LMFDB - Embedded newform 37.6.b.a.36.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,6,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, 6, names="a")

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

chi := DirichletCharacter("37.36");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 37 $
Weight:	$ k $	$=$	$ 6 $
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:	$5.93420133308$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + \cdots + 178006118400 $ x^16 + 390x^14 + 60701x^12 + 4799932x^10 + 203487156x^8 + 4519465040x^6 + 48993644736x^4 + 211923220224*x^2 + 178006118400
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			36.9
Root			$1.04262i$ of defining polynomial
Character	$\chi$	$=$	37.36
Dual form			37.6.b.a.36.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.04262i q^{2} +19.6295 q^{3} +30.9129 q^{4} -86.4713i q^{5} +20.4662i q^{6} -87.1072 q^{7} +65.5944i q^{8} +142.318 q^{9} +O(q^{10})$	\(q+1.04262i q^{2} +19.6295 q^{3} +30.9129 q^{4} -86.4713i q^{5} +20.4662i q^{6} -87.1072 q^{7} +65.5944i q^{8} +142.318 q^{9} +90.1568 q^{10} +153.498 q^{11} +606.806 q^{12} +27.6086i q^{13} -90.8199i q^{14} -1697.39i q^{15} +920.824 q^{16} +926.725i q^{17} +148.384i q^{18} +2749.94i q^{19} -2673.08i q^{20} -1709.87 q^{21} +160.041i q^{22} +1336.01i q^{23} +1287.59i q^{24} -4352.28 q^{25} -28.7854 q^{26} -1976.33 q^{27} -2692.74 q^{28} -7147.31i q^{29} +1769.74 q^{30} +4026.24i q^{31} +3059.09i q^{32} +3013.10 q^{33} -966.224 q^{34} +7532.28i q^{35} +4399.48 q^{36} +(-7808.85 - 2892.37i) q^{37} -2867.14 q^{38} +541.945i q^{39} +5672.03 q^{40} +11398.2 q^{41} -1782.75i q^{42} -8183.21i q^{43} +4745.08 q^{44} -12306.5i q^{45} -1392.95 q^{46} -13313.5 q^{47} +18075.3 q^{48} -9219.33 q^{49} -4537.78i q^{50} +18191.2i q^{51} +853.464i q^{52} +27617.4 q^{53} -2060.57i q^{54} -13273.2i q^{55} -5713.75i q^{56} +53980.0i q^{57} +7451.94 q^{58} +16008.1i q^{59} -52471.3i q^{60} -44974.5i q^{61} -4197.84 q^{62} -12397.0 q^{63} +26276.9 q^{64} +2387.35 q^{65} +3141.52i q^{66} -45824.8 q^{67} +28647.8i q^{68} +26225.2i q^{69} -7853.31 q^{70} -18749.2 q^{71} +9335.29i q^{72} -6614.81 q^{73} +(3015.65 - 8141.68i) q^{74} -85433.3 q^{75} +85008.6i q^{76} -13370.8 q^{77} -565.043 q^{78} +69916.6i q^{79} -79624.8i q^{80} -73377.8 q^{81} +11884.0i q^{82} +81375.6 q^{83} -52857.2 q^{84} +80135.1 q^{85} +8531.99 q^{86} -140298. i q^{87} +10068.6i q^{88} -120861. i q^{89} +12831.0 q^{90} -2404.91i q^{91} +41300.0i q^{92} +79033.2i q^{93} -13881.0i q^{94} +237791. q^{95} +60048.5i q^{96} +5970.34i q^{97} -9612.27i q^{98} +21845.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9}+O(q^{10}) $ 16 * q - 18 * q^3 - 268 * q^4 + 190 * q^7 + 1394 * q^9	$ 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9} - 74 q^{10} - 1110 q^{11} + 1402 q^{12} + 2900 q^{16} - 7010 q^{21} - 12052 q^{25} + 4902 q^{26} + 4266 q^{27} - 16824 q^{28} + 19280 q^{30} - 2478 q^{33} + 20556 q^{34} - 51402 q^{36} - 11400 q^{37} + 12108 q^{38} + 16966 q^{40} + 3918 q^{41} + 125394 q^{44} + 17470 q^{46} + 3822 q^{47} - 78034 q^{48} - 32618 q^{49} - 24126 q^{53} - 164718 q^{58} - 81426 q^{62} + 219268 q^{63} + 158076 q^{64} + 98976 q^{65} + 23560 q^{67} - 222404 q^{70} - 50046 q^{71} - 196274 q^{73} + 141216 q^{74} + 214054 q^{75} - 239574 q^{77} - 90822 q^{78} + 317312 q^{81} - 215814 q^{83} + 438572 q^{84} - 346472 q^{85} + 197640 q^{86} - 857612 q^{90} - 132504 q^{95} - 574860 q^{99}+O(q^{100}) $ 16 * q - 18 * q^3 - 268 * q^4 + 190 * q^7 + 1394 * q^9 - 74 * q^10 - 1110 * q^11 + 1402 * q^12 + 2900 * q^16 - 7010 * q^21 - 12052 * q^25 + 4902 * q^26 + 4266 * q^27 - 16824 * q^28 + 19280 * q^30 - 2478 * q^33 + 20556 * q^34 - 51402 * q^36 - 11400 * q^37 + 12108 * q^38 + 16966 * q^40 + 3918 * q^41 + 125394 * q^44 + 17470 * q^46 + 3822 * q^47 - 78034 * q^48 - 32618 * q^49 - 24126 * q^53 - 164718 * q^58 - 81426 * q^62 + 219268 * q^63 + 158076 * q^64 + 98976 * q^65 + 23560 * q^67 - 222404 * q^70 - 50046 * q^71 - 196274 * q^73 + 141216 * q^74 + 214054 * q^75 - 239574 * q^77 - 90822 * q^78 + 317312 * q^81 - 215814 * q^83 + 438572 * q^84 - 346472 * q^85 + 197640 * q^86 - 857612 * q^90 - 132504 * q^95 - 574860 * 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 $

$2$			1.04262i			0.184311i	0.995745	+	0.0921556i	$0.0293757\pi$
$2$			1.04262i			0.184311i	−0.995745	+	0.0921556i	$0.970624\pi$
$3$	19.6295			1.25923			0.629617	−	0.776905i	$-0.283212\pi$
$3$	19.6295			1.25923			0.629617	+	0.776905i	$0.283212\pi$
$4$	30.9129			0.966029
$5$		−	86.4713i		−	1.54685i	−0.633891	−	0.773423i	$-0.718543\pi$
$5$		−	86.4713i		−	1.54685i	0.633891	−	0.773423i	$-0.281457\pi$
$6$			20.4662i			0.232091i
$7$	−87.1072			−0.671907			−0.335954	−	0.941879i	$-0.609059\pi$
$7$	−87.1072			−0.671907			−0.335954	+	0.941879i	$0.609059\pi$
$8$			65.5944i			0.362361i
$9$	142.318			0.585673
$10$	90.1568			0.285101
$11$	153.498			0.382491			0.191246	−	0.981542i	$-0.438747\pi$
$11$	153.498			0.382491			0.191246	+	0.981542i	$0.438747\pi$
$12$	606.806			1.21646
$13$			27.6086i			0.0453092i	0.999743	+	0.0226546i	$0.00721181\pi$
$13$			27.6086i			0.0453092i	−0.999743	+	0.0226546i	$0.992788\pi$
$14$		−	90.8199i		−	0.123840i
$15$		−	1697.39i		−	1.94784i
$16$	920.824			0.899242
$17$			926.725i			0.777730i	0.921295	+	0.388865i	$0.127133\pi$
$17$			926.725i			0.777730i	−0.921295	+	0.388865i	$0.872867\pi$
$18$			148.384i			0.107946i
$19$			2749.94i			1.74759i	0.486298	+	0.873793i	$0.338347\pi$
$19$			2749.94i			1.74759i	−0.486298	+	0.873793i	$0.661653\pi$
$20$		−	2673.08i		−	1.49430i
$21$	−1709.87			−0.846089
$22$			160.041i			0.0704974i
$23$			1336.01i			0.526611i	0.964713	+	0.263305i	$0.0848127\pi$
$23$			1336.01i			0.526611i	−0.964713	+	0.263305i	$0.915187\pi$
$24$			1287.59i			0.456298i
$25$	−4352.28			−1.39273
$26$	−28.7854			−0.00835100
$27$	−1976.33			−0.521736
$28$	−2692.74			−0.649082
$29$		−	7147.31i		−	1.57815i	−0.614299	−	0.789073i	$-0.710561\pi$
$29$		−	7147.31i		−	1.57815i	0.614299	−	0.789073i	$-0.289439\pi$
$30$	1769.74			0.359009
$31$			4026.24i			0.752481i	0.926522	+	0.376240i	$0.122783\pi$
$31$			4026.24i			0.752481i	−0.926522	+	0.376240i	$0.877217\pi$
$32$			3059.09i			0.528102i
$33$	3013.10			0.481646
$34$	−966.224			−0.143344
$35$			7532.28i			1.03934i
$36$	4399.48			0.565777
$37$	−7808.85	−	2892.37i	−0.937741	−	0.347336i
$37$	−7808.85	−	2892.37i	−0.937741	−	0.347336i
$38$	−2867.14			−0.322100
$39$			541.945i			0.0570550i
$40$	5672.03			0.560517
$41$	11398.2			1.05896			0.529478	−	0.848324i	$-0.322388\pi$
$41$	11398.2			1.05896			0.529478	+	0.848324i	$0.322388\pi$
$42$		−	1782.75i		−	0.155944i
$43$		−	8183.21i		−	0.674920i	−0.941340	−	0.337460i	$-0.890432\pi$
$43$		−	8183.21i		−	0.674920i	0.941340	−	0.337460i	$-0.109568\pi$
$44$	4745.08			0.369498
$45$		−	12306.5i		−	0.905945i
$46$	−1392.95			−0.0970603
$47$	−13313.5			−0.879122			−0.439561	−	0.898213i	$-0.644866\pi$
$47$	−13313.5			−0.879122			−0.439561	+	0.898213i	$0.644866\pi$
$48$	18075.3			1.13236
$49$	−9219.33			−0.548541
$50$		−	4537.78i		−	0.256696i
$51$			18191.2i			0.979344i
$52$			853.464i			0.0437701i
$53$	27617.4			1.35050			0.675248	−	0.737591i	$-0.264037\pi$
$53$	27617.4			1.35050			0.675248	+	0.737591i	$0.264037\pi$
$54$		−	2060.57i		−	0.0961617i
$55$		−	13273.2i		−	0.591655i
$56$		−	5713.75i		−	0.243473i
$57$			53980.0i			2.20062i
$58$	7451.94			0.290870
$59$			16008.1i			0.598700i	0.954143	+	0.299350i	$0.0967699\pi$
$59$			16008.1i			0.598700i	−0.954143	+	0.299350i	$0.903230\pi$
$60$		−	52471.3i		−	1.88167i
$61$		−	44974.5i		−	1.54754i	−0.633466	−	0.773770i	$-0.718369\pi$
$61$		−	44974.5i		−	1.54754i	0.633466	−	0.773770i	$-0.281631\pi$
$62$	−4197.84			−0.138691
$63$	−12397.0			−0.393518
$64$	26276.9			0.801907
$65$	2387.35			0.0700864
$66$			3141.52i			0.0887728i
$67$	−45824.8			−1.24713			−0.623567	−	0.781770i	$-0.714317\pi$
$67$	−45824.8			−1.24713			−0.623567	+	0.781770i	$0.714317\pi$
$68$			28647.8i			0.751310i
$69$			26225.2i			0.663127i
$70$	−7853.31			−0.191561
$71$	−18749.2			−0.441404			−0.220702	−	0.975341i	$-0.570835\pi$
$71$	−18749.2			−0.441404			−0.220702	+	0.975341i	$0.570835\pi$
$72$			9335.29i			0.212225i
$73$	−6614.81			−0.145281			−0.0726407	−	0.997358i	$-0.523143\pi$
$73$	−6614.81			−0.145281			−0.0726407	+	0.997358i	$0.523143\pi$
$74$	3015.65	−	8141.68i	0.0640179	−	0.172836i
$75$	−85433.3			−1.75377
$76$			85008.6i			1.68822i
$77$	−13370.8			−0.256999
$78$	−565.043			−0.0105159
$79$			69916.6i			1.26041i	0.776428	+	0.630206i	$0.217030\pi$
$79$			69916.6i			1.26041i	−0.776428	+	0.630206i	$0.782970\pi$
$80$		−	79624.8i		−	1.39099i
$81$	−73377.8			−1.24266
$82$			11884.0i			0.195177i
$83$	81375.6			1.29658			0.648289	−	0.761394i	$-0.275485\pi$
$83$	81375.6			1.29658			0.648289	+	0.761394i	$0.275485\pi$
$84$	−52857.2			−0.817347
$85$	80135.1			1.20303
$86$	8531.99			0.124395
$87$		−	140298.i		−	1.98726i
$88$			10068.6i			0.138600i
$89$		−	120861.i		−	1.61738i	−0.588235	−	0.808690i	$-0.700177\pi$
$89$		−	120861.i		−	1.61738i	0.588235	−	0.808690i	$-0.299823\pi$
$90$	12831.0			0.166976
$91$		−	2404.91i		−	0.0304436i
$92$			41300.0i			0.508722i
$93$			79033.2i			0.947550i
$94$		−	13881.0i		−	0.162032i
$95$	237791.			2.70325
$96$			60048.5i			0.665004i
$97$			5970.34i			0.0644273i	0.999481	+	0.0322137i	$0.0102557\pi$
$97$			5970.34i			0.0644273i	−0.999481	+	0.0322137i	$0.989744\pi$
$98$		−	9612.27i		−	0.101102i
$99$	21845.6			0.224015
$100$	−134542.			−1.34542
$101$	102083.			0.995754			0.497877	−	0.867248i	$-0.334113\pi$
$101$	102083.			0.995754			0.497877	+	0.867248i	$0.334113\pi$
$102$	−18966.5			−0.180504
$103$		−	193144.i		−	1.79386i	−0.442172	−	0.896930i	$-0.645792\pi$
$103$		−	193144.i		−	1.79386i	0.442172	−	0.896930i	$-0.354208\pi$
$104$	−1810.97			−0.0164183
$105$			147855.i			1.30877i
$106$			28794.5i			0.248911i
$107$	120881.			1.02070			0.510351	−	0.859966i	$-0.329515\pi$
$107$	120881.			1.02070			0.510351	+	0.859966i	$0.329515\pi$
$108$	−61094.2			−0.504012
$109$			153803.i			1.23993i	0.784629	+	0.619966i	$0.212853\pi$
$109$			153803.i			1.23993i	−0.784629	+	0.619966i	$0.787147\pi$
$110$	13838.9			0.109049
$111$	−153284.	−	56775.9i	−1.18084	−	0.437378i
$112$	−80210.4			−0.604207
$113$		−	19713.0i		−	0.145230i	−0.997360	−	0.0726150i	$-0.976866\pi$
$113$		−	19713.0i		−	0.145230i	0.997360	−	0.0726150i	$-0.0231344\pi$
$114$	−56280.7			−0.405599
$115$	115526.			0.814586
$116$		−	220944.i		−	1.52454i
$117$			3929.22i			0.0265364i
$118$	−16690.4			−0.110347
$119$		−	80724.5i		−	0.522562i
$120$	111339.			0.705822
$121$	−137489.			−0.853700
$122$	46891.4			0.285229
$123$	223742.			1.33347
$124$			124463.i			0.726918i
$125$			106125.i			0.607493i
$126$		−	12925.3i		−	0.0725297i
$127$	−41588.5			−0.228804			−0.114402	−	0.993435i	$-0.536495\pi$
$127$	−41588.5			−0.228804			−0.114402	+	0.993435i	$0.536495\pi$
$128$			125288.i			0.675902i
$129$		−	160633.i		−	0.849883i
$130$			2489.11i			0.0129177i
$131$			167404.i			0.852289i	0.904655	+	0.426145i	$0.140129\pi$
$131$			167404.i			0.852289i	−0.904655	+	0.426145i	$0.859871\pi$
$132$	93143.7			0.465284
$133$		−	239539.i		−	1.17422i
$134$		−	47777.9i		−	0.229861i
$135$			170896.i			0.807044i
$136$	−60788.0			−0.281819
$137$	−326960.			−1.48831			−0.744155	−	0.668007i	$-0.767148\pi$
$137$	−326960.			−1.48831			−0.744155	+	0.668007i	$0.767148\pi$
$138$	−27343.0			−0.122222
$139$	62658.9			0.275071			0.137536	−	0.990497i	$-0.456082\pi$
$139$	62658.9			0.275071			0.137536	+	0.990497i	$0.456082\pi$
$140$			232845.i			1.00403i
$141$	−261339.			−1.10702
$142$		−	19548.3i		−	0.0813557i
$143$			4237.88i			0.0173304i
$144$	131050.			0.526661
$145$	−618037.			−2.44115
$146$		−	6896.74i		−	0.0267770i
$147$	−180971.			−0.690742
$148$	−241395.	−	89411.8i	−0.905885	−	0.335537i
$149$	−65982.0			−0.243478			−0.121739	−	0.992562i	$-0.538847\pi$
$149$	−65982.0			−0.243478			−0.121739	+	0.992562i	$0.538847\pi$
$150$		−	89074.6i		−	0.323240i
$151$	95460.7			0.340708			0.170354	−	0.985383i	$-0.445509\pi$
$151$	95460.7			0.340708			0.170354	+	0.985383i	$0.445509\pi$
$152$	−180380.			−0.633258
$153$			131890.i			0.455495i
$154$		−	13940.7i		−	0.0473677i
$155$	348154.			1.16397
$156$			16753.1i			0.0551168i
$157$	290753.			0.941403			0.470702	−	0.882292i	$-0.344001\pi$
$157$	290753.			0.941403			0.470702	+	0.882292i	$0.344001\pi$
$158$	−72896.6			−0.232308
$159$	542117.			1.70059
$160$	264524.			0.816892
$161$		−	116376.i		−	0.353834i
$162$		−	76505.3i		−	0.229036i
$163$			522500.i			1.54034i	0.637837	+	0.770171i	$0.279829\pi$
$163$			522500.i			1.54034i	−0.637837	+	0.770171i	$0.720171\pi$
$164$	352353.			1.02298
$165$		−	260546.i		−	0.745032i
$166$			84843.9i			0.238974i
$167$		−	320029.i		−	0.887968i	−0.896035	−	0.443984i	$-0.853565\pi$
$167$		−	320029.i		−	0.887968i	0.896035	−	0.443984i	$-0.146435\pi$
$168$		−	112158.i		−	0.306590i
$169$	370531.			0.997947
$170$			83550.6i			0.221731i
$171$			391367.i			1.02351i
$172$		−	252967.i		−	0.651993i
$173$	−386890.			−0.982816			−0.491408	−	0.870929i	$-0.663518\pi$
$173$	−386890.			−0.982816			−0.491408	+	0.870929i	$0.663518\pi$
$174$	146278.			0.366274
$175$	379115.			0.935785
$176$	141345.			0.343952
$177$			314231.i			0.753905i
$178$	126013.			0.298101
$179$		−	772439.i		−	1.80190i	−0.433918	−	0.900952i	$-0.642869\pi$
$179$		−	772439.i		−	1.80190i	0.433918	−	0.900952i	$-0.357131\pi$
$180$		−	380429.i		−	0.875169i
$181$	−219471.			−0.497944			−0.248972	−	0.968511i	$-0.580093\pi$
$181$	−219471.			−0.497944			−0.248972	+	0.968511i	$0.580093\pi$
$182$	2507.41			0.00561110
$183$		−	882829.i		−	1.94872i
$184$	−87634.7			−0.190823
$185$	−250107.	+	675241.i	−0.537275	+	1.45054i
$186$	−82401.7			−0.174644
$187$			142251.i			0.297475i
$188$	−411561.			−0.849257
$189$	172153.			0.350558
$190$			247926.i			0.498238i
$191$			119268.i			0.236559i	0.992980	+	0.118279i	$0.0377378\pi$
$191$			119268.i			0.236559i	−0.992980	+	0.118279i	$0.962262\pi$
$192$	515803.			1.00979
$193$			606131.i			1.17131i	0.810559	+	0.585657i	$0.199164\pi$
$193$			606131.i			1.17131i	−0.810559	+	0.585657i	$0.800836\pi$
$194$	−6224.81			−0.0118747
$195$	46862.7			0.0882552
$196$	−284997.			−0.529907
$197$	−48998.3			−0.0899529			−0.0449765	−	0.998988i	$-0.514321\pi$
$197$	−48998.3			−0.0899529			−0.0449765	+	0.998988i	$0.514321\pi$
$198$			22776.7i			0.0412884i
$199$		−	401960.i		−	0.719532i	−0.933043	−	0.359766i	$-0.882857\pi$
$199$		−	401960.i		−	0.719532i	0.933043	−	0.359766i	$-0.117143\pi$
$200$		−	285485.i		−	0.504671i
$201$	−899519.			−1.57044
$202$			106434.i			0.183529i
$203$			622582.i			1.06037i
$204$			562343.i			0.946075i
$205$		−	985619.i		−	1.63804i
$206$	201376.			0.330629
$207$			190139.i			0.308422i
$208$			25422.7i			0.0407440i
$209$			422110.i			0.668436i
$210$	−154157.			−0.241221
$211$	981075.			1.51704			0.758518	−	0.651652i	$-0.225924\pi$
$211$	981075.			1.51704			0.758518	+	0.651652i	$0.225924\pi$
$212$	853735.			1.30462
$213$	−368037.			−0.555831
$214$			126033.i			0.188127i
$215$	−707613.			−1.04400
$216$		−	129636.i		−	0.189057i
$217$		−	350715.i		−	0.505597i
$218$	−160358.			−0.228533
$219$	−129846.			−0.182943
$220$		−	410313.i		−	0.571556i
$221$	−25585.6			−0.0352383
$222$	59195.8	−	159817.i	0.0806136	−	0.217641i
$223$	−988479.			−1.33108			−0.665542	−	0.746361i	$-0.731799\pi$
$223$	−988479.			−1.33108			−0.665542	+	0.746361i	$0.731799\pi$
$224$		−	266469.i		−	0.354835i
$225$	−619410.			−0.815684
$226$	20553.2			0.0267675
$227$		−	1.06424e6i		−	1.37080i	−0.728168	−	0.685398i	$-0.759628\pi$
$227$		−	1.06424e6i		−	1.37080i	0.728168	−	0.685398i	$-0.240372\pi$
$228$			1.66868e6i			2.12587i
$229$	437725.			0.551586			0.275793	−	0.961217i	$-0.411060\pi$
$229$	437725.			0.551586			0.275793	+	0.961217i	$0.411060\pi$
$230$			120450.i			0.150137i
$231$	−262463.			−0.323622
$232$	468823.			0.571859
$233$	−1.01255e6			−1.22188			−0.610940	−	0.791677i	$-0.709208\pi$
$233$	−1.01255e6			−1.22188			−0.610940	+	0.791677i	$0.709208\pi$
$234$	−4096.69			−0.00489095
$235$			1.15124e6i			1.35987i
$236$			494857.i			0.578362i
$237$			1.37243e6i			1.58716i
$238$	84165.1			0.0963140
$239$			286370.i			0.324290i	0.986767	+	0.162145i	$0.0518412\pi$
$239$			286370.i			0.324290i	−0.986767	+	0.162145i	$0.948159\pi$
$240$		−	1.56300e6i		−	1.75158i
$241$		−	372657.i		−	0.413302i	−0.978415	−	0.206651i	$-0.933744\pi$
$241$		−	372657.i		−	0.413302i	0.978415	−	0.206651i	$-0.0662564\pi$
$242$		−	143349.i		−	0.157347i
$243$	−960124.			−1.04307
$244$		−	1.39029e6i		−	1.49497i
$245$			797207.i			0.848508i
$246$			233278.i			0.245774i
$247$	−75922.0			−0.0791818
$248$	−264099.			−0.272670
$249$	1.59736e6			1.63270
$250$	−110648.			−0.111968
$251$		−	807096.i		−	0.808613i	−0.914624	−	0.404306i	$-0.867513\pi$
$251$		−	807096.i		−	0.808613i	0.914624	−	0.404306i	$-0.132487\pi$
$252$	−383227.			−0.380150
$253$			205075.i			0.201424i
$254$		−	43361.1i		−	0.0421712i
$255$	1.57301e6			1.51489
$256$	710233.			0.677331
$257$			1.29694e6i			1.22486i	0.790524	+	0.612432i	$0.209809\pi$
$257$			1.29694e6i			1.22486i	−0.790524	+	0.612432i	$0.790191\pi$
$258$	167479.			0.156643
$259$	680207.	+	251947.i	0.630075	+	0.233378i
$260$	73800.2			0.0677055
$261$		−	1.01719e6i		−	0.924277i
$262$	−174539.			−0.157086
$263$	−358188.			−0.319317			−0.159658	−	0.987172i	$-0.551039\pi$
$263$	−358188.			−0.319317			−0.159658	+	0.987172i	$0.551039\pi$
$264$			197642.i			0.174530i
$265$		−	2.38811e6i		−	2.08901i
$266$	249749.			0.216421
$267$		−	2.37245e6i		−	2.03666i
$268$	−1.41658e6			−1.20477
$269$	2.26507e6			1.90853			0.954267	−	0.298955i	$-0.0966379\pi$
$269$	2.26507e6			1.90853			0.954267	+	0.298955i	$0.0966379\pi$
$270$	−178180.			−0.148747
$271$	1.06501e6			0.880910			0.440455	−	0.897775i	$-0.354817\pi$
$271$	1.06501e6			0.880910			0.440455	+	0.897775i	$0.354817\pi$
$272$			853351.i			0.699367i
$273$		−	47207.3i		−	0.0383356i
$274$		−	340896.i		−	0.274312i
$275$	−668067.			−0.532707
$276$			810699.i			0.640600i
$277$			1.07621e6i			0.842751i	0.906886	+	0.421375i	$0.138453\pi$
$277$			1.07621e6i			0.842751i	−0.906886	+	0.421375i	$0.861547\pi$
$278$			65329.5i			0.0506988i
$279$			573008.i			0.440707i
$280$	−494075.			−0.376615
$281$			540316.i			0.408208i	0.978949	+	0.204104i	$0.0654281\pi$
$281$			540316.i			0.408208i	−0.978949	+	0.204104i	$0.934572\pi$
$282$		−	272477.i		−	0.204036i
$283$			669160.i			0.496665i	0.968675	+	0.248333i	$0.0798826\pi$
$283$			669160.i			0.496665i	−0.968675	+	0.248333i	$0.920117\pi$
$284$	−579592.			−0.426409
$285$	4.66772e6			3.40402
$286$	−4418.50			−0.00319418
$287$	−992868.			−0.711520
$288$			435365.i			0.309295i
$289$	561038.			0.395137
$290$		−	644378.i		−	0.449931i
$291$			117195.i			0.0811291i
$292$	−204483.			−0.140346
$293$	−2.14251e6			−1.45799			−0.728994	−	0.684521i	$-0.760012\pi$
$293$	−2.14251e6			−1.45799			−0.728994	+	0.684521i	$0.760012\pi$
$294$		−	188684.i		−	0.127311i
$295$	1.38424e6			0.926097
$296$	189723.	−	512217.i	0.125861	−	0.339801i
$297$	−303363.			−0.199559
$298$		−	68794.2i		−	0.0448757i
$299$	−36885.4			−0.0238603
$300$	−2.64099e6			−1.69420
$301$			712817.i			0.453484i
$302$			99529.4i			0.0627963i
$303$	2.00385e6			1.25389
$304$			2.53221e6i			1.57150i
$305$	−3.88900e6			−2.39381
$306$	−137511.			−0.0839528
$307$	−1.35256e6			−0.819047			−0.409524	−	0.912299i	$-0.634305\pi$
$307$	−1.35256e6			−0.819047			−0.409524	+	0.912299i	$0.634305\pi$
$308$	−413331.			−0.248268
$309$		−	3.79133e6i		−	2.25889i
$310$			362993.i			0.214533i
$311$		−	371845.i		−	0.218002i	−0.994042	−	0.109001i	$-0.965235\pi$
$311$		−	371845.i		−	0.218002i	0.994042	−	0.109001i	$-0.0347652\pi$
$312$	−35548.5			−0.0206745
$313$		−	1.27822e6i		−	0.737470i	−0.929535	−	0.368735i	$-0.879791\pi$
$313$		−	1.27822e6i		−	0.737470i	0.929535	−	0.368735i	$-0.120209\pi$
$314$			303146.i			0.173511i
$315$			1.07198e6i			0.608711i
$316$			2.16133e6i			1.21760i
$317$	−888017.			−0.496333			−0.248166	−	0.968717i	$-0.579828\pi$
$317$	−888017.			−0.496333			−0.248166	+	0.968717i	$0.579828\pi$
$318$			565222.i			0.313438i
$319$		−	1.09710e6i		−	0.603627i
$320$		−	2.27220e6i		−	1.24043i
$321$	2.37284e6			1.28530
$322$	121336.			0.0652155
$323$	−2.54843e6			−1.35915
$324$	−2.26832e6			−1.20045
$325$		−	120161.i		−	0.0631035i
$326$	−544770.			−0.283902
$327$			3.01907e6i			1.56136i
$328$			747660.i			0.383724i
$329$	1.15971e6			0.590688
$330$	271651.			0.137318
$331$		−	1.13045e6i		−	0.567129i	−0.958953	−	0.283565i	$-0.908483\pi$
$331$		−	1.13045e6i		−	0.567129i	0.958953	−	0.283565i	$-0.0915171\pi$
$332$	2.51556e6			1.25253
$333$	−1.11134e6	−	411638.i	−0.549209	−	0.203425i
$334$	333669.			0.163663
$335$			3.96253e6i			1.92912i
$336$	−1.57449e6			−0.760839
$337$	−2.75184e6			−1.31992			−0.659961	−	0.751300i	$-0.729427\pi$
$337$	−2.75184e6			−1.31992			−0.659961	+	0.751300i	$0.729427\pi$
$338$			386323.i			0.183933i
$339$		−	386956.i		−	0.182879i
$340$	2.47721e6			1.16216
$341$			618020.i			0.287817i
$342$	−408047.			−0.188645
$343$	2.26708e6			1.04048
$344$	536773.			0.244565
$345$	2.26773e6			1.02575
$346$		−	403380.i		−	0.181144i
$347$		−	883744.i		−	0.394006i	−0.980403	−	0.197003i	$-0.936879\pi$
$347$		−	883744.i		−	0.394006i	0.980403	−	0.197003i	$-0.0631208\pi$
$348$		−	4.33703e6i		−	1.91975i
$349$	−55889.7			−0.0245623			−0.0122811	−	0.999925i	$-0.503909\pi$
$349$	−55889.7			−0.0245623			−0.0122811	+	0.999925i	$0.503909\pi$
$350$			395274.i			0.172476i
$351$		−	54563.8i		−	0.0236394i
$352$			469565.i			0.201994i
$353$			388952.i			0.166134i	0.996544	+	0.0830672i	$0.0264716\pi$
$353$			388952.i			0.166134i	−0.996544	+	0.0830672i	$0.973528\pi$
$354$	−327624.			−0.138953
$355$			1.62127e6i			0.682784i
$356$		−	3.73618e6i		−	1.56244i
$357$		−	1.58458e6i		−	0.658028i
$358$	805362.			0.332111
$359$	3.13551e6			1.28402			0.642010	−	0.766697i	$-0.278101\pi$
$359$	3.13551e6			1.28402			0.642010	+	0.766697i	$0.278101\pi$
$360$	807235.			0.328279
$361$	−5.08605e6			−2.05406
$362$		−	228825.i		−	0.0917767i
$363$	−2.69885e6			−1.07501
$364$		−	74342.9i		−	0.0294094i
$365$			571991.i			0.224728i
$366$	920456.			0.359170
$367$	−160181.			−0.0620793			−0.0310396	−	0.999518i	$-0.509882\pi$
$367$	−160181.			−0.0620793			−0.0310396	+	0.999518i	$0.509882\pi$
$368$			1.23023e6i			0.473551i
$369$	1.62218e6			0.620201
$370$	−704021.	−	260767.i	−0.267351	−	0.0990259i
$371$	−2.40568e6			−0.907407
$372$			2.44315e6i			0.915361i
$373$	3.26910e6			1.21662			0.608311	−	0.793699i	$-0.291847\pi$
$373$	3.26910e6			1.21662			0.608311	+	0.793699i	$0.291847\pi$
$374$	−148314.			−0.0548279
$375$			2.08318e6i			0.764976i
$376$		−	873294.i		−	0.318560i
$377$	197327.			0.0715046
$378$			179490.i			0.0646117i
$379$	−756174.			−0.270411			−0.135205	−	0.990818i	$-0.543169\pi$
$379$	−756174.			−0.270411			−0.135205	+	0.990818i	$0.543169\pi$
$380$	7.35080e6			2.61141
$381$	−816364.			−0.288119
$382$	−124351.			−0.0436004
$383$		−	3.40102e6i		−	1.18471i	−0.805677	−	0.592355i	$-0.798198\pi$
$383$		−	3.40102e6i		−	1.18471i	0.805677	−	0.592355i	$-0.201802\pi$
$384$			2.45934e6i			0.851120i
$385$			1.15619e6i			0.397537i
$386$	−631966.			−0.215886
$387$		−	1.16462e6i		−	0.395282i
$388$			184561.i			0.0622387i
$389$		−	3.01289e6i		−	1.00951i	−0.863263	−	0.504754i	$-0.831583\pi$
$389$		−	3.01289e6i		−	1.00951i	0.863263	−	0.504754i	$-0.168417\pi$
$390$			48860.0i			0.0162664i
$391$	−1.23811e6			−0.409561
$392$		−	604736.i		−	0.198770i
$393$			3.28606e6i			1.07323i
$394$		−	51086.7i		−	0.0165793i
$395$	6.04578e6			1.94966
$396$	675312.			0.216405
$397$	2.29374e6			0.730413			0.365207	−	0.930926i	$-0.380998\pi$
$397$	2.29374e6			0.730413			0.365207	+	0.930926i	$0.380998\pi$
$398$	419092.			0.132618
$399$		−	4.70205e6i		−	1.47861i
$400$	−4.00769e6			−1.25240
$401$			3.73275e6i			1.15922i	0.814892	+	0.579612i	$0.196796\pi$
$401$			3.73275e6i			1.15922i	−0.814892	+	0.579612i	$0.803204\pi$
$402$		−	937858.i		−	0.289449i
$403$	−111159.			−0.0340943
$404$	3.15570e6			0.961927
$405$			6.34508e6i			1.92220i
$406$	−649118.			−0.195438
$407$	−1.19864e6	−	443974.i	−0.358678	−	0.132853i
$408$	−1.19324e6			−0.354876
$409$		−	4.34834e6i		−	1.28533i	−0.766146	−	0.642667i	$-0.777828\pi$
$409$		−	4.34834e6i		−	1.28533i	0.766146	−	0.642667i	$-0.222172\pi$
$410$	1.02763e6			0.301909
$411$	−6.41807e6			−1.87413
$412$		−	5.97066e6i		−	1.73292i
$413$		−	1.39442e6i		−	0.402271i
$414$	−198243.			−0.0568455
$415$		−	7.03665e6i		−	2.00561i
$416$	−84457.4			−0.0239279
$417$	1.22996e6			0.346380
$418$	−440101.			−0.123200
$419$	3.92059e6			1.09098			0.545490	−	0.838117i	$-0.316344\pi$
$419$	3.92059e6			1.09098			0.545490	+	0.838117i	$0.316344\pi$
$420$			4.57063e6i			1.26431i
$421$			252803.i			0.0695146i	0.999396	+	0.0347573i	$0.0110658\pi$
$421$			252803.i			0.0695146i	−0.999396	+	0.0347573i	$0.988934\pi$
$422$			1.02289e6i			0.279607i
$423$	−1.89476e6			−0.514877
$424$			1.81155e6i			0.489367i
$425$		−	4.03337e6i		−	1.08317i
$426$		−	383724.i		−	0.102446i
$427$			3.91761e6i			1.03980i
$428$	3.73679e6			0.986029
$429$			83187.5i			0.0218230i
$430$		−	737772.i		−	0.192420i
$431$			5.98928e6i			1.55304i	0.630095	+	0.776518i	$0.283016\pi$
$431$			5.98928e6i			1.55304i	−0.630095	+	0.776518i	$0.716984\pi$
$432$	−1.81985e6			−0.469167
$433$	2.64435e6			0.677795			0.338898	−	0.940823i	$-0.389946\pi$
$433$	2.64435e6			0.677795			0.338898	+	0.940823i	$0.389946\pi$
$434$	365663.			0.0931872
$435$	−1.21318e7			−3.07398
$436$			4.75449e6i			1.19781i
$437$	−3.67394e6			−0.920298
$438$		−	135380.i		−	0.0337185i
$439$			6.50479e6i			1.61091i	0.592655	+	0.805457i	$0.298080\pi$
$439$			6.50479e6i			1.61091i	−0.592655	+	0.805457i	$0.701920\pi$
$440$	870646.			0.214393
$441$	−1.31208e6			−0.321265
$442$		−	26676.1i		−	0.00649482i
$443$	1.83180e6			0.443475			0.221738	−	0.975106i	$-0.428827\pi$
$443$	1.83180e6			0.443475			0.221738	+	0.975106i	$0.428827\pi$
$444$	−4.73846e6	−	1.75511e6i	−1.14072	−	0.422520i
$445$	−1.04510e7			−2.50184
$446$		−	1.03061e6i		−	0.245334i
$447$	−1.29520e6			−0.306596
$448$	−2.28891e6			−0.538807
$449$			3.59888e6i			0.842465i	0.906953	+	0.421233i	$0.138402\pi$
$449$			3.59888e6i			0.842465i	−0.906953	+	0.421233i	$0.861598\pi$
$450$		−	645810.i		−	0.150340i
$451$	1.74961e6			0.405041
$452$		−	609386.i		−	0.140296i
$453$	1.87385e6			0.429031
$454$	1.10959e6			0.252653
$455$	−207956.			−0.0470915
$456$	−3.54078e6			−0.797420
$457$			2.96825e6i			0.664829i	0.943133	+	0.332415i	$0.107863\pi$
$457$			2.96825e6i			0.664829i	−0.943133	+	0.332415i	$0.892137\pi$
$458$			456382.i			0.101663i
$459$		−	1.83152e6i		−	0.405769i
$460$	3.57126e6			0.786914
$461$		−	271586.i		−	0.0595190i	−0.999557	−	0.0297595i	$-0.990526\pi$
$461$		−	271586.i		−	0.0595190i	0.999557	−	0.0297595i	$-0.00947415\pi$
$462$		−	273649.i		−	0.0596471i
$463$			5.47928e6i			1.18788i	0.804511	+	0.593938i	$0.202427\pi$
$463$			5.47928e6i			1.18788i	−0.804511	+	0.593938i	$0.797573\pi$
$464$		−	6.58141e6i		−	1.41914i
$465$	6.83410e6			1.46571
$466$		−	1.05571e6i		−	0.225206i
$467$			361304.i			0.0766621i	0.999265	+	0.0383311i	$0.0122041\pi$
$467$			361304.i			0.0766621i	−0.999265	+	0.0383311i	$0.987796\pi$
$468$			121464.i			0.0256349i
$469$	3.99167e6			0.837959
$470$	−1.20031e6			−0.250638
$471$	5.70735e6			1.18545
$472$	−1.05004e6			−0.216946
$473$		−	1.25611e6i		−	0.258151i
$474$	−1.43093e6			−0.292531
$475$		−	1.19685e7i		−	2.43392i
$476$		−	2.49543e6i		−	0.504810i
$477$	3.93046e6			0.790948
$478$	−298576.			−0.0597703
$479$		−	6.39822e6i		−	1.27415i	−0.770802	−	0.637074i	$-0.780144\pi$
$479$		−	6.39822e6i		−	1.27415i	0.770802	−	0.637074i	$-0.219856\pi$
$480$	5.19247e6			1.02866
$481$	79854.5	−	215592.i	0.0157375	−	0.0424883i
$482$	388541.			0.0761761
$483$		−	2.28441e6i		−	0.445560i
$484$	−4.25020e6			−0.824700
$485$	516263.			0.0996591
$486$		−	1.00105e6i		−	0.192249i
$487$			5.35237e6i			1.02264i	0.859390	+	0.511321i	$0.170844\pi$
$487$			5.35237e6i			1.02264i	−0.859390	+	0.511321i	$0.829156\pi$
$488$	2.95008e6			0.560769
$489$			1.02564e7i			1.93965i
$490$	−831185.			−0.156390
$491$	−6.11966e6			−1.14557			−0.572787	−	0.819704i	$-0.694138\pi$
$491$	−6.11966e6			−1.14557			−0.572787	+	0.819704i	$0.694138\pi$
$492$	6.91652e6			1.28817
$493$	6.62359e6			1.22737
$494$		−	79157.9i		−	0.0145941i
$495$		−	1.88902e6i		−	0.346516i
$496$			3.70746e6i			0.676662i
$497$	1.63319e6			0.296582
$498$			1.66545e6i			0.300924i
$499$			9.16295e6i			1.64734i	0.567067	+	0.823672i	$0.308078\pi$
$499$			9.16295e6i			1.64734i	−0.567067	+	0.823672i	$0.691922\pi$
$500$			3.28063e6i			0.586856i
$501$		−	6.28201e6i		−	1.11816i
$502$	841495.			0.149036
$503$			434586.i			0.0765871i	0.999267	+	0.0382935i	$0.0121922\pi$
$503$			434586.i			0.0765871i	−0.999267	+	0.0382935i	$0.987808\pi$
$504$		−	813172.i		−	0.142596i
$505$		−	8.82729e6i		−	1.54028i
$506$	−213816.			−0.0371247
$507$	7.27334e6			1.25665
$508$	−1.28562e6			−0.221032
$509$	770690.			0.131852			0.0659258	−	0.997825i	$-0.479000\pi$
$509$	770690.			0.131852			0.0659258	+	0.997825i	$0.479000\pi$
$510$			1.64006e6i			0.279212i
$511$	576198.			0.0976156
$512$			4.74971e6i			0.800742i
$513$		−	5.43479e6i		−	0.911778i
$514$	−1.35222e6			−0.225756
$515$	−1.67014e7			−2.77482
$516$		−	4.96562e6i		−	0.821012i
$517$	−2.04360e6			−0.336256
$518$	−262685.	+	709199.i	−0.0430141	+	0.116130i
$519$	−7.59447e6			−1.23760
$520$			156597.i			0.0253966i
$521$	401386.			0.0647841			0.0323920	−	0.999475i	$-0.489687\pi$
$521$	401386.			0.0647841			0.0323920	+	0.999475i	$0.489687\pi$
$522$	1.06055e6			0.170355
$523$			2.01101e6i			0.321485i	0.986996	+	0.160743i	$0.0513889\pi$
$523$			2.01101e6i			0.321485i	−0.986996	+	0.160743i	$0.948611\pi$
$524$			5.17494e6i			0.823336i
$525$	7.44186e6			1.17837
$526$		−	373455.i		−	0.0588537i
$527$	−3.73122e6			−0.585226
$528$	2.77453e6			0.433117
$529$	4.65142e6			0.722681
$530$	2.48990e6			0.385027
$531$			2.27825e6i			0.350642i
$532$		−	7.40487e6i		−	1.13433i
$533$			314690.i			0.0479805i
$534$	2.47357e6			0.375380
$535$		−	1.04528e7i		−	1.57887i
$536$		−	3.00585e6i		−	0.451913i
$537$		−	1.51626e7i		−	2.26902i
$538$			2.36161e6i			0.351764i
$539$	−1.41515e6			−0.209812
$540$			5.28290e6i			0.779628i
$541$			1.06718e7i			1.56764i	0.620990	+	0.783818i	$0.286731\pi$
$541$			1.06718e7i			1.56764i	−0.620990	+	0.783818i	$0.713269\pi$
$542$			1.11041e6i			0.162362i
$543$	−4.30811e6			−0.627029
$544$	−2.83494e6			−0.410720
$545$	1.32995e7			1.91798
$546$	49219.4			0.00706569
$547$			8.72544e6i			1.24686i	0.781878	+	0.623432i	$0.214262\pi$
$547$			8.72544e6i			1.24686i	−0.781878	+	0.623432i	$0.785738\pi$
$548$	−1.01073e7			−1.43775
$549$		−	6.40070e6i		−	0.906352i
$550$		−	696541.i		−	0.0981839i
$551$	1.96546e7			2.75795
$552$	−1.72023e6			−0.240291
$553$		−	6.09024e6i		−	0.846880i
$554$	−1.12208e6			−0.155328
$555$	−4.90949e6	+	1.32547e7i	−0.676556	+	1.82657i
$556$	1.93697e6			0.265727
$557$			258227.i			0.0352667i	0.999845	+	0.0176333i	$0.00561316\pi$
$557$			258227.i			0.0352667i	−0.999845	+	0.0176333i	$0.994387\pi$
$558$	−597431.			−0.0812273
$559$	225927.			0.0305801
$560$			6.93590e6i			0.934615i
$561$			2.79231e6i			0.374591i
$562$	−563345.			−0.0752374
$563$			1.26414e6i			0.168083i	0.996462	+	0.0840413i	$0.0267828\pi$
$563$			1.26414e6i			0.168083i	−0.996462	+	0.0840413i	$0.973217\pi$
$564$	−8.07874e6			−1.06941
$565$	−1.70461e6			−0.224648
$566$	−697681.			−0.0915410
$567$	6.39174e6			0.834952
$568$		−	1.22984e6i		−	0.159948i
$569$		−	2.30163e6i		−	0.298027i	−0.988835	−	0.149014i	$-0.952390\pi$
$569$		−	2.30163e6i		−	0.298027i	0.988835	−	0.149014i	$-0.0476098\pi$
$570$			4.86666e6i			0.627399i
$571$	1.11302e6			0.142861			0.0714305	−	0.997446i	$-0.477244\pi$
$571$	1.11302e6			0.142861			0.0714305	+	0.997446i	$0.477244\pi$
$572$			131005.i			0.0167417i
$573$			2.34117e6i			0.297883i
$574$		−	1.03519e6i		−	0.131141i
$575$		−	5.81469e6i		−	0.733427i
$576$	3.73969e6			0.469655
$577$		−	5.49305e6i		−	0.686869i	−0.939177	−	0.343434i	$-0.888410\pi$
$577$		−	5.49305e6i		−	0.686869i	0.939177	−	0.343434i	$-0.111590\pi$
$578$			584950.i			0.0728281i
$579$			1.18981e7i			1.47496i
$580$	−1.91053e7			−2.35822
$581$	−7.08840e6			−0.871180
$582$	−122190.			−0.0149530
$583$	4.23922e6			0.516553
$584$		−	433894.i		−	0.0526443i
$585$	339765.			0.0410477
$586$		−	2.23383e6i		−	0.268723i
$587$		−	4.66139e6i		−	0.558367i	−0.960238	−	0.279184i	$-0.909936\pi$
$587$		−	4.66139e6i		−	0.558367i	0.960238	−	0.279184i	$-0.0900638\pi$
$588$	−5.59435e6			−0.667277
$589$	−1.10719e7			−1.31502
$590$			1.44324e6i			0.170690i
$591$	−961813.			−0.113272
$592$	−7.19058e6	−	2.66337e6i	−0.843256	−	0.312339i
$593$	−8.26345e6			−0.964994			−0.482497	−	0.875898i	$-0.660270\pi$
$593$	−8.26345e6			−0.964994			−0.482497	+	0.875898i	$0.660270\pi$
$594$		−	316293.i		−	0.0367810i
$595$	−6.98035e6			−0.808323
$596$	−2.03970e6			−0.235207
$597$		−	7.89028e6i		−	0.906060i
$598$		−	38457.5i		−	0.00439773i
$599$	−3.84507e6			−0.437862			−0.218931	−	0.975740i	$-0.570257\pi$
$599$	−3.84507e6			−0.437862			−0.218931	+	0.975740i	$0.570257\pi$
$600$		−	5.60394e6i		−	0.635500i
$601$	1.18174e7			1.33456			0.667278	−	0.744809i	$-0.267459\pi$
$601$	1.18174e7			1.33456			0.667278	+	0.744809i	$0.267459\pi$
$602$	−743198.			−0.0835821
$603$	−6.52171e6			−0.730413
$604$	2.95097e6			0.329134
$605$			1.18889e7i			1.32054i
$606$			2.08926e6i			0.231106i
$607$			8.13937e6i			0.896642i	0.893873	+	0.448321i	$0.147978\pi$
$607$			8.13937e6i			0.896642i	−0.893873	+	0.448321i	$0.852022\pi$
$608$	−8.41231e6			−0.922903
$609$			1.22210e7i			1.33525i
$610$		−	4.05476e6i		−	0.441205i
$611$		−	367569.i		−	0.0398323i
$612$			4.07711e6i			0.440021i
$613$	6.23345e6			0.670003			0.335002	−	0.942218i	$-0.391263\pi$
$613$	6.23345e6			0.670003			0.335002	+	0.942218i	$0.391263\pi$
$614$		−	1.41020e6i		−	0.150960i
$615$		−	1.93472e7i		−	2.06268i
$616$		−	877050.i		−	0.0931263i
$617$	1.61232e6			0.170506			0.0852530	−	0.996359i	$-0.472830\pi$
$617$	1.61232e6			0.170506			0.0852530	+	0.996359i	$0.472830\pi$
$618$	3.95292e6			0.416339
$619$	1.33080e7			1.39600			0.697999	−	0.716099i	$-0.254074\pi$
$619$	1.33080e7			1.39600			0.697999	+	0.716099i	$0.254074\pi$
$620$	1.07625e7			1.12443
$621$		−	2.64040e6i		−	0.274752i
$622$	387694.			0.0401803
$623$			1.05279e7i			1.08673i
$624$			499036.i			0.0513062i
$625$	−4.42415e6			−0.453033
$626$	1.33270e6			0.135924
$627$			8.28582e6i			0.841719i
$628$	8.98805e6			0.909423
$629$	2.68043e6	−	7.23666e6i	0.270134	−	0.729309i
$630$	−1.11767e6			−0.112192
$631$		−	1.16451e7i		−	1.16431i	−0.813078	−	0.582154i	$-0.802210\pi$
$631$		−	1.16451e7i		−	1.16431i	0.813078	−	0.582154i	$-0.197790\pi$
$632$	−4.58614e6			−0.456725
$633$	1.92580e7			1.91031
$634$		−	925866.i		−	0.0914797i
$635$			3.59621e6i			0.353925i
$636$	1.67584e7			1.64282
$637$		−	254533.i		−	0.0248540i
$638$	1.14386e6			0.111255
$639$	−2.66835e6			−0.258518
$640$	1.08338e7			1.04552
$641$	1.89809e6			0.182462			0.0912309	−	0.995830i	$-0.470920\pi$
$641$	1.89809e6			0.182462			0.0912309	+	0.995830i	$0.470920\pi$
$642$			2.47398e6i			0.236896i
$643$		−	3.21597e6i		−	0.306750i	−0.988168	−	0.153375i	$-0.950986\pi$
$643$		−	3.21597e6i		−	0.306750i	0.988168	−	0.153375i	$-0.0490143\pi$
$644$		−	3.59753e6i		−	0.341814i
$645$	−1.38901e7			−1.31464
$646$		−	2.65705e6i		−	0.250506i
$647$		−	9.14794e6i		−	0.859137i	−0.903034	−	0.429568i	$-0.858666\pi$
$647$		−	9.14794e6i		−	0.859137i	0.903034	−	0.429568i	$-0.141334\pi$
$648$		−	4.81318e6i		−	0.450292i
$649$			2.45721e6i			0.228998i
$650$	125282.			0.0116307
$651$		−	6.88436e6i		−	0.636665i
$652$			1.61520e7i			1.48802i
$653$			2.31944e6i			0.212863i	0.994320	+	0.106432i	$0.0339425\pi$
$653$			2.31944e6i			0.212863i	−0.994320	+	0.106432i	$0.966058\pi$
$654$	−3.14775e6			−0.287777
$655$	1.44756e7			1.31836
$656$	1.04958e7			0.952257
$657$	−941409.			−0.0850873
$658$			1.20913e6i			0.108870i
$659$	916434.			0.0822030			0.0411015	−	0.999155i	$-0.486913\pi$
$659$	916434.			0.0822030			0.0411015	+	0.999155i	$0.486913\pi$
$660$		−	8.05425e6i		−	0.719723i
$661$			2.37664e6i			0.211572i	0.994389	+	0.105786i	$0.0337359\pi$
$661$			2.37664e6i			0.211572i	−0.994389	+	0.105786i	$0.966264\pi$
$662$	1.17863e6			0.104528
$663$	−502234.			−0.0443733
$664$			5.33778e6i			0.469830i
$665$	−2.07133e7			−1.81633
$666$	429183.	−	1.15871e6i	0.0374936	−	0.101225i
$667$	9.54887e6			0.831069
$668$		−	9.89302e6i		−	0.857804i
$669$	−1.94034e7			−1.67615
$670$	−4.13142e6			−0.355559
$671$		−	6.90351e6i		−	0.591921i
$672$		−	5.23066e6i		−	0.446821i
$673$	5.10059e6			0.434093			0.217047	−	0.976161i	$-0.430358\pi$
$673$	5.10059e6			0.434093			0.217047	+	0.976161i	$0.430358\pi$
$674$		−	2.86912e6i		−	0.243276i
$675$	8.60155e6			0.726637
$676$	1.14542e7			0.964046
$677$	−1.83738e7			−1.54073			−0.770365	−	0.637603i	$-0.779926\pi$
$677$	−1.83738e7			−1.54073			−0.770365	+	0.637603i	$0.779926\pi$
$678$	403449.			0.0337066
$679$		−	520060.i		−	0.0432892i
$680$			5.25641e6i			0.435930i
$681$		−	2.08904e7i		−	1.72615i
$682$	−644361.			−0.0530479
$683$			1.34013e7i			1.09925i	0.835413	+	0.549623i	$0.185229\pi$
$683$			1.34013e7i			1.09925i	−0.835413	+	0.549623i	$0.814771\pi$
$684$			1.20983e7i			0.988744i
$685$			2.82727e7i			2.30219i
$686$			2.36371e6i			0.191771i
$687$	8.59234e6			0.694576
$688$		−	7.53530e6i		−	0.606917i
$689$			762479.i			0.0611899i
$690$			2.36438e6i			0.189058i
$691$	−5.66245e6			−0.451138			−0.225569	−	0.974227i	$-0.572424\pi$
$691$	−5.66245e6			−0.451138			−0.225569	+	0.974227i	$0.572424\pi$
$692$	−1.19599e7			−0.949430
$693$	−1.90291e6			−0.150517
$694$	921410.			0.0726197
$695$		−	5.41819e6i		−	0.425493i
$696$	9.20278e6			0.720105
$697$			1.05630e7i			0.823581i
$698$		−	58271.9i		−	0.00452710i
$699$	−1.98760e7			−1.53863
$700$	1.17196e7			0.903996
$701$			4.15435e6i			0.319306i	0.987173	+	0.159653i	$0.0510376\pi$
$701$			4.15435e6i			0.319306i	−0.987173	+	0.159653i	$0.948962\pi$
$702$	56889.4			0.00435701
$703$	7.95384e6	−	2.14738e7i	0.607000	−	1.63878i
$704$	4.03345e6			0.306722
$705$			2.25983e7i			1.71239i
$706$	−405530.			−0.0306204
$707$	−8.89221e6			−0.669054
$708$			9.71381e6i			0.728294i
$709$		−	2.41883e7i		−	1.80713i	−0.428449	−	0.903566i	$-0.640940\pi$
$709$		−	2.41883e7i		−	1.80713i	0.428449	−	0.903566i	$-0.359060\pi$
$710$	−1.69037e6			−0.125845
$711$			9.95042e6i			0.738189i
$712$	7.92782e6			0.586076
$713$	−5.37909e6			−0.396265
$714$	1.65212e6			0.121282
$715$	366455.			0.0268074
$716$		−	2.38784e7i		−	1.74069i
$717$			5.62132e6i			0.408357i
$718$			3.26915e6i			0.236659i
$719$	−1.98486e6			−0.143188			−0.0715941	−	0.997434i	$-0.522809\pi$
$719$	−1.98486e6			−0.143188			−0.0715941	+	0.997434i	$0.522809\pi$
$720$		−	1.13321e7i		−	0.814664i
$721$			1.68243e7i			1.20531i
$722$		−	5.30283e6i		−	0.378586i
$723$		−	7.31509e6i		−	0.520444i
$724$	−6.78449e6			−0.481029
$725$			3.11071e7i			2.19793i
$726$		−	2.81388e6i		−	0.198136i
$727$		−	2.20799e7i		−	1.54939i	−0.632333	−	0.774697i	$-0.717902\pi$
$727$		−	2.20799e7i		−	1.54939i	0.632333	−	0.774697i	$-0.282098\pi$
$728$	157749.			0.0110316
$729$	−1.01597e6			−0.0708044
$730$	−596370.			−0.0414199
$731$	7.58359e6			0.524906
$732$		−	2.72908e7i		−	1.88252i
$733$	−1.72118e7			−1.18322			−0.591612	−	0.806223i	$-0.701508\pi$
$733$	−1.72118e7			−1.18322			−0.591612	+	0.806223i	$0.701508\pi$
$734$		−	167008.i		−	0.0114419i
$735$			1.56488e7i			1.06847i
$736$	−4.08697e6			−0.278104
$737$	−7.03402e6			−0.477018
$738$			1.69132e6i			0.114310i
$739$	−2.51693e7			−1.69535			−0.847675	−	0.530515i	$-0.821998\pi$
$739$	−2.51693e7			−1.69535			−0.847675	+	0.530515i	$0.821998\pi$
$740$	−7.73155e6	+	2.08737e7i	−0.519024	+	1.40126i
$741$	−1.49031e6			−0.0997085
$742$		−	2.50821e6i		−	0.167245i
$743$	2.36324e7			1.57049			0.785245	−	0.619185i	$-0.212537\pi$
$743$	2.36324e7			1.57049			0.785245	+	0.619185i	$0.212537\pi$
$744$	−5.18413e6			−0.343355
$745$			5.70555e6i			0.376623i
$746$			3.40843e6i			0.224237i
$747$	1.15812e7			0.759370
$748$			4.39738e6i			0.287369i
$749$	−1.05296e7			−0.685817
$750$	−2.17197e6			−0.140994
$751$	761070.			0.0492408			0.0246204	−	0.999697i	$-0.492162\pi$
$751$	761070.			0.0492408			0.0246204	+	0.999697i	$0.492162\pi$
$752$	−1.22594e7			−0.790543
$753$		−	1.58429e7i		−	1.01823i
$754$			205738.i			0.0131791i
$755$		−	8.25461e6i		−	0.527022i
$756$	5.32175e6			0.338649
$757$		−	2.12851e7i		−	1.35001i	−0.737814	−	0.675004i	$-0.764142\pi$
$757$		−	2.12851e7i		−	1.35001i	0.737814	−	0.675004i	$-0.235858\pi$
$758$		−	788403.i		−	0.0498397i
$759$			4.02552e6i			0.253640i
$760$			1.55977e7i			0.979551i
$761$	−1.99836e7			−1.25087			−0.625435	−	0.780277i	$-0.715078\pi$
$761$	−1.99836e7			−1.25087			−0.625435	+	0.780277i	$0.715078\pi$
$762$		−	851158.i		−	0.0531035i
$763$		−	1.33973e7i		−	0.833118i
$764$			3.68691e6i			0.228523i
$765$	1.14047e7			0.704580
$766$	3.54597e6			0.218355
$767$	−441962.			−0.0271267
$768$	1.39415e7			0.852919
$769$			1.94814e7i			1.18797i	0.804477	+	0.593984i	$0.202446\pi$
$769$			1.94814e7i			1.18797i	−0.804477	+	0.593984i	$0.797554\pi$
$770$	−1.20547e6			−0.0732705
$771$			2.54584e7i			1.54239i
$772$			1.87373e7i			1.13152i
$773$	1.13110e7			0.680849			0.340425	−	0.940272i	$-0.389429\pi$
$773$	1.13110e7			0.680849			0.340425	+	0.940272i	$0.389429\pi$
$774$	1.21426e6			0.0728550
$775$		−	1.75233e7i		−	1.04800i
$776$	−391621.			−0.0233460
$777$	1.33522e7	+	4.94559e6i	0.793412	+	0.293877i
$778$	3.14131e6			0.186064
$779$			3.13444e7i			1.85062i
$780$	1.44866e6			0.0852571
$781$	−2.87796e6			−0.168833
$782$		−	1.29088e6i		−	0.0754867i
$783$			1.41254e7i			0.823375i
$784$	−8.48938e6			−0.493271
$785$		−	2.51418e7i		−	1.45621i
$786$	−3.42611e6			−0.197809
$787$	−7.02612e6			−0.404370			−0.202185	−	0.979347i	$-0.564804\pi$
$787$	−7.02612e6			−0.404370			−0.202185	+	0.979347i	$0.564804\pi$
$788$	−1.51468e6			−0.0868972
$789$	−7.03106e6			−0.402095
$790$			6.30346e6i			0.359345i
$791$			1.71714e6i			0.0975810i
$792$			1.43295e6i			0.0811742i
$793$	1.24169e6			0.0701179
$794$			2.39151e6i			0.134623i
$795$		−	4.68775e7i		−	2.63055i
$796$		−	1.24258e7i		−	0.695089i
$797$			2.61247e7i			1.45682i	0.685143	+	0.728409i	$0.259740\pi$
$797$			2.61247e7i			1.45682i	−0.685143	+	0.728409i	$0.740260\pi$
$798$	4.90245e6			0.272525
$799$		−	1.23380e7i		−	0.683719i
$800$		−	1.33140e7i		−	0.735503i
$801$		−	1.72008e7i		−	0.947255i
$802$	−3.89184e6			−0.213658
$803$	−1.01536e6			−0.0555689
$804$	−2.78068e7			−1.51709
$805$	−1.00632e7			−0.547326
$806$		−	115897.i		−	0.00628397i
$807$	4.44622e7			2.40329
$808$			6.69610e6i			0.360823i
$809$		−	2.91750e7i		−	1.56725i	−0.621233	−	0.783626i	$-0.713368\pi$
$809$		−	2.91750e7i		−	1.56725i	0.621233	−	0.783626i	$-0.286632\pi$
$810$	−6.61551e6			−0.354284
$811$	3.56190e6			0.190164			0.0950822	−	0.995469i	$-0.469689\pi$
$811$	3.56190e6			0.190164			0.0950822	+	0.995469i	$0.469689\pi$
$812$			1.92458e7i			1.02435i
$813$	2.09057e7			1.10927
$814$	462897.	−	1.24973e6i	0.0244863	−	0.0661083i
$815$	4.51812e7			2.38267
$816$			1.67509e7i			0.880668i
$817$	2.25033e7			1.17948
$818$	4.53368e6			0.236901
$819$		−	342263.i		−	0.0178300i
$820$		−	3.04684e7i		−	1.58239i
$821$	−1.48511e7			−0.768954			−0.384477	−	0.923135i	$-0.625618\pi$
$821$	−1.48511e7			−0.768954			−0.384477	+	0.923135i	$0.625618\pi$
$822$		−	6.69162e6i		−	0.345424i
$823$	4.45575e6			0.229309			0.114654	−	0.993405i	$-0.463424\pi$
$823$	4.45575e6			0.229309			0.114654	+	0.993405i	$0.463424\pi$
$824$	1.26692e7			0.650026
$825$	−1.31138e7			−0.670803
$826$	1.45385e6			0.0741431
$827$		−	9.37369e6i		−	0.476592i	−0.971193	−	0.238296i	$-0.923411\pi$
$827$		−	9.37369e6i		−	0.476592i	0.971193	−	0.238296i	$-0.0765889\pi$
$828$			5.87775e6i			0.297944i
$829$		−	4.05029e6i		−	0.204692i	−0.994749	−	0.102346i	$-0.967365\pi$
$829$		−	4.05029e6i		−	0.204692i	0.994749	−	0.102346i	$-0.0326348\pi$
$830$	7.33656e6			0.369656
$831$			2.11256e7i			1.06122i
$832$			725469.i			0.0363338i
$833$		−	8.54378e6i		−	0.426617i
$834$			1.28239e6i			0.0638416i
$835$	−2.76733e7			−1.37355
$836$			1.30487e7i			0.645729i
$837$		−	7.95718e6i		−	0.392596i
$838$			4.08769e6i			0.201080i
$839$	2.21153e7			1.08465			0.542323	−	0.840170i	$-0.317545\pi$
$839$	2.21153e7			1.08465			0.542323	+	0.840170i	$0.317545\pi$
$840$	−9.69846e6			−0.474247
$841$	−3.05728e7			−1.49055
$842$	−263577.			−0.0128123
$843$			1.06061e7i			0.514030i
$844$	3.03279e7			1.46550
$845$		−	3.20403e7i		−	1.54367i
$846$		−	1.97552e6i		−	0.0948977i
$847$	1.19763e7			0.573607
$848$	2.54308e7			1.21442
$849$			1.31353e7i			0.625419i
$850$	4.20528e6			0.199640
$851$	3.86424e6	−	1.04327e7i	0.182911	−	0.493824i
$852$	−1.13771e7			−0.536949
$853$		−	1.41295e7i		−	0.664898i	−0.943121	−	0.332449i	$-0.892125\pi$
$853$		−	1.41295e7i		−	0.664898i	0.943121	−	0.332449i	$-0.107875\pi$
$854$	−4.08458e6			−0.191647
$855$	3.38420e7			1.58322
$856$			7.92913e6i			0.369863i
$857$			2.53580e7i			1.17940i	0.807621	+	0.589702i	$0.200755\pi$
$857$			2.53580e7i			1.17940i	−0.807621	+	0.589702i	$0.799245\pi$
$858$	−86733.1			−0.00402223
$859$			2.22970e7i			1.03101i	0.856886	+	0.515506i	$0.172396\pi$
$859$			2.22970e7i			1.03101i	−0.856886	+	0.515506i	$0.827604\pi$
$860$	−2.18744e7			−1.00853
$861$	−1.94895e7			−0.895970
$862$	−6.24456e6			−0.286242
$863$	−125737.			−0.00574694			−0.00287347	−	0.999996i	$-0.500915\pi$
$863$	−125737.			−0.00574694			−0.00287347	+	0.999996i	$0.500915\pi$
$864$		−	6.04578e6i		−	0.275529i
$865$			3.34549e7i			1.52026i
$866$			2.75705e6i			0.124925i
$867$	1.10129e7			0.497570
$868$		−	1.08416e7i		−	0.488422i
$869$			1.07321e7i			0.482097i
$870$		−	1.26488e7i		−	0.566569i
$871$		−	1.26516e6i		−	0.0565067i
$872$	−1.00886e7			−0.449303
$873$			849690.i			0.0377333i
$874$		−	3.83053e6i		−	0.169621i
$875$		−	9.24423e6i		−	0.408179i
$876$	−4.01391e6			−0.176729
$877$	−6.73604e6			−0.295737			−0.147869	−	0.989007i	$-0.547241\pi$
$877$	−6.73604e6			−0.295737			−0.147869	+	0.989007i	$0.547241\pi$
$878$	−6.78204e6			−0.296909
$879$	−4.20564e7			−1.83595
$880$		−	1.22223e7i		−	0.532041i
$881$	−2.67472e7			−1.16102			−0.580509	−	0.814254i	$-0.697146\pi$
$881$	−2.67472e7			−1.16102			−0.580509	+	0.814254i	$0.697146\pi$
$882$		−	1.36800e6i		−	0.0592128i
$883$			1.30359e7i			0.562652i	0.959612	+	0.281326i	$0.0907742\pi$
$883$			1.30359e7i			0.562652i	−0.959612	+	0.281326i	$0.909226\pi$
$884$	−790927.			−0.0340413
$885$	2.71720e7			1.16617
$886$			1.90988e6i			0.0817375i
$887$	3.76423e7			1.60645			0.803225	−	0.595675i	$-0.203115\pi$
$887$	3.76423e7			1.60645			0.803225	+	0.595675i	$0.203115\pi$
$888$	3.72418e6	−	1.00546e7i	0.158489	−	0.427889i
$889$	3.62266e6			0.153735
$890$		−	1.08965e7i		−	0.461117i
$891$	−1.12634e7			−0.475307
$892$	−3.05568e7			−1.28587
$893$		−	3.66114e7i		−	1.53634i
$894$		−	1.35040e6i		−	0.0565091i
$895$	−6.67938e7			−2.78727
$896$		−	1.09135e7i		−	0.454143i
$897$	−724043.			−0.0300458
$898$	−3.75227e6			−0.155276
$899$	2.87768e7			1.18752
$900$	−1.91478e7			−0.787975
$901$			2.55937e7i			1.05032i
$902$			1.82418e6i			0.0746536i
$903$			1.39923e7i			0.571043i
$904$	1.29306e6			0.0526257
$905$			1.89779e7i			0.770242i
$906$			1.95371e6i			0.0790753i
$907$			3.29270e7i			1.32903i	0.747275	+	0.664514i	$0.231362\pi$
$907$			3.29270e7i			1.32903i	−0.747275	+	0.664514i	$0.768638\pi$
$908$		−	3.28986e7i		−	1.32423i
$909$	1.45284e7			0.583186
$910$		−	216819.i		−	0.00867950i
$911$			2.83567e7i			1.13203i	0.824394	+	0.566017i	$0.191516\pi$
$911$			2.83567e7i			1.13203i	−0.824394	+	0.566017i	$0.808484\pi$
$912$			4.97060e7i			1.97889i
$913$	1.24910e7			0.495930
$914$	−3.09476e6			−0.122535
$915$	−7.63393e7			−3.01436
$916$	1.35314e7			0.532848
$917$		−	1.45821e7i		−	0.572659i
$918$	1.90958e6			0.0747878
$919$		−	4.60773e6i		−	0.179969i	−0.995943	−	0.0899846i	$-0.971318\pi$
$919$		−	4.60773e6i		−	0.179969i	0.995943	−	0.0899846i	$-0.0286818\pi$
$920$			7.57789e6i			0.295174i
$921$	−2.65500e7			−1.03137
$922$	283162.			0.0109700
$923$		−	517639.i		−	0.0199997i
$924$	−8.11349e6			−0.312628
$925$	3.39863e7	+	1.25884e7i	1.30602	+	0.483746i
$926$	−5.71281e6			−0.218939
$927$		−	2.74880e7i		−	1.05061i
$928$	2.18643e7			0.833422
$929$	2.22109e7			0.844360			0.422180	−	0.906512i	$-0.361265\pi$
$929$	2.22109e7			0.844360			0.422180	+	0.906512i	$0.361265\pi$
$930$			7.12538e6i			0.270147i
$931$		−	2.53526e7i		−	0.958623i
$932$	−3.13010e7			−1.18037
$933$		−	7.29915e6i		−	0.274516i
$934$	−376704.			−0.0141297
$935$	1.23006e7			0.460147
$936$	−257735.			−0.00961575
$937$	−2.95907e7			−1.10105			−0.550523	−	0.834820i	$-0.685572\pi$
$937$	−2.95907e7			−1.10105			−0.550523	+	0.834820i	$0.685572\pi$
$938$			4.16180e6i			0.154445i
$939$		−	2.50908e7i		−	0.928647i
$940$			3.55882e7i			1.31367i
$941$	2.05391e7			0.756148			0.378074	−	0.925775i	$-0.376586\pi$
$941$	2.05391e7			0.756148			0.378074	+	0.925775i	$0.376586\pi$
$942$			5.95061e6i			0.218491i
$943$			1.52281e7i			0.557657i
$944$			1.47406e7i			0.538377i
$945$		−	1.48863e7i		−	0.542259i
$946$	1.30965e6			0.0475801
$947$		−	4.21333e7i		−	1.52669i	−0.645991	−	0.763345i	$-0.723556\pi$
$947$		−	4.21333e7i		−	1.52669i	0.645991	−	0.763345i	$-0.276444\pi$
$948$			4.24259e7i			1.53324i
$949$		−	182626.i		−	0.00658259i
$950$	1.24786e7			0.448598
$951$	−1.74314e7			−0.625000
$952$	5.29507e6			0.189356
$953$	4.48612e7			1.60007			0.800034	−	0.599954i	$-0.204815\pi$
$953$	4.48612e7			1.60007			0.800034	+	0.599954i	$0.204815\pi$
$954$			4.09799e6i			0.145781i
$955$	1.03132e7			0.365920
$956$			8.85255e6i			0.313274i
$957$		−	2.15355e7i		−	0.760109i
$958$	6.67092e6			0.234840
$959$	2.84806e7			1.00001
$960$		−	4.46021e7i		−	1.56199i
$961$	1.24185e7			0.433773
$962$	224781.	+	83258.0i	0.00783107	+	0.00290060i
$963$	1.72036e7			0.597798
$964$		−	1.15199e7i		−	0.399262i
$965$	5.24130e7			1.81184
$966$	2.38177e6			0.0821216
$967$			1.77799e7i			0.611453i	0.952119	+	0.305727i	$0.0988994\pi$
$967$			1.77799e7i			0.611453i	−0.952119	+	0.305727i	$0.901101\pi$
$968$		−	9.01853e6i		−	0.309348i
$969$	−5.00246e7			−1.71149
$970$			538267.i			0.0183683i
$971$	1.29448e7			0.440603			0.220302	−	0.975432i	$-0.429296\pi$
$971$	1.29448e7			0.440603			0.220302	+	0.975432i	$0.429296\pi$
$972$	−2.96803e7			−1.00763
$973$	−5.45804e6			−0.184822
$974$	−5.58049e6			−0.188484
$975$		−	2.35870e6i		−	0.0794622i
$976$		−	4.14136e7i		−	1.39161i
$977$		−	4.56506e7i		−	1.53007i	−0.643990	−	0.765034i	$-0.722722\pi$
$977$		−	4.56506e7i		−	1.53007i	0.643990	−	0.765034i	$-0.277278\pi$
$978$	−1.06936e7			−0.357500
$979$		−	1.85520e7i		−	0.618634i
$980$			2.46440e7i			0.819684i
$981$			2.18890e7i			0.726194i
$982$		−	6.38049e6i		−	0.211142i
$983$	−5.14288e7			−1.69755			−0.848775	−	0.528755i	$-0.822659\pi$
$983$	−5.14288e7			−1.69755			−0.848775	+	0.528755i	$0.822659\pi$
$984$			1.46762e7i			0.483199i
$985$			4.23694e6i			0.139143i
$986$			6.90590e6i			0.226218i
$987$	2.27645e7			0.743815
$988$	−2.34697e6			−0.0764919
$989$	1.09328e7			0.355420
$990$	1.96953e6			0.0638668
$991$			2.27943e6i			0.0737295i	0.999320	+	0.0368648i	$0.0117371\pi$
$991$			2.27943e6i			0.0737295i	−0.999320	+	0.0368648i	$0.988263\pi$
$992$	−1.23166e7			−0.397386
$993$		−	2.21902e7i		−	0.714149i
$994$			1.70280e6i			0.0546635i
$995$	−3.47580e7			−1.11300
$996$	4.93792e7			1.57723
$997$		−	2.09171e7i		−	0.666444i	−0.942848	−	0.333222i	$-0.891864\pi$
$997$		−	2.09171e7i		−	0.666444i	0.942848	−	0.333222i	$-0.108136\pi$
$998$	−9.55349e6			−0.303624
$999$	1.54329e7	+	5.71629e6i	0.489253	+	0.181218i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.b.a.36.9	yes	16
3.2	odd	2		333.6.c.d.73.8		16
4.3	odd	2		592.6.g.c.369.3		16
37.36	even	2	inner	37.6.b.a.36.8	✓	16
111.110	odd	2		333.6.c.d.73.9		16
148.147	odd	2		592.6.g.c.369.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.b.a.36.8	✓	16	37.36	even	2	inner
37.6.b.a.36.9	yes	16	1.1	even	1	trivial
333.6.c.d.73.8		16	3.2	odd	2
333.6.c.d.73.9		16	111.110	odd	2
592.6.g.c.369.3		16	4.3	odd	2
592.6.g.c.369.4		16	148.147	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:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.04262i q^{2} +19.6295 q^{3} +30.9129 q^{4} -86.4713i q^{5} +20.4662i q^{6} -87.1072 q^{7} +65.5944i q^{8} +142.318 q^{9} +O(q^{10})\)	\(q+1.04262i q^{2} +19.6295 q^{3} +30.9129 q^{4} -86.4713i q^{5} +20.4662i q^{6} -87.1072 q^{7} +65.5944i q^{8} +142.318 q^{9} +90.1568 q^{10} +153.498 q^{11} +606.806 q^{12} +27.6086i q^{13} -90.8199i q^{14} -1697.39i q^{15} +920.824 q^{16} +926.725i q^{17} +148.384i q^{18} +2749.94i q^{19} -2673.08i q^{20} -1709.87 q^{21} +160.041i q^{22} +1336.01i q^{23} +1287.59i q^{24} -4352.28 q^{25} -28.7854 q^{26} -1976.33 q^{27} -2692.74 q^{28} -7147.31i q^{29} +1769.74 q^{30} +4026.24i q^{31} +3059.09i q^{32} +3013.10 q^{33} -966.224 q^{34} +7532.28i q^{35} +4399.48 q^{36} +(-7808.85 - 2892.37i) q^{37} -2867.14 q^{38} +541.945i q^{39} +5672.03 q^{40} +11398.2 q^{41} -1782.75i q^{42} -8183.21i q^{43} +4745.08 q^{44} -12306.5i q^{45} -1392.95 q^{46} -13313.5 q^{47} +18075.3 q^{48} -9219.33 q^{49} -4537.78i q^{50} +18191.2i q^{51} +853.464i q^{52} +27617.4 q^{53} -2060.57i q^{54} -13273.2i q^{55} -5713.75i q^{56} +53980.0i q^{57} +7451.94 q^{58} +16008.1i q^{59} -52471.3i q^{60} -44974.5i q^{61} -4197.84 q^{62} -12397.0 q^{63} +26276.9 q^{64} +2387.35 q^{65} +3141.52i q^{66} -45824.8 q^{67} +28647.8i q^{68} +26225.2i q^{69} -7853.31 q^{70} -18749.2 q^{71} +9335.29i q^{72} -6614.81 q^{73} +(3015.65 - 8141.68i) q^{74} -85433.3 q^{75} +85008.6i q^{76} -13370.8 q^{77} -565.043 q^{78} +69916.6i q^{79} -79624.8i q^{80} -73377.8 q^{81} +11884.0i q^{82} +81375.6 q^{83} -52857.2 q^{84} +80135.1 q^{85} +8531.99 q^{86} -140298. i q^{87} +10068.6i q^{88} -120861. i q^{89} +12831.0 q^{90} -2404.91i q^{91} +41300.0i q^{92} +79033.2i q^{93} -13881.0i q^{94} +237791. q^{95} +60048.5i q^{96} +5970.34i q^{97} -9612.27i q^{98} +21845.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9}+O(q^{10}) \) 16 * q - 18 * q^3 - 268 * q^4 + 190 * q^7 + 1394 * q^9	\( 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9} - 74 q^{10} - 1110 q^{11} + 1402 q^{12} + 2900 q^{16} - 7010 q^{21} - 12052 q^{25} + 4902 q^{26} + 4266 q^{27} - 16824 q^{28} + 19280 q^{30} - 2478 q^{33} + 20556 q^{34} - 51402 q^{36} - 11400 q^{37} + 12108 q^{38} + 16966 q^{40} + 3918 q^{41} + 125394 q^{44} + 17470 q^{46} + 3822 q^{47} - 78034 q^{48} - 32618 q^{49} - 24126 q^{53} - 164718 q^{58} - 81426 q^{62} + 219268 q^{63} + 158076 q^{64} + 98976 q^{65} + 23560 q^{67} - 222404 q^{70} - 50046 q^{71} - 196274 q^{73} + 141216 q^{74} + 214054 q^{75} - 239574 q^{77} - 90822 q^{78} + 317312 q^{81} - 215814 q^{83} + 438572 q^{84} - 346472 q^{85} + 197640 q^{86} - 857612 q^{90} - 132504 q^{95} - 574860 q^{99}+O(q^{100}) \) 16 * q - 18 * q^3 - 268 * q^4 + 190 * q^7 + 1394 * q^9 - 74 * q^10 - 1110 * q^11 + 1402 * q^12 + 2900 * q^16 - 7010 * q^21 - 12052 * q^25 + 4902 * q^26 + 4266 * q^27 - 16824 * q^28 + 19280 * q^30 - 2478 * q^33 + 20556 * q^34 - 51402 * q^36 - 11400 * q^37 + 12108 * q^38 + 16966 * q^40 + 3918 * q^41 + 125394 * q^44 + 17470 * q^46 + 3822 * q^47 - 78034 * q^48 - 32618 * q^49 - 24126 * q^53 - 164718 * q^58 - 81426 * q^62 + 219268 * q^63 + 158076 * q^64 + 98976 * q^65 + 23560 * q^67 - 222404 * q^70 - 50046 * q^71 - 196274 * q^73 + 141216 * q^74 + 214054 * q^75 - 239574 * q^77 - 90822 * q^78 + 317312 * q^81 - 215814 * q^83 + 438572 * q^84 - 346472 * q^85 + 197640 * q^86 - 857612 * q^90 - 132504 * q^95 - 574860 * q^99

Introduction

L-functions

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