LMFDB - Embedded newform 37.8.b.a.36.5

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			36.5
Root			$-12.3896i$ of defining polynomial
Character	$\chi$	$=$	37.36
Dual form			37.8.b.a.36.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.3896i q^{2} +76.2286 q^{3} -25.5018 q^{4} -93.6237i q^{5} -944.440i q^{6} -1237.82 q^{7} -1269.91i q^{8} +3623.79 q^{9} +O(q^{10})$	\(q-12.3896i q^{2} +76.2286 q^{3} -25.5018 q^{4} -93.6237i q^{5} -944.440i q^{6} -1237.82 q^{7} -1269.91i q^{8} +3623.79 q^{9} -1159.96 q^{10} +6532.18 q^{11} -1943.96 q^{12} -14175.4i q^{13} +15336.1i q^{14} -7136.80i q^{15} -18997.9 q^{16} +14117.3i q^{17} -44897.3i q^{18} +27336.1i q^{19} +2387.57i q^{20} -94357.4 q^{21} -80931.0i q^{22} -42138.6i q^{23} -96803.4i q^{24} +69359.6 q^{25} -175628. q^{26} +109525. q^{27} +31566.7 q^{28} +245987. i q^{29} -88421.9 q^{30} +22417.9i q^{31} +72827.4i q^{32} +497939. q^{33} +174908. q^{34} +115889. i q^{35} -92413.1 q^{36} +(-124119. + 282004. i) q^{37} +338683. q^{38} -1.08057e6i q^{39} -118894. q^{40} -163897. q^{41} +1.16905e6i q^{42} +618344. i q^{43} -166582. q^{44} -339273. i q^{45} -522079. q^{46} +484454. q^{47} -1.44818e6 q^{48} +708662. q^{49} -859337. i q^{50} +1.07614e6i q^{51} +361499. i q^{52} +91799.4 q^{53} -1.35696e6i q^{54} -611567. i q^{55} +1.57192e6i q^{56} +2.08379e6i q^{57} +3.04768e6 q^{58} -184734. i q^{59} +182001. i q^{60} -1.40871e6i q^{61} +277748. q^{62} -4.48561e6 q^{63} -1.52943e6 q^{64} -1.32716e6 q^{65} -6.16925e6i q^{66} +4.56160e6 q^{67} -360017. i q^{68} -3.21216e6i q^{69} +1.43582e6 q^{70} -1.11311e6 q^{71} -4.60189e6i q^{72} +1.33448e6 q^{73} +(3.49392e6 + 1.53778e6i) q^{74} +5.28718e6 q^{75} -697120. i q^{76} -8.08568e6 q^{77} -1.33879e7 q^{78} +4.36984e6i q^{79} +1.77865e6i q^{80} +423666. q^{81} +2.03061e6i q^{82} -411488. q^{83} +2.40628e6 q^{84} +1.32172e6 q^{85} +7.66102e6 q^{86} +1.87512e7i q^{87} -8.29529e6i q^{88} -1.40992e6i q^{89} -4.20345e6 q^{90} +1.75467e7i q^{91} +1.07461e6i q^{92} +1.70888e6i q^{93} -6.00219e6i q^{94} +2.55931e6 q^{95} +5.55153e6i q^{96} -9.97879e6i q^{97} -8.78002e6i q^{98} +2.36713e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	12.3896i		−	1.09509i	−0.836775	−	0.547547i	$-0.815562\pi$
$2$		−	12.3896i		−	1.09509i	0.836775	−	0.547547i	$-0.184438\pi$
$3$	76.2286			1.63002			0.815011	−	0.579446i	$-0.196731\pi$
$3$	76.2286			1.63002			0.815011	+	0.579446i	$0.196731\pi$
$4$	−25.5018			−0.199233
$5$		−	93.6237i		−	0.334958i	−0.985876	−	0.167479i	$-0.946437\pi$
$5$		−	93.6237i		−	0.334958i	0.985876	−	0.167479i	$-0.0535627\pi$
$6$		−	944.440i		−	1.78503i
$7$	−1237.82			−1.36400			−0.682001	−	0.731351i	$-0.738890\pi$
$7$	−1237.82			−1.36400			−0.682001	+	0.731351i	$0.738890\pi$
$8$		−	1269.91i		−	0.876916i
$9$	3623.79			1.65697
$10$	−1159.96			−0.366811
$11$	6532.18			1.47973			0.739867	−	0.672753i	$-0.234888\pi$
$11$	6532.18			1.47973			0.739867	+	0.672753i	$0.234888\pi$
$12$	−1943.96			−0.324753
$13$		−	14175.4i		−	1.78951i	−0.446554	−	0.894757i	$-0.647349\pi$
$13$		−	14175.4i		−	1.78951i	0.446554	−	0.894757i	$-0.352651\pi$
$14$			15336.1i			1.49371i
$15$		−	7136.80i		−	0.545989i
$16$	−18997.9			−1.15954
$17$			14117.3i			0.696918i	0.937324	+	0.348459i	$0.113295\pi$
$17$			14117.3i			0.696918i	−0.937324	+	0.348459i	$0.886705\pi$
$18$		−	44897.3i		−	1.81454i
$19$			27336.1i			0.914323i	0.889384	+	0.457161i	$0.151134\pi$
$19$			27336.1i			0.914323i	−0.889384	+	0.457161i	$0.848866\pi$
$20$			2387.57i			0.0667346i
$21$	−94357.4			−2.22335
$22$		−	80931.0i		−	1.62045i
$23$		−	42138.6i		−	0.722158i	−0.932535	−	0.361079i	$-0.882408\pi$
$23$		−	42138.6i		−	0.722158i	0.932535	−	0.361079i	$-0.117592\pi$
$24$		−	96803.4i		−	1.42939i
$25$	69359.6			0.887803
$26$	−175628.			−1.95969
$27$	109525.			1.07087
$28$	31566.7			0.271754
$29$			245987.i			1.87292i	0.350775	+	0.936460i	$0.385918\pi$
$29$			245987.i			1.87292i	−0.350775	+	0.936460i	$0.614082\pi$
$30$	−88421.9			−0.597910
$31$			22417.9i			0.135154i	0.997714	+	0.0675769i	$0.0215268\pi$
$31$			22417.9i			0.135154i	−0.997714	+	0.0675769i	$0.978473\pi$
$32$			72827.4i			0.392889i
$33$	497939.			2.41200
$34$	174908.			0.763191
$35$			115889.i			0.456884i
$36$	−92413.1			−0.330122
$37$	−124119.	+	282004.i	−0.402838	+	0.915271i
$37$	−124119.	+	282004.i	−0.402838	+	0.915271i
$38$	338683.			1.00127
$39$		−	1.08057e6i		−	2.91694i
$40$	−118894.			−0.293730
$41$	−163897.			−0.371387			−0.185693	−	0.982608i	$-0.559453\pi$
$41$	−163897.			−0.371387			−0.185693	+	0.982608i	$0.559453\pi$
$42$			1.16905e6i			2.43478i
$43$			618344.i			1.18602i	0.805197	+	0.593008i	$0.202060\pi$
$43$			618344.i			1.18602i	−0.805197	+	0.593008i	$0.797940\pi$
$44$	−166582.			−0.294812
$45$		−	339273.i		−	0.555015i
$46$	−522079.			−0.790832
$47$	484454.			0.680629			0.340314	−	0.940312i	$-0.389466\pi$
$47$	484454.			0.680629			0.340314	+	0.940312i	$0.389466\pi$
$48$	−1.44818e6			−1.89007
$49$	708662.			0.860504
$50$		−	859337.i		−	0.972228i
$51$			1.07614e6i			1.13599i
$52$			361499.i			0.356529i
$53$	91799.4			0.0846982			0.0423491	−	0.999103i	$-0.486516\pi$
$53$	91799.4			0.0846982			0.0423491	+	0.999103i	$0.486516\pi$
$54$		−	1.35696e6i		−	1.17271i
$55$		−	611567.i		−	0.495649i
$56$			1.57192e6i			1.19612i
$57$			2.08379e6i			1.49037i
$58$	3.04768e6			2.05102
$59$		−	184734.i		−	0.117102i	−0.998284	−	0.0585510i	$-0.981352\pi$
$59$		−	184734.i		−	0.117102i	0.998284	−	0.0585510i	$-0.0186480\pi$
$60$			182001.i			0.108779i
$61$		−	1.40871e6i		−	0.794632i	−0.917682	−	0.397316i	$-0.869942\pi$
$61$		−	1.40871e6i		−	0.794632i	0.917682	−	0.397316i	$-0.130058\pi$
$62$	277748.			0.148006
$63$	−4.48561e6			−2.26011
$64$	−1.52943e6			−0.729288
$65$	−1.32716e6			−0.599412
$66$		−	6.16925e6i		−	2.64137i
$67$	4.56160e6			1.85291			0.926456	−	0.376402i	$-0.122839\pi$
$67$	4.56160e6			1.85291			0.926456	+	0.376402i	$0.122839\pi$
$68$		−	360017.i		−	0.138849i
$69$		−	3.21216e6i		−	1.17713i
$70$	1.43582e6			0.500331
$71$	−1.11311e6			−0.369091			−0.184545	−	0.982824i	$-0.559081\pi$
$71$	−1.11311e6			−0.369091			−0.184545	+	0.982824i	$0.559081\pi$
$72$		−	4.60189e6i		−	1.45302i
$73$	1.33448e6			0.401497			0.200748	−	0.979643i	$-0.435663\pi$
$73$	1.33448e6			0.401497			0.200748	+	0.979643i	$0.435663\pi$
$74$	3.49392e6	+	1.53778e6i	1.00231	+	0.441146i
$75$	5.28718e6			1.44714
$76$		−	697120.i		−	0.182163i
$77$	−8.08568e6			−2.01836
$78$	−1.33879e7			−3.19433
$79$			4.36984e6i			0.997173i	0.866840	+	0.498587i	$0.166147\pi$
$79$			4.36984e6i			0.997173i	−0.866840	+	0.498587i	$0.833853\pi$
$80$			1.77865e6i			0.388397i
$81$	423666.			0.0885781
$82$			2.03061e6i			0.406704i
$83$	−411488.			−0.0789922			−0.0394961	−	0.999220i	$-0.512575\pi$
$83$	−411488.			−0.0789922			−0.0394961	+	0.999220i	$0.512575\pi$
$84$	2.40628e6			0.442965
$85$	1.32172e6			0.233438
$86$	7.66102e6			1.29880
$87$			1.87512e7i			3.05290i
$88$		−	8.29529e6i		−	1.29760i
$89$		−	1.40992e6i		−	0.211997i	−0.994366	−	0.105999i	$-0.966196\pi$
$89$		−	1.40992e6i		−	0.211997i	0.994366	−	0.105999i	$-0.0338039\pi$
$90$	−4.20345e6			−0.607795
$91$			1.75467e7i			2.44090i
$92$			1.07461e6i			0.143877i
$93$			1.70888e6i			0.220303i
$94$		−	6.00219e6i		−	0.745353i
$95$	2.55931e6			0.306260
$96$			5.55153e6i			0.640417i
$97$		−	9.97879e6i		−	1.11014i	−0.831804	−	0.555069i	$-0.812692\pi$
$97$		−	9.97879e6i		−	1.11014i	0.831804	−	0.555069i	$-0.187308\pi$
$98$		−	8.78002e6i		−	0.942333i
$99$	2.36713e7			2.45188
$100$	−1.76879e6			−0.176879
$101$	−1.26404e7			−1.22077			−0.610387	−	0.792103i	$-0.708986\pi$
$101$	−1.26404e7			−1.22077			−0.610387	+	0.792103i	$0.708986\pi$
$102$	1.33330e7			1.24402
$103$		−	9.19596e6i		−	0.829214i	−0.910001	−	0.414607i	$-0.863919\pi$
$103$		−	9.19596e6i		−	0.829214i	0.910001	−	0.414607i	$-0.136081\pi$
$104$	−1.80015e7			−1.56925
$105$			8.83409e6i			0.744730i
$106$		−	1.13736e6i		−	0.0927526i
$107$	2.40901e6			0.190106			0.0950529	−	0.995472i	$-0.469698\pi$
$107$	2.40901e6			0.190106			0.0950529	+	0.995472i	$0.469698\pi$
$108$	−2.79307e6			−0.213353
$109$			7.18615e6i			0.531500i	0.964042	+	0.265750i	$0.0856196\pi$
$109$			7.18615e6i			0.531500i	−0.964042	+	0.265750i	$0.914380\pi$
$110$	−7.57706e6			−0.542783
$111$	−9.46138e6	+	2.14968e7i	−0.656635	+	1.49191i
$112$	2.35160e7			1.58161
$113$			1.16407e6i			0.0758932i	0.999280	+	0.0379466i	$0.0120817\pi$
$113$			1.16407e6i			0.0758932i	−0.999280	+	0.0379466i	$0.987918\pi$
$114$	2.58173e7			1.63209
$115$	−3.94517e6			−0.241893
$116$		−	6.27311e6i		−	0.373147i
$117$		−	5.13689e7i		−	2.96517i
$118$	−2.28878e6			−0.128238
$119$		−	1.74748e7i		−	0.950598i
$120$	−9.06309e6			−0.478787
$121$	2.31822e7			1.18962
$122$	−1.74533e7			−0.870198
$123$	−1.24936e7			−0.605368
$124$		−	571695.i		−	0.0269270i
$125$		−	1.38080e7i		−	0.632335i
$126$			5.55749e7i			2.47504i
$127$	−3.11966e7			−1.35143			−0.675717	−	0.737161i	$-0.736166\pi$
$127$	−3.11966e7			−1.35143			−0.675717	+	0.737161i	$0.736166\pi$
$128$			2.82709e7i			1.19153i
$129$			4.71355e7i			1.93323i
$130$			1.64429e7i			0.656413i
$131$		−	1.46412e7i		−	0.569018i	−0.958673	−	0.284509i	$-0.908169\pi$
$131$		−	1.46412e7i		−	0.569018i	0.958673	−	0.284509i	$-0.0918306\pi$
$132$	−1.26983e7			−0.480549
$133$		−	3.38373e7i		−	1.24714i
$134$		−	5.65163e7i		−	2.02912i
$135$		−	1.02541e7i		−	0.358698i
$136$	1.79277e7			0.611138
$137$	566196.			0.0188124			0.00940621	−	0.999956i	$-0.497006\pi$
$137$	566196.			0.0188124			0.00940621	+	0.999956i	$0.497006\pi$
$138$	−3.97973e7			−1.28907
$139$	−4.98455e7			−1.57425			−0.787126	−	0.616792i	$-0.788432\pi$
$139$	−4.98455e7			−1.57425			−0.787126	+	0.616792i	$0.788432\pi$
$140$		−	2.95539e6i		−	0.0910262i
$141$	3.69293e7			1.10944
$142$			1.37909e7i			0.404189i
$143$		−	9.25966e7i		−	2.64801i
$144$	−6.88444e7			−1.92132
$145$	2.30302e7			0.627350
$146$		−	1.65337e7i		−	0.439677i
$147$	5.40203e7			1.40264
$148$	3.16524e6	−	7.19161e6i	0.0802585	−	0.182352i
$149$	−4.89706e7			−1.21278			−0.606392	−	0.795166i	$-0.707384\pi$
$149$	−4.89706e7			−1.21278			−0.606392	+	0.795166i	$0.707384\pi$
$150$		−	6.55060e7i		−	1.58475i
$151$	6.16190e7			1.45645			0.728224	−	0.685339i	$-0.240346\pi$
$151$	6.16190e7			1.45645			0.728224	+	0.685339i	$0.240346\pi$
$152$	3.47144e7			0.801784
$153$			5.11583e7i			1.15477i
$154$			1.00178e8i			2.21030i
$155$	2.09884e6			0.0452709
$156$			2.75565e7i			0.581151i
$157$	4.99975e7			1.03110			0.515548	−	0.856861i	$-0.327588\pi$
$157$	4.99975e7			1.03110			0.515548	+	0.856861i	$0.327588\pi$
$158$	5.41405e7			1.09200
$159$	6.99774e6			0.138060
$160$	6.81837e6			0.131601
$161$			5.21601e7i			0.985026i
$162$		−	5.24905e6i		−	0.0970014i
$163$		−	3.51037e7i		−	0.634886i	−0.948277	−	0.317443i	$-0.897176\pi$
$163$		−	3.51037e7i		−	0.634886i	0.948277	−	0.317443i	$-0.102824\pi$
$164$	4.17965e6			0.0739923
$165$		−	4.66189e7i		−	0.807919i
$166$			5.09817e6i			0.0865039i
$167$			3.53920e7i			0.588027i	0.955801	+	0.294013i	$0.0949910\pi$
$167$			3.53920e7i			0.588027i	−0.955801	+	0.294013i	$0.905009\pi$
$168$			1.19825e8i			1.94969i
$169$	−1.38195e8			−2.20236
$170$		−	1.63755e7i		−	0.255637i
$171$			9.90605e7i			1.51500i
$172$		−	1.57689e7i		−	0.236293i
$173$	6.05009e6			0.0888383			0.0444192	−	0.999013i	$-0.485856\pi$
$173$	6.05009e6			0.0888383			0.0444192	+	0.999013i	$0.485856\pi$
$174$	2.32320e8			3.34321
$175$	−8.58549e7			−1.21097
$176$	−1.24098e8			−1.71581
$177$		−	1.40820e7i		−	0.190879i
$178$	−1.74683e7			−0.232157
$179$			5.44937e7i			0.710168i	0.934834	+	0.355084i	$0.115548\pi$
$179$			5.44937e7i			0.710168i	−0.934834	+	0.355084i	$0.884452\pi$
$180$			8.65206e6i			0.110577i
$181$	2.42459e7			0.303923			0.151961	−	0.988386i	$-0.451441\pi$
$181$	2.42459e7			0.303923			0.151961	+	0.988386i	$0.451441\pi$
$182$	2.17396e8			2.67302
$183$		−	1.07384e8i		−	1.29527i
$184$	−5.35122e7			−0.633272
$185$	2.64023e7	+	1.16204e7i	0.306578	+	0.134934i
$186$	2.11723e7			0.241253
$187$			9.22170e7i			1.03125i
$188$	−1.23544e7			−0.135604
$189$	−1.35572e8			−1.46068
$190$		−	3.17088e7i		−	0.335384i
$191$		−	8.57690e7i		−	0.890663i	−0.895366	−	0.445332i	$-0.853086\pi$
$191$		−	8.57690e7i		−	0.890663i	0.895366	−	0.445332i	$-0.146914\pi$
$192$	−1.16586e8			−1.18876
$193$			8.85554e7i			0.886675i	0.896355	+	0.443337i	$0.146206\pi$
$193$			8.85554e7i			0.886675i	−0.896355	+	0.443337i	$0.853794\pi$
$194$	−1.23633e8			−1.21571
$195$	−1.01167e8			−0.977054
$196$	−1.80721e7			−0.171440
$197$	−1.18712e8			−1.10628			−0.553138	−	0.833089i	$-0.686570\pi$
$197$	−1.18712e8			−1.10628			−0.553138	+	0.833089i	$0.686570\pi$
$198$		−	2.93277e8i		−	2.68504i
$199$			6.22870e7i			0.560288i	0.959958	+	0.280144i	$0.0903822\pi$
$199$			6.22870e7i			0.560288i	−0.959958	+	0.280144i	$0.909618\pi$
$200$		−	8.80805e7i		−	0.778529i
$201$	3.47724e8			3.02029
$202$			1.56609e8i			1.33686i
$203$		−	3.04488e8i		−	2.55467i
$204$		−	2.74436e7i		−	0.226326i
$205$			1.53446e7i			0.124399i
$206$	−1.13934e8			−0.908068
$207$		−	1.52701e8i		−	1.19659i
$208$			2.69303e8i			2.07501i
$209$			1.78565e8i			1.35296i
$210$	1.09451e8			0.815550
$211$	−1.12568e8			−0.824949			−0.412475	−	0.910969i	$-0.635335\pi$
$211$	−1.12568e8			−0.824949			−0.412475	+	0.910969i	$0.635335\pi$
$212$	−2.34105e6			−0.0168747
$213$	−8.48506e7			−0.601626
$214$		−	2.98466e7i		−	0.208184i
$215$	5.78916e7			0.397266
$216$		−	1.39086e8i		−	0.939067i
$217$		−	2.77493e7i		−	0.184350i
$218$	8.90334e7			0.582043
$219$	1.01725e8			0.654448
$220$			1.55960e7i			0.0987495i
$221$	2.00119e8			1.24714
$222$	2.66336e8	+	1.17223e8i	1.63378	+	0.719078i
$223$	3.88820e7			0.234791			0.117396	−	0.993085i	$-0.462545\pi$
$223$	3.88820e7			0.234791			0.117396	+	0.993085i	$0.462545\pi$
$224$		−	9.01474e7i		−	0.535902i
$225$	2.51345e8			1.47106
$226$	1.44223e7			0.0831102
$227$		−	2.28159e8i		−	1.29463i	−0.762221	−	0.647317i	$-0.775891\pi$
$227$		−	2.28159e8i		−	1.29463i	0.762221	−	0.647317i	$-0.224109\pi$
$228$		−	5.31405e7i		−	0.296929i
$229$	8.88248e7			0.488776			0.244388	−	0.969677i	$-0.421413\pi$
$229$	8.88248e7			0.488776			0.244388	+	0.969677i	$0.421413\pi$
$230$			4.88790e7i			0.264896i
$231$	−6.16360e8			−3.28997
$232$	3.12381e8			1.64239
$233$	−2.31263e8			−1.19773			−0.598866	−	0.800849i	$-0.704382\pi$
$233$	−2.31263e8			−1.19773			−0.598866	+	0.800849i	$0.704382\pi$
$234$	−6.36439e8			−3.24714
$235$		−	4.53564e7i		−	0.227982i
$236$			4.71104e6i			0.0233305i
$237$			3.33107e8i			1.62541i
$238$	−2.16505e8			−1.04099
$239$		−	2.36897e8i		−	1.12245i	−0.827664	−	0.561224i	$-0.810331\pi$
$239$		−	2.36897e8i		−	1.12245i	0.827664	−	0.561224i	$-0.189669\pi$
$240$			1.35584e8i			0.633096i
$241$		−	9.93160e7i		−	0.457046i	−0.973539	−	0.228523i	$-0.926610\pi$
$241$		−	9.93160e7i		−	0.457046i	0.973539	−	0.228523i	$-0.0733896\pi$
$242$		−	2.87218e8i		−	1.30274i
$243$	−2.07235e8			−0.926490
$244$			3.59245e7i			0.158317i
$245$		−	6.63475e7i		−	0.288233i
$246$			1.54790e8i			0.662936i
$247$	3.87502e8			1.63619
$248$	2.84687e7			0.118519
$249$	−3.13672e7			−0.128759
$250$	−1.71076e8			−0.692467
$251$			3.37702e8i			1.34796i	0.738751	+	0.673979i	$0.235416\pi$
$251$			3.37702e8i			1.34796i	−0.738751	+	0.673979i	$0.764584\pi$
$252$	1.14391e8			0.450288
$253$		−	2.75257e8i		−	1.06860i
$254$			3.86513e8i			1.47995i
$255$	1.00753e8			0.380509
$256$	1.54498e8			0.575549
$257$			4.05540e8i			1.49028i	0.666909	+	0.745139i	$0.267617\pi$
$257$			4.05540e8i			1.49028i	−0.666909	+	0.745139i	$0.732383\pi$
$258$	5.83989e8			2.11707
$259$	1.53637e8	−	3.49071e8i	0.549473	−	1.24843i
$260$	3.38449e7			0.119422
$261$			8.91406e8i			3.10337i
$262$	−1.81398e8			−0.623129
$263$	−3.25369e8			−1.10289			−0.551443	−	0.834213i	$-0.685922\pi$
$263$	−3.25369e8			−1.10289			−0.551443	+	0.834213i	$0.685922\pi$
$264$		−	6.32338e8i		−	2.11512i
$265$		−	8.59460e6i		−	0.0283704i
$266$	−4.19230e8			−1.36574
$267$		−	1.07476e8i		−	0.345560i
$268$	−1.16329e8			−0.369161
$269$	9.53739e6			0.0298742			0.0149371	−	0.999888i	$-0.495245\pi$
$269$	9.53739e6			0.0298742			0.0149371	+	0.999888i	$0.495245\pi$
$270$	−1.27044e8			−0.392808
$271$	6.20616e8			1.89422			0.947109	−	0.320911i	$-0.103989\pi$
$271$	6.20616e8			1.89422			0.947109	+	0.320911i	$0.103989\pi$
$272$		−	2.68200e8i		−	0.808103i
$273$			1.33756e9i			3.97872i
$274$		−	7.01493e6i		−	0.0206014i
$275$	4.53070e8			1.31371
$276$			8.19158e7i			0.234523i
$277$			1.56265e8i			0.441755i	0.975302	+	0.220877i	$0.0708921\pi$
$277$			1.56265e8i			0.441755i	−0.975302	+	0.220877i	$0.929108\pi$
$278$			6.17565e8i			1.72396i
$279$			8.12376e7i			0.223946i
$280$	1.47169e8			0.400649
$281$		−	2.80667e8i		−	0.754604i	−0.926090	−	0.377302i	$-0.876852\pi$
$281$		−	2.80667e8i		−	0.754604i	0.926090	−	0.377302i	$-0.123148\pi$
$282$		−	4.57538e8i		−	1.21494i
$283$		−	3.01350e8i		−	0.790349i	−0.918606	−	0.395174i	$-0.870684\pi$
$283$		−	3.01350e8i		−	0.790349i	0.918606	−	0.395174i	$-0.129316\pi$
$284$	2.83862e7			0.0735349
$285$	1.95092e8			0.499210
$286$	−1.14723e9			−2.89982
$287$	2.02875e8			0.506572
$288$			2.63911e8i			0.651005i
$289$	2.11040e8			0.514306
$290$		−	2.85335e8i		−	0.687008i
$291$		−	7.60669e8i		−	1.80955i
$292$	−3.40316e7			−0.0799913
$293$	7.46744e8			1.73434			0.867172	−	0.498010i	$-0.165936\pi$
$293$	7.46744e8			1.73434			0.867172	+	0.498010i	$0.165936\pi$
$294$		−	6.69288e8i		−	1.53602i
$295$	−1.72955e7			−0.0392243
$296$	3.58120e8	+	1.57619e8i	0.802616	+	0.353255i
$297$	7.15435e8			1.58461
$298$			6.06725e8i			1.32811i
$299$	−5.97333e8			−1.29231
$300$	−1.34833e8			−0.288317
$301$		−	7.65400e8i		−	1.61773i
$302$		−	7.63433e8i		−	1.59495i
$303$	−9.63558e8			−1.98989
$304$		−	5.19329e8i		−	1.06019i
$305$	−1.31888e8			−0.266169
$306$	6.33830e8			1.26458
$307$	−2.23379e8			−0.440614			−0.220307	−	0.975431i	$-0.570706\pi$
$307$	−2.23379e8			−0.440614			−0.220307	+	0.975431i	$0.570706\pi$
$308$	2.06199e8			0.402124
$309$		−	7.00995e8i		−	1.35164i
$310$		−	2.60038e7i		−	0.0495759i
$311$			8.66751e8i			1.63393i	0.576688	+	0.816964i	$0.304345\pi$
$311$			8.66751e8i			1.63393i	−0.576688	+	0.816964i	$0.695655\pi$
$312$	−1.37223e9			−2.55792
$313$		−	2.16874e8i		−	0.399763i	−0.979820	−	0.199882i	$-0.935944\pi$
$313$		−	2.16874e8i		−	0.399763i	0.979820	−	0.199882i	$-0.0640558\pi$
$314$		−	6.19448e8i		−	1.12915i
$315$			4.19959e8i			0.757043i
$316$		−	1.11439e8i		−	0.198670i
$317$	−8.48005e8			−1.49517			−0.747586	−	0.664165i	$-0.768787\pi$
$317$	−8.48005e8			−1.49517			−0.747586	+	0.664165i	$0.768787\pi$
$318$		−	8.66991e7i		−	0.151189i
$319$			1.60683e9i			2.77142i
$320$			1.43191e8i			0.244281i
$321$	1.83635e8			0.309877
$322$	6.46241e8			1.07870
$323$	−3.85913e8			−0.637208
$324$	−1.08042e7			−0.0176476
$325$		−	9.83203e8i		−	1.58874i
$326$	−4.34920e8			−0.695261
$327$			5.47790e8i			0.866357i
$328$			2.08134e8i			0.325675i
$329$	−5.99669e8			−0.928380
$330$	−5.77588e8			−0.884748
$331$		−	1.11493e9i		−	1.68986i	−0.534880	−	0.844928i	$-0.679643\pi$
$331$		−	1.11493e9i		−	1.68986i	0.534880	−	0.844928i	$-0.320357\pi$
$332$	1.04937e7			0.0157378
$333$	−4.49780e8	+	1.02193e9i	−0.667491	+	1.51658i
$334$	4.38492e8			0.643945
$335$		−	4.27073e8i		−	0.620648i
$336$	1.79259e9			2.57807
$337$	−1.35622e8			−0.193030			−0.0965151	−	0.995332i	$-0.530770\pi$
$337$	−1.35622e8			−0.193030			−0.0965151	+	0.995332i	$0.530770\pi$
$338$			1.71217e9i			2.41179i
$339$			8.87350e7i			0.123708i
$340$	−3.37061e7			−0.0465085
$341$			1.46438e8i			0.199992i
$342$	1.22732e9			1.65907
$343$	1.42203e8			0.190274
$344$	7.85241e8			1.04004
$345$	−3.00734e8			−0.394290
$346$		−	7.49581e7i		−	0.0972864i
$347$		−	7.22652e8i		−	0.928487i	−0.885708	−	0.464244i	$-0.846326\pi$
$347$		−	7.22652e8i		−	0.928487i	0.885708	−	0.464244i	$-0.153674\pi$
$348$		−	4.78190e8i		−	0.608237i
$349$	−5.11711e8			−0.644371			−0.322186	−	0.946677i	$-0.604418\pi$
$349$	−5.11711e8			−0.644371			−0.322186	+	0.946677i	$0.604418\pi$
$350$			1.06371e9i			1.32612i
$351$		−	1.55256e9i		−	1.91634i
$352$			4.75722e8i			0.581372i
$353$			1.11622e9i			1.35063i	0.737529	+	0.675315i	$0.235993\pi$
$353$			1.11622e9i			1.35063i	−0.737529	+	0.675315i	$0.764007\pi$
$354$	−1.74470e8			−0.209030
$355$			1.04213e8i			0.123630i
$356$			3.59555e7i			0.0422368i
$357$		−	1.33208e9i		−	1.54949i
$358$	6.75155e8			0.777701
$359$	−1.27437e9			−1.45367			−0.726834	−	0.686813i	$-0.759009\pi$
$359$	−1.27437e9			−1.45367			−0.726834	+	0.686813i	$0.759009\pi$
$360$	−4.30846e8			−0.486702
$361$	1.46607e8			0.164014
$362$		−	3.00396e8i		−	0.332824i
$363$	1.76715e9			1.93910
$364$		−	4.47472e8i		−	0.486307i
$365$		−	1.24939e8i		−	0.134485i
$366$	−1.33044e9			−1.41844
$367$	−9.87404e7			−0.104271			−0.0521355	−	0.998640i	$-0.516603\pi$
$367$	−9.87404e7			−0.104271			−0.0521355	+	0.998640i	$0.516603\pi$
$368$			8.00544e8i			0.837371i
$369$	−5.93927e8			−0.615376
$370$	1.43972e8	−	3.27113e8i	0.147766	−	0.335732i
$371$	−1.13631e8			−0.115529
$372$		−	4.35795e7i		−	0.0438916i
$373$	7.68593e8			0.766859			0.383430	−	0.923570i	$-0.374743\pi$
$373$	7.68593e8			0.766859			0.383430	+	0.923570i	$0.374743\pi$
$374$	1.14253e9			1.12932
$375$		−	1.05257e9i		−	1.03072i
$376$		−	6.15214e8i		−	0.596855i
$377$	3.48697e9			3.35161
$378$			1.67968e9i			1.59958i
$379$	−1.56097e8			−0.147284			−0.0736421	−	0.997285i	$-0.523462\pi$
$379$	−1.56097e8			−0.147284			−0.0736421	+	0.997285i	$0.523462\pi$
$380$	−6.52669e7			−0.0610170
$381$	−2.37807e9			−2.20287
$382$	−1.06264e9			−0.975361
$383$		−	1.06110e9i		−	0.965078i	−0.875875	−	0.482539i	$-0.839715\pi$
$383$		−	1.06110e9i		−	0.965078i	0.875875	−	0.482539i	$-0.160285\pi$
$384$			2.15505e9i			1.94222i
$385$			7.57011e8i			0.676067i
$386$	1.09716e9			0.970993
$387$			2.24075e9i			1.96519i
$388$			2.54477e8i			0.221176i
$389$			1.49216e9i			1.28526i	0.766177	+	0.642630i	$0.222157\pi$
$389$			1.49216e9i			1.28526i	−0.766177	+	0.642630i	$0.777843\pi$
$390$			1.25342e9i			1.06997i
$391$	5.94884e8			0.503285
$392$		−	8.99937e8i		−	0.754589i
$393$		−	1.11607e9i		−	0.927511i
$394$			1.47080e9i			1.21148i
$395$	4.09120e8			0.334011
$396$	−6.03660e8			−0.488494
$397$	6.58899e8			0.528509			0.264254	−	0.964453i	$-0.414874\pi$
$397$	6.58899e8			0.528509			0.264254	+	0.964453i	$0.414874\pi$
$398$	7.71709e8			0.613569
$399$		−	2.57937e9i		−	2.03286i
$400$	−1.31769e9			−1.02944
$401$			2.05350e9i			1.59034i	0.606388	+	0.795169i	$0.292618\pi$
$401$			2.05350e9i			1.59034i	−0.606388	+	0.795169i	$0.707382\pi$
$402$		−	4.30815e9i		−	3.30750i
$403$	3.17783e8			0.241859
$404$	3.22352e8			0.243218
$405$		−	3.96652e7i		−	0.0296700i
$406$	−3.77248e9			−2.79760
$407$	−8.10765e8	+	1.84210e9i	−0.596094	+	1.35436i
$408$	1.36661e9			0.996169
$409$		−	5.59086e8i		−	0.404061i	−0.979379	−	0.202031i	$-0.935246\pi$
$409$		−	5.59086e8i		−	0.404061i	0.979379	−	0.202031i	$-0.0647540\pi$
$410$	1.90113e8			0.136229
$411$	4.31603e7			0.0306646
$412$			2.34513e8i			0.165207i
$413$			2.28668e8i			0.159728i
$414$	−1.89191e9			−1.31038
$415$			3.85250e7i			0.0264591i
$416$	1.03236e9			0.703080
$417$	−3.79965e9			−2.56606
$418$	2.21234e9			1.48161
$419$	−9.96013e8			−0.661479			−0.330739	−	0.943722i	$-0.607298\pi$
$419$	−9.96013e8			−0.661479			−0.330739	+	0.943722i	$0.607298\pi$
$420$		−	2.25285e8i		−	0.148375i
$421$		−	1.77843e8i		−	0.116158i	−0.998312	−	0.0580790i	$-0.981502\pi$
$421$		−	1.77843e8i		−	0.116158i	0.998312	−	0.0580790i	$-0.0184975\pi$
$422$			1.39467e9i			0.903397i
$423$	1.75556e9			1.12778
$424$		−	1.16577e8i		−	0.0742733i
$425$			9.79173e8i			0.618726i
$426$			1.05126e9i			0.658837i
$427$			1.74373e9i			1.08388i
$428$	−6.14340e7			−0.0378753
$429$		−	7.05850e9i		−	4.31630i
$430$		−	7.17253e8i		−	0.435044i
$431$		−	7.58652e8i		−	0.456428i	−0.973611	−	0.228214i	$-0.926711\pi$
$431$		−	7.58652e8i		−	0.456428i	0.973611	−	0.228214i	$-0.0732886\pi$
$432$	−2.08074e9			−1.24172
$433$	−8.17566e8			−0.483966			−0.241983	−	0.970280i	$-0.577798\pi$
$433$	−8.17566e8			−0.483966			−0.241983	+	0.970280i	$0.577798\pi$
$434$	−3.43803e8			−0.201881
$435$	1.75556e9			1.02259
$436$		−	1.83260e8i		−	0.105892i
$437$	1.15191e9			0.660286
$438$		−	1.26034e9i		−	0.716683i
$439$		−	1.46719e9i		−	0.827676i	−0.910351	−	0.413838i	$-0.864188\pi$
$439$		−	1.46719e9i		−	0.827676i	0.910351	−	0.413838i	$-0.135812\pi$
$440$	−7.76635e8			−0.434643
$441$	2.56804e9			1.42583
$442$		−	2.47940e9i		−	1.36574i
$443$	−1.25169e9			−0.684045			−0.342023	−	0.939692i	$-0.611112\pi$
$443$	−1.25169e9			−0.684045			−0.342023	+	0.939692i	$0.611112\pi$
$444$	2.41282e8	−	5.48206e8i	0.130823	−	0.297237i
$445$	−1.32002e8			−0.0710102
$446$		−	4.81732e8i		−	0.257119i
$447$	−3.73295e9			−1.97686
$448$	1.89316e9			0.994751
$449$		−	2.46487e9i		−	1.28509i	−0.766250	−	0.642543i	$-0.777879\pi$
$449$		−	2.46487e9i		−	1.28509i	0.766250	−	0.642543i	$-0.222121\pi$
$450$		−	3.11406e9i		−	1.61095i
$451$	−1.07060e9			−0.549554
$452$		−	2.96857e7i		−	0.0151204i
$453$	4.69712e9			2.37404
$454$	−2.82679e9			−1.41775
$455$	1.64278e9			0.817600
$456$	2.64623e9			1.30693
$457$			3.47393e9i			1.70260i	0.524675	+	0.851302i	$0.324187\pi$
$457$			3.47393e9i			1.70260i	−0.524675	+	0.851302i	$0.675813\pi$
$458$		−	1.10050e9i		−	0.535257i
$459$			1.54619e9i			0.746311i
$460$	1.00609e8			0.0481929
$461$			1.31267e9i			0.624025i	0.950078	+	0.312012i	$0.101003\pi$
$461$			1.31267e9i			0.624025i	−0.950078	+	0.312012i	$0.898997\pi$
$462$			7.63644e9i			3.60283i
$463$		−	1.98567e9i		−	0.929768i	−0.885372	−	0.464884i	$-0.846096\pi$
$463$		−	1.98567e9i		−	0.929768i	0.885372	−	0.464884i	$-0.153904\pi$
$464$		−	4.67323e9i		−	2.17172i
$465$	1.59992e8			0.0737925
$466$			2.86525e9i			1.31163i
$467$		−	2.35188e9i		−	1.06858i	−0.845301	−	0.534290i	$-0.820579\pi$
$467$		−	2.35188e9i		−	1.06858i	0.845301	−	0.534290i	$-0.179421\pi$
$468$			1.31000e9i			0.590758i
$469$	−5.64645e9			−2.52738
$470$	−5.61947e8			−0.249662
$471$	3.81123e9			1.68071
$472$	−2.34595e8			−0.102689
$473$			4.03913e9i			1.75499i
$474$	4.12705e9			1.77998
$475$			1.89602e9i			0.811739i
$476$			4.45637e8i			0.189390i
$477$	3.32662e8			0.140342
$478$	−2.93505e9			−1.22919
$479$			2.31267e9i			0.961479i	0.876864	+	0.480739i	$0.159632\pi$
$479$			2.31267e9i			0.961479i	−0.876864	+	0.480739i	$0.840368\pi$
$480$	5.19754e8			0.214513
$481$	3.99754e9	+	1.75944e9i	1.63789	+	0.720884i
$482$	−1.23048e9			−0.500508
$483$			3.97609e9i			1.60561i
$484$	−5.91188e8			−0.237010
$485$	−9.34251e8			−0.371850
$486$			2.56755e9i			1.01459i
$487$		−	2.20951e9i		−	0.866850i	−0.901190	−	0.433425i	$-0.857305\pi$
$487$		−	2.20951e9i		−	0.866850i	0.901190	−	0.433425i	$-0.142695\pi$
$488$	−1.78893e9			−0.696826
$489$		−	2.67590e9i		−	1.03488i
$490$	−8.22018e8			−0.315642
$491$	2.19601e9			0.837240			0.418620	−	0.908162i	$-0.362514\pi$
$491$	2.19601e9			0.837240			0.418620	+	0.908162i	$0.362514\pi$
$492$	3.18609e8			0.120609
$493$	−3.47268e9			−1.30527
$494$		−	4.80099e9i		−	1.79179i
$495$		−	2.21619e9i		−	0.821276i
$496$		−	4.25892e8i		−	0.156716i
$497$	1.37783e9			0.503441
$498$			3.88626e8i			0.141003i
$499$		−	5.33867e9i		−	1.92345i	−0.274013	−	0.961726i	$-0.588351\pi$
$499$		−	5.33867e9i		−	1.92345i	0.274013	−	0.961726i	$-0.411649\pi$
$500$			3.52130e8i			0.125982i
$501$			2.69788e9i			0.958496i
$502$	4.18399e9			1.47614
$503$			1.58131e9i			0.554024i	0.960866	+	0.277012i	$0.0893442\pi$
$503$			1.58131e9i			0.554024i	−0.960866	+	0.277012i	$0.910656\pi$
$504$			5.69632e9i			1.98193i
$505$			1.18344e9i			0.408908i
$506$	−3.41032e9			−1.17022
$507$	−1.05344e10			−3.58989
$508$	7.95570e8			0.269250
$509$	4.20495e9			1.41335			0.706673	−	0.707540i	$-0.250195\pi$
$509$	4.20495e9			1.41335			0.706673	+	0.707540i	$0.250195\pi$
$510$		−	1.24828e9i		−	0.416694i
$511$	−1.65185e9			−0.547643
$512$			1.70451e9i			0.561248i
$513$			2.99398e9i			0.979125i
$514$	5.02447e9			1.63200
$515$	−8.60959e8			−0.277752
$516$		−	1.20204e9i		−	0.385163i
$517$	3.16454e9			1.00715
$518$	−4.32485e9	−	1.90350e9i	−1.36715	−	0.601725i
$519$	4.61189e8			0.144808
$520$			1.68537e9i			0.525634i
$521$	−6.25956e8			−0.193915			−0.0969576	−	0.995289i	$-0.530911\pi$
$521$	−6.25956e8			−0.193915			−0.0969576	+	0.995289i	$0.530911\pi$
$522$	1.10441e10			3.39849
$523$		−	5.23690e9i		−	1.60073i	−0.599512	−	0.800366i	$-0.704639\pi$
$523$		−	5.23690e9i		−	1.60073i	0.599512	−	0.800366i	$-0.295361\pi$
$524$			3.73376e8i			0.113367i
$525$	−6.54459e9			−1.97390
$526$			4.03118e9i			1.20776i
$527$	−3.16480e8			−0.0941910
$528$	−9.45979e9			−2.79681
$529$	1.62917e9			0.478488
$530$	−1.06483e8			−0.0310682
$531$		−	6.69437e8i		−	0.194035i
$532$			8.62911e8i			0.248471i
$533$			2.32331e9i			0.664601i
$534$	−1.33159e9			−0.378421
$535$		−	2.25540e8i		−	0.0636775i
$536$		−	5.79282e9i		−	1.62485i
$537$			4.15398e9i			1.15759i
$538$		−	1.18164e8i		−	0.0327151i
$539$	4.62911e9			1.27332
$540$			2.61498e8i			0.0714644i
$541$			4.25620e9i			1.15566i	0.816156	+	0.577832i	$0.196101\pi$
$541$			4.25620e9i			1.15566i	−0.816156	+	0.577832i	$0.803899\pi$
$542$		−	7.68917e9i		−	2.07435i
$543$	1.84823e9			0.495400
$544$	−1.02813e9			−0.273811
$545$	6.72794e8			0.178030
$546$	1.65718e10			4.35708
$547$			3.81186e9i			0.995820i	0.867229	+	0.497910i	$0.165899\pi$
$547$			3.81186e9i			0.995820i	−0.867229	+	0.497910i	$0.834101\pi$
$548$	−1.44390e7			−0.00374805
$549$		−	5.10486e9i		−	1.31668i
$550$		−	5.61334e9i		−	1.43864i
$551$	−6.72434e9			−1.71245
$552$	−4.07916e9			−1.03225
$553$		−	5.40909e9i		−	1.36015i
$554$	1.93605e9			0.483763
$555$	2.01261e9	+	8.85809e8i	0.499728	+	0.219945i
$556$	1.27115e9			0.313642
$557$		−	9.70937e8i		−	0.238066i	−0.992890	−	0.119033i	$-0.962021\pi$
$557$		−	9.70937e8i		−	0.238066i	0.992890	−	0.119033i	$-0.0379795\pi$
$558$	1.00650e9			0.245242
$559$	8.76530e9			2.12239
$560$		−	2.20166e9i		−	0.529775i
$561$			7.02957e9i			1.68097i
$562$	−3.47735e9			−0.826363
$563$			3.27548e9i			0.773563i	0.922171	+	0.386781i	$0.126413\pi$
$563$			3.27548e9i			0.773563i	−0.922171	+	0.386781i	$0.873587\pi$
$564$	−9.41762e8			−0.221037
$565$	1.08984e8			0.0254210
$566$	−3.73360e9			−0.865507
$567$	−5.24424e8			−0.120821
$568$			1.41355e9i			0.323662i
$569$			1.53122e9i			0.348453i	0.984706	+	0.174227i	$0.0557425\pi$
$569$			1.53122e9i			0.348453i	−0.984706	+	0.174227i	$0.944258\pi$
$570$		−	2.41711e9i		−	0.546682i
$571$	−1.21431e9			−0.272962			−0.136481	−	0.990643i	$-0.543579\pi$
$571$	−1.21431e9			−0.272962			−0.136481	+	0.990643i	$0.543579\pi$
$572$			2.36138e9i			0.527569i
$573$		−	6.53804e9i		−	1.45180i
$574$		−	2.51353e9i		−	0.554745i
$575$		−	2.92271e9i		−	0.641134i
$576$	−5.54233e9			−1.20841
$577$			5.83992e9i			1.26559i	0.774321	+	0.632793i	$0.218092\pi$
$577$			5.83992e9i			1.26559i	−0.774321	+	0.632793i	$0.781908\pi$
$578$		−	2.61469e9i		−	0.563214i
$579$			6.75045e9i			1.44530i
$580$	−5.87311e8			−0.124989
$581$	5.09350e8			0.107746
$582$	−9.42437e9			−1.98163
$583$	5.99651e8			0.125331
$584$		−	1.69467e9i		−	0.352079i
$585$	−4.80934e9			−0.993207
$586$		−	9.25185e9i		−	1.89927i
$587$			3.71496e9i			0.758090i	0.925378	+	0.379045i	$0.123747\pi$
$587$			3.71496e9i			0.758090i	−0.925378	+	0.379045i	$0.876253\pi$
$588$	−1.37761e9			−0.279451
$589$	−6.12818e8			−0.123574
$590$			2.14284e8i			0.0429543i
$591$	−9.04926e9			−1.80325
$592$	2.35799e9	−	5.35749e9i	0.467107	−	1.06129i
$593$	4.20159e9			0.827412			0.413706	−	0.910410i	$-0.364234\pi$
$593$	4.20159e9			0.827412			0.413706	+	0.910410i	$0.364234\pi$
$594$		−	8.86394e9i		−	1.73530i
$595$	−1.63605e9			−0.318410
$596$	1.24884e9			0.241626
$597$			4.74804e9i			0.913281i
$598$			7.40070e9i			1.41520i
$599$	−1.84037e9			−0.349873			−0.174937	−	0.984580i	$-0.555972\pi$
$599$	−1.84037e9			−0.349873			−0.174937	+	0.984580i	$0.555972\pi$
$600$		−	6.71425e9i		−	1.26902i
$601$	1.06842e9			0.200763			0.100381	−	0.994949i	$-0.467994\pi$
$601$	1.06842e9			0.200763			0.100381	+	0.994949i	$0.467994\pi$
$602$	−9.48299e9			−1.77157
$603$	1.65303e10			3.07022
$604$	−1.57139e9			−0.290172
$605$		−	2.17041e9i		−	0.398472i
$606$			1.19381e10i			2.17912i
$607$		−	3.49160e9i		−	0.633672i	−0.948480	−	0.316836i	$-0.897380\pi$
$607$		−	3.49160e9i		−	0.633672i	0.948480	−	0.316836i	$-0.102620\pi$
$608$	−1.99082e9			−0.359227
$609$		−	2.32107e10i		−	4.16416i
$610$			1.63404e9i			0.291480i
$611$		−	6.86736e9i		−	1.21799i
$612$		−	1.30463e9i		−	0.230068i
$613$	5.81100e9			1.01892			0.509459	−	0.860495i	$-0.329846\pi$
$613$	5.81100e9			1.01892			0.509459	+	0.860495i	$0.329846\pi$
$614$			2.76757e9i			0.482514i
$615$			1.16970e9i			0.202773i
$616$			1.02681e10i			1.76993i
$617$	−2.41003e9			−0.413071			−0.206536	−	0.978439i	$-0.566219\pi$
$617$	−2.41003e9			−0.413071			−0.206536	+	0.978439i	$0.566219\pi$
$618$	−8.68503e9			−1.48017
$619$	−1.27503e9			−0.216074			−0.108037	−	0.994147i	$-0.534457\pi$
$619$	−1.27503e9			−0.216074			−0.108037	+	0.994147i	$0.534457\pi$
$620$	−5.35242e7			−0.00901943
$621$		−	4.61521e9i		−	0.773340i
$622$	1.07387e10			1.78931
$623$			1.74523e9i			0.289165i
$624$			2.05286e10i			3.38231i
$625$	4.12596e9			0.675997
$626$	−2.68698e9			−0.437779
$627$			1.36117e10i			2.20535i
$628$	−1.27502e9			−0.205428
$629$	−3.98115e9	−	1.75222e9i	−0.637869	−	0.280745i
$630$	5.20312e9			0.829033
$631$			7.59476e9i			1.20340i	0.798721	+	0.601702i	$0.205510\pi$
$631$			7.59476e9i			1.20340i	−0.798721	+	0.601702i	$0.794490\pi$
$632$	5.54930e9			0.874438
$633$	−8.58091e9			−1.34468
$634$			1.05064e10i			1.63736i
$635$			2.92074e9i			0.452674i
$636$	−1.78455e8			−0.0275060
$637$		−	1.00456e10i		−	1.53988i
$638$	1.99080e10			3.03497
$639$	−4.03367e9			−0.611572
$640$	2.64682e9			0.399112
$641$	−3.58582e9			−0.537756			−0.268878	−	0.963174i	$-0.586653\pi$
$641$	−3.58582e9			−0.537756			−0.268878	+	0.963174i	$0.586653\pi$
$642$		−	2.27517e9i		−	0.339344i
$643$			1.87810e9i			0.278600i	0.990250	+	0.139300i	$0.0444852\pi$
$643$			1.87810e9i			0.278600i	−0.990250	+	0.139300i	$0.955515\pi$
$644$		−	1.33017e9i		−	0.196249i
$645$	4.41299e9			0.647551
$646$			4.78131e9i			0.697803i
$647$			9.51275e9i			1.38083i	0.723412	+	0.690417i	$0.242573\pi$
$647$			9.51275e9i			1.38083i	−0.723412	+	0.690417i	$0.757427\pi$
$648$		−	5.38018e8i		−	0.0776755i
$649$		−	1.20672e9i		−	0.173280i
$650$	−1.21815e10			−1.73982
$651$		−	2.11529e9i		−	0.300495i
$652$			8.95206e8i			0.126490i
$653$		−	1.05284e10i		−	1.47968i	−0.672783	−	0.739840i	$-0.734901\pi$
$653$		−	1.05284e10i		−	1.47968i	0.672783	−	0.739840i	$-0.265099\pi$
$654$	6.78689e9			0.948743
$655$	−1.37076e9			−0.190597
$656$	3.11369e9			0.430637
$657$	4.83588e9			0.665268
$658$			7.42964e9i			1.01666i
$659$	5.29033e9			0.720084			0.360042	−	0.932936i	$-0.382762\pi$
$659$	5.29033e9			0.720084			0.360042	+	0.932936i	$0.382762\pi$
$660$			1.18886e9i			0.160964i
$661$		−	1.43014e9i		−	0.192607i	−0.995352	−	0.0963035i	$-0.969298\pi$
$661$		−	1.43014e9i		−	0.192607i	0.995352	−	0.0963035i	$-0.0307019\pi$
$662$	−1.38135e10			−1.85055
$663$	1.52548e10			2.03287
$664$			5.22553e8i			0.0692695i
$665$	−3.16797e9			−0.417739
$666$	1.26612e10	+	5.57259e9i	1.66079	+	0.730966i
$667$	1.03655e10			1.35254
$668$		−	9.02558e8i		−	0.117154i
$669$	2.96392e9			0.382715
$670$	−5.29126e9			−0.679669
$671$		−	9.20193e9i		−	1.17585i
$672$		−	6.87180e9i		−	0.873531i
$673$	−5.40603e9			−0.683637			−0.341819	−	0.939766i	$-0.611043\pi$
$673$	−5.40603e9			−0.683637			−0.341819	+	0.939766i	$0.611043\pi$
$674$			1.68030e9i			0.211386i
$675$	7.59658e9			0.950725
$676$	3.52421e9			0.438781
$677$	−1.25770e10			−1.55782			−0.778910	−	0.627136i	$-0.784227\pi$
$677$	−1.25770e10			−1.55782			−0.778910	+	0.627136i	$0.784227\pi$
$678$	1.09939e9			0.135471
$679$			1.23520e10i			1.51423i
$680$		−	1.67846e9i		−	0.204706i
$681$		−	1.73922e10i		−	2.11028i
$682$	1.81430e9			0.219010
$683$		−	1.42971e9i		−	0.171703i	−0.996308	−	0.0858513i	$-0.972639\pi$
$683$		−	1.42971e9i		−	0.171703i	0.996308	−	0.0858513i	$-0.0273610\pi$
$684$		−	2.52622e9i		−	0.301838i
$685$		−	5.30093e7i		−	0.00630137i
$686$		−	1.76183e9i		−	0.208368i
$687$	6.77099e9			0.796716
$688$		−	1.17472e10i		−	1.37523i
$689$		−	1.30130e9i		−	0.151569i
$690$			3.72597e9i			0.431785i
$691$	−1.34200e10			−1.54731			−0.773657	−	0.633604i	$-0.781575\pi$
$691$	−1.34200e10			−1.54731			−0.773657	+	0.633604i	$0.781575\pi$
$692$	−1.54288e8			−0.0176995
$693$	−2.93008e10			−3.34437
$694$	−8.95336e9			−1.01678
$695$			4.66672e9i			0.527309i
$696$	2.38124e10			2.67714
$697$		−	2.31378e9i		−	0.258826i
$698$			6.33989e9i			0.705648i
$699$	−1.76288e10			−1.95233
$700$	2.18945e9			0.241264
$701$		−	1.27488e9i		−	0.139784i	−0.997555	−	0.0698919i	$-0.977735\pi$
$701$		−	1.27488e9i		−	0.139784i	0.997555	−	0.0698919i	$-0.0222654\pi$
$702$	−1.92356e10			−2.09858
$703$	−7.70891e9	−	3.39292e9i	−0.836853	−	0.368324i
$704$	−9.99051e9			−1.07915
$705$		−	3.45745e9i		−	0.371616i
$706$	1.38294e10			1.47907
$707$	1.56465e10			1.66514
$708$			3.59116e8i			0.0380293i
$709$		−	4.80461e9i		−	0.506287i	−0.967429	−	0.253143i	$-0.918536\pi$
$709$		−	4.80461e9i		−	0.506287i	0.967429	−	0.253143i	$-0.0814644\pi$
$710$	1.29116e9			0.135387
$711$			1.58354e10i			1.65229i
$712$	−1.79047e9			−0.185904
$713$	9.44656e8			0.0976024
$714$	−1.65039e10			−1.69684
$715$	−8.66923e9			−0.886971
$716$		−	1.38969e9i		−	0.141489i
$717$		−	1.80583e10i		−	1.82961i
$718$			1.57889e10i			1.59190i
$719$	1.01073e10			1.01411			0.507053	−	0.861915i	$-0.330735\pi$
$719$	1.01073e10			1.01411			0.507053	+	0.861915i	$0.330735\pi$
$720$			6.44546e9i			0.643562i
$721$			1.13830e10i			1.13105i
$722$		−	1.81640e9i		−	0.179611i
$723$		−	7.57072e9i		−	0.744994i
$724$	−6.18313e8			−0.0605513
$725$			1.70616e10i			1.66278i
$726$		−	2.18942e10i		−	2.12350i
$727$		−	1.32701e10i		−	1.28087i	−0.768012	−	0.640435i	$-0.778754\pi$
$727$		−	1.32701e10i		−	1.28087i	0.768012	−	0.640435i	$-0.221246\pi$
$728$	2.22827e10			2.14047
$729$	−1.67238e10			−1.59878
$730$	−1.54794e9			−0.147273
$731$	−8.72937e9			−0.826555
$732$			2.73847e9i			0.258060i
$733$	−9.17424e9			−0.860411			−0.430206	−	0.902731i	$-0.641559\pi$
$733$	−9.17424e9			−0.860411			−0.430206	+	0.902731i	$0.641559\pi$
$734$			1.22335e9i			0.114187i
$735$		−	5.05757e9i		−	0.469825i
$736$	3.06884e9			0.283728
$737$	2.97972e10			2.74182
$738$			7.35851e9i			0.673895i
$739$	−2.28666e9			−0.208423			−0.104212	−	0.994555i	$-0.533232\pi$
$739$	−2.28666e9			−0.208423			−0.104212	+	0.994555i	$0.533232\pi$
$740$	−6.73305e8	−	2.96342e8i	−0.0610803	−	0.0268833i
$741$	2.95387e10			2.66703
$742$			1.40785e9i			0.126515i
$743$	1.03455e10			0.925314			0.462657	−	0.886537i	$-0.346896\pi$
$743$	1.03455e10			0.925314			0.462657	+	0.886537i	$0.346896\pi$
$744$	2.17012e9			0.193188
$745$			4.58480e9i			0.406232i
$746$		−	9.52255e9i		−	0.839784i
$747$	−1.49115e9			−0.130888
$748$		−	2.35170e9i		−	0.205459i
$749$	−2.98193e9			−0.259305
$750$	−1.30409e10			−1.12874
$751$	−1.48779e10			−1.28174			−0.640872	−	0.767648i	$-0.721427\pi$
$751$	−1.48779e10			−1.28174			−0.640872	+	0.767648i	$0.721427\pi$
$752$	−9.20361e9			−0.789216
$753$			2.57426e10i			2.19720i
$754$		−	4.32022e10i		−	3.67034i
$755$		−	5.76899e9i		−	0.487849i
$756$	3.45733e9			0.291014
$757$			6.27980e9i			0.526151i	0.964775	+	0.263075i	$0.0847368\pi$
$757$			6.27980e9i			0.526151i	−0.964775	+	0.263075i	$0.915263\pi$
$758$			1.93397e9i			0.161290i
$759$		−	2.09824e10i		−	1.74185i
$760$		−	3.25009e9i		−	0.268564i
$761$	−1.95252e10			−1.60602			−0.803008	−	0.595968i	$-0.796769\pi$
$761$	−1.95252e10			−1.60602			−0.803008	+	0.595968i	$0.796769\pi$
$762$			2.94633e10i			2.41235i
$763$		−	8.89518e9i		−	0.724968i
$764$			2.18726e9i			0.177449i
$765$	4.78963e9			0.386800
$766$	−1.31466e10			−1.05685
$767$	−2.61868e9			−0.209556
$768$	1.17771e10			0.938157
$769$		−	6.88547e9i		−	0.545998i	−0.962014	−	0.272999i	$-0.911984\pi$
$769$		−	6.88547e9i		−	0.545998i	0.962014	−	0.272999i	$-0.0880156\pi$
$770$	9.37905e9			0.740358
$771$			3.09137e10i			2.42919i
$772$		−	2.25832e9i		−	0.176655i
$773$	1.15500e10			0.899403			0.449702	−	0.893179i	$-0.351530\pi$
$773$	1.15500e10			0.899403			0.449702	+	0.893179i	$0.351530\pi$
$774$	2.77619e10			2.15207
$775$			1.55489e9i			0.119990i
$776$	−1.26722e10			−0.973498
$777$	1.17115e10	−	2.66092e10i	0.895652	−	2.03497i
$778$	1.84872e10			1.40748
$779$		−	4.48030e9i		−	0.339567i
$780$	2.57994e9			0.194661
$781$	−7.27103e9			−0.546157
$782$		−	7.37037e9i		−	0.551145i
$783$			2.69416e10i			2.00566i
$784$	−1.34631e10			−0.997787
$785$		−	4.68095e9i		−	0.345374i
$786$	−1.38277e10			−1.01571
$787$	2.40756e10			1.76062			0.880309	−	0.474400i	$-0.157335\pi$
$787$	2.40756e10			1.76062			0.880309	+	0.474400i	$0.157335\pi$
$788$	3.02737e9			0.220406
$789$	−2.48024e10			−1.79773
$790$		−	5.06883e9i		−	0.365774i
$791$		−	1.44091e9i		−	0.103519i
$792$		−	3.00604e10i		−	2.15009i
$793$	−1.99690e10			−1.42200
$794$		−	8.16349e9i		−	0.578767i
$795$		−	6.55154e8i		−	0.0462443i
$796$		−	1.58843e9i		−	0.111628i
$797$			1.88025e10i			1.31557i	0.753208	+	0.657783i	$0.228506\pi$
$797$			1.88025e10i			1.31557i	−0.753208	+	0.657783i	$0.771494\pi$
$798$	−3.19573e10			−2.22618
$799$			6.83920e9i			0.474342i
$800$			5.05128e9i			0.348808i
$801$		−	5.10926e9i		−	0.351273i
$802$	2.54420e10			1.74157
$803$	8.71707e9			0.594109
$804$	−8.86758e9			−0.601740
$805$	4.88342e9			0.329942
$806$		−	3.93720e9i		−	0.264859i
$807$	7.27022e8			0.0486956
$808$			1.60521e10i			1.07052i
$809$		−	4.16851e9i		−	0.276797i	−0.990377	−	0.138398i	$-0.955805\pi$
$809$		−	4.16851e9i		−	0.276797i	0.990377	−	0.138398i	$-0.0441954\pi$
$810$	−4.91435e8			−0.0324914
$811$	−7.16522e9			−0.471690			−0.235845	−	0.971791i	$-0.575786\pi$
$811$	−7.16522e9			−0.471690			−0.235845	+	0.971791i	$0.575786\pi$
$812$			7.76499e9i			0.508973i
$813$	4.73086e10			3.08762
$814$	2.28229e10	+	1.00450e10i	1.48315	+	0.652779i
$815$	−3.28654e9			−0.212660
$816$		−	2.04445e10i		−	1.31723i
$817$	−1.69031e10			−1.08440
$818$	−6.92684e9			−0.442485
$819$			6.35855e10i			4.04450i
$820$		−	3.91314e8i		−	0.0247843i
$821$	4.22462e9			0.266432			0.133216	−	0.991087i	$-0.457470\pi$
$821$	4.22462e9			0.266432			0.133216	+	0.991087i	$0.457470\pi$
$822$		−	5.34738e8i		−	0.0335807i
$823$	−2.01570e9			−0.126045			−0.0630227	−	0.998012i	$-0.520074\pi$
$823$	−2.01570e9			−0.126045			−0.0630227	+	0.998012i	$0.520074\pi$
$824$	−1.16780e10			−0.727151
$825$	3.45368e10			2.14138
$826$	2.83310e9			0.174917
$827$		−	5.82333e8i		−	0.0358016i	−0.999840	−	0.0179008i	$-0.994302\pi$
$827$		−	5.82333e8i		−	0.0358016i	0.999840	−	0.0179008i	$-0.00569830\pi$
$828$			3.89416e9i			0.238401i
$829$			2.45373e10i			1.49584i	0.663787	+	0.747922i	$0.268948\pi$
$829$			2.45373e10i			1.49584i	−0.663787	+	0.747922i	$0.731052\pi$
$830$	4.77309e8			0.0289752
$831$			1.19118e10i			0.720070i
$832$			2.16803e10i			1.30507i
$833$			1.00044e10i			0.599700i
$834$			4.70761e10i			2.81008i
$835$	3.31352e9			0.196964
$836$		−	4.55372e9i		−	0.269553i
$837$			2.45531e9i			0.144733i
$838$			1.23402e10i			0.724382i
$839$	−2.17026e10			−1.26866			−0.634329	−	0.773063i	$-0.718724\pi$
$839$	−2.17026e10			−1.26866			−0.634329	+	0.773063i	$0.718724\pi$
$840$	1.12185e10			0.653066
$841$	−4.32597e10			−2.50783
$842$	−2.20340e9			−0.127204
$843$		−	2.13948e10i		−	1.23002i
$844$	2.87069e9			0.164357
$845$			1.29383e10i			0.737697i
$846$		−	2.17507e10i		−	1.23503i
$847$	−2.86955e10			−1.62264
$848$	−1.74400e9			−0.0982109
$849$		−	2.29715e10i		−	1.28829i
$850$	1.21315e10			0.677563
$851$	1.18833e10	+	5.23018e9i	0.660971	+	0.290913i
$852$	2.16384e9			0.119864
$853$		−	3.19477e10i		−	1.76246i	−0.472692	−	0.881228i	$-0.656718\pi$
$853$		−	3.19477e10i		−	1.76246i	0.472692	−	0.881228i	$-0.343282\pi$
$854$	2.16041e10			1.18695
$855$	9.27440e9			0.507463
$856$		−	3.05923e9i		−	0.166707i
$857$		−	2.01169e10i		−	1.09176i	−0.837862	−	0.545882i	$-0.816195\pi$
$857$		−	2.01169e10i		−	1.09176i	0.837862	−	0.545882i	$-0.183805\pi$
$858$	−8.74519e10			−4.72676
$859$		−	1.50105e10i		−	0.808014i	−0.914756	−	0.404007i	$-0.867617\pi$
$859$		−	1.50105e10i		−	0.808014i	0.914756	−	0.404007i	$-0.132383\pi$
$860$	−1.47634e9			−0.0791483
$861$	1.54649e10			0.825724
$862$	−9.39939e9			−0.499832
$863$	−2.13724e10			−1.13192			−0.565961	−	0.824432i	$-0.691495\pi$
$863$	−2.13724e10			−1.13192			−0.565961	+	0.824432i	$0.691495\pi$
$864$			7.97639e9i			0.420735i
$865$		−	5.66431e8i		−	0.0297571i
$866$			1.01293e10i			0.529989i
$867$	1.60872e10			0.838329
$868$			7.07657e8i			0.0367286i
$869$			2.85446e10i			1.47555i
$870$		−	2.17506e10i		−	1.11984i
$871$		−	6.46626e10i		−	3.31581i
$872$	9.12577e9			0.466081
$873$		−	3.61611e10i		−	1.83946i
$874$		−	1.42716e10i		−	0.723075i
$875$			1.70919e10i			0.862507i
$876$	−2.59418e9			−0.130387
$877$	1.47245e10			0.737129			0.368564	−	0.929602i	$-0.379849\pi$
$877$	1.47245e10			0.737129			0.368564	+	0.929602i	$0.379849\pi$
$878$	−1.81779e10			−0.906384
$879$	5.69232e10			2.82702
$880$			1.16185e10i			0.574725i
$881$	2.34376e10			1.15478			0.577388	−	0.816470i	$-0.304072\pi$
$881$	2.34376e10			1.15478			0.577388	+	0.816470i	$0.304072\pi$
$882$		−	3.18170e10i		−	1.56142i
$883$		−	4.56179e9i		−	0.222983i	−0.993765	−	0.111492i	$-0.964437\pi$
$883$		−	4.56179e9i		−	0.222983i	0.993765	−	0.111492i	$-0.0355628\pi$
$884$	−5.10340e9			−0.248472
$885$	−1.31841e9			−0.0639364
$886$			1.55080e10i			0.749094i
$887$	9.11195e9			0.438408			0.219204	−	0.975679i	$-0.429654\pi$
$887$	9.11195e9			0.438408			0.219204	+	0.975679i	$0.429654\pi$
$888$	2.72990e10	+	1.20151e10i	1.30828	+	0.575814i
$889$	3.86159e10			1.84336
$890$			1.63545e9i			0.0777629i
$891$	2.76747e9			0.131072
$892$	−9.91561e8			−0.0467781
$893$			1.32431e10i			0.622315i
$894$			4.62498e10i			2.16485i
$895$	5.10190e9			0.237876
$896$		−	3.49943e10i		−	1.62525i
$897$	−4.55338e10			−2.10650
$898$	−3.05387e10			−1.40729
$899$	−5.51450e9			−0.253132
$900$	−6.40974e9			−0.293084
$901$			1.29596e9i			0.0590277i
$902$			1.32643e10i			0.601814i
$903$		−	5.83453e10i		−	2.63693i
$904$	1.47826e9			0.0665520
$905$		−	2.26999e9i		−	0.101801i
$906$		−	5.81954e10i		−	2.59980i
$907$			1.33646e10i			0.594743i	0.954762	+	0.297372i	$0.0961100\pi$
$907$			1.33646e10i			0.594743i	−0.954762	+	0.297372i	$0.903890\pi$
$908$			5.81846e9i			0.257933i
$909$	−4.58061e10			−2.02278
$910$		−	2.03534e10i		−	0.895349i
$911$		−	3.06846e9i		−	0.134464i	−0.997737	−	0.0672321i	$-0.978583\pi$
$911$		−	3.06846e9i		−	0.134464i	0.997737	−	0.0672321i	$-0.0214168\pi$
$912$		−	3.95877e10i		−	1.72814i
$913$	−2.68792e9			−0.116888
$914$	4.30405e10			1.86451
$915$	−1.00537e10			−0.433860
$916$	−2.26519e9			−0.0973802
$917$			1.81232e10i			0.776142i
$918$	1.91567e10			0.817281
$919$		−	7.34704e7i		−	0.00312254i	−0.999999	−	0.00156127i	$-0.999503\pi$
$919$		−	7.34704e7i		−	0.00312254i	0.999999	−	0.00156127i	$-0.000496968\pi$
$920$			5.01001e9i			0.212120i
$921$	−1.70279e10			−0.718209
$922$	1.62634e10			0.683366
$923$			1.57788e10i			0.660493i
$924$	1.57183e10			0.655470
$925$	−8.60882e9	+	1.95597e10i	−0.357641	+	0.812580i
$926$	−2.46017e10			−1.01818
$927$		−	3.33242e10i		−	1.37398i
$928$	−1.79146e10			−0.735850
$929$	1.19252e10			0.487991			0.243996	−	0.969776i	$-0.421542\pi$
$929$	1.19252e10			0.487991			0.243996	+	0.969776i	$0.421542\pi$
$930$		−	1.98223e9i		−	0.0808097i
$931$			1.93721e10i			0.786778i
$932$	5.89761e9			0.238627
$933$			6.60712e10i			2.66334i
$934$	−2.91389e10			−1.17020
$935$	8.63369e9			0.345427
$936$	−6.52338e10			−2.60020
$937$	2.23045e10			0.885734			0.442867	−	0.896587i	$-0.353961\pi$
$937$	2.23045e10			0.885734			0.442867	+	0.896587i	$0.353961\pi$
$938$			6.99571e10i			2.76772i
$939$		−	1.65320e10i		−	0.651623i
$940$			1.15667e9i			0.0454215i
$941$	2.83544e10			1.10932			0.554659	−	0.832078i	$-0.312849\pi$
$941$	2.83544e10			1.10932			0.554659	+	0.832078i	$0.312849\pi$
$942$		−	4.72196e10i		−	1.84054i
$943$			6.90637e9i			0.268200i
$944$			3.50955e9i			0.135784i
$945$			1.26927e10i			0.489265i
$946$	5.00432e10			1.92188
$947$			3.25570e10i			1.24572i	0.782335	+	0.622858i	$0.214029\pi$
$947$			3.25570e10i			1.24572i	−0.782335	+	0.622858i	$0.785971\pi$
$948$		−	8.49481e9i		−	0.323836i
$949$		−	1.89168e10i		−	0.718484i
$950$	2.34909e10			0.888931
$951$	−6.46422e10			−2.43716
$952$	−2.21914e10			−0.833594
$953$	−2.64333e10			−0.989296			−0.494648	−	0.869093i	$-0.664703\pi$
$953$	−2.64333e10			−0.989296			−0.494648	+	0.869093i	$0.664703\pi$
$954$		−	4.12154e9i		−	0.153688i
$955$	−8.03001e9			−0.298335
$956$			6.04129e9i			0.223628i
$957$			1.22486e11i			4.51748i
$958$	2.86530e10			1.05291
$959$	−7.00850e8			−0.0256602
$960$			1.09152e10i			0.398183i
$961$	2.70101e10			0.981733
$962$	2.17987e10	−	4.95278e10i	0.789437	−	1.79364i
$963$	8.72975e9			0.315000
$964$			2.53273e9i			0.0910584i
$965$	8.29088e9			0.296999
$966$	4.92621e10			1.75830
$967$		−	3.71447e10i		−	1.32101i	−0.750824	−	0.660503i	$-0.770343\pi$
$967$		−	3.71447e10i		−	1.32101i	0.750824	−	0.660503i	$-0.229657\pi$
$968$		−	2.94394e10i		−	1.04319i
$969$	−2.94176e10			−1.03866
$970$			1.15750e10i			0.407211i
$971$	−5.49049e9			−0.192461			−0.0962307	−	0.995359i	$-0.530679\pi$
$971$	−5.49049e9			−0.192461			−0.0962307	+	0.995359i	$0.530679\pi$
$972$	5.28486e9			0.184587
$973$	6.16999e10			2.14728
$974$	−2.73749e10			−0.949283
$975$		−	7.49481e10i		−	2.58967i
$976$			2.67625e10i			0.921407i
$977$		−	2.11964e10i		−	0.727162i	−0.931563	−	0.363581i	$-0.881554\pi$
$977$		−	2.11964e10i		−	0.727162i	0.931563	−	0.363581i	$-0.118446\pi$
$978$	−3.31533e10			−1.13329
$979$		−	9.20987e9i		−	0.313700i
$980$			1.69198e9i			0.0574254i
$981$			2.60411e10i			0.880680i
$982$		−	2.72077e10i		−	0.916857i
$983$	−4.30724e10			−1.44631			−0.723156	−	0.690685i	$-0.757309\pi$
$983$	−4.30724e10			−1.44631			−0.723156	+	0.690685i	$0.757309\pi$
$984$			1.58657e10i			0.530857i
$985$			1.11143e10i			0.370556i
$986$			4.30251e10i			1.42940i
$987$	−4.57119e10			−1.51328
$988$	−9.88199e9			−0.325983
$989$	2.60561e10			0.856491
$990$	−2.74577e10			−0.899375
$991$		−	5.79478e9i		−	0.189138i	−0.995518	−	0.0945691i	$-0.969853\pi$
$991$		−	5.79478e9i		−	0.189138i	0.995518	−	0.0945691i	$-0.0301473\pi$
$992$	−1.63263e9			−0.0531004
$993$		−	8.49895e10i		−	2.75450i
$994$		−	1.70707e10i		−	0.551316i
$995$	5.83153e9			0.187673
$996$	7.99918e8			0.0256530
$997$			6.79699e9i			0.217212i	0.994085	+	0.108606i	$0.0346387\pi$
$997$			6.79699e9i			0.217212i	−0.994085	+	0.108606i	$0.965361\pi$
$998$	−6.61439e10			−2.10636
$999$	−1.35940e10	+	3.08864e10i	−0.431389	+	0.980140i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.b.a.36.5	✓	20
37.36	even	2	inner	37.8.b.a.36.16	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.b.a.36.5	✓	20	1.1	even	1	trivial
37.8.b.a.36.16	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)

\(f(q)\)	\(=\)	\(q-12.3896i q^{2} +76.2286 q^{3} -25.5018 q^{4} -93.6237i q^{5} -944.440i q^{6} -1237.82 q^{7} -1269.91i q^{8} +3623.79 q^{9} +O(q^{10})\)	\(q-12.3896i q^{2} +76.2286 q^{3} -25.5018 q^{4} -93.6237i q^{5} -944.440i q^{6} -1237.82 q^{7} -1269.91i q^{8} +3623.79 q^{9} -1159.96 q^{10} +6532.18 q^{11} -1943.96 q^{12} -14175.4i q^{13} +15336.1i q^{14} -7136.80i q^{15} -18997.9 q^{16} +14117.3i q^{17} -44897.3i q^{18} +27336.1i q^{19} +2387.57i q^{20} -94357.4 q^{21} -80931.0i q^{22} -42138.6i q^{23} -96803.4i q^{24} +69359.6 q^{25} -175628. q^{26} +109525. q^{27} +31566.7 q^{28} +245987. i q^{29} -88421.9 q^{30} +22417.9i q^{31} +72827.4i q^{32} +497939. q^{33} +174908. q^{34} +115889. i q^{35} -92413.1 q^{36} +(-124119. + 282004. i) q^{37} +338683. q^{38} -1.08057e6i q^{39} -118894. q^{40} -163897. q^{41} +1.16905e6i q^{42} +618344. i q^{43} -166582. q^{44} -339273. i q^{45} -522079. q^{46} +484454. q^{47} -1.44818e6 q^{48} +708662. q^{49} -859337. i q^{50} +1.07614e6i q^{51} +361499. i q^{52} +91799.4 q^{53} -1.35696e6i q^{54} -611567. i q^{55} +1.57192e6i q^{56} +2.08379e6i q^{57} +3.04768e6 q^{58} -184734. i q^{59} +182001. i q^{60} -1.40871e6i q^{61} +277748. q^{62} -4.48561e6 q^{63} -1.52943e6 q^{64} -1.32716e6 q^{65} -6.16925e6i q^{66} +4.56160e6 q^{67} -360017. i q^{68} -3.21216e6i q^{69} +1.43582e6 q^{70} -1.11311e6 q^{71} -4.60189e6i q^{72} +1.33448e6 q^{73} +(3.49392e6 + 1.53778e6i) q^{74} +5.28718e6 q^{75} -697120. i q^{76} -8.08568e6 q^{77} -1.33879e7 q^{78} +4.36984e6i q^{79} +1.77865e6i q^{80} +423666. q^{81} +2.03061e6i q^{82} -411488. q^{83} +2.40628e6 q^{84} +1.32172e6 q^{85} +7.66102e6 q^{86} +1.87512e7i q^{87} -8.29529e6i q^{88} -1.40992e6i q^{89} -4.20345e6 q^{90} +1.75467e7i q^{91} +1.07461e6i q^{92} +1.70888e6i q^{93} -6.00219e6i q^{94} +2.55931e6 q^{95} +5.55153e6i q^{96} -9.97879e6i q^{97} -8.78002e6i q^{98} +2.36713e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9}+O(q^{10}) \) 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

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