LMFDB - Embedded newform 76.8.e.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(45,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("76.45");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.e (of order $3$, degree $2$, 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:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.2
Character	$\chi$	$=$	76.49
Dual form			76.8.e.a.45.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-34.7112 + 60.1216i) q^{3} +(55.3969 - 95.9502i) q^{5} +314.254 q^{7} +(-1316.24 - 2279.79i) q^{9} +O(q^{10})$	\(q+(-34.7112 + 60.1216i) q^{3} +(55.3969 - 95.9502i) q^{5} +314.254 q^{7} +(-1316.24 - 2279.79i) q^{9} +5498.50 q^{11} +(-6978.48 - 12087.1i) q^{13} +(3845.79 + 6661.10i) q^{15} +(7105.41 - 12306.9i) q^{17} +(4435.93 - 29566.8i) q^{19} +(-10908.1 + 18893.4i) q^{21} +(55006.9 + 95274.7i) q^{23} +(32924.9 + 57027.5i) q^{25} +30926.0 q^{27} +(81208.2 + 140657. i) q^{29} -16244.6 q^{31} +(-190860. + 330579. i) q^{33} +(17408.7 - 30152.7i) q^{35} +309940. q^{37} +968927. q^{39} +(78221.6 - 135484. i) q^{41} +(260356. - 450949. i) q^{43} -291662. q^{45} +(13723.4 + 23769.6i) q^{47} -724788. q^{49} +(493275. + 854377. i) q^{51} +(-311020. - 538702. i) q^{53} +(304600. - 527582. i) q^{55} +(1.62363e6 + 1.29299e6i) q^{57} +(435893. - 754988. i) q^{59} +(1.44211e6 + 2.49780e6i) q^{61} +(-413633. - 716433. i) q^{63} -1.54634e6 q^{65} +(1.19108e6 + 2.06301e6i) q^{67} -7.63742e6 q^{69} +(549314. - 951439. i) q^{71} +(-233183. + 403885. i) q^{73} -4.57145e6 q^{75} +1.72792e6 q^{77} +(3.50042e6 - 6.06291e6i) q^{79} +(1.80513e6 - 3.12658e6i) q^{81} +1.93376e6 q^{83} +(-787235. - 1.36353e6i) q^{85} -1.12753e7 q^{87} +(-3.27060e6 - 5.66484e6i) q^{89} +(-2.19301e6 - 3.79841e6i) q^{91} +(563870. - 976652. i) q^{93} +(-2.59120e6 - 2.06354e6i) q^{95} +(965417. - 1.67215e6i) q^{97} +(-7.23733e6 - 1.25354e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−34.7112	+	60.1216i	−0.742242	+	1.28560i	0.209230	+	0.977866i	$0.432904\pi$
$3$	−34.7112	+	60.1216i	−0.742242	+	1.28560i	−0.951472	+	0.307734i	$0.900429\pi$
$4$		0			0
$5$	55.3969	−	95.9502i	0.198194	−	0.343282i	−0.749749	−	0.661722i	$-0.769826\pi$
$5$	55.3969	−	95.9502i	0.198194	−	0.343282i	0.947943	+	0.318440i	$0.103159\pi$
$6$		0			0
$7$	314.254			0.346288			0.173144	−	0.984897i	$-0.444607\pi$
$7$	314.254			0.346288			0.173144	+	0.984897i	$0.444607\pi$
$8$		0			0
$9$	−1316.24	−	2279.79i	−0.601846	−	1.04243i
$10$		0			0
$11$	5498.50			1.24557			0.622787	−	0.782391i	$-0.286000\pi$
$11$	5498.50			1.24557			0.622787	+	0.782391i	$0.286000\pi$
$12$		0			0
$13$	−6978.48	−	12087.1i	−0.880966	−	1.52588i	−0.850268	−	0.526350i	$-0.823560\pi$
$13$	−6978.48	−	12087.1i	−0.880966	−	1.52588i	−0.0306981	−	0.999529i	$-0.509773\pi$
$14$		0			0
$15$	3845.79	+	6661.10i	0.294216	+	0.509597i
$16$		0			0
$17$	7105.41	−	12306.9i	0.350766	−	0.607545i	−0.635618	−	0.772004i	$-0.719255\pi$
$17$	7105.41	−	12306.9i	0.350766	−	0.607545i	0.986384	+	0.164459i	$0.0525878\pi$
$18$		0			0
$19$	4435.93	−	29566.8i	0.148370	−	0.988932i
$19$	4435.93	−	29566.8i	0.148370	−	0.988932i
$20$		0			0
$21$	−10908.1	+	18893.4i	−0.257029	+	0.445188i
$22$		0			0
$23$	55006.9	+	95274.7i	0.942691	+	1.63279i	0.760309	+	0.649561i	$0.225047\pi$
$23$	55006.9	+	95274.7i	0.942691	+	1.63279i	0.182382	+	0.983228i	$0.441619\pi$
$24$		0			0
$25$	32924.9	+	57027.5i	0.421438	+	0.729953i
$26$		0			0
$27$	30926.0			0.302379
$28$		0			0
$29$	81208.2	+	140657.i	0.618311	+	1.07095i	0.989794	+	0.142506i	$0.0455162\pi$
$29$	81208.2	+	140657.i	0.618311	+	1.07095i	−0.371483	+	0.928440i	$0.621151\pi$
$30$		0			0
$31$	−16244.6			−0.0979362			−0.0489681	−	0.998800i	$-0.515593\pi$
$31$	−16244.6			−0.0979362			−0.0489681	+	0.998800i	$0.515593\pi$
$32$		0			0
$33$	−190860.	+	330579.i	−0.924518	+	1.60131i
$34$		0			0
$35$	17408.7	−	30152.7i	0.0686321	−	0.118874i
$36$		0			0
$37$	309940.			1.00594			0.502970	−	0.864304i	$-0.332241\pi$
$37$	309940.			1.00594			0.502970	+	0.864304i	$0.332241\pi$
$38$		0			0
$39$	968927.			2.61556
$40$		0			0
$41$	78221.6	−	135484.i	0.177249	−	0.307004i	−0.763688	−	0.645585i	$-0.776614\pi$
$41$	78221.6	−	135484.i	0.177249	−	0.307004i	0.940937	+	0.338581i	$0.109947\pi$
$42$		0			0
$43$	260356.	−	450949.i	0.499376	−	0.864944i	−0.500624	−	0.865665i	$-0.666896\pi$
$43$	260356.	−	450949.i	0.499376	−	0.864944i	1.00000		0.000720620i	$0.000229380\pi$
$44$		0			0
$45$	−291662.			−0.477129
$46$		0			0
$47$	13723.4	+	23769.6i	0.0192805	+	0.0333948i	0.875505	−	0.483210i	$-0.160529\pi$
$47$	13723.4	+	23769.6i	0.0192805	+	0.0333948i	−0.856224	+	0.516604i	$0.827196\pi$
$48$		0			0
$49$	−724788.			−0.880085
$50$		0			0
$51$	493275.	+	854377.i	0.520707	+	0.901891i
$52$		0			0
$53$	−311020.	−	538702.i	−0.286961	−	0.497031i	0.686122	−	0.727487i	$-0.259312\pi$
$53$	−311020.	−	538702.i	−0.286961	−	0.497031i	−0.973083	+	0.230456i	$0.925978\pi$
$54$		0			0
$55$	304600.	−	527582.i	0.246865	−	0.427583i
$56$		0			0
$57$	1.62363e6	+	1.29299e6i	1.16124	+	0.924772i
$58$		0			0
$59$	435893.	−	754988.i	0.276311	−	0.478584i	−0.694154	−	0.719826i	$-0.744221\pi$
$59$	435893.	−	754988.i	0.276311	−	0.478584i	0.970465	+	0.241242i	$0.0775548\pi$
$60$		0			0
$61$	1.44211e6	+	2.49780e6i	0.813472	+	1.40898i	0.910420	+	0.413686i	$0.135759\pi$
$61$	1.44211e6	+	2.49780e6i	0.813472	+	1.40898i	−0.0969471	+	0.995290i	$0.530908\pi$
$62$		0			0
$63$	−413633.	−	716433.i	−0.208412	−	0.360980i
$64$		0			0
$65$	−1.54634e6			−0.698409
$66$		0			0
$67$	1.19108e6	+	2.06301e6i	0.483814	+	0.837991i	0.999827	−	0.0185900i	$-0.00591771\pi$
$67$	1.19108e6	+	2.06301e6i	0.483814	+	0.837991i	−0.516013	+	0.856581i	$0.672584\pi$
$68$		0			0
$69$	−7.63742e6			−2.79882
$70$		0			0
$71$	549314.	−	951439.i	0.182145	−	0.315484i	−0.760466	−	0.649378i	$-0.775029\pi$
$71$	549314.	−	951439.i	0.182145	−	0.315484i	0.942611	+	0.333894i	$0.108363\pi$
$72$		0			0
$73$	−233183.	+	403885.i	−0.0701564	+	0.121514i	−0.898970	−	0.438011i	$-0.855683\pi$
$73$	−233183.	+	403885.i	−0.0701564	+	0.121514i	0.828813	+	0.559525i	$0.189016\pi$
$74$		0			0
$75$	−4.57145e6			−1.25124
$76$		0			0
$77$	1.72792e6			0.431327
$78$		0			0
$79$	3.50042e6	−	6.06291e6i	0.798777	−	1.38352i	−0.121636	−	0.992575i	$-0.538814\pi$
$79$	3.50042e6	−	6.06291e6i	0.798777	−	1.38352i	0.920413	−	0.390948i	$-0.127853\pi$
$80$		0			0
$81$	1.80513e6	−	3.12658e6i	0.377408	−	0.653690i
$82$		0			0
$83$	1.93376e6			0.371219			0.185609	−	0.982624i	$-0.440574\pi$
$83$	1.93376e6			0.371219			0.185609	+	0.982624i	$0.440574\pi$
$84$		0			0
$85$	−787235.	−	1.36353e6i	−0.139039	−	0.240823i
$86$		0			0
$87$	−1.12753e7			−1.83575
$88$		0			0
$89$	−3.27060e6	−	5.66484e6i	−0.491770	−	0.851770i	0.508185	−	0.861248i	$-0.330317\pi$
$89$	−3.27060e6	−	5.66484e6i	−0.491770	−	0.851770i	−0.999955	+	0.00947738i	$0.996983\pi$
$90$		0			0
$91$	−2.19301e6	−	3.79841e6i	−0.305068	−	0.528393i
$92$		0			0
$93$	563870.	−	976652.i	0.0726924	−	0.125907i
$94$		0			0
$95$	−2.59120e6	−	2.06354e6i	−0.310076	−	0.246933i
$96$		0			0
$97$	965417.	−	1.67215e6i	0.107402	−	0.186026i	−0.807315	−	0.590121i	$-0.799080\pi$
$97$	965417.	−	1.67215e6i	0.107402	−	0.186026i	0.914717	+	0.404095i	$0.132413\pi$
$98$		0			0
$99$	−7.23733e6	−	1.25354e7i	−0.749645	−	1.29842i
$100$		0			0
$101$	7.19771e6	+	1.24668e7i	0.695136	+	1.20401i	0.970135	+	0.242566i	$0.0779892\pi$
$101$	7.19771e6	+	1.24668e7i	0.695136	+	1.20401i	−0.274999	+	0.961445i	$0.588677\pi$
$102$		0			0
$103$	−6.77524e6			−0.610934			−0.305467	−	0.952203i	$-0.598813\pi$
$103$	−6.77524e6			−0.610934			−0.305467	+	0.952203i	$0.598813\pi$
$104$		0			0
$105$	1.20855e6	+	2.09327e6i	0.101883	+	0.176467i
$106$		0			0
$107$	1.36682e7			1.07862			0.539310	−	0.842107i	$-0.318685\pi$
$107$	1.36682e7			1.07862			0.539310	+	0.842107i	$0.318685\pi$
$108$		0			0
$109$	4.66638e6	−	8.08240e6i	0.345134	−	0.597789i	−0.640244	−	0.768171i	$-0.721167\pi$
$109$	4.66638e6	−	8.08240e6i	0.345134	−	0.597789i	0.985378	+	0.170382i	$0.0545003\pi$
$110$		0			0
$111$	−1.07584e7	+	1.86341e7i	−0.746650	+	1.29324i
$112$		0			0
$113$	−2.27905e6			−0.148587			−0.0742933	−	0.997236i	$-0.523670\pi$
$113$	−2.27905e6			−0.148587			−0.0742933	+	0.997236i	$0.523670\pi$
$114$		0			0
$115$	1.21888e7			0.747343
$116$		0			0
$117$	−1.83707e7	+	3.18190e7i	−1.06041	+	1.83669i
$118$		0			0
$119$	2.23290e6	−	3.86750e6i	0.121466	−	0.210385i
$120$		0			0
$121$	1.07463e7			0.551456
$122$		0			0
$123$	5.43033e6	+	9.40561e6i	0.263123	+	0.455742i
$124$		0			0
$125$	1.59515e7			0.730494
$126$		0			0
$127$	−4.33713e6	−	7.51212e6i	−0.187884	−	0.325424i	0.756661	−	0.653808i	$-0.226829\pi$
$127$	−4.33713e6	−	7.51212e6i	−0.187884	−	0.325424i	−0.944544	+	0.328384i	$0.893496\pi$
$128$		0			0
$129$	1.80745e7	+	3.13060e7i	0.741315	+	1.28400i
$130$		0			0
$131$	4.26445e6	−	7.38624e6i	0.165735	−	0.287061i	−0.771181	−	0.636616i	$-0.780334\pi$
$131$	4.26445e6	−	7.38624e6i	0.165735	−	0.287061i	0.936916	+	0.349555i	$0.113667\pi$
$132$		0			0
$133$	1.39401e6	−	9.29147e6i	0.0513788	−	0.342455i
$134$		0			0
$135$	1.71321e6	−	2.96736e6i	0.0599296	−	0.103801i
$136$		0			0
$137$	1.56197e7	+	2.70541e7i	0.518980	+	0.898899i	0.999757	+	0.0220561i	$0.00702124\pi$
$137$	1.56197e7	+	2.70541e7i	0.518980	+	0.898899i	−0.480777	+	0.876843i	$0.659645\pi$
$138$		0			0
$139$	2.44957e7	+	4.24278e7i	0.773639	+	1.33998i	0.935556	+	0.353177i	$0.114899\pi$
$139$	2.44957e7	+	4.24278e7i	0.773639	+	1.33998i	−0.161918	+	0.986804i	$0.551768\pi$
$140$		0			0
$141$	−1.90542e6			−0.0572431
$142$		0			0
$143$	−3.83712e7	−	6.64608e7i	−1.09731	−	1.90060i
$144$		0			0
$145$	1.79947e7			0.490182
$146$		0			0
$147$	2.51583e7	−	4.35754e7i	0.653236	−	1.13144i
$148$		0			0
$149$	2.24089e7	−	3.88133e7i	0.554968	−	0.961233i	−0.442938	−	0.896552i	$-0.646064\pi$
$149$	2.24089e7	−	3.88133e7i	0.554968	−	0.961233i	0.997906	−	0.0646807i	$-0.0206029\pi$
$150$		0			0
$151$	2.09065e7			0.494155			0.247077	−	0.968996i	$-0.420530\pi$
$151$	2.09065e7			0.494155			0.247077	+	0.968996i	$0.420530\pi$
$152$		0			0
$153$	−3.74096e7			−0.844429
$154$		0			0
$155$	−899901.	+	1.55867e6i	−0.0194104	+	0.0336197i
$156$		0			0
$157$	796406.	−	1.37942e6i	0.0164243	−	0.0284476i	−0.857696	−	0.514156i	$-0.828105\pi$
$157$	796406.	−	1.37942e6i	0.0164243	−	0.0284476i	0.874121	+	0.485709i	$0.161438\pi$
$158$		0			0
$159$	4.31835e7			0.851977
$160$		0			0
$161$	1.72861e7	+	2.99404e7i	0.326442	+	0.565415i
$162$		0			0
$163$	1.04774e7			0.189495			0.0947475	−	0.995501i	$-0.469796\pi$
$163$	1.04774e7			0.189495			0.0947475	+	0.995501i	$0.469796\pi$
$164$		0			0
$165$	2.11461e7	+	3.66260e7i	0.366468	+	0.634741i
$166$		0			0
$167$	−4.18024e7	−	7.24038e7i	−0.694534	−	1.20297i	−0.970338	−	0.241754i	$-0.922277\pi$
$167$	−4.18024e7	−	7.24038e7i	−0.694534	−	1.20297i	0.275804	−	0.961214i	$-0.411056\pi$
$168$		0			0
$169$	−6.60242e7	+	1.14357e8i	−1.05220	+	1.82247i
$170$		0			0
$171$	−7.32448e7	+	2.88039e7i	−1.12019	+	0.440520i
$172$		0			0
$173$	−2.50533e7	+	4.33936e7i	−0.367878	+	0.637183i	−0.989234	−	0.146345i	$-0.953249\pi$
$173$	−2.50533e7	+	4.33936e7i	−0.367878	+	0.637183i	0.621356	+	0.783529i	$0.286582\pi$
$174$		0			0
$175$	1.03468e7	+	1.79211e7i	0.145939	+	0.252774i
$176$		0			0
$177$	3.02607e7	+	5.24131e7i	0.410179	+	0.710450i
$178$		0			0
$179$	1.44225e8			1.87955			0.939775	−	0.341792i	$-0.111034\pi$
$179$	1.44225e8			1.87955			0.939775	+	0.341792i	$0.111034\pi$
$180$		0			0
$181$	−5.54023e7	−	9.59596e7i	−0.694469	−	1.20285i	−0.970359	−	0.241666i	$-0.922306\pi$
$181$	−5.54023e7	−	9.59596e7i	−0.694469	−	1.20285i	0.275891	−	0.961189i	$-0.411027\pi$
$182$		0			0
$183$	−2.00229e8			−2.41517
$184$		0			0
$185$	1.71697e7	−	2.97388e7i	0.199371	−	0.345321i
$186$		0			0
$187$	3.90691e7	−	6.76696e7i	0.436905	−	0.756742i
$188$		0			0
$189$	9.71862e6			0.104710
$190$		0			0
$191$	1.15008e8			1.19429			0.597145	−	0.802133i	$-0.296302\pi$
$191$	1.15008e8			1.19429			0.597145	+	0.802133i	$0.296302\pi$
$192$		0			0
$193$	−9.46969e7	+	1.64020e8i	−0.948168	+	1.64228i	−0.198888	+	0.980022i	$0.563733\pi$
$193$	−9.46969e7	+	1.64020e8i	−0.948168	+	1.64228i	−0.749280	+	0.662253i	$0.769600\pi$
$194$		0			0
$195$	5.36755e7	−	9.29687e7i	0.518388	−	0.897875i
$196$		0			0
$197$	−1.93583e8			−1.80400			−0.901999	−	0.431739i	$-0.857900\pi$
$197$	−1.93583e8			−1.80400			−0.901999	+	0.431739i	$0.857900\pi$
$198$		0			0
$199$	−7.42173e7	−	1.28548e8i	−0.667604	−	1.15632i	−0.978572	−	0.205904i	$-0.933986\pi$
$199$	−7.42173e7	−	1.28548e8i	−0.667604	−	1.15632i	0.310968	−	0.950420i	$-0.399347\pi$
$200$		0			0
$201$	−1.65375e8			−1.43643
$202$		0			0
$203$	2.55200e7	+	4.42019e7i	0.214114	+	0.370856i
$204$		0			0
$205$	−8.66646e6	−	1.50108e7i	−0.0702592	−	0.121693i
$206$		0			0
$207$	1.44804e8	−	2.50808e8i	1.13471	−	1.96538i
$208$		0			0
$209$	2.43910e7	−	1.62573e8i	0.184806	−	1.23179i
$210$		0			0
$211$	−6.62617e7	+	1.14769e8i	−0.485595	+	0.841074i	−0.999863	−	0.0165549i	$-0.994730\pi$
$211$	−6.62617e7	+	1.14769e8i	−0.485595	+	0.841074i	0.514268	+	0.857629i	$0.328064\pi$
$212$		0			0
$213$	3.81347e7	+	6.60512e7i	0.270391	+	0.468330i
$214$		0			0
$215$	−2.88458e7	−	4.99624e7i	−0.197947	−	0.342853i
$216$		0			0
$217$	−5.10493e6			−0.0339141
$218$		0			0
$219$	−1.61881e7	−	2.80387e7i	−0.104146	−	0.180386i
$220$		0			0
$221$	−1.98340e8			−1.23605
$222$		0			0
$223$	9.20560e7	−	1.59446e8i	0.555885	−	0.962821i	−0.441949	−	0.897040i	$-0.645713\pi$
$223$	9.20560e7	−	1.59446e8i	0.555885	−	0.962821i	0.997834	−	0.0657808i	$-0.0209538\pi$
$224$		0			0
$225$	8.66739e7	−	1.50124e8i	0.507282	−	0.878639i
$226$		0			0
$227$	−2.14356e8			−1.21631			−0.608157	−	0.793817i	$-0.708091\pi$
$227$	−2.14356e8			−1.21631			−0.608157	+	0.793817i	$0.708091\pi$
$228$		0			0
$229$	2.62062e8			1.44205			0.721023	−	0.692911i	$-0.243672\pi$
$229$	2.62062e8			1.44205			0.721023	+	0.692911i	$0.243672\pi$
$230$		0			0
$231$	−5.99783e7	+	1.03886e8i	−0.320149	+	0.554515i
$232$		0			0
$233$	1.47755e8	−	2.55919e8i	0.765236	−	1.32543i	−0.174885	−	0.984589i	$-0.555955\pi$
$233$	1.47755e8	−	2.55919e8i	0.765236	−	1.32543i	0.940122	−	0.340839i	$-0.110711\pi$
$234$		0			0
$235$	3.04093e6			0.0152851
$236$		0			0
$237$	2.43008e8	+	4.20902e8i	1.18577	+	2.05382i
$238$		0			0
$239$	−1.94284e8			−0.920543			−0.460272	−	0.887778i	$-0.652248\pi$
$239$	−1.94284e8			−0.920543			−0.460272	+	0.887778i	$0.652248\pi$
$240$		0			0
$241$	−4.00185e7	−	6.93140e7i	−0.184162	−	0.318978i	0.759132	−	0.650937i	$-0.225624\pi$
$241$	−4.00185e7	−	6.93140e7i	−0.184162	−	0.318978i	−0.943294	+	0.331959i	$0.892291\pi$
$242$		0			0
$243$	1.59134e8	+	2.75629e8i	0.711446	+	1.23226i
$244$		0			0
$245$	−4.01510e7	+	6.95435e7i	−0.174427	+	0.302117i
$246$		0			0
$247$	−3.88332e8	+	1.52714e8i	−1.63970	+	0.644820i
$248$		0			0
$249$	−6.71233e7	+	1.16261e8i	−0.275534	+	0.477239i
$250$		0			0
$251$	9.63863e7	+	1.66946e8i	0.384731	+	0.666373i	0.991732	−	0.128328i	$-0.0409609\pi$
$251$	9.63863e7	+	1.66946e8i	0.384731	+	0.666373i	−0.607001	+	0.794701i	$0.707628\pi$
$252$		0			0
$253$	3.02455e8	+	5.23868e8i	1.17419	+	2.03376i
$254$		0			0
$255$	1.09304e8			0.412804
$256$		0			0
$257$	−6.27865e7	−	1.08749e8i	−0.230728	−	0.399633i	0.727294	−	0.686326i	$-0.240777\pi$
$257$	−6.27865e7	−	1.08749e8i	−0.230728	−	0.399633i	−0.958023	+	0.286693i	$0.907444\pi$
$258$		0			0
$259$	9.73998e7			0.348345
$260$		0			0
$261$	2.13779e8	−	3.70276e8i	0.744257	−	1.28909i
$262$		0			0
$263$	−1.37145e8	+	2.37541e8i	−0.464872	+	0.805183i	−0.999196	−	0.0400976i	$-0.987233\pi$
$263$	−1.37145e8	+	2.37541e8i	−0.464872	+	0.805183i	0.534323	+	0.845280i	$0.320566\pi$
$264$		0			0
$265$	−6.89181e7			−0.227496
$266$		0			0
$267$	4.54106e8			1.46005
$268$		0			0
$269$	1.65754e8	−	2.87094e8i	0.519196	−	0.899273i	−0.480555	−	0.876964i	$-0.659565\pi$
$269$	1.65754e8	−	2.87094e8i	0.519196	−	0.899273i	0.999751	−	0.0223090i	$-0.00710178\pi$
$270$		0			0
$271$	−5.45037e7	+	9.44032e7i	−0.166354	+	0.288134i	−0.937135	−	0.348966i	$-0.886533\pi$
$271$	−5.45037e7	+	9.44032e7i	−0.166354	+	0.288134i	0.770781	+	0.637100i	$0.219866\pi$
$272$		0			0
$273$	3.04489e8			0.905737
$274$		0			0
$275$	1.81037e8	+	3.13566e8i	0.524933	+	0.909210i
$276$		0			0
$277$	−1.65820e8			−0.468769			−0.234384	−	0.972144i	$-0.575307\pi$
$277$	−1.65820e8			−0.468769			−0.234384	+	0.972144i	$0.575307\pi$
$278$		0			0
$279$	2.13818e7	+	3.70343e7i	0.0589426	+	0.102092i
$280$		0			0
$281$	−1.43800e8	−	2.49069e8i	−0.386623	−	0.669650i	0.605370	−	0.795944i	$-0.293025\pi$
$281$	−1.43800e8	−	2.49069e8i	−0.386623	−	0.669650i	−0.991993	+	0.126294i	$0.959692\pi$
$282$		0			0
$283$	−1.75879e8	+	3.04632e8i	−0.461278	+	0.798956i	−0.999025	−	0.0441499i	$-0.985942\pi$
$283$	−1.75879e8	+	3.04632e8i	−0.461278	+	0.798956i	0.537747	+	0.843106i	$0.319275\pi$
$284$		0			0
$285$	2.14007e8	−	8.41594e7i	0.547609	−	0.215350i
$286$		0			0
$287$	2.45814e7	−	4.25763e7i	0.0613791	−	0.106312i
$288$		0			0
$289$	1.04196e8	+	1.80472e8i	0.253926	+	0.439813i
$290$		0			0
$291$	6.70216e7	+	1.16085e8i	0.159437	+	0.276153i
$292$		0			0
$293$	−2.79948e8			−0.650190			−0.325095	−	0.945681i	$-0.605396\pi$
$293$	−2.79948e8			−0.650190			−0.325095	+	0.945681i	$0.605396\pi$
$294$		0			0
$295$	−4.82942e7	−	8.36480e7i	−0.109526	−	0.189705i
$296$		0			0
$297$	1.70047e8			0.376635
$298$		0			0
$299$	7.67729e8	−	1.32975e9i	1.66096	−	2.87686i
$300$		0			0
$301$	8.18177e7	−	1.41712e8i	0.172928	−	0.299520i
$302$		0			0
$303$	−9.99366e8			−2.06384
$304$		0			0
$305$	3.19553e8			0.644901
$306$		0			0
$307$	−4.49443e8	+	7.78457e8i	−0.886523	+	1.53550i	−0.0425647	+	0.999094i	$0.513553\pi$
$307$	−4.49443e8	+	7.78457e8i	−0.886523	+	1.53550i	−0.843958	+	0.536409i	$0.819780\pi$
$308$		0			0
$309$	2.35177e8	−	4.07339e8i	0.453461	−	0.785418i
$310$		0			0
$311$	−2.80954e7			−0.0529631			−0.0264816	−	0.999649i	$-0.508430\pi$
$311$	−2.80954e7			−0.0529631			−0.0264816	+	0.999649i	$0.508430\pi$
$312$		0			0
$313$	−2.87448e8	−	4.97874e8i	−0.529851	−	0.917729i	−0.999394	−	0.0348194i	$-0.988914\pi$
$313$	−2.87448e8	−	4.97874e8i	−0.529851	−	0.917729i	0.469542	−	0.882910i	$-0.344419\pi$
$314$		0			0
$315$	−9.16558e7			−0.165224
$316$		0			0
$317$	−2.19017e8	−	3.79349e8i	−0.386163	−	0.668854i	0.605767	−	0.795642i	$-0.292867\pi$
$317$	−2.19017e8	−	3.79349e8i	−0.386163	−	0.668854i	−0.991930	+	0.126788i	$0.959533\pi$
$318$		0			0
$319$	4.46523e8	+	7.73401e8i	0.770153	+	1.33394i
$320$		0			0
$321$	−4.74441e8	+	8.21755e8i	−0.800598	+	1.38668i
$322$		0			0
$323$	−3.32357e8	−	2.64677e8i	−0.548777	−	0.437026i
$324$		0			0
$325$	4.59531e8	−	7.95932e8i	0.742546	−	1.28613i
$326$		0			0
$327$	3.23951e8	+	5.61100e8i	0.512345	+	0.887408i
$328$		0			0
$329$	4.31262e6	+	7.46967e6i	0.00667660	+	0.0115642i
$330$		0			0
$331$	−4.39622e7			−0.0666317			−0.0333159	−	0.999445i	$-0.510607\pi$
$331$	−4.39622e7			−0.0666317			−0.0333159	+	0.999445i	$0.510607\pi$
$332$		0			0
$333$	−4.07955e8	−	7.06599e8i	−0.605421	−	1.04862i
$334$		0			0
$335$	2.63928e8			0.383556
$336$		0			0
$337$	−2.89827e8	+	5.01996e8i	−0.412510	+	0.714489i	−0.995164	−	0.0982322i	$-0.968681\pi$
$337$	−2.89827e8	+	5.01996e8i	−0.412510	+	0.714489i	0.582653	+	0.812721i	$0.302015\pi$
$338$		0			0
$339$	7.91087e7	−	1.37020e8i	0.110287	−	0.191023i
$340$		0			0
$341$	−8.93210e7			−0.121987
$342$		0			0
$343$	−4.86569e8			−0.651050
$344$		0			0
$345$	−4.23089e8	+	7.32812e8i	−0.554709	+	0.960784i
$346$		0			0
$347$	−3.03214e8	+	5.25183e8i	−0.389580	+	0.674772i	−0.992393	−	0.123110i	$-0.960713\pi$
$347$	−3.03214e8	+	5.25183e8i	−0.389580	+	0.674772i	0.602813	+	0.797882i	$0.294046\pi$
$348$		0			0
$349$	6.18510e7			0.0778857			0.0389428	−	0.999241i	$-0.487601\pi$
$349$	6.18510e7			0.0778857			0.0389428	+	0.999241i	$0.487601\pi$
$350$		0			0
$351$	−2.15817e8	−	3.73806e8i	−0.266385	−	0.461393i
$352$		0			0
$353$	−1.46377e9			−1.77117			−0.885587	−	0.464474i	$-0.846244\pi$
$353$	−1.46377e9			−1.77117			−0.885587	+	0.464474i	$0.846244\pi$
$354$		0			0
$355$	−6.08605e7	−	1.05414e8i	−0.0721999	−	0.125054i
$356$		0			0
$357$	1.55013e8	+	2.68491e8i	0.180314	+	0.312314i
$358$		0			0
$359$	8.75513e8	−	1.51643e9i	0.998693	−	1.72979i	0.455091	−	0.890445i	$-0.349607\pi$
$359$	8.75513e8	−	1.51643e9i	0.998693	−	1.72979i	0.543603	−	0.839343i	$-0.317060\pi$
$360$		0			0
$361$	−8.54517e8	−	2.62312e8i	−0.955972	−	0.293456i
$362$		0			0
$363$	−3.73018e8	+	6.46086e8i	−0.409314	+	0.708952i
$364$		0			0
$365$	2.58352e7	+	4.47479e7i	0.0278091	+	0.0481668i
$366$		0			0
$367$	1.94986e8	+	3.37726e8i	0.205908	+	0.356643i	0.950422	−	0.310964i	$-0.100652\pi$
$367$	1.94986e8	+	3.37726e8i	0.205908	+	0.356643i	−0.744514	+	0.667607i	$0.767319\pi$
$368$		0			0
$369$	−4.11833e8			−0.426706
$370$		0			0
$371$	−9.77391e7	−	1.69289e8i	−0.0993710	−	0.172116i
$372$		0			0
$373$	−9.89112e8			−0.986881			−0.493440	−	0.869780i	$-0.664261\pi$
$373$	−9.89112e8			−0.986881			−0.493440	+	0.869780i	$0.664261\pi$
$374$		0			0
$375$	−5.53696e8	+	9.59030e8i	−0.542203	+	0.939124i
$376$		0			0
$377$	1.13342e9	−	1.96314e9i	1.08942	−	1.88694i
$378$		0			0
$379$	1.73213e9			1.63434			0.817170	−	0.576396i	$-0.195541\pi$
$379$	1.73213e9			1.63434			0.817170	+	0.576396i	$0.195541\pi$
$380$		0			0
$381$	6.02188e8			0.557821
$382$		0			0
$383$	2.86636e8	−	4.96468e8i	0.260697	−	0.451540i	−0.705731	−	0.708480i	$-0.749381\pi$
$383$	2.86636e8	−	4.96468e8i	0.260697	−	0.451540i	0.966427	+	0.256941i	$0.0827145\pi$
$384$		0			0
$385$	9.57216e7	−	1.65795e8i	0.0854864	−	0.148067i
$386$		0			0
$387$	−1.37076e9			−1.20219
$388$		0			0
$389$	−6.74245e8	−	1.16783e9i	−0.580756	−	1.00590i	−0.995390	−	0.0959108i	$-0.969424\pi$
$389$	−6.74245e8	−	1.16783e9i	−0.580756	−	1.00590i	0.414634	−	0.909988i	$-0.363910\pi$
$390$		0			0
$391$	1.56338e9			1.32266
$392$		0			0
$393$	2.96048e8	+	5.12771e8i	0.246030	+	0.426137i
$394$		0			0
$395$	−3.87825e8	−	6.71733e8i	−0.316626	−	0.548412i
$396$		0			0
$397$	2.37718e8	−	4.11740e8i	0.190676	−	0.330260i	−0.754799	−	0.655957i	$-0.772265\pi$
$397$	2.37718e8	−	4.11740e8i	0.190676	−	0.330260i	0.945474	+	0.325697i	$0.105599\pi$
$398$		0			0
$399$	5.10230e8	+	4.06328e8i	0.402125	+	0.320237i
$400$		0			0
$401$	3.53999e8	−	6.13145e8i	0.274156	−	0.474851i	−0.695766	−	0.718268i	$-0.744935\pi$
$401$	3.53999e8	−	6.13145e8i	0.274156	−	0.474851i	0.969922	+	0.243417i	$0.0782684\pi$
$402$		0			0
$403$	1.13363e8	+	1.96350e8i	0.0862785	+	0.149439i
$404$		0			0
$405$	−1.99997e8	−	3.46406e8i	−0.149600	−	0.259115i
$406$		0			0
$407$	1.70421e9			1.25297
$408$		0			0
$409$	6.20281e8	+	1.07436e9i	0.448288	+	0.776457i	0.998275	−	0.0587163i	$-0.0187007\pi$
$409$	6.20281e8	+	1.07436e9i	0.448288	+	0.776457i	−0.549987	+	0.835173i	$0.685367\pi$
$410$		0			0
$411$	−2.16871e9			−1.54083
$412$		0			0
$413$	1.36981e8	−	2.37258e8i	0.0956830	−	0.165728i
$414$		0			0
$415$	1.07125e8	−	1.85545e8i	0.0735733	−	0.127433i
$416$		0			0
$417$	−3.40110e9			−2.29691
$418$		0			0
$419$	−1.51561e9			−1.00656			−0.503278	−	0.864124i	$-0.667873\pi$
$419$	−1.51561e9			−1.00656			−0.503278	+	0.864124i	$0.667873\pi$
$420$		0			0
$421$	−5.25723e8	+	9.10578e8i	−0.343375	+	0.594744i	−0.985057	−	0.172227i	$-0.944904\pi$
$421$	−5.25723e8	+	9.10578e8i	−0.343375	+	0.594744i	0.641682	+	0.766971i	$0.278237\pi$
$422$		0			0
$423$	3.61264e7	−	6.25728e7i	0.0232078	−	0.0401970i
$424$		0			0
$425$	9.35778e8			0.591305
$426$		0			0
$427$	4.53187e8	+	7.84943e8i	0.281696	+	0.487911i
$428$		0			0
$429$	5.32764e9			3.25788
$430$		0			0
$431$	1.22132e9	+	2.11539e9i	0.734783	+	1.27268i	0.954818	+	0.297190i	$0.0960493\pi$
$431$	1.22132e9	+	2.11539e9i	0.734783	+	1.27268i	−0.220035	+	0.975492i	$0.570617\pi$
$432$		0			0
$433$	9.88105e8	+	1.71145e9i	0.584919	+	1.01311i	0.994886	+	0.101009i	$0.0322069\pi$
$433$	9.88105e8	+	1.71145e9i	0.584919	+	1.01311i	−0.409967	+	0.912100i	$0.634460\pi$
$434$		0			0
$435$	−6.24619e8	+	1.08187e9i	−0.363834	+	0.630178i
$436$		0			0
$437$	3.06097e9	−	1.20374e9i	1.75458	−	0.690000i
$438$		0			0
$439$	−5.97929e8	+	1.03564e9i	−0.337306	+	0.584231i	−0.983925	−	0.178582i	$-0.942849\pi$
$439$	−5.97929e8	+	1.03564e9i	−0.337306	+	0.584231i	0.646619	+	0.762813i	$0.276182\pi$
$440$		0			0
$441$	9.53993e8	+	1.65236e9i	0.529676	+	0.917425i
$442$		0			0
$443$	−5.24364e8	−	9.08225e8i	−0.286563	−	0.496342i	0.686424	−	0.727201i	$-0.259179\pi$
$443$	−5.24364e8	−	9.08225e8i	−0.286563	−	0.496342i	−0.972987	+	0.230860i	$0.925846\pi$
$444$		0			0
$445$	−7.24723e8			−0.389863
$446$		0			0
$447$	1.55568e9	+	2.69451e9i	0.823841	+	1.42694i
$448$		0			0
$449$	1.44409e9			0.752890			0.376445	−	0.926439i	$-0.377146\pi$
$449$	1.44409e9			0.752890			0.376445	+	0.926439i	$0.377146\pi$
$450$		0			0
$451$	4.30101e8	−	7.44957e8i	0.220776	−	0.382396i
$452$		0			0
$453$	−7.25692e8	+	1.25693e9i	−0.366782	+	0.635286i
$454$		0			0
$455$	−4.85945e8			−0.241850
$456$		0			0
$457$	−1.02460e9			−0.502164			−0.251082	−	0.967966i	$-0.580786\pi$
$457$	−1.02460e9			−0.502164			−0.251082	+	0.967966i	$0.580786\pi$
$458$		0			0
$459$	2.19742e8	−	3.80604e8i	0.106064	−	0.183709i
$460$		0			0
$461$	−1.59748e9	+	2.76691e9i	−0.759418	+	1.31535i	0.183729	+	0.982977i	$0.441183\pi$
$461$	−1.59748e9	+	2.76691e9i	−0.759418	+	1.31535i	−0.943148	+	0.332374i	$0.892150\pi$
$462$		0			0
$463$	−1.94749e9			−0.911887			−0.455943	−	0.890009i	$-0.650698\pi$
$463$	−1.94749e9			−0.911887			−0.455943	+	0.890009i	$0.650698\pi$
$464$		0			0
$465$	−6.24733e7	−	1.08207e8i	−0.0288144	−	0.0499080i
$466$		0			0
$467$	−2.50560e7			−0.0113842			−0.00569209	−	0.999984i	$-0.501812\pi$
$467$	−2.50560e7			−0.0113842			−0.00569209	+	0.999984i	$0.501812\pi$
$468$		0			0
$469$	3.74301e8	+	6.48308e8i	0.167539	+	0.290186i
$470$		0			0
$471$	5.52884e7	+	9.57624e7i	0.0243815	+	0.0422301i
$472$		0			0
$473$	1.43157e9	−	2.47954e9i	0.622010	−	1.07735i
$474$		0			0
$475$	1.83217e9	−	7.20512e8i	0.784402	−	0.308470i
$476$		0			0
$477$	−8.18752e8	+	1.41812e9i	−0.345413	+	0.598272i
$478$		0			0
$479$	−7.68080e8	−	1.33035e9i	−0.319324	−	0.553086i	0.661023	−	0.750366i	$-0.270123\pi$
$479$	−7.68080e8	−	1.33035e9i	−0.319324	−	0.553086i	−0.980347	+	0.197280i	$0.936789\pi$
$480$		0			0
$481$	−2.16291e9	−	3.74627e9i	−0.886199	−	1.53494i
$482$		0			0
$483$	−2.40009e9			−0.969197
$484$		0			0
$485$	−1.06962e8	−	1.85264e8i	−0.0425730	−	0.0737386i
$486$		0			0
$487$	1.75879e9			0.690023			0.345011	−	0.938599i	$-0.387875\pi$
$487$	1.75879e9			0.690023			0.345011	+	0.938599i	$0.387875\pi$
$488$		0			0
$489$	−3.63684e8	+	6.29919e8i	−0.140651	+	0.243615i
$490$		0			0
$491$	2.78978e8	−	4.83204e8i	0.106362	−	0.184224i	−0.807932	−	0.589276i	$-0.799413\pi$
$491$	2.78978e8	−	4.83204e8i	0.106362	−	0.184224i	0.914294	+	0.405052i	$0.132747\pi$
$492$		0			0
$493$	2.30807e9			0.867531
$494$		0			0
$495$	−1.60370e9			−0.594300
$496$		0			0
$497$	1.72624e8	−	2.98993e8i	0.0630744	−	0.109248i
$498$		0			0
$499$	1.81789e9	−	3.14868e9i	0.654962	−	1.13443i	−0.326941	−	0.945045i	$-0.606018\pi$
$499$	1.81789e9	−	3.14868e9i	0.654962	−	1.13443i	0.981903	−	0.189383i	$-0.0606489\pi$
$500$		0			0
$501$	5.80405e9			2.06205
$502$		0			0
$503$	−2.65626e7	−	4.60077e7i	−0.00930642	−	0.0161192i	0.861335	−	0.508038i	$-0.169629\pi$
$503$	−2.65626e7	−	4.60077e7i	−0.00930642	−	0.0161192i	−0.870641	+	0.491919i	$0.836296\pi$
$504$		0			0
$505$	1.59492e9			0.551087
$506$		0			0
$507$	−4.58356e9	−	7.93896e9i	−1.56198	−	2.70543i
$508$		0			0
$509$	−1.35953e9	−	2.35478e9i	−0.456959	−	0.791476i	0.541840	−	0.840482i	$-0.317728\pi$
$509$	−1.35953e9	−	2.35478e9i	−0.456959	−	0.791476i	−0.998798	+	0.0490061i	$0.984395\pi$
$510$		0			0
$511$	−7.32786e7	+	1.26922e8i	−0.0242943	+	0.0420790i
$512$		0			0
$513$	1.37186e8	−	9.14383e8i	0.0448640	−	0.299032i
$514$		0			0
$515$	−3.75327e8	+	6.50086e8i	−0.121084	+	0.209723i
$516$		0			0
$517$	7.54579e7	+	1.30697e8i	0.0240153	+	0.0415957i
$518$		0			0
$519$	−1.73926e9	−	3.01249e9i	−0.546109	−	0.945889i
$520$		0			0
$521$	−6.09257e9			−1.88742			−0.943710	−	0.330773i	$-0.892691\pi$
$521$	−6.09257e9			−1.88742			−0.943710	+	0.330773i	$0.892691\pi$
$522$		0			0
$523$	1.57818e9	+	2.73348e9i	0.482391	+	0.835526i	0.999796	−	0.0202148i	$-0.00643502\pi$
$523$	1.57818e9	+	2.73348e9i	0.482391	+	0.835526i	−0.517404	+	0.855741i	$0.673102\pi$
$524$		0			0
$525$	−1.43659e9			−0.433288
$526$		0			0
$527$	−1.15425e8	+	1.99921e8i	−0.0343527	+	0.0595007i
$528$		0			0
$529$	−4.34910e9	+	7.53286e9i	−1.27733	+	2.21241i
$530$		0			0
$531$	−2.29495e9			−0.665186
$532$		0			0
$533$	−2.18347e9			−0.624601
$534$		0			0
$535$	7.57177e8	−	1.31147e9i	0.213776	−	0.370271i
$536$		0			0
$537$	−5.00622e9	+	8.67102e9i	−1.39508	+	2.41635i
$538$		0			0
$539$	−3.98524e9			−1.09621
$540$		0			0
$541$	−2.53489e9	−	4.39056e9i	−0.688286	−	1.19215i	−0.972392	−	0.233353i	$-0.925030\pi$
$541$	−2.53489e9	−	4.39056e9i	−0.688286	−	1.19215i	0.284106	−	0.958793i	$-0.408303\pi$
$542$		0			0
$543$	7.69232e9			2.06186
$544$		0			0
$545$	−5.17006e8	−	8.95480e8i	−0.136807	−	0.236956i
$546$		0			0
$547$	2.21156e9	+	3.83053e9i	0.577754	+	1.00070i	0.995736	+	0.0922439i	$0.0294039\pi$
$547$	2.21156e9	+	3.83053e9i	0.577754	+	1.00070i	−0.417983	+	0.908455i	$0.637263\pi$
$548$		0			0
$549$	3.79631e9	−	6.57540e9i	0.979171	−	1.69597i
$550$		0			0
$551$	4.51900e9	−	1.77712e9i	1.15083	−	0.452571i
$552$		0			0
$553$	1.10002e9	−	1.90529e9i	0.276607	−	0.479097i
$554$		0			0
$555$	1.19196e9	+	2.06454e9i	0.295963	+	0.512623i
$556$		0			0
$557$	−4.16049e8	−	7.20618e8i	−0.102012	−	0.176690i	0.810502	−	0.585737i	$-0.199195\pi$
$557$	−4.16049e8	−	7.20618e8i	−0.102012	−	0.176690i	−0.912513	+	0.409047i	$0.865861\pi$
$558$		0			0
$559$	−7.26755e9			−1.75973
$560$		0			0
$561$	2.71227e9	+	4.69779e9i	0.648579	+	1.12337i
$562$		0			0
$563$	6.03505e9			1.42528			0.712642	−	0.701527i	$-0.247498\pi$
$563$	6.03505e9			1.42528			0.712642	+	0.701527i	$0.247498\pi$
$564$		0			0
$565$	−1.26252e8	+	2.18676e8i	−0.0294490	+	0.0510071i
$566$		0			0
$567$	5.67269e8	−	9.82539e8i	0.130692	−	0.226365i
$568$		0			0
$569$	−3.73383e9			−0.849691			−0.424846	−	0.905266i	$-0.639672\pi$
$569$	−3.73383e9			−0.849691			−0.424846	+	0.905266i	$0.639672\pi$
$570$		0			0
$571$	−1.55028e9			−0.348484			−0.174242	−	0.984703i	$-0.555747\pi$
$571$	−1.55028e9			−0.348484			−0.174242	+	0.984703i	$0.555747\pi$
$572$		0			0
$573$	−3.99205e9	+	6.91444e9i	−0.886452	+	1.53538i
$574$		0			0
$575$	−3.62219e9	+	6.27381e9i	−0.794572	+	1.37624i
$576$		0			0
$577$	−3.99501e9			−0.865771			−0.432885	−	0.901449i	$-0.642505\pi$
$577$	−3.99501e9			−0.865771			−0.432885	+	0.901449i	$0.642505\pi$
$578$		0			0
$579$	−6.57409e9	−	1.13867e10i	−1.40754	−	2.43793i
$580$		0			0
$581$	6.07692e8			0.128549
$582$		0			0
$583$	−1.71014e9	−	2.96205e9i	−0.357431	−	0.619089i
$584$		0			0
$585$	2.03536e9	+	3.52534e9i	0.420335	+	0.728041i
$586$		0			0
$587$	−2.24584e8	+	3.88990e8i	−0.0458294	+	0.0793789i	−0.888030	−	0.459785i	$-0.847926\pi$
$587$	−2.24584e8	+	3.88990e8i	−0.0458294	+	0.0793789i	0.842201	+	0.539164i	$0.181260\pi$
$588$		0			0
$589$	−7.20600e7	+	4.80301e8i	−0.0145308	+	0.0968523i
$590$		0			0
$591$	6.71951e9	−	1.16385e10i	1.33900	−	2.31922i
$592$		0			0
$593$	4.47919e9	+	7.75819e9i	0.882080	+	1.52781i	0.849024	+	0.528355i	$0.177191\pi$
$593$	4.47919e9	+	7.75819e9i	0.882080	+	1.52781i	0.0330566	+	0.999453i	$0.489476\pi$
$594$		0			0
$595$	−2.47391e8	−	4.28495e8i	−0.0481477	−	0.0833942i
$596$		0			0
$597$	1.03047e10			1.98210
$598$		0			0
$599$	1.38947e9	+	2.40663e9i	0.264152	+	0.457525i	0.967341	−	0.253478i	$-0.0815745\pi$
$599$	1.38947e9	+	2.40663e9i	0.264152	+	0.457525i	−0.703189	+	0.711003i	$0.748241\pi$
$600$		0			0
$601$	−7.09307e9			−1.33283			−0.666413	−	0.745583i	$-0.732171\pi$
$601$	−7.09307e9			−1.33283			−0.666413	+	0.745583i	$0.732171\pi$
$602$		0			0
$603$	3.13549e9	−	5.43082e9i	0.582364	−	1.00868i
$604$		0			0
$605$	5.95312e8	−	1.03111e9i	0.109295	−	0.189305i
$606$		0			0
$607$	−3.27307e9			−0.594012			−0.297006	−	0.954876i	$-0.595988\pi$
$607$	−3.27307e9			−0.594012			−0.297006	+	0.954876i	$0.595988\pi$
$608$		0			0
$609$	−3.54332e9			−0.635696
$610$		0			0
$611$	1.91536e8	−	3.31751e8i	0.0339709	−	0.0588393i
$612$		0			0
$613$	2.29723e9	−	3.97891e9i	0.402803	−	0.697674i	−0.591260	−	0.806481i	$-0.701369\pi$
$613$	2.29723e9	−	3.97891e9i	0.402803	−	0.697674i	0.994063	+	0.108806i	$0.0347028\pi$
$614$		0			0
$615$	1.20329e9			0.208597
$616$		0			0
$617$	−4.13572e8	−	7.16327e8i	−0.0708847	−	0.122776i	0.828405	−	0.560130i	$-0.189249\pi$
$617$	−4.13572e8	−	7.16327e8i	−0.0708847	−	0.122776i	−0.899289	+	0.437354i	$0.855916\pi$
$618$		0			0
$619$	9.28064e9			1.57275			0.786376	−	0.617748i	$-0.211955\pi$
$619$	9.28064e9			1.57275			0.786376	+	0.617748i	$0.211955\pi$
$620$		0			0
$621$	1.70114e9	+	2.94647e9i	0.285050	+	0.493721i
$622$		0			0
$623$	−1.02780e9	−	1.78020e9i	−0.170294	−	0.294958i
$624$		0			0
$625$	−1.68859e9	+	2.92473e9i	−0.276659	+	0.479187i
$626$		0			0
$627$	8.92750e9	+	7.10953e9i	1.44642	+	1.15187i
$628$		0			0
$629$	2.20225e9	−	3.81441e9i	0.352850	−	0.611153i
$630$		0			0
$631$	1.91589e9	+	3.31842e9i	0.303576	+	0.525810i	0.976943	−	0.213499i	$-0.0684859\pi$
$631$	1.91589e9	+	3.31842e9i	0.303576	+	0.525810i	−0.673367	+	0.739308i	$0.735153\pi$
$632$		0			0
$633$	−4.60005e9	−	7.96752e9i	−0.720857	−	1.24856i
$634$		0			0
$635$	−9.61053e8			−0.148950
$636$		0			0
$637$	5.05792e9	+	8.76057e9i	0.775325	+	1.34290i
$638$		0			0
$639$	−2.89211e9			−0.438492
$640$		0			0
$641$	−3.22588e9	+	5.58739e9i	−0.483777	+	0.837926i	−0.999826	−	0.0186329i	$-0.994069\pi$
$641$	−3.22588e9	+	5.58739e9i	−0.483777	+	0.837926i	0.516050	+	0.856559i	$0.327402\pi$
$642$		0			0
$643$	2.69712e9	−	4.67156e9i	0.400095	−	0.692984i	−0.593642	−	0.804729i	$-0.702311\pi$
$643$	2.69712e9	−	4.67156e9i	0.400095	−	0.692984i	0.993737	+	0.111745i	$0.0356440\pi$
$644$		0			0
$645$	4.00509e9			0.587697
$646$		0			0
$647$	−8.67741e9			−1.25958			−0.629789	−	0.776766i	$-0.716859\pi$
$647$	−8.67741e9			−1.25958			−0.629789	+	0.776766i	$0.716859\pi$
$648$		0			0
$649$	2.39676e9	−	4.15130e9i	0.344165	−	0.596112i
$650$		0			0
$651$	1.77198e8	−	3.06916e8i	0.0251725	−	0.0436000i
$652$		0			0
$653$	2.73388e9			0.384223			0.192111	−	0.981373i	$-0.438467\pi$
$653$	2.73388e9			0.384223			0.192111	+	0.981373i	$0.438467\pi$
$654$		0			0
$655$	−4.72474e8	−	8.18350e8i	−0.0656952	−	0.113787i
$656$		0			0
$657$	1.22770e9			0.168893
$658$		0			0
$659$	3.11402e9	+	5.39364e9i	0.423860	+	0.734147i	0.996313	−	0.0857905i	$-0.0273416\pi$
$659$	3.11402e9	+	5.39364e9i	0.423860	+	0.734147i	−0.572453	+	0.819937i	$0.694008\pi$
$660$		0			0
$661$	−9.82484e8	−	1.70171e9i	−0.132318	−	0.229182i	0.792251	−	0.610195i	$-0.208909\pi$
$661$	−9.82484e8	−	1.70171e9i	−0.132318	−	0.229182i	−0.924570	+	0.381013i	$0.875575\pi$
$662$		0			0
$663$	6.88462e9	−	1.19245e10i	0.917450	−	1.58907i
$664$		0			0
$665$	−8.14295e8	−	6.48474e8i	−0.107376	−	0.0855099i
$666$		0			0
$667$	−8.93402e9	+	1.54742e10i	−1.16575	+	2.01914i
$668$		0			0
$669$	6.39075e9	+	1.10691e10i	0.825202	+	1.42929i
$670$		0			0
$671$	7.92942e9	+	1.37342e10i	1.01324	+	1.75498i
$672$		0			0
$673$	−1.68024e9			−0.212480			−0.106240	−	0.994340i	$-0.533881\pi$
$673$	−1.68024e9			−0.212480			−0.106240	+	0.994340i	$0.533881\pi$
$674$		0			0
$675$	1.01824e9	+	1.76364e9i	0.127434	+	0.220722i
$676$		0			0
$677$	4.67768e9			0.579389			0.289695	−	0.957119i	$-0.406446\pi$
$677$	4.67768e9			0.579389			0.289695	+	0.957119i	$0.406446\pi$
$678$		0			0
$679$	3.03386e8	−	5.25480e8i	0.0371921	−	0.0644187i
$680$		0			0
$681$	7.44056e9	−	1.28874e10i	0.902799	−	1.56369i
$682$		0			0
$683$	−1.28411e10			−1.54216			−0.771082	−	0.636736i	$-0.780284\pi$
$683$	−1.28411e10			−1.54216			−0.771082	+	0.636736i	$0.780284\pi$
$684$		0			0
$685$	3.46113e9			0.411434
$686$		0			0
$687$	−9.09648e9	+	1.57556e10i	−1.07035	+	1.85390i
$688$		0			0
$689$	−4.34089e9	+	7.51865e9i	−0.505606	+	0.875735i
$690$		0			0
$691$	5.11471e9			0.589723			0.294861	−	0.955540i	$-0.404727\pi$
$691$	5.11471e9			0.589723			0.294861	+	0.955540i	$0.404727\pi$
$692$		0			0
$693$	−2.27436e9	−	3.93930e9i	−0.259593	−	0.449628i
$694$		0			0
$695$	5.42794e9			0.613322
$696$		0			0
$697$	−1.11159e9	−	1.92533e9i	−0.124346	−	0.215373i
$698$		0			0
$699$	1.02575e10	+	1.77665e10i	1.13598	+	1.96758i
$700$		0			0
$701$	−1.25429e9	+	2.17249e9i	−0.137526	+	0.238201i	−0.926559	−	0.376148i	$-0.877248\pi$
$701$	−1.25429e9	+	2.17249e9i	−0.137526	+	0.238201i	0.789034	+	0.614350i	$0.210582\pi$
$702$		0			0
$703$	1.37487e9	−	9.16393e9i	0.149252	−	0.994805i
$704$		0			0
$705$	−1.05554e8	+	1.82825e8i	−0.0113452	+	0.0196505i
$706$		0			0
$707$	2.26191e9	+	3.91774e9i	0.240717	+	0.416934i
$708$		0			0
$709$	−8.83015e9	−	1.52943e10i	−0.930478	−	1.61164i	−0.782505	−	0.622644i	$-0.786058\pi$
$709$	−8.83015e9	−	1.52943e10i	−0.930478	−	1.61164i	−0.147973	−	0.988991i	$-0.547275\pi$
$710$		0			0
$711$	−1.84296e10			−1.92296
$712$		0			0
$713$	−8.93565e8	−	1.54770e9i	−0.0923236	−	0.159909i
$714$		0			0
$715$	−8.50258e9			−0.869920
$716$		0			0
$717$	6.74383e9	−	1.16807e10i	0.683266	−	1.18345i
$718$		0			0
$719$	3.57729e9	−	6.19605e9i	0.358925	−	0.621676i	−0.628857	−	0.777521i	$-0.716477\pi$
$719$	3.57729e9	−	6.19605e9i	0.358925	−	0.621676i	0.987781	+	0.155845i	$0.0498101\pi$
$720$		0			0
$721$	−2.12915e9			−0.211559
$722$		0			0
$723$	5.55636e9			0.546772
$724$		0			0
$725$	−5.34754e9	+	9.26221e9i	−0.521160	+	0.902676i
$726$		0			0
$727$	5.33725e9	−	9.24439e9i	0.515166	−	0.892294i	−0.484679	−	0.874692i	$-0.661064\pi$
$727$	5.33725e9	−	9.24439e9i	0.515166	−	0.892294i	0.999845	−	0.0176016i	$-0.00560305\pi$
$728$		0			0
$729$	−1.41993e10			−1.35744
$730$		0			0
$731$	−3.69987e9	−	6.40836e9i	−0.350328	−	0.606786i
$732$		0			0
$733$	1.53862e10			1.44300			0.721500	−	0.692414i	$-0.243453\pi$
$733$	1.53862e10			1.44300			0.721500	+	0.692414i	$0.243453\pi$
$734$		0			0
$735$	−2.78738e9	−	4.82788e9i	−0.258935	−	0.448488i
$736$		0			0
$737$	6.54914e9	+	1.13435e10i	0.602627	+	1.04378i
$738$		0			0
$739$	2.40465e9	−	4.16498e9i	0.219178	−	0.379627i	−0.735379	−	0.677656i	$-0.762996\pi$
$739$	2.40465e9	−	4.16498e9i	0.219178	−	0.379627i	0.954557	+	0.298029i	$0.0963292\pi$
$740$		0			0
$741$	4.29809e9	−	2.86480e10i	0.388072	−	2.58661i
$742$		0			0
$743$	−5.08132e9	+	8.80111e9i	−0.454481	+	0.787184i	−0.998658	−	0.0517861i	$-0.983509\pi$
$743$	−5.08132e9	+	8.80111e9i	−0.454481	+	0.787184i	0.544177	+	0.838970i	$0.316842\pi$
$744$		0			0
$745$	−2.48276e9	−	4.30027e9i	−0.219983	−	0.381021i
$746$		0			0
$747$	−2.54529e9	−	4.40858e9i	−0.223417	−	0.386969i
$748$		0			0
$749$	4.29529e9			0.373513
$750$		0			0
$751$	−4.96190e9	−	8.59427e9i	−0.427473	−	0.740405i	0.569175	−	0.822216i	$-0.307263\pi$
$751$	−4.96190e9	−	8.59427e9i	−0.427473	−	0.740405i	−0.996648	+	0.0818118i	$0.973929\pi$
$752$		0			0
$753$	−1.33827e10			−1.14225
$754$		0			0
$755$	1.15816e9	−	2.00599e9i	0.0979384	−	0.169634i
$756$		0			0
$757$	−1.26016e9	+	2.18266e9i	−0.105582	+	0.182873i	−0.913976	−	0.405769i	$-0.867004\pi$
$757$	−1.26016e9	+	2.18266e9i	−0.105582	+	0.182873i	0.808394	+	0.588642i	$0.200337\pi$
$758$		0			0
$759$	−4.19944e10			−3.48614
$760$		0			0
$761$	1.56876e10			1.29035			0.645177	−	0.764033i	$-0.276783\pi$
$761$	1.56876e10			1.29035			0.645177	+	0.764033i	$0.276783\pi$
$762$		0			0
$763$	1.46643e9	−	2.53992e9i	0.119516	−	0.207007i
$764$		0			0
$765$	−2.07238e9	+	3.58946e9i	−0.167361	+	0.289877i
$766$		0			0
$767$	−1.21675e10			−0.973681
$768$		0			0
$769$	1.41140e9	+	2.44462e9i	0.111920	+	0.193851i	0.916544	−	0.399933i	$-0.130967\pi$
$769$	1.41140e9	+	2.44462e9i	0.111920	+	0.193851i	−0.804624	+	0.593784i	$0.797633\pi$
$770$		0			0
$771$	8.71759e9			0.685024
$772$		0			0
$773$	−2.54404e9	−	4.40641e9i	−0.198105	−	0.343129i	0.749809	−	0.661655i	$-0.230146\pi$
$773$	−2.54404e9	−	4.40641e9i	−0.198105	−	0.343129i	−0.947914	+	0.318526i	$0.896812\pi$
$774$		0			0
$775$	−5.34852e8	−	9.26390e8i	−0.0412741	−	0.0714888i
$776$		0			0
$777$	−3.38087e9	+	5.85583e9i	−0.258556	+	0.447832i
$778$		0			0
$779$	−3.65883e9	−	2.91376e9i	−0.277307	−	0.220837i
$780$		0			0
$781$	3.02040e9	−	5.23149e9i	0.226875	−	0.392958i
$782$		0			0
$783$	2.51145e9	+	4.34996e9i	0.186964	+	0.323831i
$784$		0			0
$785$	−8.82368e7	−	1.52831e8i	−0.00651037	−	0.0112763i
$786$		0			0
$787$	1.12245e9			0.0820836			0.0410418	−	0.999157i	$-0.486932\pi$
$787$	1.12245e9			0.0820836			0.0410418	+	0.999157i	$0.486932\pi$
$788$		0			0
$789$	−9.52092e9	−	1.64907e10i	−0.690096	−	1.19528i
$790$		0			0
$791$	−7.16201e8			−0.0514537
$792$		0			0
$793$	2.01274e10	−	3.48617e10i	1.43328	−	2.48252i
$794$		0			0
$795$	2.39223e9	−	4.14347e9i	0.168857	−	0.292469i
$796$		0			0
$797$	8.63282e8			0.0604016			0.0302008	−	0.999544i	$-0.490385\pi$
$797$	8.63282e8			0.0604016			0.0302008	+	0.999544i	$0.490385\pi$
$798$		0			0
$799$	3.90040e8			0.0270518
$800$		0			0
$801$	−8.60977e9	+	1.49126e10i	−0.591940	+	1.02527i
$802$		0			0
$803$	−1.28216e9	+	2.22076e9i	−0.0873850	+	0.151355i
$804$		0			0
$805$	3.83039e9			0.258796
$806$		0			0
$807$	1.15071e10	+	1.99308e10i	0.770738	+	1.33496i
$808$		0			0
$809$	−1.47844e10			−0.981709			−0.490854	−	0.871242i	$-0.663315\pi$
$809$	−1.47844e10			−0.981709			−0.490854	+	0.871242i	$0.663315\pi$
$810$		0			0
$811$	1.35191e10	+	2.34158e10i	0.889969	+	1.54147i	0.839910	+	0.542726i	$0.182608\pi$
$811$	1.35191e10	+	2.34158e10i	0.889969	+	1.54147i	0.0500595	+	0.998746i	$0.484059\pi$
$812$		0			0
$813$	−3.78378e9	−	6.55370e9i	−0.246950	−	0.427730i
$814$		0			0
$815$	5.80416e8	−	1.00531e9i	0.0375568	−	0.0650502i
$816$		0			0
$817$	−1.21782e10	−	9.69826e9i	−0.781278	−	0.622181i
$818$		0			0
$819$	−5.77306e9	+	9.99923e9i	−0.367208	+	0.636023i
$820$		0			0
$821$	9.22142e9	+	1.59720e10i	0.581563	+	1.00730i	0.995294	+	0.0968980i	$0.0308921\pi$
$821$	9.22142e9	+	1.59720e10i	0.581563	+	1.00730i	−0.413731	+	0.910399i	$0.635775\pi$
$822$		0			0
$823$	1.26681e10	+	2.19418e10i	0.792158	+	1.37206i	0.924628	+	0.380871i	$0.124376\pi$
$823$	1.26681e10	+	2.19418e10i	0.792158	+	1.37206i	−0.132470	+	0.991187i	$0.542291\pi$
$824$		0			0
$825$	−2.51361e10			−1.55851
$826$		0			0
$827$	4.70372e9	+	8.14708e9i	0.289183	+	0.500879i	0.973615	−	0.228197i	$-0.0732832\pi$
$827$	4.70372e9	+	8.14708e9i	0.289183	+	0.500879i	−0.684432	+	0.729077i	$0.739950\pi$
$828$		0			0
$829$	−7.21771e9			−0.440006			−0.220003	−	0.975499i	$-0.570607\pi$
$829$	−7.21771e9			−0.440006			−0.220003	+	0.975499i	$0.570607\pi$
$830$		0			0
$831$	5.75583e9	−	9.96938e9i	0.347940	−	0.602649i
$832$		0			0
$833$	−5.14991e9	+	8.91991e9i	−0.308704	+	0.534691i
$834$		0			0
$835$	−9.26289e9			−0.550609
$836$		0			0
$837$	−5.02381e8			−0.0296138
$838$		0			0
$839$	−5.50244e9	+	9.53050e9i	−0.321653	+	0.557120i	−0.980829	−	0.194869i	$-0.937572\pi$
$839$	−5.50244e9	+	9.53050e9i	−0.321653	+	0.557120i	0.659176	+	0.751989i	$0.270905\pi$
$840$		0			0
$841$	−4.56462e9	+	7.90615e9i	−0.264617	+	0.458331i
$842$		0			0
$843$	1.99659e10			1.14787
$844$		0			0
$845$	7.31507e9	+	1.26701e10i	0.417081	+	0.722405i
$846$		0			0
$847$	3.37707e9			0.190962
$848$		0			0
$849$	−1.22100e10	−	2.11483e10i	−0.684759	−	1.18604i
$850$		0			0
$851$	1.70488e10	+	2.95294e10i	0.948290	+	1.64249i
$852$		0			0
$853$	−1.06199e9	+	1.83942e9i	−0.0585868	+	0.101475i	−0.893831	−	0.448404i	$-0.851993\pi$
$853$	−1.06199e9	+	1.83942e9i	−0.0585868	+	0.101475i	0.835244	+	0.549879i	$0.185326\pi$
$854$		0			0
$855$	−1.29379e9	+	8.62350e9i	−0.0707918	+	0.471848i
$856$		0			0
$857$	8.10438e9	−	1.40372e10i	0.439832	−	0.761812i	−0.557844	−	0.829946i	$-0.688371\pi$
$857$	8.10438e9	−	1.40372e10i	0.439832	−	0.761812i	0.997676	+	0.0681341i	$0.0217046\pi$
$858$		0			0
$859$	1.85012e9	+	3.20449e9i	0.0995916	+	0.172498i	0.911516	−	0.411265i	$-0.134913\pi$
$859$	1.85012e9	+	3.20449e9i	0.0995916	+	0.172498i	−0.811924	+	0.583763i	$0.801580\pi$
$860$		0			0
$861$	1.70650e9	+	2.95575e9i	0.0911162	+	0.157818i
$862$		0			0
$863$	2.20379e10			1.16716			0.583582	−	0.812054i	$-0.301651\pi$
$863$	2.20379e10			1.16716			0.583582	+	0.812054i	$0.301651\pi$
$864$		0			0
$865$	2.77575e9	+	4.80774e9i	0.145822	+	0.252572i
$866$		0			0
$867$	−1.44670e10			−0.753899
$868$		0			0
$869$	1.92471e10	−	3.33369e10i	0.994937	−	1.72328i
$870$		0			0
$871$	1.66238e10	−	2.87933e10i	0.852448	−	1.47648i
$872$		0			0
$873$	−5.08288e9			−0.258559
$874$		0			0
$875$	5.01282e9			0.252961
$876$		0			0
$877$	−1.67284e10	+	2.89745e10i	−0.837446	+	1.45050i	0.0545770	+	0.998510i	$0.482619\pi$
$877$	−1.67284e10	+	2.89745e10i	−0.837446	+	1.45050i	−0.892023	+	0.451990i	$0.850714\pi$
$878$		0			0
$879$	9.71732e9	−	1.68309e10i	0.482598	−	0.835884i
$880$		0			0
$881$	2.27326e10			1.12004			0.560020	−	0.828479i	$-0.310794\pi$
$881$	2.27326e10			1.12004			0.560020	+	0.828479i	$0.310794\pi$
$882$		0			0
$883$	−1.41748e10	−	2.45514e10i	−0.692873	−	1.20009i	−0.970892	−	0.239516i	$-0.923011\pi$
$883$	−1.41748e10	−	2.45514e10i	−0.692873	−	1.20009i	0.278020	−	0.960575i	$-0.410322\pi$
$884$		0			0
$885$	6.70540e9			0.325180
$886$		0			0
$887$	−4.78572e9	−	8.28912e9i	−0.230258	−	0.398819i	0.727626	−	0.685974i	$-0.240624\pi$
$887$	−4.78572e9	−	8.28912e9i	−0.230258	−	0.398819i	−0.957884	+	0.287155i	$0.907290\pi$
$888$		0			0
$889$	−1.36296e9	−	2.36071e9i	−0.0650618	−	0.112690i
$890$		0			0
$891$	9.92552e9	−	1.71915e10i	0.470090	−	0.814220i
$892$		0			0
$893$	7.63665e8	−	3.00315e8i	0.0358858	−	0.0141123i
$894$		0			0
$895$	7.98960e9	−	1.38384e10i	0.372516	−	0.645216i
$896$		0			0
$897$	5.32976e10	+	9.23142e10i	2.46567	+	4.27066i
$898$		0			0
$899$	−1.31920e9	−	2.28491e9i	−0.0605551	−	0.104884i
$900$		0			0
$901$	−8.83969e9			−0.402625
$902$		0			0
$903$	5.67999e9	+	9.83803e9i	0.256708	+	0.444632i
$904$		0			0
$905$	−1.22765e10			−0.550558
$906$		0			0
$907$	1.50792e10	−	2.61179e10i	0.671047	−	1.16229i	−0.306561	−	0.951851i	$-0.599178\pi$
$907$	1.50792e10	−	2.61179e10i	0.671047	−	1.16229i	0.977608	−	0.210436i	$-0.0674883\pi$
$908$		0			0
$909$	1.89478e10	−	3.28186e10i	0.836730	−	1.44926i
$910$		0			0
$911$	−4.05503e10			−1.77697			−0.888484	−	0.458908i	$-0.848241\pi$
$911$	−4.05503e10			−1.77697			−0.888484	+	0.458908i	$0.848241\pi$
$912$		0			0
$913$	1.06328e10			0.462381
$914$		0			0
$915$	−1.10921e10	+	1.92120e10i	−0.478673	+	0.829086i
$916$		0			0
$917$	1.34012e9	−	2.32115e9i	0.0573919	−	0.0994057i
$918$		0			0
$919$	−1.50007e10			−0.637539			−0.318769	−	0.947832i	$-0.603270\pi$
$919$	−1.50007e10			−0.637539			−0.318769	+	0.947832i	$0.603270\pi$
$920$		0			0
$921$	−3.12014e10	−	5.40424e10i	−1.31603	−	2.27943i
$922$		0			0
$923$	−1.53335e10			−0.641853
$924$		0			0
$925$	1.02047e10	+	1.76751e10i	0.423941	+	0.734288i
$926$		0			0
$927$	8.91783e9	+	1.54461e10i	0.367689	+	0.636856i
$928$		0			0
$929$	1.37382e9	−	2.37953e9i	0.0562181	−	0.0973725i	−0.836547	−	0.547896i	$-0.815429\pi$
$929$	1.37382e9	−	2.37953e9i	0.0562181	−	0.0973725i	0.892765	+	0.450523i	$0.148762\pi$
$930$		0			0
$931$	−3.21511e9	+	2.14296e10i	−0.130579	+	0.870344i
$932$		0			0
$933$	9.75225e8	−	1.68914e9i	0.0393115	−	0.0680895i
$934$		0			0
$935$	−4.32861e9	−	7.49737e9i	−0.173184	−	0.299964i
$936$		0			0
$937$	1.46295e10	+	2.53390e10i	0.580952	+	1.00624i	0.995367	+	0.0961505i	$0.0306530\pi$
$937$	1.46295e10	+	2.53390e10i	0.580952	+	1.00624i	−0.414415	+	0.910088i	$0.636014\pi$
$938$		0			0
$939$	3.99107e10			1.57311
$940$		0			0
$941$	−7.86560e8	−	1.36236e9i	−0.0307729	−	0.0533002i	0.850229	−	0.526413i	$-0.176464\pi$
$941$	−7.86560e8	−	1.36236e9i	−0.0307729	−	0.0533002i	−0.881002	+	0.473113i	$0.843130\pi$
$942$		0			0
$943$	1.72109e10			0.668363
$944$		0			0
$945$	5.38381e8	−	9.32504e8i	0.0207529	−	0.0359451i
$946$		0			0
$947$	−1.49985e10	+	2.59782e10i	−0.573883	+	0.993994i	0.422279	+	0.906466i	$0.361230\pi$
$947$	−1.49985e10	+	2.59782e10i	−0.573883	+	0.993994i	−0.996162	+	0.0875285i	$0.972103\pi$
$948$		0			0
$949$	6.50906e9			0.247222
$950$		0			0
$951$	3.04094e10			1.14651
$952$		0			0
$953$	−9.54278e9	+	1.65286e10i	−0.357149	+	0.618600i	−0.987483	−	0.157723i	$-0.949585\pi$
$953$	−9.54278e9	+	1.65286e10i	−0.357149	+	0.618600i	0.630334	+	0.776324i	$0.282918\pi$
$954$		0			0
$955$	6.37106e9	−	1.10350e10i	0.236701	−	0.409978i
$956$		0			0
$957$	−6.19975e10			−2.28656
$958$		0			0
$959$	4.90854e9	+	8.50184e9i	0.179716	+	0.311278i
$960$		0			0
$961$	−2.72487e10			−0.990408
$962$		0			0
$963$	−1.79906e10	−	3.11607e10i	−0.649164	−	1.12439i
$964$		0			0
$965$	1.04918e10	+	1.81724e10i	0.375842	+	0.650978i
$966$		0			0
$967$	1.09576e10	−	1.89790e10i	0.389691	−	0.674965i	−0.602717	−	0.797955i	$-0.705915\pi$
$967$	1.09576e10	−	1.89790e10i	0.389691	−	0.674965i	0.992408	+	0.122990i	$0.0392483\pi$
$968$		0			0
$969$	2.74493e10	−	1.07946e10i	0.969166	−	0.381130i
$970$		0			0
$971$	−1.86887e10	+	3.23697e10i	−0.655105	+	1.13467i	0.326763	+	0.945106i	$0.394042\pi$
$971$	−1.86887e10	+	3.23697e10i	−0.655105	+	1.13467i	−0.981868	+	0.189568i	$0.939291\pi$
$972$		0			0
$973$	7.69787e9	+	1.33331e10i	0.267902	+	0.464019i
$974$		0			0
$975$	3.19018e10	+	5.52555e10i	1.10230	+	1.90924i
$976$		0			0
$977$	−1.75318e10			−0.601443			−0.300721	−	0.953712i	$-0.597227\pi$
$977$	−1.75318e10			−0.601443			−0.300721	+	0.953712i	$0.597227\pi$
$978$		0			0
$979$	−1.79834e10	−	3.11481e10i	−0.612536	−	1.06094i
$980$		0			0
$981$	−2.45683e10			−0.830869
$982$		0			0
$983$	−3.71861e9	+	6.44082e9i	−0.124866	+	0.216274i	−0.921680	−	0.387950i	$-0.873183\pi$
$983$	−3.71861e9	+	6.44082e9i	−0.124866	+	0.216274i	0.796815	+	0.604224i	$0.206517\pi$
$984$		0			0
$985$	−1.07239e10	+	1.85744e10i	−0.357541	+	0.619280i
$986$		0			0
$987$	−5.98785e8			−0.0198226
$988$		0			0
$989$	5.72854e10			1.88303
$990$		0			0
$991$	−1.30467e10	+	2.25975e10i	−0.425836	+	0.737569i	−0.996498	−	0.0836156i	$-0.973353\pi$
$991$	−1.30467e10	+	2.25975e10i	−0.425836	+	0.737569i	0.570662	+	0.821185i	$0.306687\pi$
$992$		0			0
$993$	1.52598e9	−	2.64308e9i	0.0494569	−	0.0856618i
$994$		0			0
$995$	−1.64456e10			−0.529261
$996$		0			0
$997$	−5.14753e9	−	8.91578e9i	−0.164500	−	0.284922i	0.771978	−	0.635650i	$-0.219268\pi$
$997$	−5.14753e9	−	8.91578e9i	−0.164500	−	0.284922i	−0.936478	+	0.350727i	$0.885934\pi$
$998$		0			0
$999$	9.58522e9			0.304175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.e.a.49.2	yes	22
19.7	even	3	inner	76.8.e.a.45.2	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.e.a.45.2	✓	22	19.7	even	3	inner
76.8.e.a.49.2	yes	22	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-34.7112 + 60.1216i) q^{3} +(55.3969 - 95.9502i) q^{5} +314.254 q^{7} +(-1316.24 - 2279.79i) q^{9} +O(q^{10})\)	\(q+(-34.7112 + 60.1216i) q^{3} +(55.3969 - 95.9502i) q^{5} +314.254 q^{7} +(-1316.24 - 2279.79i) q^{9} +5498.50 q^{11} +(-6978.48 - 12087.1i) q^{13} +(3845.79 + 6661.10i) q^{15} +(7105.41 - 12306.9i) q^{17} +(4435.93 - 29566.8i) q^{19} +(-10908.1 + 18893.4i) q^{21} +(55006.9 + 95274.7i) q^{23} +(32924.9 + 57027.5i) q^{25} +30926.0 q^{27} +(81208.2 + 140657. i) q^{29} -16244.6 q^{31} +(-190860. + 330579. i) q^{33} +(17408.7 - 30152.7i) q^{35} +309940. q^{37} +968927. q^{39} +(78221.6 - 135484. i) q^{41} +(260356. - 450949. i) q^{43} -291662. q^{45} +(13723.4 + 23769.6i) q^{47} -724788. q^{49} +(493275. + 854377. i) q^{51} +(-311020. - 538702. i) q^{53} +(304600. - 527582. i) q^{55} +(1.62363e6 + 1.29299e6i) q^{57} +(435893. - 754988. i) q^{59} +(1.44211e6 + 2.49780e6i) q^{61} +(-413633. - 716433. i) q^{63} -1.54634e6 q^{65} +(1.19108e6 + 2.06301e6i) q^{67} -7.63742e6 q^{69} +(549314. - 951439. i) q^{71} +(-233183. + 403885. i) q^{73} -4.57145e6 q^{75} +1.72792e6 q^{77} +(3.50042e6 - 6.06291e6i) q^{79} +(1.80513e6 - 3.12658e6i) q^{81} +1.93376e6 q^{83} +(-787235. - 1.36353e6i) q^{85} -1.12753e7 q^{87} +(-3.27060e6 - 5.66484e6i) q^{89} +(-2.19301e6 - 3.79841e6i) q^{91} +(563870. - 976652. i) q^{93} +(-2.59120e6 - 2.06354e6i) q^{95} +(965417. - 1.67215e6i) q^{97} +(-7.23733e6 - 1.25354e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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