474$			5.19140e18i			0.965686i
$475$			6.45215e17i			0.118263i
$476$	1.61088e18			0.290947
$477$			9.06919e18i			1.61414i
$478$	9.86161e18			1.72963
$479$		−	2.57146e18i		−	0.444458i	−0.974994	−	0.222229i	$-0.928667\pi$
$479$		−	2.57146e18i		−	0.444458i	0.974994	−	0.222229i	$-0.0713333\pi$
$480$		−	1.04601e19i		−	1.78176i
$481$			2.17258e16i			0.00364720i
$482$			9.74471e18i			1.61228i
$483$	−2.71426e18	−	2.01148e18i	−0.442610	−	0.328009i
$484$	−1.71448e19			−2.75559
$485$	7.70042e18			1.21989
$486$	1.30508e19			2.03791
$487$	−1.00847e19			−1.55224			−0.776121	−	0.630584i	$-0.782815\pi$
$487$	−1.00847e19			−1.55224			−0.776121	+	0.630584i	$0.782815\pi$
$488$		−	2.66804e17i		−	0.0404812i
$489$	−1.08848e17			−0.0162801
$490$			7.10738e18i			1.04794i
$491$	7.03369e18			1.02238			0.511191	−	0.859467i	$-0.329205\pi$
$491$	7.03369e18			1.02238			0.511191	+	0.859467i	$0.329205\pi$
$492$	−2.09534e18			−0.300260
$493$		−	1.48086e18i		−	0.209211i
$494$		−	1.59891e17i		−	0.0222708i
$495$	1.35870e19			1.86589
$496$	−7.95052e18			−1.07652
$497$		−	1.40888e18i		−	0.188095i
$498$			6.77614e18i			0.892024i
$499$	2.31732e18			0.300802			0.150401	−	0.988625i	$-0.451943\pi$
$499$	2.31732e18			0.300802			0.150401	+	0.988625i	$0.451943\pi$
$500$		−	8.61051e18i		−	1.10215i
$501$	−7.54924e18			−0.952883
$502$			2.40765e18i			0.299686i
$503$		−	4.95728e18i		−	0.608510i	−0.952591	−	0.304255i	$-0.901592\pi$
$503$		−	4.95728e18i		−	0.608510i	0.952591	−	0.304255i	$-0.0984075\pi$
$504$		−	8.46046e16i		−	0.0102419i
$505$		−	1.60558e18i		−	0.191687i
$506$	1.86644e19	+	1.38318e19i	2.19766	+	1.62864i
$507$	−1.25391e19			−1.45617
$508$	9.50561e17			0.108876
$509$	1.07715e19			1.21689			0.608446	−	0.793595i	$-0.291793\pi$
$509$	1.07715e19			1.21689			0.608446	+	0.793595i	$0.291793\pi$
$510$	1.21095e19			1.34938
$511$			6.71608e18i			0.738192i
$512$	−1.30877e19			−1.41897
$513$			7.80646e17i			0.0834896i
$514$	1.09031e19			1.15030
$515$	5.44813e18			0.567018
$516$			1.90433e19i			1.95521i
$517$		−	8.62012e18i		−	0.873131i
$518$	5.70284e17			0.0569879
$519$	−1.85080e19			−1.82469
$520$			7.32004e15i			0.000712016i
$521$			7.83837e18i			0.752249i	0.926569	+	0.376124i	$0.122743\pi$
$521$			7.83837e18i			0.752249i	−0.926569	+	0.376124i	$0.877257\pi$
$522$	−4.66428e18			−0.441663
$523$		−	1.08638e19i		−	1.01501i	−0.861650	−	0.507504i	$-0.830568\pi$
$523$		−	1.08638e19i		−	1.01501i	0.861650	−	0.507504i	$-0.169432\pi$
$524$	1.18008e19			1.08791
$525$			1.56866e18i			0.142696i
$526$			1.49316e19i			1.34031i
$527$		−	9.36295e18i		−	0.829351i
$528$		−	3.15698e19i		−	2.75953i
$529$	−3.37341e18	−	1.10912e19i	−0.290991	−	0.956726i
$530$	−2.05970e19			−1.75337
$531$	1.57096e19			1.31978
$532$	−2.11615e18			−0.175455
$533$	−8.50039e16			−0.00695582
$534$			8.03985e18i			0.649321i
$535$	−1.62724e19			−1.29710
$536$			1.04575e17i			0.00822759i
$537$	−2.23685e18			−0.173707
$538$	9.57327e17			0.0733811
$539$			2.18210e19i			1.65102i
$540$		−	2.14330e18i		−	0.160076i
$541$	1.58434e18			0.116806			0.0584032	−	0.998293i	$-0.481399\pi$
$541$	1.58434e18			0.116806			0.0584032	+	0.998293i	$0.481399\pi$
$542$	2.83495e18			0.206325
$543$		−	2.24991e19i		−	1.61647i
$544$		−	1.51557e19i		−	1.07494i
$545$	−1.98360e18			−0.138892
$546$		−	3.88730e17i		−	0.0268719i
$547$	−1.22528e19			−0.836227			−0.418114	−	0.908395i	$-0.637309\pi$
$547$	−1.22528e19			−0.836227			−0.418114	+	0.908395i	$0.637309\pi$
$548$			1.89742e19i			1.27849i
$549$			2.84359e19i			1.89174i
$550$		−	1.07868e19i		−	0.708521i
$551$			1.94535e18i			0.126164i
$552$	3.26452e17	−	4.40509e17i	0.0209048	−	0.0282086i
$553$	2.78735e18			0.176244
$554$	1.71756e19			1.07237
$555$	2.16152e18			0.133263
$556$	3.04947e19			1.85653
$557$		−	1.24840e19i		−	0.750529i	−0.926918	−	0.375265i	$-0.877552\pi$
$557$		−	1.24840e19i		−	0.750529i	0.926918	−	0.375265i	$-0.122448\pi$
$558$	−2.94906e19			−1.75083
$559$			7.72551e17i			0.0452944i
$560$	−5.52096e18			−0.319668
$561$	3.71783e19			2.12593
$562$		−	3.80547e19i		−	2.14909i
$563$		−	6.62994e18i		−	0.369787i	−0.982759	−	0.184893i	$-0.940806\pi$
$563$		−	6.62994e18i		−	0.369787i	0.982759	−	0.184893i	$-0.0591939\pi$
$564$	−1.22009e19			−0.672109
$565$	−1.82933e19			−0.995297
$566$			5.11051e19i			2.74631i
$567$		−	6.11426e18i		−	0.324536i
$568$	2.28653e17			0.0119878
$569$			2.30183e19i			1.19203i	0.802973	+	0.596016i	$0.203250\pi$
$569$			2.30183e19i			1.19203i	−0.802973	+	0.596016i	$0.796750\pi$
$570$	−1.59078e19			−0.813738
$571$		−	2.65470e19i		−	1.34141i	−0.741722	−	0.670707i	$-0.765991\pi$
$571$		−	2.65470e19i		−	1.34141i	0.741722	−	0.670707i	$-0.234009\pi$
$572$			1.34778e18i			0.0672737i
$573$		−	1.99514e19i		−	0.983765i
$574$			2.23128e18i			0.108685i
$575$	−3.20497e18	+	4.32474e18i	−0.154223	+	0.208106i
$576$	−2.44702e19			−1.16327
$577$	1.31157e19			0.615973			0.307986	−	0.951391i	$-0.400345\pi$
$577$	1.31157e19			0.615973			0.307986	+	0.951391i	$0.400345\pi$
$578$	−1.30632e19			−0.606114
$579$	4.01408e19			1.84008
$580$		−	5.34104e18i		−	0.241897i
$581$	3.63822e18			0.162801
$582$		−	6.63671e19i		−	2.93421i
$583$	−6.32365e19			−2.76241
$584$	−1.08998e18			−0.0470467
$585$		−	7.80168e17i		−	0.0332734i
$586$			4.40752e18i			0.185742i
$587$	3.45212e18			0.143754			0.0718770	−	0.997414i	$-0.477101\pi$
$587$	3.45212e18			0.143754			0.0718770	+	0.997414i	$0.477101\pi$
$588$	3.08855e19			1.27090
$589$			1.22998e19i			0.500137i
$590$			3.56779e19i			1.43362i
$591$	−4.72953e19			−1.87804
$592$		−	2.65938e18i		−	0.104359i
$593$	1.95114e19			0.756668			0.378334	−	0.925669i	$-0.376497\pi$
$593$	1.95114e19			0.756668			0.378334	+	0.925669i	$0.376497\pi$
$594$		−	1.30509e19i		−	0.500191i
$595$		−	6.50178e18i		−	0.246271i
$596$		−	2.10216e19i		−	0.786942i
$597$			6.61453e19i			2.44725i
$598$	7.94226e17	−	1.07172e18i	0.0290426	−	0.0391896i
$599$	4.01011e19			1.44934			0.724668	−	0.689098i	$-0.241993\pi$
$599$	4.01011e19			1.44934			0.724668	+	0.689098i	$0.241993\pi$
$600$	−2.54584e17			−0.00909437
$601$	2.84025e19			1.00285			0.501424	−	0.865202i	$-0.332810\pi$
$601$	2.84025e19			1.00285			0.501424	+	0.865202i	$0.332810\pi$
$602$	2.02788e19			0.707730
$603$		−	1.11455e19i		−	0.384486i
$604$	−3.20197e19			−1.09184
$605$			6.91993e19i			2.33246i
$606$	−1.38379e19			−0.461064
$607$	3.09418e19			1.01912			0.509559	−	0.860436i	$-0.329809\pi$
$607$	3.09418e19			1.01912			0.509559	+	0.860436i	$0.329809\pi$
$608$			1.99095e19i			0.648238i
$609$			4.72955e18i			0.152230i
$610$	−6.45807e19			−2.05491
$611$	−4.94969e17			−0.0155700
$612$		−	2.78638e19i		−	0.866525i
$613$		−	4.17911e19i		−	1.28487i	−0.766339	−	0.642436i	$-0.777924\pi$
$613$		−	4.17911e19i		−	1.28487i	0.766339	−	0.642436i	$-0.222076\pi$
$614$	7.64337e19			2.32331
$615$			8.45713e18i			0.254154i
$616$	5.89920e17			0.0175278
$617$		−	2.17298e18i		−	0.0638350i	−0.999491	−	0.0319175i	$-0.989839\pi$
$617$		−	2.17298e18i		−	0.0638350i	0.999491	−	0.0319175i	$-0.0101614\pi$
$618$		−	4.69554e19i		−	1.36385i
$619$			2.05964e19i			0.591504i	0.955265	+	0.295752i	$0.0955702\pi$
$619$			2.05964e19i			0.591504i	−0.955265	+	0.295752i	$0.904430\pi$
$620$		−	3.37696e19i		−	0.958923i
$621$	−3.87770e18	+	5.23250e18i	−0.108876	+	0.146916i
$622$	−2.01812e19			−0.560291
$623$	4.31673e18			0.118506
$624$	−1.81275e18			−0.0492091
$625$	−2.51041e19			−0.673884
$626$		−	8.07636e19i		−	2.14386i
$627$	−4.88397e19			−1.28204
$628$			4.66408e19i			1.21073i
$629$	3.13183e18			0.0803977
$630$	−2.04788e19			−0.519900
$631$		−	5.02834e19i		−	1.26246i	−0.775594	−	0.631232i	$-0.782549\pi$
$631$		−	5.02834e19i		−	1.26246i	0.775594	−	0.631232i	$-0.217451\pi$
$632$			4.52371e17i			0.0112325i
$633$	2.50099e19			0.614166
$634$	5.06885e19			1.23107
$635$		−	3.83663e18i		−	0.0921581i
$636$			8.95051e19i			2.12642i
$637$	1.25297e18			0.0294417
$638$		−	3.25225e19i		−	0.755856i
$639$	−2.43697e19			−0.560202
$640$		−	1.82323e18i		−	0.0414554i
$641$			5.64725e19i			1.27008i	0.772480	+	0.635039i	$0.219016\pi$
$641$			5.64725e19i			1.27008i	−0.772480	+	0.635039i	$0.780984\pi$
$642$			1.40245e20i			3.11992i
$643$		−	4.15149e18i		−	0.0913540i	−0.998956	−	0.0456770i	$-0.985455\pi$
$643$		−	4.15149e18i		−	0.0913540i	0.998956	−	0.0456770i	$-0.0145445\pi$
$644$	−1.41841e19	−	1.05115e19i	−0.308745	−	0.228805i
$645$	7.68620e19			1.65498
$646$	−2.30487e19			−0.490930
$647$	2.28173e19			0.480767			0.240383	−	0.970678i	$-0.422727\pi$
$647$	2.28173e19			0.480767			0.240383	+	0.970678i	$0.422727\pi$
$648$	9.92310e17			0.0206834
$649$			1.09537e20i			2.25866i
$650$	−6.19380e17			−0.0126347
$651$			2.99033e19i			0.603466i
$652$	−5.68814e17			−0.0113563
$653$	5.92938e18			0.117116			0.0585581	−	0.998284i	$-0.481350\pi$
$653$	5.92938e18			0.117116			0.0585581	+	0.998284i	$0.481350\pi$
$654$			1.70959e19i			0.334078i
$655$		−	4.76302e19i		−	0.920860i
$656$	1.04050e19			0.199029
$657$	1.16170e20			2.19855
$658$			1.29925e19i			0.243284i
$659$			2.82894e19i			0.524113i	0.965052	+	0.262057i	$0.0844007\pi$
$659$			2.82894e19i			0.524113i	−0.965052	+	0.262057i	$0.915599\pi$
$660$	1.34092e20			2.45807
$661$		−	7.66685e19i		−	1.39061i	−0.718713	−	0.695307i	$-0.755268\pi$
$661$		−	7.66685e19i		−	1.39061i	0.718713	−	0.695307i	$-0.244732\pi$
$662$	−4.23220e19			−0.759557
$663$		−	2.13479e18i		−	0.0379105i
$664$			5.90463e17i			0.0103757i
$665$			8.54114e18i			0.148513i
$666$		−	9.86437e18i		−	0.169727i
$667$	−9.66310e18	+	1.30392e19i	−0.164526	+	0.222009i
$668$	−3.94506e19			−0.664690
$669$	6.31728e19			1.05329
$670$	2.53126e19			0.417650
$671$	−1.98274e20			−3.23749
$672$			4.84042e19i			0.782164i
$673$	3.31535e19			0.530181			0.265091	−	0.964224i	$-0.414598\pi$
$673$	3.31535e19			0.530181			0.265091	+	0.964224i	$0.414598\pi$
$674$		−	1.52508e20i		−	2.41364i
$675$	3.02403e18			0.0473653
$676$	−6.55265e19			−1.01576
$677$			3.31996e19i			0.509345i	0.967027	+	0.254673i	$0.0819677\pi$
$677$			3.31996e19i			0.509345i	−0.967027	+	0.254673i	$0.918032\pi$
$678$			1.57663e20i			2.39399i
$679$	−3.56336e19			−0.535514
$680$	1.05520e18			0.0156954
$681$		−	1.34537e19i		−	0.198068i
$682$		−	2.05629e20i		−	2.99635i
$683$	−2.63622e19			−0.380221			−0.190111	−	0.981763i	$-0.560885\pi$
$683$	−2.63622e19			−0.380221			−0.190111	+	0.981763i	$0.560885\pi$
$684$			3.66037e19i			0.522555i
$685$	7.65830e19			1.08218
$686$		−	7.12572e19i		−	0.996690i
$687$		−	1.06224e20i		−	1.47070i
$688$		−	9.45654e19i		−	1.29602i
$689$			3.63106e18i			0.0492604i
$690$	−1.06626e20	−	7.90185e19i	−1.43193	−	1.06117i
$691$	7.46375e19			0.992226			0.496113	−	0.868258i	$-0.334760\pi$
$691$	7.46375e19			0.992226			0.496113	+	0.868258i	$0.334760\pi$
$692$	−9.67187e19			−1.27282
$693$	−6.28735e19			−0.819097
$694$	−2.07446e20			−2.67540
$695$		−	1.23082e20i		−	1.57145i
$696$	−7.67580e17			−0.00970195
$697$			1.22535e19i			0.153332i
$698$	−8.45396e19			−1.04730
$699$	−1.19167e20			−1.46155
$700$			8.19745e18i			0.0995388i
$701$		−	9.51039e18i		−	0.114333i	−0.998365	−	0.0571666i	$-0.981793\pi$
$701$		−	9.51039e18i		−	0.114333i	0.998365	−	0.0571666i	$-0.0182066\pi$
$702$	−7.49387e17			−0.00891963
$703$	−4.11416e18			−0.0484835
$704$		−	1.70623e20i		−	1.99080i
$705$			4.92451e19i			0.568904i
$706$	2.02097e20			2.31168
$707$			7.42979e18i			0.0841474i
$708$	1.55040e20			1.73864
$709$			1.02145e20i			1.13421i	0.823644	+	0.567107i	$0.191937\pi$
$709$			1.02145e20i			1.13421i	−0.823644	+	0.567107i	$0.808063\pi$
$710$		−	5.53460e19i		−	0.608524i
$711$		−	4.82136e19i		−	0.524907i
$712$			7.00581e17i			0.00755264i
$713$	−6.10965e19	+	8.24426e19i	−0.652213	+	0.880086i
$714$	−5.60364e19			−0.592356
$715$	5.43985e18			0.0569436
$716$	−1.16893e19			−0.121170
$717$	−1.72966e20			−1.77553
$718$			2.22767e20i			2.26453i
$719$	1.37169e20			1.38087			0.690436	−	0.723394i	$-0.257419\pi$
$719$	1.37169e20			1.38087			0.690436	+	0.723394i	$0.257419\pi$
$720$			9.54978e19i			0.952063i
$721$	−2.52111e19			−0.248912
$722$	−1.14969e20			−1.12414
$723$		−	1.70916e20i		−	1.65506i
$724$		−	1.17575e20i		−	1.12758i
$725$	7.53580e18			0.0715753
$726$	5.96403e20			5.61027
$727$		−	9.28526e19i		−	0.865074i	−0.901616	−	0.432537i	$-0.857619\pi$
$727$		−	9.28526e19i		−	0.865074i	0.901616	−	0.432537i	$-0.142381\pi$
$728$		−	3.38733e16i		−	0.000312564i
$729$	−1.34930e20			−1.23315
$730$			2.63833e20i			2.38819i
$731$	1.11365e20			0.998454
$732$			2.80638e20i			2.49212i
$733$			1.31976e20i			1.16083i	0.814321	+	0.580415i	$0.197110\pi$
$733$			1.31976e20i			1.16083i	−0.814321	+	0.580415i	$0.802890\pi$
$734$		−	8.61666e19i		−	0.750700i
$735$		−	1.24659e20i		−	1.07575i
$736$	−9.88961e19	+	1.33449e20i	−0.845346	+	1.14070i
$737$	7.77142e19			0.658003
$738$	3.85952e19			0.323697
$739$	3.30629e19			0.274682			0.137341	−	0.990524i	$-0.456144\pi$
$739$	3.30629e19			0.274682			0.137341	+	0.990524i	$0.456144\pi$
$740$	1.12956e19			0.0929584
$741$			2.80439e18i			0.0228618i
$742$	9.53122e19			0.769700
$743$		−	1.14620e20i		−	0.916932i	−0.888712	−	0.458466i	$-0.848399\pi$
$743$		−	1.14620e20i		−	0.916932i	0.888712	−	0.458466i	$-0.151601\pi$
$744$	−4.85314e18			−0.0384603
$745$	−8.48469e19			−0.666104
$746$			2.76166e20i			2.14782i
$747$		−	6.29314e19i		−	0.484868i
$748$	1.94285e20			1.48296
$749$	7.53001e19			0.569407
$750$			2.99527e20i			2.24392i
$751$			9.89956e19i			0.734745i	0.930074	+	0.367372i	$0.119743\pi$
$751$			9.89956e19i			0.734745i	−0.930074	+	0.367372i	$0.880257\pi$
$752$	6.05876e19			0.445511
$753$		−	4.22286e19i		−	0.307639i
$754$	−1.86745e18			−0.0134787
$755$			1.29237e20i			0.924183i
$756$			9.91809e18i			0.0702709i
$757$		−	1.66167e19i		−	0.116647i	−0.998298	−	0.0583237i	$-0.981424\pi$
$757$		−	1.66167e19i		−	0.116647i	0.998298	−	0.0583237i	$-0.0185755\pi$
$758$			2.83811e20i			1.97399i
$759$	−3.27362e20	−	2.42601e20i	−2.25598	−	1.67186i
$760$	−1.38618e18			−0.00946508
$761$	−4.36239e18			−0.0295142			−0.0147571	−	0.999891i	$-0.504698\pi$
$761$	−4.36239e18			−0.0295142			−0.0147571	+	0.999891i	$0.504698\pi$
$762$	−3.30665e19			−0.221668
$763$	9.17907e18			0.0609715
$764$		−	1.04262e20i		−	0.686232i
$765$	−1.12463e20			−0.733467
$766$			1.88435e20i			1.21776i
$767$	6.28967e18			0.0402773
$768$	2.21759e20			1.40719
$769$		−	6.32088e19i		−	0.397460i	−0.980054	−	0.198730i	$-0.936318\pi$
$769$		−	6.32088e19i		−	0.397460i	0.980054	−	0.198730i	$-0.0636817\pi$
$770$		−	1.42792e20i		−	0.889749i
$771$	−1.91234e20			−1.18082
$772$	2.09767e20			1.28356
$773$		−	5.01646e19i		−	0.304187i	−0.988366	−	0.152093i	$-0.951399\pi$
$773$		−	5.01646e19i		−	0.304187i	0.988366	−	0.152093i	$-0.0486015\pi$
$774$		−	3.50769e20i		−	2.10782i
$775$	4.76463e19			0.283738
$776$		−	5.78313e18i		−	0.0341296i
$777$	−1.00024e19			−0.0585002
$778$		−	4.51749e20i		−	2.61842i
$779$		−	1.60970e19i		−	0.0924661i
$780$		−	7.69958e18i		−	0.0438334i
$781$		−	1.69922e20i		−	0.958722i
$782$	−1.54491e20	−	1.14490e20i	−0.863883	−	0.640206i
$783$	9.11756e18			0.0505297
$784$	−1.53371e20			−0.842426
$785$	1.88250e20			1.02482
$786$	−4.10507e20			−2.21494
$787$			1.07297e20i			0.573806i	0.957960	+	0.286903i	$0.0926257\pi$
$787$			1.07297e20i			0.573806i	−0.957960	+	0.286903i	$0.907374\pi$
$788$	−2.47155e20			−1.31004
$789$		−	2.61890e20i		−	1.37588i
$790$	1.09498e20			0.570184
$791$	8.46519e19			0.436919
$792$		−	1.02040e19i		−	0.0522030i
$793$			1.13850e19i			0.0577324i
$794$	−3.93477e20			−1.97777
$795$	3.61258e20			1.79990
$796$			3.45661e20i			1.70710i
$797$			4.33111e18i			0.0212027i	0.999944	+	0.0106013i	$0.00337457\pi$
$797$			4.33111e18i			0.0212027i	−0.999944	+	0.0106013i	$0.996625\pi$
$798$	7.36129e19			0.357218
$799$			7.13511e19i			0.343221i
$800$	7.71244e19			0.367758
$801$		−	7.46678e19i		−	0.352944i
$802$		−	2.62619e19i		−	0.123057i
$803$			8.10015e20i			3.76257i
$804$		−	1.09997e20i		−	0.506510i
$805$	−4.24263e19	+	5.72494e19i	−0.193671	+	0.261336i
$806$	−1.18072e19			−0.0534322
$807$	−1.67909e19			−0.0753285
$808$	−1.20581e18			−0.00536291
$809$	1.54871e20			0.682858			0.341429	−	0.939908i	$-0.389089\pi$
$809$	1.54871e20			0.682858			0.341429	+	0.939908i	$0.389089\pi$
$810$		−	2.40191e20i		−	1.04993i
$811$	−1.16638e20			−0.505471			−0.252735	−	0.967535i	$-0.581330\pi$
$811$	−1.16638e20			−0.505471			−0.252735	+	0.967535i	$0.581330\pi$
$812$			2.47156e19i			0.106189i
$813$	−4.97233e19			−0.211800
$814$	6.87810e19			0.290468
$815$			2.29583e18i			0.00961250i
$816$			2.61312e20i			1.08475i
$817$	−1.46296e20			−0.602114
$818$	4.27296e20			1.74363
$819$			3.61021e18i			0.0146065i
$820$			4.41951e19i			0.177287i
$821$	3.24156e20			1.28929			0.644647	−	0.764480i	$-0.277004\pi$
$821$	3.24156e20			1.28929			0.644647	+	0.764480i	$0.277004\pi$
$822$		−	6.60040e20i		−	2.60296i
$823$	2.23907e20			0.875526			0.437763	−	0.899090i	$-0.355771\pi$
$823$	2.23907e20			0.875526			0.437763	+	0.899090i	$0.355771\pi$
$824$		−	4.09163e18i		−	0.0158637i
$825$			1.89193e20i			0.727324i
$826$		−	1.65099e20i		−	0.629337i
$827$		−	3.03621e19i		−	0.114761i	−0.998352	−	0.0573803i	$-0.981725\pi$
$827$		−	3.03621e19i		−	0.114761i	0.998352	−	0.0573803i	$-0.0182748\pi$
$828$	−1.81821e20	+	2.45347e20i	−0.681447	+	0.919533i
$829$	2.07465e20			0.771015			0.385507	−	0.922705i	$-0.374026\pi$
$829$	2.07465e20			0.771015			0.385507	+	0.922705i	$0.374026\pi$
$830$	1.42923e20			0.526691
$831$	−3.01248e20			−1.10082
$832$	−9.79720e18			−0.0355008
$833$		−	1.80618e20i		−	0.649003i
$834$	−1.06080e21			−3.77981
$835$			1.59230e20i			0.562624i
$836$	−2.55225e20			−0.894292
$837$	5.76472e19			0.200309
$838$		−	2.86130e20i		−	0.985950i
$839$			2.40225e20i			0.820890i	0.911885	+	0.410445i	$0.134627\pi$
$839$			2.40225e20i			0.820890i	−0.911885	+	0.410445i	$0.865373\pi$
$840$	−3.37010e18			−0.0114206
$841$	−2.74838e20			−0.923643
$842$		−	7.40079e20i		−	2.46657i
$843$			6.67454e20i			2.20612i
$844$	1.30696e20			0.428415
$845$			2.64476e20i			0.859785i
$846$	2.24736e20			0.724569
$847$		−	3.20219e20i		−	1.02391i
$848$		−	4.44466e20i		−	1.40951i
$849$		−	8.96350e20i		−	2.81919i
$850$			8.92852e19i			0.278514i
$851$	−2.75764e19	−	2.04363e19i	−0.0853159	−	0.0632258i
$852$	−2.40508e20			−0.737994
$853$	2.97845e20			0.906457			0.453229	−	0.891394i	$-0.350272\pi$
$853$	2.97845e20			0.906457			0.453229	+	0.891394i	$0.350272\pi$
$854$	2.98846e20			0.902074
$855$	1.47739e20			0.442315
$856$			1.22208e19i			0.0362896i
$857$	−4.30255e20			−1.26724			−0.633621	−	0.773643i	$-0.718432\pi$
$857$	−4.30255e20			−1.26724			−0.633621	+	0.773643i	$0.718432\pi$
$858$		−	4.68840e19i		−	0.136966i
$859$	−6.49069e19			−0.188078			−0.0940392	−	0.995568i	$-0.529978\pi$
$859$	−6.49069e19			−0.188078			−0.0940392	+	0.995568i	$0.529978\pi$
$860$	4.01663e20			1.15444
$861$		−	3.91353e19i		−	0.111570i
$862$			4.36556e20i			1.23449i
$863$	−1.86631e20			−0.523490			−0.261745	−	0.965137i	$-0.584298\pi$
$863$	−1.86631e20			−0.523490			−0.261745	+	0.965137i	$0.584298\pi$
$864$	9.33128e19			0.259624
$865$			3.90373e20i			1.07738i
$866$			3.18749e20i			0.872616i
$867$	2.29120e20			0.622199
$868$			1.56268e20i			0.420952i
$869$	3.36178e20			0.898318
$870$			1.85795e20i			0.492492i
$871$		−	4.46237e18i		−	0.0117338i
$872$			1.48971e18i			0.00388586i
$873$			6.16365e20i			1.59492i
$874$	2.02948e20	+	1.50401e20i	0.520962	+	0.386074i
$875$	1.60821e20			0.409531
$876$	1.14650e21			2.89631
$877$	−6.59621e20			−1.65309			−0.826546	−	0.562869i	$-0.809698\pi$
$877$	−6.59621e20			−1.65309			−0.826546	+	0.562869i	$0.809698\pi$
$878$	−3.31371e20			−0.823858
$879$		−	7.73050e19i		−	0.190671i
$880$	−6.65874e20			−1.62935
$881$			3.37707e19i			0.0819802i	0.999160	+	0.0409901i	$0.0130512\pi$
$881$			3.37707e19i			0.0819802i	−0.999160	+	0.0409901i	$0.986949\pi$
$882$	−5.68896e20			−1.37010
$883$	3.48915e20			0.833671			0.416835	−	0.908982i	$-0.363139\pi$
$883$	3.48915e20			0.833671			0.416835	+	0.908982i	$0.363139\pi$
$884$		−	1.11559e19i		−	0.0264447i
$885$		−	6.25767e20i		−	1.47167i
$886$	−5.37701e20			−1.25460
$887$	−2.22877e20			−0.515942			−0.257971	−	0.966153i	$-0.583054\pi$
$887$	−2.22877e20			−0.515942			−0.257971	+	0.966153i	$0.583054\pi$
$888$		−	1.62334e18i		−	0.00372836i
$889$			1.77539e19i			0.0404559i
$890$	1.69578e20			0.383388
$891$		−	7.37431e20i		−	1.65416i
$892$	3.30127e20			0.734728
$893$		−	9.37313e19i		−	0.206978i
$894$			7.31264e20i			1.60218i
$895$			4.71798e19i			0.102564i
$896$			8.43695e18i			0.0181982i
$897$	−1.39302e19	+	1.87972e19i	−0.0298134	+	0.0402297i
$898$	−2.88145e20			−0.611895
$899$	1.43655e20			0.302694
$900$	1.41794e20			0.296455
$901$	5.23426e20			1.08588
$902$			2.69111e20i			0.553970i
$903$	−3.55678e20			−0.726511
$904$			1.37385e19i			0.0278459i
$905$	−4.74554e20			−0.954434
$906$	1.11384e21			2.22294
$907$			1.42330e20i			0.281869i	0.990019	+	0.140934i	$0.0450106\pi$
$907$			1.42330e20i			0.281869i	−0.990019	+	0.140934i	$0.954989\pi$
$908$		−	7.03062e19i		−	0.138163i
$909$	1.28515e20			0.250615
$910$	−8.19913e18			−0.0158664
$911$		−	3.40727e20i		−	0.654302i	−0.944972	−	0.327151i	$-0.893911\pi$
$911$		−	3.40727e20i		−	0.654302i	0.944972	−	0.327151i	$-0.106089\pi$
$912$		−	3.43276e20i		−	0.654153i
$913$	4.38800e20			0.829795
$914$			6.45908e20i			1.21212i
$915$	1.13270e21			2.10945
$916$		−	5.55102e20i		−	1.02590i
$917$			2.20408e20i			0.404243i
$918$			1.08026e20i			0.196621i
$919$		−	2.35176e19i		−	0.0424800i	−0.999774	−	0.0212400i	$-0.993239\pi$
$919$		−	2.35176e19i		−	0.0424800i	0.999774	−	0.0212400i	$-0.00676142\pi$
$920$	−9.29126e18	−	6.88556e18i	−0.0166556	−	0.0123431i
$921$	−1.34060e21			−2.38496
$922$	8.82023e20			1.55727
$923$	−9.75697e18			−0.0170963
$924$	−6.20507e20			−1.07905
$925$			1.59373e19i			0.0275056i
$926$	−6.31837e20			−1.08225
$927$			4.36085e20i			0.741332i
$928$	2.32533e20			0.392327
$929$	−5.33491e19			−0.0893341			−0.0446670	−	0.999002i	$-0.514223\pi$
$929$	−5.33491e19			−0.0893341			−0.0446670	+	0.999002i	$0.514223\pi$
$930$			1.17472e21i			1.95233i
$931$			2.37271e20i			0.391379i
$932$	−6.22738e20			−1.01952
$933$	3.53965e20			0.575159
$934$			7.68362e20i			1.23919i
$935$		−	7.84169e20i		−	1.25524i
$936$	−5.85918e17			−0.000930906
$937$		−	4.41898e20i		−	0.696859i	−0.937335	−	0.348429i	$-0.886715\pi$
$937$		−	4.41898e20i		−	0.696859i	0.937335	−	0.348429i	$-0.113285\pi$
$938$	−1.17133e20			−0.183342
$939$			1.41654e21i			2.20075i
$940$			2.57344e20i			0.396843i
$941$			3.16922e20i			0.485092i	0.970140	+	0.242546i	$0.0779826\pi$
$941$			3.16922e20i			0.485092i	−0.970140	+	0.242546i	$0.922017\pi$
$942$		−	1.62246e21i		−	2.46500i
$943$	7.99585e19	−	1.07895e20i	0.120582	−	0.162712i
$944$	−7.69898e20			−1.15247
$945$	4.00311e19			0.0594806
$946$	2.44580e21			3.60730
$947$	7.40623e20			1.08429			0.542147	−	0.840284i	$-0.317612\pi$
$947$	7.40623e20			1.08429			0.542147	+	0.840284i	$0.317612\pi$
$948$		−	4.75827e20i		−	0.691497i
$949$	4.65113e19			0.0670957
$950$		−	1.17291e20i		−	0.167957i
$951$	−8.89043e20			−1.26374
$952$	−4.88293e18			−0.00689004
$953$			2.49840e20i			0.349955i	0.984572	+	0.174977i	$0.0559852\pi$
$953$			2.49840e20i			0.349955i	−0.984572	+	0.174977i	$0.944015\pi$
$954$		−	1.64864e21i		−	2.29239i
$955$	−4.20818e20			−0.580859
$956$	−9.03883e20			−1.23853
$957$			5.70424e20i			0.775915i
$958$			4.67453e20i			0.631218i
$959$	−3.54387e20			−0.475058
$960$			9.74734e20i			1.29714i
$961$	1.51338e20			0.199933
$962$		−	3.94942e18i		−	0.00517974i
$963$		−	1.30249e21i		−	1.69586i
$964$		−	8.93168e20i		−	1.15450i
$965$		−	8.46654e20i		−	1.08646i
$966$	4.93411e20	+	3.65657e20i	0.628593	+	0.465837i
$967$	−1.26643e21			−1.60175			−0.800876	−	0.598830i	$-0.795632\pi$
$967$	−1.26643e21			−1.60175			−0.800876	+	0.598830i	$0.795632\pi$
$968$	5.19697e19			0.0652563
$969$	4.04260e20			0.503958
$970$	−1.39982e21			−1.73249
$971$			8.09753e20i			0.994988i	0.867467	+	0.497494i	$0.165746\pi$
$971$			8.09753e20i			0.994988i	−0.867467	+	0.497494i	$0.834254\pi$
$972$	−1.19620e21			−1.45928
$973$			5.69560e20i			0.689842i
$974$	1.83324e21			2.20449
$975$	1.08635e19			0.0129700
$976$		−	1.39360e21i		−	1.65192i
$977$		−	6.41724e20i		−	0.755242i	−0.925960	−	0.377621i	$-0.876742\pi$
$977$		−	6.41724e20i		−	0.755242i	0.925960	−	0.377621i	$-0.123258\pi$
$978$	1.97869e19			0.0231210
$979$	5.20634e20			0.604023
$980$		−	6.51440e20i		−	0.750398i
$981$		−	1.58773e20i		−	0.181591i
$982$	−1.27862e21			−1.45198
$983$		−	5.47363e20i		−	0.617164i	−0.951198	−	0.308582i	$-0.900146\pi$
$983$		−	5.47363e20i		−	0.617164i	0.951198	−	0.308582i	$-0.0998545\pi$
$984$	6.35143e18			0.00711060
$985$			9.97559e20i			1.10888i
$986$			2.69198e20i			0.297121i
$987$		−	2.27881e20i		−	0.249740i
$988$			1.46551e19i			0.0159474i
$989$	−9.80593e20	−	7.26696e20i	−1.05953	−	0.785198i
$990$	−2.46991e21			−2.64993
$991$	7.27342e20			0.774860			0.387430	−	0.921899i	$-0.373363\pi$
$991$	7.27342e20			0.774860			0.387430	+	0.921899i	$0.373363\pi$
$992$	1.47022e21			1.55526
$993$	7.42301e20			0.779714
$994$			2.56113e20i			0.267132i
$995$	1.39515e21			1.44497
$996$		−	6.21078e20i		−	0.638750i
$997$	4.77384e20			0.487531			0.243765	−	0.969834i	$-0.421617\pi$
$997$	4.77384e20			0.487531			0.243765	+	0.969834i	$0.421617\pi$
$998$	−4.21254e20			−0.427199
$999$			1.92825e19i			0.0194180i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.b.b.22.3	✓	24
23.22	odd	2	inner	23.15.b.b.22.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.b.b.22.3	✓	24	1.1	even	1	trivial
23.15.b.b.22.4	yes	24	23.22	odd	2	inner

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:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-181.785 q^{2} +3188.39 q^{3} +16661.8 q^{4} -67250.0i q^{5} -579603. q^{6} +311198. i q^{7} -50505.8 q^{8} +5.38289e6 q^{9} +O(q^{10})\)	\(q-181.785 q^{2} +3188.39 q^{3} +16661.8 q^{4} -67250.0i q^{5} -579603. q^{6} +311198. i q^{7} -50505.8 q^{8} +5.38289e6 q^{9} +1.22251e7i q^{10} +3.75331e7i q^{11} +5.31245e7 q^{12} +2.15516e6 q^{13} -5.65712e7i q^{14} -2.14420e8i q^{15} -2.63806e8 q^{16} -3.10672e8i q^{17} -9.78530e8 q^{18} +4.08118e8i q^{19} -1.12051e9i q^{20} +9.92223e8i q^{21} -6.82296e9i q^{22} +(-2.02724e9 + 2.73553e9i) q^{23} -1.61032e8 q^{24} +1.58095e9 q^{25} -3.91776e8 q^{26} +1.91279e9 q^{27} +5.18513e9i q^{28} +4.76662e9 q^{29} +3.89783e10i q^{30} +3.01377e10 q^{31} +4.87835e10 q^{32} +1.19670e11i q^{33} +5.64756e10i q^{34} +2.09281e10 q^{35} +8.96889e10 q^{36} +1.00808e10i q^{37} -7.41898e10i q^{38} +6.87151e9 q^{39} +3.39651e9i q^{40} -3.94420e10 q^{41} -1.80371e11i q^{42} +3.58465e11i q^{43} +6.25371e11i q^{44} -3.62000e11i q^{45} +(3.68523e11 - 4.97279e11i) q^{46} -2.29667e11 q^{47} -8.41119e11 q^{48} +5.81379e11 q^{49} -2.87393e11 q^{50} -9.90546e11i q^{51} +3.59089e10 q^{52} +1.68482e12i q^{53} -3.47717e11 q^{54} +2.52410e12 q^{55} -1.57173e10i q^{56} +1.30124e12i q^{57} -8.66501e11 q^{58} +2.91842e12 q^{59} -3.57262e12i q^{60} +5.28265e12i q^{61} -5.47859e12 q^{62} +1.67515e12i q^{63} -4.54592e12 q^{64} -1.44935e11i q^{65} -2.17543e13i q^{66} -2.07055e12i q^{67} -5.17637e12i q^{68} +(-6.46365e12 + 8.72195e12i) q^{69} -3.80442e12 q^{70} -4.52726e12 q^{71} -2.71867e11 q^{72} +2.15813e13 q^{73} -1.83254e12i q^{74} +5.04070e12 q^{75} +6.80000e12i q^{76} -1.16802e13 q^{77} -1.24914e12 q^{78} -8.95682e12i q^{79} +1.77410e13i q^{80} -1.96475e13 q^{81} +7.16997e12 q^{82} -1.16910e13i q^{83} +1.65323e13i q^{84} -2.08927e13 q^{85} -6.51637e13i q^{86} +1.51979e13 q^{87} -1.89564e12i q^{88} -1.38713e13i q^{89} +6.58061e13i q^{90} +6.70683e11i q^{91} +(-3.37776e13 + 4.55789e13i) q^{92} +9.60909e13 q^{93} +4.17500e13 q^{94} +2.74460e13 q^{95} +1.55541e14 q^{96} +1.14504e14i q^{97} -1.05686e14 q^{98} +2.02037e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10}) \) 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9} - 61408584 q^{12} - 75796032 q^{13} + 61604880 q^{16} + 2078422368 q^{18} + 11061310696 q^{23} - 31030771920 q^{24} - 46421538840 q^{25} - 2664699128 q^{26} + 53616275568 q^{27} + 34752122960 q^{29} - 66504561216 q^{31} - 119385530304 q^{32} + 268767587760 q^{35} - 826304812272 q^{36} - 531040317600 q^{39} - 586388906608 q^{41} - 219014904864 q^{46} + 552854704544 q^{47} + 1314198459696 q^{48} - 2682585958584 q^{49} + 1292374320920 q^{50} - 614830645416 q^{52} - 756511540560 q^{54} - 4886166096240 q^{55} - 4069251029352 q^{58} + 17167207160144 q^{59} + 7479493046896 q^{62} + 18607147469856 q^{64} - 1866422538000 q^{69} + 10676769015360 q^{70} + 15000981757712 q^{71} + 5949703969824 q^{72} + 17535136179168 q^{73} - 79533751619520 q^{75} + 53823905395344 q^{77} + 6433358950512 q^{78} + 149774494087944 q^{81} - 27510435600840 q^{82} - 170228194057680 q^{85} - 286575506293872 q^{87} + 133024510658856 q^{92} + 66347882278032 q^{93} - 205878296932464 q^{94} - 211992774994560 q^{95} + 494703018436320 q^{96} - 537018388090408 q^{98}+O(q^{100}) \) 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9 - 61408584 * q^12 - 75796032 * q^13 + 61604880 * q^16 + 2078422368 * q^18 + 11061310696 * q^23 - 31030771920 * q^24 - 46421538840 * q^25 - 2664699128 * q^26 + 53616275568 * q^27 + 34752122960 * q^29 - 66504561216 * q^31 - 119385530304 * q^32 + 268767587760 * q^35 - 826304812272 * q^36 - 531040317600 * q^39 - 586388906608 * q^41 - 219014904864 * q^46 + 552854704544 * q^47 + 1314198459696 * q^48 - 2682585958584 * q^49 + 1292374320920 * q^50 - 614830645416 * q^52 - 756511540560 * q^54 - 4886166096240 * q^55 - 4069251029352 * q^58 + 17167207160144 * q^59 + 7479493046896 * q^62 + 18607147469856 * q^64 - 1866422538000 * q^69 + 10676769015360 * q^70 + 15000981757712 * q^71 + 5949703969824 * q^72 + 17535136179168 * q^73 - 79533751619520 * q^75 + 53823905395344 * q^77 + 6433358950512 * q^78 + 149774494087944 * q^81 - 27510435600840 * q^82 - 170228194057680 * q^85 - 286575506293872 * q^87 + 133024510658856 * q^92 + 66347882278032 * q^93 - 205878296932464 * q^94 - 211992774994560 * q^95 + 494703018436320 * q^96 - 537018388090408 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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