LMFDB - Embedded newform 138.7.b.a.91.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,7,Mod(91,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("138.91");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	138.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:	$31.7474635395$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.3
Character	$\chi$	$=$	138.91
Dual form			138.7.b.a.91.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.65685 q^{2} -15.5885 q^{3} +32.0000 q^{4} -33.5442i q^{5} +88.1816 q^{6} -442.137i q^{7} -181.019 q^{8} +243.000 q^{9} +O(q^{10})$	\(q-5.65685 q^{2} -15.5885 q^{3} +32.0000 q^{4} -33.5442i q^{5} +88.1816 q^{6} -442.137i q^{7} -181.019 q^{8} +243.000 q^{9} +189.755i q^{10} -934.810i q^{11} -498.831 q^{12} +905.006 q^{13} +2501.10i q^{14} +522.902i q^{15} +1024.00 q^{16} -3778.41i q^{17} -1374.62 q^{18} +6057.54i q^{19} -1073.41i q^{20} +6892.23i q^{21} +5288.08i q^{22} +(11276.3 - 4569.58i) q^{23} +2821.81 q^{24} +14499.8 q^{25} -5119.49 q^{26} -3788.00 q^{27} -14148.4i q^{28} +1791.39 q^{29} -2957.98i q^{30} -3340.95 q^{31} -5792.62 q^{32} +14572.2i q^{33} +21373.9i q^{34} -14831.1 q^{35} +7776.00 q^{36} -40653.3i q^{37} -34266.6i q^{38} -14107.6 q^{39} +6072.15i q^{40} -44522.2 q^{41} -38988.3i q^{42} +10473.0i q^{43} -29913.9i q^{44} -8151.24i q^{45} +(-63788.4 + 25849.4i) q^{46} -185357. q^{47} -15962.6 q^{48} -77835.8 q^{49} -82023.2 q^{50} +58899.6i q^{51} +28960.2 q^{52} -58071.1i q^{53} +21428.1 q^{54} -31357.5 q^{55} +80035.3i q^{56} -94427.7i q^{57} -10133.6 q^{58} -362036. q^{59} +16732.9i q^{60} -380449. i q^{61} +18899.2 q^{62} -107439. i q^{63} +32768.0 q^{64} -30357.7i q^{65} -82433.1i q^{66} -131214. i q^{67} -120909. i q^{68} +(-175780. + 71232.7i) q^{69} +83897.5 q^{70} -218978. q^{71} -43987.7 q^{72} -6020.53 q^{73} +229970. i q^{74} -226029. q^{75} +193841. i q^{76} -413314. q^{77} +79804.9 q^{78} +899926. i q^{79} -34349.3i q^{80} +59049.0 q^{81} +251856. q^{82} -564276. i q^{83} +220551. i q^{84} -126744. q^{85} -59244.3i q^{86} -27925.0 q^{87} +169219. i q^{88} -770356. i q^{89} +46110.4i q^{90} -400136. i q^{91} +(360841. - 146227. i) q^{92} +52080.2 q^{93} +1.04854e6 q^{94} +203195. q^{95} +90298.0 q^{96} +1.70896e6i q^{97} +440306. q^{98} -227159. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 768 q^{4} + 5832 q^{9}+O(q^{10}) $ 24 * q + 768 * q^4 + 5832 * q^9	$ 24 q + 768 q^{4} + 5832 q^{9} - 768 q^{13} + 24576 q^{16} - 44104 q^{23} - 119448 q^{25} - 53888 q^{26} + 3456 q^{29} + 50976 q^{31} + 149008 q^{35} + 186624 q^{36} + 11664 q^{39} - 3920 q^{41} - 150720 q^{46} + 441088 q^{47} - 32472 q^{49} + 8320 q^{50} - 24576 q^{52} + 826176 q^{55} - 307200 q^{58} - 1210160 q^{59} + 783744 q^{62} + 786432 q^{64} + 361584 q^{69} - 2480064 q^{70} + 1531264 q^{71} + 593472 q^{73} + 23328 q^{75} + 1068784 q^{77} + 171072 q^{78} + 1417176 q^{81} + 1454592 q^{82} - 1318272 q^{85} + 697248 q^{87} - 1411328 q^{92} - 983664 q^{93} + 1115712 q^{94} + 4047632 q^{95} - 2409344 q^{98}+O(q^{100}) $ 24 * q + 768 * q^4 + 5832 * q^9 - 768 * q^13 + 24576 * q^16 - 44104 * q^23 - 119448 * q^25 - 53888 * q^26 + 3456 * q^29 + 50976 * q^31 + 149008 * q^35 + 186624 * q^36 + 11664 * q^39 - 3920 * q^41 - 150720 * q^46 + 441088 * q^47 - 32472 * q^49 + 8320 * q^50 - 24576 * q^52 + 826176 * q^55 - 307200 * q^58 - 1210160 * q^59 + 783744 * q^62 + 786432 * q^64 + 361584 * q^69 - 2480064 * q^70 + 1531264 * q^71 + 593472 * q^73 + 23328 * q^75 + 1068784 * q^77 + 171072 * q^78 + 1417176 * q^81 + 1454592 * q^82 - 1318272 * q^85 + 697248 * q^87 - 1411328 * q^92 - 983664 * q^93 + 1115712 * q^94 + 4047632 * q^95 - 2409344 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-1$

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 $

$2$	−5.65685			−0.707107
$2$	−5.65685			−0.707107
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	32.0000			0.500000
$5$		−	33.5442i		−	0.268354i	−0.990957	−	0.134177i	$-0.957161\pi$
$5$		−	33.5442i		−	0.268354i	0.990957	−	0.134177i	$-0.0428390\pi$
$6$	88.1816			0.408248
$7$		−	442.137i		−	1.28903i	−0.764592	−	0.644514i	$-0.777060\pi$
$7$		−	442.137i		−	1.28903i	0.764592	−	0.644514i	$-0.222940\pi$
$8$	−181.019			−0.353553
$9$	243.000			0.333333
$10$			189.755i			0.189755i
$11$		−	934.810i		−	0.702336i	−0.936312	−	0.351168i	$-0.885785\pi$
$11$		−	934.810i		−	0.702336i	0.936312	−	0.351168i	$-0.114215\pi$
$12$	−498.831			−0.288675
$13$	905.006			0.411928			0.205964	−	0.978560i	$-0.433967\pi$
$13$	905.006			0.411928			0.205964	+	0.978560i	$0.433967\pi$
$14$			2501.10i			0.911481i
$15$			522.902i			0.154934i
$16$	1024.00			0.250000
$17$		−	3778.41i		−	0.769064i	−0.923112	−	0.384532i	$-0.874363\pi$
$17$		−	3778.41i		−	0.769064i	0.923112	−	0.384532i	$-0.125637\pi$
$18$	−1374.62			−0.235702
$19$			6057.54i			0.883152i	0.897224	+	0.441576i	$0.145581\pi$
$19$			6057.54i			0.883152i	−0.897224	+	0.441576i	$0.854419\pi$
$20$		−	1073.41i		−	0.134177i
$21$			6892.23i			0.744221i
$22$			5288.08i			0.496627i
$23$	11276.3	−	4569.58i	0.926793	−	0.375572i
$23$	11276.3	−	4569.58i	0.926793	−	0.375572i
$24$	2821.81			0.204124
$25$	14499.8			0.927986
$26$	−5119.49			−0.291277
$27$	−3788.00			−0.192450
$28$		−	14148.4i		−	0.644514i
$29$	1791.39			0.0734507			0.0367253	−	0.999325i	$-0.488307\pi$
$29$	1791.39			0.0734507			0.0367253	+	0.999325i	$0.488307\pi$
$30$		−	2957.98i		−	0.109555i
$31$	−3340.95			−0.112146			−0.0560731	−	0.998427i	$-0.517858\pi$
$31$	−3340.95			−0.112146			−0.0560731	+	0.998427i	$0.517858\pi$
$32$	−5792.62			−0.176777
$33$			14572.2i			0.405494i
$34$			21373.9i			0.543810i
$35$	−14831.1			−0.345915
$36$	7776.00			0.166667
$37$		−	40653.3i		−	0.802585i	−0.915950	−	0.401293i	$-0.868561\pi$
$37$		−	40653.3i		−	0.802585i	0.915950	−	0.401293i	$-0.131439\pi$
$38$		−	34266.6i		−	0.624483i
$39$	−14107.6			−0.237827
$40$			6072.15i			0.0948774i
$41$	−44522.2			−0.645989			−0.322995	−	0.946401i	$-0.604690\pi$
$41$	−44522.2			−0.645989			−0.322995	+	0.946401i	$0.604690\pi$
$42$		−	38988.3i		−	0.526244i
$43$			10473.0i			0.131724i	0.997829	+	0.0658622i	$0.0209798\pi$
$43$			10473.0i			0.131724i	−0.997829	+	0.0658622i	$0.979020\pi$
$44$		−	29913.9i		−	0.351168i
$45$		−	8151.24i		−	0.0894512i
$46$	−63788.4	+	25849.4i	−0.655342	+	0.265569i
$47$	−185357.			−1.78532			−0.892658	−	0.450734i	$-0.851162\pi$
$47$	−185357.			−1.78532			−0.892658	+	0.450734i	$0.851162\pi$
$48$	−15962.6			−0.144338
$49$	−77835.8			−0.661593
$50$	−82023.2			−0.656185
$51$			58899.6i			0.444019i
$52$	28960.2			0.205964
$53$		−	58071.1i		−	0.390061i	−0.980797	−	0.195030i	$-0.937519\pi$
$53$		−	58071.1i		−	0.390061i	0.980797	−	0.195030i	$-0.0624806\pi$
$54$	21428.1			0.136083
$55$	−31357.5			−0.188475
$56$			80035.3i			0.455740i
$57$		−	94427.7i		−	0.509888i
$58$	−10133.6			−0.0519375
$59$	−362036.			−1.76277			−0.881385	−	0.472398i	$-0.843389\pi$
$59$	−362036.			−1.76277			−0.881385	+	0.472398i	$0.843389\pi$
$60$			16732.9i			0.0774670i
$61$		−	380449.i		−	1.67613i	−0.545574	−	0.838063i	$-0.683688\pi$
$61$		−	380449.i		−	1.67613i	0.545574	−	0.838063i	$-0.316312\pi$
$62$	18899.2			0.0792993
$63$		−	107439.i		−	0.429676i
$64$	32768.0			0.125000
$65$		−	30357.7i		−	0.110542i
$66$		−	82433.1i		−	0.286728i
$67$		−	131214.i		−	0.436271i	−0.975919	−	0.218135i	$-0.930003\pi$
$67$		−	131214.i		−	0.436271i	0.975919	−	0.218135i	$-0.0699974\pi$
$68$		−	120909.i		−	0.384532i
$69$	−175780.	+	71232.7i	−0.535084	+	0.216836i
$70$	83897.5			0.244599
$71$	−218978.			−0.611823			−0.305911	−	0.952060i	$-0.598961\pi$
$71$	−218978.			−0.611823			−0.305911	+	0.952060i	$0.598961\pi$
$72$	−43987.7			−0.117851
$73$	−6020.53			−0.0154763			−0.00773813	−	0.999970i	$-0.502463\pi$
$73$	−6020.53			−0.0154763			−0.00773813	+	0.999970i	$0.502463\pi$
$74$			229970.i			0.567513i
$75$	−226029.			−0.535773
$76$			193841.i			0.441576i
$77$	−413314.			−0.905331
$78$	79804.9			0.168169
$79$			899926.i			1.82526i	0.408782	+	0.912632i	$0.365954\pi$
$79$			899926.i			1.82526i	−0.408782	+	0.912632i	$0.634046\pi$
$80$		−	34349.3i		−	0.0670884i
$81$	59049.0			0.111111
$82$	251856.			0.456783
$83$		−	564276.i		−	0.986864i	−0.869784	−	0.493432i	$-0.835742\pi$
$83$		−	564276.i		−	0.986864i	0.869784	−	0.493432i	$-0.164258\pi$
$84$			220551.i			0.372110i
$85$	−126744.			−0.206381
$86$		−	59244.3i		−	0.0931432i
$87$	−27925.0			−0.0424068
$88$			169219.i			0.248313i
$89$		−	770356.i		−	1.09275i	−0.837540	−	0.546376i	$-0.816007\pi$
$89$		−	770356.i		−	1.09275i	0.837540	−	0.546376i	$-0.183993\pi$
$90$			46110.4i			0.0632516i
$91$		−	400136.i		−	0.530987i
$92$	360841.	−	146227.i	0.463397	−	0.187786i
$93$	52080.2			0.0647476
$94$	1.04854e6			1.26241
$95$	203195.			0.236997
$96$	90298.0			0.102062
$97$			1.70896e6i			1.87247i	0.351369	+	0.936237i	$0.385716\pi$
$97$			1.70896e6i			1.87247i	−0.351369	+	0.936237i	$0.614284\pi$
$98$	440306.			0.467817
$99$		−	227159.i		−	0.234112i
$100$	463993.			0.463993
$101$	−553006.			−0.536742			−0.268371	−	0.963316i	$-0.586485\pi$
$101$	−553006.			−0.536742			−0.268371	+	0.963316i	$0.586485\pi$
$102$		−	333187.i		−	0.313969i
$103$		−	1.07926e6i		−	0.987672i	−0.869555	−	0.493836i	$-0.835594\pi$
$103$		−	1.07926e6i		−	0.987672i	0.869555	−	0.493836i	$-0.164406\pi$
$104$	−163824.			−0.145639
$105$	231194.			0.199714
$106$			328500.i			0.275815i
$107$		−	391614.i		−	0.319674i	−0.987143	−	0.159837i	$-0.948903\pi$
$107$		−	391614.i		−	0.319674i	0.987143	−	0.159837i	$-0.0510968\pi$
$108$	−121216.			−0.0962250
$109$			358043.i			0.276475i	0.990399	+	0.138238i	$0.0441437\pi$
$109$			358043.i			0.276475i	−0.990399	+	0.138238i	$0.955856\pi$
$110$	177385.			0.133272
$111$			633723.i			0.463373i
$112$		−	452748.i		−	0.322257i
$113$			844463.i			0.585255i	0.956227	+	0.292627i	$0.0945296\pi$
$113$			844463.i			0.585255i	−0.956227	+	0.292627i	$0.905470\pi$
$114$			534164.i			0.360545i
$115$	−153283.	−	378254.i	−0.100786	−	0.248708i
$116$	57324.4			0.0367253
$117$	219916.			0.137309
$118$	2.04799e6			1.24647
$119$	−1.67057e6			−0.991345
$120$		−	94655.5i		−	0.0547775i
$121$	897692.			0.506723
$122$			2.15214e6i			1.18520i
$123$	694033.			0.372962
$124$	−106910.			−0.0560731
$125$		−	1.01051e6i		−	0.517382i
$126$			607768.i			0.303827i
$127$	1.38363e6			0.675475			0.337738	−	0.941240i	$-0.390338\pi$
$127$	1.38363e6			0.675475			0.337738	+	0.941240i	$0.390338\pi$
$128$	−185364.			−0.0883883
$129$		−	163258.i		−	0.0760511i
$130$			171729.i			0.0781653i
$131$	−258442.			−0.114961			−0.0574804	−	0.998347i	$-0.518307\pi$
$131$	−258442.			−0.114961			−0.0574804	+	0.998347i	$0.518307\pi$
$132$			466312.i			0.202747i
$133$	2.67826e6			1.13841
$134$			742259.i			0.308490i
$135$			127065.i			0.0516447i
$136$			683966.i			0.271905i
$137$			2.73504e6i			1.06366i	0.846851	+	0.531830i	$0.178495\pi$
$137$			2.73504e6i			1.06366i	−0.846851	+	0.531830i	$0.821505\pi$
$138$	994362.	−	402953.i	0.378362	−	0.153326i
$139$	816705.			0.304103			0.152051	−	0.988373i	$-0.451412\pi$
$139$	816705.			0.304103			0.152051	+	0.988373i	$0.451412\pi$
$140$	−474596.			−0.172958
$141$	2.88943e6			1.03075
$142$	1.23873e6			0.432624
$143$		−	846008.i		−	0.289312i
$144$	248832.			0.0833333
$145$		−	60090.7i		−	0.0197108i
$146$	34057.2			0.0109434
$147$	1.21334e6			0.381971
$148$		−	1.30091e6i		−	0.401293i
$149$			355203.i			0.107379i	0.998558	+	0.0536893i	$0.0170980\pi$
$149$			355203.i			0.107379i	−0.998558	+	0.0536893i	$0.982902\pi$
$150$	1.27861e6			0.378849
$151$	−2.10126e6			−0.610308			−0.305154	−	0.952303i	$-0.598708\pi$
$151$	−2.10126e6			−0.610308			−0.305154	+	0.952303i	$0.598708\pi$
$152$		−	1.09653e6i		−	0.312242i
$153$		−	918154.i		−	0.256355i
$154$	2.33806e6			0.640166
$155$			112069.i			0.0300948i
$156$	−451445.			−0.118913
$157$			4.24779e6i			1.09765i	0.835938	+	0.548825i	$0.184925\pi$
$157$			4.24779e6i			1.09765i	−0.835938	+	0.548825i	$0.815075\pi$
$158$		−	5.09075e6i		−	1.29066i
$159$			905239.i			0.225202i
$160$			194309.i			0.0474387i
$161$	−2.02038e6	−	4.98566e6i	−0.484122	−	1.19466i
$162$	−334032.			−0.0785674
$163$	878525.			0.202858			0.101429	−	0.994843i	$-0.467659\pi$
$163$	878525.			0.202858			0.101429	+	0.994843i	$0.467659\pi$
$164$	−1.42471e6			−0.322995
$165$	488814.			0.108816
$166$			3.19203e6i			0.697818i
$167$	−5.95043e6			−1.27761			−0.638806	−	0.769368i	$-0.720571\pi$
$167$	−5.95043e6			−1.27761			−0.638806	+	0.769368i	$0.720571\pi$
$168$		−	1.24763e6i		−	0.263122i
$169$	−4.00777e6			−0.830315
$170$	716971.			0.145934
$171$			1.47198e6i			0.294384i
$172$			335136.i			0.0658622i
$173$	−2.66904e6			−0.515486			−0.257743	−	0.966214i	$-0.582979\pi$
$173$	−2.66904e6			−0.515486			−0.257743	+	0.966214i	$0.582979\pi$
$174$	157968.			0.0299861
$175$		−	6.41089e6i		−	1.19620i
$176$		−	957245.i		−	0.175584i
$177$	5.64358e6			1.01774
$178$			4.35779e6i			0.772692i
$179$	−6.73227e6			−1.17382			−0.586911	−	0.809651i	$-0.699656\pi$
$179$	−6.73227e6			−1.17382			−0.586911	+	0.809651i	$0.699656\pi$
$180$		−	260840.i		−	0.0447256i
$181$		−	1.03632e6i		−	0.174766i	−0.996175	−	0.0873829i	$-0.972150\pi$
$181$		−	1.03632e6i		−	0.174766i	0.996175	−	0.0873829i	$-0.0278503\pi$
$182$			2.26351e6i			0.375464i
$183$			5.93061e6i			0.967712i
$184$	−2.04123e6	+	827182.i	−0.327671	+	0.132785i
$185$	−1.36368e6			−0.215377
$186$	−294610.			−0.0457835
$187$	−3.53210e6			−0.540142
$188$	−5.93142e6			−0.892658
$189$			1.67481e6i			0.248074i
$190$	−1.14945e6			−0.167582
$191$		−	2.29492e6i		−	0.329357i	−0.986347	−	0.164679i	$-0.947341\pi$
$191$		−	2.29492e6i		−	0.329357i	0.986347	−	0.164679i	$-0.0526587\pi$
$192$	−510803.			−0.0721688
$193$	−2.79873e6			−0.389305			−0.194652	−	0.980872i	$-0.562358\pi$
$193$	−2.79873e6			−0.389305			−0.194652	+	0.980872i	$0.562358\pi$
$194$		−	9.66732e6i		−	1.32404i
$195$			473230.i			0.0638217i
$196$	−2.49075e6			−0.330797
$197$	234382.			0.0306566			0.0153283	−	0.999883i	$-0.495121\pi$
$197$	234382.			0.0306566			0.0153283	+	0.999883i	$0.495121\pi$
$198$			1.28500e6i			0.165542i
$199$		−	1.20699e7i		−	1.53159i	−0.643084	−	0.765796i	$-0.722345\pi$
$199$		−	1.20699e7i		−	1.53159i	0.643084	−	0.765796i	$-0.277655\pi$
$200$	−2.62474e6			−0.328093
$201$			2.04542e6i			0.251881i
$202$	3.12827e6			0.379534
$203$		−	792039.i		−	0.0946800i
$204$			1.88479e6i			0.222010i
$205$			1.49346e6i			0.173354i
$206$			6.10519e6i			0.698390i
$207$	2.74014e6	−	1.11041e6i	0.308931	−	0.125191i
$208$	926726.			0.102982
$209$	5.66265e6			0.620270
$210$	−1.30783e6			−0.141219
$211$	−9.75965e6			−1.03893			−0.519466	−	0.854491i	$-0.673869\pi$
$211$	−9.75965e6			−1.03893			−0.519466	+	0.854491i	$0.673869\pi$
$212$		−	1.85827e6i		−	0.195030i
$213$	3.41353e6			0.353236
$214$			2.21530e6i			0.226044i
$215$	351309.			0.0353487
$216$	685700.			0.0680414
$217$			1.47715e6i			0.144560i
$218$		−	2.02540e6i		−	0.195497i
$219$	93850.7			0.00893522
$220$	−1.00344e6			−0.0942373
$221$		−	3.41949e6i		−	0.316799i
$222$		−	3.58488e6i		−	0.327654i
$223$	4.34029e6			0.391385			0.195692	−	0.980665i	$-0.437305\pi$
$223$	4.34029e6			0.391385			0.195692	+	0.980665i	$0.437305\pi$
$224$			2.56113e6i			0.227870i
$225$	3.52345e6			0.309329
$226$		−	4.77700e6i		−	0.413838i
$227$		−	3.60786e6i		−	0.308441i	−0.988036	−	0.154221i	$-0.950713\pi$
$227$		−	3.60786e6i		−	0.308441i	0.988036	−	0.154221i	$-0.0492867\pi$
$228$		−	3.02169e6i		−	0.254944i
$229$		−	3.02966e6i		−	0.252283i	−0.992012	−	0.126141i	$-0.959741\pi$
$229$		−	3.02966e6i		−	0.252283i	0.992012	−	0.126141i	$-0.0402593\pi$
$230$	867099.	+	2.13973e6i	0.0712665	+	0.175863i
$231$	6.44292e6			0.522693
$232$	−324276.			−0.0259687
$233$	8.26090e6			0.653070			0.326535	−	0.945185i	$-0.394119\pi$
$233$	8.26090e6			0.653070			0.326535	+	0.945185i	$0.394119\pi$
$234$	−1.24404e6			−0.0970924
$235$			6.21765e6i			0.479096i
$236$	−1.15852e7			−0.881385
$237$		−	1.40285e7i		−	1.05382i
$238$	9.45020e6			0.700987
$239$	−1.15513e7			−0.846133			−0.423067	−	0.906099i	$-0.639046\pi$
$239$	−1.15513e7			−0.846133			−0.423067	+	0.906099i	$0.639046\pi$
$240$			535452.i			0.0387335i
$241$			1.58670e7i			1.13356i	0.823870	+	0.566779i	$0.191811\pi$
$241$			1.58670e7i			1.13356i	−0.823870	+	0.566779i	$0.808189\pi$
$242$	−5.07811e6			−0.358308
$243$	−920483.			−0.0641500
$244$		−	1.21744e7i		−	0.838063i
$245$			2.61094e6i			0.177541i
$246$	−3.92604e6			−0.263724
$247$			5.48211e6i			0.363795i
$248$	604776.			0.0396497
$249$			8.79619e6i			0.569766i
$250$			5.71632e6i			0.365844i
$251$			2.91662e7i			1.84441i	0.386698	+	0.922206i	$0.373615\pi$
$251$			2.91662e7i			1.84441i	−0.386698	+	0.922206i	$0.626385\pi$
$252$		−	3.43805e6i		−	0.214838i
$253$	−4.27169e6	−	1.05412e7i	−0.263778	−	0.650921i
$254$	−7.82701e6			−0.477633
$255$	1.97574e6			0.119154
$256$	1.04858e6			0.0625000
$257$	1.38551e7			0.816225			0.408112	−	0.912932i	$-0.366187\pi$
$257$	1.38551e7			0.816225			0.408112	+	0.912932i	$0.366187\pi$
$258$			923527.i			0.0537762i
$259$	−1.79743e7			−1.03455
$260$		−	971447.i		−	0.0552712i
$261$	435307.			0.0244836
$262$	1.46197e6			0.0812895
$263$			2.31379e6i			0.127191i	0.997976	+	0.0635954i	$0.0202567\pi$
$263$			2.31379e6i			0.127191i	−0.997976	+	0.0635954i	$0.979743\pi$
$264$		−	2.63786e6i		−	0.143364i
$265$	−1.94795e6			−0.104674
$266$	−1.51505e7			−0.804976
$267$			1.20087e7i			0.630901i
$268$		−	4.19885e6i		−	0.218135i
$269$	1.36375e7			0.700611			0.350305	−	0.936636i	$-0.386078\pi$
$269$	1.36375e7			0.700611			0.350305	+	0.936636i	$0.386078\pi$
$270$		−	718790.i		−	0.0365183i
$271$	8.01260e6			0.402593			0.201296	−	0.979530i	$-0.435485\pi$
$271$	8.01260e6			0.402593			0.201296	+	0.979530i	$0.435485\pi$
$272$		−	3.86909e6i		−	0.192266i
$273$			6.23751e6i			0.306565i
$274$		−	1.54717e7i		−	0.752121i
$275$		−	1.35545e7i		−	0.651759i
$276$	−5.62496e6	+	2.27945e6i	−0.267542	+	0.108418i
$277$	1.52540e7			0.717701			0.358851	−	0.933395i	$-0.383169\pi$
$277$	1.52540e7			0.717701			0.358851	+	0.933395i	$0.383169\pi$
$278$	−4.61998e6			−0.215033
$279$	−811850.			−0.0373821
$280$	2.68472e6			0.122300
$281$		−	1.91616e7i		−	0.863602i	−0.901969	−	0.431801i	$-0.857878\pi$
$281$		−	1.91616e7i		−	0.863602i	0.901969	−	0.431801i	$-0.142122\pi$
$282$	−1.63451e7			−0.728853
$283$		−	1.35387e7i		−	0.597336i	−0.954357	−	0.298668i	$-0.903458\pi$
$283$		−	1.35387e7i		−	0.597336i	0.954357	−	0.298668i	$-0.0965424\pi$
$284$	−7.00730e6			−0.305911
$285$	−3.16750e6			−0.136830
$286$			4.78575e6i			0.204575i
$287$			1.96849e7i			0.832698i
$288$	−1.40761e6			−0.0589256
$289$	9.86117e6			0.408540
$290$			339924.i			0.0139376i
$291$		−	2.66400e7i		−	1.08107i
$292$	−192657.			−0.00773813
$293$		−	2.43156e7i		−	0.966678i	−0.875433	−	0.483339i	$-0.839424\pi$
$293$		−	2.43156e7i		−	0.966678i	0.875433	−	0.483339i	$-0.160576\pi$
$294$	−6.86369e6			−0.270094
$295$			1.21442e7i			0.473046i
$296$			7.35904e6i			0.283757i
$297$			3.54106e6i			0.135165i
$298$		−	2.00933e6i		−	0.0759281i
$299$	1.02051e7	−	4.13550e6i	0.381772	−	0.154708i
$300$	−7.23294e6			−0.267887
$301$	4.63050e6			0.169796
$302$	1.18865e7			0.431553
$303$	8.62051e6			0.309888
$304$			6.20292e6i			0.220788i
$305$	−1.27618e7			−0.449794
$306$			5.19386e6i			0.181270i
$307$	1.80406e7			0.623499			0.311749	−	0.950164i	$-0.399085\pi$
$307$	1.80406e7			0.623499			0.311749	+	0.950164i	$0.399085\pi$
$308$	−1.32260e7			−0.452666
$309$			1.68239e7i			0.570233i
$310$		−	633960.i		−	0.0212803i
$311$	2.01821e7			0.670941			0.335471	−	0.942051i	$-0.391105\pi$
$311$	2.01821e7			0.670941			0.335471	+	0.942051i	$0.391105\pi$
$312$	2.55376e6			0.0840845
$313$			2.18089e6i			0.0711216i	0.999368	+	0.0355608i	$0.0113217\pi$
$313$			2.18089e6i			0.0711216i	−0.999368	+	0.0355608i	$0.988678\pi$
$314$		−	2.40291e7i		−	0.776155i
$315$	−3.60396e6			−0.115305
$316$			2.87976e7i			0.912632i
$317$	4.95925e7			1.55682			0.778409	−	0.627757i	$-0.216027\pi$
$317$	4.95925e7			1.55682			0.778409	+	0.627757i	$0.216027\pi$
$318$		−	5.12080e6i		−	0.159242i
$319$		−	1.67461e6i		−	0.0515871i
$320$		−	1.09918e6i		−	0.0335442i
$321$			6.10466e6i			0.184564i
$322$	1.14290e7	+	2.82032e7i	0.342326	+	0.844754i
$323$	2.28879e7			0.679201
$324$	1.88957e6			0.0555556
$325$	1.31224e7			0.382264
$326$	−4.96969e6			−0.143442
$327$		−	5.58134e6i		−	0.159623i
$328$	8.05938e6			0.228392
$329$			8.19531e7i			2.30132i
$330$	−2.76515e6			−0.0769444
$331$	−4.39198e7			−1.21109			−0.605545	−	0.795811i	$-0.707045\pi$
$331$	−4.39198e7			−1.21109			−0.605545	+	0.795811i	$0.707045\pi$
$332$		−	1.80568e7i		−	0.493432i
$333$		−	9.87876e6i		−	0.267528i
$334$	3.36607e7			0.903408
$335$	−4.40147e6			−0.117075
$336$			7.05764e6i			0.186055i
$337$			5.83063e7i			1.52344i	0.647906	+	0.761720i	$0.275645\pi$
$337$			5.83063e7i			1.52344i	−0.647906	+	0.761720i	$0.724355\pi$
$338$	2.26714e7			0.587122
$339$		−	1.31639e7i		−	0.337897i
$340$	−4.05580e6			−0.103191
$341$			3.12315e6i			0.0787643i
$342$		−	8.32679e6i		−	0.208161i
$343$		−	1.76029e7i		−	0.436216i
$344$		−	1.89582e6i		−	0.0465716i
$345$	2.38944e6	+	5.89640e6i	0.0581888	+	0.143592i
$346$	1.50984e7			0.364504
$347$	2.74516e7			0.657022			0.328511	−	0.944500i	$-0.393453\pi$
$347$	2.74516e7			0.657022			0.328511	+	0.944500i	$0.393453\pi$
$348$	−893600.			−0.0212034
$349$	−2.13234e7			−0.501627			−0.250814	−	0.968035i	$-0.580698\pi$
$349$	−2.13234e7			−0.501627			−0.250814	+	0.968035i	$0.580698\pi$
$350$			3.62655e7i			0.845841i
$351$	−3.42816e6			−0.0792756
$352$			5.41500e6i			0.124157i
$353$	5.14945e7			1.17068			0.585338	−	0.810789i	$-0.300962\pi$
$353$	5.14945e7			1.17068			0.585338	+	0.810789i	$0.300962\pi$
$354$	−3.19249e7			−0.719648
$355$			7.34545e6i			0.164185i
$356$		−	2.46514e7i		−	0.546376i
$357$	2.60417e7			0.572353
$358$	3.80835e7			0.830018
$359$		−	1.63451e7i		−	0.353267i	−0.984277	−	0.176633i	$-0.943479\pi$
$359$		−	1.63451e7i		−	0.353267i	0.984277	−	0.176633i	$-0.0565207\pi$
$360$			1.47553e6i			0.0316258i
$361$	1.03521e7			0.220042
$362$			5.86229e6i			0.123578i
$363$	−1.39936e7			−0.292557
$364$		−	1.28044e7i		−	0.265493i
$365$			201954.i			0.00415311i
$366$		−	3.35486e7i		−	0.684275i
$367$		−	9.15308e7i		−	1.85169i	−0.377899	−	0.925847i	$-0.623353\pi$
$367$		−	9.15308e7i		−	1.85169i	0.377899	−	0.925847i	$-0.376647\pi$
$368$	1.15469e7	−	4.67925e6i	0.231698	−	0.0938929i
$369$	−1.08189e7			−0.215330
$370$	7.71416e6			0.152294
$371$	−2.56754e7			−0.502799
$372$	1.66657e6			0.0323738
$373$		−	4.32183e7i		−	0.832802i	−0.909181	−	0.416401i	$-0.863291\pi$
$373$		−	4.32183e7i		−	0.832802i	0.909181	−	0.416401i	$-0.136709\pi$
$374$	1.99806e7			0.381938
$375$			1.57523e7i			0.298711i
$376$	3.35532e7			0.631205
$377$	1.62122e6			0.0302564
$378$		−	9.47416e6i		−	0.175415i
$379$			3.38091e7i			0.621036i	0.950568	+	0.310518i	$0.100502\pi$
$379$			3.38091e7i			0.621036i	−0.950568	+	0.310518i	$0.899498\pi$
$380$	6.50225e6			0.118499
$381$	−2.15687e7			−0.389986
$382$			1.29820e7i			0.232891i
$383$			9.68623e7i			1.72408i	0.506837	+	0.862042i	$0.330815\pi$
$383$			9.68623e7i			1.72408i	−0.506837	+	0.862042i	$0.669185\pi$
$384$	2.88954e6			0.0510310
$385$			1.38643e7i			0.242949i
$386$	1.58320e7			0.275280
$387$			2.54494e6i			0.0439081i
$388$			5.46866e7i			0.936237i
$389$		−	7.26653e7i		−	1.23446i	−0.786781	−	0.617232i	$-0.788254\pi$
$389$		−	7.26653e7i		−	1.23446i	0.786781	−	0.617232i	$-0.211746\pi$
$390$		−	2.67699e6i		−	0.0451287i
$391$	−1.72658e7	−	4.26065e7i	−0.288839	−	0.712764i
$392$	1.40898e7			0.233909
$393$	4.02872e6			0.0663726
$394$	−1.32586e6			−0.0216775
$395$	3.01873e7			0.489816
$396$		−	7.26908e6i		−	0.117056i
$397$	−1.07291e8			−1.71472			−0.857359	−	0.514719i	$-0.827896\pi$
$397$	−1.07291e8			−1.71472			−0.857359	+	0.514719i	$0.827896\pi$
$398$			6.82774e7i			1.08300i
$399$	−4.17500e7			−0.657260
$400$	1.48478e7			0.231997
$401$			5.88999e7i			0.913443i	0.889610	+	0.456721i	$0.150976\pi$
$401$			5.88999e7i			0.913443i	−0.889610	+	0.456721i	$0.849024\pi$
$402$		−	1.15707e7i		−	0.178107i
$403$	−3.02358e6			−0.0461962
$404$	−1.76962e7			−0.268371
$405$		−	1.98075e6i		−	0.0298171i
$406$			4.48045e6i			0.0669489i
$407$	−3.80031e7			−0.563685
$408$		−	1.06620e7i		−	0.156985i
$409$	−4.04025e7			−0.590525			−0.295263	−	0.955416i	$-0.595407\pi$
$409$	−4.04025e7			−0.590525			−0.295263	+	0.955416i	$0.595407\pi$
$410$		−	8.44830e6i		−	0.122579i
$411$		−	4.26351e7i		−	0.614104i
$412$		−	3.45362e7i		−	0.493836i
$413$			1.60069e8i			2.27226i
$414$	−1.55006e7	+	6.28142e6i	−0.218447	+	0.0885231i
$415$	−1.89282e7			−0.264829
$416$	−5.24235e6			−0.0728193
$417$	−1.27312e7			−0.175574
$418$	−3.20328e7			−0.438597
$419$		−	1.08968e8i		−	1.48135i	−0.671864	−	0.740674i	$-0.734506\pi$
$419$		−	1.08968e8i		−	1.48135i	0.671864	−	0.740674i	$-0.265494\pi$
$420$	7.39822e6			0.0998572
$421$		−	1.28899e8i		−	1.72745i	−0.503967	−	0.863723i	$-0.668127\pi$
$421$		−	1.28899e8i		−	1.72745i	0.503967	−	0.863723i	$-0.331873\pi$
$422$	5.52089e7			0.734636
$423$	−4.50417e7			−0.595106
$424$			1.05120e7i			0.137907i
$425$		−	5.47862e7i		−	0.713681i
$426$	−1.93098e7			−0.249776
$427$	−1.68210e8			−2.16057
$428$		−	1.25317e7i		−	0.159837i
$429$			1.31880e7i			0.167034i
$430$	−1.98730e6			−0.0249953
$431$		−	1.24171e8i		−	1.55092i	−0.631399	−	0.775458i	$-0.717519\pi$
$431$		−	1.24171e8i		−	1.55092i	0.631399	−	0.775458i	$-0.282481\pi$
$432$	−3.87891e6			−0.0481125
$433$		−	9.57857e7i		−	1.17988i	−0.807448	−	0.589939i	$-0.799152\pi$
$433$		−	9.57857e7i		−	1.17988i	0.807448	−	0.589939i	$-0.200848\pi$
$434$		−	8.35605e6i		−	0.102219i
$435$			936722.i			0.0113800i
$436$			1.14574e7i			0.138238i
$437$	2.76804e7	+	6.83066e7i	0.331687	+	0.818500i
$438$	−530900.			−0.00631815
$439$	−1.34328e8			−1.58772			−0.793859	−	0.608102i	$-0.791931\pi$
$439$	−1.34328e8			−1.58772			−0.793859	+	0.608102i	$0.791931\pi$
$440$	5.67631e6			0.0666358
$441$	−1.89141e7			−0.220531
$442$			1.93435e7i			0.224011i
$443$	−1.27436e8			−1.46582			−0.732909	−	0.680327i	$-0.761838\pi$
$443$	−1.27436e8			−1.46582			−0.732909	+	0.680327i	$0.761838\pi$
$444$			2.02791e7i			0.231686i
$445$	−2.58410e7			−0.293244
$446$	−2.45524e7			−0.276751
$447$		−	5.53706e6i		−	0.0619950i
$448$		−	1.44879e7i		−	0.161129i
$449$	1.09130e8			1.20560			0.602801	−	0.797891i	$-0.294051\pi$
$449$	1.09130e8			1.20560			0.602801	+	0.797891i	$0.294051\pi$
$450$	−1.99316e7			−0.218728
$451$			4.16198e7i			0.453702i
$452$			2.70228e7i			0.292627i
$453$	3.27554e7			0.352362
$454$			2.04092e7i			0.218101i
$455$	−1.34223e7			−0.142492
$456$			1.70932e7i			0.180273i
$457$		−	2.47924e7i		−	0.259759i	−0.991530	−	0.129879i	$-0.958541\pi$
$457$		−	2.47924e7i		−	0.259759i	0.991530	−	0.129879i	$-0.0414590\pi$
$458$			1.71384e7i			0.178391i
$459$			1.43126e7i			0.148006i
$460$	−4.90505e6	−	1.21041e7i	−0.0503930	−	0.124354i
$461$	1.30389e7			0.133088			0.0665440	−	0.997783i	$-0.478803\pi$
$461$	1.30389e7			0.133088			0.0665440	+	0.997783i	$0.478803\pi$
$462$	−3.64467e7			−0.369600
$463$	−5.04368e7			−0.508165			−0.254083	−	0.967183i	$-0.581773\pi$
$463$	−5.04368e7			−0.508165			−0.254083	+	0.967183i	$0.581773\pi$
$464$	1.83438e6			0.0183627
$465$		−	1.74699e6i		−	0.0173753i
$466$	−4.67307e7			−0.461790
$467$		−	1.41528e8i		−	1.38961i	−0.719199	−	0.694804i	$-0.755491\pi$
$467$		−	1.41528e8i		−	1.38961i	0.719199	−	0.694804i	$-0.244509\pi$
$468$	7.03733e6			0.0686547
$469$	−5.80145e7			−0.562365
$470$		−	3.51724e7i		−	0.338772i
$471$		−	6.62164e7i		−	0.633728i
$472$	6.55355e7			0.623234
$473$	9.79027e6			0.0925148
$474$			7.93570e7i			0.745161i
$475$			8.78331e7i			0.819553i
$476$	−5.34584e7			−0.495673
$477$		−	1.41113e7i		−	0.130020i
$478$	6.53443e7			0.598306
$479$		−	1.16403e7i		−	0.105915i	−0.998597	−	0.0529575i	$-0.983135\pi$
$479$		−	1.16403e7i		−	0.105915i	0.998597	−	0.0529575i	$-0.0168648\pi$
$480$		−	3.02897e6i		−	0.0273887i
$481$		−	3.67915e7i		−	0.330607i
$482$		−	8.97573e7i		−	0.801546i
$483$	3.14946e7	+	7.77188e7i	0.279508	+	0.689739i
$484$	2.87261e7			0.253362
$485$	5.73256e7			0.502485
$486$	5.20704e6			0.0453609
$487$	−3.71781e7			−0.321885			−0.160942	−	0.986964i	$-0.551453\pi$
$487$	−3.71781e7			−0.321885			−0.160942	+	0.986964i	$0.551453\pi$
$488$			6.88686e7i			0.592600i
$489$	−1.36949e7			−0.117120
$490$		−	1.47697e7i		−	0.125540i
$491$	1.93343e7			0.163337			0.0816685	−	0.996660i	$-0.473975\pi$
$491$	1.93343e7			0.163337			0.0816685	+	0.996660i	$0.473975\pi$
$492$	2.22090e7			0.186481
$493$		−	6.76861e6i		−	0.0564883i
$494$		−	3.10115e7i		−	0.257242i
$495$	−7.61986e6			−0.0628249
$496$	−3.42113e6			−0.0280365
$497$			9.68182e7i			0.788657i
$498$		−	4.97588e7i		−	0.402886i
$499$	−1.16003e8			−0.933611			−0.466805	−	0.884360i	$-0.654595\pi$
$499$	−1.16003e8			−0.933611			−0.466805	+	0.884360i	$0.654595\pi$
$500$		−	3.23364e7i		−	0.258691i
$501$	9.27581e7			0.737630
$502$		−	1.64989e8i		−	1.30420i
$503$			1.79808e8i			1.41288i	0.707774	+	0.706439i	$0.249699\pi$
$503$			1.79808e8i			1.41288i	−0.707774	+	0.706439i	$0.750301\pi$
$504$			1.94486e7i			0.151913i
$505$			1.85501e7i			0.144037i
$506$	2.41643e7	+	5.96300e7i	0.186519	+	0.460271i
$507$	6.24750e7			0.479383
$508$	4.42762e7			0.337738
$509$	8.87698e7			0.673150			0.336575	−	0.941657i	$-0.390731\pi$
$509$	8.87698e7			0.673150			0.336575	+	0.941657i	$0.390731\pi$
$510$	−1.11765e7			−0.0842548
$511$			2.66190e6i			0.0199493i
$512$	−5.93164e6			−0.0441942
$513$		−	2.29459e7i		−	0.169963i
$514$	−7.83762e7			−0.577158
$515$	−3.62028e7			−0.265045
$516$		−	5.22426e6i		−	0.0380255i
$517$			1.73274e8i			1.25389i
$518$	1.01678e8			0.731541
$519$	4.16062e7			0.297616
$520$			5.49533e6i			0.0390826i
$521$		−	1.45342e8i		−	1.02773i	−0.857872	−	0.513863i	$-0.828214\pi$
$521$		−	1.45342e8i		−	1.02773i	0.857872	−	0.513863i	$-0.171786\pi$
$522$	−2.46247e6			−0.0173125
$523$			1.34862e8i			0.942723i	0.881940	+	0.471361i	$0.156237\pi$
$523$			1.34862e8i			0.942723i	−0.881940	+	0.471361i	$0.843763\pi$
$524$	−8.27015e6			−0.0574804
$525$			9.99358e7i			0.690627i
$526$		−	1.30888e7i		−	0.0899375i
$527$			1.26235e7i			0.0862476i
$528$			1.49220e7i			0.101374i
$529$	1.06274e8	−	1.03056e8i	0.717892	−	0.696155i
$530$	1.10193e7			0.0740159
$531$	−8.79748e7			−0.587590
$532$	8.57044e7			0.569204
$533$	−4.02929e7			−0.266101
$534$		−	6.79313e7i		−	0.446114i
$535$	−1.31364e7			−0.0857856
$536$			2.37523e7i			0.154245i
$537$	1.04946e8			0.677707
$538$	−7.71452e7			−0.495407
$539$			7.27617e7i			0.464661i
$540$			4.06609e6i			0.0258223i
$541$	1.11207e8			0.702328			0.351164	−	0.936314i	$-0.385786\pi$
$541$	1.11207e8			0.702328			0.351164	+	0.936314i	$0.385786\pi$
$542$	−4.53261e7			−0.284676
$543$			1.61546e7i			0.100901i
$544$			2.18869e7i			0.135953i
$545$	1.20103e7			0.0741931
$546$		−	3.52847e7i		−	0.216774i
$547$	1.80007e8			1.09983			0.549916	−	0.835220i	$-0.314660\pi$
$547$	1.80007e8			1.09983			0.549916	+	0.835220i	$0.314660\pi$
$548$			8.75214e7i			0.531830i
$549$		−	9.24490e7i		−	0.558709i
$550$			7.66761e7i			0.460863i
$551$			1.08514e7i			0.0648681i
$552$	3.18196e7	−	1.28945e7i	0.189181	−	0.0766632i
$553$	3.97890e8			2.35282
$554$	−8.62895e7			−0.507491
$555$	2.12577e7			0.124348
$556$	2.61346e7			0.152051
$557$			7.76980e7i			0.449619i	0.974403	+	0.224809i	$0.0721759\pi$
$557$			7.76980e7i			0.449619i	−0.974403	+	0.224809i	$0.927824\pi$
$558$	4.59252e6			0.0264331
$559$			9.47813e6i			0.0542609i
$560$	−1.51871e7			−0.0864789
$561$	5.50599e7			0.311851
$562$			1.08395e8i			0.610659i
$563$		−	1.00614e8i		−	0.563808i	−0.959443	−	0.281904i	$-0.909034\pi$
$563$		−	1.00614e8i		−	0.563808i	0.959443	−	0.281904i	$-0.0909660\pi$
$564$	9.24617e7			0.515377
$565$	2.83268e7			0.157055
$566$			7.65867e7i			0.422380i
$567$		−	2.61077e7i		−	0.143225i
$568$	3.96393e7			0.216312
$569$		−	6.69611e7i		−	0.363485i	−0.983346	−	0.181742i	$-0.941826\pi$
$569$		−	6.69611e7i		−	0.363485i	0.983346	−	0.181742i	$-0.0581737\pi$
$570$	1.79181e7			0.0967537
$571$		−	8.57505e7i		−	0.460605i	−0.973119	−	0.230302i	$-0.926028\pi$
$571$		−	8.57505e7i		−	0.460605i	0.973119	−	0.230302i	$-0.0739715\pi$
$572$		−	2.70723e7i		−	0.144656i
$573$			3.57742e7i			0.190154i
$574$		−	1.11355e8i		−	0.588807i
$575$	1.63504e8	−	6.62579e7i	0.860052	−	0.348525i
$576$	7.96262e6			0.0416667
$577$	2.24145e8			1.16681			0.583406	−	0.812181i	$-0.301720\pi$
$577$	2.24145e8			1.16681			0.583406	+	0.812181i	$0.301720\pi$
$578$	−5.57832e7			−0.288882
$579$	4.36279e7			0.224765
$580$		−	1.92290e6i		−	0.00985538i
$581$	−2.49487e8			−1.27210
$582$			1.50699e8i			0.764434i
$583$	−5.42854e7			−0.273954
$584$	1.08983e6			0.00547168
$585$		−	7.37692e6i		−	0.0368475i
$586$			1.37550e8i			0.683544i
$587$	−1.67841e8			−0.829821			−0.414910	−	0.909862i	$-0.636187\pi$
$587$	−1.67841e8			−0.829821			−0.414910	+	0.909862i	$0.636187\pi$
$588$	3.88269e7			0.190986
$589$		−	2.02379e7i		−	0.0990421i
$590$		−	6.86980e7i		−	0.334494i
$591$	−3.65365e6			−0.0176996
$592$		−	4.16290e7i		−	0.200646i
$593$	3.19387e8			1.53163			0.765813	−	0.643064i	$-0.222337\pi$
$593$	3.19387e8			1.53163			0.765813	+	0.643064i	$0.222337\pi$
$594$		−	2.00312e7i		−	0.0955759i
$595$			5.60381e7i			0.266031i
$596$			1.13665e7i			0.0536893i
$597$			1.88150e8i			0.884265i
$598$	−5.77288e7	+	2.33939e7i	−0.269954	+	0.109395i
$599$	−1.64679e7			−0.0766229			−0.0383114	−	0.999266i	$-0.512198\pi$
$599$	−1.64679e7			−0.0766229			−0.0383114	+	0.999266i	$0.512198\pi$
$600$	4.09157e7			0.189424
$601$	−4.28796e7			−0.197527			−0.0987636	−	0.995111i	$-0.531489\pi$
$601$	−4.28796e7			−0.197527			−0.0987636	+	0.995111i	$0.531489\pi$
$602$	−2.61941e7			−0.120064
$603$		−	3.18850e7i		−	0.145424i
$604$	−6.72404e7			−0.305154
$605$		−	3.01124e7i		−	0.135981i
$606$	−4.87650e7			−0.219124
$607$	3.76005e8			1.68123			0.840616	−	0.541632i	$-0.182193\pi$
$607$	3.76005e8			1.68123			0.840616	+	0.541632i	$0.182193\pi$
$608$		−	3.50890e7i		−	0.156121i
$609$			1.23467e7i			0.0546635i
$610$	7.21919e7			0.318053
$611$	−1.67749e8			−0.735422
$612$		−	2.93809e7i		−	0.128177i
$613$			2.00965e8i			0.872448i	0.899838	+	0.436224i	$0.143685\pi$
$613$			2.00965e8i			0.872448i	−0.899838	+	0.436224i	$0.856315\pi$
$614$	−1.02053e8			−0.440880
$615$		−	2.32808e7i		−	0.100086i
$616$	7.48178e7			0.320083
$617$			2.35132e8i			1.00105i	0.865721	+	0.500526i	$0.166860\pi$
$617$			2.35132e8i			1.00105i	−0.865721	+	0.500526i	$0.833140\pi$
$618$		−	9.51705e7i		−	0.403215i
$619$			3.57033e8i			1.50535i	0.658394	+	0.752674i	$0.271236\pi$
$619$			3.57033e8i			1.50535i	−0.658394	+	0.752674i	$0.728764\pi$
$620$			3.58622e6i			0.0150474i
$621$	−4.27146e7	+	1.73095e7i	−0.178361	+	0.0722788i
$622$	−1.14167e8			−0.474427
$623$	−3.40603e8			−1.40859
$624$	−1.44462e7			−0.0594567
$625$	1.92662e8			0.789145
$626$		−	1.23370e7i		−	0.0502906i
$627$	−8.82720e7			−0.358113
$628$			1.35929e8i			0.548825i
$629$	−1.53605e8			−0.617239
$630$	2.03871e7			0.0815330
$631$		−	3.29998e8i		−	1.31348i	−0.754118	−	0.656739i	$-0.771935\pi$
$631$		−	3.29998e8i		−	1.31348i	0.754118	−	0.656739i	$-0.228065\pi$
$632$		−	1.62904e8i		−	0.645328i
$633$	1.52138e8			0.599828
$634$	−2.80537e8			−1.10084
$635$		−	4.64128e7i		−	0.181266i
$636$			2.89676e7i			0.112601i
$637$	−7.04419e7			−0.272529
$638$			9.47301e6i			0.0364776i
$639$	−5.32117e7			−0.203941
$640$			6.21788e6i			0.0237193i
$641$		−	7.09804e7i		−	0.269503i	−0.990879	−	0.134752i	$-0.956976\pi$
$641$		−	7.09804e7i		−	0.269503i	0.990879	−	0.134752i	$-0.0430237\pi$
$642$		−	3.45332e7i		−	0.130506i
$643$		−	4.40900e8i		−	1.65847i	−0.558901	−	0.829234i	$-0.688777\pi$
$643$		−	4.40900e8i		−	1.65847i	0.558901	−	0.829234i	$-0.311223\pi$
$644$	−6.46521e7	−	1.59541e8i	−0.242061	−	0.597331i
$645$	−5.47636e6			−0.0204086
$646$	−1.29473e8			−0.480267
$647$	−5.91668e7			−0.218457			−0.109228	−	0.994017i	$-0.534838\pi$
$647$	−5.91668e7			−0.218457			−0.109228	+	0.994017i	$0.534838\pi$
$648$	−1.06890e7			−0.0392837
$649$			3.38435e8i			1.23806i
$650$	−7.42315e7			−0.270301
$651$		−	2.30266e7i		−	0.0834615i
$652$	2.81128e7			0.101429
$653$	9.12786e7			0.327816			0.163908	−	0.986476i	$-0.447590\pi$
$653$	9.12786e7			0.327816			0.163908	+	0.986476i	$0.447590\pi$
$654$			3.15728e7i			0.112870i
$655$			8.66924e6i			0.0308501i
$656$	−4.55908e7			−0.161497
$657$	−1.46299e6			−0.00515875
$658$		−	4.63597e8i		−	1.62728i
$659$		−	4.35911e8i		−	1.52315i	−0.648079	−	0.761573i	$-0.724428\pi$
$659$		−	4.35911e8i		−	1.52315i	0.648079	−	0.761573i	$-0.275572\pi$
$660$	1.56421e7			0.0544079
$661$			3.14795e8i			1.08999i	0.838438	+	0.544997i	$0.183469\pi$
$661$			3.14795e8i			1.08999i	−0.838438	+	0.544997i	$0.816531\pi$
$662$	2.48448e8			0.856370
$663$			5.33045e7i			0.182904i
$664$			1.02145e8i			0.348909i
$665$		−	8.98402e7i		−	0.305496i
$666$			5.58827e7i			0.189171i
$667$	2.02002e7	−	8.18589e6i	0.0680736	−	0.0275860i
$668$	−1.90414e8			−0.638806
$669$	−6.76584e7			−0.225966
$670$	2.48985e7			0.0827844
$671$	−3.55647e8			−1.17720
$672$		−	3.99240e7i		−	0.131561i
$673$	−1.37241e8			−0.450235			−0.225118	−	0.974332i	$-0.572277\pi$
$673$	−1.37241e8			−0.450235			−0.225118	+	0.974332i	$0.572277\pi$
$674$		−	3.29830e8i		−	1.07724i
$675$	−5.49251e7			−0.178591
$676$	−1.28249e8			−0.415158
$677$		−	7.80267e7i		−	0.251465i	−0.992064	−	0.125732i	$-0.959872\pi$
$677$		−	7.80267e7i		−	0.251465i	0.992064	−	0.125732i	$-0.0401281\pi$
$678$			7.44661e7i			0.238929i
$679$	7.55592e8			2.41367
$680$	2.29431e7			0.0729668
$681$			5.62410e7i			0.178079i
$682$		−	1.76672e7i		−	0.0556948i
$683$	1.03635e7			0.0325269			0.0162635	−	0.999868i	$-0.494823\pi$
$683$	1.03635e7			0.0325269			0.0162635	+	0.999868i	$0.494823\pi$
$684$			4.71034e7i			0.147192i
$685$	9.17449e7			0.285437
$686$			9.95769e7i			0.308451i
$687$			4.72277e7i			0.145656i
$688$			1.07244e7i			0.0329311i
$689$		−	5.25547e7i		−	0.160677i
$690$	−1.35167e7	−	3.33551e7i	−0.0411457	−	0.101535i
$691$	−3.56193e7			−0.107957			−0.0539786	−	0.998542i	$-0.517190\pi$
$691$	−3.56193e7			−0.107957			−0.0539786	+	0.998542i	$0.517190\pi$
$692$	−8.54093e7			−0.257743
$693$	−1.00435e8			−0.301777
$694$	−1.55290e8			−0.464585
$695$		−	2.73957e7i		−	0.0816071i
$696$	5.05496e6			0.0149931
$697$			1.68223e8i			0.496807i
$698$	1.20624e8			0.354704
$699$	−1.28775e8			−0.377050
$700$		−	2.05148e8i		−	0.598100i
$701$			8.06295e7i			0.234067i	0.993128	+	0.117033i	$0.0373385\pi$
$701$			8.06295e7i			0.234067i	−0.993128	+	0.117033i	$0.962662\pi$
$702$	1.93926e7			0.0560563
$703$	2.46259e8			0.708805
$704$		−	3.06318e7i		−	0.0877921i
$705$		−	9.69236e7i		−	0.276606i
$706$	−2.91297e8			−0.827793
$707$			2.44504e8i			0.691876i
$708$	1.80595e8			0.508868
$709$			1.38794e8i			0.389432i	0.980860	+	0.194716i	$0.0623785\pi$
$709$			1.38794e8i			0.389432i	−0.980860	+	0.194716i	$0.937621\pi$
$710$		−	4.15521e7i		−	0.116096i
$711$			2.18682e8i			0.608421i
$712$			1.39449e8i			0.386346i
$713$	−3.76735e7	+	1.52667e7i	−0.103936	+	0.0421189i
$714$	−1.47314e8			−0.404715
$715$	−2.83787e7			−0.0776380
$716$	−2.15433e8			−0.586911
$717$	1.80068e8			0.488515
$718$			9.24616e7i			0.249797i
$719$	−3.57049e8			−0.960597			−0.480299	−	0.877105i	$-0.659472\pi$
$719$	−3.57049e8			−0.960597			−0.480299	+	0.877105i	$0.659472\pi$
$720$		−	8.34687e6i		−	0.0223628i
$721$	−4.77179e8			−1.27314
$722$	−5.85601e7			−0.155593
$723$		−	2.47342e8i		−	0.654460i
$724$		−	3.31621e7i		−	0.0873829i
$725$	2.59748e7			0.0681612
$726$	7.91599e7			0.206869
$727$		−	3.59873e8i		−	0.936584i	−0.883574	−	0.468292i	$-0.844870\pi$
$727$		−	3.59873e8i		−	0.936584i	0.883574	−	0.468292i	$-0.155130\pi$
$728$			7.24324e7i			0.187732i
$729$	1.43489e7			0.0370370
$730$		−	1.14242e6i		−	0.00293669i
$731$	3.95713e7			0.101304
$732$			1.89779e8i			0.483856i
$733$		−	1.46603e8i		−	0.372247i	−0.982526	−	0.186124i	$-0.940408\pi$
$733$		−	1.46603e8i		−	0.372247i	0.982526	−	0.186124i	$-0.0595925\pi$
$734$			5.17776e8i			1.30934i
$735$		−	4.07005e7i		−	0.102503i
$736$	−6.53193e7	+	2.64698e7i	−0.163835	+	0.0663923i
$737$	−1.22660e8			−0.306409
$738$	6.12009e7			0.152261
$739$	4.58201e8			1.13533			0.567665	−	0.823259i	$-0.307847\pi$
$739$	4.58201e8			1.13533			0.567665	+	0.823259i	$0.307847\pi$
$740$	−4.36379e7			−0.107688
$741$		−	8.54577e7i		−	0.210037i
$742$	1.45242e8			0.355533
$743$			2.69220e8i			0.656359i	0.944615	+	0.328179i	$0.106435\pi$
$743$			2.69220e8i			0.656359i	−0.944615	+	0.328179i	$0.893565\pi$
$744$	−9.42752e6			−0.0228917
$745$	1.19150e7			0.0288154
$746$			2.44480e8i			0.588880i
$747$		−	1.37119e8i		−	0.328955i
$748$	−1.13027e8			−0.270071
$749$	−1.73147e8			−0.412069
$750$		−	8.91086e7i		−	0.211220i
$751$			3.67021e8i			0.866505i	0.901273	+	0.433252i	$0.142634\pi$
$751$			3.67021e8i			0.866505i	−0.901273	+	0.433252i	$0.857366\pi$
$752$	−1.89806e8			−0.446329
$753$		−	4.54655e8i		−	1.06487i
$754$	−9.17099e6			−0.0213945
$755$			7.04852e7i			0.163779i
$756$			5.35940e7i			0.124037i
$757$		−	3.77306e8i		−	0.869774i	−0.900485	−	0.434887i	$-0.856788\pi$
$757$		−	3.77306e8i		−	0.869774i	0.900485	−	0.434887i	$-0.143212\pi$
$758$		−	1.91253e8i		−	0.439138i
$759$	6.65890e7	+	1.64321e8i	0.152292	+	0.375809i
$760$	−3.67823e7			−0.0837912
$761$	−7.71458e8			−1.75049			−0.875243	−	0.483684i	$-0.839298\pi$
$761$	−7.71458e8			−1.75049			−0.875243	+	0.483684i	$0.839298\pi$
$762$	1.22011e8			0.275762
$763$	1.58304e8			0.356384
$764$		−	7.34374e7i		−	0.164679i
$765$	−3.07988e7			−0.0687937
$766$		−	5.47936e8i		−	1.21911i
$767$	−3.27645e8			−0.726135
$768$	−1.63457e7			−0.0360844
$769$		−	9.35216e7i		−	0.205652i	−0.994699	−	0.102826i	$-0.967212\pi$
$769$		−	9.35216e7i		−	0.205652i	0.994699	−	0.102826i	$-0.0327885\pi$
$770$		−	7.84282e7i		−	0.171791i
$771$	−2.15979e8			−0.471248
$772$	−8.95595e7			−0.194652
$773$		−	2.30059e8i		−	0.498082i	−0.968493	−	0.249041i	$-0.919885\pi$
$773$		−	2.30059e8i		−	0.498082i	0.968493	−	0.249041i	$-0.0801153\pi$
$774$		−	1.43964e7i		−	0.0310477i
$775$	−4.84430e7			−0.104070
$776$		−	3.09354e8i		−	0.662019i
$777$	2.80192e8			0.597300
$778$			4.11057e8i			0.872898i
$779$		−	2.69695e8i		−	0.570507i
$780$			1.51434e7i			0.0319108i
$781$			2.04703e8i			0.429705i
$782$	9.76699e7	+	2.41019e8i	0.204240	+	0.504000i
$783$	−6.78577e6			−0.0141356
$784$	−7.97039e7			−0.165398
$785$	1.42489e8			0.294558
$786$	−2.27899e7			−0.0469325
$787$			3.93660e8i			0.807602i	0.914847	+	0.403801i	$0.132311\pi$
$787$			3.93660e8i			0.807602i	−0.914847	+	0.403801i	$0.867689\pi$
$788$	7.50021e6			0.0153283
$789$		−	3.60684e7i		−	0.0734337i
$790$	−1.70765e8			−0.346352
$791$	3.73368e8			0.754410
$792$			4.11201e7i			0.0827711i
$793$		−	3.44308e8i		−	0.690443i
$794$	6.06931e8			1.21249
$795$	3.03655e7			0.0604337
$796$		−	3.86235e8i		−	0.765796i
$797$		−	8.21907e8i		−	1.62348i	−0.584017	−	0.811741i	$-0.698520\pi$
$797$		−	8.21907e8i		−	1.62348i	0.584017	−	0.811741i	$-0.301480\pi$
$798$	2.36173e8			0.464753
$799$			7.00355e8i			1.37302i
$800$	−8.39917e7			−0.164046
$801$		−	1.87197e8i		−	0.364251i
$802$		−	3.33188e8i		−	0.645902i
$803$			5.62805e6i			0.0108695i
$804$			6.54536e7i			0.125940i
$805$	−1.67240e8	+	6.77720e7i	−0.320592	+	0.129916i
$806$	1.71039e7			0.0326656
$807$	−2.12587e8			−0.404498
$808$	1.00105e8			0.189767
$809$	8.40293e8			1.58703			0.793516	−	0.608550i	$-0.208249\pi$
$809$	8.40293e8			1.58703			0.793516	+	0.608550i	$0.208249\pi$
$810$			1.12048e7i			0.0210839i
$811$	−1.05882e9			−1.98499			−0.992497	−	0.122266i	$-0.960984\pi$
$811$	−1.05882e9			−1.98499			−0.992497	+	0.122266i	$0.960984\pi$
$812$		−	2.53452e7i		−	0.0473400i
$813$	−1.24904e8			−0.232437
$814$	2.14978e8			0.398585
$815$		−	2.94694e7i		−	0.0544376i
$816$			6.03132e7i			0.111005i
$817$	−6.34407e7			−0.116333
$818$	2.28551e8			0.417564
$819$		−	9.72331e7i		−	0.176996i
$820$			4.77908e7i			0.0866768i
$821$	−3.91303e8			−0.707106			−0.353553	−	0.935415i	$-0.615027\pi$
$821$	−3.91303e8			−0.707106			−0.353553	+	0.935415i	$0.615027\pi$
$822$			2.41181e8i			0.434237i
$823$	8.78390e8			1.57575			0.787876	−	0.615834i	$-0.211181\pi$
$823$	8.78390e8			1.57575			0.787876	+	0.615834i	$0.211181\pi$
$824$			1.95366e8i			0.349195i
$825$			2.11294e8i			0.376293i
$826$		−	9.05489e8i		−	1.60673i
$827$			1.02440e8i			0.181114i	0.995891	+	0.0905571i	$0.0288648\pi$
$827$			1.02440e8i			0.181114i	−0.995891	+	0.0905571i	$0.971135\pi$
$828$	8.76845e7	−	3.55330e7i	0.154466	−	0.0625953i
$829$	−2.80924e8			−0.493090			−0.246545	−	0.969131i	$-0.579295\pi$
$829$	−2.80924e8			−0.493090			−0.246545	+	0.969131i	$0.579295\pi$
$830$	1.07074e8			0.187262
$831$	−2.37786e8			−0.414365
$832$	2.96552e7			0.0514910
$833$			2.94096e8i			0.508808i
$834$	7.20184e7			0.124150
$835$			1.99603e8i			0.342852i
$836$	1.81205e8			0.310135
$837$	1.26555e7			0.0215825
$838$			6.16417e8i			1.04747i
$839$			3.69035e8i			0.624859i	0.949941	+	0.312429i	$0.101143\pi$
$839$			3.69035e8i			0.624859i	−0.949941	+	0.312429i	$0.898857\pi$
$840$	−4.18506e7			−0.0706097
$841$	−5.91614e8			−0.994605
$842$			7.29165e8i			1.22149i
$843$			2.98700e8i			0.498601i
$844$	−3.12309e8			−0.519466
$845$			1.34438e8i			0.222818i
$846$	2.54795e8			0.420803
$847$		−	3.96902e8i		−	0.653181i
$848$		−	5.94648e7i		−	0.0975152i
$849$			2.11048e8i			0.344872i
$850$			3.09917e8i			0.504649i
$851$	−1.85769e8	−	4.58419e8i	−0.301428	−	0.743831i
$852$	1.09233e8			0.176618
$853$	−8.81446e8			−1.42020			−0.710098	−	0.704103i	$-0.751349\pi$
$853$	−8.81446e8			−1.42020			−0.710098	+	0.704103i	$0.751349\pi$
$854$	9.51541e8			1.52776
$855$	4.93765e7			0.0789991
$856$			7.08897e7i			0.113022i
$857$	−7.62975e8			−1.21218			−0.606091	−	0.795395i	$-0.707263\pi$
$857$	−7.62975e8			−1.21218			−0.606091	+	0.795395i	$0.707263\pi$
$858$		−	7.46024e7i		−	0.118111i
$859$	4.04509e8			0.638188			0.319094	−	0.947723i	$-0.396622\pi$
$859$	4.04509e8			0.638188			0.319094	+	0.947723i	$0.396622\pi$
$860$	1.12419e7			0.0176744
$861$		−	3.06857e8i		−	0.480759i
$862$			7.02418e8i			1.09666i
$863$	9.09614e8			1.41522			0.707612	−	0.706602i	$-0.249773\pi$
$863$	9.09614e8			1.41522			0.707612	+	0.706602i	$0.249773\pi$
$864$	2.19424e7			0.0340207
$865$			8.95309e7i			0.138333i
$866$			5.41846e8i			0.834299i
$867$	−1.53720e8			−0.235871
$868$			4.72690e7i			0.0722798i
$869$	8.41260e8			1.28195
$870$		−	5.29890e6i		−	0.00804688i
$871$		−	1.18750e8i		−	0.179712i
$872$		−	6.48127e7i		−	0.0977487i
$873$			4.15276e8i			0.624158i
$874$	−1.56584e8	−	3.86401e8i	−0.234538	−	0.578767i
$875$	−4.46784e8			−0.666920
$876$	3.00322e6			0.00446761
$877$	1.51423e8			0.224487			0.112244	−	0.993681i	$-0.464196\pi$
$877$	1.51423e8			0.224487			0.112244	+	0.993681i	$0.464196\pi$
$878$	7.59874e8			1.12269
$879$			3.79042e8i			0.558112i
$880$	−3.21100e7			−0.0471186
$881$			3.76759e8i			0.550980i	0.961304	+	0.275490i	$0.0888401\pi$
$881$			3.76759e8i			0.550980i	−0.961304	+	0.275490i	$0.911160\pi$
$882$	1.06994e8			0.155939
$883$	1.10173e9			1.60027			0.800137	−	0.599817i	$-0.204760\pi$
$883$	1.10173e9			1.60027			0.800137	+	0.599817i	$0.204760\pi$
$884$		−	1.09424e8i		−	0.158400i
$885$		−	1.89310e8i		−	0.273113i
$886$	7.20885e8			1.03649
$887$	8.28677e8			1.18745			0.593724	−	0.804669i	$-0.297657\pi$
$887$	8.28677e8			1.18745			0.593724	+	0.804669i	$0.297657\pi$
$888$		−	1.14716e8i		−	0.163827i
$889$		−	6.11754e8i		−	0.870707i
$890$	1.46179e8			0.207355
$891$		−	5.51996e7i		−	0.0780374i
$892$	1.38889e8			0.195692
$893$		−	1.12281e9i		−	1.57671i
$894$			3.13223e7i			0.0438371i
$895$			2.25829e8i			0.315000i
$896$			8.19561e7i			0.113935i
$897$	−1.59082e8	+	6.44660e7i	−0.220416	+	0.0893210i
$898$	−6.17331e8			−0.852490
$899$	−5.98493e6			−0.00823721
$900$	1.12750e8			0.154664
$901$	−2.19417e8			−0.299982
$902$		−	2.35437e8i		−	0.320816i
$903$	−7.21823e7			−0.0980320
$904$		−	1.52864e8i		−	0.206919i
$905$	−3.47624e7			−0.0468990
$906$	−1.85293e8			−0.249157
$907$			4.23862e8i			0.568071i	0.958814	+	0.284036i	$0.0916734\pi$
$907$			4.23862e8i			0.568071i	−0.958814	+	0.284036i	$0.908327\pi$
$908$		−	1.15452e8i		−	0.154221i
$909$	−1.34380e8			−0.178914
$910$	7.59277e7			0.100757
$911$			8.76974e8i			1.15993i	0.814641	+	0.579965i	$0.196934\pi$
$911$			8.76974e8i			1.15993i	−0.814641	+	0.579965i	$0.803066\pi$
$912$		−	9.66940e7i		−	0.127472i
$913$	−5.27491e8			−0.693111
$914$			1.40247e8i			0.183677i
$915$	1.98938e8			0.259689
$916$		−	9.69492e7i		−	0.126141i
$917$			1.14267e8i			0.148188i
$918$		−	8.09643e7i		−	0.104656i
$919$		−	3.15963e8i		−	0.407089i	−0.979066	−	0.203544i	$-0.934754\pi$
$919$		−	3.15963e8i		−	0.407089i	0.979066	−	0.203544i	$-0.0652461\pi$
$920$	2.77472e7	+	6.84714e7i	0.0356332	+	0.0879317i
$921$	−2.81225e8			−0.359977
$922$	−7.37592e7			−0.0941074
$923$	−1.98176e8			−0.252027
$924$	2.06174e8			0.261347
$925$		−	5.89465e8i		−	0.744788i
$926$	2.85314e8			0.359327
$927$		−	2.62259e8i		−	0.329224i
$928$	−1.03768e7			−0.0129844
$929$	9.46020e8			1.17992			0.589961	−	0.807432i	$-0.299143\pi$
$929$	9.46020e8			1.17992			0.589961	+	0.807432i	$0.299143\pi$
$930$			9.88246e6i			0.0122862i
$931$		−	4.71494e8i		−	0.584288i
$932$	2.64349e8			0.326535
$933$	−3.14607e8			−0.387368
$934$			8.00604e8i			0.982601i
$935$			1.18481e8i			0.144949i
$936$	−3.98091e7			−0.0485462
$937$		−	1.35792e9i		−	1.65066i	−0.564653	−	0.825328i	$-0.690990\pi$
$937$		−	1.35792e9i		−	1.65066i	0.564653	−	0.825328i	$-0.309010\pi$
$938$	3.28180e8			0.397652
$939$		−	3.39968e7i		−	0.0410621i
$940$			1.98965e8i			0.239548i
$941$			3.71001e8i			0.445253i	0.974904	+	0.222626i	$0.0714630\pi$
$941$			3.71001e8i			0.445253i	−0.974904	+	0.222626i	$0.928537\pi$
$942$			3.74577e8i			0.448113i
$943$	−5.02046e8	+	2.03448e8i	−0.598698	+	0.242615i
$944$	−3.70725e8			−0.440693
$945$	5.61802e7			0.0665715
$946$	−5.53821e7			−0.0654178
$947$	7.04611e8			0.829658			0.414829	−	0.909899i	$-0.363841\pi$
$947$	7.04611e8			0.829658			0.414829	+	0.909899i	$0.363841\pi$
$948$		−	4.48911e8i		−	0.526908i
$949$	−5.44861e6			−0.00637510
$950$		−	4.96859e8i		−	0.579512i
$951$	−7.73070e8			−0.898830
$952$	3.02406e8			0.350493
$953$			1.26599e9i			1.46269i	0.682008	+	0.731345i	$0.261107\pi$
$953$			1.26599e9i			1.46269i	−0.682008	+	0.731345i	$0.738893\pi$
$954$			7.98254e7i			0.0919382i
$955$	−7.69812e7			−0.0883842
$956$	−3.69643e8			−0.423067
$957$			2.61046e7i			0.0297838i
$958$			6.58474e7i			0.0748932i
$959$	1.20926e9			1.37109
$960$			1.71345e7i			0.0193668i
$961$	−8.76342e8			−0.987423
$962$			2.08124e8i			0.233775i
$963$		−	9.51623e7i		−	0.106558i
$964$			5.07744e8i			0.566779i
$965$			9.38813e7i			0.104471i
$966$	−1.78160e8	−	4.39644e8i	−0.197642	−	0.487719i
$967$	1.26507e9			1.39906			0.699529	−	0.714604i	$-0.253393\pi$
$967$	1.26507e9			1.39906			0.699529	+	0.714604i	$0.253393\pi$
$968$	−1.62500e8			−0.179154
$969$	−3.56787e8			−0.392137
$970$	−3.24282e8			−0.355311
$971$		−	5.63224e8i		−	0.615211i	−0.951514	−	0.307605i	$-0.900472\pi$
$971$		−	5.63224e8i		−	0.615211i	0.951514	−	0.307605i	$-0.0995276\pi$
$972$	−2.94555e7			−0.0320750
$973$		−	3.61095e8i		−	0.391997i
$974$	2.10311e8			0.227607
$975$	−2.04558e8			−0.220700
$976$		−	3.89579e8i		−	0.419031i
$977$			1.08993e9i			1.16873i	0.811490	+	0.584366i	$0.198657\pi$
$977$			1.08993e9i			1.16873i	−0.811490	+	0.584366i	$0.801343\pi$
$978$	7.74698e7			0.0828163
$979$	−7.20137e8			−0.767480
$980$			8.35501e7i			0.0887705i
$981$			8.70045e7i			0.0921584i
$982$	−1.09372e8			−0.115497
$983$			2.78908e8i			0.293630i	0.989164	+	0.146815i	$0.0469023\pi$
$983$			2.78908e8i			0.293630i	−0.989164	+	0.146815i	$0.953098\pi$
$984$	−1.25633e8			−0.131862
$985$		−	7.86214e6i		−	0.00822682i
$986$			3.82890e7i			0.0399433i
$987$		−	1.27752e9i		−	1.32867i
$988$			1.75428e8i			0.181898i
$989$	4.78572e7	+	1.18097e8i	0.0494719	+	0.122081i
$990$	4.31044e7			0.0444239
$991$	2.66809e8			0.274145			0.137072	−	0.990561i	$-0.456231\pi$
$991$	2.66809e8			0.274145			0.137072	+	0.990561i	$0.456231\pi$
$992$	1.93528e7			0.0198248
$993$	6.84642e8			0.699223
$994$		−	5.47687e8i		−	0.557665i
$995$	−4.04874e8			−0.411008
$996$			2.81478e8i			0.284883i
$997$	−3.10128e8			−0.312936			−0.156468	−	0.987683i	$-0.550011\pi$
$997$	−3.10128e8			−0.312936			−0.156468	+	0.987683i	$0.550011\pi$
$998$	6.56209e8			0.660162
$999$			1.53995e8i			0.154458i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.7.b.a.91.3	✓	24
3.2	odd	2		414.7.b.c.91.20		24
23.22	odd	2	inner	138.7.b.a.91.4	yes	24
69.68	even	2		414.7.b.c.91.17		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.7.b.a.91.3	✓	24	1.1	even	1	trivial
138.7.b.a.91.4	yes	24	23.22	odd	2	inner
414.7.b.c.91.17		24	69.68	even	2
414.7.b.c.91.20		24	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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.65685 q^{2} -15.5885 q^{3} +32.0000 q^{4} -33.5442i q^{5} +88.1816 q^{6} -442.137i q^{7} -181.019 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-5.65685 q^{2} -15.5885 q^{3} +32.0000 q^{4} -33.5442i q^{5} +88.1816 q^{6} -442.137i q^{7} -181.019 q^{8} +243.000 q^{9} +189.755i q^{10} -934.810i q^{11} -498.831 q^{12} +905.006 q^{13} +2501.10i q^{14} +522.902i q^{15} +1024.00 q^{16} -3778.41i q^{17} -1374.62 q^{18} +6057.54i q^{19} -1073.41i q^{20} +6892.23i q^{21} +5288.08i q^{22} +(11276.3 - 4569.58i) q^{23} +2821.81 q^{24} +14499.8 q^{25} -5119.49 q^{26} -3788.00 q^{27} -14148.4i q^{28} +1791.39 q^{29} -2957.98i q^{30} -3340.95 q^{31} -5792.62 q^{32} +14572.2i q^{33} +21373.9i q^{34} -14831.1 q^{35} +7776.00 q^{36} -40653.3i q^{37} -34266.6i q^{38} -14107.6 q^{39} +6072.15i q^{40} -44522.2 q^{41} -38988.3i q^{42} +10473.0i q^{43} -29913.9i q^{44} -8151.24i q^{45} +(-63788.4 + 25849.4i) q^{46} -185357. q^{47} -15962.6 q^{48} -77835.8 q^{49} -82023.2 q^{50} +58899.6i q^{51} +28960.2 q^{52} -58071.1i q^{53} +21428.1 q^{54} -31357.5 q^{55} +80035.3i q^{56} -94427.7i q^{57} -10133.6 q^{58} -362036. q^{59} +16732.9i q^{60} -380449. i q^{61} +18899.2 q^{62} -107439. i q^{63} +32768.0 q^{64} -30357.7i q^{65} -82433.1i q^{66} -131214. i q^{67} -120909. i q^{68} +(-175780. + 71232.7i) q^{69} +83897.5 q^{70} -218978. q^{71} -43987.7 q^{72} -6020.53 q^{73} +229970. i q^{74} -226029. q^{75} +193841. i q^{76} -413314. q^{77} +79804.9 q^{78} +899926. i q^{79} -34349.3i q^{80} +59049.0 q^{81} +251856. q^{82} -564276. i q^{83} +220551. i q^{84} -126744. q^{85} -59244.3i q^{86} -27925.0 q^{87} +169219. i q^{88} -770356. i q^{89} +46110.4i q^{90} -400136. i q^{91} +(360841. - 146227. i) q^{92} +52080.2 q^{93} +1.04854e6 q^{94} +203195. q^{95} +90298.0 q^{96} +1.70896e6i q^{97} +440306. q^{98} -227159. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 768 q^{4} + 5832 q^{9}+O(q^{10}) \) 24 * q + 768 * q^4 + 5832 * q^9	\( 24 q + 768 q^{4} + 5832 q^{9} - 768 q^{13} + 24576 q^{16} - 44104 q^{23} - 119448 q^{25} - 53888 q^{26} + 3456 q^{29} + 50976 q^{31} + 149008 q^{35} + 186624 q^{36} + 11664 q^{39} - 3920 q^{41} - 150720 q^{46} + 441088 q^{47} - 32472 q^{49} + 8320 q^{50} - 24576 q^{52} + 826176 q^{55} - 307200 q^{58} - 1210160 q^{59} + 783744 q^{62} + 786432 q^{64} + 361584 q^{69} - 2480064 q^{70} + 1531264 q^{71} + 593472 q^{73} + 23328 q^{75} + 1068784 q^{77} + 171072 q^{78} + 1417176 q^{81} + 1454592 q^{82} - 1318272 q^{85} + 697248 q^{87} - 1411328 q^{92} - 983664 q^{93} + 1115712 q^{94} + 4047632 q^{95} - 2409344 q^{98}+O(q^{100}) \) 24 * q + 768 * q^4 + 5832 * q^9 - 768 * q^13 + 24576 * q^16 - 44104 * q^23 - 119448 * q^25 - 53888 * q^26 + 3456 * q^29 + 50976 * q^31 + 149008 * q^35 + 186624 * q^36 + 11664 * q^39 - 3920 * q^41 - 150720 * q^46 + 441088 * q^47 - 32472 * q^49 + 8320 * q^50 - 24576 * q^52 + 826176 * q^55 - 307200 * q^58 - 1210160 * q^59 + 783744 * q^62 + 786432 * q^64 + 361584 * q^69 - 2480064 * q^70 + 1531264 * q^71 + 593472 * q^73 + 23328 * q^75 + 1068784 * q^77 + 171072 * q^78 + 1417176 * q^81 + 1454592 * q^82 - 1318272 * q^85 + 697248 * q^87 - 1411328 * q^92 - 983664 * q^93 + 1115712 * q^94 + 4047632 * q^95 - 2409344 * q^98

Introduction

L-functions

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Database

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