LMFDB - Embedded newform 43.9.b.b.42.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [43,9,Mod(42,43)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(43, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("43.42");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	43.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:	$17.5172802326$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			42.4
Character	$\chi$	$=$	43.42
Dual form			43.9.b.b.42.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-24.2897i q^{2} -13.8077i q^{3} -333.988 q^{4} +427.967i q^{5} -335.386 q^{6} +2563.83i q^{7} +1894.31i q^{8} +6370.35 q^{9} +O(q^{10})$	\(q-24.2897i q^{2} -13.8077i q^{3} -333.988 q^{4} +427.967i q^{5} -335.386 q^{6} +2563.83i q^{7} +1894.31i q^{8} +6370.35 q^{9} +10395.2 q^{10} +3299.24 q^{11} +4611.62i q^{12} +1783.00 q^{13} +62274.6 q^{14} +5909.26 q^{15} -39488.8 q^{16} +161819. q^{17} -154734. i q^{18} +188274. i q^{19} -142936. i q^{20} +35400.7 q^{21} -80137.4i q^{22} -430185. q^{23} +26156.2 q^{24} +207469. q^{25} -43308.5i q^{26} -178553. i q^{27} -856290. i q^{28} -1.23444e6i q^{29} -143534. i q^{30} +819928. q^{31} +1.44411e6i q^{32} -45555.0i q^{33} -3.93053e6i q^{34} -1.09724e6 q^{35} -2.12762e6 q^{36} +1.89464e6i q^{37} +4.57312e6 q^{38} -24619.2i q^{39} -810703. q^{40} +1.24331e6 q^{41} -859872. i q^{42} +(1.79787e6 + 2.90790e6i) q^{43} -1.10191e6 q^{44} +2.72630e6i q^{45} +1.04491e7i q^{46} +3.12874e6 q^{47} +545251. i q^{48} -808432. q^{49} -5.03936e6i q^{50} -2.23435e6i q^{51} -595502. q^{52} +6.47469e6 q^{53} -4.33699e6 q^{54} +1.41196e6i q^{55} -4.85670e6 q^{56} +2.59964e6 q^{57} -2.99841e7 q^{58} -177535. q^{59} -1.97362e6 q^{60} +1.30117e7i q^{61} -1.99158e7i q^{62} +1.63325e7i q^{63} +2.49679e7 q^{64} +763066. i q^{65} -1.10652e6 q^{66} -1.94324e7 q^{67} -5.40456e7 q^{68} +5.93988e6i q^{69} +2.66515e7i q^{70} +1.97077e7i q^{71} +1.20674e7i q^{72} -4.15914e7i q^{73} +4.60201e7 q^{74} -2.86468e6i q^{75} -6.28815e7i q^{76} +8.45869e6i q^{77} -597993. q^{78} +8.29308e6 q^{79} -1.68999e7i q^{80} +3.93304e7 q^{81} -3.01997e7i q^{82} +4.28237e7 q^{83} -1.18234e7 q^{84} +6.92532e7i q^{85} +(7.06319e7 - 4.36696e7i) q^{86} -1.70448e7 q^{87} +6.24978e6i q^{88} +2.70210e7i q^{89} +6.62209e7 q^{90} +4.57131e6i q^{91} +1.43677e8 q^{92} -1.13214e7i q^{93} -7.59960e7i q^{94} -8.05752e7 q^{95} +1.99400e7 q^{96} -1.00518e8 q^{97} +1.96366e7i q^{98} +2.10173e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 4284 q^{4} - 1794 q^{6} - 80754 q^{9}+O(q^{10}) $ 28 * q - 4284 * q^4 - 1794 * q^6 - 80754 * q^9	$ 28 q - 4284 q^{4} - 1794 q^{6} - 80754 q^{9} + 24982 q^{10} + 4538 q^{11} + 22086 q^{13} + 24732 q^{14} + 15388 q^{15} + 525812 q^{16} - 135136 q^{17} - 261352 q^{21} - 184432 q^{23} + 1770326 q^{24} - 2640434 q^{25} - 110272 q^{31} + 10947816 q^{35} + 11602066 q^{36} - 7189158 q^{38} - 21389338 q^{40} + 1301336 q^{41} + 2473420 q^{43} - 8818480 q^{44} + 1983566 q^{47} - 15560936 q^{49} + 12927876 q^{52} + 23942594 q^{53} - 13757972 q^{54} + 34967256 q^{56} + 35225148 q^{57} + 22565734 q^{58} - 5554336 q^{59} - 44902072 q^{60} - 170444572 q^{64} - 48457584 q^{66} - 130953802 q^{67} + 150021122 q^{68} + 205870278 q^{74} + 267860612 q^{78} + 7380250 q^{79} - 57601004 q^{81} - 42603970 q^{83} + 251931292 q^{84} - 45482652 q^{86} - 106687410 q^{87} - 255044692 q^{90} - 409532014 q^{92} + 123322986 q^{95} - 692987086 q^{96} - 318744840 q^{97} - 609707206 q^{99}+O(q^{100}) $ 28 * q - 4284 * q^4 - 1794 * q^6 - 80754 * q^9 + 24982 * q^10 + 4538 * q^11 + 22086 * q^13 + 24732 * q^14 + 15388 * q^15 + 525812 * q^16 - 135136 * q^17 - 261352 * q^21 - 184432 * q^23 + 1770326 * q^24 - 2640434 * q^25 - 110272 * q^31 + 10947816 * q^35 + 11602066 * q^36 - 7189158 * q^38 - 21389338 * q^40 + 1301336 * q^41 + 2473420 * q^43 - 8818480 * q^44 + 1983566 * q^47 - 15560936 * q^49 + 12927876 * q^52 + 23942594 * q^53 - 13757972 * q^54 + 34967256 * q^56 + 35225148 * q^57 + 22565734 * q^58 - 5554336 * q^59 - 44902072 * q^60 - 170444572 * q^64 - 48457584 * q^66 - 130953802 * q^67 + 150021122 * q^68 + 205870278 * q^74 + 267860612 * q^78 + 7380250 * q^79 - 57601004 * q^81 - 42603970 * q^83 + 251931292 * q^84 - 45482652 * q^86 - 106687410 * q^87 - 255044692 * q^90 - 409532014 * q^92 + 123322986 * q^95 - 692987086 * q^96 - 318744840 * q^97 - 609707206 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/43\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$		−	24.2897i		−	1.51810i	−0.651030	−	0.759052i	$-0.725663\pi$
$2$		−	24.2897i		−	1.51810i	0.651030	−	0.759052i	$-0.274337\pi$
$3$		−	13.8077i		−	0.170466i	−0.996361	−	0.0852330i	$-0.972837\pi$
$3$		−	13.8077i		−	0.170466i	0.996361	−	0.0852330i	$-0.0271635\pi$
$4$	−333.988			−1.30464
$5$			427.967i			0.684747i	0.939564	+	0.342374i	$0.111231\pi$
$5$			427.967i			0.684747i	−0.939564	+	0.342374i	$0.888769\pi$
$6$	−335.386			−0.258785
$7$			2563.83i			1.06782i	0.845542	+	0.533909i	$0.179278\pi$
$7$			2563.83i			1.06782i	−0.845542	+	0.533909i	$0.820722\pi$
$8$			1894.31i			0.462478i
$9$	6370.35			0.970941
$10$	10395.2			1.03952
$11$	3299.24			0.225342			0.112671	−	0.993632i	$-0.464059\pi$
$11$	3299.24			0.225342			0.112671	+	0.993632i	$0.464059\pi$
$12$			4611.62i			0.222397i
$13$	1783.00			0.0624278			0.0312139	−	0.999513i	$-0.490063\pi$
$13$	1783.00			0.0624278			0.0312139	+	0.999513i	$0.490063\pi$
$14$	62274.6			1.62106
$15$	5909.26			0.116726
$16$	−39488.8			−0.602551
$17$	161819.			1.93746			0.968732	−	0.248109i	$-0.0798092\pi$
$17$	161819.			1.93746			0.968732	+	0.248109i	$0.0798092\pi$
$18$		−	154734.i		−	1.47399i
$19$			188274.i			1.44470i	0.691529	+	0.722349i	$0.256937\pi$
$19$			188274.i			1.44470i	−0.691529	+	0.722349i	$0.743063\pi$
$20$		−	142936.i		−	0.893350i
$21$	35400.7			0.182027
$22$		−	80137.4i		−	0.342093i
$23$	−430185.			−1.53725			−0.768624	−	0.639701i	$-0.779058\pi$
$23$	−430185.			−1.53725			−0.768624	+	0.639701i	$0.779058\pi$
$24$	26156.2			0.0788368
$25$	207469.			0.531121
$26$		−	43308.5i		−	0.0947720i
$27$		−	178553.i		−	0.335978i
$28$		−	856290.i		−	1.39312i
$29$		−	1.23444e6i		−	1.74533i	−0.488320	−	0.872665i	$-0.662390\pi$
$29$		−	1.23444e6i		−	1.74533i	0.488320	−	0.872665i	$-0.337610\pi$
$30$		−	143534.i		−	0.177202i
$31$	819928.			0.887828			0.443914	−	0.896069i	$-0.353589\pi$
$31$	819928.			0.887828			0.443914	+	0.896069i	$0.353589\pi$
$32$			1.44411e6i			1.37721i
$33$		−	45555.0i		−	0.0384132i
$34$		−	3.93053e6i		−	2.94127i
$35$	−1.09724e6			−0.731185
$36$	−2.12762e6			−1.26673
$37$			1.89464e6i			1.01092i	0.862849	+	0.505462i	$0.168678\pi$
$37$			1.89464e6i			1.01092i	−0.862849	+	0.505462i	$0.831322\pi$
$38$	4.57312e6			2.19320
$39$		−	24619.2i		−	0.0106418i
$40$	−810703.			−0.316681
$41$	1.24331e6			0.439993			0.219996	−	0.975501i	$-0.429395\pi$
$41$	1.24331e6			0.439993			0.219996	+	0.975501i	$0.429395\pi$
$42$		−	859872.i		−	0.276336i
$43$	1.79787e6	+	2.90790e6i	0.525877	+	0.850561i
$43$	1.79787e6	+	2.90790e6i	0.525877	+	0.850561i
$44$	−1.10191e6			−0.293991
$45$			2.72630e6i			0.664849i
$46$			1.04491e7i			2.33370i
$47$	3.12874e6			0.641176			0.320588	−	0.947219i	$-0.396119\pi$
$47$	3.12874e6			0.641176			0.320588	+	0.947219i	$0.396119\pi$
$48$			545251.i			0.102714i
$49$	−808432.			−0.140236
$50$		−	5.03936e6i		−	0.806298i
$51$		−	2.23435e6i		−	0.330272i
$52$	−595502.			−0.0814460
$53$	6.47469e6			0.820570			0.410285	−	0.911957i	$-0.365429\pi$
$53$	6.47469e6			0.820570			0.410285	+	0.911957i	$0.365429\pi$
$54$	−4.33699e6			−0.510050
$55$			1.41196e6i			0.154303i
$56$	−4.85670e6			−0.493843
$57$	2.59964e6			0.246272
$58$	−2.99841e7			−2.64959
$59$	−177535.			−0.0146513			−0.00732565	−	0.999973i	$-0.502332\pi$
$59$	−177535.			−0.0146513			−0.00732565	+	0.999973i	$0.502332\pi$
$60$	−1.97362e6			−0.152286
$61$			1.30117e7i			0.939753i	0.882732	+	0.469876i	$0.155702\pi$
$61$			1.30117e7i			0.939753i	−0.882732	+	0.469876i	$0.844298\pi$
$62$		−	1.99158e7i		−	1.34782i
$63$			1.63325e7i			1.03679i
$64$	2.49679e7			1.48820
$65$			763066.i			0.0427473i
$66$	−1.10652e6			−0.0583153
$67$	−1.94324e7			−0.964335			−0.482167	−	0.876079i	$-0.660150\pi$
$67$	−1.94324e7			−0.964335			−0.482167	+	0.876079i	$0.660150\pi$
$68$	−5.40456e7			−2.52770
$69$			5.93988e6i			0.262048i
$70$			2.66515e7i			1.11002i
$71$			1.97077e7i			0.775536i	0.921757	+	0.387768i	$0.126754\pi$
$71$			1.97077e7i			0.775536i	−0.921757	+	0.387768i	$0.873246\pi$
$72$			1.20674e7i			0.449039i
$73$		−	4.15914e7i		−	1.46458i	−0.680994	−	0.732289i	$-0.738452\pi$
$73$		−	4.15914e7i		−	1.46458i	0.680994	−	0.732289i	$-0.261548\pi$
$74$	4.60201e7			1.53469
$75$		−	2.86468e6i		−	0.0905381i
$76$		−	6.28815e7i		−	1.88481i
$77$			8.45869e6i			0.240625i
$78$	−597993.			−0.0161554
$79$	8.29308e6			0.212916			0.106458	−	0.994317i	$-0.466049\pi$
$79$	8.29308e6			0.212916			0.106458	+	0.994317i	$0.466049\pi$
$80$		−	1.68999e7i		−	0.412595i
$81$	3.93304e7			0.913668
$82$		−	3.01997e7i		−	0.667955i
$83$	4.28237e7			0.902344			0.451172	−	0.892437i	$-0.351006\pi$
$83$	4.28237e7			0.902344			0.451172	+	0.892437i	$0.351006\pi$
$84$	−1.18234e7			−0.237480
$85$			6.92532e7i			1.32667i
$86$	7.06319e7	−	4.36696e7i	1.29124	−	0.798336i
$87$	−1.70448e7			−0.297519
$88$			6.24978e6i			0.104216i
$89$			2.70210e7i			0.430666i	0.976541	+	0.215333i	$0.0690838\pi$
$89$			2.70210e7i			0.430666i	−0.976541	+	0.215333i	$0.930916\pi$
$90$	6.62209e7			1.00931
$91$			4.57131e6i			0.0666616i
$92$	1.43677e8			2.00556
$93$		−	1.13214e7i		−	0.151345i
$94$		−	7.59960e7i		−	0.973373i
$95$	−8.05752e7			−0.989252
$96$	1.99400e7			0.234768
$97$	−1.00518e8			−1.13542			−0.567708	−	0.823230i	$-0.692170\pi$
$97$	−1.00518e8			−1.13542			−0.567708	+	0.823230i	$0.692170\pi$
$98$			1.96366e7i			0.212893i
$99$	2.10173e7			0.218794
$100$	−6.92923e7			−0.692923
$101$	−7.93058e7			−0.762113			−0.381057	−	0.924552i	$-0.624440\pi$
$101$	−7.93058e7			−0.762113			−0.381057	+	0.924552i	$0.624440\pi$
$102$	−5.42717e7			−0.501387
$103$	1.08221e8			0.961533			0.480767	−	0.876849i	$-0.340358\pi$
$103$	1.08221e8			0.961533			0.480767	+	0.876849i	$0.340358\pi$
$104$			3.37756e6i			0.0288715i
$105$			1.51503e7i			0.124642i
$106$		−	1.57268e8i		−	1.24571i
$107$	−1.03278e8			−0.787907			−0.393953	−	0.919130i	$-0.628893\pi$
$107$	−1.03278e8			−0.787907			−0.393953	+	0.919130i	$0.628893\pi$
$108$			5.96345e7i			0.438332i
$109$	−2.51482e8			−1.78156			−0.890782	−	0.454431i	$-0.849842\pi$
$109$	−2.51482e8			−1.78156			−0.890782	+	0.454431i	$0.849842\pi$
$110$	3.42962e7			0.234247
$111$	2.61606e7			0.172328
$112$		−	1.01243e8i		−	0.643415i
$113$		−	2.85143e8i		−	1.74884i	−0.485171	−	0.874419i	$-0.661243\pi$
$113$		−	2.85143e8i		−	1.74884i	0.485171	−	0.874419i	$-0.338757\pi$
$114$		−	6.31445e7i		−	0.373866i
$115$		−	1.84105e8i		−	1.05263i
$116$			4.12288e8i			2.27703i
$117$	1.13583e7			0.0606138
$118$			4.31227e6i			0.0222422i
$119$			4.14877e8i			2.06886i
$120$			1.11940e7i			0.0539833i
$121$	−2.03474e8			−0.949221
$122$	3.16049e8			1.42664
$123$		−	1.71674e7i		−	0.0750038i
$124$	−2.73846e8			−1.15830
$125$			2.55965e8i			1.04843i
$126$	3.96711e8			1.57395
$127$	−7.89136e7			−0.303345			−0.151673	−	0.988431i	$-0.548466\pi$
$127$	−7.89136e7			−0.303345			−0.151673	+	0.988431i	$0.548466\pi$
$128$		−	2.36770e8i		−	0.882036i
$129$	4.01515e7	−	2.48245e7i	0.144992	−	0.0896441i
$130$	1.85346e7			0.0648948
$131$			2.16717e8i			0.735882i	0.929849	+	0.367941i	$0.119937\pi$
$131$			2.16717e8i			0.735882i	−0.929849	+	0.367941i	$0.880063\pi$
$132$			1.52148e7i			0.0501155i
$133$	−4.82704e8			−1.54267
$134$			4.72007e8i			1.46396i
$135$	7.64146e7			0.230060
$136$			3.06536e8i			0.896035i
$137$			4.99384e8i			1.41759i	0.705412	+	0.708797i	$0.250762\pi$
$137$			4.99384e8i			1.41759i	−0.705412	+	0.708797i	$0.749238\pi$
$138$	1.44278e8			0.397817
$139$	−1.46570e7			−0.0392632			−0.0196316	−	0.999807i	$-0.506249\pi$
$139$	−1.46570e7			−0.0392632			−0.0196316	+	0.999807i	$0.506249\pi$
$140$	3.66464e8			0.953935
$141$		−	4.32008e7i		−	0.109299i
$142$	4.78693e8			1.17735
$143$	5.88255e6			0.0140676
$144$	−2.51557e8			−0.585042
$145$	5.28299e8			1.19511
$146$	−1.01024e9			−2.22338
$147$			1.11626e7i			0.0239054i
$148$		−	6.32786e8i		−	1.31889i
$149$		−	7.39245e7i		−	0.149983i	−0.997184	−	0.0749917i	$-0.976107\pi$
$149$		−	7.39245e7i		−	0.149983i	0.997184	−	0.0749917i	$-0.0238930\pi$
$150$	−6.95822e7			−0.137446
$151$		−	7.14958e8i		−	1.37522i	−0.726079	−	0.687611i	$-0.758660\pi$
$151$		−	7.14958e8i		−	1.37522i	0.726079	−	0.687611i	$-0.241340\pi$
$152$	−3.56650e8			−0.668141
$153$	1.03084e9			1.88116
$154$	2.05459e8			0.365294
$155$			3.50902e8i			0.607938i
$156$			8.22253e6i			0.0138838i
$157$		−	1.90914e7i		−	0.0314224i	−0.999877	−	0.0157112i	$-0.994999\pi$
$157$		−	1.90914e7i		−	0.0314224i	0.999877	−	0.0157112i	$-0.00500124\pi$
$158$		−	2.01436e8i		−	0.323228i
$159$		−	8.94008e7i		−	0.139879i
$160$	−6.18033e8			−0.943044
$161$		−	1.10292e9i		−	1.64150i
$162$		−	9.55323e8i		−	1.38704i
$163$			5.04417e8i			0.714561i	0.933997	+	0.357281i	$0.116296\pi$
$163$			5.04417e8i			0.714561i	−0.933997	+	0.357281i	$0.883704\pi$
$164$	−4.15253e8			−0.574033
$165$	1.94960e7			0.0263033
$166$		−	1.04017e9i		−	1.36985i
$167$	−6.01163e8			−0.772906			−0.386453	−	0.922309i	$-0.626300\pi$
$167$	−6.01163e8			−0.772906			−0.386453	+	0.922309i	$0.626300\pi$
$168$			6.70600e7i			0.0841834i
$169$	−8.12552e8			−0.996103
$170$	1.68214e9			2.01403
$171$			1.19937e9i			1.40272i
$172$	−6.00467e8	−	9.71204e8i	−0.686081	−	1.10968i
$173$	−8.47715e7			−0.0946380			−0.0473190	−	0.998880i	$-0.515068\pi$
$173$	−8.47715e7			−0.0946380			−0.0473190	+	0.998880i	$0.515068\pi$
$174$			4.14013e8i			0.451665i
$175$			5.31916e8i			0.567141i
$176$	−1.30283e8			−0.135780
$177$			2.45136e6i			0.00249755i
$178$	6.56331e8			0.653797
$179$			1.43087e8i			0.139376i	0.997569	+	0.0696882i	$0.0222004\pi$
$179$			1.43087e8i			0.139376i	−0.997569	+	0.0696882i	$0.977800\pi$
$180$		−	9.10552e8i		−	0.867390i
$181$	8.57397e8			0.798854			0.399427	−	0.916765i	$-0.369209\pi$
$181$	8.57397e8			0.798854			0.399427	+	0.916765i	$0.369209\pi$
$182$	1.11036e8			0.101199
$183$	1.79662e8			0.160196
$184$		−	8.14904e8i		−	0.710944i
$185$	−8.10842e8			−0.692228
$186$	−2.74992e8			−0.229757
$187$	5.33879e8			0.436593
$188$	−1.04496e9			−0.836505
$189$	4.57779e8			0.358764
$190$			1.95715e9i			1.50179i
$191$		−	1.26564e9i		−	0.950990i	−0.879718	−	0.475495i	$-0.842269\pi$
$191$		−	1.26564e9i		−	0.950990i	0.879718	−	0.475495i	$-0.157731\pi$
$192$		−	3.44751e8i		−	0.253688i
$193$	1.30174e9			0.938198			0.469099	−	0.883145i	$-0.344579\pi$
$193$	1.30174e9			0.938198			0.469099	+	0.883145i	$0.344579\pi$
$194$			2.44154e9i			1.72368i
$195$	1.05362e7			0.00728695
$196$	2.70007e8			0.182958
$197$	2.06607e9			1.37177			0.685885	−	0.727710i	$-0.259415\pi$
$197$	2.06607e9			1.37177			0.685885	+	0.727710i	$0.259415\pi$
$198$		−	5.10503e8i		−	0.332153i
$199$		−	9.02872e8i		−	0.575723i	−0.957672	−	0.287862i	$-0.907056\pi$
$199$		−	9.02872e8i		−	0.575723i	0.957672	−	0.287862i	$-0.0929443\pi$
$200$			3.93012e8i			0.245632i
$201$			2.68318e8i			0.164386i
$202$			1.92631e9i			1.15697i
$203$	3.16489e9			1.86370
$204$			7.46248e8i			0.430886i
$205$			5.32098e8i			0.301284i
$206$		−	2.62866e9i		−	1.45971i
$207$	−2.74043e9			−1.49258
$208$	−7.04086e7			−0.0376160
$209$			6.21162e8i			0.325552i
$210$	3.67997e8			0.189220
$211$		−	2.34791e9i		−	1.18454i	−0.805738	−	0.592272i	$-0.798231\pi$
$211$		−	2.34791e9i		−	1.18454i	0.805738	−	0.592272i	$-0.201769\pi$
$212$	−2.16247e9			−1.07055
$213$	2.72119e8			0.132203
$214$			2.50860e9i			1.19612i
$215$	−1.24448e9	+	7.69428e8i	−0.582419	+	0.360093i
$216$	3.38234e8			0.155383
$217$			2.10216e9i			0.948039i
$218$			6.10842e9i			2.70460i
$219$	−5.74284e8			−0.249661
$220$		−	4.71580e8i		−	0.201310i
$221$	2.88523e8			0.120952
$222$		−	6.35434e8i		−	0.261612i
$223$		−	2.77345e9i		−	1.12150i	−0.827984	−	0.560752i	$-0.810512\pi$
$223$		−	2.77345e9i		−	1.12150i	0.827984	−	0.560752i	$-0.189488\pi$
$224$	−3.70247e9			−1.47061
$225$	1.32165e9			0.515688
$226$	−6.92604e9			−2.65492
$227$		−	2.06840e8i		−	0.0778989i	−0.999241	−	0.0389494i	$-0.987599\pi$
$227$		−	2.06840e8i		−	0.0778989i	0.999241	−	0.0389494i	$-0.0124011\pi$
$228$	−8.68251e8			−0.321296
$229$	−2.78916e9			−1.01422			−0.507110	−	0.861881i	$-0.669286\pi$
$229$	−2.78916e9			−1.01422			−0.507110	+	0.861881i	$0.669286\pi$
$230$	−4.47185e9			−1.59800
$231$	1.16795e8			0.0410183
$232$	2.33841e9			0.807177
$233$		−	3.53341e9i		−	1.19887i	−0.800425	−	0.599433i	$-0.795393\pi$
$233$		−	3.53341e9i		−	1.19887i	0.800425	−	0.599433i	$-0.204607\pi$
$234$		−	2.75890e8i		−	0.0920180i
$235$			1.33900e9i			0.439044i
$236$	5.92946e7			0.0191147
$237$		−	1.14509e8i		−	0.0362949i
$238$	1.00772e10			3.14075
$239$	−2.07769e9			−0.636781			−0.318391	−	0.947960i	$-0.603142\pi$
$239$	−2.07769e9			−0.636781			−0.318391	+	0.947960i	$0.603142\pi$
$240$	−2.33349e8			−0.0703334
$241$		−	2.64446e8i		−	0.0783916i	−0.999232	−	0.0391958i	$-0.987520\pi$
$241$		−	2.64446e8i		−	0.0783916i	0.999232	−	0.0391958i	$-0.0124796\pi$
$242$			4.94232e9i			1.44102i
$243$		−	1.71455e9i		−	0.491728i
$244$		−	4.34575e9i		−	1.22604i
$245$		−	3.45982e8i		−	0.0960261i
$246$	−4.16990e8			−0.113864
$247$			3.35694e8i			0.0901893i
$248$			1.55320e9i			0.410602i
$249$		−	5.91299e8i		−	0.153819i
$250$	6.21730e9			1.59163
$251$	3.38964e9			0.854003			0.427001	−	0.904251i	$-0.359570\pi$
$251$	3.38964e9			0.854003			0.427001	+	0.904251i	$0.359570\pi$
$252$		−	5.45486e9i		−	1.35264i
$253$	−1.41928e9			−0.346407
$254$			1.91679e9i			0.460510i
$255$	9.56230e8			0.226153
$256$	6.40731e8			0.149182
$257$		−	6.06881e9i		−	1.39114i	−0.718458	−	0.695570i	$-0.755152\pi$
$257$		−	6.06881e9i		−	1.39114i	0.718458	−	0.695570i	$-0.244848\pi$
$258$	−6.02979e8	−	9.75267e8i	−0.136089	−	0.220112i
$259$	−4.85753e9			−1.07948
$260$		−	2.54855e8i		−	0.0557699i
$261$		−	7.86380e9i		−	1.69461i
$262$	5.26399e9			1.11715
$263$			7.16640e9i			1.49788i	0.662635	+	0.748942i	$0.269438\pi$
$263$			7.16640e9i			1.49788i	−0.662635	+	0.748942i	$0.730562\pi$
$264$	8.62954e7			0.0177653
$265$			2.77095e9i			0.561883i
$266$			1.17247e10i			2.34194i
$267$	3.73099e8			0.0734140
$268$	6.49020e9			1.25811
$269$	−1.04774e9			−0.200098			−0.100049	−	0.994982i	$-0.531900\pi$
$269$	−1.04774e9			−0.200098			−0.100049	+	0.994982i	$0.531900\pi$
$270$		−	1.85609e9i		−	0.349256i
$271$	−8.61683e9			−1.59761			−0.798804	−	0.601591i	$-0.794534\pi$
$271$	−8.61683e9			−1.59761			−0.798804	+	0.601591i	$0.794534\pi$
$272$	−6.39004e9			−1.16742
$273$	6.31195e7			0.0113635
$274$	1.21299e10			2.15206
$275$	6.84491e8			0.119684
$276$		−	1.98385e9i		−	0.341879i
$277$		−	8.11515e9i		−	1.37841i	−0.724568	−	0.689204i	$-0.757961\pi$
$277$		−	8.11515e9i		−	1.37841i	0.724568	−	0.689204i	$-0.242039\pi$
$278$			3.56013e8i			0.0596056i
$279$	5.22323e9			0.862029
$280$		−	2.07851e9i		−	0.338158i
$281$	−3.10778e9			−0.498454			−0.249227	−	0.968445i	$-0.580177\pi$
$281$	−3.10778e9			−0.498454			−0.249227	+	0.968445i	$0.580177\pi$
$282$	−1.04933e9			−0.165927
$283$	9.84866e9			1.53544			0.767718	−	0.640788i	$-0.221392\pi$
$283$	9.84866e9			1.53544			0.767718	+	0.640788i	$0.221392\pi$
$284$		−	6.58214e9i		−	1.01180i
$285$			1.11256e9i			0.168634i
$286$		−	1.42885e8i		−	0.0213561i
$287$			3.18765e9i			0.469832i
$288$			9.19951e9i			1.33719i
$289$	1.92096e10			2.75377
$290$		−	1.28322e10i		−	1.81430i
$291$			1.38792e9i			0.193550i
$292$			1.38911e10i			1.91075i
$293$	−1.18065e10			−1.60195			−0.800977	−	0.598695i	$-0.795686\pi$
$293$	−1.18065e10			−1.60195			−0.800977	+	0.598695i	$0.795686\pi$
$294$	2.71136e8			0.0362910
$295$		−	7.59791e7i		−	0.0100324i
$296$	−3.58903e9			−0.467531
$297$		−	5.89088e8i		−	0.0757102i
$298$	−1.79560e9			−0.227690
$299$	−7.67020e8			−0.0959670
$300$			9.56771e8i			0.118120i
$301$	−7.45536e9	+	4.60943e9i	−0.908244	+	0.561541i
$302$	−1.73661e10			−2.08773
$303$			1.09503e9i			0.129914i
$304$		−	7.43473e9i		−	0.870504i
$305$	−5.56856e9			−0.643493
$306$		−	2.50388e10i		−	2.85580i
$307$	−3.68250e9			−0.414561			−0.207281	−	0.978282i	$-0.566461\pi$
$307$	−3.68250e9			−0.414561			−0.207281	+	0.978282i	$0.566461\pi$
$308$		−	2.82510e9i		−	0.313929i
$309$		−	1.49429e9i		−	0.163909i
$310$	8.52330e9			0.922913
$311$	5.09479e9			0.544609			0.272305	−	0.962211i	$-0.412214\pi$
$311$	5.09479e9			0.544609			0.272305	+	0.962211i	$0.412214\pi$
$312$	4.66365e7			0.00492161
$313$		−	8.21077e9i		−	0.855473i	−0.903903	−	0.427736i	$-0.859311\pi$
$313$		−	8.21077e9i		−	0.855473i	0.903903	−	0.427736i	$-0.140689\pi$
$314$	−4.63725e8			−0.0477025
$315$	−6.98977e9			−0.709938
$316$	−2.76979e9			−0.277779
$317$	1.07248e10			1.06207			0.531034	−	0.847351i	$-0.321804\pi$
$317$	1.07248e10			1.06207			0.531034	+	0.847351i	$0.321804\pi$
$318$	−2.17152e9			−0.212351
$319$		−	4.07271e9i		−	0.393297i
$320$			1.06854e10i			1.01904i
$321$			1.42604e9i			0.134311i
$322$	−2.67896e10			−2.49197
$323$			3.04664e10i			2.79905i
$324$	−1.31359e10			−1.19201
$325$	3.69918e8			0.0331568
$326$	1.22521e10			1.08478
$327$			3.47240e9i			0.303696i
$328$			2.35523e9i			0.203487i
$329$			8.02155e9i			0.684660i
$330$		−	4.73553e8i		−	0.0399312i
$331$			2.06009e9i			0.171623i	0.996311	+	0.0858114i	$0.0273482\pi$
$331$			2.06009e9i			0.171623i	−0.996311	+	0.0858114i	$0.972652\pi$
$332$	−1.43026e10			−1.17724
$333$			1.20695e10i			0.981549i
$334$			1.46021e10i			1.17335i
$335$		−	8.31643e9i		−	0.660325i
$336$	−1.39793e9			−0.109680
$337$	1.19620e10			0.927436			0.463718	−	0.885983i	$-0.346515\pi$
$337$	1.19620e10			0.927436			0.463718	+	0.885983i	$0.346515\pi$
$338$			1.97366e10i			1.51219i
$339$	−3.93719e9			−0.298117
$340$		−	2.31297e10i		−	1.73083i
$341$	2.70514e9			0.200065
$342$	2.91324e10			2.12947
$343$			1.27073e10i			0.918072i
$344$	−5.50846e9	+	3.40572e9i	−0.393366	+	0.243207i
$345$	−2.54207e9			−0.179437
$346$			2.05907e9i			0.143670i
$347$			3.44498e9i			0.237612i	0.992917	+	0.118806i	$0.0379067\pi$
$347$			3.44498e9i			0.237612i	−0.992917	+	0.118806i	$0.962093\pi$
$348$	5.69277e9			0.388156
$349$		−	8.41368e9i		−	0.567132i	−0.958953	−	0.283566i	$-0.908482\pi$
$349$		−	8.41368e9i		−	0.567132i	0.958953	−	0.283566i	$-0.0915176\pi$
$350$	1.29201e10			0.860980
$351$		−	3.18360e8i		−	0.0209744i
$352$			4.76448e9i			0.310345i
$353$	1.58116e10			1.01830			0.509152	−	0.860677i	$-0.329959\pi$
$353$	1.58116e10			1.01830			0.509152	+	0.860677i	$0.329959\pi$
$354$	5.95427e7			0.00379154
$355$	−8.43424e9			−0.531046
$356$		−	9.02469e9i		−	0.561865i
$357$	5.72851e9			0.352670
$358$	3.47555e9			0.211588
$359$	−1.75353e10			−1.05568			−0.527842	−	0.849342i	$-0.676999\pi$
$359$	−1.75353e10			−1.05568			−0.527842	+	0.849342i	$0.676999\pi$
$360$	−5.16446e9			−0.307478
$361$	−1.84637e10			−1.08715
$362$		−	2.08259e10i		−	1.21274i
$363$			2.80952e9i			0.161810i
$364$		−	1.52677e9i		−	0.0869695i
$365$	1.77998e10			1.00287
$366$		−	4.36393e9i		−	0.243194i
$367$	1.96675e10			1.08414			0.542069	−	0.840334i	$-0.317641\pi$
$367$	1.96675e10			1.08414			0.542069	+	0.840334i	$0.317641\pi$
$368$	1.69875e10			0.926271
$369$	7.92034e9			0.427207
$370$			1.96951e10i			1.05087i
$371$			1.66000e10i			0.876219i
$372$			3.78120e9i			0.197450i
$373$		−	2.22528e10i		−	1.14961i	−0.818291	−	0.574804i	$-0.805078\pi$
$373$		−	2.22528e10i		−	1.14961i	0.818291	−	0.574804i	$-0.194922\pi$
$374$		−	1.29678e10i		−	0.662794i
$375$	3.53429e9			0.178722
$376$			5.92680e9i			0.296530i
$377$		−	2.20101e9i		−	0.108957i
$378$		−	1.11193e10i		−	0.544641i
$379$	−2.22479e10			−1.07828			−0.539141	−	0.842215i	$-0.681251\pi$
$379$	−2.22479e10			−1.07828			−0.539141	+	0.842215i	$0.681251\pi$
$380$	2.69112e10			1.29062
$381$			1.08962e9i			0.0517100i
$382$	−3.07419e10			−1.44370
$383$		−	1.88396e10i		−	0.875542i	−0.899087	−	0.437771i	$-0.855768\pi$
$383$		−	1.88396e10i		−	0.875542i	0.899087	−	0.437771i	$-0.144232\pi$
$384$	−3.26925e9			−0.150357
$385$	−3.62004e9			−0.164767
$386$		−	3.16188e10i		−	1.42428i
$387$	1.14530e10	+	1.85243e10i	0.510596	+	0.825845i
$388$	3.35717e10			1.48131
$389$			7.04839e9i			0.307816i	0.988085	+	0.153908i	$0.0491860\pi$
$389$			7.04839e9i			0.307816i	−0.988085	+	0.153908i	$0.950814\pi$
$390$		−	2.55921e8i		−	0.0110624i
$391$	−6.96121e10			−2.97836
$392$		−	1.53142e9i		−	0.0648561i
$393$	2.99237e9			0.125443
$394$		−	5.01843e10i		−	2.08249i
$395$			3.54917e9i			0.145793i
$396$	−7.01953e9			−0.285448
$397$	−7.77464e9			−0.312981			−0.156491	−	0.987679i	$-0.550018\pi$
$397$	−7.77464e9			−0.312981			−0.156491	+	0.987679i	$0.550018\pi$
$398$	−2.19305e10			−0.874008
$399$			6.66505e9i			0.262973i
$400$	−8.19271e9			−0.320028
$401$	1.87199e9			0.0723980			0.0361990	−	0.999345i	$-0.488475\pi$
$401$	1.87199e9			0.0723980			0.0361990	+	0.999345i	$0.488475\pi$
$402$	6.51735e9			0.249555
$403$	1.46193e9			0.0554252
$404$	2.64872e10			0.994285
$405$			1.68321e10i			0.625632i
$406$		−	7.68742e10i		−	2.82928i
$407$			6.25085e9i			0.227804i
$408$	4.23256e9			0.152744
$409$		−	1.64058e10i		−	0.586278i	−0.956070	−	0.293139i	$-0.905300\pi$
$409$		−	1.64058e10i		−	0.586278i	0.956070	−	0.293139i	$-0.0946999\pi$
$410$	1.29245e10			0.457380
$411$	6.89536e9			0.241652
$412$	−3.61447e10			−1.25446
$413$		−	4.55170e8i		−	0.0156449i
$414$			6.65641e10i			2.26589i
$415$			1.83271e10i			0.617877i
$416$			2.57486e9i			0.0859765i
$417$			2.02380e8i			0.00669304i
$418$	1.50878e10			0.494221
$419$		−	4.87108e10i		−	1.58041i	−0.612844	−	0.790204i	$-0.709975\pi$
$419$		−	4.87108e10i		−	1.58041i	0.612844	−	0.790204i	$-0.290025\pi$
$420$		−	5.06004e9i		−	0.162613i
$421$			9.06047e9i			0.288418i	0.989547	+	0.144209i	$0.0460638\pi$
$421$			9.06047e9i			0.288418i	−0.989547	+	0.144209i	$0.953936\pi$
$422$	−5.70299e10			−1.79826
$423$	1.99311e10			0.622545
$424$			1.22651e10i			0.379496i
$425$	3.35725e10			1.02903
$426$		−	6.60967e9i		−	0.200697i
$427$	−3.33597e10			−1.00349
$428$	3.44938e10			1.02794
$429$		−	8.12247e7i		−	0.00239805i
$430$	1.86892e10	+	3.02281e10i	0.546658	+	0.884173i
$431$	−4.56887e9			−0.132404			−0.0662019	−	0.997806i	$-0.521088\pi$
$431$	−4.56887e9			−0.132404			−0.0662019	+	0.997806i	$0.521088\pi$
$432$			7.05083e9i			0.202444i
$433$			1.09451e10i			0.311363i	0.987807	+	0.155681i	$0.0497573\pi$
$433$			1.09451e10i			0.311363i	−0.987807	+	0.155681i	$0.950243\pi$
$434$	5.10607e10			1.43922
$435$		−	7.29461e9i		−	0.203725i
$436$	8.39921e10			2.32430
$437$		−	8.09928e10i		−	2.22086i
$438$			1.39492e10i			0.379011i
$439$	−8.95525e9			−0.241112			−0.120556	−	0.992706i	$-0.538468\pi$
$439$	−8.95525e9			−0.241112			−0.120556	+	0.992706i	$0.538468\pi$
$440$	−2.67470e9			−0.0713616
$441$	−5.14999e9			−0.136161
$442$		−	7.00814e9i		−	0.183617i
$443$	2.28878e10			0.594277			0.297139	−	0.954834i	$-0.403968\pi$
$443$	2.28878e10			0.594277			0.297139	+	0.954834i	$0.403968\pi$
$444$	−8.73735e9			−0.224827
$445$	−1.15641e10			−0.294898
$446$	−6.73662e10			−1.70256
$447$	−1.02073e9			−0.0255671
$448$			6.40136e10i			1.58913i
$449$			1.64061e10i			0.403664i	0.979420	+	0.201832i	$0.0646894\pi$
$449$			1.64061e10i			0.403664i	−0.979420	+	0.201832i	$0.935311\pi$
$450$		−	3.21025e10i		−	0.782868i
$451$	4.10199e9			0.0991490
$452$			9.52346e10i			2.28161i
$453$	−9.87196e9			−0.234428
$454$	−5.02408e9			−0.118259
$455$	−1.95637e9			−0.0456463
$456$			4.92454e9i			0.113895i
$457$			6.73001e9i			0.154295i	0.997020	+	0.0771473i	$0.0245812\pi$
$457$			6.73001e9i			0.154295i	−0.997020	+	0.0771473i	$0.975419\pi$
$458$			6.77479e10i			1.53969i
$459$		−	2.88932e10i		−	0.650946i
$460$			6.14889e10i			1.37330i
$461$	9.84296e9			0.217932			0.108966	−	0.994045i	$-0.465246\pi$
$461$	9.84296e9			0.217932			0.108966	+	0.994045i	$0.465246\pi$
$462$		−	2.83692e9i		−	0.0622701i
$463$			6.34452e10i			1.38062i	0.723513	+	0.690311i	$0.242526\pi$
$463$			6.34452e10i			1.38062i	−0.723513	+	0.690311i	$0.757474\pi$
$464$			4.87465e10i			1.05165i
$465$	4.84517e9			0.103633
$466$	−8.58254e10			−1.82000
$467$			3.63255e10i			0.763738i	0.924217	+	0.381869i	$0.124719\pi$
$467$			3.63255e10i			0.763738i	−0.924217	+	0.381869i	$0.875281\pi$
$468$	−3.79355e9			−0.0790793
$469$		−	4.98215e10i		−	1.02973i
$470$	3.25238e10			0.666514
$471$	−2.63610e8			−0.00535645
$472$		−	3.36307e8i		−	0.00677591i
$473$	5.93160e9	+	9.59385e9i	0.118502	+	0.191667i
$474$	−2.78138e9			−0.0550994
$475$			3.90612e10i			0.767310i
$476$		−	1.38564e11i		−	2.69912i
$477$	4.12460e10			0.796725
$478$			5.04665e10i			0.966700i
$479$	−6.24570e10			−1.18642			−0.593211	−	0.805047i	$-0.702140\pi$
$479$	−6.24570e10			−1.18642			−0.593211	+	0.805047i	$0.702140\pi$
$480$			8.53364e9i			0.160757i
$481$			3.37814e9i			0.0631098i
$482$	−6.42332e9			−0.119007
$483$	−1.52289e10			−0.279820
$484$	6.79579e10			1.23839
$485$		−	4.30182e10i		−	0.777473i
$486$	−4.16458e10			−0.746494
$487$	−7.05965e10			−1.25507			−0.627534	−	0.778589i	$-0.715936\pi$
$487$	−7.05965e10			−1.25507			−0.627534	+	0.778589i	$0.715936\pi$
$488$	−2.46482e10			−0.434615
$489$	6.96486e9			0.121808
$490$	−8.40380e9			−0.145778
$491$		−	4.83027e10i		−	0.831085i	−0.909574	−	0.415542i	$-0.863592\pi$
$491$		−	4.83027e10i		−	0.831085i	0.909574	−	0.415542i	$-0.136408\pi$
$492$			5.73370e9i			0.0978531i
$493$		−	1.99756e11i		−	3.38151i
$494$	8.15389e9			0.136917
$495$			8.99470e9i			0.149819i
$496$	−3.23780e10			−0.534962
$497$	−5.05272e10			−0.828132
$498$	−1.43625e10			−0.233513
$499$			6.67014e10i			1.07580i	0.843008	+	0.537901i	$0.180783\pi$
$499$			6.67014e10i			1.07580i	−0.843008	+	0.537901i	$0.819217\pi$
$500$		−	8.54892e10i		−	1.36783i
$501$			8.30071e9i			0.131754i
$502$		−	8.23333e10i		−	1.29647i
$503$			1.08689e11i			1.69791i	0.528468	+	0.848954i	$0.322767\pi$
$503$			1.08689e11i			1.69791i	−0.528468	+	0.848954i	$0.677233\pi$
$504$	−3.09388e10			−0.479493
$505$		−	3.39403e10i		−	0.521855i
$506$			3.44739e10i			0.525882i
$507$			1.12195e10i			0.169802i
$508$	2.63562e10			0.395757
$509$	2.21280e10			0.329664			0.164832	−	0.986322i	$-0.447292\pi$
$509$	2.21280e10			0.329664			0.164832	+	0.986322i	$0.447292\pi$
$510$		−	2.32265e10i		−	0.343323i
$511$	1.06633e11			1.56390
$512$		−	7.61762e10i		−	1.10851i
$513$	3.36169e10			0.485387
$514$	−1.47409e11			−2.11190
$515$			4.63152e10i			0.658407i
$516$	−1.34101e10	+	8.29109e9i	−0.189162	+	0.116953i
$517$	1.03224e10			0.144484
$518$			1.17988e11i			1.63877i
$519$			1.17050e9i			0.0161326i
$520$	−1.44548e9			−0.0197697
$521$		−	8.18254e10i		−	1.11055i	−0.831668	−	0.555274i	$-0.812613\pi$
$521$		−	8.18254e10i		−	1.11055i	0.831668	−	0.555274i	$-0.187387\pi$
$522$	−1.91009e11			−2.57260
$523$		−	1.35833e10i		−	0.181552i	−0.995871	−	0.0907758i	$-0.971065\pi$
$523$		−	1.35833e10i		−	0.181552i	0.995871	−	0.0907758i	$-0.0289347\pi$
$524$		−	7.23810e10i		−	0.960063i
$525$	7.34456e9			0.0966783
$526$	1.74070e11			2.27395
$527$	1.32680e11			1.72014
$528$			1.79891e9i			0.0231459i
$529$	1.06748e11			1.36313
$530$	6.73056e10			0.852997
$531$	−1.13096e9			−0.0142255
$532$	1.61217e11			2.01264
$533$	2.21683e9			0.0274678
$534$		−	9.06244e9i		−	0.111450i
$535$		−	4.41998e10i		−	0.539517i
$536$		−	3.68111e10i		−	0.445984i
$537$	1.97571e9			0.0237589
$538$			2.54492e10i			0.303770i
$539$	−2.66721e9			−0.0316011
$540$	−2.55216e10			−0.300146
$541$	−2.54665e10			−0.297289			−0.148645	−	0.988891i	$-0.547491\pi$
$541$	−2.54665e10			−0.297289			−0.148645	+	0.988891i	$0.547491\pi$
$542$			2.09300e11i			2.42534i
$543$		−	1.18387e10i		−	0.136177i
$544$			2.33685e11i			2.66830i
$545$		−	1.07626e11i		−	1.21992i
$546$		−	1.53315e9i		−	0.0172510i
$547$	−1.06228e11			−1.18656			−0.593278	−	0.804998i	$-0.702167\pi$
$547$	−1.06228e11			−1.18656			−0.593278	+	0.804998i	$0.702167\pi$
$548$		−	1.66788e11i		−	1.84945i
$549$			8.28888e10i			0.912445i
$550$		−	1.66261e10i		−	0.181693i
$551$	2.32413e11			2.52147
$552$	−1.12520e10			−0.121192
$553$			2.12621e10i			0.227355i
$554$	−1.97114e11			−2.09257
$555$			1.11959e10i			0.118001i
$556$	4.89526e9			0.0512244
$557$	−5.50843e10			−0.572278			−0.286139	−	0.958188i	$-0.592372\pi$
$557$	−5.50843e10			−0.572278			−0.286139	+	0.958188i	$0.592372\pi$
$558$		−	1.26870e11i		−	1.30865i
$559$	3.20560e9	+	5.18479e9i	0.0328293	+	0.0530987i
$560$	4.33285e10			0.440577
$561$		−	7.37167e9i		−	0.0744242i
$562$			7.54871e10i			0.756706i
$563$	1.49411e11			1.48713			0.743564	−	0.668664i	$-0.233134\pi$
$563$	1.49411e11			1.48713			0.743564	+	0.668664i	$0.233134\pi$
$564$			1.44286e10i			0.142596i
$565$	1.22032e11			1.19751
$566$		−	2.39221e11i		−	2.33095i
$567$			1.00837e11i			0.975632i
$568$	−3.73325e10			−0.358669
$569$	−1.27794e11			−1.21916			−0.609582	−	0.792723i	$-0.708663\pi$
$569$	−1.27794e11			−1.21916			−0.609582	+	0.792723i	$0.708663\pi$
$570$	2.70238e10			0.256004
$571$		−	1.25011e11i		−	1.17599i	−0.808865	−	0.587995i	$-0.799918\pi$
$571$		−	1.25011e11i		−	1.17599i	0.808865	−	0.587995i	$-0.200082\pi$
$572$	−1.96470e9			−0.0183532
$573$	−1.74756e10			−0.162111
$574$	7.74270e10			0.713255
$575$	−8.92502e10			−0.816465
$576$	1.59054e11			1.44496
$577$			1.59101e11i			1.43539i	0.696360	+	0.717693i	$0.254802\pi$
$577$			1.59101e11i			1.43539i	−0.696360	+	0.717693i	$0.745198\pi$
$578$		−	4.66595e11i		−	4.18051i
$579$		−	1.79741e10i		−	0.159931i
$580$	−1.76446e11			−1.55919
$581$			1.09793e11i			0.963539i
$582$	3.37122e10			0.293829
$583$	2.13615e10			0.184909
$584$	7.87871e10			0.677336
$585$			4.86099e9i			0.0415051i
$586$			2.86776e11i			2.43193i
$587$			1.39779e10i			0.117731i	0.998266	+	0.0588654i	$0.0187483\pi$
$587$			1.39779e10i			0.117731i	−0.998266	+	0.0588654i	$0.981252\pi$
$588$		−	3.72819e9i		−	0.0311880i
$589$			1.54372e11i			1.28264i
$590$	−1.84551e9			−0.0152303
$591$		−	2.85278e10i		−	0.233840i
$592$		−	7.48169e10i		−	0.609134i
$593$		−	8.73992e10i		−	0.706787i	−0.935475	−	0.353394i	$-0.885028\pi$
$593$		−	8.73992e10i		−	0.706787i	0.935475	−	0.353394i	$-0.114972\pi$
$594$	−1.43088e10			−0.114936
$595$	−1.77553e11			−1.41665
$596$			2.46899e10i			0.195675i
$597$	−1.24666e10			−0.0981412
$598$			1.86307e10i			0.145688i
$599$	−1.57778e11			−1.22557			−0.612785	−	0.790250i	$-0.709951\pi$
$599$	−1.57778e11			−1.22557			−0.612785	+	0.790250i	$0.709951\pi$
$600$	5.42660e9			0.0418719
$601$		−	1.08858e11i		−	0.834381i	−0.908819	−	0.417190i	$-0.863015\pi$
$601$		−	1.08858e11i		−	0.834381i	0.908819	−	0.417190i	$-0.136985\pi$
$602$	1.11962e11	+	1.81088e11i	0.852478	+	1.37881i
$603$	−1.23791e11			−0.936312
$604$			2.38788e11i			1.79417i
$605$		−	8.70801e10i		−	0.649976i
$606$	2.65980e10			0.197224
$607$		−	2.80501e10i		−	0.206624i	−0.994649	−	0.103312i	$-0.967056\pi$
$607$		−	2.80501e10i		−	0.206624i	0.994649	−	0.103312i	$-0.0329440\pi$
$608$	−2.71890e11			−1.98966
$609$		−	4.37000e10i		−	0.317697i
$610$			1.35259e11i			0.976890i
$611$	5.57854e9			0.0400272
$612$	−3.44289e11			−2.45425
$613$	9.06755e10			0.642167			0.321084	−	0.947051i	$-0.395953\pi$
$613$	9.06755e10			0.642167			0.321084	+	0.947051i	$0.395953\pi$
$614$			8.94466e10i			0.629348i
$615$	7.34707e9			0.0513586
$616$	−1.60234e10			−0.111284
$617$	2.14084e11			1.47721			0.738607	−	0.674136i	$-0.235484\pi$
$617$	2.14084e11			1.47721			0.738607	+	0.674136i	$0.235484\pi$
$618$	−3.62959e10			−0.248831
$619$	−4.84848e10			−0.330250			−0.165125	−	0.986273i	$-0.552803\pi$
$619$	−4.84848e10			−0.330250			−0.165125	+	0.986273i	$0.552803\pi$
$620$		−	1.17197e11i		−	0.793141i
$621$			7.68107e10i			0.516482i
$622$		−	1.23751e11i		−	0.826774i
$623$	−6.92772e10			−0.459873
$624$			9.72183e8i			0.00641224i
$625$	−2.85017e10			−0.186789
$626$	−1.99437e11			−1.29870
$627$	8.57685e9			0.0554955
$628$			6.37631e9i			0.0409950i
$629$			3.06588e11i			1.95863i
$630$			1.69779e11i			1.07776i
$631$		−	1.62923e11i		−	1.02770i	−0.857881	−	0.513848i	$-0.828220\pi$
$631$		−	1.62923e11i		−	1.02770i	0.857881	−	0.513848i	$-0.171780\pi$
$632$			1.57097e10i			0.0984689i
$633$	−3.24193e10			−0.201924
$634$		−	2.60502e11i		−	1.61233i
$635$		−	3.37724e10i		−	0.207715i
$636$			2.98588e10i			0.182492i
$637$	−1.44144e9			−0.00875462
$638$	−9.89247e10			−0.597066
$639$			1.25545e11i			0.753000i
$640$	1.01330e11			0.603971
$641$		−	1.93130e11i		−	1.14398i	−0.820262	−	0.571988i	$-0.806172\pi$
$641$		−	1.93130e11i		−	1.14398i	0.820262	−	0.571988i	$-0.193828\pi$
$642$	3.46381e10			0.203899
$643$	−2.75508e11			−1.61172			−0.805861	−	0.592105i	$-0.798297\pi$
$643$	−2.75508e11			−1.61172			−0.805861	+	0.592105i	$0.798297\pi$
$644$			3.68363e11i			2.14157i
$645$	1.06241e10	+	1.71835e10i	0.0613835	+	0.0992826i
$646$	7.40018e11			4.24925
$647$			1.84960e11i			1.05550i	0.849398	+	0.527752i	$0.176965\pi$
$647$			1.84960e11i			1.05550i	−0.849398	+	0.527752i	$0.823035\pi$
$648$			7.45041e10i			0.422552i
$649$	−5.85730e8			−0.00330156
$650$		−	8.98519e9i		−	0.0503354i
$651$	2.90261e10			0.161608
$652$		−	1.68470e11i		−	0.932247i
$653$		−	2.03633e11i		−	1.11994i	−0.828513	−	0.559970i	$-0.810813\pi$
$653$		−	2.03633e11i		−	1.11994i	0.828513	−	0.559970i	$-0.189187\pi$
$654$	8.43435e10			0.461042
$655$	−9.27478e10			−0.503893
$656$	−4.90970e10			−0.265118
$657$		−	2.64952e11i		−	1.42202i
$658$	1.94841e11			1.03939
$659$	3.21214e11			1.70315			0.851575	−	0.524232i	$-0.175648\pi$
$659$	3.21214e11			1.70315			0.851575	+	0.524232i	$0.175648\pi$
$660$	−6.51145e9			−0.0343164
$661$	−2.59186e11			−1.35771			−0.678853	−	0.734274i	$-0.737523\pi$
$661$	−2.59186e11			−1.35771			−0.678853	+	0.734274i	$0.737523\pi$
$662$	5.00390e10			0.260541
$663$		−	3.98386e9i		−	0.0206181i
$664$			8.11215e10i			0.417315i
$665$		−	2.06581e11i		−	1.05634i
$666$	2.93164e11			1.49009
$667$			5.31037e11i			2.68300i
$668$	2.00781e11			1.00837
$669$	−3.82951e10			−0.191178
$670$	−2.02003e11			−1.00244
$671$			4.29286e10i			0.211766i
$672$			5.11227e10i			0.250690i
$673$			1.82200e11i			0.888153i	0.895989	+	0.444077i	$0.146468\pi$
$673$			1.82200e11i			0.888153i	−0.895989	+	0.444077i	$0.853532\pi$
$674$		−	2.90553e11i		−	1.40794i
$675$		−	3.70442e10i		−	0.178445i
$676$	2.71383e11			1.29956
$677$			3.72816e11i			1.77476i	0.461040	+	0.887379i	$0.347476\pi$
$677$			3.72816e11i			1.77476i	−0.461040	+	0.887379i	$0.652524\pi$
$678$			9.56330e10i			0.452573i
$679$		−	2.57710e11i		−	1.21242i
$680$	−1.31187e11			−0.613558
$681$	−2.85599e9			−0.0132791
$682$		−	6.57069e10i		−	0.303720i
$683$	−5.95037e10			−0.273439			−0.136720	−	0.990610i	$-0.543656\pi$
$683$	−5.95037e10			−0.273439			−0.136720	+	0.990610i	$0.543656\pi$
$684$		−	4.00577e11i		−	1.83004i
$685$	−2.13720e11			−0.970694
$686$	3.08656e11			1.39373
$687$			3.85121e10i			0.172890i
$688$	−7.09957e10	−	1.14829e11i	−0.316868	−	0.512506i
$689$	1.15444e10			0.0512264
$690$			6.17461e10i			0.272404i
$691$		−	1.43990e11i		−	0.631568i	−0.948831	−	0.315784i	$-0.897732\pi$
$691$		−	1.43990e11i		−	0.631568i	0.948831	−	0.315784i	$-0.102268\pi$
$692$	2.83127e10			0.123469
$693$			5.38848e10i			0.233633i
$694$	8.36775e10			0.360721
$695$		−	6.27271e9i		−	0.0268854i
$696$		−	3.22882e10i		−	0.137596i
$697$	2.01192e11			0.852470
$698$	−2.04366e11			−0.860966
$699$	−4.87884e10			−0.204366
$700$		−	1.77654e11i		−	0.739916i
$701$	3.56792e11			1.47755			0.738776	−	0.673951i	$-0.235404\pi$
$701$	3.56792e11			1.47755			0.738776	+	0.673951i	$0.235404\pi$
$702$	−7.73285e9			−0.0318413
$703$	−3.56711e11			−1.46048
$704$	8.23751e10			0.335356
$705$	1.84885e10			0.0748420
$706$		−	3.84059e11i		−	1.54589i
$707$		−	2.03327e11i		−	0.813798i
$708$		−	8.18725e8i		−	0.00325840i
$709$	−1.69458e11			−0.670622			−0.335311	−	0.942107i	$-0.608841\pi$
$709$	−1.69458e11			−0.670622			−0.335311	+	0.942107i	$0.608841\pi$
$710$			2.04865e11i			0.806184i
$711$	5.28298e10			0.206729
$712$	−5.11861e10			−0.199174
$713$	−3.52721e11			−1.36481
$714$		−	1.39144e11i		−	0.535390i
$715$			2.51753e9i			0.00963277i
$716$		−	4.77895e10i		−	0.181836i
$717$			2.86883e10i			0.108549i
$718$			4.25926e11i			1.60264i
$719$	−3.94401e11			−1.47578			−0.737892	−	0.674919i	$-0.764179\pi$
$719$	−3.94401e11			−1.47578			−0.737892	+	0.674919i	$0.764179\pi$
$720$		−	1.07658e11i		−	0.400606i
$721$			2.77462e11i			1.02674i
$722$			4.48477e11i			1.65041i
$723$	−3.65141e9			−0.0133631
$724$	−2.86360e11			−1.04222
$725$		−	2.56108e11i		−	0.926982i
$726$	6.82422e10			0.245644
$727$		−	4.40578e11i		−	1.57720i	−0.614909	−	0.788598i	$-0.710807\pi$
$727$		−	4.40578e11i		−	1.57720i	0.614909	−	0.788598i	$-0.289193\pi$
$728$	−8.65949e9			−0.0308295
$729$	2.34373e11			0.829846
$730$		−	4.32350e11i		−	1.52245i
$731$	2.90929e11	+	4.70553e11i	1.01887	+	1.64793i
$732$	−6.00049e10			−0.208998
$733$			9.56982e10i			0.331503i	0.986168	+	0.165752i	$0.0530050\pi$
$733$			9.56982e10i			0.331503i	−0.986168	+	0.165752i	$0.946995\pi$
$734$		−	4.77716e11i		−	1.64583i
$735$	−4.77723e9			−0.0163692
$736$		−	6.21236e11i		−	2.11712i
$737$	−6.41122e10			−0.217305
$738$		−	1.92383e11i		−	0.648545i
$739$			2.11867e10i			0.0710373i	0.999369	+	0.0355186i	$0.0113083\pi$
$739$			2.11867e10i			0.0710373i	−0.999369	+	0.0355186i	$0.988692\pi$
$740$	2.70812e11			0.903110
$741$	4.63517e9			0.0153742
$742$	4.03209e11			1.33019
$743$			5.43058e10i			0.178193i	0.996023	+	0.0890965i	$0.0283980\pi$
$743$			5.43058e10i			0.178193i	−0.996023	+	0.0890965i	$0.971602\pi$
$744$	2.14462e10			0.0699936
$745$	3.16372e10			0.102701
$746$	−5.40514e11			−1.74523
$747$	2.72802e11			0.876123
$748$	−1.78309e11			−0.569597
$749$		−	2.64789e11i		−	0.841341i
$750$		−	8.58468e10i		−	0.271318i
$751$			1.24175e11i			0.390367i	0.980767	+	0.195183i	$0.0625302\pi$
$751$			1.24175e11i			0.390367i	−0.980767	+	0.195183i	$0.937470\pi$
$752$	−1.23550e11			−0.386342
$753$		−	4.68033e10i		−	0.145578i
$754$	−5.34617e10			−0.165408
$755$	3.05978e11			0.941679
$756$	−1.52893e11			−0.468058
$757$			2.64168e11i			0.804447i	0.915542	+	0.402223i	$0.131762\pi$
$757$			2.64168e11i			0.804447i	−0.915542	+	0.402223i	$0.868238\pi$
$758$			5.40395e11i			1.63695i
$759$			1.95971e10i			0.0590506i
$760$		−	1.52635e11i		−	0.457508i
$761$			2.13908e11i			0.637806i	0.947787	+	0.318903i	$0.103314\pi$
$761$			2.13908e11i			0.637806i	−0.947787	+	0.318903i	$0.896686\pi$
$762$	2.64665e10			0.0785012
$763$		−	6.44758e11i		−	1.90239i
$764$			4.22708e11i			1.24070i
$765$			4.41167e11i			1.28812i
$766$	−4.57608e11			−1.32916
$767$	−3.16545e8			−0.000914649
$768$		−	8.84705e9i		−	0.0254304i
$769$	1.42773e11			0.408263			0.204132	−	0.978943i	$-0.434563\pi$
$769$	1.42773e11			0.408263			0.204132	+	0.978943i	$0.434563\pi$
$770$			8.79296e10i			0.250134i
$771$	−8.37966e10			−0.237142
$772$	−4.34766e11			−1.22401
$773$		−	6.20070e9i		−	0.0173669i	−0.999962	−	0.00868346i	$-0.997236\pi$
$773$		−	6.20070e9i		−	0.0173669i	0.999962	−	0.00868346i	$-0.00276407\pi$
$774$	4.49950e11	−	2.78191e11i	1.25372	−	0.775138i
$775$	1.70110e11			0.471545
$776$		−	1.90412e11i		−	0.525106i
$777$			6.70715e10i			0.184015i
$778$	1.71203e11			0.467297
$779$			2.34084e11i			0.635657i
$780$	−3.51897e9			−0.00950687
$781$			6.50203e10i			0.174761i
$782$			1.69085e12i			4.52147i
$783$	−2.20412e11			−0.586393
$784$	3.19240e10			0.0844993
$785$	8.17050e9			0.0215164
$786$		−	7.26838e10i		−	0.190435i
$787$	1.45567e11			0.379458			0.189729	−	0.981836i	$-0.439239\pi$
$787$	1.45567e11			0.379458			0.189729	+	0.981836i	$0.439239\pi$
$788$	−6.90045e11			−1.78967
$789$	9.89519e10			0.255338
$790$	8.62081e10			0.221330
$791$	7.31060e11			1.86744
$792$			3.98133e10i			0.101188i
$793$			2.31998e10i			0.0586667i
$794$			1.88844e11i			0.475138i
$795$	3.82606e10			0.0957819
$796$			3.01549e11i			0.751113i
$797$	−7.57891e11			−1.87834			−0.939168	−	0.343458i	$-0.888402\pi$
$797$	−7.57891e11			−1.87834			−0.939168	+	0.343458i	$0.888402\pi$
$798$	1.61892e11			0.399221
$799$	5.06289e11			1.24226
$800$			2.99609e11i			0.731468i
$801$			1.72133e11i			0.418152i
$802$		−	4.54701e10i		−	0.109908i
$803$		−	1.37220e11i		−	0.330031i
$804$		−	8.96150e10i		−	0.214465i
$805$	4.72014e11			1.12401
$806$		−	3.55099e10i		−	0.0841413i
$807$			1.44669e10i			0.0341099i
$808$		−	1.50230e11i		−	0.352461i
$809$	−4.18797e11			−0.977709			−0.488854	−	0.872365i	$-0.662585\pi$
$809$	−4.18797e11			−0.977709			−0.488854	+	0.872365i	$0.662585\pi$
$810$	4.08847e11			0.949775
$811$			2.57888e11i			0.596139i	0.954544	+	0.298069i	$0.0963426\pi$
$811$			2.57888e11i			0.596139i	−0.954544	+	0.298069i	$0.903657\pi$
$812$	−1.05704e12			−2.43145
$813$			1.18979e11i			0.272338i
$814$	1.51831e11			0.345831
$815$	−2.15874e11			−0.489294
$816$			8.82320e10i			0.199006i
$817$	−5.47483e11	+	3.38493e11i	−1.22880	+	0.759733i
$818$	−3.98491e11			−0.890032
$819$			2.91209e10i			0.0647245i
$820$		−	1.77714e11i		−	0.393068i
$821$	6.01891e11			1.32478			0.662392	−	0.749158i	$-0.269541\pi$
$821$	6.01891e11			1.32478			0.662392	+	0.749158i	$0.269541\pi$
$822$		−	1.67486e11i		−	0.366853i
$823$	8.61186e10			0.187714			0.0938572	−	0.995586i	$-0.470080\pi$
$823$	8.61186e10			0.187714			0.0938572	+	0.995586i	$0.470080\pi$
$824$			2.05005e11i			0.444688i
$825$		−	9.45127e9i		−	0.0204021i
$826$	−1.10559e10			−0.0237506
$827$	8.32732e11			1.78026			0.890129	−	0.455708i	$-0.150614\pi$
$827$	8.32732e11			1.78026			0.890129	+	0.455708i	$0.150614\pi$
$828$	9.15271e11			1.94728
$829$		−	3.87262e11i		−	0.819949i	−0.912097	−	0.409975i	$-0.865538\pi$
$829$		−	3.87262e11i		−	0.819949i	0.912097	−	0.409975i	$-0.134462\pi$
$830$	4.45160e11			0.938003
$831$	−1.12052e11			−0.234971
$832$	4.45178e10			0.0929054
$833$	−1.30820e11			−0.271702
$834$	4.91574e9			0.0101607
$835$		−	2.57278e11i		−	0.529245i
$836$		−	2.07461e11i		−	0.424728i
$837$		−	1.46400e11i		−	0.298291i
$838$	−1.18317e12			−2.39922
$839$		−	4.89599e11i		−	0.988081i	−0.869439	−	0.494040i	$-0.835519\pi$
$839$		−	4.89599e11i		−	0.988081i	0.869439	−	0.494040i	$-0.164481\pi$
$840$	−2.86995e10			−0.0576443
$841$	−1.02359e12			−2.04618
$842$	2.20076e11			0.437849
$843$			4.29115e10i			0.0849695i
$844$			7.84174e11i			1.54541i
$845$		−	3.47745e11i		−	0.682078i
$846$		−	4.84121e11i		−	0.945088i
$847$		−	5.21673e11i		−	1.01360i
$848$	−2.55678e11			−0.494435
$849$		−	1.35988e11i		−	0.261739i
$850$		−	8.15464e11i		−	1.56217i
$851$		−	8.15044e11i		−	1.55404i
$852$	−9.08844e10			−0.172477
$853$	5.16576e11			0.975749			0.487875	−	0.872914i	$-0.337772\pi$
$853$	5.16576e11			0.975749			0.487875	+	0.872914i	$0.337772\pi$
$854$			8.10297e11i			1.52340i
$855$	−5.13292e11			−0.960506
$856$		−	1.95642e11i		−	0.364390i
$857$	5.30011e11			0.982565			0.491282	−	0.871000i	$-0.336528\pi$
$857$	5.30011e11			0.982565			0.491282	+	0.871000i	$0.336528\pi$
$858$	−1.97292e9			−0.00364050
$859$			9.23019e11i			1.69527i	0.530583	+	0.847633i	$0.321973\pi$
$859$			9.23019e11i			1.69527i	−0.530583	+	0.847633i	$0.678027\pi$
$860$	4.15643e11	−	2.56980e11i	0.759848	−	0.469792i
$861$	4.40142e10			0.0800904
$862$			1.10976e11i			0.201003i
$863$			4.53159e11i			0.816972i	0.912765	+	0.408486i	$0.133943\pi$
$863$			4.53159e11i			0.816972i	−0.912765	+	0.408486i	$0.866057\pi$
$864$	2.57850e11			0.462714
$865$		−	3.62794e10i		−	0.0648031i
$866$	2.65852e11			0.472681
$867$		−	2.65241e11i		−	0.469424i
$868$		−	7.02096e11i		−	1.23685i
$869$	2.73609e10			0.0479789
$870$	−1.77184e11			−0.309277
$871$	−3.46480e10			−0.0602013
$872$		−	4.76386e11i		−	0.823935i
$873$	−6.40332e11			−1.10242
$874$	−1.96729e12			−3.37150
$875$	−6.56250e11			−1.11953
$876$	1.91804e11			0.325718
$877$	1.41919e10			0.0239907			0.0119953	−	0.999928i	$-0.496182\pi$
$877$	1.41919e10			0.0239907			0.0119953	+	0.999928i	$0.496182\pi$
$878$			2.17520e11i			0.366034i
$879$			1.63021e11i			0.273079i
$880$		−	5.57568e10i		−	0.0929752i
$881$	3.79321e11			0.629656			0.314828	−	0.949149i	$-0.398053\pi$
$881$	3.79321e11			0.629656			0.314828	+	0.949149i	$0.398053\pi$
$882$			1.25092e11i			0.206706i
$883$	9.27615e11			1.52590			0.762948	−	0.646459i	$-0.223751\pi$
$883$	9.27615e11			1.52590			0.762948	+	0.646459i	$0.223751\pi$
$884$	−9.63634e10			−0.157799
$885$	−1.04910e9			−0.00171019
$886$		−	5.55937e11i		−	0.902175i
$887$		−	1.16557e12i		−	1.88297i	−0.337058	−	0.941484i	$-0.609432\pi$
$887$		−	1.16557e12i		−	1.88297i	0.337058	−	0.941484i	$-0.390568\pi$
$888$			4.95564e10i			0.0796981i
$889$		−	2.02321e11i		−	0.323918i
$890$			2.80888e11i			0.447685i
$891$	1.29760e11			0.205888
$892$			9.26299e11i			1.46316i
$893$			5.89061e11i			0.926306i
$894$			2.47932e10i			0.0388135i
$895$	−6.12367e10			−0.0954376
$896$	6.07038e11			0.941854
$897$			1.05908e10i			0.0163591i
$898$	3.98498e11			0.612804
$899$		−	1.01215e12i		−	1.54955i
$900$	−4.41416e11			−0.672788
$901$	1.04773e12			1.58982
$902$		−	9.96360e10i		−	0.150519i
$903$	6.36458e10	+	1.02942e11i	0.0957236	+	0.154825i
$904$	5.40151e11			0.808800
$905$			3.66937e11i			0.547013i
$906$			2.39787e11i			0.355887i
$907$	−9.10561e11			−1.34549			−0.672744	−	0.739876i	$-0.734884\pi$
$907$	−9.10561e11			−1.34549			−0.672744	+	0.739876i	$0.734884\pi$
$908$			6.90822e10i			0.101630i
$909$	−5.05205e11			−0.739967
$910$			4.75196e10i			0.0692959i
$911$		−	6.72912e11i		−	0.976978i	−0.872570	−	0.488489i	$-0.837548\pi$
$911$		−	6.72912e11i		−	0.976978i	0.872570	−	0.488489i	$-0.162452\pi$
$912$	−1.02657e11			−0.148391
$913$	1.41286e11			0.203336
$914$	1.63470e11			0.234235
$915$			7.68893e10i			0.109694i
$916$	9.31548e11			1.32319
$917$	−5.55626e11			−0.785788
$918$	−7.01807e11			−0.988204
$919$	1.19782e12			1.67930			0.839652	−	0.543125i	$-0.182759\pi$
$919$	1.19782e12			1.67930			0.839652	+	0.543125i	$0.182759\pi$
$920$	3.48752e11			0.486817
$921$			5.08470e10i			0.0706686i
$922$		−	2.39082e11i		−	0.330844i
$923$			3.51388e10i			0.0484151i
$924$	−3.90083e10			−0.0535142
$925$			3.93079e11i			0.536924i
$926$	1.54106e12			2.09593
$927$	6.89408e11			0.933593
$928$	1.78267e12			2.40369
$929$		−	3.59469e11i		−	0.482612i	−0.970449	−	0.241306i	$-0.922424\pi$
$929$		−	3.59469e11i		−	0.482612i	0.970449	−	0.241306i	$-0.0775758\pi$
$930$		−	1.17688e11i		−	0.157325i
$931$		−	1.52207e11i		−	0.202598i
$932$			1.18012e12i			1.56409i
$933$		−	7.03476e10i		−	0.0928373i
$934$	8.82335e11			1.15943
$935$			2.28483e11i			0.298956i
$936$			2.15162e10i			0.0280326i
$937$			6.42002e11i			0.832871i	0.909165	+	0.416436i	$0.136721\pi$
$937$			6.42002e11i			0.832871i	−0.909165	+	0.416436i	$0.863279\pi$
$938$	−1.21015e12			−1.56324
$939$	−1.13372e11			−0.145829
$940$		−	4.47209e11i		−	0.572795i
$941$	−1.25177e12			−1.59650			−0.798248	−	0.602329i	$-0.794239\pi$
$941$	−1.25177e12			−1.59650			−0.798248	+	0.602329i	$0.794239\pi$
$942$			6.40299e9i			0.00813166i
$943$	−5.34855e11			−0.676378
$944$	7.01065e9			0.00882816
$945$			1.95914e11i			0.245663i
$946$	2.33031e11	−	1.44077e11i	0.290971	−	0.179899i
$947$	5.56633e11			0.692101			0.346050	−	0.938216i	$-0.387523\pi$
$947$	5.56633e11			0.692101			0.346050	+	0.938216i	$0.387523\pi$
$948$			3.82446e10i			0.0473518i
$949$		−	7.41576e10i		−	0.0914304i
$950$	9.48783e11			1.16486
$951$		−	1.48085e11i		−	0.181046i
$952$	−7.85905e11			−0.956803
$953$		−	8.41007e11i		−	1.01960i	−0.860294	−	0.509798i	$-0.829720\pi$
$953$		−	8.41007e11i		−	1.01960i	0.860294	−	0.509798i	$-0.170280\pi$
$954$		−	1.00185e12i		−	1.20951i
$955$	5.41651e11			0.651188
$956$	6.93926e11			0.830771
$957$	−5.62349e10			−0.0670437
$958$			1.51706e12i			1.80111i
$959$	−1.28034e12			−1.51373
$960$	1.47542e11			0.173712
$961$	−1.80609e11			−0.211761
$962$	8.20539e10			0.0958073
$963$	−6.57920e11			−0.765011
$964$			8.83220e10i			0.102273i
$965$			5.57101e11i			0.642429i
$966$			3.69904e11i			0.424796i
$967$	3.17275e11			0.362853			0.181426	−	0.983405i	$-0.441929\pi$
$967$	3.17275e11			0.362853			0.181426	+	0.983405i	$0.441929\pi$
$968$		−	3.85443e11i		−	0.438994i
$969$	4.20672e11			0.477143
$970$	−1.04490e12			−1.18029
$971$	−9.75359e11			−1.09720			−0.548602	−	0.836084i	$-0.684840\pi$
$971$	−9.75359e11			−1.09720			−0.548602	+	0.836084i	$0.684840\pi$
$972$			5.72639e11i			0.641529i
$973$		−	3.75781e10i		−	0.0419259i
$974$			1.71477e12i			1.90533i
$975$		−	5.10773e9i		−	0.00565210i
$976$		−	5.13815e11i		−	0.566249i
$977$	6.48624e11			0.711893			0.355947	−	0.934506i	$-0.384158\pi$
$977$	6.48624e11			0.711893			0.355947	+	0.934506i	$0.384158\pi$
$978$		−	1.69174e11i		−	0.184918i
$979$			8.91486e10i			0.0970474i
$980$			1.15554e11i			0.125280i
$981$	−1.60203e12			−1.72979
$982$	−1.17326e12			−1.26167
$983$		−	1.10400e12i		−	1.18238i	−0.806533	−	0.591189i	$-0.798659\pi$
$983$		−	1.10400e12i		−	1.18238i	0.806533	−	0.591189i	$-0.201341\pi$
$984$	3.25203e10			0.0346876
$985$			8.84212e11i			0.939315i
$986$	−4.85200e12			−5.13349
$987$	1.10760e11			0.116711
$988$		−	1.12118e11i		−	0.117665i
$989$	−7.73416e11	−	1.25093e12i	−0.808403	−	1.30752i
$990$	2.18478e11			0.227441
$991$		−	8.40020e11i		−	0.870954i	−0.900200	−	0.435477i	$-0.856580\pi$
$991$		−	8.40020e11i		−	0.870954i	0.900200	−	0.435477i	$-0.143420\pi$
$992$			1.18407e12i			1.22273i
$993$	2.84452e10			0.0292558
$994$			1.22729e12i			1.25719i
$995$	3.86399e11			0.394225
$996$			1.97487e11i			0.200679i
$997$			1.03692e12i			1.04946i	0.851270	+	0.524728i	$0.175833\pi$
$997$			1.03692e12i			1.04946i	−0.851270	+	0.524728i	$0.824167\pi$
$998$	1.62015e12			1.63318
$999$	3.38292e11			0.339649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.b.b.42.4	✓	28
43.42	odd	2	inner	43.9.b.b.42.25	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.b.b.42.4	✓	28	1.1	even	1	trivial
43.9.b.b.42.25	yes	28	43.42	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:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-24.2897i q^{2} -13.8077i q^{3} -333.988 q^{4} +427.967i q^{5} -335.386 q^{6} +2563.83i q^{7} +1894.31i q^{8} +6370.35 q^{9} +O(q^{10})\)	\(q-24.2897i q^{2} -13.8077i q^{3} -333.988 q^{4} +427.967i q^{5} -335.386 q^{6} +2563.83i q^{7} +1894.31i q^{8} +6370.35 q^{9} +10395.2 q^{10} +3299.24 q^{11} +4611.62i q^{12} +1783.00 q^{13} +62274.6 q^{14} +5909.26 q^{15} -39488.8 q^{16} +161819. q^{17} -154734. i q^{18} +188274. i q^{19} -142936. i q^{20} +35400.7 q^{21} -80137.4i q^{22} -430185. q^{23} +26156.2 q^{24} +207469. q^{25} -43308.5i q^{26} -178553. i q^{27} -856290. i q^{28} -1.23444e6i q^{29} -143534. i q^{30} +819928. q^{31} +1.44411e6i q^{32} -45555.0i q^{33} -3.93053e6i q^{34} -1.09724e6 q^{35} -2.12762e6 q^{36} +1.89464e6i q^{37} +4.57312e6 q^{38} -24619.2i q^{39} -810703. q^{40} +1.24331e6 q^{41} -859872. i q^{42} +(1.79787e6 + 2.90790e6i) q^{43} -1.10191e6 q^{44} +2.72630e6i q^{45} +1.04491e7i q^{46} +3.12874e6 q^{47} +545251. i q^{48} -808432. q^{49} -5.03936e6i q^{50} -2.23435e6i q^{51} -595502. q^{52} +6.47469e6 q^{53} -4.33699e6 q^{54} +1.41196e6i q^{55} -4.85670e6 q^{56} +2.59964e6 q^{57} -2.99841e7 q^{58} -177535. q^{59} -1.97362e6 q^{60} +1.30117e7i q^{61} -1.99158e7i q^{62} +1.63325e7i q^{63} +2.49679e7 q^{64} +763066. i q^{65} -1.10652e6 q^{66} -1.94324e7 q^{67} -5.40456e7 q^{68} +5.93988e6i q^{69} +2.66515e7i q^{70} +1.97077e7i q^{71} +1.20674e7i q^{72} -4.15914e7i q^{73} +4.60201e7 q^{74} -2.86468e6i q^{75} -6.28815e7i q^{76} +8.45869e6i q^{77} -597993. q^{78} +8.29308e6 q^{79} -1.68999e7i q^{80} +3.93304e7 q^{81} -3.01997e7i q^{82} +4.28237e7 q^{83} -1.18234e7 q^{84} +6.92532e7i q^{85} +(7.06319e7 - 4.36696e7i) q^{86} -1.70448e7 q^{87} +6.24978e6i q^{88} +2.70210e7i q^{89} +6.62209e7 q^{90} +4.57131e6i q^{91} +1.43677e8 q^{92} -1.13214e7i q^{93} -7.59960e7i q^{94} -8.05752e7 q^{95} +1.99400e7 q^{96} -1.00518e8 q^{97} +1.96366e7i q^{98} +2.10173e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4284 q^{4} - 1794 q^{6} - 80754 q^{9}+O(q^{10}) \) 28 * q - 4284 * q^4 - 1794 * q^6 - 80754 * q^9	\( 28 q - 4284 q^{4} - 1794 q^{6} - 80754 q^{9} + 24982 q^{10} + 4538 q^{11} + 22086 q^{13} + 24732 q^{14} + 15388 q^{15} + 525812 q^{16} - 135136 q^{17} - 261352 q^{21} - 184432 q^{23} + 1770326 q^{24} - 2640434 q^{25} - 110272 q^{31} + 10947816 q^{35} + 11602066 q^{36} - 7189158 q^{38} - 21389338 q^{40} + 1301336 q^{41} + 2473420 q^{43} - 8818480 q^{44} + 1983566 q^{47} - 15560936 q^{49} + 12927876 q^{52} + 23942594 q^{53} - 13757972 q^{54} + 34967256 q^{56} + 35225148 q^{57} + 22565734 q^{58} - 5554336 q^{59} - 44902072 q^{60} - 170444572 q^{64} - 48457584 q^{66} - 130953802 q^{67} + 150021122 q^{68} + 205870278 q^{74} + 267860612 q^{78} + 7380250 q^{79} - 57601004 q^{81} - 42603970 q^{83} + 251931292 q^{84} - 45482652 q^{86} - 106687410 q^{87} - 255044692 q^{90} - 409532014 q^{92} + 123322986 q^{95} - 692987086 q^{96} - 318744840 q^{97} - 609707206 q^{99}+O(q^{100}) \) 28 * q - 4284 * q^4 - 1794 * q^6 - 80754 * q^9 + 24982 * q^10 + 4538 * q^11 + 22086 * q^13 + 24732 * q^14 + 15388 * q^15 + 525812 * q^16 - 135136 * q^17 - 261352 * q^21 - 184432 * q^23 + 1770326 * q^24 - 2640434 * q^25 - 110272 * q^31 + 10947816 * q^35 + 11602066 * q^36 - 7189158 * q^38 - 21389338 * q^40 + 1301336 * q^41 + 2473420 * q^43 - 8818480 * q^44 + 1983566 * q^47 - 15560936 * q^49 + 12927876 * q^52 + 23942594 * q^53 - 13757972 * q^54 + 34967256 * q^56 + 35225148 * q^57 + 22565734 * q^58 - 5554336 * q^59 - 44902072 * q^60 - 170444572 * q^64 - 48457584 * q^66 - 130953802 * q^67 + 150021122 * q^68 + 205870278 * q^74 + 267860612 * q^78 + 7380250 * q^79 - 57601004 * q^81 - 42603970 * q^83 + 251931292 * q^84 - 45482652 * q^86 - 106687410 * q^87 - 255044692 * q^90 - 409532014 * q^92 + 123322986 * q^95 - 692987086 * q^96 - 318744840 * q^97 - 609707206 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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