LMFDB - Embedded newform 38.7.b.a.37.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,7,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	38.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:	$8.74205517755$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 5050x^{8} + 7354489x^{6} + 2475755792x^{4} + 232626987584x^{2} + 2900002611200 $ x^10 + 5050x^8 + 7354489x^6 + 2475755792x^4 + 232626987584x^2 + 2900002611200
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.4
Root			$11.5377i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.7.b.a.37.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.65685i q^{2} +14.3661i q^{3} -32.0000 q^{4} -88.1981 q^{5} +81.2670 q^{6} +443.109 q^{7} +181.019i q^{8} +522.615 q^{9} +O(q^{10})$	\(q-5.65685i q^{2} +14.3661i q^{3} -32.0000 q^{4} -88.1981 q^{5} +81.2670 q^{6} +443.109 q^{7} +181.019i q^{8} +522.615 q^{9} +498.924i q^{10} +843.338 q^{11} -459.716i q^{12} +2248.44i q^{13} -2506.60i q^{14} -1267.06i q^{15} +1024.00 q^{16} +8662.52 q^{17} -2956.36i q^{18} +(993.652 - 6786.64i) q^{19} +2822.34 q^{20} +6365.75i q^{21} -4770.64i q^{22} -4047.37 q^{23} -2600.54 q^{24} -7846.10 q^{25} +12719.1 q^{26} +17980.8i q^{27} -14179.5 q^{28} +12069.5i q^{29} -7167.60 q^{30} +22921.6i q^{31} -5792.62i q^{32} +12115.5i q^{33} -49002.6i q^{34} -39081.3 q^{35} -16723.7 q^{36} +71411.8i q^{37} +(-38391.1 - 5620.95i) q^{38} -32301.4 q^{39} -15965.6i q^{40} -134921. i q^{41} +36010.1 q^{42} +28498.1 q^{43} -26986.8 q^{44} -46093.6 q^{45} +22895.4i q^{46} +40848.6 q^{47} +14710.9i q^{48} +78696.4 q^{49} +44384.2i q^{50} +124447. i q^{51} -71950.2i q^{52} -153651. i q^{53} +101715. q^{54} -74380.8 q^{55} +80211.3i q^{56} +(97497.7 + 14274.9i) q^{57} +68275.4 q^{58} +187198. i q^{59} +40546.0i q^{60} +162214. q^{61} +129664. q^{62} +231575. q^{63} -32768.0 q^{64} -198308. i q^{65} +68535.6 q^{66} -51055.2i q^{67} -277201. q^{68} -58145.0i q^{69} +221077. i q^{70} -485892. i q^{71} +94603.4i q^{72} -705653. q^{73} +403966. q^{74} -112718. i q^{75} +(-31796.9 + 217173. i) q^{76} +373690. q^{77} +182724. i q^{78} -57963.0i q^{79} -90314.8 q^{80} +122671. q^{81} -763230. q^{82} -174244. q^{83} -203704. i q^{84} -764018. q^{85} -161209. i q^{86} -173392. q^{87} +152660. i q^{88} -446408. i q^{89} +260745. i q^{90} +996306. i q^{91} +129516. q^{92} -329295. q^{93} -231075. i q^{94} +(-87638.2 + 598569. i) q^{95} +83217.4 q^{96} -503915. i q^{97} -445174. i q^{98} +440741. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 320 q^{4} - 112 q^{5} + 160 q^{6} - 224 q^{7} - 2890 q^{9}+O(q^{10}) $ 10 * q - 320 * q^4 - 112 * q^5 + 160 * q^6 - 224 * q^7 - 2890 * q^9	$ 10 q - 320 q^{4} - 112 q^{5} + 160 q^{6} - 224 q^{7} - 2890 q^{9} + 3644 q^{11} + 10240 q^{16} - 10420 q^{17} - 17230 q^{19} + 3584 q^{20} + 37712 q^{23} - 5120 q^{24} - 52078 q^{25} - 7104 q^{26} + 7168 q^{28} + 94688 q^{30} - 161720 q^{35} + 92480 q^{36} + 25152 q^{38} - 78876 q^{39} + 53792 q^{42} + 6308 q^{43} - 116608 q^{44} + 309808 q^{45} + 322220 q^{47} - 235770 q^{49} - 321728 q^{54} - 377880 q^{55} + 24228 q^{57} + 445920 q^{58} + 426304 q^{61} + 59424 q^{62} - 517916 q^{63} - 327680 q^{64} - 1417312 q^{66} + 333440 q^{68} - 786076 q^{73} - 293280 q^{74} + 551360 q^{76} + 2303716 q^{77} - 114688 q^{80} + 5261090 q^{81} - 455136 q^{82} - 101500 q^{83} - 1261380 q^{85} - 2460732 q^{87} - 1206784 q^{92} - 2827032 q^{93} + 3106292 q^{95} + 163840 q^{96} + 1061428 q^{99}+O(q^{100}) $ 10 * q - 320 * q^4 - 112 * q^5 + 160 * q^6 - 224 * q^7 - 2890 * q^9 + 3644 * q^11 + 10240 * q^16 - 10420 * q^17 - 17230 * q^19 + 3584 * q^20 + 37712 * q^23 - 5120 * q^24 - 52078 * q^25 - 7104 * q^26 + 7168 * q^28 + 94688 * q^30 - 161720 * q^35 + 92480 * q^36 + 25152 * q^38 - 78876 * q^39 + 53792 * q^42 + 6308 * q^43 - 116608 * q^44 + 309808 * q^45 + 322220 * q^47 - 235770 * q^49 - 321728 * q^54 - 377880 * q^55 + 24228 * q^57 + 445920 * q^58 + 426304 * q^61 + 59424 * q^62 - 517916 * q^63 - 327680 * q^64 - 1417312 * q^66 + 333440 * q^68 - 786076 * q^73 - 293280 * q^74 + 551360 * q^76 + 2303716 * q^77 - 114688 * q^80 + 5261090 * q^81 - 455136 * q^82 - 101500 * q^83 - 1261380 * q^85 - 2460732 * q^87 - 1206784 * q^92 - 2827032 * q^93 + 3106292 * q^95 + 163840 * q^96 + 1061428 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$.

$n$	$21$
$\chi(n)$	$-1$

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 $

$2$		−	5.65685i		−	0.707107i
$2$		−	5.65685i		−	0.707107i
$3$			14.3661i			0.532078i	0.963962	+	0.266039i	$0.0857151\pi$
$3$			14.3661i			0.532078i	−0.963962	+	0.266039i	$0.914285\pi$
$4$	−32.0000			−0.500000
$5$	−88.1981			−0.705585			−0.352792	−	0.935702i	$-0.614768\pi$
$5$	−88.1981			−0.705585			−0.352792	+	0.935702i	$0.614768\pi$
$6$	81.2670			0.376236
$7$	443.109			1.29186			0.645931	−	0.763396i	$-0.276469\pi$
$7$	443.109			1.29186			0.645931	+	0.763396i	$0.276469\pi$
$8$			181.019i			0.353553i
$9$	522.615			0.716893
$10$			498.924i			0.498924i
$11$	843.338			0.633612			0.316806	−	0.948490i	$-0.397390\pi$
$11$	843.338			0.633612			0.316806	+	0.948490i	$0.397390\pi$
$12$		−	459.716i		−	0.266039i
$13$			2248.44i			1.02342i	0.859159	+	0.511708i	$0.170987\pi$
$13$			2248.44i			1.02342i	−0.859159	+	0.511708i	$0.829013\pi$
$14$		−	2506.60i		−	0.913485i
$15$		−	1267.06i		−	0.375426i
$16$	1024.00			0.250000
$17$	8662.52			1.76318			0.881592	−	0.472012i	$-0.156472\pi$
$17$	8662.52			1.76318			0.881592	+	0.472012i	$0.156472\pi$
$18$		−	2956.36i		−	0.506920i
$19$	993.652	−	6786.64i	0.144868	−	0.989451i
$19$	993.652	−	6786.64i	0.144868	−	0.989451i
$20$	2822.34			0.352792
$21$			6365.75i			0.687372i
$22$		−	4770.64i		−	0.448032i
$23$	−4047.37			−0.332651			−0.166326	−	0.986071i	$-0.553190\pi$
$23$	−4047.37			−0.332651			−0.166326	+	0.986071i	$0.553190\pi$
$24$	−2600.54			−0.188118
$25$	−7846.10			−0.502150
$26$	12719.1			0.723664
$27$			17980.8i			0.913521i
$28$	−14179.5			−0.645931
$29$			12069.5i			0.494875i	0.968904	+	0.247437i	$0.0795884\pi$
$29$			12069.5i			0.494875i	−0.968904	+	0.247437i	$0.920412\pi$
$30$	−7167.60			−0.265466
$31$			22921.6i			0.769415i	0.923039	+	0.384708i	$0.125698\pi$
$31$			22921.6i			0.769415i	−0.923039	+	0.384708i	$0.874302\pi$
$32$		−	5792.62i		−	0.176777i
$33$			12115.5i			0.337131i
$34$		−	49002.6i		−	1.24676i
$35$	−39081.3			−0.911518
$36$	−16723.7			−0.358446
$37$			71411.8i			1.40982i	0.709294	+	0.704912i	$0.249014\pi$
$37$			71411.8i			1.40982i	−0.709294	+	0.704912i	$0.750986\pi$
$38$	−38391.1	−	5620.95i	−0.699647	−	0.102437i
$39$	−32301.4			−0.544537
$40$		−	15965.6i		−	0.249462i
$41$		−	134921.i		−	1.95762i	−0.204766	−	0.978811i	$-0.565643\pi$
$41$		−	134921.i		−	1.95762i	0.204766	−	0.978811i	$-0.434357\pi$
$42$	36010.1			0.486045
$43$	28498.1			0.358435			0.179217	−	0.983810i	$-0.442643\pi$
$43$	28498.1			0.358435			0.179217	+	0.983810i	$0.442643\pi$
$44$	−26986.8			−0.316806
$45$	−46093.6			−0.505828
$46$			22895.4i			0.235220i
$47$	40848.6			0.393445			0.196723	−	0.980459i	$-0.436970\pi$
$47$	40848.6			0.393445			0.196723	+	0.980459i	$0.436970\pi$
$48$			14710.9i			0.133020i
$49$	78696.4			0.668908
$50$			44384.2i			0.355074i
$51$			124447.i			0.938152i
$52$		−	71950.2i		−	0.511708i
$53$		−	153651.i		−	1.03207i	−0.856568	−	0.516034i	$-0.827408\pi$
$53$		−	153651.i		−	1.03207i	0.856568	−	0.516034i	$-0.172592\pi$
$54$	101715.			0.645957
$55$	−74380.8			−0.447067
$56$			80211.3i			0.456742i
$57$	97497.7	+	14274.9i	0.526465	+	0.0770814i
$58$	68275.4			0.349929
$59$			187198.i			0.911477i	0.890114	+	0.455739i	$0.150625\pi$
$59$			187198.i			0.911477i	−0.890114	+	0.455739i	$0.849375\pi$
$60$			40546.0i			0.187713i
$61$	162214.			0.714658			0.357329	−	0.933979i	$-0.383688\pi$
$61$	162214.			0.714658			0.357329	+	0.933979i	$0.383688\pi$
$62$	129664.			0.544059
$63$	231575.			0.926127
$64$	−32768.0			−0.125000
$65$		−	198308.i		−	0.722106i
$66$	68535.6			0.238388
$67$		−	51055.2i		−	0.169752i	−0.996392	−	0.0848761i	$-0.972951\pi$
$67$		−	51055.2i		−	0.169752i	0.996392	−	0.0848761i	$-0.0270494\pi$
$68$	−277201.			−0.881592
$69$		−	58145.0i		−	0.176997i
$70$			221077.i			0.644541i
$71$		−	485892.i		−	1.35758i	−0.734334	−	0.678788i	$-0.762505\pi$
$71$		−	485892.i		−	1.35758i	0.734334	−	0.678788i	$-0.237495\pi$
$72$			94603.4i			0.253460i
$73$	−705653.			−1.81394			−0.906969	−	0.421197i	$-0.861610\pi$
$73$	−705653.			−1.81394			−0.906969	+	0.421197i	$0.861610\pi$
$74$	403966.			0.996897
$75$		−	112718.i		−	0.267183i
$76$	−31796.9	+	217173.i	−0.0724342	+	0.494725i
$77$	373690.			0.818540
$78$			182724.i			0.385046i
$79$		−	57963.0i		−	0.117563i	−0.998271	−	0.0587814i	$-0.981279\pi$
$79$		−	57963.0i		−	0.117563i	0.998271	−	0.0587814i	$-0.0187215\pi$
$80$	−90314.8			−0.176396
$81$	122671.			0.230828
$82$	−763230.			−1.38425
$83$	−174244.			−0.304735			−0.152368	−	0.988324i	$-0.548690\pi$
$83$	−174244.			−0.304735			−0.152368	+	0.988324i	$0.548690\pi$
$84$		−	203704.i		−	0.343686i
$85$	−764018.			−1.24408
$86$		−	161209.i		−	0.253452i
$87$	−173392.			−0.263312
$88$			152660.i			0.224016i
$89$		−	446408.i		−	0.633231i	−0.948554	−	0.316615i	$-0.897454\pi$
$89$		−	446408.i		−	0.633231i	0.948554	−	0.316615i	$-0.102546\pi$
$90$			260745.i			0.357675i
$91$			996306.i			1.32211i
$92$	129516.			0.166326
$93$	−329295.			−0.409389
$94$		−	231075.i		−	0.278208i
$95$	−87638.2	+	598569.i	−0.102217	+	0.698141i
$96$	83217.4			0.0940591
$97$		−	503915.i		−	0.552131i	−0.961139	−	0.276066i	$-0.910969\pi$
$97$		−	503915.i		−	0.552131i	0.961139	−	0.276066i	$-0.0890307\pi$
$98$		−	445174.i		−	0.472990i
$99$	440741.			0.454232
$100$	251075.			0.251075
$101$	274413.			0.266342			0.133171	−	0.991093i	$-0.457484\pi$
$101$	274413.			0.266342			0.133171	+	0.991093i	$0.457484\pi$
$102$	703977.			0.663374
$103$			372124.i			0.340546i	0.985397	+	0.170273i	$0.0544650\pi$
$103$			372124.i			0.340546i	−0.985397	+	0.170273i	$0.945535\pi$
$104$	−407012.			−0.361832
$105$		−	561447.i		−	0.484999i
$106$	−869182.			−0.729782
$107$			2.01674e6i			1.64626i	0.567850	+	0.823132i	$0.307775\pi$
$107$			2.01674e6i			1.64626i	−0.567850	+	0.823132i	$0.692225\pi$
$108$		−	575387.i		−	0.456761i
$109$			60863.7i			0.0469979i	0.999724	+	0.0234990i	$0.00748064\pi$
$109$			60863.7i			0.0469979i	−0.999724	+	0.0234990i	$0.992519\pi$
$110$			420761.i			0.316124i
$111$	−1.02591e6			−0.750137
$112$	453743.			0.322966
$113$		−	1.63566e6i		−	1.13360i	−0.823857	−	0.566798i	$-0.808182\pi$
$113$		−	1.63566e6i		−	1.13360i	0.823857	−	0.566798i	$-0.191818\pi$
$114$	80751.2	−	551530.i	0.0545047	−	0.372267i
$115$	356970.			0.234714
$116$		−	386224.i		−	0.247437i
$117$			1.17507e6i			0.733679i
$118$	1.05895e6			0.644512
$119$	3.83844e6			2.27779
$120$	229363.			0.132733
$121$	−1.06034e6			−0.598535
$122$		−	917619.i		−	0.505339i
$123$	1.93829e6			1.04161
$124$		−	733493.i		−	0.384708i
$125$	2.07011e6			1.05989
$126$		−	1.30999e6i		−	0.654870i
$127$			1.92623e6i			0.940368i	0.882568	+	0.470184i	$0.155812\pi$
$127$			1.92623e6i			0.940368i	−0.882568	+	0.470184i	$0.844188\pi$
$128$			185364.i			0.0883883i
$129$			409407.i			0.190715i
$130$	−1.12180e6			−0.510606
$131$	−1.95713e6			−0.870572			−0.435286	−	0.900292i	$-0.643353\pi$
$131$	−1.95713e6			−0.870572			−0.435286	+	0.900292i	$0.643353\pi$
$132$		−	387696.i		−	0.168566i
$133$	440296.	−	3.00722e6i	0.187150	−	1.27823i
$134$	−288812.			−0.120033
$135$		−	1.58588e6i		−	0.644567i
$136$			1.56808e6i			0.623380i
$137$	897719.			0.349123			0.174562	−	0.984646i	$-0.444149\pi$
$137$	897719.			0.349123			0.174562	+	0.984646i	$0.444149\pi$
$138$	−328918.			−0.125155
$139$	−3.75579e6			−1.39848			−0.699240	−	0.714887i	$-0.746478\pi$
$139$	−3.75579e6			−1.39848			−0.699240	+	0.714887i	$0.746478\pi$
$140$	1.25060e6			0.455759
$141$			586836.i			0.209344i
$142$	−2.74862e6			−0.959952
$143$			1.89620e6i			0.648449i
$144$	535157.			0.179223
$145$		−	1.06451e6i		−	0.349176i
$146$			3.99178e6i			1.28265i
$147$			1.13056e6i			0.355912i
$148$		−	2.28518e6i		−	0.704912i
$149$	−4.44414e6			−1.34347			−0.671736	−	0.740790i	$-0.734451\pi$
$149$	−4.44414e6			−1.34347			−0.671736	+	0.740790i	$0.734451\pi$
$150$	−637629.			−0.188927
$151$			4.71415e6i			1.36922i	0.728910	+	0.684610i	$0.240027\pi$
$151$			4.71415e6i			1.36922i	−0.728910	+	0.684610i	$0.759973\pi$
$152$	1.22851e6	+	179870.i	0.349824	+	0.0512187i
$153$	4.52716e6			1.26401
$154$		−	2.11391e6i		−	0.578795i
$155$		−	2.02164e6i		−	0.542887i
$156$	1.03365e6			0.272269
$157$	−2.23912e6			−0.578599			−0.289300	−	0.957239i	$-0.593422\pi$
$157$	−2.23912e6			−0.578599			−0.289300	+	0.957239i	$0.593422\pi$
$158$	−327888.			−0.0831294
$159$	2.20737e6			0.549141
$160$			510898.i			0.124731i
$161$	−1.79342e6			−0.429740
$162$		−	693933.i		−	0.163220i
$163$	−6.99610e6			−1.61545			−0.807725	−	0.589560i	$-0.799301\pi$
$163$	−6.99610e6			−1.61545			−0.807725	+	0.589560i	$0.799301\pi$
$164$			4.31748e6i			0.978811i
$165$		−	1.06856e6i		−	0.237875i
$166$			985671.i			0.215480i
$167$			1.64432e6i			0.353050i	0.984296	+	0.176525i	$0.0564857\pi$
$167$			1.64432e6i			0.353050i	−0.984296	+	0.176525i	$0.943514\pi$
$168$	−1.15232e6			−0.243023
$169$	−228694.			−0.0473801
$170$			4.32194e6i			0.879694i
$171$	519297.	−	3.54680e6i	0.103855	−	0.709330i
$172$	−911938.			−0.179217
$173$		−	7.04201e6i		−	1.36006i	−0.733184	−	0.680031i	$-0.761966\pi$
$173$		−	7.04201e6i		−	1.36006i	0.733184	−	0.680031i	$-0.238034\pi$
$174$			980852.i			0.186190i
$175$	−3.47668e6			−0.648709
$176$	863578.			0.158403
$177$	−2.68931e6			−0.484977
$178$	−2.52527e6			−0.447762
$179$		−	2.49268e6i		−	0.434618i	−0.976103	−	0.217309i	$-0.930272\pi$
$179$		−	2.49268e6i		−	0.434618i	0.976103	−	0.217309i	$-0.0697279\pi$
$180$	1.47500e6			0.252914
$181$		−	8.52372e6i		−	1.43745i	−0.695293	−	0.718726i	$-0.744725\pi$
$181$		−	8.52372e6i		−	1.43745i	0.695293	−	0.718726i	$-0.255275\pi$
$182$	5.63596e6			0.934875
$183$			2.33038e6i			0.380254i
$184$		−	732652.i		−	0.117610i
$185$		−	6.29839e6i		−	0.994751i
$186$			1.86277e6i			0.289482i
$187$	7.30543e6			1.11718
$188$	−1.30716e6			−0.196723
$189$			7.96747e6i			1.18014i
$190$	3.38602e6	+	495757.i	0.493660	+	0.0722783i
$191$	7.28172e6			1.04504			0.522521	−	0.852626i	$-0.324992\pi$
$191$	7.28172e6			1.04504			0.522521	+	0.852626i	$0.324992\pi$
$192$		−	470749.i		−	0.0665098i
$193$		−	881891.i		−	0.122671i	−0.998117	−	0.0613357i	$-0.980464\pi$
$193$		−	881891.i		−	0.122671i	0.998117	−	0.0613357i	$-0.0195360\pi$
$194$	−2.85057e6			−0.390416
$195$	2.84892e6			0.384217
$196$	−2.51828e6			−0.334454
$197$	1.06316e7			1.39059			0.695297	−	0.718723i	$-0.255273\pi$
$197$	1.06316e7			1.39059			0.695297	+	0.718723i	$0.255273\pi$
$198$		−	2.49321e6i		−	0.321191i
$199$	−227906.			−0.0289199			−0.0144600	−	0.999895i	$-0.504603\pi$
$199$	−227906.			−0.0289199			−0.0144600	+	0.999895i	$0.504603\pi$
$200$		−	1.42030e6i		−	0.177537i
$201$	733464.			0.0903214
$202$		−	1.55231e6i		−	0.188332i
$203$			5.34810e6i			0.639310i
$204$		−	3.98230e6i		−	0.469076i
$205$			1.18998e7i			1.38127i
$206$	2.10505e6			0.240803
$207$	−2.11521e6			−0.238475
$208$			2.30241e6i			0.255854i
$209$	837985.	−	5.72344e6i	0.0917904	−	0.626928i
$210$	−3.17602e6			−0.342946
$211$			4.71940e6i			0.502389i	0.967937	+	0.251194i	$0.0808233\pi$
$211$			4.71940e6i			0.502389i	−0.967937	+	0.251194i	$0.919177\pi$
$212$			4.91684e6i			0.516034i
$213$	6.98038e6			0.722337
$214$	1.14084e7			1.16408
$215$	−2.51347e6			−0.252906
$216$	−3.25488e6			−0.322979
$217$			1.01568e7i			0.993978i
$218$	344297.			0.0332325
$219$		−	1.01375e7i		−	0.965157i
$220$	2.38019e6			0.223534
$221$			1.94772e7i			1.80447i
$222$			5.80343e6i			0.530427i
$223$		−	2.20494e7i		−	1.98830i	−0.108021	−	0.994149i	$-0.534451\pi$
$223$		−	2.20494e7i		−	1.98830i	0.108021	−	0.994149i	$-0.465549\pi$
$224$		−	2.56676e6i		−	0.228371i
$225$	−4.10049e6			−0.359988
$226$	−9.25270e6			−0.801573
$227$		−	1.32001e7i		−	1.12849i	−0.825606	−	0.564246i	$-0.809167\pi$
$227$		−	1.32001e7i		−	1.12849i	0.825606	−	0.564246i	$-0.190833\pi$
$228$	−3.11993e6	−	456798.i	−0.263233	−	0.0385407i
$229$	−1.05934e7			−0.882121			−0.441060	−	0.897477i	$-0.645398\pi$
$229$	−1.05934e7			−0.882121			−0.441060	+	0.897477i	$0.645398\pi$
$230$		−	2.01933e6i		−	0.165968i
$231$			5.36848e6i			0.435527i
$232$	−2.18481e6			−0.174965
$233$	−8.02445e6			−0.634377			−0.317189	−	0.948362i	$-0.602739\pi$
$233$	−8.02445e6			−0.634377			−0.317189	+	0.948362i	$0.602739\pi$
$234$	6.64720e6			0.518790
$235$	−3.60277e6			−0.277609
$236$		−	5.99035e6i		−	0.455739i
$237$	832703.			0.0625526
$238$		−	2.17135e7i		−	1.61064i
$239$	−2.23585e6			−0.163776			−0.0818878	−	0.996642i	$-0.526095\pi$
$239$	−2.23585e6			−0.163776			−0.0818878	+	0.996642i	$0.526095\pi$
$240$		−	1.29747e6i		−	0.0938566i
$241$			1.51033e7i			1.07900i	0.841987	+	0.539498i	$0.181386\pi$
$241$			1.51033e7i			1.07900i	−0.841987	+	0.539498i	$0.818614\pi$
$242$			5.99820e6i			0.423228i
$243$			1.48703e7i			1.03634i
$244$	−5.19084e6			−0.357329
$245$	−6.94087e6			−0.471971
$246$		−	1.09646e7i		−	0.736528i
$247$	1.52594e7	+	2.23417e6i	1.01262	+	0.148261i
$248$	−4.14926e6			−0.272029
$249$		−	2.50320e6i		−	0.162143i
$250$		−	1.17103e7i		−	0.749458i
$251$	−2.60236e7			−1.64568			−0.822841	−	0.568271i	$-0.807612\pi$
$251$	−2.60236e7			−1.64568			−0.822841	+	0.568271i	$0.807612\pi$
$252$	−7.41041e6			−0.463063
$253$	−3.41330e6			−0.210772
$254$	1.08964e7			0.664941
$255$		−	1.09760e7i		−	0.661946i
$256$	1.04858e6			0.0625000
$257$		−	2.37806e7i		−	1.40095i	−0.713675	−	0.700477i	$-0.752971\pi$
$257$		−	2.37806e7i		−	1.40095i	0.713675	−	0.700477i	$-0.247029\pi$
$258$	2.31595e6			0.134856
$259$			3.16432e7i			1.82130i
$260$			6.34587e6i			0.361053i
$261$			6.30770e6i			0.354772i
$262$			1.10712e7i			0.615588i
$263$	1.51982e7			0.835459			0.417729	−	0.908571i	$-0.362826\pi$
$263$	1.51982e7			0.835459			0.417729	+	0.908571i	$0.362826\pi$
$264$	−2.19314e6			−0.119194
$265$			1.35517e7i			0.728211i
$266$	−1.70114e7	−	2.49069e6i	−0.903848	−	0.132335i
$267$	6.41315e6			0.336928
$268$			1.63377e6i			0.0848761i
$269$			3.09109e6i			0.158802i	0.996843	+	0.0794009i	$0.0253007\pi$
$269$			3.09109e6i			0.158802i	−0.996843	+	0.0794009i	$0.974699\pi$
$270$	−8.97107e6			−0.455777
$271$	−2.30217e7			−1.15672			−0.578362	−	0.815780i	$-0.696308\pi$
$271$	−2.30217e7			−1.15672			−0.578362	+	0.815780i	$0.696308\pi$
$272$	8.87042e6			0.440796
$273$	−1.43130e7			−0.703467
$274$		−	5.07826e6i		−	0.246867i
$275$	−6.61691e6			−0.318169
$276$			1.86064e6i			0.0884983i
$277$	1.85846e7			0.874409			0.437205	−	0.899362i	$-0.355969\pi$
$277$	1.85846e7			0.874409			0.437205	+	0.899362i	$0.355969\pi$
$278$			2.12459e7i			0.988875i
$279$			1.19792e7i			0.551588i
$280$		−	7.07448e6i		−	0.322270i
$281$			5.09402e6i			0.229584i	0.993390	+	0.114792i	$0.0366202\pi$
$281$			5.09402e6i			0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$282$	3.31965e6			0.148028
$283$	3.57694e7			1.57817			0.789083	−	0.614286i	$-0.210556\pi$
$283$	3.57694e7			1.57817			0.789083	+	0.614286i	$0.210556\pi$
$284$			1.55485e7i			0.678788i
$285$	−8.59911e6	−	1.25902e6i	−0.371466	−	0.0543874i
$286$	1.07265e7			0.458523
$287$		−	5.97848e7i		−	2.52898i
$288$		−	3.02731e6i		−	0.126730i
$289$	5.09017e7			2.10882
$290$	−6.02176e6			−0.246905
$291$	7.23930e6			0.293777
$292$	2.25809e7			0.906969
$293$		−	8.47647e6i		−	0.336986i	−0.985703	−	0.168493i	$-0.946110\pi$
$293$		−	8.47647e6i		−	0.336986i	0.985703	−	0.168493i	$-0.0538901\pi$
$294$	6.39542e6			0.251668
$295$		−	1.65105e7i		−	0.643124i
$296$	−1.29269e7			−0.498448
$297$			1.51639e7i			0.578818i
$298$			2.51399e7i			0.949979i
$299$		−	9.10028e6i		−	0.340441i
$300$			3.60698e6i			0.133592i
$301$	1.26277e7			0.463048
$302$	2.66673e7			0.968184
$303$			3.94225e6i			0.141715i
$304$	1.01750e6	−	6.94952e6i	0.0362171	−	0.247363i
$305$	−1.43069e7			−0.504251
$306$		−	2.56095e7i		−	0.893792i
$307$		−	3.21243e7i		−	1.11024i	−0.831769	−	0.555122i	$-0.812672\pi$
$307$		−	3.21243e7i		−	1.11024i	0.831769	−	0.555122i	$-0.187328\pi$
$308$	−1.19581e7			−0.409270
$309$	−5.34598e6			−0.181197
$310$	−1.14362e7			−0.383879
$311$	−3.20291e7			−1.06479			−0.532395	−	0.846496i	$-0.678708\pi$
$311$	−3.20291e7			−1.06479			−0.532395	+	0.846496i	$0.678708\pi$
$312$		−	5.84718e6i		−	0.192523i
$313$	3.71743e7			1.21230			0.606150	−	0.795350i	$-0.292713\pi$
$313$	3.71743e7			1.21230			0.606150	+	0.795350i	$0.292713\pi$
$314$			1.26664e7i			0.409132i
$315$	−2.04245e7			−0.653461
$316$			1.85482e6i			0.0587814i
$317$			4.65424e7i			1.46107i	0.682875	+	0.730535i	$0.260729\pi$
$317$			4.65424e7i			1.46107i	−0.682875	+	0.730535i	$0.739271\pi$
$318$		−	1.24868e7i		−	0.388301i
$319$			1.01787e7i			0.313559i
$320$	2.89007e6			0.0881981
$321$	−2.89728e7			−0.875941
$322$			1.01451e7i			0.303872i
$323$	8.60754e6	−	5.87895e7i	0.255430	−	1.74458i
$324$	−3.92548e6			−0.115414
$325$		−	1.76415e7i		−	0.513909i
$326$			3.95759e7i			1.14230i
$327$	−874375.			−0.0250066
$328$	2.44234e7			0.692124
$329$	1.81004e7			0.508277
$330$	−6.04471e6			−0.168203
$331$		−	9.73822e6i		−	0.268532i	−0.990945	−	0.134266i	$-0.957132\pi$
$331$		−	9.73822e6i		−	0.268532i	0.990945	−	0.134266i	$-0.0428676\pi$
$332$	5.57580e6			0.152368
$333$			3.73209e7i			1.01069i
$334$	9.30167e6			0.249644
$335$			4.50297e6i			0.119774i
$336$			6.51853e6i			0.171843i
$337$		−	5.43094e7i		−	1.41901i	−0.704701	−	0.709505i	$-0.748919\pi$
$337$		−	5.43094e7i		−	1.41901i	0.704701	−	0.709505i	$-0.251081\pi$
$338$			1.29369e6i			0.0335028i
$339$	2.34981e7			0.603161
$340$	2.44486e7			0.622038
$341$			1.93307e7i			0.487511i
$342$	−2.00637e7	−	2.93759e6i	−0.501572	−	0.0734366i
$343$	−1.72602e7			−0.427725
$344$			5.15870e6i			0.126726i
$345$			5.12827e6i			0.124886i
$346$	−3.98356e7			−0.961709
$347$	−2.27844e7			−0.545318			−0.272659	−	0.962111i	$-0.587903\pi$
$347$	−2.27844e7			−0.545318			−0.272659	+	0.962111i	$0.587903\pi$
$348$	5.54854e6			0.131656
$349$	−4.88429e7			−1.14901			−0.574507	−	0.818500i	$-0.694806\pi$
$349$	−4.88429e7			−1.14901			−0.574507	+	0.818500i	$0.694806\pi$
$350$			1.96670e7i			0.458707i
$351$	−4.04289e7			−0.934912
$352$		−	4.88514e6i		−	0.112008i
$353$	−9.20565e6			−0.209281			−0.104641	−	0.994510i	$-0.533369\pi$
$353$	−9.20565e6			−0.209281			−0.104641	+	0.994510i	$0.533369\pi$
$354$			1.52130e7i			0.342931i
$355$			4.28547e7i			0.957885i
$356$			1.42851e7i			0.316615i
$357$			5.51435e7i			1.21196i
$358$	−1.41007e7			−0.307321
$359$	−4.65298e7			−1.00565			−0.502826	−	0.864388i	$-0.667706\pi$
$359$	−4.65298e7			−1.00565			−0.502826	+	0.864388i	$0.667706\pi$
$360$		−	8.34383e6i		−	0.178837i
$361$	−4.50712e7	−	1.34871e7i	−0.958026	−	0.286680i
$362$	−4.82175e7			−1.01643
$363$		−	1.52330e7i		−	0.318468i
$364$		−	3.18818e7i		−	0.661056i
$365$	6.22372e7			1.27989
$366$	1.31826e7			0.268880
$367$	9.72438e7			1.96727			0.983635	−	0.180173i	$-0.0576658\pi$
$367$	9.72438e7			1.96727			0.983635	+	0.180173i	$0.0576658\pi$
$368$	−4.14450e6			−0.0831628
$369$		−	7.05118e7i		−	1.40340i
$370$	−3.56291e7			−0.703395
$371$		−	6.80842e7i		−	1.33329i
$372$	1.05374e7			0.204695
$373$			5.95500e7i			1.14751i	0.819028	+	0.573753i	$0.194513\pi$
$373$			5.95500e7i			1.14751i	−0.819028	+	0.573753i	$0.805487\pi$
$374$		−	4.13258e7i		−	0.789962i
$375$			2.97394e7i			0.563947i
$376$			7.39439e6i			0.139104i
$377$	−2.71376e7			−0.506463
$378$	4.50708e7			0.834488
$379$			7.91278e7i			1.45349i	0.686909	+	0.726744i	$0.258967\pi$
$379$			7.91278e7i			1.45349i	−0.686909	+	0.726744i	$0.741033\pi$
$380$	2.80442e6	−	1.91542e7i	0.0511085	−	0.349071i
$381$	−2.76725e7			−0.500349
$382$		−	4.11916e7i		−	0.738956i
$383$			6.73426e7i			1.19865i	0.800505	+	0.599327i	$0.204565\pi$
$383$			6.73426e7i			1.19865i	−0.800505	+	0.599327i	$0.795435\pi$
$384$	−2.66296e6			−0.0470295
$385$	−3.29588e7			−0.577549
$386$	−4.98873e6			−0.0867418
$387$	1.48935e7			0.256959
$388$			1.61253e7i			0.276066i
$389$	6.05262e7			1.02824			0.514120	−	0.857718i	$-0.328119\pi$
$389$	6.05262e7			1.02824			0.514120	+	0.857718i	$0.328119\pi$
$390$		−	1.61159e7i		−	0.271683i
$391$	−3.50604e7			−0.586525
$392$			1.42456e7i			0.236495i
$393$		−	2.81163e7i		−	0.463213i
$394$		−	6.01415e7i		−	0.983298i
$395$			5.11222e6i			0.0829504i
$396$	−1.41037e7			−0.227116
$397$	8.45812e7			1.35177			0.675885	−	0.737007i	$-0.263762\pi$
$397$	8.45812e7			1.35177			0.675885	+	0.737007i	$0.263762\pi$
$398$			1.28923e6i			0.0204495i
$399$	4.32021e7	+	6.32535e6i	0.680121	+	0.0995785i
$400$	−8.03441e6			−0.125538
$401$		−	4.02141e7i		−	0.623656i	−0.950139	−	0.311828i	$-0.899059\pi$
$401$		−	4.02141e7i		−	0.623656i	0.950139	−	0.311828i	$-0.100941\pi$
$402$		−	4.14910e6i		−	0.0638669i
$403$	−5.15381e7			−0.787432
$404$	−8.78121e6			−0.133171
$405$	−1.08194e7			−0.162868
$406$	3.02534e7			0.452060
$407$			6.02243e7i			0.893282i
$408$	−2.25273e7			−0.331687
$409$		−	1.25079e8i		−	1.82816i	−0.405529	−	0.914082i	$-0.632913\pi$
$409$		−	1.25079e8i		−	1.82816i	0.405529	−	0.914082i	$-0.367087\pi$
$410$	6.73154e7			0.976704
$411$			1.28967e7i			0.185761i
$412$		−	1.19080e7i		−	0.170273i
$413$			8.29492e7i			1.17750i
$414$			1.19655e7i			0.168627i
$415$	1.53680e7			0.215016
$416$	1.30244e7			0.180916
$417$		−	5.39560e7i		−	0.744101i
$418$	−3.23766e7	−	4.74036e6i	−0.443305	−	0.0649056i
$419$	1.43907e8			1.95631			0.978157	−	0.207867i	$-0.0666523\pi$
$419$	1.43907e8			1.95631			0.978157	+	0.207867i	$0.0666523\pi$
$420$			1.79663e7i			0.242500i
$421$		−	2.37670e6i		−	0.0318514i	−0.999873	−	0.0159257i	$-0.994930\pi$
$421$		−	2.37670e6i		−	0.0318514i	0.999873	−	0.0159257i	$-0.00506952\pi$
$422$	2.66970e7			0.355242
$423$	2.13481e7			0.282058
$424$	2.78138e7			0.364891
$425$	−6.79670e7			−0.885383
$426$		−	3.94870e7i		−	0.510770i
$427$	7.18783e7			0.923239
$428$		−	6.45358e7i		−	0.823132i
$429$	−2.72410e7			−0.345026
$430$			1.42184e7i			0.178832i
$431$			5.20048e7i			0.649548i	0.945792	+	0.324774i	$0.105288\pi$
$431$			5.20048e7i			0.649548i	−0.945792	+	0.324774i	$0.894712\pi$
$432$			1.84124e7i			0.228380i
$433$		−	2.80831e7i		−	0.345925i	−0.984928	−	0.172962i	$-0.944666\pi$
$433$		−	2.80831e7i		−	0.345925i	0.984928	−	0.172962i	$-0.0553339\pi$
$434$	5.74554e7			0.702849
$435$	1.52928e7			0.185789
$436$		−	1.94764e6i		−	0.0234990i
$437$	−4.02168e6	+	2.74680e7i	−0.0481907	+	0.329142i
$438$	−5.73463e7			−0.682469
$439$			1.34570e8i			1.59058i	0.606228	+	0.795291i	$0.292682\pi$
$439$			1.34570e8i			1.59058i	−0.606228	+	0.795291i	$0.707318\pi$
$440$		−	1.34644e7i		−	0.158062i
$441$	4.11279e7			0.479535
$442$	1.10180e8			1.27595
$443$	−8.52003e7			−0.980009			−0.490005	−	0.871720i	$-0.663005\pi$
$443$	−8.52003e7			−0.980009			−0.490005	+	0.871720i	$0.663005\pi$
$444$	3.28292e7			0.375069
$445$			3.93723e7i			0.446798i
$446$	−1.24730e8			−1.40594
$447$		−	6.38450e7i		−	0.714833i
$448$	−1.45198e7			−0.161483
$449$		−	1.98487e6i		−	0.0219277i	−0.999940	−	0.0109639i	$-0.996510\pi$
$449$		−	1.98487e6i		−	0.0219277i	0.999940	−	0.0109639i	$-0.00348998\pi$
$450$			2.31959e7i			0.254550i
$451$		−	1.13784e8i		−	1.24037i
$452$			5.23412e7i			0.566798i
$453$	−6.77241e7			−0.728532
$454$	−7.46709e7			−0.797965
$455$		−	8.78722e7i		−	0.932862i
$456$	−2.58404e6	+	1.76490e7i	−0.0272524	+	0.186134i
$457$	1.17624e8			1.23238			0.616192	−	0.787596i	$-0.288674\pi$
$457$	1.17624e8			1.23238			0.616192	+	0.787596i	$0.288674\pi$
$458$			5.99252e7i			0.623753i
$459$			1.55759e8i			1.61071i
$460$	−1.14230e7			−0.117357
$461$	−9.91086e7			−1.01160			−0.505800	−	0.862651i	$-0.668803\pi$
$461$	−9.91086e7			−1.01160			−0.505800	+	0.862651i	$0.668803\pi$
$462$	3.03687e7			0.307964
$463$	−2.50690e7			−0.252577			−0.126288	−	0.991994i	$-0.540306\pi$
$463$	−2.50690e7			−0.252577			−0.126288	+	0.991994i	$0.540306\pi$
$464$			1.23592e7i			0.123719i
$465$	2.90432e7			0.288859
$466$			4.53932e7i			0.448572i
$467$	−9.78891e7			−0.961133			−0.480567	−	0.876958i	$-0.659569\pi$
$467$	−9.78891e7			−0.961133			−0.480567	+	0.876958i	$0.659569\pi$
$468$		−	3.76022e7i		−	0.366840i
$469$		−	2.26230e7i		−	0.219296i
$470$			2.03804e7i			0.196299i
$471$		−	3.21674e7i		−	0.307860i
$472$	−3.38865e7			−0.322256
$473$	2.40335e7			0.227109
$474$		−	4.71048e6i		−	0.0442314i
$475$	−7.79630e6	+	5.32487e7i	−0.0727457	+	0.496853i
$476$	−1.22830e8			−1.13890
$477$		−	8.03003e7i		−	0.739882i
$478$			1.26479e7i			0.115807i
$479$	−960933.			−0.00874352			−0.00437176	−	0.999990i	$-0.501392\pi$
$479$	−960933.			−0.00874352			−0.00437176	+	0.999990i	$0.501392\pi$
$480$	−7.33962e6			−0.0663666
$481$	−1.60566e8			−1.44284
$482$	8.54370e7			0.762965
$483$		−	2.57645e7i		−	0.228655i
$484$	3.39309e7			0.299268
$485$			4.44443e7i			0.389575i
$486$	8.41194e7			0.732803
$487$		−	1.11694e8i		−	0.967039i	−0.875334	−	0.483520i	$-0.839358\pi$
$487$		−	1.11694e8i		−	0.967039i	0.875334	−	0.483520i	$-0.160642\pi$
$488$			2.93638e7i			0.252670i
$489$		−	1.00507e8i		−	0.859546i
$490$			3.92635e7i			0.333734i
$491$	1.22509e8			1.03496			0.517481	−	0.855694i	$-0.326870\pi$
$491$	1.22509e8			1.03496			0.517481	+	0.855694i	$0.326870\pi$
$492$	−6.20254e7			−0.520804
$493$			1.04552e8i			0.872555i
$494$	1.26384e7	−	8.63202e7i	0.104836	−	0.716030i
$495$	−3.88725e7			−0.320499
$496$			2.34718e7i			0.192354i
$497$		−	2.15303e8i		−	1.75380i
$498$	−1.41603e7			−0.114652
$499$	−1.28023e8			−1.03035			−0.515176	−	0.857084i	$-0.672274\pi$
$499$	−1.28023e8			−1.03035			−0.515176	+	0.857084i	$0.672274\pi$
$500$	−6.62434e7			−0.529947
$501$	−2.36225e7			−0.187850
$502$			1.47212e8i			1.16367i
$503$	6.43740e7			0.505832			0.252916	−	0.967488i	$-0.418610\pi$
$503$	6.43740e7			0.505832			0.252916	+	0.967488i	$0.418610\pi$
$504$			4.19196e7i			0.327435i
$505$	−2.42027e7			−0.187927
$506$			1.93085e7i			0.149038i
$507$		−	3.28545e6i		−	0.0252099i
$508$		−	6.16395e7i		−	0.470184i
$509$		−	9.29852e7i		−	0.705116i	−0.935790	−	0.352558i	$-0.885312\pi$
$509$		−	9.29852e7i		−	0.705116i	0.935790	−	0.352558i	$-0.114688\pi$
$510$	−6.20895e7			−0.468066
$511$	−3.12681e8			−2.34336
$512$		−	5.93164e6i		−	0.0441942i
$513$	1.22030e8	+	1.78667e7i	0.903885	+	0.132340i
$514$	−1.34524e8			−0.990624
$515$		−	3.28206e7i		−	0.240284i
$516$		−	1.31010e7i		−	0.0953577i
$517$	3.44492e7			0.249292
$518$	1.79001e8			1.28785
$519$	1.01166e8			0.723659
$520$	3.58977e7			0.255303
$521$		−	7.12432e7i		−	0.503767i	−0.967758	−	0.251884i	$-0.918950\pi$
$521$		−	7.12432e7i		−	0.503767i	0.967758	−	0.251884i	$-0.0810500\pi$
$522$	3.56817e7			0.250862
$523$			1.96941e8i			1.37668i	0.725390	+	0.688338i	$0.241659\pi$
$523$			1.96941e8i			1.37668i	−0.725390	+	0.688338i	$0.758341\pi$
$524$	6.26280e7			0.435286
$525$		−	4.99463e7i		−	0.345164i
$526$		−	8.59740e7i		−	0.590759i
$527$			1.98559e8i			1.35662i
$528$			1.24063e7i			0.0842829i
$529$	−1.31655e8			−0.889343
$530$	7.66602e7			0.514923
$531$			9.78326e7i			0.653431i
$532$	−1.40895e7	+	9.62311e7i	−0.0935750	+	0.639117i
$533$	3.03363e8			2.00346
$534$		−	3.62783e7i		−	0.238244i
$535$		−	1.77873e8i		−	1.16158i
$536$	9.24197e6			0.0600164
$537$	3.58101e7			0.231251
$538$	1.74859e7			0.112290
$539$	6.63677e7			0.423829
$540$			5.07480e7i			0.322283i
$541$	−1.91247e8			−1.20782			−0.603909	−	0.797053i	$-0.706391\pi$
$541$	−1.91247e8			−1.20782			−0.603909	+	0.797053i	$0.706391\pi$
$542$			1.30231e8i			0.817928i
$543$	1.22453e8			0.764838
$544$		−	5.01787e7i		−	0.311690i
$545$		−	5.36806e6i		−	0.0331610i
$546$			8.09668e7i			0.497427i
$547$			1.37101e7i			0.0837682i	0.999122	+	0.0418841i	$0.0133360\pi$
$547$			1.37101e7i			0.0837682i	−0.999122	+	0.0418841i	$0.986664\pi$
$548$	−2.87270e7			−0.174562
$549$	8.47753e7			0.512333
$550$			3.74309e7i			0.224979i
$551$	8.19114e7	+	1.19929e7i	0.489654	+	0.0716917i
$552$	1.05254e7			0.0625777
$553$		−	2.56839e7i		−	0.151875i
$554$		−	1.05131e8i		−	0.618301i
$555$	9.04834e7			0.529285
$556$	1.20185e8			0.699240
$557$	2.61178e7			0.151137			0.0755686	−	0.997141i	$-0.475923\pi$
$557$	2.61178e7			0.151137			0.0755686	+	0.997141i	$0.475923\pi$
$558$	6.77645e7			0.390032
$559$			6.40763e7i			0.366828i
$560$	−4.00193e7			−0.227880
$561$			1.04951e8i			0.594425i
$562$	2.88161e7			0.162341
$563$			8.77820e6i			0.0491904i	0.999697	+	0.0245952i	$0.00782968\pi$
$563$			8.77820e6i			0.0491904i	−0.999697	+	0.0245952i	$0.992170\pi$
$564$		−	1.87788e7i		−	0.104672i
$565$			1.44262e8i			0.799847i
$566$		−	2.02343e8i		−	1.11593i
$567$	5.43567e7			0.298197
$568$	8.79558e7			0.479976
$569$		−	1.21946e8i		−	0.661959i	−0.943638	−	0.330980i	$-0.892621\pi$
$569$		−	1.21946e8i		−	0.661959i	0.943638	−	0.330980i	$-0.107379\pi$
$570$	−7.12210e6	+	4.86439e7i	−0.0384577	+	0.262666i
$571$	2.03510e8			1.09314			0.546572	−	0.837412i	$-0.315932\pi$
$571$	2.03510e8			1.09314			0.546572	+	0.837412i	$0.315932\pi$
$572$		−	6.06784e7i		−	0.324224i
$573$			1.04610e8i			0.556044i
$574$	−3.38194e8			−1.78826
$575$	3.17561e7			0.167041
$576$	−1.71250e7			−0.0896116
$577$	1.06556e8			0.554688			0.277344	−	0.960771i	$-0.410546\pi$
$577$	1.06556e8			0.554688			0.277344	+	0.960771i	$0.410546\pi$
$578$		−	2.87944e8i		−	1.49116i
$579$	1.26694e7			0.0652708
$580$			3.40642e7i			0.174588i
$581$	−7.72089e7			−0.393676
$582$		−	4.09517e7i		−	0.207732i
$583$		−	1.29580e8i		−	0.653931i
$584$		−	1.27737e8i		−	0.641324i
$585$		−	1.03639e8i		−	0.517673i
$586$	−4.79501e7			−0.238285
$587$	−2.13383e8			−1.05498			−0.527492	−	0.849560i	$-0.676867\pi$
$587$	−2.13383e8			−1.05498			−0.527492	+	0.849560i	$0.676867\pi$
$588$		−	3.61780e7i		−	0.177956i
$589$	1.55561e8	+	2.27761e7i	0.761299	+	0.111464i
$590$	−9.33977e7			−0.454758
$591$			1.52735e8i			0.739905i
$592$			7.31257e7i			0.352456i
$593$	−8.55599e7			−0.410304			−0.205152	−	0.978730i	$-0.565769\pi$
$593$	−8.55599e7			−0.410304			−0.205152	+	0.978730i	$0.565769\pi$
$594$	8.57801e7			0.409286
$595$	−3.38543e8			−1.60717
$596$	1.42212e8			0.671736
$597$		−	3.27413e6i		−	0.0153877i
$598$	−5.14790e7			−0.240728
$599$			3.85400e8i			1.79321i	0.442831	+	0.896605i	$0.353974\pi$
$599$			3.85400e8i			1.79321i	−0.442831	+	0.896605i	$0.646026\pi$
$600$	2.04041e7			0.0944636
$601$		−	1.89176e8i		−	0.871449i	−0.900080	−	0.435725i	$-0.856492\pi$
$601$		−	1.89176e8i		−	0.871449i	0.900080	−	0.435725i	$-0.143508\pi$
$602$		−	7.14333e7i		−	0.327425i
$603$		−	2.66822e7i		−	0.121694i
$604$		−	1.50853e8i		−	0.684610i
$605$	9.35201e7			0.422317
$606$	2.23007e7			0.100208
$607$		−	2.73759e8i		−	1.22406i	−0.790836	−	0.612028i	$-0.790354\pi$
$607$		−	2.73759e8i		−	1.22406i	0.790836	−	0.612028i	$-0.209646\pi$
$608$	−3.93124e7	−	5.75585e6i	−0.174912	−	0.0256094i
$609$	−7.68314e7			−0.340163
$610$			8.09323e7i			0.356560i
$611$			9.18459e7i			0.402658i
$612$	−1.44869e8			−0.632007
$613$	−1.29168e8			−0.560758			−0.280379	−	0.959889i	$-0.590460\pi$
$613$	−1.29168e8			−0.560758			−0.280379	+	0.959889i	$0.590460\pi$
$614$	−1.81722e8			−0.785061
$615$	−1.70954e8			−0.734943
$616$			6.76452e7i			0.289398i
$617$	7.84207e7			0.333868			0.166934	−	0.985968i	$-0.446613\pi$
$617$	7.84207e7			0.333868			0.166934	+	0.985968i	$0.446613\pi$
$618$			3.02414e7i			0.128126i
$619$	−1.21305e8			−0.511456			−0.255728	−	0.966749i	$-0.582315\pi$
$619$	−1.21305e8			−0.511456			−0.255728	+	0.966749i	$0.582315\pi$
$620$			6.46926e7i			0.271444i
$621$		−	7.27751e7i		−	0.303884i
$622$			1.81184e8i			0.752920i
$623$		−	1.97807e8i		−	0.818047i
$624$	−3.30767e7			−0.136134
$625$	−5.99840e7			−0.245695
$626$		−	2.10290e8i		−	0.857225i
$627$	8.22235e7	+	1.20386e7i	0.333575	+	0.0488397i
$628$	7.16518e7			0.289300
$629$			6.18607e8i			2.48578i
$630$			1.15538e8i			0.462066i
$631$	3.52334e8			1.40238			0.701191	−	0.712973i	$-0.252652\pi$
$631$	3.52334e8			1.40238			0.701191	+	0.712973i	$0.252652\pi$
$632$	1.04924e7			0.0415647
$633$	−6.77995e7			−0.267310
$634$	2.63284e8			1.03313
$635$		−	1.69890e8i		−	0.663509i
$636$	−7.06358e7			−0.274570
$637$			1.76944e8i			0.684571i
$638$	5.75792e7			0.221719
$639$		−	2.53934e8i		−	0.973237i
$640$		−	1.63487e7i		−	0.0623655i
$641$			4.58136e8i			1.73948i	0.493507	+	0.869742i	$0.335715\pi$
$641$			4.58136e8i			1.73948i	−0.493507	+	0.869742i	$0.664285\pi$
$642$			1.63895e8i			0.619384i
$643$	−5.78444e7			−0.217585			−0.108792	−	0.994064i	$-0.534698\pi$
$643$	−5.78444e7			−0.217585			−0.108792	+	0.994064i	$0.534698\pi$
$644$	5.73896e7			0.214870
$645$		−	3.61089e7i		−	0.134566i
$646$	−3.32563e8	−	4.86916e7i	−1.23361	−	0.180616i
$647$	−7.21979e7			−0.266570			−0.133285	−	0.991078i	$-0.542553\pi$
$647$	−7.21979e7			−0.266570			−0.133285	+	0.991078i	$0.542553\pi$
$648$			2.22059e7i			0.0816099i
$649$			1.57871e8i			0.577523i
$650$	−9.97955e7			−0.363388
$651$	−1.45914e8			−0.528874
$652$	2.23875e8			0.807725
$653$	−4.68602e8			−1.68293			−0.841463	−	0.540316i	$-0.818305\pi$
$653$	−4.68602e8			−1.68293			−0.841463	+	0.540316i	$0.818305\pi$
$654$			4.94621e6i			0.0176823i
$655$	1.72615e8			0.614262
$656$		−	1.38159e8i		−	0.489405i
$657$	−3.68785e8			−1.30040
$658$		−	1.02391e8i		−	0.359406i
$659$		−	1.19637e8i		−	0.418032i	−0.977912	−	0.209016i	$-0.932974\pi$
$659$		−	1.19637e8i		−	0.418032i	0.977912	−	0.209016i	$-0.0670261\pi$
$660$			3.41940e7i			0.118937i
$661$			7.97327e7i			0.276078i	0.990427	+	0.138039i	$0.0440799\pi$
$661$			7.97327e7i			0.276078i	−0.990427	+	0.138039i	$0.955920\pi$
$662$	−5.50877e7			−0.189881
$663$	−2.79812e8			−0.960120
$664$		−	3.15415e7i		−	0.107740i
$665$	−3.88333e7	+	2.65231e8i	−0.132050	+	0.901902i
$666$	2.11119e8			0.714668
$667$		−	4.88497e7i		−	0.164621i
$668$		−	5.26182e7i		−	0.176525i
$669$	3.16764e8			1.05793
$670$	2.54726e7			0.0846934
$671$	1.36801e8			0.452816
$672$	3.68744e7			0.121511
$673$			2.04103e8i			0.669584i	0.942292	+	0.334792i	$0.108666\pi$
$673$			2.04103e8i			0.669584i	−0.942292	+	0.334792i	$0.891334\pi$
$674$	−3.07220e8			−1.00339
$675$		−	1.41079e8i		−	0.458725i
$676$	7.31822e6			0.0236900
$677$			1.53931e7i			0.0496090i	0.999692	+	0.0248045i	$0.00789633\pi$
$677$			1.53931e7i			0.0496090i	−0.999692	+	0.0248045i	$0.992104\pi$
$678$		−	1.32925e8i		−	0.426500i
$679$		−	2.23289e8i		−	0.713277i
$680$		−	1.38302e8i		−	0.439847i
$681$	1.89634e8			0.600447
$682$	1.09351e8			0.344722
$683$		−	3.74454e8i		−	1.17527i	−0.809127	−	0.587633i	$-0.800060\pi$
$683$		−	3.74454e8i		−	1.17527i	0.809127	−	0.587633i	$-0.199940\pi$
$684$	−1.66175e7	+	1.13498e8i	−0.0519275	+	0.354665i
$685$	−7.91771e7			−0.246336
$686$			9.76387e7i			0.302447i
$687$		−	1.52186e8i		−	0.469357i
$688$	2.91820e7			0.0896087
$689$	3.45476e8			1.05623
$690$	2.90099e7			0.0883078
$691$	2.10626e7			0.0638377			0.0319189	−	0.999490i	$-0.489838\pi$
$691$	2.10626e7			0.0638377			0.0319189	+	0.999490i	$0.489838\pi$
$692$			2.25344e8i			0.680031i
$693$	1.95296e8			0.586805
$694$			1.28888e8i			0.385598i
$695$	3.31253e8			0.986746
$696$		−	3.13873e7i		−	0.0930949i
$697$		−	1.16876e9i		−	3.45165i
$698$			2.76297e8i			0.812475i
$699$		−	1.15280e8i		−	0.337538i
$700$	1.11254e8			0.324355
$701$	4.79784e7			0.139281			0.0696405	−	0.997572i	$-0.477815\pi$
$701$	4.79784e7			0.139281			0.0696405	+	0.997572i	$0.477815\pi$
$702$			2.28701e8i			0.661083i
$703$	4.84647e8	+	7.09586e7i	1.39495	+	0.204239i
$704$	−2.76345e7			−0.0792015
$705$		−	5.17578e7i		−	0.147710i
$706$			5.20750e7i			0.147984i
$707$	1.21595e8			0.344078
$708$	8.60580e7			0.242489
$709$	2.94752e8			0.827023			0.413511	−	0.910499i	$-0.364302\pi$
$709$	2.94752e8			0.827023			0.413511	+	0.910499i	$0.364302\pi$
$710$	2.42423e8			0.677327
$711$		−	3.02923e7i		−	0.0842798i
$712$	8.08085e7			0.223881
$713$		−	9.27723e7i		−	0.255947i
$714$	3.11939e8			0.856987
$715$		−	1.67241e8i		−	0.457536i
$716$			7.97658e7i			0.217309i
$717$		−	3.21205e7i		−	0.0871415i
$718$			2.63212e8i			0.711103i
$719$	−4.44179e8			−1.19501			−0.597505	−	0.801865i	$-0.703841\pi$
$719$	−4.44179e8			−1.19501			−0.597505	+	0.801865i	$0.703841\pi$
$720$	−4.71999e7			−0.126457
$721$			1.64891e8i			0.439939i
$722$	−7.62947e7	+	2.54961e8i	−0.202714	+	0.677427i
$723$	−2.16975e8			−0.574110
$724$			2.72759e8i			0.718726i
$725$		−	9.46985e7i		−	0.248502i
$726$	−8.61708e7			−0.225191
$727$	−5.00461e8			−1.30247			−0.651234	−	0.758877i	$-0.725748\pi$
$727$	−5.00461e8			−1.30247			−0.651234	+	0.758877i	$0.725748\pi$
$728$	−1.80351e8			−0.467437
$729$	−1.24202e8			−0.320586
$730$		−	3.52067e8i		−	0.905017i
$731$	2.46865e8			0.631986
$732$		−	7.45722e7i		−	0.190127i
$733$	−3.49394e8			−0.887163			−0.443581	−	0.896234i	$-0.646292\pi$
$733$	−3.49394e8			−0.887163			−0.443581	+	0.896234i	$0.646292\pi$
$734$		−	5.50094e8i		−	1.39107i
$735$		−	9.97133e7i		−	0.251126i
$736$			2.34449e7i			0.0588050i
$737$		−	4.30568e7i		−	0.107557i
$738$	−3.98875e8			−0.992357
$739$	6.49894e8			1.61031			0.805154	−	0.593065i	$-0.202082\pi$
$739$	6.49894e8			1.61031			0.805154	+	0.593065i	$0.202082\pi$
$740$			2.01548e8i			0.497375i
$741$	−3.20964e7	+	2.19218e8i	−0.0788863	+	0.538793i
$742$	−3.85142e8			−0.942778
$743$			3.83317e8i			0.934525i	0.884119	+	0.467263i	$0.154760\pi$
$743$			3.83317e8i			0.934525i	−0.884119	+	0.467263i	$0.845240\pi$
$744$		−	5.96088e7i		−	0.144741i
$745$	3.91965e8			0.947934
$746$	3.36866e8			0.811410
$747$	−9.10623e7			−0.218462
$748$	−2.33774e8			−0.558588
$749$			8.93637e8i			2.12675i
$750$	1.68231e8			0.398771
$751$			6.34895e8i			1.49893i	0.662043	+	0.749466i	$0.269690\pi$
$751$			6.34895e8i			1.49893i	−0.662043	+	0.749466i	$0.730310\pi$
$752$	4.18290e7			0.0983613
$753$		−	3.73858e8i		−	0.875632i
$754$			1.53513e8i			0.358123i
$755$		−	4.15779e8i		−	0.966100i
$756$		−	2.54959e8i		−	0.590072i
$757$	4.31098e8			0.993775			0.496888	−	0.867815i	$-0.334476\pi$
$757$	4.31098e8			0.993775			0.496888	+	0.867815i	$0.334476\pi$
$758$	4.47614e8			1.02777
$759$		−	4.90359e7i		−	0.112147i
$760$	−1.08353e8	−	1.58642e7i	−0.246830	−	0.0361391i
$761$	4.43879e8			1.00719			0.503594	−	0.863940i	$-0.332011\pi$
$761$	4.43879e8			1.00719			0.503594	+	0.863940i	$0.332011\pi$
$762$			1.56539e8i			0.353801i
$763$			2.69692e7i			0.0607148i
$764$	−2.33015e8			−0.522521
$765$	−3.99287e8			−0.891868
$766$	3.80947e8			0.847576
$767$	−4.20905e8			−0.932820
$768$			1.50640e7i			0.0332549i
$769$	−4.67263e8			−1.02750			−0.513751	−	0.857939i	$-0.671745\pi$
$769$	−4.67263e8			−1.02750			−0.513751	+	0.857939i	$0.671745\pi$
$770$			1.86443e8i			0.408389i
$771$	3.41635e8			0.745417
$772$			2.82205e7i			0.0613357i
$773$			1.87256e8i			0.405413i	0.979240	+	0.202706i	$0.0649737\pi$
$773$			1.87256e8i			0.405413i	−0.979240	+	0.202706i	$0.935026\pi$
$774$		−	8.42504e7i		−	0.181698i
$775$		−	1.79846e8i		−	0.386362i
$776$	9.12184e7			0.195208
$777$	−4.54590e8			−0.969074
$778$		−	3.42388e8i		−	0.727075i
$779$	−9.15662e8	−	1.34065e8i	−1.93697	−	0.283598i
$780$	−9.11655e7			−0.192109
$781$		−	4.09771e8i		−	0.860178i
$782$			1.98332e8i			0.414736i
$783$	−2.17020e8			−0.452079
$784$	8.05851e7			0.167227
$785$	1.97486e8			0.408251
$786$	−1.59050e8			−0.327541
$787$			2.29826e8i			0.471492i	0.971815	+	0.235746i	$0.0757533\pi$
$787$			2.29826e8i			0.471492i	−0.971815	+	0.235746i	$0.924247\pi$
$788$	−3.40211e8			−0.695297
$789$			2.18339e8i			0.444530i
$790$	2.89191e7			0.0586548
$791$		−	7.24776e8i		−	1.46445i
$792$			7.97826e7i			0.160595i
$793$			3.64729e8i			0.731392i
$794$		−	4.78464e8i		−	0.955845i
$795$	−1.94686e8			−0.387465
$796$	7.29300e6			0.0144600
$797$			7.30396e8i			1.44273i	0.692558	+	0.721363i	$0.256484\pi$
$797$			7.30396e8i			1.44273i	−0.692558	+	0.721363i	$0.743516\pi$
$798$	3.57816e7	−	2.44388e8i	0.0704126	−	0.480918i
$799$	3.53852e8			0.693716
$800$			4.54495e7i			0.0887685i
$801$		−	2.33299e8i		−	0.453958i
$802$	−2.27485e8			−0.440992
$803$	−5.95104e8			−1.14933
$804$	−2.34709e7			−0.0451607
$805$	1.58177e8			0.303218
$806$			2.91543e8i			0.556798i
$807$	−4.44070e7			−0.0844950
$808$			4.96740e7i			0.0941662i
$809$	−7.36585e8			−1.39116			−0.695580	−	0.718448i	$-0.744853\pi$
$809$	−7.36585e8			−1.39116			−0.695580	+	0.718448i	$0.744853\pi$
$810$			6.12036e7i			0.115165i
$811$			5.64694e8i			1.05865i	0.848421	+	0.529323i	$0.177554\pi$
$811$			5.64694e8i			1.05865i	−0.848421	+	0.529323i	$0.822446\pi$
$812$		−	1.71139e8i		−	0.319655i
$813$		−	3.30733e8i		−	0.615468i
$814$	3.40680e8			0.631646
$815$	6.17043e8			1.13984
$816$			1.27434e8i			0.234538i
$817$	2.83172e7	−	1.93406e8i	0.0519259	−	0.354654i
$818$	−7.07555e8			−1.29271
$819$			5.20684e8i			0.947813i
$820$		−	3.80793e8i		−	0.690634i
$821$	−4.47879e8			−0.809340			−0.404670	−	0.914463i	$-0.632614\pi$
$821$	−4.47879e8			−0.809340			−0.404670	+	0.914463i	$0.632614\pi$
$822$	7.29549e7			0.131353
$823$	6.68232e7			0.119875			0.0599374	−	0.998202i	$-0.480910\pi$
$823$	6.68232e7			0.119875			0.0599374	+	0.998202i	$0.480910\pi$
$824$	−6.73617e7			−0.120401
$825$		−	9.50594e7i		−	0.169291i
$826$	4.69232e8			0.832620
$827$		−	3.98025e8i		−	0.703710i	−0.936055	−	0.351855i	$-0.885551\pi$
$827$		−	3.98025e8i		−	0.703710i	0.936055	−	0.351855i	$-0.114449\pi$
$828$	6.76868e7			0.119238
$829$		−	1.22758e8i		−	0.215469i	−0.994180	−	0.107735i	$-0.965640\pi$
$829$		−	1.22758e8i		−	0.215469i	0.994180	−	0.107735i	$-0.0343597\pi$
$830$		−	8.69343e7i		−	0.152040i
$831$			2.66989e8i			0.465254i
$832$		−	7.36770e7i		−	0.127927i
$833$	6.81709e8			1.17941
$834$	−3.05222e8			−0.526159
$835$		−	1.45026e8i		−	0.249107i
$836$	−2.68155e7	+	1.83150e8i	−0.0458952	+	0.313464i
$837$	−4.12151e8			−0.702877
$838$		−	8.14059e8i		−	1.38332i
$839$			3.01242e8i			0.510071i	0.966932	+	0.255035i	$0.0820871\pi$
$839$			3.01242e8i			0.510071i	−0.966932	+	0.255035i	$0.917913\pi$
$840$	1.01633e8			0.171473
$841$	4.49151e8			0.755099
$842$	−1.34446e7			−0.0225223
$843$	−7.31813e7			−0.122157
$844$		−	1.51021e8i		−	0.251194i
$845$	2.01704e7			0.0334306
$846$		−	1.20763e8i		−	0.199445i
$847$	−4.69847e8			−0.773225
$848$		−	1.57339e8i		−	0.258017i
$849$			5.13868e8i			0.839708i
$850$			3.84480e8i			0.626061i
$851$		−	2.89030e8i		−	0.468980i
$852$	−2.23372e8			−0.361169
$853$	5.69751e8			0.917991			0.458995	−	0.888439i	$-0.348209\pi$
$853$	5.69751e8			0.917991			0.458995	+	0.888439i	$0.348209\pi$
$854$		−	4.06605e8i		−	0.652829i
$855$	−4.58010e7	+	3.12821e8i	−0.0732786	+	0.500492i
$856$	−3.65070e8			−0.582042
$857$		−	3.43485e7i		−	0.0545714i	−0.999628	−	0.0272857i	$-0.991314\pi$
$857$		−	3.43485e7i		−	0.0545714i	0.999628	−	0.0272857i	$-0.00868639\pi$
$858$			1.54098e8i			0.243970i
$859$	2.02023e8			0.318729			0.159365	−	0.987220i	$-0.449055\pi$
$859$	2.02023e8			0.318729			0.159365	+	0.987220i	$0.449055\pi$
$860$	8.04312e7			0.126453
$861$	8.58875e8			1.34561
$862$	2.94183e8			0.459300
$863$			1.48089e8i			0.230404i	0.993342	+	0.115202i	$0.0367515\pi$
$863$			1.48089e8i			0.230404i	−0.993342	+	0.115202i	$0.963249\pi$
$864$	1.04156e8			0.161489
$865$			6.21092e8i			0.959638i
$866$	−1.58862e8			−0.244606
$867$			7.31260e8i			1.12206i
$868$		−	3.25017e8i		−	0.496989i
$869$		−	4.88824e7i		−	0.0744892i
$870$		−	8.65093e7i		−	0.131373i
$871$	1.14795e8			0.173727
$872$	−1.10175e7			−0.0166163
$873$		−	2.63353e8i		−	0.395819i
$874$	1.55383e8	+	2.27500e7i	0.232739	+	0.0340759i
$875$	9.17282e8			1.36924
$876$			3.24400e8i			0.482579i
$877$		−	8.48597e8i		−	1.25806i	−0.777379	−	0.629032i	$-0.783451\pi$
$877$		−	8.48597e8i		−	1.25806i	0.777379	−	0.629032i	$-0.216549\pi$
$878$	7.61245e8			1.12471
$879$	1.21774e8			0.179303
$880$	−7.61659e7			−0.111767
$881$	−2.29592e8			−0.335760			−0.167880	−	0.985807i	$-0.553692\pi$
$881$	−2.29592e8			−0.335760			−0.167880	+	0.985807i	$0.553692\pi$
$882$		−	2.32654e8i		−	0.339083i
$883$	6.05960e8			0.880160			0.440080	−	0.897958i	$-0.354950\pi$
$883$	6.05960e8			0.880160			0.440080	+	0.897958i	$0.354950\pi$
$884$		−	6.23270e8i		−	0.902235i
$885$	2.37192e8			0.342193
$886$			4.81966e8i			0.692971i
$887$		−	1.01673e9i		−	1.45692i	−0.685086	−	0.728462i	$-0.740235\pi$
$887$		−	1.01673e9i		−	1.45692i	0.685086	−	0.728462i	$-0.259765\pi$
$888$		−	1.85710e8i		−	0.265214i
$889$			8.53531e8i			1.21483i
$890$	2.22724e8			0.315934
$891$	1.03453e8			0.146255
$892$			7.05579e8i			0.994149i
$893$	4.05894e7	−	2.77225e8i	0.0569978	−	0.389295i
$894$	−3.61162e8			−0.505463
$895$			2.19850e8i			0.306660i
$896$			8.21363e7i			0.114186i
$897$	1.30736e8			0.181141
$898$	−1.12281e7			−0.0155053
$899$	−2.76653e8			−0.380764
$900$	1.31216e8			0.179994
$901$		−	1.33101e9i		−	1.81972i
$902$	−6.43661e8			−0.877076
$903$			1.81412e8i			0.246378i
$904$	2.96086e8			0.400786
$905$			7.51776e8i			1.01424i
$906$			3.83105e8i			0.515150i
$907$			9.83840e8i			1.31857i	0.751894	+	0.659284i	$0.229141\pi$
$907$			9.83840e8i			1.31857i	−0.751894	+	0.659284i	$0.770859\pi$
$908$			4.22402e8i			0.564246i
$909$	1.43412e8			0.190939
$910$	−4.97080e8			−0.659633
$911$			1.30290e7i			0.0172328i	0.999963	+	0.00861641i	$0.00274272\pi$
$911$			1.30290e7i			0.0172328i	−0.999963	+	0.00861641i	$0.997257\pi$
$912$	9.98377e7	+	1.46175e7i	0.131616	+	0.0192703i
$913$	−1.46946e8			−0.193084
$914$		−	6.65380e8i		−	0.871428i
$915$		−	2.05535e8i		−	0.268301i
$916$	3.38988e8			0.441060
$917$	−8.67220e8			−1.12466
$918$	8.81108e8			1.13894
$919$	−2.98731e8			−0.384888			−0.192444	−	0.981308i	$-0.561641\pi$
$919$	−2.98731e8			−0.384888			−0.192444	+	0.981308i	$0.561641\pi$
$920$			6.46185e7i			0.0829838i
$921$	4.61501e8			0.590737
$922$			5.60643e8i			0.715309i
$923$	1.09250e9			1.38937
$924$		−	1.71791e8i		−	0.217764i
$925$		−	5.60305e8i		−	0.707944i
$926$			1.41812e8i			0.178599i
$927$			1.94478e8i			0.244135i
$928$	6.99140e7			0.0874823
$929$	−9.87802e8			−1.23203			−0.616017	−	0.787733i	$-0.711255\pi$
$929$	−9.87802e8			−1.23203			−0.616017	+	0.787733i	$0.711255\pi$
$930$		−	1.64293e8i		−	0.204254i
$931$	7.81969e7	−	5.34084e8i	0.0969037	−	0.661852i
$932$	2.56782e8			0.317189
$933$		−	4.60134e8i		−	0.566552i
$934$			5.53744e8i			0.679624i
$935$	−6.44325e8			−0.788261
$936$	−2.12710e8			−0.259395
$937$	−8.49634e8			−1.03279			−0.516396	−	0.856350i	$-0.672727\pi$
$937$	−8.49634e8			−1.03279			−0.516396	+	0.856350i	$0.672727\pi$
$938$	−1.27975e8			−0.155066
$939$			5.34051e8i			0.645039i
$940$	1.15289e8			0.138804
$941$		−	8.83763e8i		−	1.06064i	−0.847798	−	0.530319i	$-0.822072\pi$
$941$		−	8.83763e8i		−	1.06064i	0.847798	−	0.530319i	$-0.177928\pi$
$942$	−1.81966e8			−0.217690
$943$			5.46076e8i			0.651205i
$944$			1.91691e8i			0.227869i
$945$		−	7.02715e8i		−	0.832691i
$946$		−	1.35954e8i		−	0.160590i
$947$	1.85948e8			0.218948			0.109474	−	0.993990i	$-0.465083\pi$
$947$	1.85948e8			0.218948			0.109474	+	0.993990i	$0.465083\pi$
$948$	−2.66465e7			−0.0312763
$949$		−	1.58662e9i		−	1.85641i
$950$	3.01220e8	+	4.41025e7i	0.351328	+	0.0514390i
$951$	−6.68634e8			−0.777404
$952$			6.94832e8i			0.805321i
$953$			9.07378e8i			1.04836i	0.851608	+	0.524179i	$0.175628\pi$
$953$			9.07378e8i			1.04836i	−0.851608	+	0.524179i	$0.824372\pi$
$954$	−4.54247e8			−0.523175
$955$	−6.42233e8			−0.737366
$956$	7.15473e7			0.0818878
$957$	−1.46228e8			−0.166838
$958$			5.43586e6i			0.00618260i
$959$	3.97787e8			0.451019
$960$			4.15191e7i			0.0469283i
$961$	3.62102e8			0.408000
$962$			9.08296e8i			1.02024i
$963$			1.05398e9i			1.18019i
$964$		−	4.83304e8i		−	0.539498i
$965$			7.77811e7i			0.0865550i
$966$	−1.45746e8			−0.161684
$967$	4.77936e8			0.528555			0.264277	−	0.964447i	$-0.414867\pi$
$967$	4.77936e8			0.528555			0.264277	+	0.964447i	$0.414867\pi$
$968$		−	1.91942e8i		−	0.211614i
$969$	8.44576e8	+	1.23657e8i	0.928255	+	0.135909i
$970$	2.51415e8			0.275471
$971$		−	6.23290e8i		−	0.680820i	−0.940277	−	0.340410i	$-0.889434\pi$
$971$		−	6.23290e8i		−	0.680820i	0.940277	−	0.340410i	$-0.110566\pi$
$972$		−	4.75851e8i		−	0.518170i
$973$	−1.66422e9			−1.80664
$974$	−6.31838e8			−0.683800
$975$	2.53440e8			0.273440
$976$	1.66107e8			0.178664
$977$		−	1.47715e9i		−	1.58395i	−0.610554	−	0.791975i	$-0.709053\pi$
$977$		−	1.47715e9i		−	1.58395i	0.610554	−	0.791975i	$-0.290947\pi$
$978$	−5.68553e8			−0.607791
$979$		−	3.76473e8i		−	0.401223i
$980$	2.22108e8			0.235986
$981$			3.18082e7i			0.0336925i
$982$		−	6.93018e8i		−	0.731829i
$983$		−	1.78294e9i		−	1.87705i	−0.345206	−	0.938527i	$-0.612191\pi$
$983$		−	1.78294e9i		−	1.87705i	0.345206	−	0.938527i	$-0.387809\pi$
$984$			3.50869e8i			0.368264i
$985$	−9.37687e8			−0.981182
$986$	5.91437e8			0.616990
$987$			2.60032e8i			0.270443i
$988$	−4.88301e8	−	7.14935e7i	−0.506310	−	0.0741303i
$989$	−1.15342e8			−0.119234
$990$			2.19896e8i			0.226627i
$991$			1.44206e9i			1.48171i	0.671668	+	0.740853i	$0.265578\pi$
$991$			1.44206e9i			1.48171i	−0.671668	+	0.740853i	$0.734422\pi$
$992$	1.32776e8			0.136015
$993$	1.39900e8			0.142880
$994$	−1.21794e9			−1.24013
$995$	2.01009e7			0.0204054
$996$			8.01025e7i			0.0810715i
$997$	−6.78975e8			−0.685123			−0.342561	−	0.939495i	$-0.611294\pi$
$997$	−6.78975e8			−0.685123			−0.342561	+	0.939495i	$0.611294\pi$
$998$			7.24207e8i			0.728570i
$999$	−1.28405e9			−1.28791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.7.b.a.37.4	✓	10
3.2	odd	2		342.7.d.a.37.9		10
4.3	odd	2		304.7.e.e.113.3		10
19.18	odd	2	inner	38.7.b.a.37.7	yes	10
57.56	even	2		342.7.d.a.37.4		10
76.75	even	2		304.7.e.e.113.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.7.b.a.37.4	✓	10	1.1	even	1	trivial
38.7.b.a.37.7	yes	10	19.18	odd	2	inner
304.7.e.e.113.3		10	4.3	odd	2
304.7.e.e.113.8		10	76.75	even	2
342.7.d.a.37.4		10	57.56	even	2
342.7.d.a.37.9		10	3.2	odd	2

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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.65685i q^{2} +14.3661i q^{3} -32.0000 q^{4} -88.1981 q^{5} +81.2670 q^{6} +443.109 q^{7} +181.019i q^{8} +522.615 q^{9} +O(q^{10})\)	\(q-5.65685i q^{2} +14.3661i q^{3} -32.0000 q^{4} -88.1981 q^{5} +81.2670 q^{6} +443.109 q^{7} +181.019i q^{8} +522.615 q^{9} +498.924i q^{10} +843.338 q^{11} -459.716i q^{12} +2248.44i q^{13} -2506.60i q^{14} -1267.06i q^{15} +1024.00 q^{16} +8662.52 q^{17} -2956.36i q^{18} +(993.652 - 6786.64i) q^{19} +2822.34 q^{20} +6365.75i q^{21} -4770.64i q^{22} -4047.37 q^{23} -2600.54 q^{24} -7846.10 q^{25} +12719.1 q^{26} +17980.8i q^{27} -14179.5 q^{28} +12069.5i q^{29} -7167.60 q^{30} +22921.6i q^{31} -5792.62i q^{32} +12115.5i q^{33} -49002.6i q^{34} -39081.3 q^{35} -16723.7 q^{36} +71411.8i q^{37} +(-38391.1 - 5620.95i) q^{38} -32301.4 q^{39} -15965.6i q^{40} -134921. i q^{41} +36010.1 q^{42} +28498.1 q^{43} -26986.8 q^{44} -46093.6 q^{45} +22895.4i q^{46} +40848.6 q^{47} +14710.9i q^{48} +78696.4 q^{49} +44384.2i q^{50} +124447. i q^{51} -71950.2i q^{52} -153651. i q^{53} +101715. q^{54} -74380.8 q^{55} +80211.3i q^{56} +(97497.7 + 14274.9i) q^{57} +68275.4 q^{58} +187198. i q^{59} +40546.0i q^{60} +162214. q^{61} +129664. q^{62} +231575. q^{63} -32768.0 q^{64} -198308. i q^{65} +68535.6 q^{66} -51055.2i q^{67} -277201. q^{68} -58145.0i q^{69} +221077. i q^{70} -485892. i q^{71} +94603.4i q^{72} -705653. q^{73} +403966. q^{74} -112718. i q^{75} +(-31796.9 + 217173. i) q^{76} +373690. q^{77} +182724. i q^{78} -57963.0i q^{79} -90314.8 q^{80} +122671. q^{81} -763230. q^{82} -174244. q^{83} -203704. i q^{84} -764018. q^{85} -161209. i q^{86} -173392. q^{87} +152660. i q^{88} -446408. i q^{89} +260745. i q^{90} +996306. i q^{91} +129516. q^{92} -329295. q^{93} -231075. i q^{94} +(-87638.2 + 598569. i) q^{95} +83217.4 q^{96} -503915. i q^{97} -445174. i q^{98} +440741. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 320 q^{4} - 112 q^{5} + 160 q^{6} - 224 q^{7} - 2890 q^{9}+O(q^{10}) \) 10 * q - 320 * q^4 - 112 * q^5 + 160 * q^6 - 224 * q^7 - 2890 * q^9	\( 10 q - 320 q^{4} - 112 q^{5} + 160 q^{6} - 224 q^{7} - 2890 q^{9} + 3644 q^{11} + 10240 q^{16} - 10420 q^{17} - 17230 q^{19} + 3584 q^{20} + 37712 q^{23} - 5120 q^{24} - 52078 q^{25} - 7104 q^{26} + 7168 q^{28} + 94688 q^{30} - 161720 q^{35} + 92480 q^{36} + 25152 q^{38} - 78876 q^{39} + 53792 q^{42} + 6308 q^{43} - 116608 q^{44} + 309808 q^{45} + 322220 q^{47} - 235770 q^{49} - 321728 q^{54} - 377880 q^{55} + 24228 q^{57} + 445920 q^{58} + 426304 q^{61} + 59424 q^{62} - 517916 q^{63} - 327680 q^{64} - 1417312 q^{66} + 333440 q^{68} - 786076 q^{73} - 293280 q^{74} + 551360 q^{76} + 2303716 q^{77} - 114688 q^{80} + 5261090 q^{81} - 455136 q^{82} - 101500 q^{83} - 1261380 q^{85} - 2460732 q^{87} - 1206784 q^{92} - 2827032 q^{93} + 3106292 q^{95} + 163840 q^{96} + 1061428 q^{99}+O(q^{100}) \) 10 * q - 320 * q^4 - 112 * q^5 + 160 * q^6 - 224 * q^7 - 2890 * q^9 + 3644 * q^11 + 10240 * q^16 - 10420 * q^17 - 17230 * q^19 + 3584 * q^20 + 37712 * q^23 - 5120 * q^24 - 52078 * q^25 - 7104 * q^26 + 7168 * q^28 + 94688 * q^30 - 161720 * q^35 + 92480 * q^36 + 25152 * q^38 - 78876 * q^39 + 53792 * q^42 + 6308 * q^43 - 116608 * q^44 + 309808 * q^45 + 322220 * q^47 - 235770 * q^49 - 321728 * q^54 - 377880 * q^55 + 24228 * q^57 + 445920 * q^58 + 426304 * q^61 + 59424 * q^62 - 517916 * q^63 - 327680 * q^64 - 1417312 * q^66 + 333440 * q^68 - 786076 * q^73 - 293280 * q^74 + 551360 * q^76 + 2303716 * q^77 - 114688 * q^80 + 5261090 * q^81 - 455136 * q^82 - 101500 * q^83 - 1261380 * q^85 - 2460732 * q^87 - 1206784 * q^92 - 2827032 * q^93 + 3106292 * q^95 + 163840 * q^96 + 1061428 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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