LMFDB - Embedded newform 37.8.b.a.36.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,8,Mod(36,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("37.36");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 37 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	37.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:	$11.5582459429$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 1702 x^{18} + 1194509 x^{16} + 450999516 x^{14} + 100204783492 x^{12} + 13461378480848 x^{10} + \cdots + 13\!\cdots\!96 $ x^20 + 1702x^18 + 1194509x^16 + 450999516x^14 + 100204783492x^12 + 13461378480848x^10 + 1081011973644416x^8 + 49304995250225664x^6 + 1131877572418003968x^4 + 9402469145336696832*x^2 + 130757963535876096
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			36.8
Root			$-7.52533i$ of defining polynomial
Character	$\chi$	$=$	37.36
Dual form			37.8.b.a.36.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.52533i q^{2} -79.9440 q^{3} +71.3694 q^{4} -468.997i q^{5} +601.605i q^{6} -1534.47 q^{7} -1500.32i q^{8} +4204.05 q^{9} +O(q^{10})$	\(q-7.52533i q^{2} -79.9440 q^{3} +71.3694 q^{4} -468.997i q^{5} +601.605i q^{6} -1534.47 q^{7} -1500.32i q^{8} +4204.05 q^{9} -3529.36 q^{10} +1242.09 q^{11} -5705.56 q^{12} +9018.86i q^{13} +11547.4i q^{14} +37493.5i q^{15} -2155.12 q^{16} -10548.3i q^{17} -31636.8i q^{18} +46646.5i q^{19} -33472.0i q^{20} +122672. q^{21} -9347.12i q^{22} -32095.0i q^{23} +119942. i q^{24} -141833. q^{25} +67869.9 q^{26} -161251. q^{27} -109514. q^{28} -29625.3i q^{29} +282151. q^{30} +102768. i q^{31} -175823. i q^{32} -99297.5 q^{33} -79379.7 q^{34} +719661. i q^{35} +300040. q^{36} +(19692.1 - 307480. i) q^{37} +351030. q^{38} -721004. i q^{39} -703646. q^{40} -647947. q^{41} -923144. i q^{42} +300560. i q^{43} +88647.1 q^{44} -1.97169e6i q^{45} -241525. q^{46} +372716. q^{47} +172289. q^{48} +1.53105e6 q^{49} +1.06734e6i q^{50} +843276. i q^{51} +643671. i q^{52} +308300. q^{53} +1.21347e6i q^{54} -582535. i q^{55} +2.30219e6i q^{56} -3.72911e6i q^{57} -222940. q^{58} +470589. i q^{59} +2.67589e6i q^{60} +916513. i q^{61} +773364. q^{62} -6.45098e6 q^{63} -1.59898e6 q^{64} +4.22982e6 q^{65} +747246. i q^{66} -2.61073e6 q^{67} -752828. i q^{68} +2.56580e6i q^{69} +5.41569e6 q^{70} -3.34388e6 q^{71} -6.30742e6i q^{72} +725900. q^{73} +(-2.31389e6 - 148189. i) q^{74} +1.13387e7 q^{75} +3.32913e6i q^{76} -1.90594e6 q^{77} -5.42579e6 q^{78} +165767. i q^{79} +1.01075e6i q^{80} +3.69679e6 q^{81} +4.87602e6i q^{82} -5.82996e6 q^{83} +8.75500e6 q^{84} -4.94714e6 q^{85} +2.26182e6 q^{86} +2.36836e6i q^{87} -1.86353e6i q^{88} +4.14004e6i q^{89} -1.48376e7 q^{90} -1.38392e7i q^{91} -2.29060e6i q^{92} -8.21570e6i q^{93} -2.80481e6i q^{94} +2.18771e7 q^{95} +1.40560e7i q^{96} +4.09598e6i q^{97} -1.15217e7i q^{98} +5.22180e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9}+O(q^{10}) $ 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9	$ 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9} - 882 q^{10} + 3498 q^{11} - 30374 q^{12} + 36116 q^{16} + 113482 q^{21} - 108112 q^{25} + 49278 q^{26} - 304110 q^{27} - 41192 q^{28} + 429776 q^{30} + 305646 q^{33} - 960356 q^{34} + 484758 q^{36} + 108732 q^{37} + 1049916 q^{38} - 496346 q^{40} - 1577742 q^{41} + 685266 q^{44} - 2906298 q^{46} - 1512786 q^{47} + 1522958 q^{48} + 3269246 q^{49} + 2999358 q^{53} + 405946 q^{58} + 3728310 q^{62} - 11995292 q^{63} - 11109700 q^{64} + 4251792 q^{65} + 3562224 q^{67} + 21605644 q^{70} - 15259086 q^{71} + 11088018 q^{73} - 2036544 q^{74} + 14882062 q^{75} - 2419122 q^{77} - 12178734 q^{78} - 17764972 q^{81} - 12873822 q^{83} + 9944396 q^{84} - 2698920 q^{85} + 15345336 q^{86} - 13219100 q^{90} + 48981192 q^{95} + 43111380 q^{99}+O(q^{100}) $ 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9 - 882 * q^10 + 3498 * q^11 - 30374 * q^12 + 36116 * q^16 + 113482 * q^21 - 108112 * q^25 + 49278 * q^26 - 304110 * q^27 - 41192 * q^28 + 429776 * q^30 + 305646 * q^33 - 960356 * q^34 + 484758 * q^36 + 108732 * q^37 + 1049916 * q^38 - 496346 * q^40 - 1577742 * q^41 + 685266 * q^44 - 2906298 * q^46 - 1512786 * q^47 + 1522958 * q^48 + 3269246 * q^49 + 2999358 * q^53 + 405946 * q^58 + 3728310 * q^62 - 11995292 * q^63 - 11109700 * q^64 + 4251792 * q^65 + 3562224 * q^67 + 21605644 * q^70 - 15259086 * q^71 + 11088018 * q^73 - 2036544 * q^74 + 14882062 * q^75 - 2419122 * q^77 - 12178734 * q^78 - 17764972 * q^81 - 12873822 * q^83 + 9944396 * q^84 - 2698920 * q^85 + 15345336 * q^86 - 13219100 * q^90 + 48981192 * q^95 + 43111380 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$		−	7.52533i		−	0.665151i	−0.943077	−	0.332576i	$-0.892082\pi$
$2$		−	7.52533i		−	0.665151i	0.943077	−	0.332576i	$-0.107918\pi$
$3$	−79.9440			−1.70947			−0.854735	−	0.519064i	$-0.826281\pi$
$3$	−79.9440			−1.70947			−0.854735	+	0.519064i	$0.826281\pi$
$4$	71.3694			0.557574
$5$		−	468.997i		−	1.67793i	−0.544182	−	0.838967i	$-0.683160\pi$
$5$		−	468.997i		−	1.67793i	0.544182	−	0.838967i	$-0.316840\pi$
$6$			601.605i			1.13706i
$7$	−1534.47			−1.69089			−0.845444	−	0.534064i	$-0.820664\pi$
$7$	−1534.47			−1.69089			−0.845444	+	0.534064i	$0.820664\pi$
$8$		−	1500.32i		−	1.03602i
$9$	4204.05			1.92229
$10$	−3529.36			−1.11608
$11$	1242.09			0.281370			0.140685	−	0.990054i	$-0.455070\pi$
$11$	1242.09			0.281370			0.140685	+	0.990054i	$0.455070\pi$
$12$	−5705.56			−0.953156
$13$			9018.86i			1.13854i	0.822149	+	0.569272i	$0.192775\pi$
$13$			9018.86i			1.13854i	−0.822149	+	0.569272i	$0.807225\pi$
$14$			11547.4i			1.12470i
$15$			37493.5i			2.86838i
$16$	−2155.12			−0.131538
$17$		−	10548.3i		−	0.520730i	−0.965510	−	0.260365i	$-0.916157\pi$
$17$		−	10548.3i		−	0.520730i	0.965510	−	0.260365i	$-0.0838429\pi$
$18$		−	31636.8i		−	1.27861i
$19$			46646.5i			1.56020i	0.625652	+	0.780102i	$0.284833\pi$
$19$			46646.5i			1.56020i	−0.625652	+	0.780102i	$0.715167\pi$
$20$		−	33472.0i		−	0.935572i
$21$	122672.			2.89052
$22$		−	9347.12i		−	0.187154i
$23$		−	32095.0i		−	0.550034i	−0.961439	−	0.275017i	$-0.911317\pi$
$23$		−	32095.0i		−	0.550034i	0.961439	−	0.275017i	$-0.0886835\pi$
$24$			119942.i			1.77105i
$25$	−141833.			−1.81546
$26$	67869.9			0.757304
$27$	−161251.			−1.57663
$28$	−109514.			−0.942794
$29$		−	29625.3i		−	0.225564i	−0.993620	−	0.112782i	$-0.964024\pi$
$29$		−	29625.3i		−	0.225564i	0.993620	−	0.112782i	$-0.0359761\pi$
$30$	282151.			1.90791
$31$			102768.i			0.619573i	0.950806	+	0.309787i	$0.100258\pi$
$31$			102768.i			0.619573i	−0.950806	+	0.309787i	$0.899742\pi$
$32$		−	175823.i		−	0.948529i
$33$	−99297.5			−0.480994
$34$	−79379.7			−0.346364
$35$			719661.i			2.83720i
$36$	300040.			1.07182
$37$	19692.1	−	307480.i	0.0639125	−	0.997956i
$37$	19692.1	−	307480.i	0.0639125	−	0.997956i
$38$	351030.			1.03777
$39$		−	721004.i		−	1.94631i
$40$	−703646.			−1.73838
$41$	−647947.			−1.46824			−0.734119	−	0.679021i	$-0.762404\pi$
$41$	−647947.			−1.46824			−0.734119	+	0.679021i	$0.762404\pi$
$42$		−	923144.i		−	1.92264i
$43$			300560.i			0.576490i	0.957557	+	0.288245i	$0.0930718\pi$
$43$			300560.i			0.576490i	−0.957557	+	0.288245i	$0.906928\pi$
$44$	88647.1			0.156884
$45$		−	1.97169e6i		−	3.22548i
$46$	−241525.			−0.365856
$47$	372716.			0.523644			0.261822	−	0.965116i	$-0.415677\pi$
$47$	372716.			0.523644			0.261822	+	0.965116i	$0.415677\pi$
$48$	172289.			0.224861
$49$	1.53105e6			1.85910
$50$			1.06734e6i			1.20756i
$51$			843276.i			0.890173i
$52$			643671.i			0.634822i
$53$	308300.			0.284451			0.142225	−	0.989834i	$-0.454574\pi$
$53$	308300.			0.284451			0.142225	+	0.989834i	$0.454574\pi$
$54$			1.21347e6i			1.04870i
$55$		−	582535.i		−	0.472120i
$56$			2.30219e6i			1.75180i
$57$		−	3.72911e6i		−	2.66712i
$58$	−222940.			−0.150034
$59$			470589.i			0.298305i	0.988814	+	0.149152i	$0.0476545\pi$
$59$			470589.i			0.298305i	−0.988814	+	0.149152i	$0.952346\pi$
$60$			2.67589e6i			1.59933i
$61$			916513.i			0.516993i	0.966012	+	0.258496i	$0.0832270\pi$
$61$			916513.i			0.516993i	−0.966012	+	0.258496i	$0.916773\pi$
$62$	773364.			0.412110
$63$	−6.45098e6			−3.25038
$64$	−1.59898e6			−0.762454
$65$	4.22982e6			1.91040
$66$			747246.i			0.319934i
$67$	−2.61073e6			−1.06047			−0.530237	−	0.847850i	$-0.677897\pi$
$67$	−2.61073e6			−1.06047			−0.530237	+	0.847850i	$0.677897\pi$
$68$		−	752828.i		−	0.290345i
$69$			2.56580e6i			0.940267i
$70$	5.41569e6			1.88717
$71$	−3.34388e6			−1.10878			−0.554391	−	0.832256i	$-0.687049\pi$
$71$	−3.34388e6			−1.10878			−0.554391	+	0.832256i	$0.687049\pi$
$72$		−	6.30742e6i		−	1.99153i
$73$	725900.			0.218397			0.109199	−	0.994020i	$-0.465172\pi$
$73$	725900.			0.218397			0.109199	+	0.994020i	$0.465172\pi$
$74$	−2.31389e6	−	148189.i	−0.663792	−	0.0425115i
$75$	1.13387e7			3.10348
$76$			3.32913e6i			0.869928i
$77$	−1.90594e6			−0.475765
$78$	−5.42579e6			−1.29459
$79$			165767.i			0.0378272i	0.999821	+	0.0189136i	$0.00602075\pi$
$79$			165767.i			0.0378272i	−0.999821	+	0.0189136i	$0.993979\pi$
$80$			1.01075e6i			0.220713i
$81$	3.69679e6			0.772907
$82$			4.87602e6i			0.976600i
$83$	−5.82996e6			−1.11916			−0.559580	−	0.828776i	$-0.689038\pi$
$83$	−5.82996e6			−1.11916			−0.559580	+	0.828776i	$0.689038\pi$
$84$	8.75500e6			1.61168
$85$	−4.94714e6			−0.873751
$86$	2.26182e6			0.383453
$87$			2.36836e6i			0.385595i
$88$		−	1.86353e6i		−	0.291506i
$89$			4.14004e6i			0.622500i	0.950328	+	0.311250i	$0.100748\pi$
$89$			4.14004e6i			0.622500i	−0.950328	+	0.311250i	$0.899252\pi$
$90$	−1.48376e7			−2.14543
$91$		−	1.38392e7i		−	1.92515i
$92$		−	2.29060e6i		−	0.306684i
$93$		−	8.21570e6i		−	1.05914i
$94$		−	2.80481e6i		−	0.348303i
$95$	2.18771e7			2.61792
$96$			1.40560e7i			1.62148i
$97$			4.09598e6i			0.455677i	0.973699	+	0.227838i	$0.0731658\pi$
$97$			4.09598e6i			0.455677i	−0.973699	+	0.227838i	$0.926834\pi$
$98$		−	1.15217e7i		−	1.23658i
$99$	5.22180e6			0.540875
$100$	−1.01225e7			−1.01225
$101$	−4.88454e6			−0.471736			−0.235868	−	0.971785i	$-0.575793\pi$
$101$	−4.88454e6			−0.471736			−0.235868	+	0.971785i	$0.575793\pi$
$102$	6.34593e6			0.592100
$103$			1.24677e7i			1.12424i	0.827057	+	0.562118i	$0.190013\pi$
$103$			1.24677e7i			1.12424i	−0.827057	+	0.562118i	$0.809987\pi$
$104$	1.35312e7			1.17956
$105$		−	5.75326e7i		−	4.85011i
$106$		−	2.32006e6i		−	0.189203i
$107$	−2.16328e7			−1.70714			−0.853572	−	0.520974i	$-0.825569\pi$
$107$	−2.16328e7			−1.70714			−0.853572	+	0.520974i	$0.825569\pi$
$108$	−1.15084e7			−0.879085
$109$		−	1.44707e7i		−	1.07028i	−0.844763	−	0.535140i	$-0.820259\pi$
$109$		−	1.44707e7i		−	1.07028i	0.844763	−	0.535140i	$-0.179741\pi$
$110$	−4.38377e6			−0.314032
$111$	−1.57426e6	+	2.45812e7i	−0.109256	+	1.70598i
$112$	3.30697e6			0.222416
$113$			7.08230e6i			0.461742i	0.972984	+	0.230871i	$0.0741576\pi$
$113$			7.08230e6i			0.461742i	−0.972984	+	0.230871i	$0.925842\pi$
$114$	−2.80628e7			−1.77404
$115$	−1.50524e7			−0.922921
$116$		−	2.11434e6i		−	0.125768i
$117$			3.79157e7i			2.18861i
$118$	3.54134e6			0.198418
$119$			1.61861e7i			0.880496i
$120$	5.62523e7			2.97171
$121$	−1.79444e7			−0.920831
$122$	6.89706e6			0.343878
$123$	5.17995e7			2.50991
$124$			7.33450e6i			0.345458i
$125$			2.98789e7i			1.36829i
$126$			4.85457e7i			2.16199i
$127$	−1.77853e7			−0.770458			−0.385229	−	0.922821i	$-0.625878\pi$
$127$	−1.77853e7			−0.770458			−0.385229	+	0.922821i	$0.625878\pi$
$128$		−	1.04725e7i		−	0.441382i
$129$		−	2.40280e7i		−	0.985493i
$130$		−	3.18308e7i		−	1.27071i
$131$		−	2.34022e7i		−	0.909509i	−0.890617	−	0.454754i	$-0.849727\pi$
$131$		−	2.34022e7i		−	0.909509i	0.890617	−	0.454754i	$-0.150273\pi$
$132$	−7.08680e6			−0.268189
$133$		−	7.15776e7i		−	2.63813i
$134$			1.96466e7i			0.705375i
$135$			7.56262e7i			2.64548i
$136$	−1.58259e7			−0.539488
$137$	1.71962e7			0.571360			0.285680	−	0.958325i	$-0.407780\pi$
$137$	1.71962e7			0.571360			0.285680	+	0.958325i	$0.407780\pi$
$138$	1.93085e7			0.625420
$139$	3.50229e7			1.10611			0.553057	−	0.833143i	$-0.313461\pi$
$139$	3.50229e7			1.10611			0.553057	+	0.833143i	$0.313461\pi$
$140$			5.13618e7i			1.58195i
$141$	−2.97965e7			−0.895154
$142$			2.51638e7i			0.737508i
$143$			1.12022e7i			0.320352i
$144$	−9.06024e6			−0.252855
$145$	−1.38942e7			−0.378481
$146$		−	5.46264e6i		−	0.145267i
$147$	−1.22398e8			−3.17808
$148$	1.40541e6	−	2.19447e7i	0.0356359	−	0.556434i
$149$	−2.54751e7			−0.630905			−0.315453	−	0.948941i	$-0.602156\pi$
$149$	−2.54751e7			−0.630905			−0.315453	+	0.948941i	$0.602156\pi$
$150$		−	8.53275e7i		−	2.06429i
$151$	6.24028e7			1.47498			0.737488	−	0.675361i	$-0.236012\pi$
$151$	6.24028e7			1.47498			0.737488	+	0.675361i	$0.236012\pi$
$152$	6.99847e7			1.61641
$153$		−	4.43457e7i		−	1.00099i
$154$			1.43429e7i			0.316456i
$155$	4.81980e7			1.03960
$156$		−	5.14576e7i		−	1.08521i
$157$	3.20801e7			0.661587			0.330794	−	0.943703i	$-0.392684\pi$
$157$	3.20801e7			0.661587			0.330794	+	0.943703i	$0.392684\pi$
$158$	1.24745e6			0.0251608
$159$	−2.46467e7			−0.486261
$160$	−8.24605e7			−1.59157
$161$			4.92487e7i			0.930045i
$162$		−	2.78196e7i		−	0.514101i
$163$		−	5.82054e7i		−	1.05270i	−0.850267	−	0.526352i	$-0.823559\pi$
$163$		−	5.82054e7i		−	1.05270i	0.850267	−	0.526352i	$-0.176441\pi$
$164$	−4.62436e7			−0.818650
$165$			4.65702e7i			0.807076i
$166$			4.38724e7i			0.744412i
$167$			7.57508e7i			1.25858i	0.777172	+	0.629288i	$0.216653\pi$
$167$			7.57508e7i			1.25858i	−0.777172	+	0.629288i	$0.783347\pi$
$168$		−	1.84047e8i		−	2.99465i
$169$	−1.85913e7			−0.296283
$170$			3.72288e7i			0.581177i
$171$			1.96104e8i			2.99916i
$172$			2.14508e7i			0.321436i
$173$	−8.38791e7			−1.23166			−0.615832	−	0.787877i	$-0.711180\pi$
$173$	−8.38791e7			−1.23166			−0.615832	+	0.787877i	$0.711180\pi$
$174$	1.78227e7			0.256479
$175$	2.17638e8			3.06975
$176$	−2.67685e6			−0.0370109
$177$		−	3.76208e7i		−	0.509943i
$178$	3.11552e7			0.414057
$179$		−	9.55331e7i		−	1.24500i	−0.782621	−	0.622498i	$-0.786118\pi$
$179$		−	9.55331e7i		−	1.24500i	0.782621	−	0.622498i	$-0.213882\pi$
$180$		−	1.40718e8i		−	1.79844i
$181$	4.96787e7			0.622723			0.311362	−	0.950291i	$-0.399215\pi$
$181$	4.96787e7			0.622723			0.311362	+	0.950291i	$0.399215\pi$
$182$	−1.04144e8			−1.28052
$183$		−	7.32698e7i		−	0.883784i
$184$	−4.81527e7			−0.569847
$185$	−1.44207e8	−	9.23552e6i	−1.67450	−	0.107241i
$186$	−6.18259e7			−0.704490
$187$		−	1.31020e7i		−	0.146518i
$188$	2.66006e7			0.291970
$189$	2.47434e8			2.66590
$190$		−	1.64632e8i		−	1.74131i
$191$			4.61185e7i			0.478915i	0.970907	+	0.239458i	$0.0769696\pi$
$191$			4.61185e7i			0.478915i	−0.970907	+	0.239458i	$0.923030\pi$
$192$	1.27829e8			1.30339
$193$			1.44124e8i			1.44306i	0.692383	+	0.721530i	$0.256561\pi$
$193$			1.44124e8i			1.44306i	−0.692383	+	0.721530i	$0.743439\pi$
$194$	3.08236e7			0.303094
$195$	−3.38149e8			−3.26578
$196$	1.09270e8			1.03659
$197$	−2.61703e7			−0.243881			−0.121940	−	0.992537i	$-0.538912\pi$
$197$	−2.61703e7			−0.243881			−0.121940	+	0.992537i	$0.538912\pi$
$198$		−	3.92957e7i		−	0.359764i
$199$		−	2.08209e8i		−	1.87290i	−0.350807	−	0.936448i	$-0.614093\pi$
$199$		−	2.08209e8i		−	1.87290i	0.350807	−	0.936448i	$-0.385907\pi$
$200$			2.12795e8i			1.88086i
$201$	2.08712e8			1.81285
$202$			3.67578e7i			0.313776i
$203$			4.54590e7i			0.381403i
$204$			6.01841e7i			0.496337i
$205$			3.03885e8i			2.46361i
$206$	9.38239e7			0.747788
$207$		−	1.34929e8i		−	1.05732i
$208$		−	1.94368e7i		−	0.149762i
$209$			5.79390e7i			0.438995i
$210$	−4.32952e8			−3.22606
$211$	−1.53528e8			−1.12512			−0.562561	−	0.826756i	$-0.690184\pi$
$211$	−1.53528e8			−1.12512			−0.562561	+	0.826756i	$0.690184\pi$
$212$	2.20032e7			0.158602
$213$	2.67323e8			1.89543
$214$			1.62794e8i			1.13551i
$215$	1.40962e8			0.967313
$216$			2.41928e8i			1.63342i
$217$		−	1.57694e8i		−	1.04763i
$218$	−1.08897e8			−0.711899
$219$	−5.80314e7			−0.373343
$220$		−	4.15752e7i		−	0.263242i
$221$	9.51339e7			0.592874
$222$	1.84982e8	+	1.18469e7i	1.13473	+	0.0726721i
$223$	−1.71918e8			−1.03813			−0.519067	−	0.854733i	$-0.673721\pi$
$223$	−1.71918e8			−1.03813			−0.519067	+	0.854733i	$0.673721\pi$
$224$			2.69795e8i			1.60386i
$225$	−5.96273e8			−3.48985
$226$	5.32966e7			0.307129
$227$			3.23307e8i			1.83453i	0.398276	+	0.917266i	$0.369609\pi$
$227$			3.23307e8i			1.83453i	−0.398276	+	0.917266i	$0.630391\pi$
$228$		−	2.66144e8i		−	1.48712i
$229$	−3.15171e7			−0.173429			−0.0867146	−	0.996233i	$-0.527637\pi$
$229$	−3.15171e7			−0.173429			−0.0867146	+	0.996233i	$0.527637\pi$
$230$			1.13275e8i			0.613882i
$231$	1.52369e8			0.813307
$232$	−4.44474e7			−0.233689
$233$	1.42302e8			0.736996			0.368498	−	0.929629i	$-0.379872\pi$
$233$	1.42302e8			0.736996			0.368498	+	0.929629i	$0.379872\pi$
$234$	2.85328e8			1.45576
$235$		−	1.74803e8i		−	0.878640i
$236$			3.35857e7i			0.166327i
$237$		−	1.32521e7i		−	0.0646645i
$238$	1.21806e8			0.585663
$239$			2.99145e7i			0.141739i	0.997486	+	0.0708694i	$0.0225774\pi$
$239$			2.99145e7i			0.141739i	−0.997486	+	0.0708694i	$0.977423\pi$
$240$		−	8.08031e7i		−	0.377302i
$241$		−	2.32484e8i		−	1.06988i	−0.844891	−	0.534938i	$-0.820335\pi$
$241$		−	2.32484e8i		−	1.06988i	0.844891	−	0.534938i	$-0.179665\pi$
$242$			1.35037e8i			0.612492i
$243$	5.71192e7			0.255364
$244$			6.54110e7i			0.288261i
$245$		−	7.18058e8i		−	3.11945i
$246$		−	3.89808e8i		−	1.66947i
$247$	−4.20698e8			−1.77636
$248$	1.54185e8			0.641892
$249$	4.66071e8			1.91317
$250$	2.24849e8			0.910123
$251$		−	2.24634e8i		−	0.896638i	−0.893874	−	0.448319i	$-0.852023\pi$
$251$		−	2.24634e8i		−	0.896638i	0.893874	−	0.448319i	$-0.147977\pi$
$252$	−4.60402e8			−1.81232
$253$		−	3.98648e7i		−	0.154763i
$254$			1.33841e8i			0.512471i
$255$	3.95494e8			1.49365
$256$	−2.83479e8			−1.05604
$257$		−	1.80173e8i		−	0.662100i	−0.943613	−	0.331050i	$-0.892597\pi$
$257$		−	1.80173e8i		−	0.662100i	0.943613	−	0.331050i	$-0.107403\pi$
$258$	−1.80819e8			−0.655502
$259$	−3.02169e7	+	4.71819e8i	−0.108069	+	1.68743i
$260$	3.01880e8			1.06519
$261$		−	1.24546e8i		−	0.433599i
$262$	−1.76109e8			−0.604961
$263$	5.36369e7			0.181810			0.0909052	−	0.995860i	$-0.471024\pi$
$263$	5.36369e7			0.181810			0.0909052	+	0.995860i	$0.471024\pi$
$264$			1.48978e8i			0.498320i
$265$		−	1.44592e8i		−	0.477290i
$266$	−5.38645e8			−1.75476
$267$		−	3.30971e8i		−	1.06415i
$268$	−1.86326e8			−0.591292
$269$	−6.39643e7			−0.200357			−0.100179	−	0.994969i	$-0.531941\pi$
$269$	−6.39643e7			−0.200357			−0.100179	+	0.994969i	$0.531941\pi$
$270$	5.69112e8			1.75964
$271$	−1.60403e8			−0.489576			−0.244788	−	0.969577i	$-0.578718\pi$
$271$	−1.60403e8			−0.489576			−0.244788	+	0.969577i	$0.578718\pi$
$272$			2.27329e7i			0.0684959i
$273$			1.10636e9i			3.29099i
$274$		−	1.29407e8i		−	0.380041i
$275$	−1.76169e8			−0.510817
$276$			1.83120e8i			0.524268i
$277$			9.77098e7i			0.276222i	0.990417	+	0.138111i	$0.0441031\pi$
$277$			9.77098e7i			0.276222i	−0.990417	+	0.138111i	$0.955897\pi$
$278$		−	2.63559e8i		−	0.735734i
$279$			4.32042e8i			1.19100i
$280$	1.07972e9			2.93940
$281$		−	5.08343e8i		−	1.36674i	−0.730074	−	0.683368i	$-0.760514\pi$
$281$		−	5.08343e8i		−	1.36674i	0.730074	−	0.683368i	$-0.239486\pi$
$282$			2.24228e8i			0.595413i
$283$		−	3.25504e8i		−	0.853697i	−0.904323	−	0.426848i	$-0.859624\pi$
$283$		−	3.25504e8i		−	0.853697i	0.904323	−	0.426848i	$-0.140376\pi$
$284$	−2.38651e8			−0.618228
$285$	−1.74894e9			−4.47526
$286$	8.43004e7			0.213083
$287$	9.94255e8			2.48262
$288$		−	7.39168e8i		−	1.82335i
$289$	2.99071e8			0.728840
$290$			1.04558e8i			0.251747i
$291$		−	3.27449e8i		−	0.778966i
$292$	5.18071e7			0.121772
$293$	−6.51658e8			−1.51350			−0.756751	−	0.653703i	$-0.773214\pi$
$293$	−6.51658e8			−1.51350			−0.756751	+	0.653703i	$0.773214\pi$
$294$			9.21087e8i			2.11390i
$295$	2.20705e8			0.500536
$296$	−4.61319e8	−	2.95444e7i	−1.03390	−	0.0662147i
$297$	−2.00288e8			−0.443616
$298$			1.91709e8i			0.419648i
$299$	2.89460e8			0.626238
$300$	8.09237e8			1.73042
$301$		−	4.61200e8i		−	0.974781i
$302$		−	4.69602e8i		−	0.981082i
$303$	3.90490e8			0.806419
$304$		−	1.00529e8i		−	0.205226i
$305$	4.29842e8			0.867480
$306$	−3.33716e8			−0.665812
$307$	−9.51288e8			−1.87641			−0.938205	−	0.346081i	$-0.887512\pi$
$307$	−9.51288e8			−1.87641			−0.938205	+	0.346081i	$0.887512\pi$
$308$	−1.36026e8			−0.265274
$309$		−	9.96722e8i		−	1.92185i
$310$		−	3.62706e8i		−	0.691494i
$311$		−	3.95170e8i		−	0.744942i	−0.928044	−	0.372471i	$-0.878511\pi$
$311$		−	3.95170e8i		−	0.744942i	0.928044	−	0.372471i	$-0.121489\pi$
$312$	−1.08174e9			−2.01642
$313$			1.05621e9i			1.94691i	0.228888	+	0.973453i	$0.426491\pi$
$313$			1.05621e9i			1.94691i	−0.228888	+	0.973453i	$0.573509\pi$
$314$		−	2.41413e8i		−	0.440056i
$315$			3.02549e9i			5.45392i
$316$			1.18307e7i			0.0210914i
$317$	1.09980e8			0.193912			0.0969562	−	0.995289i	$-0.469089\pi$
$317$	1.09980e8			0.193912			0.0969562	+	0.995289i	$0.469089\pi$
$318$			1.85475e8i			0.323437i
$319$		−	3.67972e7i		−	0.0634669i
$320$			7.49918e8i			1.27935i
$321$	1.72942e9			2.91831
$322$	3.70613e8			0.618621
$323$	4.92043e8			0.812445
$324$	2.63838e8			0.430953
$325$		−	1.27917e9i		−	2.06699i
$326$	−4.38015e8			−0.700208
$327$			1.15685e9i			1.82961i
$328$			9.72129e8i			1.52113i
$329$	−5.71922e8			−0.885423
$330$	3.50456e8			0.536828
$331$		−	2.77960e8i		−	0.421294i	−0.977562	−	0.210647i	$-0.932443\pi$
$331$		−	2.77960e8i		−	0.421294i	0.977562	−	0.210647i	$-0.0675570\pi$
$332$	−4.16081e8			−0.624015
$333$	8.27864e7	−	1.29266e9i	0.122858	−	1.91836i
$334$	5.70050e8			0.837144
$335$			1.22442e9i			1.77940i
$336$	−2.64372e8			−0.380214
$337$	−5.78029e8			−0.822706			−0.411353	−	0.911476i	$-0.634944\pi$
$337$	−5.78029e8			−0.822706			−0.411353	+	0.911476i	$0.634944\pi$
$338$			1.39906e8i			0.197073i
$339$		−	5.66188e8i		−	0.789335i
$340$	−3.53074e8			−0.487180
$341$			1.27647e8i			0.174329i
$342$	1.47575e9			1.99490
$343$	−1.08565e9			−1.45264
$344$	4.50937e8			0.597257
$345$	1.20335e9			1.57771
$346$			6.31218e8i			0.819243i
$347$			6.16495e8i			0.792093i	0.918230	+	0.396047i	$0.129618\pi$
$347$			6.16495e8i			0.792093i	−0.918230	+	0.396047i	$0.870382\pi$
$348$			1.69029e8i			0.214997i
$349$	3.67910e8			0.463289			0.231645	−	0.972800i	$-0.425589\pi$
$349$	3.67910e8			0.463289			0.231645	+	0.972800i	$0.425589\pi$
$350$		−	1.63780e9i		−	2.04185i
$351$		−	1.45430e9i		−	1.79506i
$352$		−	2.18388e8i		−	0.266888i
$353$		−	8.69751e8i		−	1.05241i	−0.850359	−	0.526203i	$-0.823615\pi$
$353$		−	8.69751e8i		−	1.05241i	0.850359	−	0.526203i	$-0.176385\pi$
$354$	−2.83109e8			−0.339189
$355$			1.56827e9i			1.86046i
$356$			2.95472e8i			0.347090i
$357$		−	1.29398e9i		−	1.50518i
$358$	−7.18918e8			−0.828111
$359$	−1.48050e7			−0.0168880			−0.00844401	−	0.999964i	$-0.502688\pi$
$359$	−1.48050e7			−0.0168880			−0.00844401	+	0.999964i	$0.502688\pi$
$360$	−2.95816e9			−3.34166
$361$	−1.28202e9			−1.43424
$362$		−	3.73849e8i		−	0.414205i
$363$	1.43455e9			1.57413
$364$		−	9.87692e8i		−	1.07341i
$365$		−	3.40445e8i		−	0.366456i
$366$	−5.51379e8			−0.587850
$367$	1.15329e8			0.121788			0.0608941	−	0.998144i	$-0.480605\pi$
$367$	1.15329e8			0.121788			0.0608941	+	0.998144i	$0.480605\pi$
$368$			6.91686e7i			0.0723505i
$369$	−2.72400e9			−2.82238
$370$	−6.95004e7	+	1.08521e9i	−0.0713314	+	1.11380i
$371$	−4.73076e8			−0.480975
$372$		−	5.86350e8i		−	0.590550i
$373$	1.44953e9			1.44626			0.723132	−	0.690710i	$-0.242702\pi$
$373$	1.44953e9			1.44626			0.723132	+	0.690710i	$0.242702\pi$
$374$	−9.85965e7			−0.0974565
$375$		−	2.38864e9i		−	2.33906i
$376$		−	5.59194e8i		−	0.542507i
$377$	2.67186e8			0.256814
$378$		−	1.86203e9i		−	1.77323i
$379$	1.14753e9			1.08275			0.541374	−	0.840782i	$-0.317904\pi$
$379$	1.14753e9			1.08275			0.541374	+	0.840782i	$0.317904\pi$
$380$	1.56135e9			1.45968
$381$	1.42183e9			1.31708
$382$	3.47057e8			0.318551
$383$		−	9.56008e8i		−	0.869493i	−0.900553	−	0.434746i	$-0.856838\pi$
$383$		−	9.56008e8i		−	0.869493i	0.900553	−	0.434746i	$-0.143162\pi$
$384$			8.37212e8i			0.754530i
$385$			8.93882e8i			0.798303i
$386$	1.08458e9			0.959853
$387$			1.26357e9i			1.10818i
$388$			2.92328e8i			0.254073i
$389$		−	1.71414e9i		−	1.47647i	−0.674546	−	0.738233i	$-0.735661\pi$
$389$		−	1.71414e9i		−	1.47647i	0.674546	−	0.738233i	$-0.264339\pi$
$390$			2.54468e9i			2.17224i
$391$	−3.38548e8			−0.286419
$392$		−	2.29707e9i		−	1.92607i
$393$			1.87086e9i			1.55478i
$394$			1.96940e8i			0.162218i
$395$	7.77444e7			0.0634716
$396$	3.72676e8			0.301577
$397$	−8.71594e7			−0.0699113			−0.0349557	−	0.999389i	$-0.511129\pi$
$397$	−8.71594e7			−0.0699113			−0.0349557	+	0.999389i	$0.511129\pi$
$398$	−1.56684e9			−1.24576
$399$			5.72220e9i			4.50980i
$400$	3.05668e8			0.238803
$401$		−	4.38409e8i		−	0.339527i	−0.985485	−	0.169763i	$-0.945700\pi$
$401$		−	4.38409e8i		−	0.339527i	0.985485	−	0.169763i	$-0.0543004\pi$
$402$		−	1.57063e9i		−	1.20582i
$403$	−9.26852e8			−0.705412
$404$	−3.48607e8			−0.263028
$405$		−	1.73378e9i		−	1.29689i
$406$	3.42094e8			0.253691
$407$	2.44593e7	−	3.81917e8i	0.0179831	−	0.280795i
$408$	1.26518e9			0.922239
$409$			2.42054e9i			1.74936i	0.484698	+	0.874682i	$0.338930\pi$
$409$			2.42054e9i			1.74936i	−0.484698	+	0.874682i	$0.661070\pi$
$410$	2.28684e9			1.63867
$411$	−1.37473e9			−0.976724
$412$			8.89816e8i			0.626845i
$413$		−	7.22105e8i		−	0.504400i
$414$	−1.01538e9			−0.703281
$415$			2.73424e9i			1.87788i
$416$	1.58572e9			1.07994
$417$	−2.79987e9			−1.89087
$418$	4.36010e8			0.291998
$419$	−2.28794e9			−1.51948			−0.759741	−	0.650226i	$-0.774674\pi$
$419$	−2.28794e9			−1.51948			−0.759741	+	0.650226i	$0.774674\pi$
$420$		−	4.10607e9i		−	2.70429i
$421$			9.42286e8i			0.615454i	0.951475	+	0.307727i	$0.0995683\pi$
$421$			9.42286e8i			0.615454i	−0.951475	+	0.307727i	$0.900432\pi$
$422$			1.15535e9i			0.748376i
$423$	1.56692e9			1.00660
$424$		−	4.62548e8i		−	0.294698i
$425$			1.49610e9i			0.945366i
$426$		−	2.01170e9i		−	1.26075i
$427$		−	1.40636e9i		−	0.874176i
$428$	−1.54392e9			−0.951859
$429$		−	8.95550e8i		−	0.547633i
$430$		−	1.06078e9i		−	0.643410i
$431$			1.47592e9i			0.887956i	0.896038	+	0.443978i	$0.146433\pi$
$431$			1.47592e9i			0.887956i	−0.896038	+	0.443978i	$0.853567\pi$
$432$	3.47515e8			0.207387
$433$	−9.96879e8			−0.590112			−0.295056	−	0.955480i	$-0.595338\pi$
$433$	−9.96879e8			−0.590112			−0.295056	+	0.955480i	$0.595338\pi$
$434$	−1.18670e9			−0.696832
$435$	1.11076e9			0.647002
$436$		−	1.03277e9i		−	0.596760i
$437$	1.49712e9			0.858165
$438$			4.36705e8i			0.248330i
$439$		−	1.12739e8i		−	0.0635989i	−0.999494	−	0.0317995i	$-0.989876\pi$
$439$		−	1.12739e8i		−	0.0635989i	0.999494	−	0.0317995i	$-0.0101238\pi$
$440$	−8.73990e8			−0.489127
$441$	6.43661e9			3.57373
$442$		−	7.15914e8i		−	0.394351i
$443$	9.54128e8			0.521427			0.260714	−	0.965416i	$-0.416042\pi$
$443$	9.54128e8			0.521427			0.260714	+	0.965416i	$0.416042\pi$
$444$	−1.12354e8	+	1.75435e9i	−0.0609185	+	0.951207i
$445$	1.94167e9			1.04451
$446$			1.29374e9i			0.690517i
$447$	2.03658e9			1.07851
$448$	2.45359e9			1.28922
$449$		−	1.27258e8i		−	0.0663473i	−0.999450	−	0.0331737i	$-0.989439\pi$
$449$		−	1.27258e8i		−	0.0663473i	0.999450	−	0.0331737i	$-0.0105614\pi$
$450$			4.48715e9i			2.32128i
$451$	−8.04807e8			−0.413118
$452$			5.05460e8i			0.257455i
$453$	−4.98873e9			−2.52143
$454$	2.43299e9			1.22024
$455$	−6.49052e9			−3.23028
$456$	−5.59486e9			−2.76320
$457$			1.77023e9i			0.867606i	0.901008	+	0.433803i	$0.142829\pi$
$457$			1.77023e9i			0.867606i	−0.901008	+	0.433803i	$0.857171\pi$
$458$			2.37177e8i			0.115357i
$459$			1.70093e9i			0.820997i
$460$	−1.07428e9			−0.514596
$461$		−	5.62131e8i		−	0.267229i	−0.991033	−	0.133615i	$-0.957342\pi$
$461$		−	5.62131e8i		−	0.267229i	0.991033	−	0.133615i	$-0.0426585\pi$
$462$		−	1.14663e9i		−	0.540972i
$463$		−	5.93481e7i		−	0.0277890i	−0.999903	−	0.0138945i	$-0.995577\pi$
$463$		−	5.93481e7i		−	0.0277890i	0.999903	−	0.0138945i	$-0.00442290\pi$
$464$			6.38461e7i			0.0296703i
$465$	−3.85314e9			−1.77717
$466$		−	1.07087e9i		−	0.490214i
$467$		−	2.30665e9i		−	1.04803i	−0.851710	−	0.524014i	$-0.824434\pi$
$467$		−	2.30665e9i		−	1.04803i	0.851710	−	0.524014i	$-0.175566\pi$
$468$			2.70602e9i			1.22031i
$469$	4.00608e9			1.79314
$470$	−1.31545e9			−0.584429
$471$	−2.56461e9			−1.13096
$472$	7.06035e8			0.309050
$473$			3.73322e8i			0.162207i
$474$	−9.97265e7			−0.0430117
$475$		−	6.61602e9i		−	2.83249i
$476$			1.15519e9i			0.490941i
$477$	1.29611e9			0.546797
$478$	2.25116e8			0.0942778
$479$			1.44991e8i			0.0602790i	0.999546	+	0.0301395i	$0.00959515\pi$
$479$			1.44991e8i			0.0602790i	−0.999546	+	0.0301395i	$0.990405\pi$
$480$	6.59222e9			2.72074
$481$	2.77312e9	+	1.77600e8i	1.13622	+	0.0727672i
$482$	−1.74952e9			−0.711630
$483$		−	3.93714e9i		−	1.58989i
$484$	−1.28068e9			−0.513431
$485$	1.92100e9			0.764596
$486$		−	4.29841e8i		−	0.169856i
$487$			1.74737e8i			0.0685543i	0.999412	+	0.0342772i	$0.0109129\pi$
$487$			1.74737e8i			0.0685543i	−0.999412	+	0.0342772i	$0.989087\pi$
$488$	1.37506e9			0.535616
$489$			4.65317e9i			1.79957i
$490$	−5.40362e9			−2.07491
$491$	2.03634e8			0.0776364			0.0388182	−	0.999246i	$-0.487641\pi$
$491$	2.03634e8			0.0776364			0.0388182	+	0.999246i	$0.487641\pi$
$492$	3.69690e9			1.39946
$493$	−3.12497e8			−0.117458
$494$			3.16589e9i			1.18155i
$495$		−	2.44901e9i		−	0.907552i
$496$		−	2.21478e8i		−	0.0814976i
$497$	5.13108e9			1.87483
$498$		−	3.50734e9i		−	1.27255i
$499$		−	8.95989e8i		−	0.322813i	−0.986888	−	0.161406i	$-0.948397\pi$
$499$		−	8.95989e8i		−	0.322813i	0.986888	−	0.161406i	$-0.0516030\pi$
$500$			2.13244e9i			0.762925i
$501$		−	6.05582e9i		−	2.15150i
$502$	−1.69044e9			−0.596400
$503$		−	5.20251e9i		−	1.82274i	−0.411589	−	0.911370i	$-0.635026\pi$
$503$		−	5.20251e9i		−	1.82274i	0.411589	−	0.911370i	$-0.364974\pi$
$504$			9.67853e9i			3.36746i
$505$			2.29084e9i			0.791542i
$506$	−2.99995e8			−0.102941
$507$	1.48627e9			0.506487
$508$	−1.26933e9			−0.429587
$509$	−1.83206e9			−0.615782			−0.307891	−	0.951422i	$-0.599623\pi$
$509$	−1.83206e9			−0.615782			−0.307891	+	0.951422i	$0.599623\pi$
$510$		−	2.97622e9i		−	0.993504i
$511$	−1.11387e9			−0.369285
$512$			7.92792e8i			0.261044i
$513$		−	7.52179e9i		−	2.45986i
$514$	−1.35586e9			−0.440397
$515$	5.84734e9			1.88640
$516$		−	1.71486e9i		−	0.549485i
$517$	4.62947e8			0.147338
$518$	3.55059e9	+	2.27392e8i	1.12240	+	0.0718821i
$519$	6.70563e9			2.10549
$520$		−	6.34608e9i		−	1.97922i
$521$	−6.38354e9			−1.97756			−0.988780	−	0.149382i	$-0.952272\pi$
$521$	−6.38354e9			−1.97756			−0.988780	+	0.149382i	$0.952272\pi$
$522$	−9.37250e8			−0.288409
$523$		−	4.56025e9i		−	1.39391i	−0.717117	−	0.696953i	$-0.754539\pi$
$523$		−	4.56025e9i		−	1.39391i	0.717117	−	0.696953i	$-0.245461\pi$
$524$		−	1.67020e9i		−	0.507118i
$525$	−1.73989e10			−5.24764
$526$		−	4.03636e8i		−	0.120931i
$527$	1.08403e9			0.322630
$528$	2.13998e8			0.0632691
$529$	2.37474e9			0.697463
$530$	−1.08810e9			−0.317470
$531$			1.97838e9i			0.573428i
$532$		−	5.10845e9i		−	1.47095i
$533$		−	5.84375e9i		−	1.67165i
$534$	−2.49067e9			−0.707818
$535$			1.01457e10i			2.86448i
$536$			3.91693e9i			1.09867i
$537$			7.63730e9i			2.12829i
$538$			4.81353e8i			0.133268i
$539$	1.90170e9			0.523095
$540$			5.39739e9i			1.47505i
$541$			4.90530e9i			1.33191i	0.745991	+	0.665956i	$0.231976\pi$
$541$			4.90530e9i			1.33191i	−0.745991	+	0.665956i	$0.768024\pi$
$542$			1.20709e9i			0.325642i
$543$	−3.97152e9			−1.06453
$544$	−1.85464e9			−0.493928
$545$	−6.78673e9			−1.79586
$546$	8.32571e9			2.18901
$547$		−	1.57457e9i		−	0.411346i	−0.978621	−	0.205673i	$-0.934062\pi$
$547$		−	1.57457e9i		−	0.411346i	0.978621	−	0.205673i	$-0.0659384\pi$
$548$	1.22728e9			0.318575
$549$			3.85306e9i			0.993809i
$550$			1.32573e9i			0.339771i
$551$	1.38191e9			0.351925
$552$	3.84952e9			0.974137
$553$		−	2.54365e8i		−	0.0639615i
$554$	7.35299e8			0.183730
$555$	1.15285e10	+	7.38325e8i	2.86252	+	0.183325i
$556$	2.49956e9			0.616740
$557$			9.70284e8i			0.237906i	0.992900	+	0.118953i	$0.0379538\pi$
$557$			9.70284e8i			0.237906i	−0.992900	+	0.118953i	$0.962046\pi$
$558$	3.25126e9			0.792195
$559$	−2.71071e9			−0.656360
$560$		−	1.55096e9i		−	0.373200i
$561$			1.04742e9i			0.250468i
$562$	−3.82545e9			−0.909087
$563$		−	3.94458e9i		−	0.931584i	−0.884894	−	0.465792i	$-0.845770\pi$
$563$		−	3.94458e9i		−	0.931584i	0.884894	−	0.465792i	$-0.154230\pi$
$564$	−2.12656e9			−0.499114
$565$	3.32158e9			0.774773
$566$	−2.44952e9			−0.567838
$567$	−5.67261e9			−1.30690
$568$			5.01689e9i			1.14872i
$569$			5.38883e9i			1.22631i	0.789961	+	0.613157i	$0.210101\pi$
$569$			5.38883e9i			1.22631i	−0.789961	+	0.613157i	$0.789899\pi$
$570$			1.31614e10i			2.97672i
$571$	−5.30348e9			−1.19216			−0.596080	−	0.802925i	$-0.703276\pi$
$571$	−5.30348e9			−1.19216			−0.596080	+	0.802925i	$0.703276\pi$
$572$			7.99496e8i			0.178620i
$573$		−	3.68690e9i		−	0.818691i
$574$		−	7.48209e9i		−	1.65132i
$575$			4.55213e9i			0.998566i
$576$	−6.72220e9			−1.46566
$577$		−	4.56886e9i		−	0.990130i	−0.868856	−	0.495065i	$-0.835144\pi$
$577$		−	4.56886e9i		−	0.990130i	0.868856	−	0.495065i	$-0.164856\pi$
$578$		−	2.25061e9i		−	0.484789i
$579$		−	1.15218e10i		−	2.46687i
$580$	−9.91618e8			−0.211031
$581$	8.94589e9			1.89238
$582$	−2.46416e9			−0.518131
$583$	3.82935e8			0.0800360
$584$		−	1.08908e9i		−	0.226264i
$585$	1.77824e10			3.67235
$586$			4.90394e9i			1.00671i
$587$			5.39359e9i			1.10064i	0.834955	+	0.550319i	$0.185494\pi$
$587$			5.39359e9i			1.10064i	−0.834955	+	0.550319i	$0.814506\pi$
$588$	−8.73549e9			−1.77201
$589$	−4.79377e9			−0.966661
$590$		−	1.66088e9i		−	0.332932i
$591$	2.09216e9			0.416907
$592$	−4.24388e7	+	6.62658e8i	−0.00840693	+	0.131269i
$593$	−4.69994e9			−0.925551			−0.462776	−	0.886475i	$-0.653146\pi$
$593$	−4.69994e9			−0.925551			−0.462776	+	0.886475i	$0.653146\pi$
$594$			1.50723e9i			0.295072i
$595$	7.59122e9			1.47741
$596$	−1.81814e9			−0.351776
$597$			1.66451e10i			3.20166i
$598$		−	2.17828e9i		−	0.416543i
$599$	4.08534e9			0.776666			0.388333	−	0.921519i	$-0.373051\pi$
$599$	4.08534e9			0.776666			0.388333	+	0.921519i	$0.373051\pi$
$600$		−	1.70117e10i		−	3.21528i
$601$	−2.51324e9			−0.472252			−0.236126	−	0.971722i	$-0.575878\pi$
$601$	−2.51324e9			−0.472252			−0.236126	+	0.971722i	$0.575878\pi$
$602$	−3.47068e9			−0.648377
$603$	−1.09756e10			−2.03854
$604$	4.45365e9			0.822407
$605$			8.41586e9i			1.54509i
$606$		−	2.93857e9i		−	0.536391i
$607$		−	4.50460e7i		−	0.00817515i	−0.999992	−	0.00408758i	$-0.998699\pi$
$607$		−	4.50460e7i		−	0.00817515i	0.999992	−	0.00408758i	$-0.00130112\pi$
$608$	8.20153e9			1.47990
$609$		−	3.63418e9i		−	0.651997i
$610$		−	3.23470e9i		−	0.577005i
$611$			3.36148e9i			0.596192i
$612$		−	3.16493e9i		−	0.558128i
$613$	−1.76519e9			−0.309514			−0.154757	−	0.987953i	$-0.549459\pi$
$613$	−1.76519e9			−0.309514			−0.154757	+	0.987953i	$0.549459\pi$
$614$			7.15875e9i			1.24810i
$615$		−	2.42938e10i		−	4.21146i
$616$			2.85953e9i			0.492903i
$617$	8.62546e9			1.47837			0.739186	−	0.673501i	$-0.235210\pi$
$617$	8.62546e9			1.47837			0.739186	+	0.673501i	$0.235210\pi$
$618$	−7.50066e9			−1.27832
$619$	−8.69059e9			−1.47276			−0.736380	−	0.676568i	$-0.763466\pi$
$619$	−8.69059e9			−1.47276			−0.736380	+	0.676568i	$0.763466\pi$
$620$	3.43986e9			0.579655
$621$			5.17534e9i			0.867198i
$622$	−2.97378e9			−0.495499
$623$		−	6.35276e9i		−	1.05258i
$624$			1.55385e9i			0.256014i
$625$	2.93240e9			0.480445
$626$	7.94832e9			1.29499
$627$		−	4.63188e9i		−	0.750448i
$628$	2.28954e9			0.368884
$629$	−3.24340e9	−	2.07719e8i	−0.519665	−	0.0332811i
$630$	2.27678e10			3.62768
$631$		−	2.17823e9i		−	0.345144i	−0.984997	−	0.172572i	$-0.944792\pi$
$631$		−	2.17823e9i		−	0.345144i	0.984997	−	0.172572i	$-0.0552078\pi$
$632$	2.48704e8			0.0391898
$633$	1.22737e10			1.92336
$634$		−	8.27634e8i		−	0.128981i
$635$			8.34127e9i			1.29278i
$636$	−1.75902e9			−0.271126
$637$			1.38083e10i			2.11667i
$638$	−2.76911e8			−0.0422151
$639$	−1.40578e10			−2.13140
$640$	−4.91156e9			−0.740610
$641$	−8.15460e9			−1.22292			−0.611462	−	0.791274i	$-0.709418\pi$
$641$	−8.15460e9			−1.22292			−0.611462	+	0.791274i	$0.709418\pi$
$642$		−	1.30144e10i		−	1.94112i
$643$		−	2.89885e9i		−	0.430019i	−0.976612	−	0.215010i	$-0.931022\pi$
$643$		−	2.89885e9i		−	0.430019i	0.976612	−	0.215010i	$-0.0689783\pi$
$644$			3.51485e9i			0.518569i
$645$	−1.12691e10			−1.65359
$646$		−	3.70278e9i		−	0.540399i
$647$			9.61918e9i			1.39628i	0.715960	+	0.698141i	$0.245989\pi$
$647$			9.61918e9i			1.39628i	−0.715960	+	0.698141i	$0.754011\pi$
$648$		−	5.54637e9i		−	0.800749i
$649$			5.84513e8i			0.0839340i
$650$	−9.62620e9			−1.37486
$651$			1.26067e10i			1.79089i
$652$		−	4.15408e9i		−	0.586960i
$653$		−	7.94020e9i		−	1.11593i	−0.829866	−	0.557963i	$-0.811583\pi$
$653$		−	7.94020e9i		−	1.11593i	0.829866	−	0.557963i	$-0.188417\pi$
$654$	8.70567e9			1.21697
$655$	−1.09756e10			−1.52610
$656$	1.39641e9			0.193129
$657$	3.05172e9			0.419822
$658$			4.30390e9i			0.588941i
$659$	5.16365e9			0.702843			0.351421	−	0.936217i	$-0.385698\pi$
$659$	5.16365e9			0.702843			0.351421	+	0.936217i	$0.385698\pi$
$660$			3.32369e9i			0.450004i
$661$		−	1.43498e10i		−	1.93260i	−0.257424	−	0.966299i	$-0.582874\pi$
$661$		−	1.43498e10i		−	1.93260i	0.257424	−	0.966299i	$-0.417126\pi$
$662$	−2.09174e9			−0.280224
$663$	−7.60539e9			−1.01350
$664$			8.74681e9i			1.15948i
$665$	−3.35697e10			−4.42661
$666$	−9.72770e9	−	6.22995e8i	−1.27600	−	0.0817193i
$667$	−9.50822e8			−0.124068
$668$			5.40629e9i			0.701749i
$669$	1.37438e10			1.77466
$670$	9.21419e9			1.18357
$671$			1.13839e9i			0.145466i
$672$		−	2.15685e10i		−	2.74175i
$673$	8.54404e9			1.08046			0.540232	−	0.841516i	$-0.318336\pi$
$673$	8.54404e9			1.08046			0.540232	+	0.841516i	$0.318336\pi$
$674$			4.34986e9i			0.547224i
$675$	2.28707e10			2.86231
$676$	−1.32685e9			−0.165200
$677$	1.44234e10			1.78652			0.893258	−	0.449544i	$-0.148414\pi$
$677$	1.44234e10			1.78652			0.893258	+	0.449544i	$0.148414\pi$
$678$	−4.26075e9			−0.525027
$679$		−	6.28516e9i		−	0.770498i
$680$			7.42229e9i			0.905225i
$681$		−	2.58465e10i		−	3.13608i
$682$	9.60586e8			0.115955
$683$		−	4.64398e9i		−	0.557722i	−0.960331	−	0.278861i	$-0.910043\pi$
$683$		−	4.64398e9i		−	0.557722i	0.960331	−	0.278861i	$-0.0899569\pi$
$684$			1.39958e10i			1.67225i
$685$		−	8.06496e9i		−	0.958705i
$686$			8.16985e9i			0.966228i
$687$	2.51961e9			0.296472
$688$		−	6.47744e8i		−	0.0758305i
$689$			2.78051e9i			0.323860i
$690$		−	9.05562e9i		−	1.04941i
$691$	−4.85178e9			−0.559407			−0.279703	−	0.960086i	$-0.590236\pi$
$691$	−4.85178e9			−0.559407			−0.279703	+	0.960086i	$0.590236\pi$
$692$	−5.98640e9			−0.686744
$693$	−8.01268e9			−0.914558
$694$	4.63933e9			0.526862
$695$		−	1.64256e10i		−	1.85599i
$696$	3.55330e9			0.399485
$697$			6.83476e9i			0.764555i
$698$		−	2.76864e9i		−	0.308158i
$699$	−1.13762e10			−1.25987
$700$	1.55327e10			1.71161
$701$			1.21949e10i			1.33710i	0.743667	+	0.668550i	$0.233085\pi$
$701$			1.21949e10i			1.33710i	−0.743667	+	0.668550i	$0.766915\pi$
$702$	−1.09441e10			−1.19399
$703$	1.43429e10	+	9.18566e8i	1.55701	+	0.0997165i
$704$	−1.98608e9			−0.214532
$705$			1.39744e10i			1.50201i
$706$	−6.54516e9			−0.700009
$707$	7.49518e9			0.797653
$708$		−	2.68498e9i		−	0.284331i
$709$			1.48259e10i			1.56228i	0.624354	+	0.781141i	$0.285362\pi$
$709$			1.48259e10i			1.56228i	−0.624354	+	0.781141i	$0.714638\pi$
$710$	1.18017e10			1.23749
$711$			6.96894e8i			0.0727148i
$712$	6.21139e9			0.644924
$713$	3.29834e9			0.340786
$714$	−9.73763e9			−1.00117
$715$	5.25380e9			0.537530
$716$		−	6.81814e9i		−	0.694177i
$717$		−	2.39148e9i		−	0.242298i
$718$			1.11413e8i			0.0112331i
$719$	−1.15949e9			−0.116336			−0.0581682	−	0.998307i	$-0.518526\pi$
$719$	−1.15949e9			−0.116336			−0.0581682	+	0.998307i	$0.518526\pi$
$720$			4.24922e9i			0.424273i
$721$		−	1.91314e10i		−	1.90096i
$722$			9.64765e9i			0.953984i
$723$			1.85857e10i			1.82892i
$724$	3.54554e9			0.347214
$725$			4.20184e9i			0.409503i
$726$		−	1.07954e10i		−	1.04704i
$727$			1.41921e10i			1.36986i	0.728607	+	0.684932i	$0.240168\pi$
$727$			1.41921e10i			1.36986i	−0.728607	+	0.684932i	$0.759832\pi$
$728$	−2.07632e10			−1.99450
$729$	−1.26512e10			−1.20945
$730$	−2.56196e9			−0.243749
$731$	3.17041e9			0.300196
$732$		−	5.22922e9i		−	0.492774i
$733$	7.29800e9			0.684447			0.342224	−	0.939619i	$-0.388820\pi$
$733$	7.29800e9			0.684447			0.342224	+	0.939619i	$0.388820\pi$
$734$		−	8.67885e8i		−	0.0810076i
$735$			5.74044e10i			5.33261i
$736$	−5.64303e9			−0.521723
$737$	−3.24275e9			−0.298385
$738$			2.04990e10i			1.87731i
$739$	−1.25280e10			−1.14190			−0.570950	−	0.820985i	$-0.693425\pi$
$739$	−1.25280e10			−1.14190			−0.570950	+	0.820985i	$0.693425\pi$
$740$	−1.02920e10	−	6.59134e8i	−0.933659	−	0.0597947i
$741$	3.36323e10			3.03664
$742$			3.56005e9i			0.319921i
$743$	1.58557e10			1.41816			0.709081	−	0.705127i	$-0.249110\pi$
$743$	1.58557e10			1.41816			0.709081	+	0.705127i	$0.249110\pi$
$744$	−1.23262e10			−1.09730
$745$			1.19477e10i			1.05862i
$746$		−	1.09082e10i		−	0.961985i
$747$	−2.45094e10			−2.15135
$748$		−	9.35079e8i		−	0.0816945i
$749$	3.31949e10			2.88659
$750$	−1.79753e10			−1.55583
$751$	1.83850e10			1.58389			0.791943	−	0.610595i	$-0.209070\pi$
$751$	1.83850e10			1.58389			0.791943	+	0.610595i	$0.209070\pi$
$752$	−8.03250e8			−0.0688792
$753$			1.79581e10i			1.53278i
$754$		−	2.01066e9i		−	0.170820i
$755$		−	2.92667e10i		−	2.47491i
$756$	1.76592e10			1.48643
$757$		−	8.36176e9i		−	0.700587i	−0.936640	−	0.350294i	$-0.886082\pi$
$757$		−	8.36176e9i		−	0.700587i	0.936640	−	0.350294i	$-0.113918\pi$
$758$		−	8.63556e9i		−	0.720192i
$759$			3.18695e9i			0.264563i
$760$		−	3.28226e10i		−	2.71222i
$761$	−8.66729e9			−0.712913			−0.356457	−	0.934312i	$-0.616015\pi$
$761$	−8.66729e9			−0.712913			−0.356457	+	0.934312i	$0.616015\pi$
$762$		−	1.06998e10i		−	0.876055i
$763$			2.22049e10i			1.80972i
$764$			3.29145e9i			0.267030i
$765$	−2.07980e10			−1.67960
$766$	−7.19428e9			−0.578344
$767$	−4.24418e9			−0.339633
$768$	2.26624e10			1.80527
$769$		−	4.66322e9i		−	0.369780i	−0.982759	−	0.184890i	$-0.940807\pi$
$769$		−	4.66322e9i		−	0.369780i	0.982759	−	0.184890i	$-0.0591929\pi$
$770$	6.72676e9			0.530992
$771$			1.44037e10i			1.13184i
$772$			1.02860e10i			0.804612i
$773$	−2.26664e10			−1.76504			−0.882520	−	0.470275i	$-0.844155\pi$
$773$	−2.26664e10			−1.76504			−0.882520	+	0.470275i	$0.844155\pi$
$774$	9.50878e9			0.737109
$775$		−	1.45759e10i		−	1.12481i
$776$	6.14529e9			0.472091
$777$	2.41566e9	−	3.77191e10i	0.184740	−	2.88461i
$778$	−1.28995e10			−0.982074
$779$		−	3.02245e10i		−	2.29075i
$780$	−2.41335e10			−1.82091
$781$	−4.15339e9			−0.311978
$782$			2.54769e9i			0.190512i
$783$			4.77710e9i			0.355630i
$784$	−3.29960e9			−0.244543
$785$		−	1.50455e10i		−	1.11010i
$786$	1.40789e10			1.03416
$787$	−1.70076e10			−1.24374			−0.621871	−	0.783120i	$-0.713627\pi$
$787$	−1.70076e10			−1.24374			−0.621871	+	0.783120i	$0.713627\pi$
$788$	−1.86776e9			−0.135981
$789$	−4.28795e9			−0.310800
$790$		−	5.85052e8i		−	0.0422182i
$791$		−	1.08676e10i		−	0.780754i
$792$		−	7.83437e9i		−	0.560358i
$793$	−8.26591e9			−0.588619
$794$			6.55903e8i			0.0465016i
$795$			1.15592e10i			0.815913i
$796$		−	1.48597e10i		−	1.04428i
$797$			1.27977e10i			0.895423i	0.894178	+	0.447711i	$0.147761\pi$
$797$			1.27977e10i			0.895423i	−0.894178	+	0.447711i	$0.852239\pi$
$798$	4.30614e10			2.99970
$799$		−	3.93154e9i		−	0.272677i
$800$			2.49375e10i			1.72202i
$801$			1.74049e10i			1.19663i
$802$	−3.29917e9			−0.225837
$803$	9.01631e8			0.0614504
$804$	1.48957e10			1.01080
$805$	2.30975e10			1.56056
$806$			6.97487e9i			0.469206i
$807$	5.11356e9			0.342505
$808$			7.32838e9i			0.488729i
$809$		−	1.10312e10i		−	0.732491i	−0.930518	−	0.366246i	$-0.880643\pi$
$809$		−	1.10312e10i		−	0.732491i	0.930518	−	0.366246i	$-0.119357\pi$
$810$	−1.30473e10			−0.862627
$811$	−1.57848e10			−1.03912			−0.519559	−	0.854434i	$-0.673904\pi$
$811$	−1.57848e10			−1.03912			−0.519559	+	0.854434i	$0.673904\pi$
$812$			3.24438e9i			0.212660i
$813$	1.28233e10			0.836916
$814$	−2.87405e9	−	1.84064e8i	−0.186771	−	0.0119615i
$815$	−2.72981e10			−1.76637
$816$		−	1.81736e9i		−	0.117092i
$817$	−1.40201e10			−0.899443
$818$	1.82153e10			1.16359
$819$		−	5.81805e10i		−	3.70070i
$820$			2.16881e10i			1.37364i
$821$	−2.11593e10			−1.33444			−0.667221	−	0.744860i	$-0.732516\pi$
$821$	−2.11593e10			−1.33444			−0.667221	+	0.744860i	$0.732516\pi$
$822$			1.03453e10i			0.649669i
$823$	−1.88231e10			−1.17704			−0.588521	−	0.808482i	$-0.700289\pi$
$823$	−1.88231e10			−1.17704			−0.588521	+	0.808482i	$0.700289\pi$
$824$	1.87056e10			1.16473
$825$	1.40837e10			0.873227
$826$	−5.43408e9			−0.335502
$827$		−	2.87534e10i		−	1.76774i	−0.467729	−	0.883872i	$-0.654928\pi$
$827$		−	2.87534e10i		−	1.76774i	0.467729	−	0.883872i	$-0.345072\pi$
$828$		−	9.62978e9i		−	0.589536i
$829$			1.31626e10i			0.802418i	0.915986	+	0.401209i	$0.131410\pi$
$829$			1.31626e10i			0.802418i	−0.915986	+	0.401209i	$0.868590\pi$
$830$	2.05760e10			1.24907
$831$		−	7.81132e9i		−	0.472194i
$832$		−	1.44210e10i		−	0.868088i
$833$		−	1.61500e10i		−	0.968090i
$834$			2.10699e10i			1.25771i
$835$	3.55269e10			2.11181
$836$			4.13507e9i			0.244772i
$837$		−	1.65715e10i		−	0.976836i
$838$			1.72175e10i			1.01069i
$839$	8.67223e9			0.506949			0.253474	−	0.967342i	$-0.418427\pi$
$839$	8.67223e9			0.506949			0.253474	+	0.967342i	$0.418427\pi$
$840$	−8.63173e10			−5.02482
$841$	1.63722e10			0.949121
$842$	7.09101e9			0.409370
$843$			4.06390e10i			2.33640i
$844$	−1.09572e10			−0.627338
$845$			8.71928e9i			0.497144i
$846$		−	1.17916e10i		−	0.669538i
$847$	2.75351e10			1.55702
$848$	−6.64423e8			−0.0374162
$849$			2.60221e10i			1.45937i
$850$	1.12587e10			0.628812
$851$	−9.86857e9	−	6.32016e8i	−0.548909	−	0.0351540i
$852$	1.90787e10			1.05684
$853$			8.04477e8i			0.0443805i	0.999754	+	0.0221902i	$0.00706395\pi$
$853$			8.04477e8i			0.0443805i	−0.999754	+	0.0221902i	$0.992936\pi$
$854$	−1.05833e10			−0.581460
$855$	9.19722e10			5.03240
$856$			3.24562e10i			1.76864i
$857$		−	8.50199e9i		−	0.461411i	−0.973024	−	0.230705i	$-0.925897\pi$
$857$		−	8.50199e9i		−	0.461411i	0.973024	−	0.230705i	$-0.0741034\pi$
$858$	−6.73931e9			−0.364259
$859$		−	2.60384e10i		−	1.40165i	−0.713335	−	0.700823i	$-0.752816\pi$
$859$		−	2.60384e10i		−	1.40165i	0.713335	−	0.700823i	$-0.247184\pi$
$860$	1.00604e10			0.539348
$861$	−7.94847e10			−4.24397
$862$	1.11068e10			0.590625
$863$	−3.33767e9			−0.176769			−0.0883844	−	0.996086i	$-0.528170\pi$
$863$	−3.33767e9			−0.176769			−0.0883844	+	0.996086i	$0.528170\pi$
$864$			2.83516e10i			1.49548i
$865$			3.93390e10i			2.06665i
$866$			7.50184e9i			0.392514i
$867$	−2.39090e10			−1.24593
$868$		−	1.12546e10i		−	0.584130i
$869$			2.05898e8i			0.0106434i
$870$		−	8.35880e9i		−	0.430355i
$871$		−	2.35458e10i		−	1.20740i
$872$	−2.17107e10			−1.10883
$873$			1.72197e10i			0.875943i
$874$		−	1.12663e10i		−	0.570810i
$875$		−	4.58482e10i		−	2.31363i
$876$	−4.14166e9			−0.208166
$877$	3.14148e9			0.157266			0.0786331	−	0.996904i	$-0.474944\pi$
$877$	3.14148e9			0.157266			0.0786331	+	0.996904i	$0.474944\pi$
$878$	−8.48401e8			−0.0423029
$879$	5.20961e10			2.58729
$880$			1.25544e9i			0.0621019i
$881$	−2.00427e10			−0.987508			−0.493754	−	0.869602i	$-0.664376\pi$
$881$	−2.00427e10			−0.987508			−0.493754	+	0.869602i	$0.664376\pi$
$882$		−	4.84376e10i		−	2.37707i
$883$		−	3.39989e10i		−	1.66189i	−0.556354	−	0.830946i	$-0.687800\pi$
$883$		−	3.39989e10i		−	1.66189i	0.556354	−	0.830946i	$-0.312200\pi$
$884$	6.78965e9			0.330571
$885$	−1.76440e10			−0.855651
$886$		−	7.18013e9i		−	0.346828i
$887$	−2.33774e10			−1.12477			−0.562384	−	0.826876i	$-0.690116\pi$
$887$	−2.33774e10			−1.12477			−0.562384	+	0.826876i	$0.690116\pi$
$888$	3.68797e10	+	2.36190e9i	1.76743	+	0.113192i
$889$	2.72910e10			1.30276
$890$		−	1.46117e10i		−	0.694760i
$891$	4.59174e9			0.217473
$892$	−1.22697e10			−0.578837
$893$			1.73859e10i			0.816991i
$894$		−	1.53260e10i		−	0.717375i
$895$	−4.48047e10			−2.08902
$896$			1.60697e10i			0.746327i
$897$	−2.31406e10			−1.07054
$898$	−9.57660e8			−0.0441310
$899$	3.04453e9			0.139753
$900$	−4.25557e10			−1.94585
$901$		−	3.25205e9i		−	0.148122i
$902$			6.05644e9i			0.274786i
$903$			3.68702e10i			1.66636i
$904$	1.06257e10			0.478375
$905$		−	2.32992e10i		−	1.04489i
$906$			3.75418e10i			1.67713i
$907$			2.15141e10i			0.957409i	0.877976	+	0.478704i	$0.158893\pi$
$907$			2.15141e10i			0.957409i	−0.877976	+	0.478704i	$0.841107\pi$
$908$			2.30743e10i			1.02289i
$909$	−2.05348e10			−0.906813
$910$			4.88433e10i			2.14862i
$911$		−	1.58486e10i		−	0.694509i	−0.937771	−	0.347254i	$-0.887114\pi$
$911$		−	1.58486e10i		−	0.694509i	0.937771	−	0.347254i	$-0.112886\pi$
$912$			8.03669e9i			0.350829i
$913$	−7.24133e9			−0.314898
$914$	1.33216e10			0.577090
$915$	−3.43633e10			−1.48293
$916$	−2.24936e9			−0.0966996
$917$			3.59099e10i			1.53788i
$918$	1.28000e10			0.546087
$919$		−	2.79556e10i		−	1.18813i	−0.804416	−	0.594066i	$-0.797522\pi$
$919$		−	2.79556e10i		−	1.18813i	0.804416	−	0.594066i	$-0.202478\pi$
$920$			2.25835e10i			0.956166i
$921$	7.60498e10			3.20767
$922$	−4.23022e9			−0.177748
$923$		−	3.01580e10i		−	1.26240i
$924$	1.08745e10			0.453478
$925$	−2.79299e9	+	4.36109e10i	−0.116031	+	1.81175i
$926$	−4.46614e8			−0.0184839
$927$			5.24150e10i			2.16111i
$928$	−5.20880e9			−0.213954
$929$	1.18412e10			0.484554			0.242277	−	0.970207i	$-0.422106\pi$
$929$	1.18412e10			0.484554			0.242277	+	0.970207i	$0.422106\pi$
$930$			2.89961e10i			1.18209i
$931$			7.14181e10i			2.90058i
$932$	1.01560e10			0.410930
$933$			3.15915e10i			1.27346i
$934$	−1.73583e10			−0.697097
$935$	−6.14478e9			−0.245847
$936$	5.68857e10			2.26745
$937$	3.16939e10			1.25860			0.629299	−	0.777163i	$-0.283342\pi$
$937$	3.16939e10			1.25860			0.629299	+	0.777163i	$0.283342\pi$
$938$		−	3.01471e10i		−	1.19271i
$939$		−	8.44376e10i		−	3.32818i
$940$		−	1.24756e10i		−	0.489907i
$941$	1.04957e10			0.410629			0.205314	−	0.978696i	$-0.434178\pi$
$941$	1.04957e10			0.410629			0.205314	+	0.978696i	$0.434178\pi$
$942$			1.92996e10i			0.752262i
$943$			2.07958e10i			0.807580i
$944$		−	1.01418e9i		−	0.0392385i
$945$		−	1.16046e11i		−	4.47320i
$946$	2.80937e9			0.107892
$947$		−	4.70002e9i		−	0.179835i	−0.995949	−	0.0899175i	$-0.971340\pi$
$947$		−	4.70002e9i		−	0.179835i	0.995949	−	0.0899175i	$-0.0286603\pi$
$948$		−	9.45795e8i		−	0.0360552i
$949$			6.54679e9i			0.248655i
$950$	−4.97877e10			−1.88404
$951$	−8.79223e9			−0.331487
$952$	2.42843e10			0.912213
$953$	8.49720e9			0.318017			0.159009	−	0.987277i	$-0.449170\pi$
$953$	8.49720e9			0.318017			0.159009	+	0.987277i	$0.449170\pi$
$954$		−	9.75363e9i		−	0.363703i
$955$	2.16294e10			0.803588
$956$			2.13498e9i			0.0790298i
$957$			2.94172e9i			0.108495i
$958$	1.09110e9			0.0400947
$959$	−2.63870e10			−0.966106
$960$		−	5.99514e10i		−	2.18701i
$961$	1.69513e10			0.616129
$962$	1.33650e9	−	2.08687e10i	0.0484012	−	0.755756i
$963$	−9.09455e10			−3.28163
$964$		−	1.65923e10i		−	0.596535i
$965$	6.75935e10			2.42136
$966$	−2.96283e10			−1.05751
$967$			6.46868e9i			0.230050i	0.993363	+	0.115025i	$0.0366949\pi$
$967$			6.46868e9i			0.230050i	−0.993363	+	0.115025i	$0.963305\pi$
$968$			2.69223e10i			0.954001i
$969$	−3.93359e10			−1.38885
$970$		−	1.44562e10i		−	0.508572i
$971$	4.34152e10			1.52186			0.760929	−	0.648835i	$-0.224743\pi$
$971$	4.34152e10			1.52186			0.760929	+	0.648835i	$0.224743\pi$
$972$	4.07656e9			0.142384
$973$	−5.37415e10			−1.87031
$974$	1.31496e9			0.0455990
$975$			1.02262e11i			3.53345i
$976$		−	1.97520e9i		−	0.0680043i
$977$			3.58936e10i			1.23136i	0.787995	+	0.615681i	$0.211119\pi$
$977$			3.58936e10i			1.23136i	−0.787995	+	0.615681i	$0.788881\pi$
$978$	3.50167e10			1.19699
$979$			5.14229e9i			0.175153i
$980$		−	5.12473e10i		−	1.73932i
$981$		−	6.08356e10i		−	2.05739i
$982$		−	1.53241e9i		−	0.0516400i
$983$	1.63713e10			0.549727			0.274863	−	0.961483i	$-0.411367\pi$
$983$	1.63713e10			0.549727			0.274863	+	0.961483i	$0.411367\pi$
$984$		−	7.77159e10i		−	2.60032i
$985$			1.22738e10i			0.409216i
$986$			2.35164e9i			0.0781272i
$987$	4.57217e10			1.51360
$988$	−3.00250e10			−0.990452
$989$	9.64647e9			0.317089
$990$	−1.84296e10			−0.603660
$991$		−	2.30328e10i		−	0.751777i	−0.926665	−	0.375889i	$-0.877338\pi$
$991$		−	2.30328e10i		−	0.751777i	0.926665	−	0.375889i	$-0.122662\pi$
$992$	1.80690e10			0.587684
$993$			2.22213e10i			0.720189i
$994$		−	3.86131e10i		−	1.24704i
$995$	−9.76494e10			−3.14260
$996$	3.32632e10			1.06673
$997$		−	1.45517e10i		−	0.465030i	−0.972593	−	0.232515i	$-0.925305\pi$
$997$		−	1.45517e10i		−	0.465030i	0.972593	−	0.232515i	$-0.0746954\pi$
$998$	−6.74262e9			−0.214719
$999$	−3.17536e9	+	4.95815e10i	−0.100766	+	1.57340i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.b.a.36.8	✓	20
37.36	even	2	inner	37.8.b.a.36.13	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.b.a.36.8	✓	20	1.1	even	1	trivial
37.8.b.a.36.13	yes	20	37.36	even	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:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.52533i q^{2} -79.9440 q^{3} +71.3694 q^{4} -468.997i q^{5} +601.605i q^{6} -1534.47 q^{7} -1500.32i q^{8} +4204.05 q^{9} +O(q^{10})\)	\(q-7.52533i q^{2} -79.9440 q^{3} +71.3694 q^{4} -468.997i q^{5} +601.605i q^{6} -1534.47 q^{7} -1500.32i q^{8} +4204.05 q^{9} -3529.36 q^{10} +1242.09 q^{11} -5705.56 q^{12} +9018.86i q^{13} +11547.4i q^{14} +37493.5i q^{15} -2155.12 q^{16} -10548.3i q^{17} -31636.8i q^{18} +46646.5i q^{19} -33472.0i q^{20} +122672. q^{21} -9347.12i q^{22} -32095.0i q^{23} +119942. i q^{24} -141833. q^{25} +67869.9 q^{26} -161251. q^{27} -109514. q^{28} -29625.3i q^{29} +282151. q^{30} +102768. i q^{31} -175823. i q^{32} -99297.5 q^{33} -79379.7 q^{34} +719661. i q^{35} +300040. q^{36} +(19692.1 - 307480. i) q^{37} +351030. q^{38} -721004. i q^{39} -703646. q^{40} -647947. q^{41} -923144. i q^{42} +300560. i q^{43} +88647.1 q^{44} -1.97169e6i q^{45} -241525. q^{46} +372716. q^{47} +172289. q^{48} +1.53105e6 q^{49} +1.06734e6i q^{50} +843276. i q^{51} +643671. i q^{52} +308300. q^{53} +1.21347e6i q^{54} -582535. i q^{55} +2.30219e6i q^{56} -3.72911e6i q^{57} -222940. q^{58} +470589. i q^{59} +2.67589e6i q^{60} +916513. i q^{61} +773364. q^{62} -6.45098e6 q^{63} -1.59898e6 q^{64} +4.22982e6 q^{65} +747246. i q^{66} -2.61073e6 q^{67} -752828. i q^{68} +2.56580e6i q^{69} +5.41569e6 q^{70} -3.34388e6 q^{71} -6.30742e6i q^{72} +725900. q^{73} +(-2.31389e6 - 148189. i) q^{74} +1.13387e7 q^{75} +3.32913e6i q^{76} -1.90594e6 q^{77} -5.42579e6 q^{78} +165767. i q^{79} +1.01075e6i q^{80} +3.69679e6 q^{81} +4.87602e6i q^{82} -5.82996e6 q^{83} +8.75500e6 q^{84} -4.94714e6 q^{85} +2.26182e6 q^{86} +2.36836e6i q^{87} -1.86353e6i q^{88} +4.14004e6i q^{89} -1.48376e7 q^{90} -1.38392e7i q^{91} -2.29060e6i q^{92} -8.21570e6i q^{93} -2.80481e6i q^{94} +2.18771e7 q^{95} +1.40560e7i q^{96} +4.09598e6i q^{97} -1.15217e7i q^{98} +5.22180e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9}+O(q^{10}) \) 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9	\( 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9} - 882 q^{10} + 3498 q^{11} - 30374 q^{12} + 36116 q^{16} + 113482 q^{21} - 108112 q^{25} + 49278 q^{26} - 304110 q^{27} - 41192 q^{28} + 429776 q^{30} + 305646 q^{33} - 960356 q^{34} + 484758 q^{36} + 108732 q^{37} + 1049916 q^{38} - 496346 q^{40} - 1577742 q^{41} + 685266 q^{44} - 2906298 q^{46} - 1512786 q^{47} + 1522958 q^{48} + 3269246 q^{49} + 2999358 q^{53} + 405946 q^{58} + 3728310 q^{62} - 11995292 q^{63} - 11109700 q^{64} + 4251792 q^{65} + 3562224 q^{67} + 21605644 q^{70} - 15259086 q^{71} + 11088018 q^{73} - 2036544 q^{74} + 14882062 q^{75} - 2419122 q^{77} - 12178734 q^{78} - 17764972 q^{81} - 12873822 q^{83} + 9944396 q^{84} - 2698920 q^{85} + 15345336 q^{86} - 13219100 q^{90} + 48981192 q^{95} + 43111380 q^{99}+O(q^{100}) \) 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9 - 882 * q^10 + 3498 * q^11 - 30374 * q^12 + 36116 * q^16 + 113482 * q^21 - 108112 * q^25 + 49278 * q^26 - 304110 * q^27 - 41192 * q^28 + 429776 * q^30 + 305646 * q^33 - 960356 * q^34 + 484758 * q^36 + 108732 * q^37 + 1049916 * q^38 - 496346 * q^40 - 1577742 * q^41 + 685266 * q^44 - 2906298 * q^46 - 1512786 * q^47 + 1522958 * q^48 + 3269246 * q^49 + 2999358 * q^53 + 405946 * q^58 + 3728310 * q^62 - 11995292 * q^63 - 11109700 * q^64 + 4251792 * q^65 + 3562224 * q^67 + 21605644 * q^70 - 15259086 * q^71 + 11088018 * q^73 - 2036544 * q^74 + 14882062 * q^75 - 2419122 * q^77 - 12178734 * q^78 - 17764972 * q^81 - 12873822 * q^83 + 9944396 * q^84 - 2698920 * q^85 + 15345336 * q^86 - 13219100 * q^90 + 48981192 * q^95 + 43111380 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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