LMFDB - Embedded newform 23.15.b.b.22.8

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			22.8
Character	$\chi$	$=$	23.22
Dual form			23.15.b.b.22.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-115.892 q^{2} -980.720 q^{3} -2953.11 q^{4} +117312. i q^{5} +113657. q^{6} -1.26769e6i q^{7} +2.24101e6 q^{8} -3.82116e6 q^{9} +O(q^{10})$	\(q-115.892 q^{2} -980.720 q^{3} -2953.11 q^{4} +117312. i q^{5} +113657. q^{6} -1.26769e6i q^{7} +2.24101e6 q^{8} -3.82116e6 q^{9} -1.35955e7i q^{10} +3.63062e6i q^{11} +2.89617e6 q^{12} -1.04665e8 q^{13} +1.46914e8i q^{14} -1.15050e8i q^{15} -2.11331e8 q^{16} -6.51567e8i q^{17} +4.42840e8 q^{18} -4.41893e8i q^{19} -3.46435e8i q^{20} +1.24325e9i q^{21} -4.20759e8i q^{22} +(-2.61802e9 + 2.17688e9i) q^{23} -2.19780e9 q^{24} -7.65859e9 q^{25} +1.21298e10 q^{26} +8.43824e9 q^{27} +3.74361e9i q^{28} +3.11517e10 q^{29} +1.33334e10i q^{30} -1.09847e10 q^{31} -1.22252e10 q^{32} -3.56062e9i q^{33} +7.55112e10i q^{34} +1.48715e11 q^{35} +1.12843e10 q^{36} +2.69163e10i q^{37} +5.12117e10i q^{38} +1.02647e11 q^{39} +2.62897e11i q^{40} +7.90506e10 q^{41} -1.44082e11i q^{42} +3.19759e11i q^{43} -1.07216e10i q^{44} -4.48267e11i q^{45} +(3.03406e11 - 2.52283e11i) q^{46} -1.98345e11 q^{47} +2.07257e11 q^{48} -9.28806e11 q^{49} +8.87567e11 q^{50} +6.39005e11i q^{51} +3.09086e11 q^{52} +9.69766e11i q^{53} -9.77922e11 q^{54} -4.25915e11 q^{55} -2.84090e12i q^{56} +4.33373e11i q^{57} -3.61022e12 q^{58} +2.50513e12 q^{59} +3.39756e11i q^{60} +2.77472e11i q^{61} +1.27304e12 q^{62} +4.84403e12i q^{63} +4.87925e12 q^{64} -1.22784e13i q^{65} +4.12646e11i q^{66} +1.37767e12i q^{67} +1.92415e12i q^{68} +(2.56754e12 - 2.13491e12i) q^{69} -1.72348e13 q^{70} -1.25001e13 q^{71} -8.56325e12 q^{72} -1.16864e12 q^{73} -3.11937e12i q^{74} +7.51093e12 q^{75} +1.30496e12i q^{76} +4.60249e12 q^{77} -1.18959e13 q^{78} -1.99534e13i q^{79} -2.47916e13i q^{80} +1.00009e13 q^{81} -9.16132e12 q^{82} +2.82637e13i q^{83} -3.67144e12i q^{84} +7.64366e13 q^{85} -3.70574e13i q^{86} -3.05511e13 q^{87} +8.13625e12i q^{88} +6.73528e13i q^{89} +5.19505e13i q^{90} +1.32682e14i q^{91} +(7.73128e12 - 6.42857e12i) q^{92} +1.07729e13 q^{93} +2.29865e13 q^{94} +5.18393e13 q^{95} +1.19895e13 q^{96} -1.04972e14i q^{97} +1.07641e14 q^{98} -1.38732e13i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−115.892			−0.905404			−0.452702	−	0.891662i	$-0.649540\pi$
$2$	−115.892			−0.905404			−0.452702	+	0.891662i	$0.649540\pi$
$3$	−980.720			−0.448432			−0.224216	−	0.974539i	$-0.571982\pi$
$3$	−980.720			−0.448432			−0.224216	+	0.974539i	$0.571982\pi$
$4$	−2953.11			−0.180243
$5$			117312.i			1.50159i	0.660533	+	0.750797i	$0.270330\pi$
$5$			117312.i			1.50159i	−0.660533	+	0.750797i	$0.729670\pi$
$6$	113657.			0.406012
$7$		−	1.26769e6i		−	1.53931i	−0.638461	−	0.769654i	$-0.720429\pi$
$7$		−	1.26769e6i		−	1.53931i	0.638461	−	0.769654i	$-0.279571\pi$
$8$	2.24101e6			1.06860
$9$	−3.82116e6			−0.798909
$10$		−	1.35955e7i		−	1.35955i
$11$			3.63062e6i			0.186308i	0.995652	+	0.0931541i	$0.0296949\pi$
$11$			3.63062e6i			0.186308i	−0.995652	+	0.0931541i	$0.970305\pi$
$12$	2.89617e6			0.0808269
$13$	−1.04665e8			−1.66800			−0.834002	−	0.551761i	$-0.813956\pi$
$13$	−1.04665e8			−1.66800			−0.834002	+	0.551761i	$0.813956\pi$
$14$			1.46914e8i			1.39370i
$15$		−	1.15050e8i		−	0.673362i
$16$	−2.11331e8			−0.787269
$17$		−	6.51567e8i		−	1.58788i	−0.607999	−	0.793938i	$-0.708027\pi$
$17$		−	6.51567e8i		−	1.58788i	0.607999	−	0.793938i	$-0.291973\pi$
$18$	4.42840e8			0.723335
$19$		−	4.41893e8i		−	0.494358i	−0.968970	−	0.247179i	$-0.920496\pi$
$19$		−	4.41893e8i		−	0.494358i	0.968970	−	0.247179i	$-0.0795035\pi$
$20$		−	3.46435e8i		−	0.270652i
$21$			1.24325e9i			0.690275i
$22$		−	4.20759e8i		−	0.168684i
$23$	−2.61802e9	+	2.17688e9i	−0.768914	+	0.639353i
$23$	−2.61802e9	+	2.17688e9i	−0.768914	+	0.639353i
$24$	−2.19780e9			−0.479193
$25$	−7.65859e9			−1.25478
$26$	1.21298e10			1.51022
$27$	8.43824e9			0.806688
$28$			3.74361e9i			0.277450i
$29$	3.11517e10			1.80591			0.902954	−	0.429738i	$-0.141394\pi$
$29$	3.11517e10			1.80591			0.902954	+	0.429738i	$0.141394\pi$
$30$			1.33334e10i			0.609665i
$31$	−1.09847e10			−0.399261			−0.199630	−	0.979871i	$-0.563974\pi$
$31$	−1.09847e10			−0.399261			−0.199630	+	0.979871i	$0.563974\pi$
$32$	−1.22252e10			−0.355801
$33$		−	3.56062e9i		−	0.0835465i
$34$			7.55112e10i			1.43767i
$35$	1.48715e11			2.31141
$36$	1.12843e10			0.143998
$37$			2.69163e10i			0.283532i	0.989900	+	0.141766i	$0.0452781\pi$
$37$			2.69163e10i			0.283532i	−0.989900	+	0.141766i	$0.954722\pi$
$38$			5.12117e10i			0.447594i
$39$	1.02647e11			0.747986
$40$			2.62897e11i			1.60460i
$41$	7.90506e10			0.405899			0.202950	−	0.979189i	$-0.434947\pi$
$41$	7.90506e10			0.405899			0.202950	+	0.979189i	$0.434947\pi$
$42$		−	1.44082e11i		−	0.624978i
$43$			3.19759e11i			1.17637i	0.808727	+	0.588185i	$0.200157\pi$
$43$			3.19759e11i			1.17637i	−0.808727	+	0.588185i	$0.799843\pi$
$44$		−	1.07216e10i		−	0.0335808i
$45$		−	4.48267e11i		−	1.19964i
$46$	3.03406e11	−	2.52283e11i	0.696178	−	0.578873i
$47$	−1.98345e11			−0.391504			−0.195752	−	0.980653i	$-0.562715\pi$
$47$	−1.98345e11			−0.391504			−0.195752	+	0.980653i	$0.562715\pi$
$48$	2.07257e11			0.353036
$49$	−9.28806e11			−1.36947
$50$	8.87567e11			1.13609
$51$			6.39005e11i			0.712054i
$52$	3.09086e11			0.300647
$53$			9.69766e11i			0.825536i	0.910836	+	0.412768i	$0.135438\pi$
$53$			9.69766e11i			0.825536i	−0.910836	+	0.412768i	$0.864562\pi$
$54$	−9.77922e11			−0.730379
$55$	−4.25915e11			−0.279759
$56$		−	2.84090e12i		−	1.64490i
$57$			4.33373e11i			0.221686i
$58$	−3.61022e12			−1.63508
$59$	2.50513e12			1.00662			0.503310	−	0.864106i	$-0.332115\pi$
$59$	2.50513e12			1.00662			0.503310	+	0.864106i	$0.332115\pi$
$60$			3.39756e11i			0.121369i
$61$			2.77472e11i			0.0882897i	0.999025	+	0.0441448i	$0.0140563\pi$
$61$			2.77472e11i			0.0882897i	−0.999025	+	0.0441448i	$0.985944\pi$
$62$	1.27304e12			0.361492
$63$			4.84403e12i			1.22977i
$64$	4.87925e12			1.10941
$65$		−	1.22784e13i		−	2.50466i
$66$			4.12646e11i			0.0756433i
$67$			1.37767e12i			0.227312i	0.993520	+	0.113656i	$0.0362562\pi$
$67$			1.37767e12i			0.227312i	−0.993520	+	0.113656i	$0.963744\pi$
$68$			1.92415e12i			0.286204i
$69$	2.56754e12	−	2.13491e12i	0.344805	−	0.286706i
$70$	−1.72348e13			−2.09276
$71$	−1.25001e13			−1.37437			−0.687186	−	0.726481i	$-0.741154\pi$
$71$	−1.25001e13			−1.37437			−0.687186	+	0.726481i	$0.741154\pi$
$72$	−8.56325e12			−0.853712
$73$	−1.16864e12			−0.105785			−0.0528923	−	0.998600i	$-0.516844\pi$
$73$	−1.16864e12			−0.105785			−0.0528923	+	0.998600i	$0.516844\pi$
$74$		−	3.11937e12i		−	0.256711i
$75$	7.51093e12			0.562685
$76$			1.30496e12i			0.0891047i
$77$	4.60249e12			0.286786
$78$	−1.18959e13			−0.677230
$79$		−	1.99534e13i		−	1.03903i	−0.854461	−	0.519515i	$-0.826113\pi$
$79$		−	1.99534e13i		−	1.03903i	0.854461	−	0.519515i	$-0.173887\pi$
$80$		−	2.47916e13i		−	1.18216i
$81$	1.00009e13			0.437164
$82$	−9.16132e12			−0.367503
$83$			2.82637e13i			1.04155i	0.853693	+	0.520777i	$0.174358\pi$
$83$			2.82637e13i			1.04155i	−0.853693	+	0.520777i	$0.825642\pi$
$84$		−	3.67144e12i		−	0.124417i
$85$	7.64366e13			2.38434
$86$		−	3.70574e13i		−	1.06509i
$87$	−3.05511e13			−0.809826
$88$			8.13625e12i			0.199088i
$89$			6.73528e13i			1.52274i	0.648318	+	0.761369i	$0.275473\pi$
$89$			6.73528e13i			1.52274i	−0.648318	+	0.761369i	$0.724527\pi$
$90$			5.19505e13i			1.08616i
$91$			1.32682e14i			2.56757i
$92$	7.73128e12	−	6.42857e12i	0.138592	−	0.115239i
$93$	1.07729e13			0.179041
$94$	2.29865e13			0.354469
$95$	5.18393e13			0.742324
$96$	1.19895e13			0.159552
$97$		−	1.04972e14i		−	1.29919i	−0.760281	−	0.649594i	$-0.774939\pi$
$97$		−	1.04972e14i		−	1.29919i	0.760281	−	0.649594i	$-0.225061\pi$
$98$	1.07641e14			1.23992
$99$		−	1.38732e13i		−	0.148843i
$100$	2.26166e13			0.226166
$101$	1.53995e14			1.43634			0.718170	−	0.695868i	$-0.244980\pi$
$101$	1.53995e14			1.43634			0.718170	+	0.695868i	$0.244980\pi$
$102$		−	7.40554e13i		−	0.644697i
$103$			1.07721e14i			0.875874i	0.899006	+	0.437937i	$0.144291\pi$
$103$			1.07721e14i			0.875874i	−0.899006	+	0.437937i	$0.855709\pi$
$104$	−2.34555e14			−1.78242
$105$	−1.45848e14			−1.03651
$106$		−	1.12388e14i		−	0.747443i
$107$			2.64768e14i			1.64884i	0.565978	+	0.824421i	$0.308499\pi$
$107$			2.64768e14i			1.64884i	−0.565978	+	0.824421i	$0.691501\pi$
$108$	−2.49190e13			−0.145400
$109$		−	2.49896e14i		−	1.36702i	−0.729942	−	0.683509i	$-0.760453\pi$
$109$		−	2.49896e14i		−	1.36702i	0.729942	−	0.683509i	$-0.239547\pi$
$110$	4.93600e13			0.253295
$111$		−	2.63973e13i		−	0.127145i
$112$			2.67901e14i			1.21185i
$113$		−	6.76463e13i		−	0.287538i	−0.989611	−	0.143769i	$-0.954078\pi$
$113$		−	6.76463e13i		−	0.287538i	0.989611	−	0.143769i	$-0.0459222\pi$
$114$		−	5.02243e13i		−	0.200715i
$115$	−2.55375e14	−	3.07125e14i	−0.960048	−	1.15460i
$116$	−9.19943e13			−0.325503
$117$	3.99941e14			1.33258
$118$	−2.90323e14			−0.911398
$119$	−8.25982e14			−2.44423
$120$		−	2.57829e14i		−	0.719553i
$121$	3.66568e14			0.965289
$122$		−	3.21567e13i		−	0.0799378i
$123$	−7.75266e13			−0.182018
$124$	3.24390e13			0.0719641
$125$		−	1.82428e14i		−	0.382580i
$126$		−	5.61383e14i		−	1.11344i
$127$	7.32189e14			1.37403			0.687017	−	0.726641i	$-0.258920\pi$
$127$	7.32189e14			1.37403			0.687017	+	0.726641i	$0.258920\pi$
$128$	−3.65166e14			−0.648666
$129$		−	3.13594e14i		−	0.527521i
$130$			1.42297e15i			2.26773i
$131$	1.79551e14			0.271199			0.135599	−	0.990764i	$-0.456704\pi$
$131$	1.79551e14			0.271199			0.135599	+	0.990764i	$0.456704\pi$
$132$			1.05149e13i			0.0150587i
$133$	−5.60181e14			−0.760969
$134$		−	1.59661e14i		−	0.205809i
$135$			9.89907e14i			1.21132i
$136$		−	1.46017e15i		−	1.69680i
$137$		−	2.43031e14i		−	0.268298i	−0.990961	−	0.134149i	$-0.957170\pi$
$137$		−	2.43031e14i		−	0.268298i	0.990961	−	0.134149i	$-0.0428300\pi$
$138$	−2.97557e14	+	2.47419e14i	−0.312188	+	0.259585i
$139$	−5.85870e14			−0.584383			−0.292191	−	0.956360i	$-0.594384\pi$
$139$	−5.85870e14			−0.584383			−0.292191	+	0.956360i	$0.594384\pi$
$140$	−4.39171e14			−0.416617
$141$	1.94521e14			0.175563
$142$	1.44866e15			1.24436
$143$		−	3.79998e14i		−	0.310763i
$144$	8.07528e14			0.628956
$145$			3.65447e15i			2.71174i
$146$	1.35436e14			0.0957778
$147$	9.10899e14			0.614114
$148$		−	7.94866e13i		−	0.0511048i
$149$		−	2.20705e15i		−	1.35365i	−0.736142	−	0.676827i	$-0.763354\pi$
$149$		−	2.20705e15i		−	1.35365i	0.736142	−	0.676827i	$-0.236646\pi$
$150$	−8.70455e14			−0.509457
$151$	1.91288e15			1.06869			0.534343	−	0.845268i	$-0.320559\pi$
$151$	1.91288e15			1.06869			0.534343	+	0.845268i	$0.320559\pi$
$152$		−	9.90286e14i		−	0.528269i
$153$			2.48974e15i			1.26857i
$154$	−5.33390e14			−0.259657
$155$		−	1.28864e15i		−	0.599527i
$156$	−3.03127e14			−0.134820
$157$		−	8.98298e14i		−	0.382053i	−0.981585	−	0.191026i	$-0.938818\pi$
$157$		−	8.98298e14i		−	0.382053i	0.981585	−	0.191026i	$-0.0611816\pi$
$158$			2.31244e15i			0.940742i
$159$		−	9.51069e14i		−	0.370196i
$160$		−	1.43416e15i		−	0.534268i
$161$	2.75961e15	+	3.31882e15i	0.984161	+	1.18359i
$162$	−1.15902e15			−0.395810
$163$	−3.32854e14			−0.108878			−0.0544390	−	0.998517i	$-0.517337\pi$
$163$	−3.32854e14			−0.108878			−0.0544390	+	0.998517i	$0.517337\pi$
$164$	−2.33445e14			−0.0731607
$165$	4.17703e14			0.125453
$166$		−	3.27552e15i		−	0.943027i
$167$	−2.56655e15			−0.708492			−0.354246	−	0.935152i	$-0.615262\pi$
$167$	−2.56655e15			−0.708492			−0.354246	+	0.935152i	$0.615262\pi$
$168$			2.78613e15i			0.737626i
$169$	7.01735e15			1.78224
$170$	−8.85837e15			−2.15879
$171$			1.68854e15i			0.394947i
$172$		−	9.44283e14i		−	0.212033i
$173$	6.88020e14			0.148347			0.0741735	−	0.997245i	$-0.476368\pi$
$173$	6.88020e14			0.148347			0.0741735	+	0.997245i	$0.476368\pi$
$174$	3.54062e15			0.733220
$175$			9.70869e15i			1.93150i
$176$		−	7.67262e14i		−	0.146675i
$177$	−2.45683e15			−0.451401
$178$		−	7.80563e15i		−	1.37869i
$179$	9.26970e15			1.57433			0.787163	−	0.616745i	$-0.211549\pi$
$179$	9.26970e15			1.57433			0.787163	+	0.616745i	$0.211549\pi$
$180$			1.32378e15i			0.216226i
$181$			2.87765e15i			0.452156i	0.974109	+	0.226078i	$0.0725904\pi$
$181$			2.87765e15i			0.452156i	−0.974109	+	0.226078i	$0.927410\pi$
$182$		−	1.53768e16i		−	2.32469i
$183$		−	2.72122e14i		−	0.0395919i
$184$	−5.86700e15	+	4.87842e15i	−0.821659	+	0.683211i
$185$	−3.15760e15			−0.425750
$186$	−1.24849e15			−0.162105
$187$	2.36559e15			0.295834
$188$	5.85734e14			0.0705660
$189$		−	1.06970e16i		−	1.24174i
$190$	−6.00774e15			−0.672104
$191$		−	1.22686e16i		−	1.32300i	−0.749944	−	0.661502i	$-0.769919\pi$
$191$		−	1.22686e16i		−	1.32300i	0.749944	−	0.661502i	$-0.230081\pi$
$192$	−4.78518e15			−0.497496
$193$	−2.44733e15			−0.245353			−0.122677	−	0.992447i	$-0.539148\pi$
$193$	−2.44733e15			−0.245353			−0.122677	+	0.992447i	$0.539148\pi$
$194$			1.21654e16i			1.17629i
$195$			1.20417e16i			1.12317i
$196$	2.74286e15			0.246838
$197$	−8.27701e15			−0.718803			−0.359402	−	0.933183i	$-0.617019\pi$
$197$	−8.27701e15			−0.718803			−0.359402	+	0.933183i	$0.617019\pi$
$198$			1.60778e15i			0.134763i
$199$			1.74506e16i			1.41201i	0.708205	+	0.706007i	$0.249505\pi$
$199$			1.74506e16i			1.41201i	−0.708205	+	0.706007i	$0.750495\pi$
$200$	−1.71630e16			−1.34086
$201$		−	1.35111e15i		−	0.101934i
$202$	−1.78468e16			−1.30047
$203$		−	3.94906e16i		−	2.77985i
$204$		−	1.88705e15i		−	0.128343i
$205$			9.27359e15i			0.609496i
$206$		−	1.24840e16i		−	0.793020i
$207$	1.00039e16	−	8.31822e15i	0.614292	−	0.510785i
$208$	2.21189e16			1.31317
$209$	1.60434e15			0.0921029
$210$	1.69025e16			0.938462
$211$	8.18326e14			0.0439490			0.0219745	−	0.999759i	$-0.493005\pi$
$211$	8.18326e14			0.0439490			0.0219745	+	0.999759i	$0.493005\pi$
$212$		−	2.86382e15i		−	0.148797i
$213$	1.22591e16			0.616312
$214$		−	3.06844e16i		−	1.49287i
$215$	−3.75116e16			−1.76643
$216$	1.89102e16			0.862024
$217$			1.39252e16i			0.614585i
$218$			2.89609e16i			1.23770i
$219$	1.14611e15			0.0474372
$220$	1.25777e15			0.0504247
$221$			6.81961e16i			2.64858i
$222$			3.05923e15i			0.115118i
$223$	−2.76618e15			−0.100866			−0.0504332	−	0.998727i	$-0.516060\pi$
$223$	−2.76618e15			−0.100866			−0.0504332	+	0.998727i	$0.516060\pi$
$224$			1.54977e16i			0.547687i
$225$	2.92647e16			1.00246
$226$			7.83965e15i			0.260338i
$227$		−	1.00477e16i		−	0.323509i	−0.986831	−	0.161755i	$-0.948285\pi$
$227$		−	1.00477e16i		−	0.323509i	0.986831	−	0.161755i	$-0.0517153\pi$
$228$		−	1.27980e15i		−	0.0399574i
$229$			7.76693e15i			0.235180i	0.993062	+	0.117590i	$0.0375169\pi$
$229$			7.76693e15i			0.235180i	−0.993062	+	0.117590i	$0.962483\pi$
$230$	2.95958e16	+	3.55932e16i	0.869231	+	1.04538i
$231$	−4.51375e15			−0.128604
$232$	6.98113e16			1.92979
$233$	−1.05571e15			−0.0283174			−0.0141587	−	0.999900i	$-0.504507\pi$
$233$	−1.05571e15			−0.0283174			−0.0141587	+	0.999900i	$0.504507\pi$
$234$	−4.63498e16			−1.20653
$235$		−	2.32682e16i		−	0.587880i
$236$	−7.39791e15			−0.181437
$237$			1.95688e16i			0.465934i
$238$	9.57245e16			2.21302
$239$	2.58779e16			0.580957			0.290479	−	0.956881i	$-0.406186\pi$
$239$	2.58779e16			0.580957			0.290479	+	0.956881i	$0.406186\pi$
$240$			2.43137e16i			0.530117i
$241$		−	4.33455e16i		−	0.917961i	−0.888446	−	0.458981i	$-0.848215\pi$
$241$		−	4.33455e16i		−	0.917961i	0.888446	−	0.458981i	$-0.151785\pi$
$242$	−4.24823e16			−0.873977
$243$	−5.01679e16			−1.00273
$244$		−	8.19404e14i		−	0.0159136i
$245$		−	1.08960e17i		−	2.05639i
$246$	8.98469e15			0.164800
$247$			4.62506e16i			0.824591i
$248$	−2.46168e16			−0.426649
$249$		−	2.77187e16i		−	0.467066i
$250$			2.11419e16i			0.346390i
$251$		−	1.26102e16i		−	0.200913i	−0.994941	−	0.100456i	$-0.967970\pi$
$251$		−	1.26102e16i		−	0.200913i	0.994941	−	0.100456i	$-0.0320303\pi$
$252$		−	1.43049e16i		−	0.221657i
$253$	−7.90344e15	−	9.50502e15i	−0.119117	−	0.143255i
$254$	−8.48547e16			−1.24406
$255$	−7.49629e16			−1.06922
$256$	−3.76218e16			−0.522108
$257$	2.24910e16			0.303723			0.151861	−	0.988402i	$-0.451473\pi$
$257$	2.24910e16			0.303723			0.151861	+	0.988402i	$0.451473\pi$
$258$			3.63430e16i			0.477620i
$259$	3.41214e16			0.436444
$260$			3.62595e16i			0.451449i
$261$	−1.19035e17			−1.44276
$262$	−2.08084e16			−0.245544
$263$		−	1.26838e17i		−	1.45734i	−0.684867	−	0.728668i	$-0.740140\pi$
$263$		−	1.26838e17i		−	1.45734i	0.684867	−	0.728668i	$-0.259860\pi$
$264$		−	7.97939e15i		−	0.0892775i
$265$	−1.13765e17			−1.23962
$266$	6.49204e16			0.688985
$267$		−	6.60542e16i		−	0.682844i
$268$		−	4.06841e15i		−	0.0409715i
$269$	1.32297e17			1.29803			0.649016	−	0.760775i	$-0.275181\pi$
$269$	1.32297e17			1.29803			0.649016	+	0.760775i	$0.275181\pi$
$270$		−	1.14722e17i		−	1.09673i
$271$	−7.98842e16			−0.744176			−0.372088	−	0.928197i	$-0.621358\pi$
$271$	−7.98842e16			−0.744176			−0.372088	+	0.928197i	$0.621358\pi$
$272$			1.37696e17i			1.25009i
$273$		−	1.30124e17i		−	1.15138i
$274$			2.81652e16i			0.242918i
$275$		−	2.78054e16i		−	0.233776i
$276$	−7.58223e15	+	6.30463e15i	−0.0621489	+	0.0516769i
$277$	−1.06183e17			−0.848589			−0.424295	−	0.905524i	$-0.639478\pi$
$277$	−1.06183e17			−0.848589			−0.424295	+	0.905524i	$0.639478\pi$
$278$	6.78974e16			0.529102
$279$	4.19743e16			0.318973
$280$	3.33272e17			2.46997
$281$			1.95710e17i			1.41471i	0.706856	+	0.707357i	$0.250113\pi$
$281$			1.95710e17i			1.41471i	−0.706856	+	0.707357i	$0.749887\pi$
$282$	−2.25434e16			−0.158955
$283$			1.47938e17i			1.01760i	0.860886	+	0.508799i	$0.169910\pi$
$283$			1.47938e17i			1.01760i	−0.860886	+	0.508799i	$0.830090\pi$
$284$	3.69141e16			0.247722
$285$	−5.08398e16			−0.332882
$286$			4.40386e16i			0.281366i
$287$		−	1.00211e17i		−	0.624804i
$288$	4.67145e16			0.284252
$289$	−2.56162e17			−1.52135
$290$		−	4.23522e17i		−	2.45522i
$291$			1.02948e17i			0.582597i
$292$	3.45113e15			0.0190670
$293$			1.39303e17i			0.751427i	0.926736	+	0.375713i	$0.122602\pi$
$293$			1.39303e17i			0.751427i	−0.926736	+	0.375713i	$0.877398\pi$
$294$	−1.05566e17			−0.556021
$295$			2.93881e17i			1.51153i
$296$			6.03196e16i			0.302982i
$297$			3.06360e16i			0.150293i
$298$			2.55779e17i			1.22560i
$299$	2.74014e17	−	2.27843e17i	1.28255	−	1.06644i
$300$	−2.21806e16			−0.101420
$301$	4.05354e17			1.81079
$302$	−2.21687e17			−0.967592
$303$	−1.51026e17			−0.644100
$304$			9.33855e16i			0.389193i
$305$	−3.25508e16			−0.132575
$306$		−	2.88540e17i		−	1.14857i
$307$	2.72833e17			1.06152			0.530759	−	0.847523i	$-0.321907\pi$
$307$	2.72833e17			1.06152			0.530759	+	0.847523i	$0.321907\pi$
$308$	−1.35916e16			−0.0516912
$309$		−	1.05645e17i		−	0.392770i
$310$			1.49342e17i			0.542814i
$311$	8.39815e16			0.298443			0.149221	−	0.988804i	$-0.452323\pi$
$311$	8.39815e16			0.298443			0.149221	+	0.988804i	$0.452323\pi$
$312$	2.30033e17			0.799296
$313$		−	3.15766e17i		−	1.07289i	−0.843936	−	0.536444i	$-0.819767\pi$
$313$		−	3.15766e17i		−	1.07289i	0.843936	−	0.536444i	$-0.180233\pi$
$314$			1.04105e17i			0.345912i
$315$	−5.68263e17			−1.84661
$316$			5.89247e16i			0.187278i
$317$	1.83104e17			0.569223			0.284612	−	0.958643i	$-0.408135\pi$
$317$	1.83104e17			0.569223			0.284612	+	0.958643i	$0.408135\pi$
$318$			1.10221e17i			0.335177i
$319$			1.13100e17i			0.336455i
$320$			5.72394e17i			1.66589i
$321$		−	2.59663e17i		−	0.739393i
$322$	−3.19816e17	−	3.84624e17i	−0.891063	−	1.07163i
$323$	−2.87923e17			−0.784979
$324$	−2.95338e16			−0.0787960
$325$	8.01584e17			2.09298
$326$	3.85750e16			0.0985786
$327$			2.45078e17i			0.613014i
$328$	1.77153e17			0.433743
$329$			2.51439e17i			0.602645i
$330$	−4.84084e16			−0.113586
$331$	3.96227e17			0.910225			0.455112	−	0.890434i	$-0.349599\pi$
$331$	3.96227e17			0.910225			0.455112	+	0.890434i	$0.349599\pi$
$332$		−	8.34656e16i		−	0.187733i
$333$		−	1.02851e17i		−	0.226516i
$334$	2.97442e17			0.641471
$335$	−1.61618e17			−0.341330
$336$		−	2.62736e17i		−	0.543432i
$337$			1.35773e16i			0.0275045i	0.999905	+	0.0137523i	$0.00437762\pi$
$337$			1.35773e16i			0.0275045i	−0.999905	+	0.0137523i	$0.995622\pi$
$338$	−8.13252e17			−1.61365
$339$			6.63421e16i			0.128941i
$340$	−2.25725e17			−0.429762
$341$		−	3.98813e16i		−	0.0743855i
$342$		−	1.95688e17i		−	0.357587i
$343$			3.17660e17i			0.568727i
$344$			7.16583e17i			1.25706i
$345$	2.50451e17	+	3.01203e17i	0.430516	+	0.517757i
$346$	−7.97359e16			−0.134314
$347$	1.65318e15			0.00272906			0.00136453	−	0.999999i	$-0.499566\pi$
$347$	1.65318e15			0.00272906			0.00136453	+	0.999999i	$0.499566\pi$
$348$	9.02206e16			0.145966
$349$	−6.62050e16			−0.104982			−0.0524908	−	0.998621i	$-0.516716\pi$
$349$	−6.62050e16			−0.104982			−0.0524908	+	0.998621i	$0.516716\pi$
$350$		−	1.12516e18i		−	1.74879i
$351$	−8.83187e17			−1.34556
$352$		−	4.43851e16i		−	0.0662885i
$353$	−5.66954e17			−0.830089			−0.415044	−	0.909801i	$-0.636234\pi$
$353$	−5.66954e17			−0.830089			−0.415044	+	0.909801i	$0.636234\pi$
$354$	2.84726e17			0.408700
$355$		−	1.46641e18i		−	2.06375i
$356$		−	1.98900e17i		−	0.274464i
$357$	8.10058e17			1.09607
$358$	−1.07428e18			−1.42540
$359$			1.11872e18i			1.45566i	0.685760	+	0.727828i	$0.259470\pi$
$359$			1.11872e18i			1.45566i	−0.685760	+	0.727828i	$0.740530\pi$
$360$		−	1.00457e18i		−	1.28193i
$361$	6.03738e17			0.755610
$362$		−	3.33496e17i		−	0.409384i
$363$	−3.59501e17			−0.432866
$364$		−	3.91825e17i		−	0.462788i
$365$		−	1.37096e17i		−	0.158845i
$366$			3.15367e16i			0.0358467i
$367$			6.49618e17i			0.724428i	0.932095	+	0.362214i	$0.117979\pi$
$367$			6.49618e17i			0.724428i	−0.932095	+	0.362214i	$0.882021\pi$
$368$	5.53268e17	−	4.60043e17i	0.605342	−	0.503343i
$369$	−3.02065e17			−0.324277
$370$	3.65940e17			0.385476
$371$	1.22936e18			1.27075
$372$	−3.18136e16			−0.0322710
$373$			1.09191e18i			1.08699i	0.839413	+	0.543494i	$0.182899\pi$
$373$			1.09191e18i			1.08699i	−0.839413	+	0.543494i	$0.817101\pi$
$374$	−2.74152e17			−0.267849
$375$			1.78911e17i			0.171561i
$376$	−4.44493e17			−0.418360
$377$	−3.26048e18			−3.01226
$378$			1.23970e18i			1.12428i
$379$			1.09410e17i			0.0974053i	0.998813	+	0.0487026i	$0.0155087\pi$
$379$			1.09410e17i			0.0974053i	−0.998813	+	0.0487026i	$0.984491\pi$
$380$	−1.53087e17			−0.133799
$381$	−7.18073e17			−0.616160
$382$			1.42183e18i			1.19785i
$383$		−	1.89689e18i		−	1.56910i	−0.620065	−	0.784551i	$-0.712894\pi$
$383$		−	1.89689e18i		−	1.56910i	0.620065	−	0.784551i	$-0.287106\pi$
$384$	3.58126e17			0.290882
$385$			5.39927e17i			0.430635i
$386$	2.83626e17			0.222144
$387$		−	1.22185e18i		−	0.939812i
$388$			3.09994e17i			0.234170i
$389$		−	1.92410e18i		−	1.42752i	−0.700391	−	0.713759i	$-0.746991\pi$
$389$		−	1.92410e18i		−	1.42752i	0.700391	−	0.713759i	$-0.253009\pi$
$390$		−	1.39553e18i		−	1.01692i
$391$	1.41839e18	+	1.70581e18i	1.01521	+	1.22094i
$392$	−2.08146e18			−1.46341
$393$	−1.76089e17			−0.121614
$394$	9.59237e17			0.650807
$395$	2.34078e18			1.56020
$396$			4.09689e16i			0.0268280i
$397$	2.45903e18			1.58209			0.791045	−	0.611758i	$-0.209537\pi$
$397$	2.45903e18			1.58209			0.791045	+	0.611758i	$0.209537\pi$
$398$		−	2.02238e18i		−	1.27844i
$399$	5.49381e17			0.341243
$400$	1.61850e18			0.987852
$401$			2.33941e18i			1.40313i	0.712608	+	0.701563i	$0.247514\pi$
$401$			2.33941e18i			1.40313i	−0.712608	+	0.701563i	$0.752486\pi$
$402$			1.56583e17i			0.0922914i
$403$	1.14971e18			0.665968
$404$	−4.54764e17			−0.258891
$405$			1.17323e18i			0.656443i
$406$			4.57663e18i			2.51689i
$407$	−9.77226e16			−0.0528244
$408$			1.43202e18i			0.760899i
$409$	9.41268e17			0.491643			0.245822	−	0.969315i	$-0.420942\pi$
$409$	9.41268e17			0.491643			0.245822	+	0.969315i	$0.420942\pi$
$410$		−	1.07473e18i		−	0.551840i
$411$			2.38345e17i			0.120313i
$412$		−	3.18113e17i		−	0.157871i
$413$		−	3.17572e18i		−	1.54950i
$414$	−1.15936e18	+	9.64012e17i	−0.556182	+	0.462466i
$415$	−3.31567e18			−1.56399
$416$	1.27955e18			0.593477
$417$	5.74574e17			0.262056
$418$	−1.85930e17			−0.0833903
$419$			6.14822e17i			0.271176i	0.990765	+	0.135588i	$0.0432923\pi$
$419$			6.14822e17i			0.271176i	−0.990765	+	0.135588i	$0.956708\pi$
$420$	4.30704e17			0.186824
$421$			7.62816e17i			0.325421i	0.986674	+	0.162710i	$0.0520236\pi$
$421$			7.62816e17i			0.325421i	−0.986674	+	0.162710i	$0.947976\pi$
$422$	−9.48372e16			−0.0397916
$423$	7.57907e17			0.312776
$424$			2.17326e18i			0.882165i
$425$			4.99008e18i			1.99244i
$426$	−1.42073e18			−0.558012
$427$	3.51747e17			0.135905
$428$		−	7.81888e17i		−	0.297193i
$429$			3.72672e17i			0.139356i
$430$	4.34728e18			1.59933
$431$		−	4.62890e18i		−	1.67547i	−0.546077	−	0.837735i	$-0.683879\pi$
$431$		−	4.62890e18i		−	1.67547i	0.546077	−	0.837735i	$-0.316121\pi$
$432$	−1.78326e18			−0.635080
$433$			7.60758e17i			0.266582i	0.991077	+	0.133291i	$0.0425545\pi$
$433$			7.60758e17i			0.266582i	−0.991077	+	0.133291i	$0.957445\pi$
$434$		−	1.61381e18i		−	0.556448i
$435$		−	3.58401e18i		−	1.21603i
$436$			7.37970e17i			0.246396i
$437$	9.61949e17	+	1.15688e18i	0.316069	+	0.380118i
$438$	−1.32825e17			−0.0429498
$439$	4.72923e17			0.150501			0.0752504	−	0.997165i	$-0.476024\pi$
$439$	4.72923e17			0.150501			0.0752504	+	0.997165i	$0.476024\pi$
$440$	−9.54480e17			−0.298950
$441$	3.54911e18			1.09408
$442$		−	7.90337e18i		−	2.39804i
$443$	−2.31008e18			−0.689923			−0.344961	−	0.938617i	$-0.612108\pi$
$443$	−2.31008e18			−0.689923			−0.344961	+	0.938617i	$0.612108\pi$
$444$			7.79541e16i			0.0229170i
$445$	−7.90129e18			−2.28653
$446$	3.20577e17			0.0913248
$447$			2.16450e18i			0.607022i
$448$		−	6.18536e18i		−	1.70773i
$449$	5.61677e18			1.52673			0.763365	−	0.645967i	$-0.223546\pi$
$449$	5.61677e18			1.52673			0.763365	+	0.645967i	$0.223546\pi$
$450$	−3.39153e18			−0.907629
$451$			2.87003e17i			0.0756223i
$452$			1.99767e17i			0.0518268i
$453$	−1.87600e18			−0.479233
$454$			1.16445e18i			0.292907i
$455$	−1.55652e19			−3.85545
$456$			9.71193e17i			0.236893i
$457$			2.17125e18i			0.521551i	0.965400	+	0.260775i	$0.0839782\pi$
$457$			2.17125e18i			0.521551i	−0.965400	+	0.260775i	$0.916022\pi$
$458$		−	9.00123e17i		−	0.212933i
$459$		−	5.49808e18i		−	1.28092i
$460$	7.54149e17	+	9.06972e17i	0.173042	+	0.208108i
$461$	−6.06885e17			−0.137151			−0.0685756	−	0.997646i	$-0.521845\pi$
$461$	−6.06885e17			−0.137151			−0.0685756	+	0.997646i	$0.521845\pi$
$462$	5.23106e17			0.116438
$463$	4.78270e18			1.04859			0.524295	−	0.851537i	$-0.324329\pi$
$463$	4.78270e18			1.04859			0.524295	+	0.851537i	$0.324329\pi$
$464$	−6.58331e18			−1.42174
$465$			1.26379e18i			0.268847i
$466$	1.22348e17			0.0256387
$467$		−	1.31013e17i		−	0.0270455i	−0.999909	−	0.0135228i	$-0.995695\pi$
$467$		−	1.31013e17i		−	0.0270455i	0.999909	−	0.0135228i	$-0.00430456\pi$
$468$	−1.18107e18			−0.240189
$469$	1.74646e18			0.349903
$470$			2.69660e18i			0.532269i
$471$			8.80980e17i			0.171324i
$472$	5.61402e18			1.07567
$473$	−1.16092e18			−0.219167
$474$		−	2.26786e18i		−	0.421859i
$475$			3.38427e18i			0.620312i
$476$	2.43921e18			0.440556
$477$		−	3.70563e18i		−	0.659528i
$478$	−2.99904e18			−0.526001
$479$		−	8.91649e17i		−	0.154115i	−0.997027	−	0.0770576i	$-0.975447\pi$
$479$		−	8.91649e17i		−	0.154115i	0.997027	−	0.0770576i	$-0.0245525\pi$
$480$			1.40651e18i			0.239583i
$481$		−	2.81718e18i		−	0.472933i
$482$			5.02338e18i			0.831126i
$483$	−2.70640e18	−	3.25484e18i	−0.441329	−	0.530762i
$484$	−1.08252e18			−0.173987
$485$	1.23145e19			1.95085
$486$	5.81405e18			0.907873
$487$	9.73942e18			1.49910			0.749550	−	0.661948i	$-0.230270\pi$
$487$	9.73942e18			1.49910			0.749550	+	0.661948i	$0.230270\pi$
$488$			6.21817e17i			0.0943461i
$489$	3.26436e17			0.0488243
$490$			1.26276e19i			1.86186i
$491$	6.77949e17			0.0985432			0.0492716	−	0.998785i	$-0.484310\pi$
$491$	6.77949e17			0.0985432			0.0492716	+	0.998785i	$0.484310\pi$
$492$	2.28944e17			0.0328076
$493$		−	2.02974e19i		−	2.86756i
$494$		−	5.36006e18i		−	0.746588i
$495$	1.62749e18			0.223502
$496$	2.32141e18			0.314325
$497$			1.58462e19i			2.11558i
$498$			3.21237e18i			0.422883i
$499$	−1.40037e19			−1.81777			−0.908884	−	0.417050i	$-0.863064\pi$
$499$	−1.40037e19			−1.81777			−0.908884	+	0.417050i	$0.863064\pi$
$500$			5.38731e17i			0.0689575i
$501$	2.51707e18			0.317710
$502$			1.46142e18i			0.181907i
$503$			6.08988e17i			0.0747537i	0.999301	+	0.0373769i	$0.0119002\pi$
$503$			6.08988e17i			0.0747537i	−0.999301	+	0.0373769i	$0.988100\pi$
$504$			1.08555e19i			1.31413i
$505$			1.80655e19i			2.15680i
$506$	9.15943e17	+	1.10155e18i	0.107849	+	0.129704i
$507$	−6.88205e18			−0.799213
$508$	−2.16223e18			−0.247660
$509$	9.50588e18			1.07391			0.536955	−	0.843611i	$-0.319575\pi$
$509$	9.50588e18			1.07391			0.536955	+	0.843611i	$0.319575\pi$
$510$	8.68758e18			0.968072
$511$			1.48147e18i			0.162835i
$512$	1.03429e19			1.12138
$513$		−	3.72880e18i		−	0.398793i
$514$	−2.60652e18			−0.274992
$515$	−1.26370e19			−1.31521
$516$			9.26077e17i			0.0950822i
$517$		−	7.20115e17i		−	0.0729403i
$518$	−3.95438e18			−0.395158
$519$	−6.74756e17			−0.0665235
$520$		−	2.75161e19i		−	2.67648i
$521$		−	9.43734e18i		−	0.905703i	−0.891586	−	0.452851i	$-0.850407\pi$
$521$		−	9.43734e18i		−	0.905703i	0.891586	−	0.452851i	$-0.149593\pi$
$522$	1.37952e19			1.30628
$523$		−	5.31856e17i		−	0.0496915i	−0.999691	−	0.0248458i	$-0.992091\pi$
$523$		−	5.31856e17i		−	0.0496915i	0.999691	−	0.0248458i	$-0.00790947\pi$
$524$	−5.30232e17			−0.0488818
$525$		−	9.52151e18i		−	0.866145i
$526$			1.46995e19i			1.31948i
$527$			7.15727e18i			0.633976i
$528$			7.52469e17i			0.0657735i
$529$	2.11519e18	−	1.13982e19i	0.182456	−	0.983214i
$530$	1.31844e19			1.12236
$531$	−9.57248e18			−0.804198
$532$	1.65427e18			0.137160
$533$	−8.27382e18			−0.677042
$534$			7.65514e18i			0.618250i
$535$	−3.10604e19			−2.47589
$536$			3.08738e18i			0.242905i
$537$	−9.09099e18			−0.705978
$538$	−1.53322e19			−1.17524
$539$		−	3.37214e18i		−	0.255143i
$540$		−	2.92330e18i		−	0.218332i
$541$	2.29776e19			1.69404			0.847022	−	0.531558i	$-0.178393\pi$
$541$	2.29776e19			1.69404			0.847022	+	0.531558i	$0.178393\pi$
$542$	9.25791e18			0.673780
$543$		−	2.82217e18i		−	0.202761i
$544$			7.96555e18i			0.564967i
$545$	2.93158e19			2.05271
$546$			1.50803e19i			1.04247i
$547$	3.55870e17			0.0242874			0.0121437	−	0.999926i	$-0.496134\pi$
$547$	3.55870e17			0.0242874			0.0121437	+	0.999926i	$0.496134\pi$
$548$			7.17695e17i			0.0483589i
$549$		−	1.06026e18i		−	0.0705354i
$550$			3.22242e18i			0.211662i
$551$		−	1.37657e19i		−	0.892765i
$552$	5.75389e18	−	4.78437e18i	0.368458	−	0.306373i
$553$	−2.52947e19			−1.59939
$554$	1.23058e19			0.768316
$555$	3.09672e18			0.190920
$556$	1.73014e18			0.105331
$557$			1.89229e19i			1.13763i	0.822466	+	0.568815i	$0.192598\pi$
$557$			1.89229e19i			1.13763i	−0.822466	+	0.568815i	$0.807402\pi$
$558$	−4.86447e18			−0.288799
$559$		−	3.34675e19i		−	1.96219i
$560$	−3.14280e19			−1.81971
$561$	−2.31998e18			−0.132661
$562$		−	2.26812e19i		−	1.28089i
$563$			2.10175e19i			1.17226i	0.810218	+	0.586129i	$0.199349\pi$
$563$			2.10175e19i			1.17226i	−0.810218	+	0.586129i	$0.800651\pi$
$564$	−5.74441e17			−0.0316440
$565$	7.93573e18			0.431765
$566$		−	1.71448e19i		−	0.921337i
$567$		−	1.26780e19i		−	0.672931i
$568$	−2.80128e19			−1.46865
$569$			9.55774e18i			0.494959i	0.968893	+	0.247480i	$0.0796024\pi$
$569$			9.55774e18i			0.494959i	−0.968893	+	0.247480i	$0.920398\pi$
$570$	5.89192e18			0.301393
$571$			3.78824e19i			1.91419i	0.289777	+	0.957094i	$0.406419\pi$
$571$			3.78824e19i			1.91419i	−0.289777	+	0.957094i	$0.593581\pi$
$572$			1.12217e18i			0.0560129i
$573$			1.20320e19i			0.593277i
$574$			1.16137e19i			0.565700i
$575$	2.00503e19	−	1.66719e19i	0.964820	−	0.802249i
$576$	−1.86444e19			−0.886319
$577$	−3.08347e19			−1.44813			−0.724066	−	0.689730i	$-0.757729\pi$
$577$	−3.08347e19			−1.44813			−0.724066	+	0.689730i	$0.757729\pi$
$578$	2.96870e19			1.37744
$579$	2.40015e18			0.110024
$580$		−	1.07920e19i		−	0.488773i
$581$	3.58295e19			1.60327
$582$		−	1.19309e19i		−	0.527486i
$583$	−3.52085e18			−0.153804
$584$	−2.61894e18			−0.113041
$585$			4.69178e19i			2.00100i
$586$		−	1.61440e19i		−	0.680345i
$587$	−1.38541e19			−0.576916			−0.288458	−	0.957493i	$-0.593143\pi$
$587$	−1.38541e19			−0.576916			−0.288458	+	0.957493i	$0.593143\pi$
$588$	−2.68998e18			−0.110690
$589$			4.85406e18i			0.197378i
$590$		−	3.40584e19i		−	1.36855i
$591$	8.11743e18			0.322334
$592$		−	5.68824e18i		−	0.223216i
$593$	−1.07259e19			−0.415959			−0.207979	−	0.978133i	$-0.566689\pi$
$593$	−1.07259e19			−0.415959			−0.207979	+	0.978133i	$0.566689\pi$
$594$		−	3.55046e18i		−	0.136075i
$595$		−	9.68976e19i		−	3.67024i
$596$			6.51765e18i			0.243987i
$597$		−	1.71142e19i		−	0.633192i
$598$	−3.17560e19	+	2.64051e19i	−1.16123	+	0.965562i
$599$	−1.04668e19			−0.378292			−0.189146	−	0.981949i	$-0.560572\pi$
$599$	−1.04668e19			−0.378292			−0.189146	+	0.981949i	$0.560572\pi$
$600$	1.68321e19			0.601283
$601$	2.40165e19			0.847986			0.423993	−	0.905665i	$-0.360628\pi$
$601$	2.40165e19			0.847986			0.423993	+	0.905665i	$0.360628\pi$
$602$	−4.69772e19			−1.63950
$603$		−	5.26430e18i		−	0.181602i
$604$	−5.64895e18			−0.192623
$605$			4.30029e19i			1.44947i
$606$	1.75027e19			0.583171
$607$	3.74937e18			0.123492			0.0617458	−	0.998092i	$-0.480333\pi$
$607$	3.74937e18			0.123492			0.0617458	+	0.998092i	$0.480333\pi$
$608$			5.40223e18i			0.175893i
$609$			3.87292e19i			1.24657i
$610$	3.77236e18			0.120034
$611$	2.07597e19			0.653030
$612$		−	7.35247e18i		−	0.228651i
$613$		−	1.39428e19i		−	0.428674i	−0.976760	−	0.214337i	$-0.931241\pi$
$613$		−	1.39428e19i		−	0.428674i	0.976760	−	0.214337i	$-0.0687590\pi$
$614$	−3.16190e19			−0.961103
$615$		−	9.09480e18i		−	0.273317i
$616$	1.03142e19			0.306458
$617$			5.35871e19i			1.57422i	0.616815	+	0.787108i	$0.288423\pi$
$617$			5.35871e19i			1.57422i	−0.616815	+	0.787108i	$0.711577\pi$
$618$			1.22433e19i			0.355615i
$619$		−	2.23612e19i		−	0.642185i	−0.947048	−	0.321093i	$-0.895950\pi$
$619$		−	2.23612e19i		−	0.642185i	0.947048	−	0.321093i	$-0.104050\pi$
$620$			3.80548e18i			0.108061i
$621$	−2.20915e19	+	1.83691e19i	−0.620273	+	0.515758i
$622$	−9.73276e18			−0.270211
$623$	8.53822e19			2.34396
$624$	−2.16925e19			−0.588866
$625$	−2.53433e19			−0.680303
$626$			3.65946e19i			0.971397i
$627$	−1.57341e18			−0.0413019
$628$			2.65277e18i			0.0688624i
$629$	1.75377e19			0.450214
$630$	6.58569e19			1.67193
$631$			3.38659e19i			0.850271i	0.905130	+	0.425135i	$0.139774\pi$
$631$			3.38659e19i			0.850271i	−0.905130	+	0.425135i	$0.860226\pi$
$632$		−	4.47159e19i		−	1.11031i
$633$	−8.02549e17			−0.0197081
$634$	−2.12202e19			−0.515377
$635$			8.58946e19i			2.06324i
$636$			2.80861e18i			0.0667254i
$637$	9.72133e19			2.28428
$638$		−	1.31073e19i		−	0.304628i
$639$	4.77648e19			1.09800
$640$		−	4.28384e19i		−	0.974032i
$641$		−	1.08329e19i		−	0.243633i	−0.992553	−	0.121817i	$-0.961128\pi$
$641$		−	1.08329e19i		−	0.243633i	0.992553	−	0.121817i	$-0.0388720\pi$
$642$			3.00928e19i			0.669449i
$643$		−	3.64162e19i		−	0.801342i	−0.916222	−	0.400671i	$-0.868777\pi$
$643$		−	3.64162e19i		−	0.801342i	0.916222	−	0.400671i	$-0.131223\pi$
$644$	−8.14941e18	−	9.80084e18i	−0.177388	−	0.213335i
$645$	3.67884e19			0.792123
$646$	3.33678e19			0.710723
$647$	−2.20342e19			−0.464267			−0.232133	−	0.972684i	$-0.574571\pi$
$647$	−2.20342e19			−0.464267			−0.232133	+	0.972684i	$0.574571\pi$
$648$	2.24122e19			0.467153
$649$			9.09516e18i			0.187542i
$650$	−9.28970e19			−1.89500
$651$		−	1.36567e19i		−	0.275599i
$652$	9.82952e17			0.0196245
$653$	1.39521e19			0.275579			0.137789	−	0.990462i	$-0.456000\pi$
$653$	1.39521e19			0.275579			0.137789	+	0.990462i	$0.456000\pi$
$654$		−	2.84025e19i		−	0.555026i
$655$			2.10634e19i			0.407230i
$656$	−1.67058e19			−0.319552
$657$	4.46557e18			0.0845122
$658$		−	2.91397e19i		−	0.545637i
$659$			7.29680e18i			0.135187i	0.997713	+	0.0675935i	$0.0215321\pi$
$659$			7.29680e18i			0.135187i	−0.997713	+	0.0675935i	$0.978468\pi$
$660$	−1.23352e18			−0.0226120
$661$			9.05186e19i			1.64183i	0.571052	+	0.820914i	$0.306535\pi$
$661$			9.05186e19i			1.64183i	−0.571052	+	0.820914i	$0.693465\pi$
$662$	−4.59195e19			−0.824121
$663$		−	6.68813e19i		−	1.18771i
$664$			6.33392e19i			1.11300i
$665$		−	6.57160e19i		−	1.14267i
$666$			1.19196e19i			0.205089i
$667$	−8.15556e19	+	6.78136e19i	−1.38859	+	1.15461i
$668$	7.57930e18			0.127701
$669$	2.71285e18			0.0452317
$670$	1.87301e19			0.309042
$671$	−1.00739e18			−0.0164491
$672$		−	1.51989e19i		−	0.245600i
$673$	1.11396e19			0.178142			0.0890708	−	0.996025i	$-0.471610\pi$
$673$	1.11396e19			0.178142			0.0890708	+	0.996025i	$0.471610\pi$
$674$		−	1.57350e18i		−	0.0249027i
$675$	−6.46250e19			−1.01222
$676$	−2.07230e19			−0.321237
$677$		−	4.97814e18i		−	0.0763742i	−0.999271	−	0.0381871i	$-0.987842\pi$
$677$		−	4.97814e18i		−	0.0763742i	0.999271	−	0.0381871i	$-0.0121583\pi$
$678$		−	7.68851e18i		−	0.116744i
$679$	−1.33072e20			−1.99985
$680$	1.71295e20			2.54790
$681$			9.85401e18i			0.145072i
$682$			4.62191e18i			0.0673489i
$683$	−4.38741e19			−0.632795			−0.316398	−	0.948627i	$-0.602473\pi$
$683$	−4.38741e19			−0.632795			−0.316398	+	0.948627i	$0.602473\pi$
$684$		−	4.98644e18i		−	0.0711866i
$685$	2.85104e19			0.402874
$686$		−	3.68142e19i		−	0.514928i
$687$		−	7.61718e18i		−	0.105462i
$688$		−	6.75750e19i		−	0.926119i
$689$		−	1.01500e20i		−	1.37700i
$690$	−2.90252e19	−	3.49070e19i	−0.389791	−	0.468780i
$691$	−1.38111e20			−1.83604			−0.918022	−	0.396530i	$-0.870214\pi$
$691$	−1.38111e20			−1.83604			−0.918022	+	0.396530i	$0.870214\pi$
$692$	−2.03180e18			−0.0267385
$693$	−1.75868e19			−0.229116
$694$	−1.91590e17			−0.00247090
$695$		−	6.87295e19i		−	0.877505i
$696$	−6.84653e19			−0.865378
$697$		−	5.15068e19i		−	0.644518i
$698$	7.67261e18			0.0950507
$699$	1.03536e18			0.0126984
$700$		−	2.86708e19i		−	0.348140i
$701$			4.81174e19i			0.578463i	0.957259	+	0.289232i	$0.0933998\pi$
$701$			4.81174e19i			0.578463i	−0.957259	+	0.289232i	$0.906600\pi$
$702$	1.02354e20			1.21827
$703$	1.18941e19			0.140166
$704$			1.77147e19i			0.206693i
$705$			2.28196e19i			0.263624i
$706$	6.57052e19			0.751566
$707$		−	1.95217e20i		−	2.21097i
$708$	7.25528e18			0.0813619
$709$		−	1.48033e20i		−	1.64374i	−0.569674	−	0.821871i	$-0.692930\pi$
$709$		−	1.48033e20i		−	1.64374i	0.569674	−	0.821871i	$-0.307070\pi$
$710$			1.69945e20i			1.86853i
$711$			7.62452e19i			0.830091i
$712$			1.50938e20i			1.62719i
$713$	2.87581e19	−	2.39124e19i	0.306997	−	0.255268i
$714$	−9.38790e19			−0.992387
$715$	4.45783e19			0.466639
$716$	−2.73744e19			−0.283762
$717$	−2.53790e19			−0.260520
$718$		−	1.29650e20i		−	1.31796i
$719$	−1.09390e20			−1.10122			−0.550611	−	0.834762i	$-0.685605\pi$
$719$	−1.09390e20			−1.10122			−0.550611	+	0.834762i	$0.685605\pi$
$720$			9.47328e19i			0.944437i
$721$	1.36557e20			1.34824
$722$	−6.99682e19			−0.684133
$723$			4.25098e19i			0.411643i
$724$		−	8.49802e18i		−	0.0814981i
$725$	−2.38578e20			−2.26602
$726$	4.16632e19			0.391919
$727$			2.47147e19i			0.230258i	0.993351	+	0.115129i	$0.0367282\pi$
$727$			2.47147e19i			0.230258i	−0.993351	+	0.115129i	$0.963272\pi$
$728$			2.97342e20i			2.74370i
$729$	1.36664e18			0.0124900
$730$			1.58883e19i			0.143819i
$731$	2.08344e20			1.86793
$732$			8.03606e17i			0.00713618i
$733$			8.59753e19i			0.756216i	0.925762	+	0.378108i	$0.123425\pi$
$733$			8.59753e19i			0.756216i	−0.925762	+	0.378108i	$0.876575\pi$
$734$		−	7.52853e19i		−	0.655900i
$735$			1.06859e20i			0.922149i
$736$	3.20058e19	−	2.66129e19i	0.273580	−	0.227482i
$737$	−5.00180e18			−0.0423501
$738$	3.50068e19			0.293601
$739$	−5.15895e19			−0.428598			−0.214299	−	0.976768i	$-0.568747\pi$
$739$	−5.15895e19			−0.428598			−0.214299	+	0.976768i	$0.568747\pi$
$740$	9.32473e18			0.0767387
$741$		−	4.53589e19i		−	0.369773i
$742$	−1.42473e20			−1.15055
$743$		−	1.49056e20i		−	1.19241i	−0.802831	−	0.596207i	$-0.796674\pi$
$743$		−	1.49056e20i		−	1.19241i	0.802831	−	0.596207i	$-0.203326\pi$
$744$	2.41422e19			0.191323
$745$	2.58913e20			2.03264
$746$		−	1.26543e20i		−	0.984164i
$747$		−	1.08000e20i		−	0.832107i
$748$	−6.98584e18			−0.0533221
$749$	3.35643e20			2.53807
$750$		−	2.07343e19i		−	0.155332i
$751$			1.65239e19i			0.122640i	0.998118	+	0.0613202i	$0.0195311\pi$
$751$			1.65239e19i			0.122640i	−0.998118	+	0.0613202i	$0.980469\pi$
$752$	4.19164e19			0.308219
$753$			1.23671e19i			0.0900957i
$754$	3.77863e20			2.72731
$755$			2.24404e20i			1.60473i
$756$			3.15895e19i			0.223816i
$757$		−	1.84168e19i		−	0.129283i	−0.997909	−	0.0646416i	$-0.979410\pi$
$757$		−	1.84168e19i		−	0.129283i	0.997909	−	0.0646416i	$-0.0205904\pi$
$758$		−	1.26797e19i		−	0.0881911i
$759$	7.75106e18	+	9.32176e18i	0.0534157	+	0.0642400i
$760$	1.16172e20			0.793246
$761$	2.35170e19			0.159107			0.0795536	−	0.996831i	$-0.474651\pi$
$761$	2.35170e19			0.159107			0.0795536	+	0.996831i	$0.474651\pi$
$762$	8.32187e19			0.557874
$763$	−3.16790e20			−2.10426
$764$			3.62304e19i			0.238463i
$765$	−2.92076e20			−1.90487
$766$			2.19834e20i			1.42067i
$767$	−2.62199e20			−1.67905
$768$	3.68965e19			0.234130
$769$			1.37379e20i			0.863846i	0.901911	+	0.431923i	$0.142165\pi$
$769$			1.37379e20i			0.863846i	−0.901911	+	0.431923i	$0.857835\pi$
$770$		−	6.25730e19i		−	0.389899i
$771$	−2.20574e19			−0.136199
$772$	7.22724e18			0.0442233
$773$			1.45925e20i			0.884857i	0.896804	+	0.442429i	$0.145883\pi$
$773$			1.45925e20i			0.884857i	−0.896804	+	0.442429i	$0.854117\pi$
$774$			1.41602e20i			0.850910i
$775$	8.41273e19			0.500985
$776$		−	2.35244e20i		−	1.38831i
$777$	−3.34635e19			−0.195715
$778$			2.22988e20i			1.29248i
$779$		−	3.49319e19i		−	0.200660i
$780$		−	3.55605e19i		−	0.202444i
$781$		−	4.53830e19i		−	0.256057i
$782$	−1.64379e20	−	1.97690e20i	−0.919178	−	1.10544i
$783$	2.62865e20			1.45680
$784$	1.96285e20			1.07814
$785$	1.05381e20			0.573688
$786$	2.04072e19			0.110110
$787$		−	6.16391e19i		−	0.329635i	−0.986324	−	0.164818i	$-0.947296\pi$
$787$		−	6.16391e19i		−	0.329635i	0.986324	−	0.164818i	$-0.0527035\pi$
$788$	2.44429e19			0.129560
$789$			1.24393e20i			0.653516i
$790$	−2.71277e20			−1.41261
$791$	−8.57543e19			−0.442610
$792$		−	3.10899e19i		−	0.159053i
$793$		−	2.90415e19i		−	0.147268i
$794$	−2.84982e20			−1.43243
$795$	1.11572e20			0.555884
$796$		−	5.15335e19i		−	0.254506i
$797$		−	3.71931e20i		−	1.82076i	−0.413769	−	0.910382i	$-0.635788\pi$
$797$		−	3.71931e20i		−	1.82076i	0.413769	−	0.910382i	$-0.364212\pi$
$798$	−6.36687e19			−0.308963
$799$			1.29235e20i			0.621660i
$800$	9.36279e19			0.446453
$801$		−	2.57365e20i		−	1.21653i
$802$		−	2.71119e20i		−	1.27040i
$803$		−	4.24290e18i		−	0.0197085i
$804$			3.98998e18i			0.0183729i
$805$	−3.89338e20	+	3.23735e20i	−1.77728	+	1.47781i
$806$	−1.33242e20			−0.602971
$807$	−1.29747e20			−0.582079
$808$	3.45105e20			1.53487
$809$	−1.88012e20			−0.828986			−0.414493	−	0.910053i	$-0.636041\pi$
$809$	−1.88012e20			−0.828986			−0.414493	+	0.910053i	$0.636041\pi$
$810$		−	1.35967e20i		−	0.594346i
$811$	−1.18296e20			−0.512653			−0.256326	−	0.966590i	$-0.582512\pi$
$811$	−1.18296e20			−0.512653			−0.256326	+	0.966590i	$0.582512\pi$
$812$			1.16620e20i			0.501049i
$813$	7.83440e19			0.333712
$814$	1.13252e19			0.0478274
$815$		−	3.90477e19i		−	0.163490i
$816$		−	1.35041e20i		−	0.560578i
$817$	1.41299e20			0.581547
$818$	−1.09085e20			−0.445136
$819$		−	5.06999e20i		−	2.05126i
$820$		−	2.73859e19i		−	0.109858i
$821$	1.61051e20			0.640562			0.320281	−	0.947323i	$-0.396223\pi$
$821$	1.61051e20			0.640562			0.320281	+	0.947323i	$0.396223\pi$
$822$		−	2.76222e19i		−	0.108932i
$823$	2.15754e20			0.843645			0.421823	−	0.906678i	$-0.361390\pi$
$823$	2.15754e20			0.843645			0.421823	+	0.906678i	$0.361390\pi$
$824$			2.41405e20i			0.935957i
$825$			2.72693e19i			0.104833i
$826$			3.68039e20i			1.40292i
$827$		−	1.74924e20i		−	0.661168i	−0.943777	−	0.330584i	$-0.892754\pi$
$827$		−	1.74924e20i		−	0.661168i	0.943777	−	0.330584i	$-0.107246\pi$
$828$	−2.95424e19	+	2.45646e19i	−0.110722	+	0.0920655i
$829$	−4.17300e20			−1.55084			−0.775419	−	0.631447i	$-0.782461\pi$
$829$	−4.17300e20			−1.55084			−0.775419	+	0.631447i	$0.782461\pi$
$830$	3.84258e20			1.41604
$831$	1.04136e20			0.380534
$832$	−5.10685e20			−1.85050
$833$			6.05179e20i			2.17455i
$834$	−6.65884e19			−0.237266
$835$		−	3.01087e20i		−	1.06387i
$836$	−4.73780e18			−0.0166009
$837$	−9.26916e19			−0.322079
$838$		−	7.12528e19i		−	0.245524i
$839$			2.33145e20i			0.796697i	0.917234	+	0.398349i	$0.130417\pi$
$839$			2.33145e20i			0.796697i	−0.917234	+	0.398349i	$0.869583\pi$
$840$	−3.26846e20			−1.10761
$841$	6.72869e20			2.26130
$842$		−	8.84041e19i		−	0.294637i
$843$		−	1.91937e20i		−	0.634403i
$844$	−2.41660e18			−0.00792152
$845$			8.23219e20i			2.67620i
$846$	−8.78351e19			−0.283189
$847$		−	4.64694e20i		−	1.48588i
$848$		−	2.04941e20i		−	0.649918i
$849$		−	1.45086e20i		−	0.456323i
$850$		−	5.78309e20i		−	1.80396i
$851$	−5.85936e19	−	7.04672e19i	−0.181277	−	0.218012i
$852$	−3.62024e19			−0.111086
$853$	5.41136e20			1.64689			0.823443	−	0.567400i	$-0.192051\pi$
$853$	5.41136e20			1.64689			0.823443	+	0.567400i	$0.192051\pi$
$854$	−4.07646e19			−0.123049
$855$	−1.98086e20			−0.593050
$856$			5.93348e20i			1.76195i
$857$	−3.30312e20			−0.972877			−0.486439	−	0.873715i	$-0.661704\pi$
$857$	−3.30312e20			−0.972877			−0.486439	+	0.873715i	$0.661704\pi$
$858$		−	4.31896e19i		−	0.126173i
$859$	3.26471e20			0.946004			0.473002	−	0.881061i	$-0.343170\pi$
$859$	3.26471e20			0.946004			0.473002	+	0.881061i	$0.343170\pi$
$860$	1.10776e20			0.318387
$861$			9.82794e19i			0.280182i
$862$			5.36451e20i			1.51698i
$863$	2.33452e19			0.0654819			0.0327410	−	0.999464i	$-0.489576\pi$
$863$	2.33452e19			0.0654819			0.0327410	+	0.999464i	$0.489576\pi$
$864$	−1.03159e20			−0.287020
$865$			8.07130e19i			0.222757i
$866$		−	8.81655e19i		−	0.241365i
$867$	2.51223e20			0.682222
$868$		−	4.11225e19i		−	0.110775i
$869$	7.24433e19			0.193580
$870$			4.15357e20i			1.10100i
$871$		−	1.44194e20i		−	0.379157i
$872$		−	5.60020e20i		−	1.46079i
$873$			4.01115e20i			1.03793i
$874$	−1.11482e20	−	1.34073e20i	−0.286170	−	0.344161i
$875$	−2.31262e20			−0.588909
$876$	−3.38460e18			−0.00855023
$877$	−4.71863e20			−1.18255			−0.591274	−	0.806471i	$-0.701375\pi$
$877$	−4.71863e20			−1.18255			−0.591274	+	0.806471i	$0.701375\pi$
$878$	−5.48078e19			−0.136264
$879$		−	1.36617e20i		−	0.336964i
$880$	9.00090e19			0.220246
$881$		−	2.83673e20i		−	0.688632i	−0.938854	−	0.344316i	$-0.888111\pi$
$881$		−	2.83673e20i		−	0.688632i	0.938854	−	0.344316i	$-0.111889\pi$
$882$	−4.11313e20			−0.990586
$883$	3.55270e19			0.0848856			0.0424428	−	0.999099i	$-0.486486\pi$
$883$	3.55270e19			0.0848856			0.0424428	+	0.999099i	$0.486486\pi$
$884$		−	2.01390e20i		−	0.477390i
$885$		−	2.88215e20i		−	0.677820i
$886$	2.67719e20			0.624659
$887$	6.98818e20			1.61770			0.808851	−	0.588013i	$-0.200090\pi$
$887$	6.98818e20			1.61770			0.808851	+	0.588013i	$0.200090\pi$
$888$		−	5.91567e19i		−	0.135867i
$889$		−	9.28187e20i		−	2.11506i
$890$	9.15694e20			2.07024
$891$			3.63095e19i			0.0814473i
$892$	8.16882e18			0.0181805
$893$			8.76471e19i			0.193543i
$894$		−	2.50847e20i		−	0.549600i
$895$			1.08745e21i			2.36400i
$896$			4.62917e20i			0.998497i
$897$	−2.68731e20	+	2.23450e20i	−0.575137	+	0.478227i
$898$	−6.50937e20			−1.38231
$899$	−3.42192e20			−0.721028
$900$	−8.64217e19			−0.180686
$901$	6.31867e20			1.31085
$902$		−	3.32612e19i		−	0.0684688i
$903$	−3.97539e20			−0.812018
$904$		−	1.51596e20i		−	0.307262i
$905$	−3.37583e20			−0.678955
$906$	2.17413e20			0.433899
$907$			8.04447e20i			1.59311i	0.604564	+	0.796556i	$0.293347\pi$
$907$			8.04447e20i			1.59311i	−0.604564	+	0.796556i	$0.706653\pi$
$908$			2.96720e19i			0.0583104i
$909$	−5.88439e20			−1.14750
$910$	1.80388e21			3.49074
$911$			8.73242e20i			1.67690i	0.544982	+	0.838448i	$0.316537\pi$
$911$			8.73242e20i			1.67690i	−0.544982	+	0.838448i	$0.683463\pi$
$912$		−	9.15851e19i		−	0.174526i
$913$	−1.02615e20			−0.194050
$914$		−	2.51630e20i		−	0.472214i
$915$	3.19232e19			0.0594509
$916$		−	2.29366e19i		−	0.0423897i
$917$		−	2.27614e20i		−	0.417458i
$918$			6.37182e20i			1.15975i
$919$			4.79433e20i			0.866002i	0.901393	+	0.433001i	$0.142545\pi$
$919$			4.79433e20i			0.866002i	−0.901393	+	0.433001i	$0.857455\pi$
$920$	−5.72297e20	−	6.88270e20i	−1.02590	−	1.23380i
$921$	−2.67572e20			−0.476019
$922$	7.03329e19			0.124177
$923$	1.30832e21			2.29246
$924$	1.33296e19			0.0231800
$925$		−	2.06140e20i		−	0.355771i
$926$	−5.54275e20			−0.949398
$927$		−	4.11621e20i		−	0.699744i
$928$	−3.80836e20			−0.642543
$929$	5.16450e20			0.864805			0.432402	−	0.901681i	$-0.357666\pi$
$929$	5.16450e20			0.864805			0.432402	+	0.901681i	$0.357666\pi$
$930$		−	1.46463e20i		−	0.243415i
$931$			4.10432e20i			0.677008i
$932$	3.11763e18			0.00510402
$933$	−8.23624e19			−0.133831
$934$			1.51833e19i			0.0244871i
$935$			2.77512e20i			0.444223i
$936$	8.96271e20			1.42400
$937$		−	1.34678e20i		−	0.212383i	−0.994346	−	0.106192i	$-0.966134\pi$
$937$		−	1.34678e20i		−	0.212383i	0.994346	−	0.106192i	$-0.0338657\pi$
$938$	−2.02400e20			−0.316804
$939$			3.09678e20i			0.481117i
$940$			6.87136e19i			0.105961i
$941$		−	2.78243e20i		−	0.425889i	−0.977064	−	0.212945i	$-0.931695\pi$
$941$		−	2.78243e20i		−	0.425889i	0.977064	−	0.212945i	$-0.0683054\pi$
$942$		−	1.02098e20i		−	0.155118i
$943$	−2.06956e20	+	1.72084e20i	−0.312102	+	0.259513i
$944$	−5.29411e20			−0.792481
$945$	1.25489e21			1.86459
$946$	1.34541e20			0.198435
$947$	−1.19835e20			−0.175442			−0.0877211	−	0.996145i	$-0.527958\pi$
$947$	−1.19835e20			−0.175442			−0.0877211	+	0.996145i	$0.527958\pi$
$948$		−	5.77886e19i		−	0.0839816i
$949$	1.22316e20			0.176449
$950$		−	3.92209e20i		−	0.561633i
$951$	−1.79574e20			−0.255258
$952$	−1.85104e21			−2.61190
$953$		−	9.97236e20i		−	1.39684i	−0.715687	−	0.698421i	$-0.753886\pi$
$953$		−	9.97236e20i		−	1.39684i	0.715687	−	0.698421i	$-0.246114\pi$
$954$			4.29452e20i			0.597139i
$955$	1.43925e21			1.98661
$956$	−7.64203e19			−0.104714
$957$		−	1.10919e20i		−	0.150877i
$958$			1.03335e20i			0.139537i
$959$	−3.08087e20			−0.412993
$960$		−	5.61359e20i		−	0.747036i
$961$	−6.36280e20			−0.840591
$962$			3.26488e20i			0.428196i
$963$		−	1.01172e21i		−	1.31727i
$964$			1.28004e20i			0.165456i
$965$		−	2.87101e20i		−	0.368421i
$966$	3.13650e20	+	3.77209e20i	0.399581	+	0.480554i
$967$	−4.62443e20			−0.584888			−0.292444	−	0.956283i	$-0.594468\pi$
$967$	−4.62443e20			−0.584888			−0.292444	+	0.956283i	$0.594468\pi$
$968$	8.21484e20			1.03151
$969$	2.82371e20			0.352009
$970$	−1.42715e21			−1.76631
$971$		−	4.00926e20i		−	0.492640i	−0.969189	−	0.246320i	$-0.920779\pi$
$971$		−	4.00926e20i		−	0.492640i	0.969189	−	0.246320i	$-0.0792214\pi$
$972$	1.48151e20			0.180735
$973$			7.42699e20i			0.899545i
$974$	−1.12872e21			−1.35729
$975$	−7.86130e20			−0.938560
$976$		−	5.86384e19i		−	0.0695077i
$977$		−	1.58166e21i		−	1.86145i	−0.365718	−	0.930726i	$-0.619177\pi$
$977$		−	1.58166e21i		−	1.86145i	0.365718	−	0.930726i	$-0.380823\pi$
$978$	−3.78313e19			−0.0442058
$979$	−2.44532e20			−0.283699
$980$			3.21771e20i			0.370650i
$981$			9.54893e20i			1.09212i
$982$	−7.85687e19			−0.0892214
$983$			4.83576e20i			0.545244i	0.962121	+	0.272622i	$0.0878908\pi$
$983$			4.83576e20i			0.545244i	−0.962121	+	0.272622i	$0.912109\pi$
$984$	−1.73738e20			−0.194504
$985$		−	9.70993e20i		−	1.07935i
$986$			2.35230e21i			2.59630i
$987$		−	2.46591e20i		−	0.270245i
$988$		−	1.36583e20i		−	0.148627i
$989$	−6.96078e20	−	8.37134e20i	−0.752115	−	0.904526i
$990$	−1.88612e20			−0.202360
$991$	1.84068e20			0.196093			0.0980467	−	0.995182i	$-0.468741\pi$
$991$	1.84068e20			0.196093			0.0980467	+	0.995182i	$0.468741\pi$
$992$	1.34290e20			0.142057
$993$	−3.88588e20			−0.408174
$994$		−	1.83644e21i		−	1.91546i
$995$	−2.04716e21			−2.12027
$996$			8.18564e19i			0.0841855i
$997$	−1.41490e21			−1.44497			−0.722487	−	0.691385i	$-0.757001\pi$
$997$	−1.41490e21			−1.44497			−0.722487	+	0.691385i	$0.757001\pi$
$998$	1.62291e21			1.64581
$999$			2.27126e20i			0.228722i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.b.b.22.8	yes	24
23.22	odd	2	inner	23.15.b.b.22.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.b.b.22.7	✓	24	23.22	odd	2	inner
23.15.b.b.22.8	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-115.892 q^{2} -980.720 q^{3} -2953.11 q^{4} +117312. i q^{5} +113657. q^{6} -1.26769e6i q^{7} +2.24101e6 q^{8} -3.82116e6 q^{9} +O(q^{10})\)	\(q-115.892 q^{2} -980.720 q^{3} -2953.11 q^{4} +117312. i q^{5} +113657. q^{6} -1.26769e6i q^{7} +2.24101e6 q^{8} -3.82116e6 q^{9} -1.35955e7i q^{10} +3.63062e6i q^{11} +2.89617e6 q^{12} -1.04665e8 q^{13} +1.46914e8i q^{14} -1.15050e8i q^{15} -2.11331e8 q^{16} -6.51567e8i q^{17} +4.42840e8 q^{18} -4.41893e8i q^{19} -3.46435e8i q^{20} +1.24325e9i q^{21} -4.20759e8i q^{22} +(-2.61802e9 + 2.17688e9i) q^{23} -2.19780e9 q^{24} -7.65859e9 q^{25} +1.21298e10 q^{26} +8.43824e9 q^{27} +3.74361e9i q^{28} +3.11517e10 q^{29} +1.33334e10i q^{30} -1.09847e10 q^{31} -1.22252e10 q^{32} -3.56062e9i q^{33} +7.55112e10i q^{34} +1.48715e11 q^{35} +1.12843e10 q^{36} +2.69163e10i q^{37} +5.12117e10i q^{38} +1.02647e11 q^{39} +2.62897e11i q^{40} +7.90506e10 q^{41} -1.44082e11i q^{42} +3.19759e11i q^{43} -1.07216e10i q^{44} -4.48267e11i q^{45} +(3.03406e11 - 2.52283e11i) q^{46} -1.98345e11 q^{47} +2.07257e11 q^{48} -9.28806e11 q^{49} +8.87567e11 q^{50} +6.39005e11i q^{51} +3.09086e11 q^{52} +9.69766e11i q^{53} -9.77922e11 q^{54} -4.25915e11 q^{55} -2.84090e12i q^{56} +4.33373e11i q^{57} -3.61022e12 q^{58} +2.50513e12 q^{59} +3.39756e11i q^{60} +2.77472e11i q^{61} +1.27304e12 q^{62} +4.84403e12i q^{63} +4.87925e12 q^{64} -1.22784e13i q^{65} +4.12646e11i q^{66} +1.37767e12i q^{67} +1.92415e12i q^{68} +(2.56754e12 - 2.13491e12i) q^{69} -1.72348e13 q^{70} -1.25001e13 q^{71} -8.56325e12 q^{72} -1.16864e12 q^{73} -3.11937e12i q^{74} +7.51093e12 q^{75} +1.30496e12i q^{76} +4.60249e12 q^{77} -1.18959e13 q^{78} -1.99534e13i q^{79} -2.47916e13i q^{80} +1.00009e13 q^{81} -9.16132e12 q^{82} +2.82637e13i q^{83} -3.67144e12i q^{84} +7.64366e13 q^{85} -3.70574e13i q^{86} -3.05511e13 q^{87} +8.13625e12i q^{88} +6.73528e13i q^{89} +5.19505e13i q^{90} +1.32682e14i q^{91} +(7.73128e12 - 6.42857e12i) q^{92} +1.07729e13 q^{93} +2.29865e13 q^{94} +5.18393e13 q^{95} +1.19895e13 q^{96} -1.04972e14i q^{97} +1.07641e14 q^{98} -1.38732e13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10}) \) 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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