LMFDB - Embedded newform 26.8.b.a.25.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,8,Mod(25,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("26.25");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 26 = 2 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	26.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.12201066259$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 17770x^{8} + 98320641x^{6} + 176057788072x^{4} + 109845194658832x^{2} + 14762086704451584 $ x^10 + 17770x^8 + 98320641x^6 + 176057788072x^4 + 109845194658832x^2 + 14762086704451584
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			25.8
Root			$13.5100i$ of defining polynomial
Character	$\chi$	$=$	26.25
Dual form			26.8.b.a.25.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000i q^{2} +18.5100 q^{3} -64.0000 q^{4} -6.61448i q^{5} +148.080i q^{6} +1604.67i q^{7} -512.000i q^{8} -1844.38 q^{9} +O(q^{10})$	\(q+8.00000i q^{2} +18.5100 q^{3} -64.0000 q^{4} -6.61448i q^{5} +148.080i q^{6} +1604.67i q^{7} -512.000i q^{8} -1844.38 q^{9} +52.9158 q^{10} -1173.55i q^{11} -1184.64 q^{12} +(-3780.46 + 6961.08i) q^{13} -12837.3 q^{14} -122.434i q^{15} +4096.00 q^{16} -28053.1 q^{17} -14755.1i q^{18} +13825.5i q^{19} +423.327i q^{20} +29702.3i q^{21} +9388.37 q^{22} +94424.9 q^{23} -9477.10i q^{24} +78081.2 q^{25} +(-55688.6 - 30243.7i) q^{26} -74620.7 q^{27} -102699. i q^{28} +159421. q^{29} +979.470 q^{30} +113805. i q^{31} +32768.0i q^{32} -21722.3i q^{33} -224425. i q^{34} +10614.0 q^{35} +118040. q^{36} -402800. i q^{37} -110604. q^{38} +(-69976.1 + 128849. i) q^{39} -3386.61 q^{40} +583062. i q^{41} -237618. q^{42} +88507.4 q^{43} +75107.0i q^{44} +12199.6i q^{45} +755399. i q^{46} -112810. i q^{47} +75816.8 q^{48} -1.75141e6 q^{49} +624650. i q^{50} -519262. q^{51} +(241949. - 445509. i) q^{52} +548744. q^{53} -596966. i q^{54} -7762.40 q^{55} +821589. q^{56} +255910. i q^{57} +1.27536e6i q^{58} -1.69495e6i q^{59} +7835.76i q^{60} +1.33588e6 q^{61} -910442. q^{62} -2.95961e6i q^{63} -262144. q^{64} +(46043.9 + 25005.8i) q^{65} +173778. q^{66} -719239. i q^{67} +1.79540e6 q^{68} +1.74780e6 q^{69} +84912.2i q^{70} +1.85217e6i q^{71} +944323. i q^{72} +1.50153e6i q^{73} +3.22240e6 q^{74} +1.44528e6 q^{75} -884833. i q^{76} +1.88315e6 q^{77} +(-1.03079e6 - 559809. i) q^{78} -6.07672e6 q^{79} -27092.9i q^{80} +2.65244e6 q^{81} -4.66450e6 q^{82} -661334. i q^{83} -1.90095e6i q^{84} +185557. i q^{85} +708059. i q^{86} +2.95087e6 q^{87} -600856. q^{88} +1.03091e7i q^{89} -97597.0 q^{90} +(-1.11702e7 - 6.06637e6i) q^{91} -6.04319e6 q^{92} +2.10653e6i q^{93} +902482. q^{94} +91448.6 q^{95} +606534. i q^{96} +6.14220e6i q^{97} -1.40113e7i q^{98} +2.16447e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 54 q^{3} - 640 q^{4} + 13960 q^{9}+O(q^{10}) $ 10 * q + 54 * q^3 - 640 * q^4 + 13960 * q^9	$ 10 q + 54 q^{3} - 640 q^{4} + 13960 q^{9} - 1136 q^{10} - 3456 q^{12} + 3432 q^{13} + 3792 q^{14} + 40960 q^{16} - 6918 q^{17} + 17280 q^{22} + 94164 q^{23} - 330788 q^{25} + 118896 q^{26} - 53550 q^{27} - 131304 q^{29} - 434992 q^{30} + 873450 q^{35} - 893440 q^{36} + 602976 q^{38} - 1569460 q^{39} + 72704 q^{40} + 1800592 q^{42} + 2740774 q^{43} + 221184 q^{48} - 3083904 q^{49} + 1103186 q^{51} - 219648 q^{52} - 3605916 q^{53} + 3999840 q^{55} - 242688 q^{56} + 11727656 q^{61} - 4210560 q^{62} - 2621440 q^{64} - 7481034 q^{65} - 5557248 q^{66} + 442752 q^{68} + 2035796 q^{69} + 18145392 q^{74} - 32855720 q^{75} - 7147920 q^{77} - 8343632 q^{78} - 28449228 q^{79} + 49311466 q^{81} + 17868800 q^{82} + 50156320 q^{87} - 1105920 q^{88} - 23059744 q^{90} - 20225194 q^{91} - 6026496 q^{92} + 8240976 q^{94} + 16547484 q^{95}+O(q^{100}) $ 10 * q + 54 * q^3 - 640 * q^4 + 13960 * q^9 - 1136 * q^10 - 3456 * q^12 + 3432 * q^13 + 3792 * q^14 + 40960 * q^16 - 6918 * q^17 + 17280 * q^22 + 94164 * q^23 - 330788 * q^25 + 118896 * q^26 - 53550 * q^27 - 131304 * q^29 - 434992 * q^30 + 873450 * q^35 - 893440 * q^36 + 602976 * q^38 - 1569460 * q^39 + 72704 * q^40 + 1800592 * q^42 + 2740774 * q^43 + 221184 * q^48 - 3083904 * q^49 + 1103186 * q^51 - 219648 * q^52 - 3605916 * q^53 + 3999840 * q^55 - 242688 * q^56 + 11727656 * q^61 - 4210560 * q^62 - 2621440 * q^64 - 7481034 * q^65 - 5557248 * q^66 + 442752 * q^68 + 2035796 * q^69 + 18145392 * q^74 - 32855720 * q^75 - 7147920 * q^77 - 8343632 * q^78 - 28449228 * q^79 + 49311466 * q^81 + 17868800 * q^82 + 50156320 * q^87 - 1105920 * q^88 - 23059744 * q^90 - 20225194 * q^91 - 6026496 * q^92 + 8240976 * q^94 + 16547484 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/26\mathbb{Z}\right)^\times$.

$n$	$15$
$\chi(n)$	$-1$

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 $

$2$			8.00000i			0.707107i
$2$			8.00000i			0.707107i
$3$	18.5100			0.395805			0.197902	−	0.980222i	$-0.436587\pi$
$3$	18.5100			0.395805			0.197902	+	0.980222i	$0.436587\pi$
$4$	−64.0000			−0.500000
$5$		−	6.61448i		−	0.0236647i	−0.999930	−	0.0118323i	$-0.996234\pi$
$5$		−	6.61448i		−	0.0236647i	0.999930	−	0.0118323i	$-0.00376644\pi$
$6$			148.080i			0.279876i
$7$			1604.67i			1.76824i	0.467260	+	0.884120i	$0.345241\pi$
$7$			1604.67i			1.76824i	−0.467260	+	0.884120i	$0.654759\pi$
$8$		−	512.000i		−	0.353553i
$9$	−1844.38			−0.843339
$10$	52.9158			0.0167335
$11$		−	1173.55i		−	0.265843i	−0.991127	−	0.132922i	$-0.957564\pi$
$11$		−	1173.55i		−	0.265843i	0.991127	−	0.132922i	$-0.0424359\pi$
$12$	−1184.64			−0.197902
$13$	−3780.46	+	6961.08i	−0.477247	+	0.878769i
$13$	−3780.46	+	6961.08i	−0.477247	+	0.878769i
$14$	−12837.3			−1.25033
$15$		−	122.434i		−	0.00936659i
$16$	4096.00			0.250000
$17$	−28053.1			−1.38487			−0.692437	−	0.721478i	$-0.743463\pi$
$17$	−28053.1			−1.38487			−0.692437	+	0.721478i	$0.743463\pi$
$18$		−	14755.1i		−	0.596330i
$19$			13825.5i			0.462428i	0.972903	+	0.231214i	$0.0742697\pi$
$19$			13825.5i			0.462428i	−0.972903	+	0.231214i	$0.925730\pi$
$20$			423.327i			0.0118323i
$21$			29702.3i			0.699878i
$22$	9388.37			0.187980
$23$	94424.9			1.61823			0.809113	−	0.587653i	$-0.199948\pi$
$23$	94424.9			1.61823			0.809113	+	0.587653i	$0.199948\pi$
$24$		−	9477.10i		−	0.139938i
$25$	78081.2			0.999440
$26$	−55688.6	−	30243.7i	−0.621384	−	0.337464i
$27$	−74620.7			−0.729602
$28$		−	102699.i		−	0.884120i
$29$	159421.			1.21381			0.606906	−	0.794774i	$-0.292410\pi$
$29$	159421.			1.21381			0.606906	+	0.794774i	$0.292410\pi$
$30$	979.470			0.00662318
$31$			113805.i			0.686114i	0.939315	+	0.343057i	$0.111462\pi$
$31$			113805.i			0.686114i	−0.939315	+	0.343057i	$0.888538\pi$
$32$			32768.0i			0.176777i
$33$		−	21722.3i		−	0.105222i
$34$		−	224425.i		−	0.979254i
$35$	10614.0			0.0418448
$36$	118040.			0.421669
$37$		−	402800.i		−	1.30732i	−0.756787	−	0.653662i	$-0.773232\pi$
$37$		−	402800.i		−	1.30732i	0.756787	−	0.653662i	$-0.226768\pi$
$38$	−110604.			−0.326986
$39$	−69976.1	+	128849.i	−0.188896	+	0.347821i
$40$	−3386.61			−0.00836673
$41$			583062.i			1.32121i	0.750734	+	0.660604i	$0.229700\pi$
$41$			583062.i			1.32121i	−0.750734	+	0.660604i	$0.770300\pi$
$42$	−237618.			−0.494888
$43$	88507.4			0.169762			0.0848809	−	0.996391i	$-0.472949\pi$
$43$	88507.4			0.169762			0.0848809	+	0.996391i	$0.472949\pi$
$44$			75107.0i			0.132922i
$45$			12199.6i			0.0199573i
$46$			755399.i			1.14426i
$47$		−	112810.i		−	0.158491i	−0.996855	−	0.0792457i	$-0.974749\pi$
$47$		−	112810.i		−	0.158491i	0.996855	−	0.0792457i	$-0.0252512\pi$
$48$	75816.8			0.0989512
$49$	−1.75141e6			−2.12667
$50$			624650.i			0.706711i
$51$	−519262.			−0.548140
$52$	241949.	−	445509.i	0.238623	−	0.439385i
$53$	548744.			0.506296			0.253148	−	0.967428i	$-0.418534\pi$
$53$	548744.			0.506296			0.253148	+	0.967428i	$0.418534\pi$
$54$		−	596966.i		−	0.515907i
$55$	−7762.40			−0.00629110
$56$	821589.			0.625167
$57$			255910.i			0.183031i
$58$			1.27536e6i			0.858294i
$59$		−	1.69495e6i		−	1.07442i	−0.843448	−	0.537211i	$-0.819478\pi$
$59$		−	1.69495e6i		−	1.07442i	0.843448	−	0.537211i	$-0.180522\pi$
$60$			7835.76i			0.00468330i
$61$	1.33588e6			0.753554			0.376777	−	0.926304i	$-0.377032\pi$
$61$	1.33588e6			0.753554			0.376777	+	0.926304i	$0.377032\pi$
$62$	−910442.			−0.485156
$63$		−	2.95961e6i		−	1.49123i
$64$	−262144.			−0.125000
$65$	46043.9	+	25005.8i	0.0207958	+	0.0112939i
$66$	173778.			0.0744033
$67$		−	719239.i		−	0.292154i	−0.989273	−	0.146077i	$-0.953335\pi$
$67$		−	719239.i		−	0.292154i	0.989273	−	0.146077i	$-0.0466647\pi$
$68$	1.79540e6			0.692437
$69$	1.74780e6			0.640501
$70$			84912.2i			0.0295888i
$71$			1.85217e6i			0.614153i	0.951685	+	0.307077i	$0.0993508\pi$
$71$			1.85217e6i			0.614153i	−0.951685	+	0.307077i	$0.900649\pi$
$72$			944323.i			0.298165i
$73$			1.50153e6i			0.451756i	0.974156	+	0.225878i	$0.0725250\pi$
$73$			1.50153e6i			0.451756i	−0.974156	+	0.225878i	$0.927475\pi$
$74$	3.22240e6			0.924417
$75$	1.44528e6			0.395583
$76$		−	884833.i		−	0.231214i
$77$	1.88315e6			0.470075
$78$	−1.03079e6	−	559809.i	−0.245947	−	0.133570i
$79$	−6.07672e6			−1.38667			−0.693337	−	0.720614i	$-0.743860\pi$
$79$	−6.07672e6			−1.38667			−0.693337	+	0.720614i	$0.743860\pi$
$80$		−	27092.9i		−	0.00591617i
$81$	2.65244e6			0.554559
$82$	−4.66450e6			−0.934235
$83$		−	661334.i		−	0.126954i	−0.997983	−	0.0634771i	$-0.979781\pi$
$83$		−	661334.i		−	0.126954i	0.997983	−	0.0634771i	$-0.0202190\pi$
$84$		−	1.90095e6i		−	0.349939i
$85$			185557.i			0.0327726i
$86$			708059.i			0.120040i
$87$	2.95087e6			0.480432
$88$	−600856.			−0.0939898
$89$			1.03091e7i			1.55008i	0.631910	+	0.775041i	$0.282271\pi$
$89$			1.03091e7i			1.55008i	−0.631910	+	0.775041i	$0.717729\pi$
$90$	−97597.0			−0.0141120
$91$	−1.11702e7	−	6.06637e6i	−1.55388	−	0.843887i
$92$	−6.04319e6			−0.809113
$93$			2.10653e6i			0.271567i
$94$	902482.			0.112070
$95$	91448.6			0.0109432
$96$			606534.i			0.0699691i
$97$			6.14220e6i			0.683318i	0.939824	+	0.341659i	$0.110989\pi$
$97$			6.14220e6i			0.683318i	−0.939824	+	0.341659i	$0.889011\pi$
$98$		−	1.40113e7i		−	1.50379i
$99$			2.16447e6i			0.224196i
$100$	−4.99720e6			−0.499720
$101$	−5.85424e6			−0.565387			−0.282693	−	0.959210i	$-0.591228\pi$
$101$	−5.85424e6			−0.565387			−0.282693	+	0.959210i	$0.591228\pi$
$102$		−	4.15410e6i		−	0.387593i
$103$	1.80551e6			0.162806			0.0814031	−	0.996681i	$-0.474060\pi$
$103$	1.80551e6			0.162806			0.0814031	+	0.996681i	$0.474060\pi$
$104$	3.56407e6	+	1.93559e6i	0.310692	+	0.168732i
$105$	196465.			0.0165624
$106$			4.38996e6i			0.358005i
$107$	1.32024e7			1.04186			0.520931	−	0.853599i	$-0.325585\pi$
$107$	1.32024e7			1.04186			0.520931	+	0.853599i	$0.325585\pi$
$108$	4.77572e6			0.364801
$109$			2.36217e7i			1.74710i	0.486734	+	0.873550i	$0.338188\pi$
$109$			2.36217e7i			1.74710i	−0.486734	+	0.873550i	$0.661812\pi$
$110$		−	62099.2i		−	0.00444848i
$111$		−	7.45581e6i		−	0.517445i
$112$			6.57271e6i			0.442060i
$113$	1.03738e7			0.676340			0.338170	−	0.941085i	$-0.390192\pi$
$113$	1.03738e7			0.676340			0.338170	+	0.941085i	$0.390192\pi$
$114$	−2.04728e6			−0.129423
$115$		−	624571.i		−	0.0382948i
$116$	−1.02029e7			−0.606906
$117$	6.97261e6	−	1.28389e7i	0.402480	−	0.741100i
$118$	1.35596e7			0.759732
$119$		−	4.50159e7i		−	2.44879i
$120$	−62686.1			−0.00331159
$121$	1.81100e7			0.929327
$122$			1.06871e7i			0.532843i
$123$			1.07925e7i			0.522941i
$124$		−	7.28354e6i		−	0.343057i
$125$		−	1.03322e6i		−	0.0473161i
$126$	2.36769e7			1.05446
$127$	−3.07208e7			−1.33082			−0.665409	−	0.746479i	$-0.731743\pi$
$127$	−3.07208e7			−1.33082			−0.665409	+	0.746479i	$0.731743\pi$
$128$		−	2.09715e6i		−	0.0883883i
$129$	1.63827e6			0.0671925
$130$	−200046.	+	368351.i	−0.00798598	+	0.0147048i
$131$	1.62980e7			0.633411			0.316706	−	0.948524i	$-0.397423\pi$
$131$	1.62980e7			0.633411			0.316706	+	0.948524i	$0.397423\pi$
$132$			1.39023e6i			0.0526110i
$133$	−2.21853e7			−0.817683
$134$	5.75391e6			0.206584
$135$			493577.i			0.0172658i
$136$			1.43632e7i			0.489627i
$137$		−	3.22644e7i		−	1.07202i	−0.844213	−	0.536008i	$-0.819932\pi$
$137$		−	3.22644e7i		−	1.07202i	0.844213	−	0.536008i	$-0.180068\pi$
$138$			1.39824e7i			0.452903i
$139$	−3.77383e7			−1.19188			−0.595938	−	0.803030i	$-0.703220\pi$
$139$	−3.77383e7			−1.19188			−0.595938	+	0.803030i	$0.703220\pi$
$140$	−679298.			−0.0209224
$141$		−	2.08811e6i		−	0.0627317i
$142$	−1.48174e7			−0.434272
$143$	8.16915e6	+	4.43654e6i	0.233615	+	0.126873i
$144$	−7.55459e6			−0.210835
$145$		−	1.05448e6i		−	0.0287245i
$146$	−1.20122e7			−0.319439
$147$	−3.24185e7			−0.841748
$148$			2.57792e7i			0.653662i
$149$			1.69205e7i			0.419045i	0.977804	+	0.209523i	$0.0671910\pi$
$149$			1.69205e7i			0.419045i	−0.977804	+	0.209523i	$0.932809\pi$
$150$			1.15622e7i			0.279720i
$151$			3.87829e7i			0.916687i	0.888775	+	0.458343i	$0.151557\pi$
$151$			3.87829e7i			0.916687i	−0.888775	+	0.458343i	$0.848443\pi$
$152$	7.07867e6			0.163493
$153$	5.17407e7			1.16792
$154$			1.50652e7i			0.332393i
$155$	752763.			0.0162367
$156$	4.47847e6	−	8.24636e6i	0.0944482	−	0.173911i
$157$	−4.86546e7			−1.00340			−0.501701	−	0.865041i	$-0.667292\pi$
$157$	−4.86546e7			−1.00340			−0.501701	+	0.865041i	$0.667292\pi$
$158$		−	4.86137e7i		−	0.980526i
$159$	1.01572e7			0.200394
$160$	216743.			0.00418336
$161$			1.51520e8i			2.86141i
$162$			2.12195e7i			0.392132i
$163$		−	8.82607e7i		−	1.59629i	−0.602468	−	0.798143i	$-0.705816\pi$
$163$		−	8.82607e7i		−	1.59629i	0.602468	−	0.798143i	$-0.294184\pi$
$164$		−	3.73160e7i		−	0.660604i
$165$	−143682.			−0.00249005
$166$	5.29067e6			0.0897702
$167$		−	7.18201e7i		−	1.19327i	−0.802513	−	0.596634i	$-0.796504\pi$
$167$		−	7.18201e7i		−	1.19327i	0.802513	−	0.596634i	$-0.203496\pi$
$168$	1.52076e7			0.247444
$169$	−3.41648e7	−	5.26322e7i	−0.544471	−	0.838779i
$170$	−1.48446e6			−0.0231737
$171$		−	2.54995e7i		−	0.389983i
$172$	−5.66447e6			−0.0848809
$173$	−6.00357e7			−0.881553			−0.440776	−	0.897617i	$-0.645297\pi$
$173$	−6.00357e7			−0.881553			−0.440776	+	0.897617i	$0.645297\pi$
$174$			2.36069e7i			0.339717i
$175$			1.25294e8i			1.76725i
$176$		−	4.80685e6i		−	0.0664609i
$177$		−	3.13735e7i		−	0.425262i
$178$	−8.24726e7			−1.09607
$179$	1.34440e8			1.75203			0.876015	−	0.482284i	$-0.160193\pi$
$179$	1.34440e8			1.75203			0.876015	+	0.482284i	$0.160193\pi$
$180$		−	780776.i		−	0.00997867i
$181$	1.55862e8			1.95374			0.976868	−	0.213843i	$-0.0685980\pi$
$181$	1.55862e8			1.95374			0.976868	+	0.213843i	$0.0685980\pi$
$182$	4.85310e7	−	8.93616e7i	0.596718	−	1.09876i
$183$	2.47272e7			0.298260
$184$		−	4.83455e7i		−	0.572129i
$185$	−2.66431e6			−0.0309374
$186$	−1.68522e7			−0.192027
$187$			3.29217e7i			0.368160i
$188$			7.21985e6i			0.0792457i
$189$		−	1.19741e8i		−	1.29011i
$190$			731589.i			0.00773801i
$191$	−2.23323e7			−0.231909			−0.115954	−	0.993255i	$-0.536993\pi$
$191$	−2.23323e7			−0.231909			−0.115954	+	0.993255i	$0.536993\pi$
$192$	−4.85227e6			−0.0494756
$193$		−	2.69730e6i		−	0.0270072i	−0.999909	−	0.0135036i	$-0.995702\pi$
$193$		−	2.69730e6i		−	0.0270072i	0.999909	−	0.0135036i	$-0.00429845\pi$
$194$	−4.91376e7			−0.483179
$195$	852271.	+	462856.i	0.00823108	+	0.00447017i
$196$	1.12090e8			1.06334
$197$			7.30124e7i			0.680401i	0.940353	+	0.340200i	$0.110495\pi$
$197$			7.30124e7i			0.680401i	−0.940353	+	0.340200i	$0.889505\pi$
$198$	−1.73157e7			−0.158531
$199$	1.10593e8			0.994817			0.497409	−	0.867516i	$-0.334285\pi$
$199$	1.10593e8			0.994817			0.497409	+	0.867516i	$0.334285\pi$
$200$		−	3.99776e7i		−	0.353355i
$201$		−	1.33131e7i		−	0.115636i
$202$		−	4.68339e7i		−	0.399789i
$203$			2.55817e8i			2.14631i
$204$	3.32328e7			0.274070
$205$	3.85665e6			0.0312660
$206$			1.44441e7i			0.115121i
$207$	−1.74156e8			−1.36471
$208$	−1.54848e7	+	2.85126e7i	−0.119312	+	0.219692i
$209$	1.62249e7			0.122933
$210$			1.57172e6i			0.0117114i
$211$	−7.79932e7			−0.571568			−0.285784	−	0.958294i	$-0.592254\pi$
$211$	−7.79932e7			−0.571568			−0.285784	+	0.958294i	$0.592254\pi$
$212$	−3.51196e7			−0.253148
$213$			3.42836e7i			0.243085i
$214$			1.05619e8i			0.736708i
$215$		−	585430.i		−	0.00401736i
$216$			3.82058e7i			0.257953i
$217$	−1.82619e8			−1.21322
$218$	−1.88973e8			−1.23539
$219$			2.77932e7i			0.178807i
$220$	496793.			0.00314555
$221$	1.06054e8	−	1.95280e8i	0.660926	−	1.21698i
$222$	5.96464e7			0.365889
$223$		−	2.81334e8i		−	1.69885i	−0.527709	−	0.849425i	$-0.676949\pi$
$223$		−	2.81334e8i		−	1.69885i	0.527709	−	0.849425i	$-0.323051\pi$
$224$	−5.25817e7			−0.312584
$225$	−1.44012e8			−0.842866
$226$			8.29908e7i			0.478245i
$227$			2.87156e8i			1.62940i	0.579882	+	0.814700i	$0.303099\pi$
$227$			2.87156e8i			1.62940i	−0.579882	+	0.814700i	$0.696901\pi$
$228$		−	1.63782e7i		−	0.0915156i
$229$		−	1.77698e8i		−	0.977819i	−0.872335	−	0.488909i	$-0.837395\pi$
$229$		−	1.77698e8i		−	0.977819i	0.872335	−	0.488909i	$-0.162605\pi$
$230$	4.99657e6			0.0270785
$231$	3.48570e7			0.186058
$232$		−	8.16233e7i		−	0.429147i
$233$	−1.38051e8			−0.714979			−0.357490	−	0.933917i	$-0.616367\pi$
$233$	−1.38051e8			−0.714979			−0.357490	+	0.933917i	$0.616367\pi$
$234$	1.02711e8	+	5.57809e7i	0.524037	+	0.284597i
$235$	−746181.			−0.00375065
$236$			1.08477e8i			0.537211i
$237$	−1.12480e8			−0.548852
$238$	3.60127e8			1.73156
$239$			6.38380e7i			0.302473i	0.988498	+	0.151237i	$0.0483255\pi$
$239$			6.38380e7i			0.302473i	−0.988498	+	0.151237i	$0.951674\pi$
$240$		−	501489.i		−	0.00234165i
$241$		−	1.77813e8i		−	0.818285i	−0.912470	−	0.409143i	$-0.865828\pi$
$241$		−	1.77813e8i		−	0.818285i	0.912470	−	0.409143i	$-0.134172\pi$
$242$			1.44880e8i			0.657134i
$243$	2.12292e8			0.949099
$244$	−8.54966e7			−0.376777
$245$			1.15846e7i			0.0503271i
$246$	−8.63396e7			−0.369775
$247$	−9.62406e7	−	5.22668e7i	−0.406367	−	0.220692i
$248$	5.82683e7			0.242578
$249$		−	1.22413e7i		−	0.0502491i
$250$	8.26578e6			0.0334575
$251$	−2.57165e8			−1.02649			−0.513244	−	0.858243i	$-0.671556\pi$
$251$	−2.57165e8			−1.02649			−0.513244	+	0.858243i	$0.671556\pi$
$252$			1.89415e8i			0.745613i
$253$		−	1.10812e8i		−	0.430195i
$254$		−	2.45766e8i		−	0.941031i
$255$			3.43465e6i			0.0129716i
$256$	1.67772e7			0.0625000
$257$	2.24906e8			0.826486			0.413243	−	0.910621i	$-0.364396\pi$
$257$	2.24906e8			0.826486			0.413243	+	0.910621i	$0.364396\pi$
$258$			1.31061e7i			0.0475123i
$259$	6.46359e8			2.31166
$260$	−2.94681e6	−	1.60037e6i	−0.0103979	−	0.00564694i
$261$	−2.94032e8			−1.02365
$262$			1.30384e8i			0.447890i
$263$	4.78020e8			1.62032			0.810161	−	0.586208i	$-0.199380\pi$
$263$	4.78020e8			1.62032			0.810161	+	0.586208i	$0.199380\pi$
$264$	−1.11218e7			−0.0372016
$265$		−	3.62966e6i		−	0.0119813i
$266$		−	1.77483e8i		−	0.578189i
$267$			1.90821e8i			0.613530i
$268$			4.60313e7i			0.146077i
$269$	1.76243e8			0.552049			0.276025	−	0.961151i	$-0.410983\pi$
$269$	1.76243e8			0.552049			0.276025	+	0.961151i	$0.410983\pi$
$270$	−3.94862e6			−0.0122088
$271$			1.88337e8i			0.574834i	0.957806	+	0.287417i	$0.0927966\pi$
$271$			1.88337e8i			0.574834i	−0.957806	+	0.287417i	$0.907203\pi$
$272$	−1.14906e8			−0.346219
$273$	−2.06760e8	−	1.12288e8i	−0.615031	−	0.334014i
$274$	2.58115e8			0.758030
$275$		−	9.16320e7i		−	0.265695i
$276$	−1.11859e8			−0.320251
$277$	−3.15057e7			−0.0890656			−0.0445328	−	0.999008i	$-0.514180\pi$
$277$	−3.15057e7			−0.0890656			−0.0445328	+	0.999008i	$0.514180\pi$
$278$		−	3.01907e8i		−	0.842784i
$279$		−	2.09900e8i		−	0.578627i
$280$		−	5.43438e6i		−	0.0147944i
$281$		−	2.67569e8i		−	0.719388i	−0.933070	−	0.359694i	$-0.882881\pi$
$281$		−	2.67569e8i		−	0.719388i	0.933070	−	0.359694i	$-0.117119\pi$
$282$	1.67049e7			0.0443580
$283$	1.31279e8			0.344303			0.172152	−	0.985070i	$-0.444928\pi$
$283$	1.31279e8			0.344303			0.172152	+	0.985070i	$0.444928\pi$
$284$		−	1.18539e8i		−	0.307077i
$285$	1.69271e6			0.00433137
$286$	−3.54924e7	+	6.53532e7i	−0.0897126	+	0.165191i
$287$	−9.35620e8			−2.33621
$288$		−	6.04367e7i		−	0.149083i
$289$	3.76640e8			0.917876
$290$	8.43587e6			0.0203113
$291$			1.13692e8i			0.270460i
$292$		−	9.60978e7i		−	0.225878i
$293$			5.02530e8i			1.16715i	0.812060	+	0.583574i	$0.198346\pi$
$293$			5.02530e8i			1.16715i	−0.812060	+	0.583574i	$0.801654\pi$
$294$		−	2.59348e8i		−	0.595206i
$295$	−1.12112e7			−0.0254259
$296$	−2.06233e8			−0.462209
$297$			8.75709e7i			0.193960i
$298$	−1.35364e8			−0.296310
$299$	−3.56969e8	+	6.57299e8i	−0.772293	+	1.42205i
$300$	−9.24980e7			−0.197792
$301$			1.42025e8i			0.300180i
$302$	−3.10263e8			−0.648196
$303$	−1.08362e8			−0.223783
$304$			5.66293e7i			0.115607i
$305$		−	8.83618e6i		−	0.0178326i
$306$			4.13926e8i			0.825843i
$307$			4.04464e7i			0.0797802i	0.999204	+	0.0398901i	$0.0127008\pi$
$307$			4.04464e7i			0.0797802i	−0.999204	+	0.0398901i	$0.987299\pi$
$308$	−1.20522e8			−0.235038
$309$	3.34200e7			0.0644394
$310$			6.02210e6i			0.0114811i
$311$	5.80479e7			0.109427			0.0547136	−	0.998502i	$-0.482575\pi$
$311$	5.80479e7			0.109427			0.0547136	+	0.998502i	$0.482575\pi$
$312$	6.59708e7	+	3.58278e7i	0.122973	+	0.0667850i
$313$	1.87305e8			0.345259			0.172629	−	0.984987i	$-0.444774\pi$
$313$	1.87305e8			0.345259			0.172629	+	0.984987i	$0.444774\pi$
$314$		−	3.89237e8i		−	0.709513i
$315$	−1.95763e7			−0.0352894
$316$	3.88910e8			0.693337
$317$			6.09607e7i			0.107484i	0.998555	+	0.0537419i	$0.0171148\pi$
$317$			6.09607e7i			0.107484i	−0.998555	+	0.0537419i	$0.982885\pi$
$318$			8.12579e7i			0.141700i
$319$		−	1.87087e8i		−	0.322684i
$320$			1.73395e6i			0.00295808i
$321$	2.44376e8			0.412374
$322$	−1.21216e9			−2.02332
$323$		−	3.87849e8i		−	0.640404i
$324$	−1.69756e8			−0.277279
$325$	−2.95183e8	+	5.43530e8i	−0.476979	+	0.878277i
$326$	7.06085e8			1.12874
$327$			4.37236e8i			0.691511i
$328$	2.98528e8			0.467118
$329$	1.81023e8			0.280251
$330$		−	1.14945e6i		−	0.00176073i
$331$		−	4.54468e8i		−	0.688820i	−0.938819	−	0.344410i	$-0.888079\pi$
$331$		−	4.54468e8i		−	0.688820i	0.938819	−	0.344410i	$-0.111921\pi$
$332$			4.23254e7i			0.0634771i
$333$			7.42916e8i			1.10252i
$334$	5.74561e8			0.843769
$335$	−4.75739e6			−0.00691372
$336$			1.21661e8i			0.174969i
$337$	−2.05246e8			−0.292125			−0.146063	−	0.989275i	$-0.546660\pi$
$337$	−2.05246e8			−0.292125			−0.146063	+	0.989275i	$0.546660\pi$
$338$	4.21057e8	−	2.73318e8i	0.593107	−	0.384999i
$339$	1.92019e8			0.267699
$340$		−	1.18756e7i		−	0.0163863i
$341$	1.33556e8			0.182399
$342$	2.03996e8			0.275760
$343$		−	1.48891e9i		−	1.99223i
$344$		−	4.53158e7i		−	0.0600198i
$345$		−	1.15608e7i		−	0.0151573i
$346$		−	4.80286e8i		−	0.623352i
$347$	2.99289e8			0.384537			0.192268	−	0.981342i	$-0.438416\pi$
$347$	2.99289e8			0.384537			0.192268	+	0.981342i	$0.438416\pi$
$348$	−1.88856e8			−0.240216
$349$		−	1.03098e8i		−	0.129826i	−0.997891	−	0.0649128i	$-0.979323\pi$
$349$		−	1.03098e8i		−	0.129826i	0.997891	−	0.0649128i	$-0.0206769\pi$
$350$	−1.00235e9			−1.24963
$351$	2.82100e8	−	5.19441e8i	0.348200	−	0.641152i
$352$	3.84548e7			0.0469949
$353$			3.79231e8i			0.458873i	0.973324	+	0.229436i	$0.0736883\pi$
$353$			3.79231e8i			0.458873i	−0.973324	+	0.229436i	$0.926312\pi$
$354$	2.50988e8			0.300705
$355$	1.22511e7			0.0145337
$356$		−	6.59781e8i		−	0.775041i
$357$		−	8.33242e8i		−	0.969243i
$358$			1.07552e9i			1.23887i
$359$		−	1.29204e9i		−	1.47382i	−0.675988	−	0.736912i	$-0.736283\pi$
$359$		−	1.29204e9i		−	1.47382i	0.675988	−	0.736912i	$-0.263717\pi$
$360$	6.24621e6			0.00705598
$361$	7.02727e8			0.786161
$362$			1.24690e9i			1.38150i
$363$	3.35215e8			0.367832
$364$	7.14893e8	+	3.88248e8i	0.776938	+	0.421943i
$365$	9.93183e6			0.0106907
$366$			1.97817e8i			0.210902i
$367$	−3.21664e8			−0.339681			−0.169841	−	0.985472i	$-0.554325\pi$
$367$	−3.21664e8			−0.339681			−0.169841	+	0.985472i	$0.554325\pi$
$368$	3.86764e8			0.404556
$369$		−	1.07539e9i		−	1.11423i
$370$		−	2.13145e7i		−	0.0218760i
$371$			8.80551e8i			0.895253i
$372$		−	1.34818e8i		−	0.135784i
$373$	−5.95386e8			−0.594043			−0.297021	−	0.954871i	$-0.595993\pi$
$373$	−5.95386e8			−0.594043			−0.297021	+	0.954871i	$0.595993\pi$
$374$	−2.63373e8			−0.260328
$375$		−	1.91249e7i		−	0.0187279i
$376$	−5.77588e7			−0.0560352
$377$	−6.02683e8	+	1.10974e9i	−0.579287	+	1.06666i
$378$	9.57930e8			0.912247
$379$			1.84954e9i			1.74512i	0.488503	+	0.872562i	$0.337543\pi$
$379$			1.84954e9i			1.74512i	−0.488503	+	0.872562i	$0.662457\pi$
$380$	−5.85271e6			−0.00547160
$381$	−5.68640e8			−0.526744
$382$		−	1.78659e8i		−	0.163984i
$383$		−	1.94010e9i		−	1.76453i	−0.470757	−	0.882263i	$-0.656019\pi$
$383$		−	1.94010e9i		−	1.76453i	0.470757	−	0.882263i	$-0.343981\pi$
$384$		−	3.88182e7i		−	0.0349845i
$385$		−	1.24561e7i		−	0.0111242i
$386$	2.15784e7			0.0190969
$387$	−1.63241e8			−0.143167
$388$		−	3.93101e8i		−	0.341659i
$389$	1.63159e9			1.40536			0.702680	−	0.711506i	$-0.251986\pi$
$389$	1.63159e9			1.40536			0.702680	+	0.711506i	$0.251986\pi$
$390$	−3.70285e6	+	6.81817e6i	−0.00316089	+	0.00582025i
$391$	−2.64891e9			−2.24104
$392$			8.96721e8i			0.751893i
$393$	3.01676e8			0.250707
$394$	−5.84099e8			−0.481116
$395$			4.01943e7i			0.0328152i
$396$		−	1.38526e8i		−	0.112098i
$397$			1.46137e9i			1.17218i	0.810246	+	0.586090i	$0.199334\pi$
$397$			1.46137e9i			1.17218i	−0.810246	+	0.586090i	$0.800666\pi$
$398$			8.84747e8i			0.703442i
$399$	−4.10650e8			−0.323643
$400$	3.19821e8			0.249860
$401$		−	2.15957e8i		−	0.167248i	−0.996497	−	0.0836240i	$-0.973351\pi$
$401$		−	2.15957e8i		−	0.167248i	0.996497	−	0.0836240i	$-0.0266495\pi$
$402$	1.06505e8			0.0817669
$403$	−7.92208e8	−	4.30236e8i	−0.602936	−	0.327446i
$404$	3.74671e8			0.282693
$405$		−	1.75445e7i		−	0.0131234i
$406$	−2.04653e9			−1.51767
$407$	−4.72704e8			−0.347543
$408$			2.65862e8i			0.193797i
$409$			2.10034e9i			1.51795i	0.651120	+	0.758975i	$0.274300\pi$
$409$			2.10034e9i			1.51795i	−0.651120	+	0.758975i	$0.725700\pi$
$410$			3.08532e7i			0.0221084i
$411$		−	5.97213e8i		−	0.424309i
$412$	−1.15553e8			−0.0814031
$413$	2.71983e9			1.89984
$414$		−	1.39324e9i		−	0.964997i
$415$	−4.37438e6			−0.00300433
$416$	−2.28101e8	−	1.23878e8i	−0.155346	−	0.0843661i
$417$	−6.98535e8			−0.471750
$418$			1.29799e8i			0.0869270i
$419$	4.33365e8			0.287809			0.143905	−	0.989592i	$-0.454034\pi$
$419$	4.33365e8			0.287809			0.143905	+	0.989592i	$0.454034\pi$
$420$	−1.25738e7			−0.00828119
$421$		−	2.60776e9i		−	1.70326i	−0.524145	−	0.851629i	$-0.675615\pi$
$421$		−	2.60776e9i		−	1.70326i	0.524145	−	0.851629i	$-0.324385\pi$
$422$		−	6.23946e8i		−	0.404160i
$423$			2.08065e8i			0.133662i
$424$		−	2.80957e8i		−	0.179003i
$425$	−2.19042e9			−1.38410
$426$	−2.74269e8			−0.171887
$427$			2.14365e9i			1.33246i
$428$	−8.44954e8			−0.520931
$429$	1.51211e8	+	8.21202e7i	0.0924660	+	0.0502169i
$430$	4.68344e6			0.00284070
$431$		−	6.82317e8i		−	0.410503i	−0.978709	−	0.205251i	$-0.934199\pi$
$431$		−	6.82317e8i		−	0.410503i	0.978709	−	0.205251i	$-0.0658012\pi$
$432$	−3.05646e8			−0.182401
$433$	−2.49592e9			−1.47749			−0.738743	−	0.673987i	$-0.764580\pi$
$433$	−2.49592e9			−1.47749			−0.738743	+	0.673987i	$0.764580\pi$
$434$		−	1.46096e9i		−	0.857873i
$435$		−	1.95185e7i		−	0.0113693i
$436$		−	1.51179e9i		−	0.873550i
$437$			1.30547e9i			0.748312i
$438$	−2.22346e8			−0.126436
$439$	1.51778e9			0.856216			0.428108	−	0.903727i	$-0.359180\pi$
$439$	1.51778e9			0.856216			0.428108	+	0.903727i	$0.359180\pi$
$440$			3.97435e6i			0.00222424i
$441$	3.23026e9			1.79351
$442$	1.56224e9	+	8.48430e8i	0.860538	+	0.467345i
$443$	2.56533e9			1.40194			0.700971	−	0.713190i	$-0.252750\pi$
$443$	2.56533e9			1.40194			0.700971	+	0.713190i	$0.252750\pi$
$444$			4.77172e8i			0.258722i
$445$	6.81892e7			0.0366822
$446$	2.25067e9			1.20127
$447$			3.13197e8i			0.165860i
$448$		−	4.20653e8i		−	0.221030i
$449$			2.22454e9i			1.15978i	0.814693	+	0.579892i	$0.196905\pi$
$449$			2.22454e9i			1.15978i	−0.814693	+	0.579892i	$0.803095\pi$
$450$		−	1.15209e9i		−	0.595996i
$451$	6.84251e8			0.351235
$452$	−6.63926e8			−0.338170
$453$			7.17870e8i			0.362829i
$454$	−2.29725e9			−1.15216
$455$	−4.01259e7	+	7.38851e7i	−0.0199703	+	0.0367720i
$456$	1.31026e8			0.0647113
$457$		−	1.58220e9i		−	0.775453i	−0.921775	−	0.387726i	$-0.873261\pi$
$457$		−	1.58220e9i		−	0.775453i	0.921775	−	0.387726i	$-0.126739\pi$
$458$	1.42158e9			0.691422
$459$	2.09335e9			1.01041
$460$			3.99726e7i			0.0191474i
$461$		−	1.33292e9i		−	0.633652i	−0.948484	−	0.316826i	$-0.897383\pi$
$461$		−	1.33292e9i		−	0.633652i	0.948484	−	0.316826i	$-0.102617\pi$
$462$			2.78856e8i			0.131563i
$463$		−	2.03271e9i		−	0.951790i	−0.879502	−	0.475895i	$-0.842124\pi$
$463$		−	2.03271e9i		−	0.951790i	0.879502	−	0.475895i	$-0.157876\pi$
$464$	6.52987e8			0.303453
$465$	1.39336e7			0.00642655
$466$		−	1.10441e9i		−	0.505567i
$467$	3.10675e7			0.0141155			0.00705777	−	0.999975i	$-0.497753\pi$
$467$	3.10675e7			0.0141155			0.00705777	+	0.999975i	$0.497753\pi$
$468$	−4.46247e8	+	8.21689e8i	−0.201240	+	0.370550i
$469$	1.15414e9			0.516598
$470$		−	5.96945e6i		−	0.00265211i
$471$	−9.00595e8			−0.397152
$472$	−8.67815e8			−0.379866
$473$		−	1.03867e8i		−	0.0451300i
$474$		−	8.99838e8i		−	0.388097i
$475$			1.07951e9i			0.462169i
$476$			2.88102e9i			1.22440i
$477$	−1.01209e9			−0.426979
$478$	−5.10704e8			−0.213881
$479$		−	2.95620e9i		−	1.22902i	−0.788908	−	0.614512i	$-0.789353\pi$
$479$		−	2.95620e9i		−	1.22902i	0.788908	−	0.614512i	$-0.210647\pi$
$480$	4.01191e6			0.00165580
$481$	2.80392e9	+	1.52277e9i	1.14884	+	0.623916i
$482$	1.42251e9			0.578615
$483$			2.80464e9i			1.13256i
$484$	−1.15904e9			−0.464664
$485$	4.06274e7			0.0161705
$486$			1.69834e9i			0.671114i
$487$			3.14356e8i			0.123330i	0.998097	+	0.0616652i	$0.0196411\pi$
$487$			3.14356e8i			0.123330i	−0.998097	+	0.0616652i	$0.980359\pi$
$488$		−	6.83973e8i		−	0.266422i
$489$		−	1.63370e9i		−	0.631817i
$490$	−9.26772e7			−0.0355866
$491$	−1.74300e9			−0.664525			−0.332262	−	0.943187i	$-0.607812\pi$
$491$	−1.74300e9			−0.664525			−0.332262	+	0.943187i	$0.607812\pi$
$492$		−	6.90717e8i		−	0.261470i
$493$	−4.47225e9			−1.68098
$494$	4.18134e8	−	7.69925e8i	0.156053	−	0.287345i
$495$	1.43168e7			0.00530553
$496$			4.66146e8i			0.171529i
$497$	−2.97211e9			−1.08597
$498$	9.79301e7			0.0355315
$499$			3.47846e9i			1.25324i	0.779324	+	0.626621i	$0.215563\pi$
$499$			3.47846e9i			1.25324i	−0.779324	+	0.626621i	$0.784437\pi$
$500$			6.61263e7i			0.0236581i
$501$		−	1.32939e9i		−	0.472302i
$502$		−	2.05732e9i		−	0.725837i
$503$	−2.26923e9			−0.795043			−0.397521	−	0.917593i	$-0.630130\pi$
$503$	−2.26923e9			−0.795043			−0.397521	+	0.917593i	$0.630130\pi$
$504$	−1.51532e9			−0.527228
$505$			3.87227e7i			0.0133797i
$506$	8.86496e8			0.304194
$507$	−6.32389e8	−	9.74219e8i	−0.215504	−	0.331993i
$508$	1.96613e9			0.665409
$509$		−	3.74316e9i		−	1.25813i	−0.777352	−	0.629065i	$-0.783438\pi$
$509$		−	3.74316e9i		−	1.25813i	0.777352	−	0.629065i	$-0.216562\pi$
$510$	−2.74772e7			−0.00917227
$511$	−2.40945e9			−0.798812
$512$			1.34218e8i			0.0441942i
$513$		−	1.03167e9i		−	0.337388i
$514$			1.79925e9i			0.584414i
$515$		−	1.19425e7i		−	0.00385275i
$516$	−1.04849e8			−0.0335962
$517$	−1.32388e8			−0.0421339
$518$			5.17087e9i			1.63459i
$519$	−1.11126e9			−0.348923
$520$	1.28030e7	−	2.35745e7i	0.00399299	−	0.00735242i
$521$	−1.68654e9			−0.522475			−0.261238	−	0.965275i	$-0.584131\pi$
$521$	−1.68654e9			−0.522475			−0.261238	+	0.965275i	$0.584131\pi$
$522$		−	2.35226e9i		−	0.723833i
$523$	4.11403e9			1.25751			0.628755	−	0.777604i	$-0.283565\pi$
$523$	4.11403e9			1.25751			0.628755	+	0.777604i	$0.283565\pi$
$524$	−1.04307e9			−0.316706
$525$			2.31919e9i			0.699486i
$526$			3.82416e9i			1.14574i
$527$		−	3.19260e9i		−	0.950182i
$528$		−	8.89745e7i		−	0.0263055i
$529$	5.51124e9			1.61865
$530$	2.90373e7			0.00847208
$531$			3.12614e9i			0.906102i
$532$	1.41986e9			0.408842
$533$	−4.05874e9	−	2.20424e9i	−1.16104	−	0.630542i
$534$	−1.52657e9			−0.433831
$535$		−	8.73271e7i		−	0.0246553i
$536$	−3.68250e8			−0.103292
$537$	2.48847e9			0.693462
$538$			1.40994e9i			0.390358i
$539$			2.05536e9i			0.565362i
$540$		−	3.15889e7i		−	0.00863290i
$541$			5.21199e9i			1.41519i	0.706620	+	0.707593i	$0.250219\pi$
$541$			5.21199e9i			1.41519i	−0.706620	+	0.707593i	$0.749781\pi$
$542$	−1.50669e9			−0.406469
$543$	2.88500e9			0.773298
$544$		−	9.19245e8i		−	0.244813i
$545$	1.56245e8			0.0413446
$546$	8.98306e8	−	1.65408e9i	0.236184	−	0.434893i
$547$	−5.06174e9			−1.32234			−0.661172	−	0.750235i	$-0.729941\pi$
$547$	−5.06174e9			−1.32234			−0.661172	+	0.750235i	$0.729941\pi$
$548$			2.06492e9i			0.536008i
$549$	−2.46388e9			−0.635501
$550$	7.33056e8			0.187874
$551$			2.20407e9i			0.561300i
$552$		−	8.94874e8i		−	0.226451i
$553$		−	9.75110e9i		−	2.45197i
$554$		−	2.52046e8i		−	0.0629789i
$555$	−4.93163e7			−0.0122452
$556$	2.41525e9			0.595938
$557$		−	5.64001e9i		−	1.38289i	−0.722430	−	0.691444i	$-0.756975\pi$
$557$		−	5.64001e9i		−	1.38289i	0.722430	−	0.691444i	$-0.243025\pi$
$558$	1.67920e9			0.409151
$559$	−3.34598e8	+	6.16107e8i	−0.0810182	+	0.149181i
$560$	4.34750e7			0.0104612
$561$			6.09379e8i			0.145719i
$562$	2.14055e9			0.508684
$563$	1.03881e9			0.245334			0.122667	−	0.992448i	$-0.460855\pi$
$563$	1.03881e9			0.245334			0.122667	+	0.992448i	$0.460855\pi$
$564$			1.33639e8i			0.0313658i
$565$		−	6.86176e7i		−	0.0160054i
$566$			1.05023e9i			0.243459i
$567$			4.25627e9i			0.980593i
$568$	9.48311e8			0.217136
$569$	−1.36295e9			−0.310160			−0.155080	−	0.987902i	$-0.549564\pi$
$569$	−1.36295e9			−0.310160			−0.155080	+	0.987902i	$0.549564\pi$
$570$			1.35417e7i			0.00306274i
$571$	7.59504e9			1.70727			0.853637	−	0.520868i	$-0.174392\pi$
$571$	7.59504e9			1.70727			0.853637	+	0.520868i	$0.174392\pi$
$572$	−5.22826e8	−	2.83939e8i	−0.116808	−	0.0634364i
$573$	−4.13370e8			−0.0917906
$574$		−	7.48496e9i		−	1.65195i
$575$	7.37281e9			1.61732
$576$	4.83494e8			0.105417
$577$		−	9.14980e8i		−	0.198288i	−0.995073	−	0.0991439i	$-0.968390\pi$
$577$		−	9.14980e8i		−	0.198288i	0.995073	−	0.0991439i	$-0.0316104\pi$
$578$			3.01312e9i			0.649036i
$579$		−	4.99269e7i		−	0.0106896i
$580$			6.74870e7i			0.0143622i
$581$	1.06122e9			0.224486
$582$	−9.09534e8			−0.191244
$583$		−	6.43977e8i		−	0.134595i
$584$	7.68782e8			0.159720
$585$	−8.49226e7	−	4.61202e7i	−0.0175379	−	0.00952457i
$586$	−4.02024e9			−0.825298
$587$		−	3.69625e9i		−	0.754272i	−0.926158	−	0.377136i	$-0.876909\pi$
$587$		−	3.69625e9i		−	0.754272i	0.926158	−	0.377136i	$-0.123091\pi$
$588$	2.07478e9			0.420874
$589$	−1.57342e9			−0.317278
$590$		−	8.96898e7i		−	0.0179788i
$591$			1.35146e9i			0.269306i
$592$		−	1.64987e9i		−	0.326831i
$593$		−	4.56755e9i		−	0.899480i	−0.893159	−	0.449740i	$-0.851517\pi$
$593$		−	4.56755e9i		−	0.899480i	0.893159	−	0.449740i	$-0.148483\pi$
$594$	−7.00567e8			−0.137150
$595$	−2.97757e8			−0.0579498
$596$		−	1.08291e9i		−	0.209523i
$597$	2.04708e9			0.393753
$598$	−5.25839e9	−	2.85576e9i	−1.00554	−	0.546093i
$599$	1.63940e9			0.311668			0.155834	−	0.987783i	$-0.450194\pi$
$599$	1.63940e9			0.311668			0.155834	+	0.987783i	$0.450194\pi$
$600$		−	7.39984e8i		−	0.139860i
$601$	−4.92573e9			−0.925572			−0.462786	−	0.886470i	$-0.653150\pi$
$601$	−4.92573e9			−0.925572			−0.462786	+	0.886470i	$0.653150\pi$
$602$	−1.13620e9			−0.212259
$603$			1.32655e9i			0.246384i
$604$		−	2.48211e9i		−	0.458343i
$605$		−	1.19788e8i		−	0.0219922i
$606$		−	8.66894e8i		−	0.158238i
$607$	7.26751e9			1.31894			0.659470	−	0.751731i	$-0.270781\pi$
$607$	7.26751e9			1.31894			0.659470	+	0.751731i	$0.270781\pi$
$608$	−4.53035e8			−0.0817464
$609$			4.73515e9i			0.849520i
$610$	7.06894e7			0.0126096
$611$	7.85281e8	+	4.26474e8i	0.139277	+	0.0756395i
$612$	−3.31140e9			−0.583959
$613$			3.16466e9i			0.554900i	0.960740	+	0.277450i	$0.0894893\pi$
$613$			3.16466e9i			0.554900i	−0.960740	+	0.277450i	$0.910511\pi$
$614$	−3.23571e8			−0.0564131
$615$	7.13865e7			0.0123752
$616$		−	9.64172e8i		−	0.166197i
$617$			1.11473e10i			1.91060i	0.295634	+	0.955301i	$0.404469\pi$
$617$			1.11473e10i			1.91060i	−0.295634	+	0.955301i	$0.595531\pi$
$618$			2.67360e8i			0.0455656i
$619$			8.69140e9i			1.47290i	0.676493	+	0.736449i	$0.263499\pi$
$619$			8.69140e9i			1.47290i	−0.676493	+	0.736449i	$0.736501\pi$
$620$	−4.81768e7			−0.00811834
$621$	−7.04605e9			−1.18066
$622$			4.64384e8i			0.0773767i
$623$	−1.65426e10			−2.74092
$624$	−2.86622e8	+	5.27767e8i	−0.0472241	+	0.0869553i
$625$	6.09326e9			0.998320
$626$			1.49844e9i			0.244135i
$627$	3.00322e8			0.0486576
$628$	3.11390e9			0.501701
$629$			1.12998e10i			1.81048i
$630$		−	1.56610e8i		−	0.0249534i
$631$		−	3.41348e9i		−	0.540872i	−0.962738	−	0.270436i	$-0.912832\pi$
$631$		−	3.41348e9i		−	0.540872i	0.962738	−	0.270436i	$-0.0871679\pi$
$632$			3.11128e9i			0.490263i
$633$	−1.44365e9			−0.226229
$634$	−4.87686e8			−0.0760025
$635$			2.03202e8i			0.0314934i
$636$	−6.50063e8			−0.100197
$637$	6.62112e9	−	1.21917e10i	1.01495	−	1.86886i
$638$	1.49670e9			0.228172
$639$		−	3.41611e9i		−	0.517939i
$640$	−1.38716e7			−0.00209168
$641$	−1.93523e9			−0.290221			−0.145110	−	0.989415i	$-0.546354\pi$
$641$	−1.93523e9			−0.290221			−0.145110	+	0.989415i	$0.546354\pi$
$642$			1.95501e9i			0.291592i
$643$			7.98936e9i			1.18515i	0.805515	+	0.592575i	$0.201889\pi$
$643$			7.98936e9i			1.18515i	−0.805515	+	0.592575i	$0.798111\pi$
$644$		−	9.69730e9i		−	1.43071i
$645$		−	1.08363e7i		−	0.00159009i
$646$	3.10279e9			0.452834
$647$	−8.47178e9			−1.22973			−0.614865	−	0.788632i	$-0.710789\pi$
$647$	−8.47178e9			−1.22973			−0.614865	+	0.788632i	$0.710789\pi$
$648$		−	1.35805e9i		−	0.196066i
$649$	−1.98910e9			−0.285628
$650$	−4.34824e9	−	2.36146e9i	−0.621036	−	0.337275i
$651$	−3.38028e9			−0.480196
$652$			5.64868e9i			0.798143i
$653$	8.72038e9			1.22557			0.612787	−	0.790248i	$-0.290048\pi$
$653$	8.72038e9			1.22557			0.612787	+	0.790248i	$0.290048\pi$
$654$	−3.49789e9			−0.488972
$655$		−	1.07803e8i		−	0.0149895i
$656$			2.38822e9i			0.330302i
$657$		−	2.76939e9i		−	0.380983i
$658$			1.44818e9i			0.198167i
$659$	−4.03937e9			−0.549813			−0.274907	−	0.961471i	$-0.588647\pi$
$659$	−4.03937e9			−0.549813			−0.274907	+	0.961471i	$0.588647\pi$
$660$	9.19563e6			0.00124502
$661$		−	3.29803e9i		−	0.444170i	−0.975027	−	0.222085i	$-0.928714\pi$
$661$		−	3.29803e9i		−	0.444170i	0.975027	−	0.222085i	$-0.0712862\pi$
$662$	3.63575e9			0.487069
$663$	1.96305e9	−	3.61463e9i	0.261598	−	0.481688i
$664$	−3.38603e8			−0.0448851
$665$			1.46744e8i			0.0193502i
$666$	−5.94333e9			−0.779597
$667$	1.50533e10			1.96422
$668$			4.59649e9i			0.596634i
$669$		−	5.20748e9i		−	0.672413i
$670$		−	3.80591e7i		−	0.00488874i
$671$		−	1.56772e9i		−	0.200327i
$672$	−9.73284e8			−0.123722
$673$	−5.27449e9			−0.667003			−0.333502	−	0.942750i	$-0.608230\pi$
$673$	−5.27449e9			−0.667003			−0.333502	+	0.942750i	$0.608230\pi$
$674$		−	1.64196e9i		−	0.206564i
$675$	−5.82648e9			−0.729194
$676$	2.18655e9	+	3.36846e9i	0.272236	+	0.419390i
$677$	1.02661e10			1.27158			0.635791	−	0.771861i	$-0.280674\pi$
$677$	1.02661e10			1.27158			0.635791	+	0.771861i	$0.280674\pi$
$678$			1.53616e9i			0.189292i
$679$	−9.85617e9			−1.20827
$680$	9.50051e7			0.0115869
$681$			5.31525e9i			0.644925i
$682$			1.06845e9i			0.128976i
$683$		−	3.81750e9i		−	0.458465i	−0.973372	−	0.229233i	$-0.926378\pi$
$683$		−	3.81750e9i		−	0.458465i	0.973372	−	0.229233i	$-0.0736216\pi$
$684$			1.63197e9i			0.194992i
$685$	−2.13412e8			−0.0253689
$686$	1.19113e10			1.40872
$687$		−	3.28918e9i		−	0.387025i
$688$	3.62526e8			0.0424404
$689$	−2.07451e9	+	3.81985e9i	−0.241628	+	0.444918i
$690$	9.24863e7			0.0107178
$691$		−	1.41493e10i		−	1.63140i	−0.578475	−	0.815700i	$-0.696352\pi$
$691$		−	1.41493e10i		−	1.63140i	0.578475	−	0.815700i	$-0.303648\pi$
$692$	3.84229e9			0.440776
$693$	−3.47325e9			−0.396432
$694$			2.39431e9i			0.271909i
$695$			2.49619e8i			0.0282054i
$696$		−	1.51084e9i		−	0.169859i
$697$		−	1.63567e10i		−	1.82971i
$698$	8.24782e8			0.0918006
$699$	−2.55532e9			−0.282992
$700$		−	8.01883e9i		−	0.883625i
$701$	2.63084e9			0.288457			0.144228	−	0.989544i	$-0.453930\pi$
$701$	2.63084e9			0.288457			0.144228	+	0.989544i	$0.453930\pi$
$702$	4.15553e9	+	2.25680e9i	0.453363	+	0.246215i
$703$	5.56892e9			0.604543
$704$			3.07638e8i			0.0332304i
$705$	−1.38118e7			−0.00148453
$706$	−3.03385e9			−0.324472
$707$		−	9.39409e9i		−	0.999740i
$708$			2.00790e9i			0.212631i
$709$			1.48321e10i			1.56293i	0.623947	+	0.781466i	$0.285528\pi$
$709$			1.48321e10i			1.56293i	−0.623947	+	0.781466i	$0.714472\pi$
$710$			9.80092e7i			0.0102769i
$711$	1.12078e10			1.16944
$712$	5.27825e9			0.548037
$713$			1.07461e10i			1.11029i
$714$	6.66594e9			0.685358
$715$	2.93454e7	−	5.40347e7i	0.00300241	−	0.00552843i
$716$	−8.60413e9			−0.876015
$717$			1.18164e9i			0.119720i
$718$	1.03363e10			1.04215
$719$	−4.09811e9			−0.411180			−0.205590	−	0.978638i	$-0.565911\pi$
$719$	−4.09811e9			−0.411180			−0.205590	+	0.978638i	$0.565911\pi$
$720$			4.99697e7i			0.00498933i
$721$			2.89725e9i			0.287880i
$722$			5.62181e9i			0.555899i
$723$		−	3.29132e9i		−	0.323881i
$724$	−9.97518e9			−0.976868
$725$	1.24478e10			1.21313
$726$			2.68172e9i			0.260097i
$727$	1.07923e10			1.04171			0.520853	−	0.853646i	$-0.325614\pi$
$727$	1.07923e10			1.04171			0.520853	+	0.853646i	$0.325614\pi$
$728$	−3.10598e9	+	5.71914e9i	−0.298359	+	0.549378i
$729$	−1.87136e9			−0.178901
$730$			7.94546e7i			0.00755943i
$731$	−2.48291e9			−0.235099
$732$	−1.58254e9			−0.149130
$733$			4.31755e9i			0.404924i	0.979290	+	0.202462i	$0.0648942\pi$
$733$			4.31755e9i			0.404924i	−0.979290	+	0.202462i	$0.935106\pi$
$734$		−	2.57331e9i		−	0.240191i
$735$			2.14431e8i			0.0199197i
$736$			3.09411e9i			0.286065i
$737$	−8.44060e8			−0.0776671
$738$	8.60311e9			0.787877
$739$		−	1.59219e10i		−	1.45124i	−0.688097	−	0.725619i	$-0.741554\pi$
$739$		−	1.59219e10i		−	1.45124i	0.688097	−	0.725619i	$-0.258446\pi$
$740$	1.70516e8			0.0154687
$741$	−1.78141e9	−	9.67456e8i	−0.160842	−	0.0873510i
$742$	−7.04441e9			−0.633040
$743$		−	7.46206e9i		−	0.667418i	−0.942676	−	0.333709i	$-0.891700\pi$
$743$		−	7.46206e9i		−	0.667418i	0.942676	−	0.333709i	$-0.108300\pi$
$744$	1.07854e9			0.0960136
$745$	1.11920e8			0.00991657
$746$		−	4.76309e9i		−	0.420052i
$747$			1.21975e9i			0.107065i
$748$		−	2.10699e9i		−	0.184080i
$749$			2.11855e10i			1.84226i
$750$	1.52999e8			0.0132427
$751$	−7.78292e9			−0.670507			−0.335253	−	0.942128i	$-0.608822\pi$
$751$	−7.78292e9			−0.670507			−0.335253	+	0.942128i	$0.608822\pi$
$752$		−	4.62071e8i		−	0.0396229i
$753$	−4.76011e9			−0.406289
$754$	−8.87792e9	−	4.82146e9i	−0.754243	−	0.409618i
$755$	2.56529e8			0.0216931
$756$			7.66344e9i			0.645056i
$757$	−1.36187e10			−1.14104			−0.570518	−	0.821285i	$-0.693257\pi$
$757$	−1.36187e10			−1.14104			−0.570518	+	0.821285i	$0.693257\pi$
$758$	−1.47963e10			−1.23399
$759$		−	2.05113e9i		−	0.170273i
$760$		−	4.68217e7i		−	0.00386901i
$761$		−	7.93048e9i		−	0.652309i	−0.945317	−	0.326154i	$-0.894247\pi$
$761$		−	7.93048e9i		−	0.652309i	0.945317	−	0.326154i	$-0.105753\pi$
$762$		−	4.54912e9i		−	0.372464i
$763$	−3.79049e10			−3.08929
$764$	1.42927e9			0.115954
$765$		−	3.42238e8i		−	0.0276384i
$766$	1.55208e10			1.24771
$767$	1.17987e10	+	6.40769e9i	0.944170	+	0.512765i
$768$	3.10546e8			0.0247378
$769$			1.80037e10i			1.42764i	0.700329	+	0.713820i	$0.253036\pi$
$769$			1.80037e10i			1.42764i	−0.700329	+	0.713820i	$0.746964\pi$
$770$	9.96484e7			0.00786598
$771$	4.16300e9			0.327127
$772$			1.72627e8i			0.0135036i
$773$		−	1.79933e10i		−	1.40114i	−0.713583	−	0.700571i	$-0.752929\pi$
$773$		−	1.79933e10i		−	1.40114i	0.713583	−	0.700571i	$-0.247071\pi$
$774$		−	1.30593e9i		−	0.101234i
$775$			8.88606e9i			0.685730i
$776$	3.14480e9			0.241589
$777$	1.19641e10			0.914967
$778$			1.30527e10i			0.993740i
$779$	−8.06114e9			−0.610963
$780$	−5.45453e7	−	2.96228e7i	−0.00411554	−	0.00223509i
$781$	2.17361e9			0.163269
$782$		−	2.11913e10i		−	1.58465i
$783$	−1.18961e10			−0.885600
$784$	−7.17377e9			−0.531669
$785$			3.21825e8i			0.0237452i
$786$			2.41341e9i			0.177277i
$787$			4.61318e9i			0.337356i	0.985671	+	0.168678i	$0.0539498\pi$
$787$			4.61318e9i			0.337356i	−0.985671	+	0.168678i	$0.946050\pi$
$788$		−	4.67279e9i		−	0.340200i
$789$	8.84813e9			0.641331
$790$	−3.21554e8			−0.0232038
$791$			1.66465e10i			1.19593i
$792$	1.10821e9			0.0792653
$793$	−5.05026e9	+	9.29920e9i	−0.359631	+	0.662200i
$794$	−1.16910e10			−0.828857
$795$		−	6.71848e7i		−	0.00474227i
$796$	−7.07798e9			−0.497409
$797$	2.80281e9			0.196106			0.0980528	−	0.995181i	$-0.468739\pi$
$797$	2.80281e9			0.196106			0.0980528	+	0.995181i	$0.468739\pi$
$798$		−	3.28520e9i		−	0.228850i
$799$			3.16468e9i			0.219491i
$800$			2.55857e9i			0.176678i
$801$		−	1.90139e10i		−	1.30724i
$802$	1.72765e9			0.118262
$803$	1.76211e9			0.120096
$804$			8.52037e8i			0.0578179i
$805$	1.00223e9			0.0677144
$806$	3.44189e9	−	6.33766e9i	0.231539	−	0.426340i
$807$	3.26224e9			0.218504
$808$			2.99737e9i			0.199894i
$809$	8.36628e9			0.555537			0.277768	−	0.960648i	$-0.410405\pi$
$809$	8.36628e9			0.555537			0.277768	+	0.960648i	$0.410405\pi$
$810$	1.40356e8			0.00927968
$811$		−	4.54358e9i		−	0.299106i	−0.988754	−	0.149553i	$-0.952217\pi$
$811$		−	4.54358e9i		−	0.299106i	0.988754	−	0.149553i	$-0.0477835\pi$
$812$		−	1.63723e10i		−	1.07316i
$813$			3.48611e9i			0.227522i
$814$		−	3.78163e9i		−	0.245750i
$815$	−5.83798e8			−0.0377756
$816$	−2.12690e9			−0.137035
$817$			1.22366e9i			0.0785025i
$818$	−1.68027e10			−1.07335
$819$	2.06021e10	+	1.11887e10i	1.31044	+	0.711682i
$820$	−2.46826e8			−0.0156330
$821$			1.50748e10i			0.950717i	0.879792	+	0.475358i	$0.157682\pi$
$821$			1.50748e10i			0.950717i	−0.879792	+	0.475358i	$0.842318\pi$
$822$	4.77770e9			0.300032
$823$	1.67438e10			1.04702			0.523508	−	0.852021i	$-0.324623\pi$
$823$	1.67438e10			1.04702			0.523508	+	0.852021i	$0.324623\pi$
$824$		−	9.24424e8i		−	0.0575607i
$825$		−	1.69610e9i		−	0.105163i
$826$			2.17586e10i			1.34339i
$827$		−	1.05833e10i		−	0.650656i	−0.945601	−	0.325328i	$-0.894525\pi$
$827$		−	1.05833e10i		−	0.650656i	0.945601	−	0.325328i	$-0.105475\pi$
$828$	1.11460e10			0.682356
$829$	−2.53716e10			−1.54670			−0.773352	−	0.633976i	$-0.781422\pi$
$829$	−2.53716e10			−1.54670			−0.773352	+	0.633976i	$0.781422\pi$
$830$		−	3.49950e7i		−	0.00212438i
$831$	−5.83169e8			−0.0352526
$832$	9.91025e8	−	1.82481e9i	0.0596558	−	0.109846i
$833$	4.91325e10			2.94518
$834$		−	5.58828e9i		−	0.333578i
$835$	−4.75053e8			−0.0282383
$836$	−1.03839e9			−0.0614667
$837$		−	8.49223e9i		−	0.500591i
$838$			3.46692e9i			0.203512i
$839$			8.37014e9i			0.489290i	0.969613	+	0.244645i	$0.0786714\pi$
$839$			8.37014e9i			0.489290i	−0.969613	+	0.244645i	$0.921329\pi$
$840$		−	1.00590e8i		−	0.00585569i
$841$	8.16504e9			0.473339
$842$	2.08621e10			1.20439
$843$		−	4.95268e9i		−	0.284737i
$844$	4.99156e9			0.285784
$845$	−3.48134e8	+	2.25982e8i	−0.0198494	+	0.0128847i
$846$	−1.66452e9			−0.0945133
$847$			2.90604e10i			1.64327i
$848$	2.24766e9			0.126574
$849$	2.42996e9			0.136277
$850$		−	1.75234e10i		−	0.978705i
$851$		−	3.80343e10i		−	2.11554i
$852$		−	2.19415e9i		−	0.121542i
$853$			8.05268e9i			0.444241i	0.975019	+	0.222121i	$0.0712979\pi$
$853$			8.05268e9i			0.444241i	−0.975019	+	0.222121i	$0.928702\pi$
$854$	−1.71492e10			−0.942195
$855$	−1.68666e8			−0.00922883
$856$		−	6.75964e9i		−	0.368354i
$857$	−2.21962e10			−1.20461			−0.602304	−	0.798267i	$-0.705751\pi$
$857$	−2.21962e10			−1.20461			−0.602304	+	0.798267i	$0.705751\pi$
$858$	−6.56962e8	+	1.20969e9i	−0.0355087	+	0.0653833i
$859$	−4.25915e8			−0.0229270			−0.0114635	−	0.999934i	$-0.503649\pi$
$859$	−4.25915e8			−0.0229270			−0.0114635	+	0.999934i	$0.503649\pi$
$860$			3.74675e7i			0.00200868i
$861$	−1.73183e10			−0.924685
$862$	5.45854e9			0.290269
$863$		−	1.84502e10i		−	0.977156i	−0.872520	−	0.488578i	$-0.837516\pi$
$863$		−	1.84502e10i		−	0.977156i	0.872520	−	0.488578i	$-0.162484\pi$
$864$		−	2.44517e9i		−	0.128977i
$865$			3.97105e8i			0.0208617i
$866$		−	1.99674e10i		−	1.04474i
$867$	6.97159e9			0.363300
$868$	1.16876e10			0.606608
$869$			7.13131e9i			0.368638i
$870$	1.56148e8			0.00803930
$871$	5.00668e9	+	2.71905e9i	0.256736	+	0.139429i
$872$	1.20943e10			0.617693
$873$		−	1.13286e10i		−	0.576268i
$874$	−1.04438e10			−0.529137
$875$	1.65798e9			0.0836662
$876$		−	1.77877e9i		−	0.0894035i
$877$		−	1.81523e10i		−	0.908725i	−0.890817	−	0.454363i	$-0.849867\pi$
$877$		−	1.81523e10i		−	0.908725i	0.890817	−	0.454363i	$-0.150133\pi$
$878$			1.21423e10i			0.605436i
$879$			9.30182e9i			0.461962i
$880$	−3.17948e7			−0.00157277
$881$	−2.80509e10			−1.38207			−0.691037	−	0.722819i	$-0.742846\pi$
$881$	−2.80509e10			−1.38207			−0.691037	+	0.722819i	$0.742846\pi$
$882$			2.58421e10i			1.26820i
$883$	−2.12689e9			−0.103964			−0.0519819	−	0.998648i	$-0.516554\pi$
$883$	−2.12689e9			−0.103964			−0.0519819	+	0.998648i	$0.516554\pi$
$884$	−6.78744e9	+	1.24979e10i	−0.330463	+	0.608492i
$885$	−2.07519e8			−0.0100637
$886$			2.05226e10i			0.991323i
$887$	2.61618e10			1.25874			0.629368	−	0.777107i	$-0.283314\pi$
$887$	2.61618e10			1.25874			0.629368	+	0.777107i	$0.283314\pi$
$888$	−3.81737e9			−0.182944
$889$		−	4.92965e10i		−	2.35321i
$890$			5.45514e8i			0.0259382i
$891$		−	3.11276e9i		−	0.147426i
$892$			1.80054e10i			0.849425i
$893$	1.55966e9			0.0732909
$894$	−2.50558e9			−0.117281
$895$		−	8.89248e8i		−	0.0414612i
$896$	3.36523e9			0.156292
$897$	−6.60749e9	+	1.21666e10i	−0.305677	+	0.562853i
$898$	−1.77963e10			−0.820091
$899$			1.81429e10i			0.832814i
$900$	9.21674e9			0.421433
$901$	−1.53940e10			−0.701156
$902$			5.47400e9i			0.248360i
$903$			2.62887e9i			0.118812i
$904$		−	5.31141e9i		−	0.239122i
$905$		−	1.03095e9i		−	0.0462345i
$906$	−5.74296e9			−0.256559
$907$	−1.57121e10			−0.699213			−0.349607	−	0.936897i	$-0.613685\pi$
$907$	−1.57121e10			−0.699213			−0.349607	+	0.936897i	$0.613685\pi$
$908$		−	1.83780e10i		−	0.814700i
$909$	1.07974e10			0.476812
$910$	−5.91081e8	−	3.21007e8i	−0.0260017	−	0.0141211i
$911$	−3.17054e10			−1.38937			−0.694687	−	0.719312i	$-0.744457\pi$
$911$	−3.17054e10			−1.38937			−0.694687	+	0.719312i	$0.744457\pi$
$912$			1.04821e9i			0.0457578i
$913$	−7.76106e8			−0.0337500
$914$	1.26576e10			0.548328
$915$		−	1.63557e8i		−	0.00705823i
$916$			1.13727e10i			0.488909i
$917$			2.61529e10i			1.12002i
$918$			1.67468e10i			0.714466i
$919$	−7.15392e9			−0.304047			−0.152023	−	0.988377i	$-0.548579\pi$
$919$	−7.15392e9			−0.304047			−0.152023	+	0.988377i	$0.548579\pi$
$920$	−3.19781e8			−0.0135393
$921$			7.48660e8i			0.0315774i
$922$	1.06634e10			0.448059
$923$	−1.28931e10	−	7.00206e9i	−0.539699	−	0.293103i
$924$	−2.23085e9			−0.0930290
$925$		−	3.14511e10i		−	1.30659i
$926$	1.62617e10			0.673017
$927$	−3.33006e9			−0.137301
$928$			5.22389e9i			0.214574i
$929$			3.59995e10i			1.47313i	0.676365	+	0.736566i	$0.263554\pi$
$929$			3.59995e10i			1.47313i	−0.676365	+	0.736566i	$0.736446\pi$
$930$			1.11469e8i			0.00454426i
$931$		−	2.42141e10i		−	0.983433i
$932$	8.83526e9			0.357490
$933$	1.07446e9			0.0433118
$934$			2.48540e8i			0.00998119i
$935$	2.17760e8			0.00871238
$936$	−6.57351e9	−	3.56998e9i	−0.262018	−	0.142298i
$937$	−1.38764e10			−0.551047			−0.275523	−	0.961294i	$-0.588851\pi$
$937$	−1.38764e10			−0.551047			−0.275523	+	0.961294i	$0.588851\pi$
$938$			9.23310e9i			0.365290i
$939$	3.46701e9			0.136655
$940$	4.77556e7			0.00187532
$941$		−	1.24319e9i		−	0.0486379i	−0.999704	−	0.0243190i	$-0.992258\pi$
$941$		−	1.24319e9i		−	0.0486379i	0.999704	−	0.0243190i	$-0.00774173\pi$
$942$		−	7.20476e9i		−	0.280829i
$943$			5.50556e10i			2.13801i
$944$		−	6.94252e9i		−	0.268606i
$945$	−7.92026e8			−0.0305301
$946$	8.30940e8			0.0319118
$947$			1.08695e10i			0.415896i	0.978140	+	0.207948i	$0.0666785\pi$
$947$			1.08695e10i			0.415896i	−0.978140	+	0.207948i	$0.933321\pi$
$948$	7.19870e9			0.274426
$949$	−1.04523e10	−	5.67647e9i	−0.396989	−	0.215599i
$950$	−8.63611e9			−0.326803
$951$			1.12838e9i			0.0425426i
$952$	−2.30481e10			−0.865778
$953$	−4.97595e9			−0.186230			−0.0931152	−	0.995655i	$-0.529682\pi$
$953$	−4.97595e9			−0.186230			−0.0931152	+	0.995655i	$0.529682\pi$
$954$		−	8.09675e9i		−	0.301920i
$955$			1.47717e8i			0.00548804i
$956$		−	4.08563e9i		−	0.151237i
$957$		−	3.46298e9i		−	0.127720i
$958$	2.36496e10			0.869051
$959$	5.17736e10			1.89558
$960$			3.20953e7i			0.00117082i
$961$	1.45610e10			0.529247
$962$	−1.21821e10	+	2.24314e10i	−0.441175	+	0.812350i
$963$	−2.43503e10			−0.878642
$964$			1.13801e10i			0.409143i
$965$	−1.78412e7			−0.000639116
$966$	−2.24371e10			−0.800841
$967$			1.95659e10i			0.695838i	0.937525	+	0.347919i	$0.113112\pi$
$967$			1.95659e10i			0.695838i	−0.937525	+	0.347919i	$0.886888\pi$
$968$		−	9.27230e9i		−	0.328567i
$969$		−	7.17907e9i		−	0.253475i
$970$			3.25019e8i			0.0114343i
$971$	−6.28877e9			−0.220444			−0.110222	−	0.993907i	$-0.535156\pi$
$971$	−6.28877e9			−0.220444			−0.110222	+	0.993907i	$0.535156\pi$
$972$	−1.35867e10			−0.474550
$973$		−	6.05574e10i		−	2.10752i
$974$	−2.51485e9			−0.0872078
$975$	−5.46382e9	+	1.00607e10i	−0.188791	+	0.347626i
$976$	5.47178e9			0.188389
$977$		−	1.75153e9i		−	0.0600878i	−0.999549	−	0.0300439i	$-0.990435\pi$
$977$		−	1.75153e9i		−	0.0600878i	0.999549	−	0.0300439i	$-0.00956470\pi$
$978$	1.30696e10			0.446762
$979$	1.20982e10			0.412079
$980$		−	7.41417e8i		−	0.0251635i
$981$		−	4.35674e10i		−	1.47340i
$982$		−	1.39440e10i		−	0.469890i
$983$		−	3.03137e10i		−	1.01789i	−0.860799	−	0.508946i	$-0.830035\pi$
$983$		−	3.03137e10i		−	1.01789i	0.860799	−	0.508946i	$-0.169965\pi$
$984$	5.52574e9			0.184887
$985$	4.82939e8			0.0161015
$986$		−	3.57780e10i		−	1.18863i
$987$	3.35072e9			0.110925
$988$	6.15940e9	+	3.34508e9i	0.203184	+	0.110346i
$989$	8.35730e9			0.274713
$990$			1.14535e8i			0.00375157i
$991$	4.08606e10			1.33367			0.666833	−	0.745207i	$-0.267649\pi$
$991$	4.08606e10			1.33367			0.666833	+	0.745207i	$0.267649\pi$
$992$	−3.72917e9			−0.121289
$993$		−	8.41219e9i		−	0.272638i
$994$		−	2.37769e10i		−	0.767897i
$995$		−	7.31517e8i		−	0.0235420i
$996$			7.83441e8i			0.0251246i
$997$	5.95647e10			1.90351			0.951757	−	0.306853i	$-0.0992759\pi$
$997$	5.95647e10			1.90351			0.951757	+	0.306853i	$0.0992759\pi$
$998$	−2.78277e10			−0.886176
$999$			3.00572e10i			0.953826i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	26.8.b.a.25.8	yes	10
3.2	odd	2		234.8.b.c.181.3		10
4.3	odd	2		208.8.f.c.129.5		10
13.5	odd	4		338.8.a.l.1.3		5
13.8	odd	4		338.8.a.k.1.3		5
13.12	even	2	inner	26.8.b.a.25.3	✓	10
39.38	odd	2		234.8.b.c.181.8		10
52.51	odd	2		208.8.f.c.129.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.b.a.25.3	✓	10	13.12	even	2	inner
26.8.b.a.25.8	yes	10	1.1	even	1	trivial
208.8.f.c.129.5		10	4.3	odd	2
208.8.f.c.129.6		10	52.51	odd	2
234.8.b.c.181.3		10	3.2	odd	2
234.8.b.c.181.8		10	39.38	odd	2
338.8.a.k.1.3		5	13.8	odd	4
338.8.a.l.1.3		5	13.5	odd	4

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:	\( N \)	\(=\)	\( 26 = 2 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	26.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} +18.5100 q^{3} -64.0000 q^{4} -6.61448i q^{5} +148.080i q^{6} +1604.67i q^{7} -512.000i q^{8} -1844.38 q^{9} +O(q^{10})\)	\(q+8.00000i q^{2} +18.5100 q^{3} -64.0000 q^{4} -6.61448i q^{5} +148.080i q^{6} +1604.67i q^{7} -512.000i q^{8} -1844.38 q^{9} +52.9158 q^{10} -1173.55i q^{11} -1184.64 q^{12} +(-3780.46 + 6961.08i) q^{13} -12837.3 q^{14} -122.434i q^{15} +4096.00 q^{16} -28053.1 q^{17} -14755.1i q^{18} +13825.5i q^{19} +423.327i q^{20} +29702.3i q^{21} +9388.37 q^{22} +94424.9 q^{23} -9477.10i q^{24} +78081.2 q^{25} +(-55688.6 - 30243.7i) q^{26} -74620.7 q^{27} -102699. i q^{28} +159421. q^{29} +979.470 q^{30} +113805. i q^{31} +32768.0i q^{32} -21722.3i q^{33} -224425. i q^{34} +10614.0 q^{35} +118040. q^{36} -402800. i q^{37} -110604. q^{38} +(-69976.1 + 128849. i) q^{39} -3386.61 q^{40} +583062. i q^{41} -237618. q^{42} +88507.4 q^{43} +75107.0i q^{44} +12199.6i q^{45} +755399. i q^{46} -112810. i q^{47} +75816.8 q^{48} -1.75141e6 q^{49} +624650. i q^{50} -519262. q^{51} +(241949. - 445509. i) q^{52} +548744. q^{53} -596966. i q^{54} -7762.40 q^{55} +821589. q^{56} +255910. i q^{57} +1.27536e6i q^{58} -1.69495e6i q^{59} +7835.76i q^{60} +1.33588e6 q^{61} -910442. q^{62} -2.95961e6i q^{63} -262144. q^{64} +(46043.9 + 25005.8i) q^{65} +173778. q^{66} -719239. i q^{67} +1.79540e6 q^{68} +1.74780e6 q^{69} +84912.2i q^{70} +1.85217e6i q^{71} +944323. i q^{72} +1.50153e6i q^{73} +3.22240e6 q^{74} +1.44528e6 q^{75} -884833. i q^{76} +1.88315e6 q^{77} +(-1.03079e6 - 559809. i) q^{78} -6.07672e6 q^{79} -27092.9i q^{80} +2.65244e6 q^{81} -4.66450e6 q^{82} -661334. i q^{83} -1.90095e6i q^{84} +185557. i q^{85} +708059. i q^{86} +2.95087e6 q^{87} -600856. q^{88} +1.03091e7i q^{89} -97597.0 q^{90} +(-1.11702e7 - 6.06637e6i) q^{91} -6.04319e6 q^{92} +2.10653e6i q^{93} +902482. q^{94} +91448.6 q^{95} +606534. i q^{96} +6.14220e6i q^{97} -1.40113e7i q^{98} +2.16447e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 54 q^{3} - 640 q^{4} + 13960 q^{9}+O(q^{10}) \) 10 * q + 54 * q^3 - 640 * q^4 + 13960 * q^9	\( 10 q + 54 q^{3} - 640 q^{4} + 13960 q^{9} - 1136 q^{10} - 3456 q^{12} + 3432 q^{13} + 3792 q^{14} + 40960 q^{16} - 6918 q^{17} + 17280 q^{22} + 94164 q^{23} - 330788 q^{25} + 118896 q^{26} - 53550 q^{27} - 131304 q^{29} - 434992 q^{30} + 873450 q^{35} - 893440 q^{36} + 602976 q^{38} - 1569460 q^{39} + 72704 q^{40} + 1800592 q^{42} + 2740774 q^{43} + 221184 q^{48} - 3083904 q^{49} + 1103186 q^{51} - 219648 q^{52} - 3605916 q^{53} + 3999840 q^{55} - 242688 q^{56} + 11727656 q^{61} - 4210560 q^{62} - 2621440 q^{64} - 7481034 q^{65} - 5557248 q^{66} + 442752 q^{68} + 2035796 q^{69} + 18145392 q^{74} - 32855720 q^{75} - 7147920 q^{77} - 8343632 q^{78} - 28449228 q^{79} + 49311466 q^{81} + 17868800 q^{82} + 50156320 q^{87} - 1105920 q^{88} - 23059744 q^{90} - 20225194 q^{91} - 6026496 q^{92} + 8240976 q^{94} + 16547484 q^{95}+O(q^{100}) \) 10 * q + 54 * q^3 - 640 * q^4 + 13960 * q^9 - 1136 * q^10 - 3456 * q^12 + 3432 * q^13 + 3792 * q^14 + 40960 * q^16 - 6918 * q^17 + 17280 * q^22 + 94164 * q^23 - 330788 * q^25 + 118896 * q^26 - 53550 * q^27 - 131304 * q^29 - 434992 * q^30 + 873450 * q^35 - 893440 * q^36 + 602976 * q^38 - 1569460 * q^39 + 72704 * q^40 + 1800592 * q^42 + 2740774 * q^43 + 221184 * q^48 - 3083904 * q^49 + 1103186 * q^51 - 219648 * q^52 - 3605916 * q^53 + 3999840 * q^55 - 242688 * q^56 + 11727656 * q^61 - 4210560 * q^62 - 2621440 * q^64 - 7481034 * q^65 - 5557248 * q^66 + 442752 * q^68 + 2035796 * q^69 + 18145392 * q^74 - 32855720 * q^75 - 7147920 * q^77 - 8343632 * q^78 - 28449228 * q^79 + 49311466 * q^81 + 17868800 * q^82 + 50156320 * q^87 - 1105920 * q^88 - 23059744 * q^90 - 20225194 * q^91 - 6026496 * q^92 + 8240976 * q^94 + 16547484 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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