LMFDB - Embedded newform 76.8.e.a.49.1

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			49.1
Character	$\chi$	$=$	76.49
Dual form			76.8.e.a.45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.1712 + 69.5786i) q^{3} +(-97.7376 + 169.287i) q^{5} -587.742 q^{7} +(-2133.95 - 3696.12i) q^{9} +O(q^{10})$	\(q+(-40.1712 + 69.5786i) q^{3} +(-97.7376 + 169.287i) q^{5} -587.742 q^{7} +(-2133.95 - 3696.12i) q^{9} -6546.65 q^{11} +(2812.44 + 4871.29i) q^{13} +(-7852.48 - 13600.9i) q^{15} +(-11254.1 + 19492.6i) q^{17} +(29897.2 + 169.581i) q^{19} +(23610.3 - 40894.2i) q^{21} +(-42909.8 - 74322.0i) q^{23} +(19957.2 + 34566.9i) q^{25} +167185. q^{27} +(74347.9 + 128774. i) q^{29} +178184. q^{31} +(262987. - 455507. i) q^{33} +(57444.5 - 99496.8i) q^{35} -111603. q^{37} -451916. q^{39} +(112180. - 194301. i) q^{41} +(78377.6 - 135754. i) q^{43} +834271. q^{45} +(-335551. - 581192. i) q^{47} -478103. q^{49} +(-904180. - 1.56609e6i) q^{51} +(718815. + 1.24502e6i) q^{53} +(639854. - 1.10826e6i) q^{55} +(-1.21281e6 + 2.07339e6i) q^{57} +(1.19760e6 - 2.07431e6i) q^{59} +(-11810.8 - 20456.8i) q^{61} +(1.25421e6 + 2.17236e6i) q^{63} -1.09952e6 q^{65} +(212375. + 367843. i) q^{67} +6.89496e6 q^{69} +(635552. - 1.10081e6i) q^{71} +(-3.00652e6 + 5.20745e6i) q^{73} -3.20682e6 q^{75} +3.84774e6 q^{77} +(-1.79747e6 + 3.11331e6i) q^{79} +(-2.04908e6 + 3.54911e6i) q^{81} -8.04455e6 q^{83} +(-2.19989e6 - 3.81033e6i) q^{85} -1.19466e7 q^{87} +(-4.96526e6 - 8.60008e6i) q^{89} +(-1.65299e6 - 2.86306e6i) q^{91} +(-7.15788e6 + 1.23978e7i) q^{93} +(-2.95079e6 + 5.04462e6i) q^{95} +(5.27599e6 - 9.13828e6i) q^{97} +(1.39702e7 + 2.41972e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−40.1712	+	69.5786i	−0.858995	+	1.48782i	0.0138935	+	0.999903i	$0.495577\pi$
$3$	−40.1712	+	69.5786i	−0.858995	+	1.48782i	−0.872889	+	0.487920i	$0.837756\pi$
$4$		0			0
$5$	−97.7376	+	169.287i	−0.349677	+	0.605658i	−0.986192	−	0.165606i	$-0.947042\pi$
$5$	−97.7376	+	169.287i	−0.349677	+	0.605658i	0.636515	+	0.771264i	$0.280375\pi$
$6$		0			0
$7$	−587.742			−0.647654			−0.323827	−	0.946116i	$-0.604970\pi$
$7$	−587.742			−0.647654			−0.323827	+	0.946116i	$0.604970\pi$
$8$		0			0
$9$	−2133.95	−	3696.12i	−0.975745	−	1.69004i
$10$		0			0
$11$	−6546.65			−1.48301			−0.741506	−	0.670946i	$-0.765888\pi$
$11$	−6546.65			−1.48301			−0.741506	+	0.670946i	$0.765888\pi$
$12$		0			0
$13$	2812.44	+	4871.29i	0.355043	+	0.614953i	0.987125	−	0.159948i	$-0.0511328\pi$
$13$	2812.44	+	4871.29i	0.355043	+	0.614953i	−0.632082	+	0.774901i	$0.717799\pi$
$14$		0			0
$15$	−7852.48	−	13600.9i	−0.600741	−	1.04051i
$16$		0			0
$17$	−11254.1	+	19492.6i	−0.555570	+	0.962275i	0.442289	+	0.896873i	$0.354167\pi$
$17$	−11254.1	+	19492.6i	−0.555570	+	0.962275i	−0.997859	+	0.0654028i	$0.979167\pi$
$18$		0			0
$19$	29897.2	+	169.581i	0.999984	+	0.00567205i
$19$	29897.2	+	169.581i	0.999984	+	0.00567205i
$20$		0			0
$21$	23610.3	−	40894.2i	0.556332	−	0.963595i
$22$		0			0
$23$	−42909.8	−	74322.0i	−0.735376	−	1.27371i	−0.954558	−	0.298025i	$-0.903672\pi$
$23$	−42909.8	−	74322.0i	−0.735376	−	1.27371i	0.219182	−	0.975684i	$-0.429661\pi$
$24$		0			0
$25$	19957.2	+	34566.9i	0.255452	+	0.442456i
$26$		0			0
$27$	167185.			1.63465
$28$		0			0
$29$	74347.9	+	128774.i	0.566077	+	0.980475i	0.996949	+	0.0780613i	$0.0248730\pi$
$29$	74347.9	+	128774.i	0.566077	+	0.980475i	−0.430871	+	0.902413i	$0.641794\pi$
$30$		0			0
$31$	178184.			1.07424			0.537122	−	0.843504i	$-0.319511\pi$
$31$	178184.			1.07424			0.537122	+	0.843504i	$0.319511\pi$
$32$		0			0
$33$	262987.	−	455507.i	1.27390	−	2.20646i
$34$		0			0
$35$	57444.5	−	99496.8i	0.226470	−	0.392257i
$36$		0			0
$37$	−111603.			−0.362217			−0.181108	−	0.983463i	$-0.557968\pi$
$37$	−111603.			−0.362217			−0.181108	+	0.983463i	$0.557968\pi$
$38$		0			0
$39$	−451916.			−1.21992
$40$		0			0
$41$	112180.	−	194301.i	0.254198	−	0.440284i	−0.710479	−	0.703718i	$-0.751522\pi$
$41$	112180.	−	194301.i	0.254198	−	0.440284i	0.964677	+	0.263434i	$0.0848552\pi$
$42$		0			0
$43$	78377.6	−	135754.i	0.150332	−	0.260383i	−0.781017	−	0.624509i	$-0.785299\pi$
$43$	78377.6	−	135754.i	0.150332	−	0.260383i	0.931350	+	0.364126i	$0.118632\pi$
$44$		0			0
$45$	834271.			1.36478
$46$		0			0
$47$	−335551.	−	581192.i	−0.471429	−	0.816539i	0.528037	−	0.849222i	$-0.322928\pi$
$47$	−335551.	−	581192.i	−0.471429	−	0.816539i	−0.999466	+	0.0326824i	$0.989595\pi$
$48$		0			0
$49$	−478103.			−0.580544
$50$		0			0
$51$	−904180.	−	1.56609e6i	−0.954464	−	1.65318i
$52$		0			0
$53$	718815.	+	1.24502e6i	0.663211	+	1.14871i	0.979767	+	0.200141i	$0.0641400\pi$
$53$	718815.	+	1.24502e6i	0.663211	+	1.14871i	−0.316556	+	0.948574i	$0.602527\pi$
$54$		0			0
$55$	639854.	−	1.10826e6i	0.518575	−	0.898198i
$56$		0			0
$57$	−1.21281e6	+	2.07339e6i	−0.867420	+	1.48293i
$58$		0			0
$59$	1.19760e6	−	2.07431e6i	0.759156	−	1.31490i	−0.184125	−	0.982903i	$-0.558945\pi$
$59$	1.19760e6	−	2.07431e6i	0.759156	−	1.31490i	0.943281	−	0.331994i	$-0.107721\pi$
$60$		0			0
$61$	−11810.8	−	20456.8i	−0.00666228	−	0.0115394i	0.862675	−	0.505759i	$-0.168787\pi$
$61$	−11810.8	−	20456.8i	−0.00666228	−	0.0115394i	−0.869337	+	0.494219i	$0.835454\pi$
$62$		0			0
$63$	1.25421e6	+	2.17236e6i	0.631946	+	1.09456i
$64$		0			0
$65$	−1.09952e6			−0.496602
$66$		0			0
$67$	212375.	+	367843.i	0.0862662	+	0.149417i	0.905930	−	0.423427i	$-0.139173\pi$
$67$	212375.	+	367843.i	0.0862662	+	0.149417i	−0.819664	+	0.572845i	$0.805840\pi$
$68$		0			0
$69$	6.89496e6			2.52674
$70$		0			0
$71$	635552.	−	1.10081e6i	0.210740	−	0.365012i	−0.741206	−	0.671277i	$-0.765746\pi$
$71$	635552.	−	1.10081e6i	0.210740	−	0.365012i	0.951946	+	0.306265i	$0.0990794\pi$
$72$		0			0
$73$	−3.00652e6	+	5.20745e6i	−0.904553	+	1.56673i	−0.0830371	+	0.996546i	$0.526462\pi$
$73$	−3.00652e6	+	5.20745e6i	−0.904553	+	1.56673i	−0.821516	+	0.570185i	$0.806871\pi$
$74$		0			0
$75$	−3.20682e6			−0.877729
$76$		0			0
$77$	3.84774e6			0.960479
$78$		0			0
$79$	−1.79747e6	+	3.11331e6i	−0.410173	+	0.710441i	−0.994908	−	0.100783i	$-0.967865\pi$
$79$	−1.79747e6	+	3.11331e6i	−0.410173	+	0.710441i	0.584735	+	0.811224i	$0.301199\pi$
$80$		0			0
$81$	−2.04908e6	+	3.54911e6i	−0.428412	+	0.742031i
$82$		0			0
$83$	−8.04455e6			−1.54429			−0.772144	−	0.635447i	$-0.780816\pi$
$83$	−8.04455e6			−1.54429			−0.772144	+	0.635447i	$0.780816\pi$
$84$		0			0
$85$	−2.19989e6	−	3.81033e6i	−0.388540	−	0.672971i
$86$		0			0
$87$	−1.19466e7			−1.94503
$88$		0			0
$89$	−4.96526e6	−	8.60008e6i	−0.746581	−	1.29312i	−0.949452	−	0.313911i	$-0.898361\pi$
$89$	−4.96526e6	−	8.60008e6i	−0.746581	−	1.29312i	0.202871	−	0.979205i	$-0.434973\pi$
$90$		0			0
$91$	−1.65299e6	−	2.86306e6i	−0.229945	−	0.398277i
$92$		0			0
$93$	−7.15788e6	+	1.23978e7i	−0.922771	+	1.59829i
$94$		0			0
$95$	−2.95079e6	+	5.04462e6i	−0.353106	+	0.603665i
$96$		0			0
$97$	5.27599e6	−	9.13828e6i	0.586952	−	1.01663i	−0.407677	−	0.913126i	$-0.633661\pi$
$97$	5.27599e6	−	9.13828e6i	0.586952	−	1.01663i	0.994629	−	0.103505i	$-0.0330057\pi$
$98$		0			0
$99$	1.39702e7	+	2.41972e7i	1.44704	+	2.50635i
$100$		0			0
$101$	−1.26434e6	−	2.18991e6i	−0.122107	−	0.211496i	0.798491	−	0.602006i	$-0.205632\pi$
$101$	−1.26434e6	−	2.18991e6i	−0.122107	−	0.211496i	−0.920598	+	0.390511i	$0.872298\pi$
$102$		0			0
$103$	−5.50819e6			−0.496683			−0.248341	−	0.968673i	$-0.579885\pi$
$103$	−5.50819e6			−0.496683			−0.248341	+	0.968673i	$0.579885\pi$
$104$		0			0
$105$	4.61523e6	+	7.99381e6i	0.389073	+	0.673894i
$106$		0			0
$107$	−6.33943e6			−0.500273			−0.250137	−	0.968211i	$-0.580476\pi$
$107$	−6.33943e6			−0.500273			−0.250137	+	0.968211i	$0.580476\pi$
$108$		0			0
$109$	8.15947e6	−	1.41326e7i	0.603489	−	1.04527i	−0.388800	−	0.921322i	$-0.627110\pi$
$109$	8.15947e6	−	1.41326e7i	0.603489	−	1.04527i	0.992288	−	0.123951i	$-0.0395565\pi$
$110$		0			0
$111$	4.48322e6	−	7.76516e6i	0.311143	−	0.538915i
$112$		0			0
$113$	2.70526e6			0.176374			0.0881869	−	0.996104i	$-0.471893\pi$
$113$	2.70526e6			0.176374			0.0881869	+	0.996104i	$0.471893\pi$
$114$		0			0
$115$	1.67756e7			1.02858
$116$		0			0
$117$	1.20032e7	−	2.07902e7i	0.692863	−	1.20007i
$118$		0			0
$119$	6.61449e6	−	1.14566e7i	0.359817	−	0.623222i
$120$		0			0
$121$	2.33714e7			1.19932
$122$		0			0
$123$	9.01282e6	+	1.56107e7i	0.436709	+	0.756403i
$124$		0			0
$125$	−2.30738e7			−1.05666
$126$		0			0
$127$	7.57057e6	+	1.31126e7i	0.327956	+	0.568037i	0.982106	−	0.188328i	$-0.0603069\pi$
$127$	7.57057e6	+	1.31126e7i	0.327956	+	0.568037i	−0.654150	+	0.756365i	$0.726974\pi$
$128$		0			0
$129$	6.29705e6	+	1.09068e7i	0.258270	+	0.447336i
$130$		0			0
$131$	1.00582e7	−	1.74214e7i	0.390906	−	0.677069i	−0.601663	−	0.798750i	$-0.705495\pi$
$131$	1.00582e7	−	1.74214e7i	0.390906	−	0.677069i	0.992569	+	0.121681i	$0.0388284\pi$
$132$		0			0
$133$	−1.75718e7	−	99669.9i	−0.647644	−	0.00367353i
$134$		0			0
$135$	−1.63403e7	+	2.83022e7i	−0.571599	+	0.990039i
$136$		0			0
$137$	−1.30548e7	−	2.26115e7i	−0.433757	−	0.751290i	0.563436	−	0.826160i	$-0.309479\pi$
$137$	−1.30548e7	−	2.26115e7i	−0.433757	−	0.751290i	−0.997193	+	0.0748701i	$0.976146\pi$
$138$		0			0
$139$	−2.76210e7	−	4.78410e7i	−0.872345	−	1.51095i	−0.859564	−	0.511027i	$-0.829265\pi$
$139$	−2.76210e7	−	4.78410e7i	−0.872345	−	1.51095i	−0.0127804	−	0.999918i	$-0.504068\pi$
$140$		0			0
$141$	5.39180e7			1.61982
$142$		0			0
$143$	−1.84120e7	−	3.18906e7i	−0.526533	−	0.911983i
$144$		0			0
$145$	−2.90664e7			−0.791776
$146$		0			0
$147$	1.92060e7	−	3.32657e7i	0.498684	−	0.863746i
$148$		0			0
$149$	2.69765e7	−	4.67246e7i	0.668087	−	1.15716i	−0.310351	−	0.950622i	$-0.600447\pi$
$149$	2.69765e7	−	4.67246e7i	0.668087	−	1.15716i	0.978438	−	0.206539i	$-0.0662200\pi$
$150$		0			0
$151$	−5.19945e7			−1.22896			−0.614480	−	0.788932i	$-0.710634\pi$
$151$	−5.19945e7			−1.22896			−0.614480	+	0.788932i	$0.710634\pi$
$152$		0			0
$153$	9.60628e7			2.16838
$154$		0			0
$155$	−1.74153e7	+	3.01642e7i	−0.375638	+	0.650625i
$156$		0			0
$157$	1.56977e7	−	2.71892e7i	0.323732	−	0.560721i	−0.657523	−	0.753435i	$-0.728396\pi$
$157$	1.56977e7	−	2.71892e7i	0.323732	−	0.560721i	0.981255	+	0.192714i	$0.0617289\pi$
$158$		0			0
$159$	−1.15503e8			−2.27878
$160$		0			0
$161$	2.52199e7	+	4.36821e7i	0.476270	+	0.824923i
$162$		0			0
$163$	1.22478e7			0.221515			0.110757	−	0.993847i	$-0.464672\pi$
$163$	1.22478e7			0.221515			0.110757	+	0.993847i	$0.464672\pi$
$164$		0			0
$165$	5.14074e7	+	8.90403e7i	0.890906	+	1.54310i
$166$		0			0
$167$	2.77101e7	+	4.79952e7i	0.460394	+	0.797426i	0.998980	−	0.0451441i	$-0.0143747\pi$
$167$	2.77101e7	+	4.79952e7i	0.460394	+	0.797426i	−0.538586	+	0.842570i	$0.681041\pi$
$168$		0			0
$169$	1.55546e7	−	2.69414e7i	0.247889	−	0.429356i
$170$		0			0
$171$	−6.31725e7	−	1.10865e8i	−0.966143	−	1.69555i
$172$		0			0
$173$	−3.58681e7	+	6.21254e7i	−0.526680	+	0.912237i	0.472837	+	0.881150i	$0.343230\pi$
$173$	−3.58681e7	+	6.21254e7i	−0.526680	+	0.912237i	−0.999517	+	0.0310865i	$0.990103\pi$
$174$		0			0
$175$	−1.17297e7	−	2.03164e7i	−0.165445	−	0.286559i
$176$		0			0
$177$	9.62184e7	+	1.66655e8i	1.30422	+	2.25898i
$178$		0			0
$179$	4.30973e7			0.561648			0.280824	−	0.959759i	$-0.409392\pi$
$179$	4.30973e7			0.561648			0.280824	+	0.959759i	$0.409392\pi$
$180$		0			0
$181$	−1.27395e7	−	2.20655e7i	−0.159690	−	0.276591i	0.775067	−	0.631879i	$-0.217716\pi$
$181$	−1.27395e7	−	2.20655e7i	−0.159690	−	0.276591i	−0.934757	+	0.355288i	$0.884383\pi$
$182$		0			0
$183$	1.89781e6			0.0228915
$184$		0			0
$185$	1.09078e7	−	1.88928e7i	0.126659	−	0.219380i
$186$		0			0
$187$	7.36765e7	−	1.27611e8i	0.823917	−	1.42707i
$188$		0			0
$189$	−9.82618e7			−1.05869
$190$		0			0
$191$	−1.32571e8			−1.37667			−0.688335	−	0.725393i	$-0.741658\pi$
$191$	−1.32571e8			−1.37667			−0.688335	+	0.725393i	$0.741658\pi$
$192$		0			0
$193$	−6.06106e7	+	1.04981e8i	−0.606873	+	1.05113i	0.384879	+	0.922967i	$0.374243\pi$
$193$	−6.06106e7	+	1.04981e8i	−0.606873	+	1.05113i	−0.991752	+	0.128168i	$0.959090\pi$
$194$		0			0
$195$	4.41692e7	−	7.65034e7i	0.426578	−	0.738855i
$196$		0			0
$197$	7.77611e7			0.724654			0.362327	−	0.932051i	$-0.381982\pi$
$197$	7.77611e7			0.724654			0.362327	+	0.932051i	$0.381982\pi$
$198$		0			0
$199$	1.05896e8	+	1.83416e8i	0.952559	+	1.64988i	0.739858	+	0.672763i	$0.234893\pi$
$199$	1.05896e8	+	1.83416e8i	0.952559	+	1.64988i	0.212701	+	0.977117i	$0.431774\pi$
$200$		0			0
$201$	−3.41254e7			−0.296409
$202$		0			0
$203$	−4.36974e7	−	7.56861e7i	−0.366623	−	0.635009i
$204$		0			0
$205$	2.19284e7	+	3.79811e7i	0.177774	+	0.307914i
$206$		0			0
$207$	−1.83135e8	+	3.17200e8i	−1.43508	+	2.48563i
$208$		0			0
$209$	−1.95727e8	−	1.11019e6i	−1.48299	−	0.00841172i
$210$		0			0
$211$	−9.04501e7	+	1.56664e8i	−0.662858	+	1.14810i	0.317003	+	0.948424i	$0.397323\pi$
$211$	−9.04501e7	+	1.56664e8i	−0.662858	+	1.14810i	−0.979861	+	0.199679i	$0.936010\pi$
$212$		0			0
$213$	5.10618e7	+	8.84416e7i	0.362049	+	0.627087i
$214$		0			0
$215$	1.53209e7	+	2.65366e7i	0.105135	+	0.182100i
$216$		0			0
$217$	−1.04726e8			−0.695740
$218$		0			0
$219$	−2.41551e8	−	4.18379e8i	−1.55401	−	2.69163i
$220$		0			0
$221$	−1.26606e8			−0.789006
$222$		0			0
$223$	1.20392e8	−	2.08524e8i	0.726991	−	1.25919i	−0.231158	−	0.972916i	$-0.574251\pi$
$223$	1.20392e8	−	2.08524e8i	0.726991	−	1.25919i	0.958149	−	0.286270i	$-0.0924153\pi$
$224$		0			0
$225$	8.51756e7	−	1.47528e8i	0.498513	−	0.863449i
$226$		0			0
$227$	2.55493e7			0.144974			0.0724868	−	0.997369i	$-0.476906\pi$
$227$	2.55493e7			0.144974			0.0724868	+	0.997369i	$0.476906\pi$
$228$		0			0
$229$	2.37655e8			1.30774			0.653871	−	0.756606i	$-0.273144\pi$
$229$	2.37655e8			1.30774			0.653871	+	0.756606i	$0.273144\pi$
$230$		0			0
$231$	−1.54568e8	+	2.67720e8i	−0.825047	+	1.42902i
$232$		0			0
$233$	3.33490e7	−	5.77621e7i	0.172718	−	0.299156i	−0.766651	−	0.642064i	$-0.778079\pi$
$233$	3.33490e7	−	5.77621e7i	0.172718	−	0.299156i	0.939369	+	0.342908i	$0.111412\pi$
$234$		0			0
$235$	1.31184e8			0.659391
$236$		0			0
$237$	−1.44413e8	−	2.50131e8i	−0.704674	−	1.22053i
$238$		0			0
$239$	−3.05408e8			−1.44707			−0.723533	−	0.690290i	$-0.757483\pi$
$239$	−3.05408e8			−1.44707			−0.723533	+	0.690290i	$0.757483\pi$
$240$		0			0
$241$	1.70135e8	+	2.94682e8i	0.782949	+	1.35611i	0.930217	+	0.367011i	$0.119619\pi$
$241$	1.70135e8	+	2.94682e8i	0.782949	+	1.35611i	−0.147268	+	0.989097i	$0.547048\pi$
$242$		0			0
$243$	1.81891e7	+	3.15044e7i	0.0813184	+	0.140848i
$244$		0			0
$245$	4.67286e7	−	8.09363e7i	0.203003	−	0.351611i
$246$		0			0
$247$	8.32580e7	+	1.46115e8i	0.351550	+	0.616957i
$248$		0			0
$249$	3.23159e8	−	5.59729e8i	1.32654	−	2.29763i
$250$		0			0
$251$	2.17250e8	+	3.76288e8i	0.867165	+	1.50197i	0.864882	+	0.501976i	$0.167393\pi$
$251$	2.17250e8	+	3.76288e8i	0.867165	+	1.50197i	0.00228301	+	0.999997i	$0.499273\pi$
$252$		0			0
$253$	2.80916e8	+	4.86560e8i	1.09057	+	1.88892i
$254$		0			0
$255$	3.53490e8			1.33502
$256$		0			0
$257$	−1.83911e8	−	3.18543e8i	−0.675837	−	1.17058i	−0.976224	−	0.216766i	$-0.930449\pi$
$257$	−1.83911e8	−	3.18543e8i	−0.675837	−	1.17058i	0.300387	−	0.953817i	$-0.402884\pi$
$258$		0			0
$259$	6.55936e7			0.234591
$260$		0			0
$261$	3.17310e8	−	5.49597e8i	1.10469	−	1.91339i
$262$		0			0
$263$	−6.01410e7	+	1.04167e8i	−0.203857	+	0.353091i	−0.949768	−	0.312955i	$-0.898681\pi$
$263$	−6.01410e7	+	1.04167e8i	−0.203857	+	0.353091i	0.745911	+	0.666046i	$0.232014\pi$
$264$		0			0
$265$	−2.81021e8			−0.927638
$266$		0			0
$267$	7.97842e8			2.56524
$268$		0			0
$269$	−1.10308e8	+	1.91059e8i	−0.345520	+	0.598459i	−0.985448	−	0.169976i	$-0.945631\pi$
$269$	−1.10308e8	+	1.91059e8i	−0.345520	+	0.598459i	0.639928	+	0.768435i	$0.278964\pi$
$270$		0			0
$271$	1.53166e8	−	2.65292e8i	0.467488	−	0.809713i	−0.531822	−	0.846856i	$-0.678492\pi$
$271$	1.53166e8	−	2.65292e8i	0.467488	−	0.809713i	0.999310	+	0.0371432i	$0.0118258\pi$
$272$		0			0
$273$	2.65610e8			0.790088
$274$		0			0
$275$	−1.30653e8	−	2.26297e8i	−0.378839	−	0.656168i
$276$		0			0
$277$	−5.45194e8			−1.54125			−0.770623	−	0.637291i	$-0.780055\pi$
$277$	−5.45194e8			−1.54125			−0.770623	+	0.637291i	$0.780055\pi$
$278$		0			0
$279$	−3.80237e8	−	6.58590e8i	−1.04819	−	1.81552i
$280$		0			0
$281$	1.54420e8	+	2.67462e8i	0.415174	+	0.719102i	0.995447	−	0.0953198i	$-0.0303874\pi$
$281$	1.54420e8	+	2.67462e8i	0.415174	+	0.719102i	−0.580273	+	0.814422i	$0.697054\pi$
$282$		0			0
$283$	1.38464e8	−	2.39827e8i	0.363149	−	0.628993i	−0.625328	−	0.780362i	$-0.715035\pi$
$283$	1.38464e8	−	2.39827e8i	0.363149	−	0.628993i	0.988477	+	0.151369i	$0.0483682\pi$
$284$		0			0
$285$	−2.32461e8	−	4.07960e8i	−0.594830	−	1.04391i
$286$		0			0
$287$	−6.59329e7	+	1.14199e8i	−0.164632	+	0.285152i
$288$		0			0
$289$	−4.81393e7	−	8.33797e7i	−0.117316	−	0.203197i
$290$		0			0
$291$	4.23886e8	+	7.34192e8i	1.00838	+	1.74656i
$292$		0			0
$293$	−5.45142e8			−1.26611			−0.633057	−	0.774105i	$-0.718200\pi$
$293$	−5.45142e8			−1.26611			−0.633057	+	0.774105i	$0.718200\pi$
$294$		0			0
$295$	2.34102e8	+	4.05476e8i	0.530919	+	0.919578i
$296$		0			0
$297$	−1.09450e9			−2.42421
$298$		0			0
$299$	2.41363e8	−	4.18052e8i	0.522181	−	0.904443i
$300$		0			0
$301$	−4.60658e7	+	7.97883e7i	−0.0973634	+	0.168638i
$302$		0			0
$303$	2.03161e8			0.419557
$304$		0			0
$305$	4.61742e6			0.00931858
$306$		0			0
$307$	9.36949e7	−	1.62284e8i	0.184813	−	0.320105i	−0.758701	−	0.651439i	$-0.774166\pi$
$307$	9.36949e7	−	1.62284e8i	0.184813	−	0.320105i	0.943513	+	0.331334i	$0.107499\pi$
$308$		0			0
$309$	2.21271e8	−	3.83252e8i	0.426648	−	0.738976i
$310$		0			0
$311$	−7.41614e8			−1.39803			−0.699015	−	0.715107i	$-0.746378\pi$
$311$	−7.41614e8			−1.39803			−0.699015	+	0.715107i	$0.746378\pi$
$312$		0			0
$313$	4.02671e8	+	6.97446e8i	0.742241	+	1.28560i	0.951473	+	0.307733i	$0.0995704\pi$
$313$	4.02671e8	+	6.97446e8i	0.742241	+	1.28560i	−0.209232	+	0.977866i	$0.567096\pi$
$314$		0			0
$315$	−4.90336e8			−0.883907
$316$		0			0
$317$	−1.26492e8	−	2.19090e8i	−0.223026	−	0.386292i	0.732699	−	0.680552i	$-0.238260\pi$
$317$	−1.26492e8	−	2.19090e8i	−0.223026	−	0.386292i	−0.955725	+	0.294260i	$0.904927\pi$
$318$		0			0
$319$	−4.86730e8	−	8.43041e8i	−0.839499	−	1.45406i
$320$		0			0
$321$	2.54663e8	−	4.41089e8i	0.429732	−	0.744318i
$322$		0			0
$323$	−3.39771e8	+	5.80867e8i	−0.561019	+	0.959109i
$324$		0			0
$325$	−1.12257e8	+	1.94435e8i	−0.181393	+	0.314182i
$326$		0			0
$327$	6.55552e8	+	1.13545e9i	1.03679	+	1.79577i
$328$		0			0
$329$	1.97217e8	+	3.41591e8i	0.305323	+	0.528835i
$330$		0			0
$331$	6.99203e8			1.05975			0.529877	−	0.848075i	$-0.322238\pi$
$331$	6.99203e8			1.05975			0.529877	+	0.848075i	$0.322238\pi$
$332$		0			0
$333$	2.38155e8	+	4.12497e8i	0.353431	+	0.612161i
$334$		0			0
$335$	−8.30279e7			−0.120661
$336$		0			0
$337$	1.82957e8	−	3.16891e8i	0.260402	−	0.451030i	−0.705946	−	0.708265i	$-0.749478\pi$
$337$	1.82957e8	−	3.16891e8i	0.260402	−	0.451030i	0.966349	+	0.257235i	$0.0828114\pi$
$338$		0			0
$339$	−1.08674e8	+	1.88228e8i	−0.151504	+	0.262413i
$340$		0			0
$341$	−1.16651e9			−1.59312
$342$		0			0
$343$	7.65031e8			1.02365
$344$		0			0
$345$	−6.73897e8	+	1.16722e9i	−0.883541	+	1.53034i
$346$		0			0
$347$	3.85810e8	−	6.68243e8i	0.495702	−	0.858581i	−0.504286	−	0.863537i	$-0.668244\pi$
$347$	3.85810e8	−	6.68243e8i	0.495702	−	0.858581i	0.999988	+	0.00495575i	$0.00157747\pi$
$348$		0			0
$349$	5.86744e8			0.738855			0.369428	−	0.929260i	$-0.379554\pi$
$349$	5.86744e8			0.738855			0.369428	+	0.929260i	$0.379554\pi$
$350$		0			0
$351$	4.70198e8	+	8.14407e8i	0.580372	+	1.00523i
$352$		0			0
$353$	3.92757e8			0.475240			0.237620	−	0.971358i	$-0.423633\pi$
$353$	3.92757e8			0.475240			0.237620	+	0.971358i	$0.423633\pi$
$354$		0			0
$355$	1.24235e8	+	2.15181e8i	0.147382	+	0.255273i
$356$		0			0
$357$	5.31424e8	+	9.20454e8i	0.618163	+	1.07069i
$358$		0			0
$359$	−3.12113e6	+	5.40596e6i	−0.00356026	+	0.00616655i	−0.867800	−	0.496914i	$-0.834467\pi$
$359$	−3.12113e6	+	5.40596e6i	−0.00356026	+	0.00616655i	0.864240	+	0.503080i	$0.167800\pi$
$360$		0			0
$361$	8.93814e8	+	1.01400e7i	0.999936	+	0.0113439i
$362$		0			0
$363$	−9.38859e8	+	1.62615e9i	−1.03021	+	1.78438i
$364$		0			0
$365$	−5.87700e8	−	1.01793e9i	−0.632602	−	1.09570i
$366$		0			0
$367$	−6.09976e8	−	1.05651e9i	−0.644142	−	1.11569i	−0.984499	−	0.175390i	$-0.943881\pi$
$367$	−6.09976e8	−	1.05651e9i	−0.644142	−	1.11569i	0.340358	−	0.940296i	$-0.389452\pi$
$368$		0			0
$369$	−9.57548e8			−0.992129
$370$		0			0
$371$	−4.22478e8	−	7.31752e8i	−0.429531	−	0.743970i
$372$		0			0
$373$	−9.77023e8			−0.974819			−0.487409	−	0.873174i	$-0.662058\pi$
$373$	−9.77023e8			−0.974819			−0.487409	+	0.873174i	$0.662058\pi$
$374$		0			0
$375$	9.26902e8	−	1.60544e9i	0.907663	−	1.57212i
$376$		0			0
$377$	−4.18198e8	+	7.24340e8i	−0.401964	+	0.696222i
$378$		0			0
$379$	−1.84386e9			−1.73977			−0.869884	−	0.493256i	$-0.835807\pi$
$379$	−1.84386e9			−1.73977			−0.869884	+	0.493256i	$0.835807\pi$
$380$		0			0
$381$	−1.21648e9			−1.12685
$382$		0			0
$383$	−3.76105e8	+	6.51433e8i	−0.342069	+	0.592480i	−0.984817	−	0.173597i	$-0.944461\pi$
$383$	−3.76105e8	+	6.51433e8i	−0.342069	+	0.592480i	0.642748	+	0.766078i	$0.277794\pi$
$384$		0			0
$385$	−3.76069e8	+	6.51370e8i	−0.335857	+	0.581722i
$386$		0			0
$387$	−6.69017e8			−0.586744
$388$		0			0
$389$	−8.05480e8	−	1.39513e9i	−0.693795	−	1.20169i	−0.970585	−	0.240758i	$-0.922604\pi$
$389$	−8.05480e8	−	1.39513e9i	−0.693795	−	1.20169i	0.276790	−	0.960930i	$-0.410729\pi$
$390$		0			0
$391$	1.93164e9			1.63421
$392$		0			0
$393$	8.08104e8	+	1.39968e9i	0.671573	+	1.16320i
$394$		0			0
$395$	−3.51361e8	−	6.08576e8i	−0.286856	−	0.496850i
$396$		0			0
$397$	−4.98608e8	+	8.63614e8i	−0.399938	+	0.692713i	−0.993718	−	0.111915i	$-0.964302\pi$
$397$	−4.98608e8	+	8.63614e8i	−0.399938	+	0.692713i	0.593780	+	0.804627i	$0.297635\pi$
$398$		0			0
$399$	7.12817e8	−	1.21862e9i	0.561789	−	0.960424i
$400$		0			0
$401$	−5.66157e8	+	9.80612e8i	−0.438461	+	0.759437i	−0.997571	−	0.0696563i	$-0.977810\pi$
$401$	−5.66157e8	+	9.80612e8i	−0.438461	+	0.759437i	0.559110	+	0.829094i	$0.311143\pi$
$402$		0			0
$403$	5.01132e8	+	8.67986e8i	0.381403	+	0.660610i
$404$		0			0
$405$	−4.00544e8	−	6.93763e8i	−0.299611	−	0.518942i
$406$		0			0
$407$	7.30624e8			0.537172
$408$		0			0
$409$	−1.36333e9	−	2.36136e9i	−0.985303	−	1.70659i	−0.640582	−	0.767890i	$-0.721307\pi$
$409$	−1.36333e9	−	2.36136e9i	−0.985303	−	1.70659i	−0.344721	−	0.938705i	$-0.612027\pi$
$410$		0			0
$411$	2.09770e9			1.49038
$412$		0			0
$413$	−7.03882e8	+	1.21916e9i	−0.491671	+	0.851599i
$414$		0			0
$415$	7.86255e8	−	1.36183e9i	0.540002	−	0.935311i
$416$		0			0
$417$	4.43828e9			2.99736
$418$		0			0
$419$	−3.05021e8			−0.202573			−0.101286	−	0.994857i	$-0.532296\pi$
$419$	−3.05021e8			−0.202573			−0.101286	+	0.994857i	$0.532296\pi$
$420$		0			0
$421$	1.14477e9	−	1.98280e9i	0.747708	−	1.29507i	−0.201211	−	0.979548i	$-0.564488\pi$
$421$	1.14477e9	−	1.98280e9i	0.747708	−	1.29507i	0.948919	−	0.315520i	$-0.102179\pi$
$422$		0			0
$423$	−1.43210e9	+	2.48047e9i	−0.919989	+	1.59347i
$424$		0			0
$425$	−8.98400e8			−0.567687
$426$		0			0
$427$	6.94167e6	+	1.20233e7i	0.00431486	+	0.00747355i
$428$		0			0
$429$	2.95854e9			1.80916
$430$		0			0
$431$	−1.28151e9	−	2.21965e9i	−0.770997	−	1.33541i	−0.937017	−	0.349283i	$-0.886425\pi$
$431$	−1.28151e9	−	2.21965e9i	−0.770997	−	1.33541i	0.166021	−	0.986122i	$-0.446908\pi$
$432$		0			0
$433$	3.98356e8	+	6.89973e8i	0.235811	+	0.408437i	0.959508	−	0.281681i	$-0.0908920\pi$
$433$	3.98356e8	+	6.89973e8i	0.235811	+	0.408437i	−0.723697	+	0.690118i	$0.757559\pi$
$434$		0			0
$435$	1.16763e9	−	2.02240e9i	0.680132	−	1.17802i
$436$		0			0
$437$	−1.27028e9	−	2.22930e9i	−0.728140	−	1.27786i
$438$		0			0
$439$	4.19687e8	−	7.26919e8i	0.236755	−	0.410072i	−0.723026	−	0.690821i	$-0.757249\pi$
$439$	4.19687e8	−	7.26919e8i	0.236755	−	0.410072i	0.959781	+	0.280749i	$0.0905827\pi$
$440$		0			0
$441$	1.02025e9	+	1.76712e9i	0.566463	+	0.981142i
$442$		0			0
$443$	−1.01908e9	−	1.76510e9i	−0.556923	−	0.964618i	−0.997751	−	0.0670270i	$-0.978649\pi$
$443$	−1.01908e9	−	1.76510e9i	−0.556923	−	0.964618i	0.440828	−	0.897591i	$-0.354685\pi$
$444$		0			0
$445$	1.94117e9			1.04425
$446$		0			0
$447$	2.16736e9	+	3.75397e9i	1.14777	+	1.98799i
$448$		0			0
$449$	−7.61644e8			−0.397091			−0.198545	−	0.980092i	$-0.563622\pi$
$449$	−7.61644e8			−0.397091			−0.198545	+	0.980092i	$0.563622\pi$
$450$		0			0
$451$	−7.34403e8	+	1.27202e9i	−0.376978	+	0.652946i
$452$		0			0
$453$	2.08868e9	−	3.61770e9i	1.05567	−	1.82847i
$454$		0			0
$455$	6.46236e8			0.321626
$456$		0			0
$457$	−1.81761e9			−0.890831			−0.445415	−	0.895324i	$-0.646944\pi$
$457$	−1.81761e9			−0.890831			−0.445415	+	0.895324i	$0.646944\pi$
$458$		0			0
$459$	−1.88152e9	+	3.25888e9i	−0.908163	+	1.57298i
$460$		0			0
$461$	−1.31270e8	+	2.27367e8i	−0.0624040	+	0.108087i	−0.895540	−	0.444982i	$-0.853210\pi$
$461$	−1.31270e8	+	2.27367e8i	−0.0624040	+	0.108087i	0.833135	+	0.553069i	$0.186543\pi$
$462$		0			0
$463$	2.42693e9			1.13638			0.568190	−	0.822897i	$-0.307644\pi$
$463$	2.42693e9			1.13638			0.568190	+	0.822897i	$0.307644\pi$
$464$		0			0
$465$	−1.39919e9	−	2.42346e9i	−0.645343	−	1.11777i
$466$		0			0
$467$	2.94879e9			1.33978			0.669892	−	0.742459i	$-0.266341\pi$
$467$	2.94879e9			1.33978			0.669892	+	0.742459i	$0.266341\pi$
$468$		0			0
$469$	−1.24821e8	−	2.16197e8i	−0.0558707	−	0.0967709i
$470$		0			0
$471$	1.26119e9	+	2.18444e9i	0.556169	+	0.963313i
$472$		0			0
$473$	−5.13111e8	+	8.88734e8i	−0.222945	+	0.386152i
$474$		0			0
$475$	5.90803e8	+	1.03684e9i	0.252939	+	0.443898i
$476$		0			0
$477$	3.06784e9	−	5.31365e9i	1.29425	−	2.24171i
$478$		0			0
$479$	−1.50006e9	−	2.59818e9i	−0.623642	−	1.08018i	−0.988802	−	0.149234i	$-0.952319\pi$
$479$	−1.50006e9	−	2.59818e9i	−0.623642	−	1.08018i	0.365160	−	0.930945i	$-0.381014\pi$
$480$		0			0
$481$	−3.13876e8	−	5.43649e8i	−0.128603	−	0.222746i
$482$		0			0
$483$	−4.05246e9			−1.63645
$484$		0			0
$485$	1.03133e9	+	1.78631e9i	0.410487	+	0.710984i
$486$		0			0
$487$	1.75101e9			0.686971			0.343485	−	0.939158i	$-0.388392\pi$
$487$	1.75101e9			0.686971			0.343485	+	0.939158i	$0.388392\pi$
$488$		0			0
$489$	−4.92011e8	+	8.52187e8i	−0.190280	+	0.329575i
$490$		0			0
$491$	−8.41667e8	+	1.45781e9i	−0.320889	+	0.555796i	−0.980672	−	0.195660i	$-0.937315\pi$
$491$	−8.41667e8	+	1.45781e9i	−0.320889	+	0.555796i	0.659783	+	0.751457i	$0.270648\pi$
$492$		0			0
$493$	−3.34687e9			−1.25798
$494$		0			0
$495$	−5.46168e9			−2.02399
$496$		0			0
$497$	−3.73540e8	+	6.46991e8i	−0.136487	+	0.236402i
$498$		0			0
$499$	−1.77928e9	+	3.08180e9i	−0.641050	+	1.11033i	0.344149	+	0.938915i	$0.388167\pi$
$499$	−1.77928e9	+	3.08180e9i	−0.641050	+	1.11033i	−0.985199	+	0.171416i	$0.945166\pi$
$500$		0			0
$501$	−4.45259e9			−1.58191
$502$		0			0
$503$	−2.28643e9	−	3.96022e9i	−0.801070	−	1.38749i	−0.918913	−	0.394461i	$-0.870931\pi$
$503$	−2.28643e9	−	3.96022e9i	−0.801070	−	1.38749i	0.117843	−	0.993032i	$-0.462402\pi$
$504$		0			0
$505$	4.94296e8			0.170792
$506$		0			0
$507$	1.24970e9	+	2.16454e9i	0.425870	+	0.737629i
$508$		0			0
$509$	5.86983e8	+	1.01668e9i	0.197294	+	0.341722i	0.947650	−	0.319311i	$-0.103451\pi$
$509$	5.86983e8	+	1.01668e9i	0.197294	+	0.341722i	−0.750356	+	0.661033i	$0.770118\pi$
$510$		0			0
$511$	1.76706e9	−	3.06063e9i	0.585838	−	1.01470i
$512$		0			0
$513$	4.99837e9	+	2.83515e7i	1.63462	+	0.00927182i
$514$		0			0
$515$	5.38358e8	−	9.32463e8i	0.173678	−	0.300820i
$516$		0			0
$517$	2.19674e9	+	3.80486e9i	0.699135	+	1.21094i
$518$		0			0
$519$	−2.88173e9	−	4.99130e9i	−0.904831	−	1.56721i
$520$		0			0
$521$	−4.34807e9			−1.34699			−0.673495	−	0.739192i	$-0.735208\pi$
$521$	−4.34807e9			−1.34699			−0.673495	+	0.739192i	$0.735208\pi$
$522$		0			0
$523$	1.94511e9	+	3.36902e9i	0.594549	+	1.02979i	0.993610	+	0.112864i	$0.0360026\pi$
$523$	1.94511e9	+	3.36902e9i	0.594549	+	1.02979i	−0.399062	+	0.916924i	$0.630664\pi$
$524$		0			0
$525$	1.88478e9			0.568465
$526$		0			0
$527$	−2.00530e9	+	3.47328e9i	−0.596818	+	1.03372i
$528$		0			0
$529$	−1.98010e9	+	3.42963e9i	−0.581556	+	1.00728i
$530$		0			0
$531$	−1.02225e10			−2.96297
$532$		0			0
$533$	1.26200e9			0.361005
$534$		0			0
$535$	6.19601e8	−	1.07318e9i	0.174934	−	0.302994i
$536$		0			0
$537$	−1.73127e9	+	2.99865e9i	−0.482453	+	0.835632i
$538$		0			0
$539$	3.12997e9			0.860953
$540$		0			0
$541$	−2.79256e8	−	4.83686e8i	−0.0758250	−	0.131333i	0.825620	−	0.564227i	$-0.190826\pi$
$541$	−2.79256e8	−	4.83686e8i	−0.0758250	−	0.131333i	−0.901445	+	0.432894i	$0.857492\pi$
$542$		0			0
$543$	2.04705e9			0.548692
$544$		0			0
$545$	1.59497e9	+	2.76258e9i	0.422052	+	0.731015i
$546$		0			0
$547$	8.62746e8	+	1.49432e9i	0.225386	+	0.390380i	0.956435	−	0.291945i	$-0.0943023\pi$
$547$	8.62746e8	+	1.49432e9i	0.225386	+	0.390380i	−0.731049	+	0.682325i	$0.760969\pi$
$548$		0			0
$549$	−5.04072e7	+	8.73079e7i	−0.0130014	+	0.0225191i
$550$		0			0
$551$	2.20096e9	+	3.86260e9i	0.560507	+	0.983670i
$552$		0			0
$553$	1.05645e9	−	1.82982e9i	0.265651	−	0.460120i
$554$		0			0
$555$	8.76358e8	+	1.51790e9i	0.217599	+	0.376892i
$556$		0			0
$557$	−9.83462e8	−	1.70341e9i	−0.241137	−	0.417662i	0.719901	−	0.694077i	$-0.244187\pi$
$557$	−9.83462e8	−	1.70341e9i	−0.241137	−	0.417662i	−0.961039	+	0.276414i	$0.910854\pi$
$558$		0			0
$559$	8.81729e8			0.213498
$560$		0			0
$561$	5.91935e9	+	1.02526e10i	1.41548	+	2.45169i
$562$		0			0
$563$	−3.53287e9			−0.834350			−0.417175	−	0.908826i	$-0.636980\pi$
$563$	−3.53287e9			−0.834350			−0.417175	+	0.908826i	$0.636980\pi$
$564$		0			0
$565$	−2.64406e8	+	4.57964e8i	−0.0616739	+	0.106822i
$566$		0			0
$567$	1.20433e9	−	2.08596e9i	0.277463	−	0.480579i
$568$		0			0
$569$	7.17320e8			0.163237			0.0816187	−	0.996664i	$-0.473991\pi$
$569$	7.17320e8			0.163237			0.0816187	+	0.996664i	$0.473991\pi$
$570$		0			0
$571$	−4.98985e9			−1.12166			−0.560830	−	0.827931i	$-0.689518\pi$
$571$	−4.98985e9			−1.12166			−0.560830	+	0.827931i	$0.689518\pi$
$572$		0			0
$573$	5.32552e9	−	9.22407e9i	1.18255	−	2.04824i
$574$		0			0
$575$	1.71272e9	−	2.96652e9i	0.375707	−	0.650744i
$576$		0			0
$577$	1.71470e9			0.371598			0.185799	−	0.982588i	$-0.440513\pi$
$577$	1.71470e9			0.371598			0.185799	+	0.982588i	$0.440513\pi$
$578$		0			0
$579$	−4.86960e9	−	8.43440e9i	−1.04260	−	1.80584i
$580$		0			0
$581$	4.72812e9			1.00017
$582$		0			0
$583$	−4.70583e9	−	8.15073e9i	−0.983549	−	1.70356i
$584$		0			0
$585$	2.34633e9	+	4.06397e9i	0.484556	+	0.839276i
$586$		0			0
$587$	5.08541e8	−	8.80819e8i	0.103775	−	0.179743i	−0.809462	−	0.587172i	$-0.800241\pi$
$587$	5.08541e8	−	8.80819e8i	0.103775	−	0.179743i	0.913237	+	0.407429i	$0.133574\pi$
$588$		0			0
$589$	5.32721e9	+	3.02167e7i	1.07423	+	0.00609317i
$590$		0			0
$591$	−3.12376e9	+	5.41051e9i	−0.622474	+	1.07816i
$592$		0			0
$593$	2.54928e9	+	4.41548e9i	0.502026	+	0.869535i	0.999997	+	0.00234107i	$0.000745185\pi$
$593$	2.54928e9	+	4.41548e9i	0.502026	+	0.869535i	−0.497971	+	0.867194i	$0.665921\pi$
$594$		0			0
$595$	1.29297e9	+	2.23949e9i	0.251640	+	0.435853i
$596$		0			0
$597$	−1.70158e10			−3.27297
$598$		0			0
$599$	3.58478e9	+	6.20901e9i	0.681504	+	1.18040i	0.974522	+	0.224292i	$0.0720070\pi$
$599$	3.58478e9	+	6.20901e9i	0.681504	+	1.18040i	−0.293018	+	0.956107i	$0.594660\pi$
$600$		0			0
$601$	5.78544e9			1.08711			0.543557	−	0.839372i	$-0.317077\pi$
$601$	5.78544e9			1.08711			0.543557	+	0.839372i	$0.317077\pi$
$602$		0			0
$603$	9.06395e8	−	1.56992e9i	0.168348	−	0.291587i
$604$		0			0
$605$	−2.28427e9	+	3.95647e9i	−0.419376	+	0.726380i
$606$		0			0
$607$	−5.62010e9			−1.01996			−0.509980	−	0.860186i	$-0.670347\pi$
$607$	−5.62010e9			−1.01996			−0.509980	+	0.860186i	$0.670347\pi$
$608$		0			0
$609$	7.02151e9			1.25971
$610$		0			0
$611$	1.88743e9	−	3.26913e9i	0.334756	−	0.579814i
$612$		0			0
$613$	−5.33281e9	+	9.23670e9i	−0.935071	+	1.61959i	−0.160564	+	0.987025i	$0.551331\pi$
$613$	−5.33281e9	+	9.23670e9i	−0.935071	+	1.61959i	−0.774507	+	0.632565i	$0.782002\pi$
$614$		0			0
$615$	−3.52356e9			−0.610829
$616$		0			0
$617$	1.78954e9	+	3.09958e9i	0.306722	+	0.531257i	0.977643	−	0.210271i	$-0.0674346\pi$
$617$	1.78954e9	+	3.09958e9i	0.306722	+	0.531257i	−0.670922	+	0.741528i	$0.734101\pi$
$618$		0			0
$619$	−4.38355e9			−0.742862			−0.371431	−	0.928461i	$-0.621133\pi$
$619$	−4.38355e9			−0.742862			−0.371431	+	0.928461i	$0.621133\pi$
$620$		0			0
$621$	−7.17389e9	−	1.24255e10i	−1.20208	−	2.08207i
$622$		0			0
$623$	2.91829e9	+	5.05463e9i	0.483527	+	0.837493i
$624$		0			0
$625$	6.96020e8	−	1.20554e9i	0.114036	−	0.197516i
$626$		0			0
$627$	7.93982e9	−	1.35738e10i	1.28639	−	2.19920i
$628$		0			0
$629$	1.25599e9	−	2.17543e9i	0.201237	−	0.348552i
$630$		0			0
$631$	−3.10917e9	−	5.38524e9i	−0.492654	−	0.853301i	0.507311	−	0.861763i	$-0.330640\pi$
$631$	−3.10917e9	−	5.38524e9i	−0.492654	−	0.853301i	−0.999964	+	0.00846212i	$0.997306\pi$
$632$		0			0
$633$	−7.26698e9	−	1.25868e10i	−1.13878	−	1.97243i
$634$		0			0
$635$	−2.95972e9			−0.458715
$636$		0			0
$637$	−1.34463e9	−	2.32898e9i	−0.206118	−	0.357007i
$638$		0			0
$639$	−5.42495e9			−0.822514
$640$		0			0
$641$	4.70080e9	−	8.14203e9i	0.704967	−	1.22104i	−0.261736	−	0.965140i	$-0.584295\pi$
$641$	4.70080e9	−	8.14203e9i	0.704967	−	1.22104i	0.966703	−	0.255900i	$-0.0823717\pi$
$642$		0			0
$643$	3.05317e9	−	5.28824e9i	0.452911	−	0.784464i	−0.545655	−	0.838010i	$-0.683719\pi$
$643$	3.05317e9	−	5.28824e9i	0.452911	−	0.784464i	0.998565	+	0.0535459i	$0.0170523\pi$
$644$		0			0
$645$	−2.46184e9			−0.361243
$646$		0			0
$647$	1.75861e8			0.0255272			0.0127636	−	0.999919i	$-0.495937\pi$
$647$	1.75861e8			0.0255272			0.0127636	+	0.999919i	$0.495937\pi$
$648$		0			0
$649$	−7.84029e9	+	1.35798e10i	−1.12584	+	1.95001i
$650$		0			0
$651$	4.20698e9	−	7.28671e9i	0.597637	−	1.03514i
$652$		0			0
$653$	1.43607e9			0.201828			0.100914	−	0.994895i	$-0.467823\pi$
$653$	1.43607e9			0.201828			0.100914	+	0.994895i	$0.467823\pi$
$654$		0			0
$655$	1.96614e9	+	3.40545e9i	0.273382	+	0.473511i
$656$		0			0
$657$	2.56631e10			3.53045
$658$		0			0
$659$	6.47537e9	+	1.12157e10i	0.881385	+	1.52660i	0.849802	+	0.527102i	$0.176722\pi$
$659$	6.47537e9	+	1.12157e10i	0.881385	+	1.52660i	0.0315832	+	0.999501i	$0.489945\pi$
$660$		0			0
$661$	−3.16773e9	−	5.48666e9i	−0.426621	−	0.738930i	0.569949	−	0.821680i	$-0.306963\pi$
$661$	−3.16773e9	−	5.48666e9i	−0.426621	−	0.738930i	−0.996570	+	0.0827502i	$0.973630\pi$
$662$		0			0
$663$	5.08590e9	−	8.80904e9i	0.677752	−	1.17390i
$664$		0			0
$665$	1.73430e9	−	2.96493e9i	0.228691	−	0.390966i
$666$		0			0
$667$	6.38051e9	−	1.10514e10i	0.832559	−	1.44204i
$668$		0			0
$669$	9.67256e9	+	1.67534e10i	1.24896	+	2.16327i
$670$		0			0
$671$	7.73208e7	+	1.33924e8i	0.00988025	+	0.0171131i
$672$		0			0
$673$	4.43656e9			0.561040			0.280520	−	0.959848i	$-0.409493\pi$
$673$	4.43656e9			0.561040			0.280520	+	0.959848i	$0.409493\pi$
$674$		0			0
$675$	3.33655e9	+	5.77908e9i	0.417575	+	0.723261i
$676$		0			0
$677$	−4.63687e9			−0.574334			−0.287167	−	0.957881i	$-0.592713\pi$
$677$	−4.63687e9			−0.574334			−0.287167	+	0.957881i	$0.592713\pi$
$678$		0			0
$679$	−3.10092e9	+	5.37095e9i	−0.380142	+	0.658426i
$680$		0			0
$681$	−1.02635e9	+	1.77769e9i	−0.124532	+	0.215695i
$682$		0			0
$683$	6.36907e9			0.764898			0.382449	−	0.923977i	$-0.375081\pi$
$683$	6.36907e9			0.764898			0.382449	+	0.923977i	$0.375081\pi$
$684$		0			0
$685$	5.10376e9			0.606699
$686$		0			0
$687$	−9.54688e9	+	1.65357e10i	−1.12334	+	1.94569i
$688$		0			0
$689$	−4.04325e9	+	7.00311e9i	−0.470937	+	0.815687i
$690$		0			0
$691$	−4.91509e8			−0.0566707			−0.0283353	−	0.999598i	$-0.509021\pi$
$691$	−4.91509e8			−0.0566707			−0.0283353	+	0.999598i	$0.509021\pi$
$692$		0			0
$693$	−8.21090e9	−	1.42217e10i	−0.937183	−	1.62325i
$694$		0			0
$695$	1.07985e10			1.22015
$696$		0			0
$697$	2.52497e9	+	4.37337e9i	0.282449	+	0.489217i
$698$		0			0
$699$	2.67934e9	+	4.64075e9i	0.296727	+	0.513947i
$700$		0			0
$701$	4.98265e9	−	8.63021e9i	0.546321	−	0.946255i	−0.452202	−	0.891916i	$-0.649361\pi$
$701$	4.98265e9	−	8.63021e9i	0.546321	−	0.946255i	0.998523	−	0.0543394i	$-0.0173053\pi$
$702$		0			0
$703$	−3.33661e9	−	1.89257e7i	−0.362211	−	0.00205451i
$704$		0			0
$705$	−5.26982e9	+	9.12760e9i	−0.566414	+	0.981058i
$706$		0			0
$707$	7.43108e8	+	1.28710e9i	0.0790831	+	0.136976i
$708$		0			0
$709$	−8.76843e9	−	1.51874e10i	−0.923974	−	1.60037i	−0.793203	−	0.608957i	$-0.791588\pi$
$709$	−8.76843e9	−	1.51874e10i	−0.923974	−	1.60037i	−0.130771	−	0.991413i	$-0.541745\pi$
$710$		0			0
$711$	1.53429e10			1.60090
$712$		0			0
$713$	−7.64586e9	−	1.32430e10i	−0.789974	−	1.36827i
$714$		0			0
$715$	7.19820e9			0.736466
$716$		0			0
$717$	1.22686e10	−	2.12499e10i	1.24302	−	2.15298i
$718$		0			0
$719$	−7.14817e8	+	1.23810e9i	−0.0717206	+	0.124224i	−0.899655	−	0.436601i	$-0.856182\pi$
$719$	−7.14817e8	+	1.23810e9i	−0.0717206	+	0.124224i	0.827935	+	0.560824i	$0.189516\pi$
$720$		0			0
$721$	3.23740e9			0.321679
$722$		0			0
$723$	−2.73381e10			−2.69020
$724$		0			0
$725$	−2.96755e9	+	5.13995e9i	−0.289212	+	0.500929i
$726$		0			0
$727$	−5.06457e9	+	8.77209e9i	−0.488846	+	0.846706i	−0.999918	−	0.0128318i	$-0.995915\pi$
$727$	−5.06457e9	+	8.77209e9i	−0.488846	+	0.846706i	0.511071	+	0.859538i	$0.329249\pi$
$728$		0			0
$729$	−1.18854e10			−1.13623
$730$		0			0
$731$	1.76414e9	+	3.05557e9i	0.167040	+	0.289322i
$732$		0			0
$733$	−1.05164e10			−0.986288			−0.493144	−	0.869948i	$-0.664152\pi$
$733$	−1.05164e10			−0.986288			−0.493144	+	0.869948i	$0.664152\pi$
$734$		0			0
$735$	3.75429e9	+	6.50262e9i	0.348757	+	0.604064i
$736$		0			0
$737$	−1.39034e9	−	2.40814e9i	−0.127934	−	0.221588i
$738$		0			0
$739$	−5.74786e8	+	9.95558e8i	−0.0523902	+	0.0907425i	−0.891031	−	0.453942i	$-0.850017\pi$
$739$	−5.74786e8	+	9.95558e8i	−0.0523902	+	0.0907425i	0.838641	+	0.544685i	$0.183351\pi$
$740$		0			0
$741$	−1.35110e10	−	7.66365e7i	−1.21990	−	0.00691946i
$742$		0			0
$743$	−7.74585e9	+	1.34162e10i	−0.692801	+	1.19997i	0.278116	+	0.960548i	$0.410290\pi$
$743$	−7.74585e9	+	1.34162e10i	−0.692801	+	1.19997i	−0.970916	+	0.239418i	$0.923043\pi$
$744$		0			0
$745$	5.27323e9	+	9.13351e9i	0.467229	+	0.809265i
$746$		0			0
$747$	1.71667e10	+	2.97336e10i	1.50683	+	2.60991i
$748$		0			0
$749$	3.72595e9			0.324004
$750$		0			0
$751$	−5.29979e9	−	9.17950e9i	−0.456582	−	0.790823i	0.542196	−	0.840252i	$-0.317593\pi$
$751$	−5.29979e9	−	9.17950e9i	−0.456582	−	0.790823i	−0.998778	+	0.0494293i	$0.984260\pi$
$752$		0			0
$753$	−3.49088e10			−2.97956
$754$		0			0
$755$	5.08181e9	−	8.80196e9i	0.429739	−	0.744329i
$756$		0			0
$757$	7.07575e9	−	1.22556e10i	0.592839	−	1.02683i	−0.401009	−	0.916074i	$-0.631340\pi$
$757$	7.07575e9	−	1.22556e10i	0.592839	−	1.02683i	0.993848	−	0.110754i	$-0.0353264\pi$
$758$		0			0
$759$	−4.51389e10			−3.74718
$760$		0			0
$761$	−1.14521e10			−0.941976			−0.470988	−	0.882140i	$-0.656102\pi$
$761$	−1.14521e10			−0.941976			−0.470988	+	0.882140i	$0.656102\pi$
$762$		0			0
$763$	−4.79566e9	+	8.30633e9i	−0.390852	+	0.676976i
$764$		0			0
$765$	−9.38895e9	+	1.62621e10i	−0.758232	+	1.31330i
$766$		0			0
$767$	1.34727e10			1.07813
$768$		0			0
$769$	−2.44799e9	−	4.24005e9i	−0.194119	−	0.336224i	0.752492	−	0.658601i	$-0.228851\pi$
$769$	−2.44799e9	−	4.24005e9i	−0.194119	−	0.336224i	−0.946611	+	0.322377i	$0.895518\pi$
$770$		0			0
$771$	2.95517e10			2.32216
$772$		0			0
$773$	−5.73818e9	−	9.93882e9i	−0.446834	−	0.773939i	0.551344	−	0.834278i	$-0.314115\pi$
$773$	−5.73818e9	−	9.93882e9i	−0.446834	−	0.773939i	−0.998178	+	0.0603392i	$0.980782\pi$
$774$		0			0
$775$	3.55606e9	+	6.15928e9i	0.274418	+	0.475307i
$776$		0			0
$777$	−2.63497e9	+	4.56391e9i	−0.201513	+	0.349030i
$778$		0			0
$779$	3.38682e9	−	5.79005e9i	0.256691	−	0.438835i
$780$		0			0
$781$	−4.16073e9	+	7.20660e9i	−0.312530	+	0.541317i
$782$		0			0
$783$	1.24299e10	+	2.15292e10i	0.925339	+	1.60273i
$784$		0			0
$785$	3.06851e9	+	5.31481e9i	0.226403	+	0.392142i
$786$		0			0
$787$	−1.22404e10			−0.895126			−0.447563	−	0.894252i	$-0.647708\pi$
$787$	−1.22404e10			−0.895126			−0.447563	+	0.894252i	$0.647708\pi$
$788$		0			0
$789$	−4.83187e9	−	8.36905e9i	−0.350224	−	0.606606i
$790$		0			0
$791$	−1.58999e9			−0.114229
$792$		0			0
$793$	6.64340e7	−	1.15067e8i	0.00473080	−	0.00819398i
$794$		0			0
$795$	1.12890e10	−	1.95531e10i	0.796836	−	1.38016i
$796$		0			0
$797$	2.63386e10			1.84285			0.921423	−	0.388562i	$-0.127028\pi$
$797$	2.63386e10			1.84285			0.921423	+	0.388562i	$0.127028\pi$
$798$		0			0
$799$	1.51053e10			1.04765
$800$		0			0
$801$	−2.11913e10	+	3.67044e10i	−1.45695	+	2.52350i
$802$		0			0
$803$	1.96826e10	−	3.40913e10i	1.34146	−	2.32348i
$804$		0			0
$805$	−9.85973e9			−0.666162
$806$		0			0
$807$	−8.86240e9	−	1.53501e10i	−0.593600	−	1.02815i
$808$		0			0
$809$	1.65242e9			0.109723			0.0548617	−	0.998494i	$-0.482528\pi$
$809$	1.65242e9			0.109723			0.0548617	+	0.998494i	$0.482528\pi$
$810$		0			0
$811$	−1.38134e10	−	2.39255e10i	−0.909344	−	1.57503i	−0.814978	−	0.579492i	$-0.803251\pi$
$811$	−1.38134e10	−	2.39255e10i	−0.909344	−	1.57503i	−0.0943662	−	0.995538i	$-0.530082\pi$
$812$		0			0
$813$	1.23057e10	+	2.13142e10i	0.803140	+	1.39108i
$814$		0			0
$815$	−1.19707e9	+	2.07339e9i	−0.0774586	+	0.134162i
$816$		0			0
$817$	2.36629e9	−	4.04538e9i	0.151807	−	0.259526i
$818$		0			0
$819$	−7.05480e9	+	1.22193e10i	−0.448736	+	0.777234i
$820$		0			0
$821$	8.83232e9	+	1.52980e10i	0.557024	+	0.964793i	0.997743	+	0.0671483i	$0.0213901\pi$
$821$	8.83232e9	+	1.52980e10i	0.557024	+	0.964793i	−0.440719	+	0.897645i	$0.645277\pi$
$822$		0			0
$823$	8.76539e9	+	1.51821e10i	0.548115	+	0.949363i	0.998404	+	0.0564803i	$0.0179878\pi$
$823$	8.76539e9	+	1.51821e10i	0.548115	+	0.949363i	−0.450288	+	0.892883i	$0.648679\pi$
$824$		0			0
$825$	2.09939e10			1.30168
$826$		0			0
$827$	7.65936e9	+	1.32664e10i	0.470894	+	0.815612i	0.999446	−	0.0332888i	$-0.0105981\pi$
$827$	7.65936e9	+	1.32664e10i	0.470894	+	0.815612i	−0.528552	+	0.848901i	$0.677265\pi$
$828$		0			0
$829$	−1.42058e10			−0.866013			−0.433007	−	0.901391i	$-0.642547\pi$
$829$	−1.42058e10			−0.866013			−0.433007	+	0.901391i	$0.642547\pi$
$830$		0			0
$831$	2.19011e10	−	3.79339e10i	1.32392	−	2.29310i
$832$		0			0
$833$	5.38061e9	−	9.31948e9i	0.322533	−	0.558643i
$834$		0			0
$835$	−1.08333e10			−0.643957
$836$		0			0
$837$	2.97898e10			1.75601
$838$		0			0
$839$	−1.06248e10	+	1.84027e10i	−0.621090	+	1.07576i	0.368193	+	0.929749i	$0.379977\pi$
$839$	−1.06248e10	+	1.84027e10i	−0.621090	+	1.07576i	−0.989283	+	0.146010i	$0.953357\pi$
$840$		0			0
$841$	−2.43029e9	+	4.20938e9i	−0.140887	+	0.244024i
$842$		0			0
$843$	−2.48129e10			−1.42653
$844$		0			0
$845$	3.04055e9	+	5.26638e9i	0.173362	+	0.300271i
$846$		0			0
$847$	−1.37364e10			−0.776748
$848$		0			0
$849$	1.11246e10	+	1.92683e10i	0.623887	+	1.08060i
$850$		0			0
$851$	4.78885e9	+	8.29454e9i	0.266366	+	0.461359i
$852$		0			0
$853$	5.58386e9	−	9.67152e9i	0.308044	−	0.533548i	−0.669891	−	0.742460i	$-0.733659\pi$
$853$	5.58386e9	−	9.67152e9i	0.308044	−	0.533548i	0.977934	+	0.208912i	$0.0669923\pi$
$854$		0			0
$855$	2.49424e10	+	1.41477e8i	1.36476	+	0.00774111i
$856$		0			0
$857$	6.90893e9	−	1.19666e10i	0.374954	−	0.649440i	−0.615366	−	0.788241i	$-0.710992\pi$
$857$	6.90893e9	−	1.19666e10i	0.374954	−	0.649440i	0.990320	+	0.138802i	$0.0443251\pi$
$858$		0			0
$859$	−6.60532e9	−	1.14408e10i	−0.355564	−	0.615855i	0.631650	−	0.775253i	$-0.282378\pi$
$859$	−6.60532e9	−	1.14408e10i	−0.355564	−	0.615855i	−0.987214	+	0.159398i	$0.949045\pi$
$860$		0			0
$861$	−5.29721e9	−	9.17503e9i	−0.282837	−	0.489888i
$862$		0			0
$863$	−1.07663e10			−0.570203			−0.285101	−	0.958497i	$-0.592027\pi$
$863$	−1.07663e10			−0.570203			−0.285101	+	0.958497i	$0.592027\pi$
$864$		0			0
$865$	−7.01132e9	−	1.21440e10i	−0.368336	−	0.637976i
$866$		0			0
$867$	7.73526e9			0.403096
$868$		0			0
$869$	1.17674e10	−	2.03818e10i	0.608292	−	1.05359i
$870$		0			0
$871$	−1.19458e9	+	2.06907e9i	−0.0612565	+	0.106099i
$872$		0			0
$873$	−4.50349e10			−2.29086
$874$		0			0
$875$	1.35614e10			0.684348
$876$		0			0
$877$	2.06151e9	−	3.57064e9i	0.103202	−	0.178751i	−0.809800	−	0.586706i	$-0.800425\pi$
$877$	2.06151e9	−	3.57064e9i	0.103202	−	0.178751i	0.913002	+	0.407955i	$0.133758\pi$
$878$		0			0
$879$	2.18990e10	−	3.79302e10i	1.08759	−	1.88375i
$880$		0			0
$881$	2.52274e9			0.124296			0.0621479	−	0.998067i	$-0.480205\pi$
$881$	2.52274e9			0.124296			0.0621479	+	0.998067i	$0.480205\pi$
$882$		0			0
$883$	−2.12039e9	−	3.67263e9i	−0.103646	−	0.179521i	0.809538	−	0.587068i	$-0.199718\pi$
$883$	−2.12039e9	−	3.67263e9i	−0.103646	−	0.179521i	−0.913184	+	0.407547i	$0.866384\pi$
$884$		0			0
$885$	−3.76166e10			−1.82423
$886$		0			0
$887$	−1.13120e10	−	1.95930e10i	−0.544263	−	0.942690i	−0.998653	−	0.0518877i	$-0.983476\pi$
$887$	−1.13120e10	−	1.95930e10i	−0.544263	−	0.942690i	0.454390	−	0.890803i	$-0.349857\pi$
$888$		0			0
$889$	−4.44954e9	−	7.70683e9i	−0.212402	−	0.367891i
$890$		0			0
$891$	1.34146e10	−	2.32348e10i	0.635339	−	1.10044i
$892$		0			0
$893$	−9.93349e9	−	1.74329e10i	−0.466790	−	0.819200i
$894$		0			0
$895$	−4.21222e9	+	7.29579e9i	−0.196395	+	0.340166i
$896$		0			0
$897$	1.93917e10	+	3.35873e10i	0.897101	+	1.55382i
$898$		0			0
$899$	1.32476e10	+	2.29456e10i	0.608106	+	1.05327i
$900$		0			0
$901$	−3.23584e10			−1.47384
$902$		0			0
$903$	−3.70104e9	−	6.41039e9i	−0.167269	−	0.289719i
$904$		0			0
$905$	4.98052e9			0.223360
$906$		0			0
$907$	1.47628e10	−	2.55700e10i	0.656969	−	1.13790i	−0.324428	−	0.945911i	$-0.605172\pi$
$907$	1.47628e10	−	2.55700e10i	0.656969	−	1.13790i	0.981396	−	0.191993i	$-0.0614950\pi$
$908$		0			0
$909$	−5.39611e9	+	9.34633e9i	−0.238291	+	0.412731i
$910$		0			0
$911$	−2.91259e10			−1.27633			−0.638167	−	0.769898i	$-0.720307\pi$
$911$	−2.91259e10			−1.27633			−0.638167	+	0.769898i	$0.720307\pi$
$912$		0			0
$913$	5.26648e10			2.29020
$914$		0			0
$915$	−1.85487e8	+	3.21274e8i	−0.00800462	+	0.0138644i
$916$		0			0
$917$	−5.91165e9	+	1.02393e10i	−0.253172	+	0.438507i
$918$		0			0
$919$	2.15568e10			0.916178			0.458089	−	0.888906i	$-0.348534\pi$
$919$	2.15568e10			0.916178			0.458089	+	0.888906i	$0.348534\pi$
$920$		0			0
$921$	7.52768e9	+	1.30383e10i	0.317506	+	0.549937i
$922$		0			0
$923$	7.14980e9			0.299287
$924$		0			0
$925$	−2.22728e9	−	3.85776e9i	−0.0925291	−	0.160265i
$926$		0			0
$927$	1.17542e10	+	2.03589e10i	0.484636	+	0.839413i
$928$		0			0
$929$	−4.54074e9	+	7.86478e9i	−0.185811	+	0.321834i	−0.943849	−	0.330376i	$-0.892825\pi$
$929$	−4.54074e9	+	7.86478e9i	−0.185811	+	0.321834i	0.758039	+	0.652210i	$0.226158\pi$
$930$		0			0
$931$	−1.42939e10	−	8.10772e7i	−0.580534	−	0.00329287i
$932$		0			0
$933$	2.97915e10	−	5.16005e10i	1.20090	−	2.08002i
$934$		0			0
$935$	1.44019e10	+	2.49449e10i	0.576209	+	0.998024i
$936$		0			0
$937$	7.66570e9	+	1.32774e10i	0.304413	+	0.527259i	0.977131	−	0.212640i	$-0.0682063\pi$
$937$	7.66570e9	+	1.32774e10i	0.304413	+	0.527259i	−0.672717	+	0.739900i	$0.734873\pi$
$938$		0			0
$939$	−6.47031e10			−2.55033
$940$		0			0
$941$	−4.20535e9	−	7.28389e9i	−0.164528	−	0.284970i	0.771960	−	0.635671i	$-0.219277\pi$
$941$	−4.20535e9	−	7.28389e9i	−0.164528	−	0.284970i	−0.936487	+	0.350701i	$0.885943\pi$
$942$		0			0
$943$	−1.92545e10			−0.747724
$944$		0			0
$945$	9.60387e9	−	1.66344e10i	0.370199	−	0.641203i
$946$		0			0
$947$	−3.29580e9	+	5.70850e9i	−0.126106	+	0.218422i	−0.922165	−	0.386797i	$-0.873581\pi$
$947$	−3.29580e9	+	5.70850e9i	−0.126106	+	0.218422i	0.796059	+	0.605220i	$0.206915\pi$
$948$		0			0
$949$	−3.38226e10			−1.28462
$950$		0			0
$951$	2.03253e10			0.766312
$952$		0			0
$953$	1.13013e10	−	1.95744e10i	0.422964	−	0.732594i	−0.573264	−	0.819371i	$-0.694323\pi$
$953$	1.13013e10	−	1.95744e10i	0.422964	−	0.732594i	0.996228	+	0.0867762i	$0.0276565\pi$
$954$		0			0
$955$	1.29571e10	−	2.24424e10i	0.481390	−	0.833792i
$956$		0			0
$957$	7.82101e10			2.88450
$958$		0			0
$959$	7.67282e9	+	1.32897e10i	0.280925	+	0.486576i
$960$		0			0
$961$	4.23700e9			0.154002
$962$		0			0
$963$	1.35281e10	+	2.34313e10i	0.488139	+	0.845481i
$964$		0			0
$965$	−1.18479e10	−	2.05211e10i	−0.424419	−	0.735115i
$966$		0			0
$967$	−5.01080e9	+	8.67896e9i	−0.178203	+	0.308656i	−0.941265	−	0.337669i	$-0.890362\pi$
$967$	−5.01080e9	+	8.67896e9i	−0.178203	+	0.308656i	0.763062	+	0.646325i	$0.223695\pi$
$968$		0			0
$969$	−2.67669e10	−	4.69749e10i	−0.945071	−	1.65857i
$970$		0			0
$971$	−8.79847e9	+	1.52394e10i	−0.308418	+	0.534196i	−0.978017	−	0.208527i	$-0.933133\pi$
$971$	−8.79847e9	+	1.52394e10i	−0.308418	+	0.534196i	0.669598	+	0.742723i	$0.266466\pi$
$972$		0			0
$973$	1.62340e10	+	2.81182e10i	0.564978	+	0.978571i
$974$		0			0
$975$	−9.01899e9	−	1.56214e10i	−0.311632	−	0.539762i
$976$		0			0
$977$	4.05117e10			1.38979			0.694896	−	0.719110i	$-0.255450\pi$
$977$	4.05117e10			1.38979			0.694896	+	0.719110i	$0.255450\pi$
$978$		0			0
$979$	3.25058e10	+	5.63017e10i	1.10719	+	1.91771i
$980$		0			0
$981$	−6.96477e10			−2.35540
$982$		0			0
$983$	1.30892e10	−	2.26711e10i	0.439517	−	0.761265i	−0.558136	−	0.829750i	$-0.688483\pi$
$983$	1.30892e10	−	2.26711e10i	0.439517	−	0.761265i	0.997652	+	0.0684847i	$0.0218164\pi$
$984$		0			0
$985$	−7.60019e9	+	1.31639e10i	−0.253395	+	0.438893i
$986$		0			0
$987$	−3.16899e10			−1.04908
$988$		0			0
$989$	−1.34527e10			−0.442203
$990$		0			0
$991$	5.72185e9	−	9.91054e9i	0.186758	−	0.323474i	−0.757410	−	0.652940i	$-0.773535\pi$
$991$	5.72185e9	−	9.91054e9i	0.186758	−	0.323474i	0.944167	+	0.329466i	$0.106869\pi$
$992$		0			0
$993$	−2.80878e10	+	4.86495e10i	−0.910323	+	1.57673i
$994$		0			0
$995$	−4.13999e10			−1.33235
$996$		0			0
$997$	1.18786e10	+	2.05743e10i	0.379605	+	0.657495i	0.991005	−	0.133827i	$-0.0427265\pi$
$997$	1.18786e10	+	2.05743e10i	0.379605	+	0.657495i	−0.611400	+	0.791322i	$0.709393\pi$
$998$		0			0
$999$	−1.86583e10			−0.592098

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.e.a.49.1	yes	22
19.7	even	3	inner	76.8.e.a.45.1	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.e.a.45.1	✓	22	19.7	even	3	inner
76.8.e.a.49.1	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)

\(f(q)\)	\(=\)	\(q+(-40.1712 + 69.5786i) q^{3} +(-97.7376 + 169.287i) q^{5} -587.742 q^{7} +(-2133.95 - 3696.12i) q^{9} +O(q^{10})\)	\(q+(-40.1712 + 69.5786i) q^{3} +(-97.7376 + 169.287i) q^{5} -587.742 q^{7} +(-2133.95 - 3696.12i) q^{9} -6546.65 q^{11} +(2812.44 + 4871.29i) q^{13} +(-7852.48 - 13600.9i) q^{15} +(-11254.1 + 19492.6i) q^{17} +(29897.2 + 169.581i) q^{19} +(23610.3 - 40894.2i) q^{21} +(-42909.8 - 74322.0i) q^{23} +(19957.2 + 34566.9i) q^{25} +167185. q^{27} +(74347.9 + 128774. i) q^{29} +178184. q^{31} +(262987. - 455507. i) q^{33} +(57444.5 - 99496.8i) q^{35} -111603. q^{37} -451916. q^{39} +(112180. - 194301. i) q^{41} +(78377.6 - 135754. i) q^{43} +834271. q^{45} +(-335551. - 581192. i) q^{47} -478103. q^{49} +(-904180. - 1.56609e6i) q^{51} +(718815. + 1.24502e6i) q^{53} +(639854. - 1.10826e6i) q^{55} +(-1.21281e6 + 2.07339e6i) q^{57} +(1.19760e6 - 2.07431e6i) q^{59} +(-11810.8 - 20456.8i) q^{61} +(1.25421e6 + 2.17236e6i) q^{63} -1.09952e6 q^{65} +(212375. + 367843. i) q^{67} +6.89496e6 q^{69} +(635552. - 1.10081e6i) q^{71} +(-3.00652e6 + 5.20745e6i) q^{73} -3.20682e6 q^{75} +3.84774e6 q^{77} +(-1.79747e6 + 3.11331e6i) q^{79} +(-2.04908e6 + 3.54911e6i) q^{81} -8.04455e6 q^{83} +(-2.19989e6 - 3.81033e6i) q^{85} -1.19466e7 q^{87} +(-4.96526e6 - 8.60008e6i) q^{89} +(-1.65299e6 - 2.86306e6i) q^{91} +(-7.15788e6 + 1.23978e7i) q^{93} +(-2.95079e6 + 5.04462e6i) q^{95} +(5.27599e6 - 9.13828e6i) q^{97} +(1.39702e7 + 2.41972e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) \) 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

Representations

Groups

Database

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