LMFDB - Embedded newform 80.9.p.b.33.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,9,Mod(17,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("80.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	80.p (of order $4$, degree $2$, not 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:	$32.5902888049$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{249})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 125x^{2} + 3844 $ x^4 + 125*x^2 + 3844
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			33.1
Root			$8.38987i$ of defining polynomial
Character	$\chi$	$=$	80.33
Dual form			80.9.p.b.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-52.9493 - 52.9493i) q^{3} +(-490.341 + 387.544i) q^{5} +(2624.01 - 2624.01i) q^{7} -953.736i q^{9} -5372.73 q^{11} +(27972.7 + 27972.7i) q^{13} +(46483.4 + 5443.05i) q^{15} +(53444.2 - 53444.2i) q^{17} -144788. i q^{19} -277879. q^{21} +(-52148.7 - 52148.7i) q^{23} +(90244.3 - 380058. i) q^{25} +(-397900. + 397900. i) q^{27} +41723.4i q^{29} -244148. q^{31} +(284482. + 284482. i) q^{33} +(-269741. + 2.30358e6i) q^{35} +(-2.00653e6 + 2.00653e6i) q^{37} -2.96228e6i q^{39} -4.17322e6 q^{41} +(-4.01446e6 - 4.01446e6i) q^{43} +(369615. + 467656. i) q^{45} +(-2.26940e6 + 2.26940e6i) q^{47} -8.00606e6i q^{49} -5.65967e6 q^{51} +(-3.72734e6 - 3.72734e6i) q^{53} +(2.63447e6 - 2.08217e6i) q^{55} +(-7.66643e6 + 7.66643e6i) q^{57} +1.12165e7i q^{59} +2.02002e7 q^{61} +(-2.50261e6 - 2.50261e6i) q^{63} +(-2.45569e7 - 2.87552e6i) q^{65} +(1.30857e7 - 1.30857e7i) q^{67} +5.52248e6i q^{69} -2.61041e7 q^{71} +(-1.85930e7 - 1.85930e7i) q^{73} +(-2.49022e7 + 1.53454e7i) q^{75} +(-1.40981e7 + 1.40981e7i) q^{77} -3.07594e6i q^{79} +3.58796e7 q^{81} +(-2.41522e7 - 2.41522e7i) q^{83} +(-5.49393e6 + 4.69179e7i) q^{85} +(2.20923e6 - 2.20923e6i) q^{87} -5.76618e6i q^{89} +1.46802e8 q^{91} +(1.29275e7 + 1.29275e7i) q^{93} +(5.61117e7 + 7.09955e7i) q^{95} +(2.86891e7 - 2.86891e7i) q^{97} +5.12416e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 54 q^{3} + 90 q^{5} + 1186 q^{7} + 19852 q^{11} + 73704 q^{13} + 137490 q^{15} + 198944 q^{17} - 766572 q^{21} - 631334 q^{23} + 545600 q^{25} - 184680 q^{27} + 2329892 q^{31} + 1362948 q^{33} - 1403870 q^{35}+ \cdots + 139631604 q^{97}+O(q^{100}) $ 4 * q - 54 * q^3 + 90 * q^5 + 1186 * q^7 + 19852 * q^11 + 73704 * q^13 + 137490 * q^15 + 198944 * q^17 - 766572 * q^21 - 631334 * q^23 + 545600 * q^25 - 184680 * q^27 + 2329892 * q^31 + 1362948 * q^33 - 1403870 * q^35 - 976356 * q^37 - 4284652 * q^41 - 7146534 * q^43 - 2624430 * q^45 + 3300186 * q^47 - 6541788 * q^51 + 9989164 * q^53 + 21649020 * q^55 - 19136400 * q^57 + 85377108 * q^61 + 16175226 * q^63 - 33878940 * q^65 + 57195466 * q^67 - 113004028 * q^71 - 60875756 * q^73 - 22407150 * q^75 - 90339932 * q^77 + 35054064 * q^81 - 71051214 * q^83 + 8601320 * q^85 - 58152480 * q^87 + 221467572 * q^91 + 98985108 * q^93 + 159500600 * q^95 + 139631604 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−52.9493	−	52.9493i	−0.653695	−	0.653695i	0.300185	−	0.953881i	$-0.402951\pi$
$3$	−52.9493	−	52.9493i	−0.653695	−	0.653695i	−0.953881	+	0.300185i	$0.902951\pi$
$4$		0			0
$5$	−490.341	+	387.544i	−0.784546	+	0.620070i
$5$	−490.341	+	387.544i	−0.784546	+	0.620070i
$6$		0			0
$7$	2624.01	−	2624.01i	1.09288	−	1.09288i	0.0976629	−	0.995220i	$-0.468863\pi$
$7$	2624.01	−	2624.01i	1.09288	−	1.09288i	0.995220	−	0.0976629i	$-0.0311367\pi$
$8$		0			0
$9$		−	953.736i		−	0.145364i
$10$		0			0
$11$	−5372.73			−0.366964			−0.183482	−	0.983023i	$-0.558737\pi$
$11$	−5372.73			−0.366964			−0.183482	+	0.983023i	$0.558737\pi$
$12$		0			0
$13$	27972.7	+	27972.7i	0.979403	+	0.979403i	0.999792	−	0.0203888i	$-0.00649040\pi$
$13$	27972.7	+	27972.7i	0.979403	+	0.979403i	−0.0203888	+	0.999792i	$0.506490\pi$
$14$		0			0
$15$	46483.4	+	5443.05i	0.918192	+	0.107517i
$16$		0			0
$17$	53444.2	−	53444.2i	0.639890	−	0.639890i	−0.310638	−	0.950528i	$-0.600543\pi$
$17$	53444.2	−	53444.2i	0.639890	−	0.639890i	0.950528	+	0.310638i	$0.100543\pi$
$18$		0			0
$19$		−	144788.i		−	1.11101i	−0.831513	−	0.555505i	$-0.812525\pi$
$19$		−	144788.i		−	1.11101i	0.831513	−	0.555505i	$-0.187475\pi$
$20$		0			0
$21$	−277879.			−1.42882
$22$		0			0
$23$	−52148.7	−	52148.7i	−0.186351	−	0.186351i	0.607765	−	0.794117i	$-0.292066\pi$
$23$	−52148.7	−	52148.7i	−0.186351	−	0.186351i	−0.794117	+	0.607765i	$0.792066\pi$
$24$		0			0
$25$	90244.3	−	380058.i	0.231025	−	0.972948i
$26$		0			0
$27$	−397900.	+	397900.i	−0.748720	+	0.748720i
$28$		0			0
$29$			41723.4i			0.0589913i	0.999565	+	0.0294956i	$0.00939012\pi$
$29$			41723.4i			0.0589913i	−0.999565	+	0.0294956i	$0.990610\pi$
$30$		0			0
$31$	−244148.			−0.264367			−0.132183	−	0.991225i	$-0.542199\pi$
$31$	−244148.			−0.264367			−0.132183	+	0.991225i	$0.542199\pi$
$32$		0			0
$33$	284482.	+	284482.i	0.239883	+	0.239883i
$34$		0			0
$35$	−269741.	+	2.30358e6i	−0.179753	+	1.53508i
$36$		0			0
$37$	−2.00653e6	+	2.00653e6i	−1.07063	+	1.07063i	−0.0733185	+	0.997309i	$0.523359\pi$
$37$	−2.00653e6	+	2.00653e6i	−1.07063	+	1.07063i	−0.997309	+	0.0733185i	$0.976641\pi$
$38$		0			0
$39$		−	2.96228e6i		−	1.28046i
$40$		0			0
$41$	−4.17322e6			−1.47685			−0.738424	−	0.674336i	$-0.764430\pi$
$41$	−4.17322e6			−1.47685			−0.738424	+	0.674336i	$0.764430\pi$
$42$		0			0
$43$	−4.01446e6	−	4.01446e6i	−1.17423	−	1.17423i	−0.981191	−	0.193038i	$-0.938166\pi$
$43$	−4.01446e6	−	4.01446e6i	−1.17423	−	1.17423i	−0.193038	−	0.981191i	$-0.561834\pi$
$44$		0			0
$45$	369615.	+	467656.i	0.0901362	+	0.114045i
$46$		0			0
$47$	−2.26940e6	+	2.26940e6i	−0.465071	+	0.465071i	−0.900313	−	0.435242i	$-0.856663\pi$
$47$	−2.26940e6	+	2.26940e6i	−0.465071	+	0.465071i	0.435242	+	0.900313i	$0.356663\pi$
$48$		0			0
$49$		−	8.00606e6i		−	1.38878i
$50$		0			0
$51$	−5.65967e6			−0.836586
$52$		0			0
$53$	−3.72734e6	−	3.72734e6i	−0.472384	−	0.472384i	0.430301	−	0.902685i	$-0.358407\pi$
$53$	−3.72734e6	−	3.72734e6i	−0.472384	−	0.472384i	−0.902685	+	0.430301i	$0.858407\pi$
$54$		0			0
$55$	2.63447e6	−	2.08217e6i	0.287901	−	0.227544i
$56$		0			0
$57$	−7.66643e6	+	7.66643e6i	−0.726262	+	0.726262i
$58$		0			0
$59$			1.12165e7i			0.925653i	0.886449	+	0.462826i	$0.153165\pi$
$59$			1.12165e7i			0.925653i	−0.886449	+	0.462826i	$0.846835\pi$
$60$		0			0
$61$	2.02002e7			1.45893			0.729467	−	0.684016i	$-0.239768\pi$
$61$	2.02002e7			1.45893			0.729467	+	0.684016i	$0.239768\pi$
$62$		0			0
$63$	−2.50261e6	−	2.50261e6i	−0.158866	−	0.158866i
$64$		0			0
$65$	−2.45569e7	−	2.87552e6i	−1.37569	−	0.161088i
$66$		0			0
$67$	1.30857e7	−	1.30857e7i	0.649377	−	0.649377i	−0.303465	−	0.952843i	$-0.598144\pi$
$67$	1.30857e7	−	1.30857e7i	0.649377	−	0.649377i	0.952843	+	0.303465i	$0.0981436\pi$
$68$		0			0
$69$			5.52248e6i			0.243634i
$70$		0			0
$71$	−2.61041e7			−1.02725			−0.513624	−	0.858015i	$-0.671697\pi$
$71$	−2.61041e7			−1.02725			−0.513624	+	0.858015i	$0.671697\pi$
$72$		0			0
$73$	−1.85930e7	−	1.85930e7i	−0.654722	−	0.654722i	0.299404	−	0.954126i	$-0.403212\pi$
$73$	−1.85930e7	−	1.85930e7i	−0.654722	−	0.654722i	−0.954126	+	0.299404i	$0.903212\pi$
$74$		0			0
$75$	−2.49022e7	+	1.53454e7i	−0.787032	+	0.484991i
$76$		0			0
$77$	−1.40981e7	+	1.40981e7i	−0.401049	+	0.401049i
$78$		0			0
$79$		−	3.07594e6i		−	0.0789714i	−0.999220	−	0.0394857i	$-0.987428\pi$
$79$		−	3.07594e6i		−	0.0789714i	0.999220	−	0.0394857i	$-0.0125720\pi$
$80$		0			0
$81$	3.58796e7			0.833505
$82$		0			0
$83$	−2.41522e7	−	2.41522e7i	−0.508914	−	0.508914i	0.405279	−	0.914193i	$-0.367174\pi$
$83$	−2.41522e7	−	2.41522e7i	−0.508914	−	0.508914i	−0.914193	+	0.405279i	$0.867174\pi$
$84$		0			0
$85$	−5.49393e6	+	4.69179e7i	−0.105246	+	0.898800i
$86$		0			0
$87$	2.20923e6	−	2.20923e6i	0.0385623	−	0.0385623i
$88$		0			0
$89$		−	5.76618e6i		−	0.0919027i	−0.998944	−	0.0459513i	$-0.985368\pi$
$89$		−	5.76618e6i		−	0.0919027i	0.998944	−	0.0459513i	$-0.0146319\pi$
$90$		0			0
$91$	1.46802e8			2.14075
$92$		0			0
$93$	1.29275e7	+	1.29275e7i	0.172815	+	0.172815i
$94$		0			0
$95$	5.61117e7	+	7.09955e7i	0.688905	+	0.871639i
$96$		0			0
$97$	2.86891e7	−	2.86891e7i	0.324063	−	0.324063i	−0.526260	−	0.850324i	$-0.676406\pi$
$97$	2.86891e7	−	2.86891e7i	0.324063	−	0.324063i	0.850324	+	0.526260i	$0.176406\pi$
$98$		0			0
$99$			5.12416e6i			0.0533436i
$100$		0			0
$101$	7.95260e7			0.764229			0.382115	−	0.924115i	$-0.375196\pi$
$101$	7.95260e7			0.764229			0.382115	+	0.924115i	$0.375196\pi$
$102$		0			0
$103$	−4.96462e7	−	4.96462e7i	−0.441100	−	0.441100i	0.451282	−	0.892382i	$-0.350967\pi$
$103$	−4.96462e7	−	4.96462e7i	−0.441100	−	0.441100i	−0.892382	+	0.451282i	$0.850967\pi$
$104$		0			0
$105$	1.36256e8	−	1.07690e8i	1.12098	−	0.885972i
$106$		0			0
$107$	−6.12534e7	+	6.12534e7i	−0.467300	+	0.467300i	−0.901039	−	0.433739i	$-0.857194\pi$
$107$	−6.12534e7	+	6.12534e7i	−0.467300	+	0.467300i	0.433739	+	0.901039i	$0.357194\pi$
$108$		0			0
$109$		−	9.33557e7i		−	0.661355i	−0.943744	−	0.330678i	$-0.892723\pi$
$109$		−	9.33557e7i		−	0.661355i	0.943744	−	0.330678i	$-0.107277\pi$
$110$		0			0
$111$	2.12489e8			1.39973
$112$		0			0
$113$	1.17246e8	+	1.17246e8i	0.719093	+	0.719093i	0.968419	−	0.249327i	$-0.0802093\pi$
$113$	1.17246e8	+	1.17246e8i	0.719093	+	0.719093i	−0.249327	+	0.968419i	$0.580209\pi$
$114$		0			0
$115$	4.57806e7	+	5.36075e6i	0.261752	+	0.0306503i
$116$		0			0
$117$	2.66786e7	−	2.66786e7i	0.142370	−	0.142370i
$118$		0			0
$119$		−	2.80477e8i		−	1.39865i
$120$		0			0
$121$	−1.85493e8			−0.865337
$122$		0			0
$123$	2.20969e8	+	2.20969e8i	0.965409	+	0.965409i
$124$		0			0
$125$	1.03039e8	+	2.21332e8i	0.422046	+	0.906574i
$126$		0			0
$127$	−1.16482e7	+	1.16482e7i	−0.0447759	+	0.0447759i	−0.729140	−	0.684364i	$-0.760080\pi$
$127$	−1.16482e7	+	1.16482e7i	−0.0447759	+	0.0447759i	0.684364	+	0.729140i	$0.260080\pi$
$128$		0			0
$129$			4.25126e8i			1.53518i
$130$		0			0
$131$	−3.83537e8			−1.30233			−0.651167	−	0.758934i	$-0.725720\pi$
$131$	−3.83537e8			−1.30233			−0.651167	+	0.758934i	$0.725720\pi$
$132$		0			0
$133$	−3.79925e8	−	3.79925e8i	−1.21420	−	1.21420i
$134$		0			0
$135$	4.09031e7	−	3.49311e8i	0.123146	−	1.05166i
$136$		0			0
$137$	−1.46786e7	+	1.46786e7i	−0.0416681	+	0.0416681i	−0.727634	−	0.685966i	$-0.759380\pi$
$137$	−1.46786e7	+	1.46786e7i	−0.0416681	+	0.0416681i	0.685966	+	0.727634i	$0.259380\pi$
$138$		0			0
$139$			4.00314e8i			1.07236i	0.844103	+	0.536181i	$0.180134\pi$
$139$			4.00314e8i			1.07236i	−0.844103	+	0.536181i	$0.819866\pi$
$140$		0			0
$141$	2.40326e8			0.608030
$142$		0			0
$143$	−1.50290e8	−	1.50290e8i	−0.359406	−	0.359406i
$144$		0			0
$145$	−1.61697e7	−	2.04587e7i	−0.0365788	−	0.0462814i
$146$		0			0
$147$	−4.23916e8	+	4.23916e8i	−0.907842	+	0.907842i
$148$		0			0
$149$		−	5.62457e8i		−	1.14115i	−0.821244	−	0.570577i	$-0.806719\pi$
$149$		−	5.62457e8i		−	1.14115i	0.821244	−	0.570577i	$-0.193281\pi$
$150$		0			0
$151$	2.75254e7			0.0529452			0.0264726	−	0.999650i	$-0.491573\pi$
$151$	2.75254e7			0.0529452			0.0264726	+	0.999650i	$0.491573\pi$
$152$		0			0
$153$	−5.09717e7	−	5.09717e7i	−0.0930172	−	0.0930172i
$154$		0			0
$155$	1.19716e8	−	9.46182e7i	0.207408	−	0.163926i
$156$		0			0
$157$	−3.22628e8	+	3.22628e8i	−0.531010	+	0.531010i	−0.920873	−	0.389863i	$-0.872522\pi$
$157$	−3.22628e8	+	3.22628e8i	−0.531010	+	0.531010i	0.389863	+	0.920873i	$0.372522\pi$
$158$		0			0
$159$			3.94720e8i			0.617591i
$160$		0			0
$161$	−2.73678e8			−0.407320
$162$		0			0
$163$	−2.36679e8	−	2.36679e8i	−0.335281	−	0.335281i	0.519307	−	0.854588i	$-0.326190\pi$
$163$	−2.36679e8	−	2.36679e8i	−0.335281	−	0.335281i	−0.854588	+	0.519307i	$0.826190\pi$
$164$		0			0
$165$	−2.49743e8	−	2.92440e7i	−0.336944	−	0.0394549i
$166$		0			0
$167$	5.45436e8	−	5.45436e8i	0.701258	−	0.701258i	−0.263422	−	0.964681i	$-0.584851\pi$
$167$	5.45436e8	−	5.45436e8i	0.701258	−	0.701258i	0.964681	+	0.263422i	$0.0848512\pi$
$168$		0			0
$169$			7.49218e8i			0.918462i
$170$		0			0
$171$	−1.38089e8			−0.161501
$172$		0			0
$173$	−6.41655e8	−	6.41655e8i	−0.716337	−	0.716337i	0.251516	−	0.967853i	$-0.419071\pi$
$173$	−6.41655e8	−	6.41655e8i	−0.716337	−	0.716337i	−0.967853	+	0.251516i	$0.919071\pi$
$174$		0			0
$175$	−7.60474e8	−	1.23408e9i	−0.810834	−	1.31580i
$176$		0			0
$177$	5.93905e8	−	5.93905e8i	0.605095	−	0.605095i
$178$		0			0
$179$			9.41081e8i			0.916673i	0.888779	+	0.458337i	$0.151555\pi$
$179$			9.41081e8i			0.916673i	−0.888779	+	0.458337i	$0.848445\pi$
$180$		0			0
$181$	−4.09543e8			−0.381580			−0.190790	−	0.981631i	$-0.561105\pi$
$181$	−4.09543e8			−0.381580			−0.190790	+	0.981631i	$0.561105\pi$
$182$		0			0
$183$	−1.06959e9	−	1.06959e9i	−0.953699	−	0.953699i
$184$		0			0
$185$	2.06266e8	−	1.76150e9i	0.176092	−	1.50382i
$186$		0			0
$187$	−2.87141e8	+	2.87141e8i	−0.234817	+	0.234817i
$188$		0			0
$189$			2.08819e9i			1.63652i
$190$		0			0
$191$	−5.56054e8			−0.417814			−0.208907	−	0.977935i	$-0.566991\pi$
$191$	−5.56054e8			−0.417814			−0.208907	+	0.977935i	$0.566991\pi$
$192$		0			0
$193$	1.72776e8	+	1.72776e8i	0.124524	+	0.124524i	0.766622	−	0.642098i	$-0.221936\pi$
$193$	1.72776e8	+	1.72776e8i	0.124524	+	0.124524i	−0.642098	+	0.766622i	$0.721936\pi$
$194$		0			0
$195$	1.14801e9	+	1.45253e9i	0.793977	+	1.00458i
$196$		0			0
$197$	−1.39026e9	+	1.39026e9i	−0.923060	+	0.923060i	−0.997245	−	0.0741846i	$-0.976365\pi$
$197$	−1.39026e9	+	1.39026e9i	−0.923060	+	0.923060i	0.0741846	+	0.997245i	$0.476365\pi$
$198$		0			0
$199$		−	1.80023e8i		−	0.114793i	−0.998351	−	0.0573964i	$-0.981720\pi$
$199$		−	1.80023e8i		−	0.114793i	0.998351	−	0.0573964i	$-0.0182799\pi$
$200$		0			0
$201$	−1.38576e9			−0.848990
$202$		0			0
$203$	1.09483e8	+	1.09483e8i	0.0644705	+	0.0644705i
$204$		0			0
$205$	2.04630e9	−	1.61731e9i	1.15866	−	0.915750i
$206$		0			0
$207$	−4.97361e7	+	4.97361e7i	−0.0270888	+	0.0270888i
$208$		0			0
$209$			7.77906e8i			0.407701i
$210$		0			0
$211$	4.17651e8			0.210709			0.105355	−	0.994435i	$-0.466402\pi$
$211$	4.17651e8			0.210709			0.105355	+	0.994435i	$0.466402\pi$
$212$		0			0
$213$	1.38219e9	+	1.38219e9i	0.671507	+	0.671507i
$214$		0			0
$215$	3.52423e9	+	4.12675e8i	1.64934	+	0.193132i
$216$		0			0
$217$	−6.40648e8	+	6.40648e8i	−0.288922	+	0.288922i
$218$		0			0
$219$			1.96897e9i			0.855978i
$220$		0			0
$221$	2.98996e9			1.25342
$222$		0			0
$223$	−1.19002e9	−	1.19002e9i	−0.481210	−	0.481210i	0.424308	−	0.905518i	$-0.360517\pi$
$223$	−1.19002e9	−	1.19002e9i	−0.481210	−	0.481210i	−0.905518	+	0.424308i	$0.860517\pi$
$224$		0			0
$225$	−3.62475e8	−	8.60692e7i	−0.141432	−	0.0335829i
$226$		0			0
$227$	−2.69749e9	+	2.69749e9i	−1.01591	+	1.01591i	−0.0160418	+	0.999871i	$0.505106\pi$
$227$	−2.69749e9	+	2.69749e9i	−1.01591	+	1.01591i	−0.999871	+	0.0160418i	$0.994894\pi$
$228$		0			0
$229$		−	4.90322e9i		−	1.78295i	−0.453069	−	0.891475i	$-0.649671\pi$
$229$		−	4.90322e9i		−	1.78295i	0.453069	−	0.891475i	$-0.350329\pi$
$230$		0			0
$231$	1.49297e9			0.524328
$232$		0			0
$233$	−2.60187e9	−	2.60187e9i	−0.882798	−	0.882798i	0.111020	−	0.993818i	$-0.464588\pi$
$233$	−2.60187e9	−	2.60187e9i	−0.882798	−	0.882798i	−0.993818	+	0.111020i	$0.964588\pi$
$234$		0			0
$235$	2.33288e8	−	1.99227e9i	0.0764929	−	0.653247i
$236$		0			0
$237$	−1.62869e8	+	1.62869e8i	−0.0516232	+	0.0516232i
$238$		0			0
$239$		−	4.57968e9i		−	1.40360i	−0.712374	−	0.701800i	$-0.752380\pi$
$239$		−	4.57968e9i		−	1.40360i	0.712374	−	0.701800i	$-0.247620\pi$
$240$		0			0
$241$	5.82501e9			1.72675			0.863373	−	0.504566i	$-0.168348\pi$
$241$	5.82501e9			1.72675			0.863373	+	0.504566i	$0.168348\pi$
$242$		0			0
$243$	7.10820e8	+	7.10820e8i	0.203861	+	0.203861i
$244$		0			0
$245$	3.10270e9	+	3.92570e9i	0.861144	+	1.08957i
$246$		0			0
$247$	4.05012e9	−	4.05012e9i	1.08813	−	1.08813i
$248$		0			0
$249$			2.55769e9i			0.665350i
$250$		0			0
$251$	4.01845e9			1.01243			0.506214	−	0.862408i	$-0.331045\pi$
$251$	4.01845e9			1.01243			0.506214	+	0.862408i	$0.331045\pi$
$252$		0			0
$253$	2.80181e8	+	2.80181e8i	0.0683843	+	0.0683843i
$254$		0			0
$255$	2.77517e9	−	2.19337e9i	0.656340	−	0.518742i
$256$		0			0
$257$	−1.77795e9	+	1.77795e9i	−0.407556	+	0.407556i	−0.880885	−	0.473330i	$-0.843052\pi$
$257$	−1.77795e9	+	1.77795e9i	−0.407556	+	0.407556i	0.473330	+	0.880885i	$0.343052\pi$
$258$		0			0
$259$			1.05303e10i			2.34014i
$260$		0			0
$261$	3.97931e7			0.00857523
$262$		0			0
$263$	−1.87568e9	−	1.87568e9i	−0.392046	−	0.392046i	0.483370	−	0.875416i	$-0.339412\pi$
$263$	−1.87568e9	−	1.87568e9i	−0.392046	−	0.392046i	−0.875416	+	0.483370i	$0.839412\pi$
$264$		0			0
$265$	3.27218e9	+	3.83161e8i	0.663519	+	0.0776958i
$266$		0			0
$267$	−3.05315e8	+	3.05315e8i	−0.0600764	+	0.0600764i
$268$		0			0
$269$			5.91528e9i			1.12971i	0.825191	+	0.564854i	$0.191068\pi$
$269$			5.91528e9i			1.12971i	−0.825191	+	0.564854i	$0.808932\pi$
$270$		0			0
$271$	1.60432e9			0.297449			0.148725	−	0.988879i	$-0.452483\pi$
$271$	1.60432e9			0.297449			0.148725	+	0.988879i	$0.452483\pi$
$272$		0			0
$273$	−7.77304e9	−	7.77304e9i	−1.39940	−	1.39940i
$274$		0			0
$275$	−4.84858e8	+	2.04195e9i	−0.0847781	+	0.357037i
$276$		0			0
$277$	−1.67266e9	+	1.67266e9i	−0.284111	+	0.284111i	−0.834746	−	0.550635i	$-0.814386\pi$
$277$	−1.67266e9	+	1.67266e9i	−0.284111	+	0.284111i	0.550635	+	0.834746i	$0.314386\pi$
$278$		0			0
$279$			2.32853e8i			0.0384295i
$280$		0			0
$281$	9.29346e9			1.49057			0.745285	−	0.666747i	$-0.232314\pi$
$281$	9.29346e9			1.49057			0.745285	+	0.666747i	$0.232314\pi$
$282$		0			0
$283$	−4.61061e9	−	4.61061e9i	−0.718808	−	0.718808i	0.249553	−	0.968361i	$-0.419716\pi$
$283$	−4.61061e9	−	4.61061e9i	−0.718808	−	0.718808i	−0.968361	+	0.249553i	$0.919716\pi$
$284$		0			0
$285$	7.88088e8	−	6.73024e9i	0.119453	−	1.02012i
$286$		0			0
$287$	−1.09506e10	+	1.09506e10i	−1.61402	+	1.61402i
$288$		0			0
$289$			1.26318e9i			0.181082i
$290$		0			0
$291$	−3.03814e9			−0.423678
$292$		0			0
$293$	−6.99292e9	−	6.99292e9i	−0.948829	−	0.948829i	0.0499241	−	0.998753i	$-0.484102\pi$
$293$	−6.99292e9	−	6.99292e9i	−0.948829	−	0.948829i	−0.998753	+	0.0499241i	$0.984102\pi$
$294$		0			0
$295$	−4.34688e9	−	5.49990e9i	−0.573970	−	0.726217i
$296$		0			0
$297$	2.13781e9	−	2.13781e9i	0.274753	−	0.274753i
$298$		0			0
$299$		−	2.91749e9i		−	0.365026i
$300$		0			0
$301$	−2.10680e10			−2.56659
$302$		0			0
$303$	−4.21085e9	−	4.21085e9i	−0.499573	−	0.499573i
$304$		0			0
$305$	−9.90498e9	+	7.82845e9i	−1.14460	+	0.904642i
$306$		0			0
$307$	−2.31844e9	+	2.31844e9i	−0.261001	+	0.261001i	−0.825461	−	0.564459i	$-0.809085\pi$
$307$	−2.31844e9	+	2.31844e9i	−0.261001	+	0.261001i	0.564459	+	0.825461i	$0.309085\pi$
$308$		0			0
$309$			5.25746e9i			0.576690i
$310$		0			0
$311$	1.45111e10			1.55117			0.775584	−	0.631245i	$-0.217456\pi$
$311$	1.45111e10			1.55117			0.775584	+	0.631245i	$0.217456\pi$
$312$		0			0
$313$	2.03082e9	+	2.03082e9i	0.211590	+	0.211590i	0.804943	−	0.593353i	$-0.202196\pi$
$313$	2.03082e9	+	2.03082e9i	0.211590	+	0.211590i	−0.593353	+	0.804943i	$0.702196\pi$
$314$		0			0
$315$	2.19701e9	+	2.57262e8i	0.223146	+	0.0261296i
$316$		0			0
$317$	9.69018e8	−	9.69018e8i	0.0959611	−	0.0959611i	−0.657497	−	0.753458i	$-0.728385\pi$
$317$	9.69018e8	−	9.69018e8i	0.0959611	−	0.0959611i	0.753458	+	0.657497i	$0.228385\pi$
$318$		0			0
$319$		−	2.24168e8i		−	0.0216477i
$320$		0			0
$321$	6.48666e9			0.610943
$322$		0			0
$323$	−7.73808e9	−	7.73808e9i	−0.710924	−	0.710924i
$324$		0			0
$325$	1.31556e10	−	8.10688e9i	1.17918	−	0.726641i
$326$		0			0
$327$	−4.94312e9	+	4.94312e9i	−0.432325	+	0.432325i
$328$		0			0
$329$			1.19099e10i			1.01654i
$330$		0			0
$331$	7.63072e9			0.635702			0.317851	−	0.948141i	$-0.397039\pi$
$331$	7.63072e9			0.635702			0.317851	+	0.948141i	$0.397039\pi$
$332$		0			0
$333$	1.91370e9	+	1.91370e9i	0.155631	+	0.155631i
$334$		0			0
$335$	−1.34517e9	+	1.14877e10i	−0.106807	+	0.912126i
$336$		0			0
$337$	−2.47412e9	+	2.47412e9i	−0.191824	+	0.191824i	−0.796484	−	0.604660i	$-0.793309\pi$
$337$	−2.47412e9	+	2.47412e9i	−0.191824	+	0.191824i	0.604660	+	0.796484i	$0.293309\pi$
$338$		0			0
$339$		−	1.24162e10i		−	0.940135i
$340$		0			0
$341$	1.31174e9			0.0970132
$342$		0			0
$343$	−5.88110e9	−	5.88110e9i	−0.424895	−	0.424895i
$344$		0			0
$345$	−2.14020e9	−	2.70790e9i	−0.151070	−	0.191142i
$346$		0			0
$347$	2.63557e9	−	2.63557e9i	0.181785	−	0.181785i	−0.610348	−	0.792133i	$-0.708971\pi$
$347$	2.63557e9	−	2.63557e9i	0.181785	−	0.181785i	0.792133	+	0.610348i	$0.208971\pi$
$348$		0			0
$349$		−	2.46965e10i		−	1.66469i	−0.554257	−	0.832346i	$-0.686998\pi$
$349$		−	2.46965e10i		−	1.66469i	0.554257	−	0.832346i	$-0.313002\pi$
$350$		0			0
$351$	−2.22607e10			−1.46660
$352$		0			0
$353$	−1.25084e10	−	1.25084e10i	−0.805566	−	0.805566i	0.178393	−	0.983959i	$-0.442910\pi$
$353$	−1.25084e10	−	1.25084e10i	−0.805566	−	0.805566i	−0.983959	+	0.178393i	$0.942910\pi$
$354$		0			0
$355$	1.27999e10	−	1.01165e10i	0.805923	−	0.636966i
$356$		0			0
$357$	−1.48510e10	+	1.48510e10i	−0.914290	+	0.914290i
$358$		0			0
$359$			2.30279e10i			1.38636i	0.720763	+	0.693181i	$0.243792\pi$
$359$			2.30279e10i			1.38636i	−0.720763	+	0.693181i	$0.756208\pi$
$360$		0			0
$361$	−3.97999e9			−0.234344
$362$		0			0
$363$	9.82172e9	+	9.82172e9i	0.565667	+	0.565667i
$364$		0			0
$365$	1.63225e10	+	1.91131e9i	0.919634	+	0.107686i
$366$		0			0
$367$	9.47400e9	−	9.47400e9i	0.522239	−	0.522239i	−0.396008	−	0.918247i	$-0.629605\pi$
$367$	9.47400e9	−	9.47400e9i	0.522239	−	0.522239i	0.918247	+	0.396008i	$0.129605\pi$
$368$		0			0
$369$			3.98015e9i			0.214681i
$370$		0			0
$371$	−1.95612e10			−1.03252
$372$		0			0
$373$	6.05089e9	+	6.05089e9i	0.312597	+	0.312597i	0.845915	−	0.533318i	$-0.179055\pi$
$373$	6.05089e9	+	6.05089e9i	0.312597	+	0.312597i	−0.533318	+	0.845915i	$0.679055\pi$
$374$		0			0
$375$	6.26354e9	−	1.71752e10i	0.316734	−	0.868513i
$376$		0			0
$377$	−1.16712e9	+	1.16712e9i	−0.0577763	+	0.0577763i
$378$		0			0
$379$		−	4.27189e9i		−	0.207044i	−0.994627	−	0.103522i	$-0.966989\pi$
$379$		−	4.27189e9i		−	0.207044i	0.994627	−	0.103522i	$-0.0330113\pi$
$380$		0			0
$381$	1.23353e9			0.0585397
$382$		0			0
$383$	2.68406e10	+	2.68406e10i	1.24738	+	1.24738i	0.956874	+	0.290502i	$0.0938224\pi$
$383$	2.68406e10	+	2.68406e10i	1.24738	+	1.24738i	0.290502	+	0.956874i	$0.406178\pi$
$384$		0			0
$385$	1.44925e9	−	1.23765e10i	0.0659628	−	0.563320i
$386$		0			0
$387$	−3.82873e9	+	3.82873e9i	−0.170691	+	0.170691i
$388$		0			0
$389$			9.16592e8i			0.0400293i	0.999800	+	0.0200146i	$0.00637128\pi$
$389$			9.16592e8i			0.0400293i	−0.999800	+	0.0200146i	$0.993629\pi$
$390$		0			0
$391$	−5.57410e9			−0.238489
$392$		0			0
$393$	2.03080e10	+	2.03080e10i	0.851330	+	0.851330i
$394$		0			0
$395$	1.19206e9	+	1.50826e9i	0.0489678	+	0.0619567i
$396$		0			0
$397$	5.24090e9	−	5.24090e9i	0.210981	−	0.210981i	−0.593703	−	0.804684i	$-0.702335\pi$
$397$	5.24090e9	−	5.24090e9i	0.210981	−	0.210981i	0.804684	+	0.593703i	$0.202335\pi$
$398$		0			0
$399$			4.02336e10i			1.58744i
$400$		0			0
$401$	1.76862e10			0.684000			0.342000	−	0.939700i	$-0.388896\pi$
$401$	1.76862e10			0.684000			0.342000	+	0.939700i	$0.388896\pi$
$402$		0			0
$403$	−6.82950e9	−	6.82950e9i	−0.258922	−	0.258922i
$404$		0			0
$405$	−1.75933e10	+	1.39049e10i	−0.653923	+	0.516832i
$406$		0			0
$407$	1.07805e10	−	1.07805e10i	0.392882	−	0.392882i
$408$		0			0
$409$			2.18253e9i			0.0779950i	0.999239	+	0.0389975i	$0.0124164\pi$
$409$			2.18253e9i			0.0779950i	−0.999239	+	0.0389975i	$0.987584\pi$
$410$		0			0
$411$	1.55445e9			0.0544765
$412$		0			0
$413$	2.94321e10	+	2.94321e10i	1.01163	+	1.01163i
$414$		0			0
$415$	2.12029e10	+	2.48278e9i	0.714829	+	0.0837040i
$416$		0			0
$417$	2.11964e10	−	2.11964e10i	0.700998	−	0.700998i
$418$		0			0
$419$		−	3.02443e10i		−	0.981268i	−0.871366	−	0.490634i	$-0.836765\pi$
$419$		−	3.02443e10i		−	0.981268i	0.871366	−	0.490634i	$-0.163235\pi$
$420$		0			0
$421$	1.77840e10			0.566109			0.283055	−	0.959104i	$-0.408652\pi$
$421$	1.77840e10			0.566109			0.283055	+	0.959104i	$0.408652\pi$
$422$		0			0
$423$	2.16441e9	+	2.16441e9i	0.0676048	+	0.0676048i
$424$		0			0
$425$	−1.54889e10	−	2.51349e10i	−0.474749	−	0.770410i
$426$		0			0
$427$	5.30055e10	−	5.30055e10i	1.59444	−	1.59444i
$428$		0			0
$429$			1.59155e10i			0.469884i
$430$		0			0
$431$	2.12717e10			0.616443			0.308221	−	0.951315i	$-0.400266\pi$
$431$	2.12717e10			0.616443			0.308221	+	0.951315i	$0.400266\pi$
$432$		0			0
$433$	−2.24808e10	−	2.24808e10i	−0.639529	−	0.639529i	0.310910	−	0.950439i	$-0.399366\pi$
$433$	−2.24808e10	−	2.24808e10i	−0.639529	−	0.639529i	−0.950439	+	0.310910i	$0.899366\pi$
$434$		0			0
$435$	−2.27103e8	+	1.93945e9i	−0.00634257	+	0.0541653i
$436$		0			0
$437$	−7.55051e9	+	7.55051e9i	−0.207038	+	0.207038i
$438$		0			0
$439$		−	1.46904e10i		−	0.395526i	−0.980250	−	0.197763i	$-0.936632\pi$
$439$		−	1.46904e10i		−	0.395526i	0.980250	−	0.197763i	$-0.0633676\pi$
$440$		0			0
$441$	−7.63567e9			−0.201880
$442$		0			0
$443$	−1.79953e10	−	1.79953e10i	−0.467245	−	0.467245i	0.433776	−	0.901021i	$-0.357181\pi$
$443$	−1.79953e10	−	1.79953e10i	−0.467245	−	0.467245i	−0.901021	+	0.433776i	$0.857181\pi$
$444$		0			0
$445$	2.23465e9	+	2.82740e9i	0.0569861	+	0.0721019i
$446$		0			0
$447$	−2.97817e10	+	2.97817e10i	−0.745968	+	0.745968i
$448$		0			0
$449$			2.81226e10i			0.691943i	0.938245	+	0.345972i	$0.112451\pi$
$449$			2.81226e10i			0.691943i	−0.938245	+	0.345972i	$0.887549\pi$
$450$		0			0
$451$	2.24216e10			0.541951
$452$		0			0
$453$	−1.45745e9	−	1.45745e9i	−0.0346100	−	0.0346100i
$454$		0			0
$455$	−7.19829e10	+	5.68921e10i	−1.67951	+	1.32741i
$456$		0			0
$457$	5.32217e10	−	5.32217e10i	1.22018	−	1.22018i	0.252614	−	0.967567i	$-0.418710\pi$
$457$	5.32217e10	−	5.32217e10i	1.22018	−	1.22018i	0.967567	−	0.252614i	$-0.0812901\pi$
$458$		0			0
$459$			4.25310e10i			0.958196i
$460$		0			0
$461$	2.80695e9			0.0621485			0.0310743	−	0.999517i	$-0.490107\pi$
$461$	2.80695e9			0.0621485			0.0310743	+	0.999517i	$0.490107\pi$
$462$		0			0
$463$	1.84228e10	+	1.84228e10i	0.400896	+	0.400896i	0.878549	−	0.477653i	$-0.158512\pi$
$463$	1.84228e10	+	1.84228e10i	0.400896	+	0.400896i	−0.477653	+	0.878549i	$0.658512\pi$
$464$		0			0
$465$	−1.13489e10	−	1.32891e9i	−0.242739	−	0.0284239i
$466$		0			0
$467$	1.19286e10	−	1.19286e10i	0.250796	−	0.250796i	−0.570501	−	0.821297i	$-0.693251\pi$
$467$	1.19286e10	−	1.19286e10i	0.250796	−	0.250796i	0.821297	+	0.570501i	$0.193251\pi$
$468$		0			0
$469$		−	6.86739e10i		−	1.41939i
$470$		0			0
$471$	3.41658e10			0.694238
$472$		0			0
$473$	2.15686e10	+	2.15686e10i	0.430900	+	0.430900i
$474$		0			0
$475$	−5.50278e10	−	1.30663e10i	−1.08095	−	0.256672i
$476$		0			0
$477$	−3.55490e9	+	3.55490e9i	−0.0686679	+	0.0686679i
$478$		0			0
$479$		−	8.08280e10i		−	1.53539i	−0.640813	−	0.767697i	$-0.721403\pi$
$479$		−	8.08280e10i		−	1.53539i	0.640813	−	0.767697i	$-0.278597\pi$
$480$		0			0
$481$	−1.12256e11			−2.09715
$482$		0			0
$483$	1.44911e10	+	1.44911e10i	0.266263	+	0.266263i
$484$		0			0
$485$	−2.94916e9	+	2.51857e10i	−0.0533006	+	0.455185i
$486$		0			0
$487$	−7.93140e10	+	7.93140e10i	−1.41005	+	1.41005i	−0.650797	+	0.759252i	$0.725565\pi$
$487$	−7.93140e10	+	7.93140e10i	−1.41005	+	1.41005i	−0.759252	+	0.650797i	$0.774435\pi$
$488$		0			0
$489$			2.50640e10i			0.438343i
$490$		0			0
$491$	−6.00911e10			−1.03391			−0.516957	−	0.856011i	$-0.672935\pi$
$491$	−6.00911e10			−1.03391			−0.516957	+	0.856011i	$0.672935\pi$
$492$		0			0
$493$	2.22988e9	+	2.22988e9i	0.0377479	+	0.0377479i
$494$		0			0
$495$	−1.98584e9	−	2.51259e9i	−0.0330768	−	0.0418505i
$496$		0			0
$497$	−6.84974e10	+	6.84974e10i	−1.12266	+	1.12266i
$498$		0			0
$499$		−	1.08749e11i		−	1.75397i	−0.480516	−	0.876986i	$-0.659551\pi$
$499$		−	1.08749e11i		−	1.75397i	0.480516	−	0.876986i	$-0.340449\pi$
$500$		0			0
$501$	−5.77610e10			−0.916819
$502$		0			0
$503$	4.36650e10	+	4.36650e10i	0.682122	+	0.682122i	0.960478	−	0.278356i	$-0.0897896\pi$
$503$	4.36650e10	+	4.36650e10i	0.682122	+	0.682122i	−0.278356	+	0.960478i	$0.589790\pi$
$504$		0			0
$505$	−3.89949e10	+	3.08198e10i	−0.599573	+	0.473876i
$506$		0			0
$507$	3.96706e10	−	3.96706e10i	0.600394	−	0.600394i
$508$		0			0
$509$		−	6.95461e10i		−	1.03610i	−0.855350	−	0.518050i	$-0.826658\pi$
$509$		−	6.95461e10i		−	1.03610i	0.855350	−	0.518050i	$-0.173342\pi$
$510$		0			0
$511$	−9.75763e10			−1.43107
$512$		0			0
$513$	5.76112e10	+	5.76112e10i	0.831835	+	0.831835i
$514$		0			0
$515$	4.35836e10	+	5.10349e9i	0.619576	+	0.0725502i
$516$		0			0
$517$	1.21929e10	−	1.21929e10i	0.170665	−	0.170665i
$518$		0			0
$519$			6.79504e10i			0.936533i
$520$		0			0
$521$	1.18226e10			0.160459			0.0802294	−	0.996776i	$-0.474435\pi$
$521$	1.18226e10			0.160459			0.0802294	+	0.996776i	$0.474435\pi$
$522$		0			0
$523$	−5.44052e10	−	5.44052e10i	−0.727166	−	0.727166i	0.242888	−	0.970054i	$-0.421905\pi$
$523$	−5.44052e10	−	5.44052e10i	−0.727166	−	0.727166i	−0.970054	+	0.242888i	$0.921905\pi$
$524$		0			0
$525$	−2.50770e10	+	1.05610e11i	−0.330095	+	1.39017i
$526$		0			0
$527$	−1.30483e10	+	1.30483e10i	−0.169166	+	0.169166i
$528$		0			0
$529$		−	7.28720e10i		−	0.930546i
$530$		0			0
$531$	1.06975e10			0.134557
$532$		0			0
$533$	−1.16736e11	−	1.16736e11i	−1.44643	−	1.44643i
$534$		0			0
$535$	6.29669e9	−	5.37735e10i	0.0768595	−	0.656377i
$536$		0			0
$537$	4.98296e10	−	4.98296e10i	0.599225	−	0.599225i
$538$		0			0
$539$			4.30144e10i			0.509634i
$540$		0			0
$541$	1.29671e11			1.51375			0.756873	−	0.653562i	$-0.226726\pi$
$541$	1.29671e11			1.51375			0.756873	+	0.653562i	$0.226726\pi$
$542$		0			0
$543$	2.16850e10	+	2.16850e10i	0.249437	+	0.249437i
$544$		0			0
$545$	3.61794e10	+	4.57761e10i	0.410087	+	0.518864i
$546$		0			0
$547$	−2.70067e10	+	2.70067e10i	−0.301664	+	0.301664i	−0.841664	−	0.540001i	$-0.818424\pi$
$547$	−2.70067e10	+	2.70067e10i	−0.301664	+	0.301664i	0.540001	+	0.841664i	$0.318424\pi$
$548$		0			0
$549$		−	1.92656e10i		−	0.212077i
$550$		0			0
$551$	6.04105e9			0.0655399
$552$		0			0
$553$	−8.07130e9	−	8.07130e9i	−0.0863064	−	0.0863064i
$554$		0			0
$555$	−1.04192e11	+	8.23487e10i	−1.09815	+	0.867930i
$556$		0			0
$557$	1.51528e9	−	1.51528e9i	0.0157425	−	0.0157425i	−0.699192	−	0.714934i	$-0.746457\pi$
$557$	1.51528e9	−	1.51528e9i	0.0157425	−	0.0157425i	0.714934	+	0.699192i	$0.246457\pi$
$558$		0			0
$559$		−	2.24591e11i		−	2.30009i
$560$		0			0
$561$	3.04079e10			0.306997
$562$		0			0
$563$	1.16988e11	+	1.16988e11i	1.16441	+	1.16441i	0.983500	+	0.180911i	$0.0579045\pi$
$563$	1.16988e11	+	1.16988e11i	1.16441	+	1.16441i	0.180911	+	0.983500i	$0.442095\pi$
$564$		0			0
$565$	−1.02929e11	−	1.20526e10i	−1.01005	−	0.118273i
$566$		0			0
$567$	9.41486e10	−	9.41486e10i	0.910923	−	0.910923i
$568$		0			0
$569$		−	6.87794e10i		−	0.656159i	−0.944650	−	0.328080i	$-0.893599\pi$
$569$		−	6.87794e10i		−	0.656159i	0.944650	−	0.328080i	$-0.106401\pi$
$570$		0			0
$571$	1.03730e10			0.0975795			0.0487897	−	0.998809i	$-0.484464\pi$
$571$	1.03730e10			0.0975795			0.0487897	+	0.998809i	$0.484464\pi$
$572$		0			0
$573$	2.94427e10	+	2.94427e10i	0.273123	+	0.273123i
$574$		0			0
$575$	−2.45257e10	+	1.51134e10i	−0.224362	+	0.138258i
$576$		0			0
$577$	5.84784e10	−	5.84784e10i	0.527584	−	0.527584i	−0.392267	−	0.919851i	$-0.628309\pi$
$577$	5.84784e10	−	5.84784e10i	0.527584	−	0.527584i	0.919851	+	0.392267i	$0.128309\pi$
$578$		0			0
$579$		−	1.82967e10i		−	0.162802i
$580$		0			0
$581$	−1.26751e11			−1.11237
$582$		0			0
$583$	2.00260e10	+	2.00260e10i	0.173348	+	0.173348i
$584$		0			0
$585$	−2.74249e9	+	2.34208e10i	−0.0234165	+	0.199976i
$586$		0			0
$587$	−2.36584e10	+	2.36584e10i	−0.199266	+	0.199266i	−0.799685	−	0.600420i	$-0.795000\pi$
$587$	−2.36584e10	+	2.36584e10i	−0.199266	+	0.199266i	0.600420	+	0.799685i	$0.295000\pi$
$588$		0			0
$589$			3.53497e10i			0.293714i
$590$		0			0
$591$	1.47226e11			1.20680
$592$		0			0
$593$	1.22246e11	+	1.22246e11i	0.988589	+	0.988589i	0.999936	−	0.0113464i	$-0.00361175\pi$
$593$	1.22246e11	+	1.22246e11i	0.988589	+	0.988589i	−0.0113464	+	0.999936i	$0.503612\pi$
$594$		0			0
$595$	1.08697e11	+	1.37529e11i	0.867261	+	1.09730i
$596$		0			0
$597$	−9.53208e9	+	9.53208e9i	−0.0750396	+	0.0750396i
$598$		0			0
$599$			9.59446e10i			0.745269i	0.927978	+	0.372635i	$0.121546\pi$
$599$			9.59446e10i			0.745269i	−0.927978	+	0.372635i	$0.878454\pi$
$600$		0			0
$601$	1.94971e11			1.49442			0.747210	−	0.664588i	$-0.231393\pi$
$601$	1.94971e11			1.49442			0.747210	+	0.664588i	$0.231393\pi$
$602$		0			0
$603$	−1.24803e10	−	1.24803e10i	−0.0943964	−	0.0943964i
$604$		0			0
$605$	9.09547e10	−	7.18866e10i	0.678897	−	0.536570i
$606$		0			0
$607$	1.26277e11	−	1.26277e11i	0.930188	−	0.930188i	−0.0675290	−	0.997717i	$-0.521512\pi$
$607$	1.26277e11	−	1.26277e11i	0.930188	−	0.930188i	0.997717	+	0.0675290i	$0.0215115\pi$
$608$		0			0
$609$		−	1.15941e10i		−	0.0842882i
$610$		0			0
$611$	−1.26963e11			−0.910984
$612$		0			0
$613$	1.84169e11	+	1.84169e11i	1.30429	+	1.30429i	0.925468	+	0.378826i	$0.123672\pi$
$613$	1.84169e11	+	1.84169e11i	1.30429	+	1.30429i	0.378826	+	0.925468i	$0.376328\pi$
$614$		0			0
$615$	−1.93986e11	−	2.27151e10i	−1.35603	−	0.158786i
$616$		0			0
$617$	−1.20679e11	+	1.20679e11i	−0.832707	+	0.832707i	−0.987886	−	0.155180i	$-0.950404\pi$
$617$	−1.20679e11	+	1.20679e11i	−0.832707	+	0.832707i	0.155180	+	0.987886i	$0.450404\pi$
$618$		0			0
$619$			1.43750e11i			0.979143i	0.871963	+	0.489572i	$0.162847\pi$
$619$			1.43750e11i			0.979143i	−0.871963	+	0.489572i	$0.837153\pi$
$620$		0			0
$621$	4.15000e10			0.279050
$622$		0			0
$623$	−1.51305e10	−	1.51305e10i	−0.100439	−	0.100439i
$624$		0			0
$625$	−1.36300e11	−	6.85961e10i	−0.893255	−	0.449551i
$626$		0			0
$627$	4.11896e10	−	4.11896e10i	0.266512	−	0.266512i
$628$		0			0
$629$			2.14475e11i			1.37017i
$630$		0			0
$631$	1.08715e11			0.685760			0.342880	−	0.939379i	$-0.388598\pi$
$631$	1.08715e11			0.685760			0.342880	+	0.939379i	$0.388598\pi$
$632$		0			0
$633$	−2.21144e10	−	2.21144e10i	−0.137740	−	0.137740i
$634$		0			0
$635$	1.19741e9	−	1.02258e10i	0.00736456	−	0.0628930i
$636$		0			0
$637$	2.23952e11	−	2.23952e11i	1.36018	−	1.36018i
$638$		0			0
$639$			2.48964e10i			0.149325i
$640$		0			0
$641$	−3.25454e11			−1.92778			−0.963890	−	0.266299i	$-0.914199\pi$
$641$	−3.25454e11			−1.92778			−0.963890	+	0.266299i	$0.914199\pi$
$642$		0			0
$643$	−4.11926e10	−	4.11926e10i	−0.240977	−	0.240977i	0.576278	−	0.817254i	$-0.304505\pi$
$643$	−4.11926e10	−	4.11926e10i	−0.240977	−	0.240977i	−0.817254	+	0.576278i	$0.804505\pi$
$644$		0			0
$645$	−1.64755e11	−	2.08457e11i	−0.951918	−	1.20442i
$646$		0			0
$647$	8.13749e10	−	8.13749e10i	0.464380	−	0.464380i	−0.435708	−	0.900088i	$-0.643502\pi$
$647$	8.13749e10	−	8.13749e10i	0.464380	−	0.464380i	0.900088	+	0.435708i	$0.143502\pi$
$648$		0			0
$649$		−	6.02630e10i		−	0.339682i
$650$		0			0
$651$	6.78438e10			0.377734
$652$		0			0
$653$	1.29147e11	+	1.29147e11i	0.710285	+	0.710285i	0.966595	−	0.256310i	$-0.0825067\pi$
$653$	1.29147e11	+	1.29147e11i	0.710285	+	0.710285i	−0.256310	+	0.966595i	$0.582507\pi$
$654$		0			0
$655$	1.88064e11	−	1.48638e11i	1.02174	−	0.807539i
$656$		0			0
$657$	−1.77328e10	+	1.77328e10i	−0.0951733	+	0.0951733i
$658$		0			0
$659$			2.74018e10i			0.145291i	0.997358	+	0.0726453i	$0.0231441\pi$
$659$			2.74018e10i			0.145291i	−0.997358	+	0.0726453i	$0.976856\pi$
$660$		0			0
$661$	−1.19547e11			−0.626228			−0.313114	−	0.949716i	$-0.601372\pi$
$661$	−1.19547e11			−0.626228			−0.313114	+	0.949716i	$0.601372\pi$
$662$		0			0
$663$	−1.58317e11	−	1.58317e11i	−0.819355	−	0.819355i
$664$		0			0
$665$	3.33531e11	+	3.90553e10i	1.70549	+	0.199707i
$666$		0			0
$667$	2.17582e9	−	2.17582e9i	0.0109931	−	0.0109931i
$668$		0			0
$669$			1.26021e11i			0.629129i
$670$		0			0
$671$	−1.08530e11			−0.535377
$672$		0			0
$673$	−2.17592e11	−	2.17592e11i	−1.06068	−	1.06068i	−0.998036	−	0.0626416i	$-0.980047\pi$
$673$	−2.17592e11	−	2.17592e11i	−1.06068	−	1.06068i	−0.0626416	−	0.998036i	$-0.519953\pi$
$674$		0			0
$675$	1.15317e11	+	1.87133e11i	0.555492	+	0.901438i
$676$		0			0
$677$	1.34337e11	−	1.34337e11i	0.639503	−	0.639503i	−0.310930	−	0.950433i	$-0.600640\pi$
$677$	1.34337e11	−	1.34337e11i	0.639503	−	0.639503i	0.950433	+	0.310930i	$0.100640\pi$
$678$		0			0
$679$		−	1.50561e11i		−	0.708327i
$680$		0			0
$681$	2.85661e11			1.32820
$682$		0			0
$683$	2.62510e11	+	2.62510e11i	1.20632	+	1.20632i	0.972210	+	0.234110i	$0.0752177\pi$
$683$	2.62510e11	+	2.62510e11i	1.20632	+	1.20632i	0.234110	+	0.972210i	$0.424782\pi$
$684$		0			0
$685$	1.50893e9	−	1.28862e10i	0.00685339	−	0.0585277i
$686$		0			0
$687$	−2.59622e11	+	2.59622e11i	−1.16551	+	1.16551i
$688$		0			0
$689$		−	2.08528e11i		−	0.925310i
$690$		0			0
$691$	−3.13746e11			−1.37615			−0.688075	−	0.725639i	$-0.741544\pi$
$691$	−3.13746e11			−1.37615			−0.688075	+	0.725639i	$0.741544\pi$
$692$		0			0
$693$	1.34459e10	+	1.34459e10i	0.0582982	+	0.0582982i
$694$		0			0
$695$	−1.55139e11	−	1.96290e11i	−0.664940	−	0.841318i
$696$		0			0
$697$	−2.23035e11	+	2.23035e11i	−0.945021	+	0.945021i
$698$		0			0
$699$			2.75534e11i			1.15416i
$700$		0			0
$701$	2.56197e11			1.06097			0.530483	−	0.847696i	$-0.322011\pi$
$701$	2.56197e11			1.06097			0.530483	+	0.847696i	$0.322011\pi$
$702$		0			0
$703$	2.90521e11	+	2.90521e11i	1.18948	+	1.18948i
$704$		0			0
$705$	−1.17842e11	+	9.31370e10i	−0.477027	+	0.377021i
$706$		0			0
$707$	2.08677e11	−	2.08677e11i	0.835213	−	0.835213i
$708$		0			0
$709$			1.51234e10i			0.0598499i	0.999552	+	0.0299250i	$0.00952683\pi$
$709$			1.51234e10i			0.0598499i	−0.999552	+	0.0299250i	$0.990473\pi$
$710$		0			0
$711$	−2.93364e9			−0.0114796
$712$		0			0
$713$	1.27320e10	+	1.27320e10i	0.0492651	+	0.0492651i
$714$		0			0
$715$	1.31937e11	+	1.54494e10i	0.504828	+	0.0591136i
$716$		0			0
$717$	−2.42491e11	+	2.42491e11i	−0.917527	+	0.917527i
$718$		0			0
$719$		−	4.20879e11i		−	1.57486i	−0.616404	−	0.787430i	$-0.711411\pi$
$719$		−	4.20879e11i		−	1.57486i	0.616404	−	0.787430i	$-0.288589\pi$
$720$		0			0
$721$	−2.60544e11			−0.964140
$722$		0			0
$723$	−3.08430e11	−	3.08430e11i	−1.12877	−	1.12877i
$724$		0			0
$725$	1.58573e10	+	3.76530e9i	0.0573954	+	0.0136285i
$726$		0			0
$727$	−2.08188e11	+	2.08188e11i	−0.745277	+	0.745277i	−0.973588	−	0.228312i	$-0.926680\pi$
$727$	−2.08188e11	+	2.08188e11i	−0.745277	+	0.745277i	0.228312	+	0.973588i	$0.426680\pi$
$728$		0			0
$729$		−	3.10681e11i		−	1.10003i
$730$		0			0
$731$	−4.29099e11			−1.50275
$732$		0			0
$733$	−2.48791e11	−	2.48791e11i	−0.861826	−	0.861826i	0.129725	−	0.991550i	$-0.458591\pi$
$733$	−2.48791e11	−	2.48791e11i	−0.861826	−	0.861826i	−0.991550	+	0.129725i	$0.958591\pi$
$734$		0			0
$735$	4.35774e10	−	3.72149e11i	0.149318	−	1.27517i
$736$		0			0
$737$	−7.03058e10	+	7.03058e10i	−0.238298	+	0.238298i
$738$		0			0
$739$			1.15406e11i			0.386946i	0.981106	+	0.193473i	$0.0619752\pi$
$739$			1.15406e11i			0.386946i	−0.981106	+	0.193473i	$0.938025\pi$
$740$		0			0
$741$	−4.28902e11			−1.42261
$742$		0			0
$743$	−1.60556e11	−	1.60556e11i	−0.526833	−	0.526833i	0.392794	−	0.919627i	$-0.371509\pi$
$743$	−1.60556e11	−	1.60556e11i	−0.526833	−	0.526833i	−0.919627	+	0.392794i	$0.871509\pi$
$744$		0			0
$745$	2.17977e11	+	2.75796e11i	0.707596	+	0.895289i
$746$		0			0
$747$	−2.30348e10	+	2.30348e10i	−0.0739780	+	0.0739780i
$748$		0			0
$749$			3.21459e11i			1.02141i
$750$		0			0
$751$	5.98277e10			0.188080			0.0940400	−	0.995568i	$-0.470022\pi$
$751$	5.98277e10			0.188080			0.0940400	+	0.995568i	$0.470022\pi$
$752$		0			0
$753$	−2.12774e11	−	2.12774e11i	−0.661819	−	0.661819i
$754$		0			0
$755$	−1.34969e10	+	1.06673e10i	−0.0415379	+	0.0328297i
$756$		0			0
$757$	−2.88999e11	+	2.88999e11i	−0.880060	+	0.880060i	−0.993540	−	0.113480i	$-0.963800\pi$
$757$	−2.88999e11	+	2.88999e11i	−0.880060	+	0.880060i	0.113480	+	0.993540i	$0.463800\pi$
$758$		0			0
$759$		−	2.96708e10i		−	0.0894050i
$760$		0			0
$761$	−1.15311e11			−0.343819			−0.171910	−	0.985113i	$-0.554994\pi$
$761$	−1.15311e11			−0.343819			−0.171910	+	0.985113i	$0.554994\pi$
$762$		0			0
$763$	−2.44966e11	−	2.44966e11i	−0.722783	−	0.722783i
$764$		0			0
$765$	4.47473e10	+	5.23975e9i	0.130654	+	0.0152991i
$766$		0			0
$767$	−3.13755e11	+	3.13755e11i	−0.906587	+	0.906587i
$768$		0			0
$769$			1.71633e11i			0.490791i	0.969423	+	0.245395i	$0.0789178\pi$
$769$			1.71633e11i			0.490791i	−0.969423	+	0.245395i	$0.921082\pi$
$770$		0			0
$771$	1.88283e11			0.532835
$772$		0			0
$773$	−3.60143e11	−	3.60143e11i	−1.00869	−	1.00869i	−0.999962	−	0.00872501i	$-0.997223\pi$
$773$	−3.60143e11	−	3.60143e11i	−1.00869	−	1.00869i	−0.00872501	−	0.999962i	$-0.502777\pi$
$774$		0			0
$775$	−2.20330e10	+	9.27905e10i	−0.0610754	+	0.257215i
$776$		0			0
$777$	5.57572e11	−	5.57572e11i	1.52974	−	1.52974i
$778$		0			0
$779$			6.04232e11i			1.64079i
$780$		0			0
$781$	1.40250e11			0.376963
$782$		0			0
$783$	−1.66018e10	−	1.66018e10i	−0.0441679	−	0.0441679i
$784$		0			0
$785$	3.31653e10	−	2.83230e11i	0.0873383	−	0.745866i
$786$		0			0
$787$	−3.81822e11	+	3.81822e11i	−0.995318	+	0.995318i	−0.999989	−	0.00467136i	$-0.998513\pi$
$787$	−3.81822e11	+	3.81822e11i	−0.995318	+	0.995318i	0.00467136	+	0.999989i	$0.498513\pi$
$788$		0			0
$789$			1.98632e11i			0.512557i
$790$		0			0
$791$	6.15310e11			1.57177
$792$		0			0
$793$	5.65054e11	+	5.65054e11i	1.42888	+	1.42888i
$794$		0			0
$795$	−1.52972e11	−	1.93548e11i	−0.382950	−	0.484529i
$796$		0			0
$797$	4.32433e11	−	4.32433e11i	1.07173	−	1.07173i	0.0745090	−	0.997220i	$-0.476261\pi$
$797$	4.32433e11	−	4.32433e11i	1.07173	−	1.07173i	0.997220	−	0.0745090i	$-0.0237389\pi$
$798$		0			0
$799$			2.42573e11i			0.595189i
$800$		0			0
$801$	−5.49941e9			−0.0133594
$802$		0			0
$803$	9.98949e10	+	9.98949e10i	0.240260	+	0.240260i
$804$		0			0
$805$	1.34195e11	−	1.06062e11i	0.319561	−	0.252567i
$806$		0			0
$807$	3.13210e11	−	3.13210e11i	0.738485	−	0.738485i
$808$		0			0
$809$			4.04427e11i			0.944160i	0.881556	+	0.472080i	$0.156497\pi$
$809$			4.04427e11i			0.944160i	−0.881556	+	0.472080i	$0.843503\pi$
$810$		0			0
$811$	6.23463e11			1.44121			0.720605	−	0.693346i	$-0.243864\pi$
$811$	6.23463e11			1.44121			0.720605	+	0.693346i	$0.243864\pi$
$812$		0			0
$813$	−8.49475e10	−	8.49475e10i	−0.194441	−	0.194441i
$814$		0			0
$815$	2.07777e11	+	2.43299e10i	0.470941	+	0.0551456i
$816$		0			0
$817$	−5.81245e11	+	5.81245e11i	−1.30458	+	1.30458i
$818$		0			0
$819$		−	1.40010e11i		−	0.311188i
$820$		0			0
$821$	3.67266e11			0.808366			0.404183	−	0.914678i	$-0.367556\pi$
$821$	3.67266e11			0.808366			0.404183	+	0.914678i	$0.367556\pi$
$822$		0			0
$823$	−1.17460e11	−	1.17460e11i	−0.256030	−	0.256030i	0.567407	−	0.823437i	$-0.307947\pi$
$823$	−1.17460e11	−	1.17460e11i	−0.256030	−	0.256030i	−0.823437	+	0.567407i	$0.807947\pi$
$824$		0			0
$825$	1.33793e11	−	8.24468e10i	0.288813	−	0.177975i
$826$		0			0
$827$	5.29740e11	−	5.29740e11i	1.13251	−	1.13251i	0.142748	−	0.989759i	$-0.454406\pi$
$827$	5.29740e11	−	5.29740e11i	1.13251	−	1.13251i	0.989759	−	0.142748i	$-0.0455939\pi$
$828$		0			0
$829$		−	2.39704e11i		−	0.507525i	−0.967267	−	0.253762i	$-0.918332\pi$
$829$		−	2.39704e11i		−	0.507525i	0.967267	−	0.253762i	$-0.0816681\pi$
$830$		0			0
$831$	1.77132e11			0.371444
$832$		0			0
$833$	−4.27878e11	−	4.27878e11i	−0.888669	−	0.888669i
$834$		0			0
$835$	−5.60694e10	+	4.78831e11i	−0.115340	+	0.984999i
$836$		0			0
$837$	9.71467e10	−	9.71467e10i	0.197937	−	0.197937i
$838$		0			0
$839$		−	5.22857e11i		−	1.05520i	−0.849493	−	0.527601i	$-0.823092\pi$
$839$		−	5.22857e11i		−	1.05520i	0.849493	−	0.527601i	$-0.176908\pi$
$840$		0			0
$841$	4.98506e11			0.996520
$842$		0			0
$843$	−4.92082e11	−	4.92082e11i	−0.974378	−	0.974378i
$844$		0			0
$845$	−2.90355e11	−	3.67372e11i	−0.569511	−	0.720576i
$846$		0			0
$847$	−4.86735e11	+	4.86735e11i	−0.945712	+	0.945712i
$848$		0			0
$849$			4.88258e11i			0.939764i
$850$		0			0
$851$	2.09276e11			0.399025
$852$		0			0
$853$	−2.94110e11	−	2.94110e11i	−0.555537	−	0.555537i	0.372496	−	0.928034i	$-0.378502\pi$
$853$	−2.94110e11	−	2.94110e11i	−0.555537	−	0.555537i	−0.928034	+	0.372496i	$0.878502\pi$
$854$		0			0
$855$	6.77110e10	−	5.35158e10i	0.126705	−	0.100142i
$856$		0			0
$857$	−1.69558e11	+	1.69558e11i	−0.314336	+	0.314336i	−0.846587	−	0.532251i	$-0.821346\pi$
$857$	−1.69558e11	+	1.69558e11i	−0.314336	+	0.314336i	0.532251	+	0.846587i	$0.321346\pi$
$858$		0			0
$859$		−	3.81187e11i		−	0.700109i	−0.936729	−	0.350055i	$-0.886163\pi$
$859$		−	3.81187e11i		−	0.700109i	0.936729	−	0.350055i	$-0.113837\pi$
$860$		0			0
$861$	1.15965e12			2.11016
$862$		0			0
$863$	−8.42734e10	−	8.42734e10i	−0.151931	−	0.151931i	0.627049	−	0.778980i	$-0.284263\pi$
$863$	−8.42734e10	−	8.42734e10i	−0.151931	−	0.151931i	−0.778980	+	0.627049i	$0.784263\pi$
$864$		0			0
$865$	5.63300e11	+	6.59605e10i	1.00618	+	0.117820i
$866$		0			0
$867$	6.68848e10	−	6.68848e10i	0.118373	−	0.118373i
$868$		0			0
$869$			1.65262e10i			0.0289797i
$870$		0			0
$871$	7.32085e11			1.27200
$872$		0			0
$873$	−2.73618e10	−	2.73618e10i	−0.0471073	−	0.0471073i
$874$		0			0
$875$	8.51151e11	+	3.10402e11i	1.45203	+	0.529533i
$876$		0			0
$877$	3.72176e10	−	3.72176e10i	0.0629144	−	0.0629144i	−0.674950	−	0.737864i	$-0.735835\pi$
$877$	3.72176e10	−	3.72176e10i	0.0629144	−	0.0629144i	0.737864	+	0.674950i	$0.235835\pi$
$878$		0			0
$879$			7.40541e11i			1.24049i
$880$		0			0
$881$	−4.91908e10			−0.0816545			−0.0408273	−	0.999166i	$-0.512999\pi$
$881$	−4.91908e10			−0.0816545			−0.0408273	+	0.999166i	$0.512999\pi$
$882$		0			0
$883$	−4.82360e11	−	4.82360e11i	−0.793467	−	0.793467i	0.188589	−	0.982056i	$-0.439609\pi$
$883$	−4.82360e11	−	4.82360e11i	−0.793467	−	0.793467i	−0.982056	+	0.188589i	$0.939609\pi$
$884$		0			0
$885$	−6.10518e10	+	5.21380e11i	−0.0995235	+	0.849927i
$886$		0			0
$887$	−7.58872e11	+	7.58872e11i	−1.22595	+	1.22595i	−0.260472	+	0.965481i	$0.583878\pi$
$887$	−7.58872e11	+	7.58872e11i	−1.22595	+	1.22595i	−0.965481	+	0.260472i	$0.916122\pi$
$888$		0			0
$889$			6.11301e10i			0.0978697i
$890$		0			0
$891$	−1.92772e11			−0.305867
$892$		0			0
$893$	3.28582e11	+	3.28582e11i	0.516699	+	0.516699i
$894$		0			0
$895$	−3.64710e11	−	4.61451e11i	−0.568402	−	0.719173i
$896$		0			0
$897$	−1.54479e11	+	1.54479e11i	−0.238616	+	0.238616i
$898$		0			0
$899$		−	1.01867e10i		−	0.0155953i
$900$		0			0
$901$	−3.98410e11			−0.604548
$902$		0			0
$903$	1.11553e12	+	1.11553e12i	1.67777	+	1.67777i
$904$		0			0
$905$	2.00816e11	−	1.58716e11i	0.299367	−	0.236606i
$906$		0			0
$907$	8.25250e11	−	8.25250e11i	1.21943	−	1.21943i	0.251597	−	0.967832i	$-0.419044\pi$
$907$	8.25250e11	−	8.25250e11i	1.21943	−	1.21943i	0.967832	−	0.251597i	$-0.0809558\pi$
$908$		0			0
$909$		−	7.58468e10i		−	0.111092i
$910$		0			0
$911$	−5.86650e11			−0.851737			−0.425869	−	0.904785i	$-0.640031\pi$
$911$	−5.86650e11			−0.851737			−0.425869	+	0.904785i	$0.640031\pi$
$912$		0			0
$913$	1.29763e11	+	1.29763e11i	0.186753	+	0.186753i
$914$		0			0
$915$	9.38973e11	+	1.09951e11i	1.33958	+	0.156860i
$916$		0			0
$917$	−1.00641e12	+	1.00641e12i	−1.42330	+	1.42330i
$918$		0			0
$919$			1.27566e12i			1.78843i	0.447640	+	0.894214i	$0.352265\pi$
$919$			1.27566e12i			1.78843i	−0.447640	+	0.894214i	$0.647735\pi$
$920$		0			0
$921$	2.45520e11			0.341231
$922$		0			0
$923$	−7.30203e11	−	7.30203e11i	−1.00609	−	1.00609i
$924$		0			0
$925$	5.81519e11	+	9.43674e11i	0.794322	+	1.28901i
$926$		0			0
$927$	−4.73493e10	+	4.73493e10i	−0.0641202	+	0.0641202i
$928$		0			0
$929$		−	1.11916e12i		−	1.50255i	−0.659991	−	0.751274i	$-0.729440\pi$
$929$		−	1.11916e12i		−	1.50255i	0.659991	−	0.751274i	$-0.270560\pi$
$930$		0			0
$931$	−1.15918e12			−1.54295
$932$		0			0
$933$	−7.68353e11	−	7.68353e11i	−1.01399	−	1.01399i
$934$		0			0
$935$	2.95174e10	−	2.52077e11i	0.0386217	−	0.329828i
$936$		0			0
$937$	3.48571e11	−	3.48571e11i	0.452202	−	0.452202i	−0.443883	−	0.896085i	$-0.646399\pi$
$937$	3.48571e11	−	3.48571e11i	0.452202	−	0.452202i	0.896085	+	0.443883i	$0.146399\pi$
$938$		0			0
$939$		−	2.15061e11i		−	0.276631i
$940$		0			0
$941$	3.10729e11			0.396299			0.198150	−	0.980172i	$-0.436507\pi$
$941$	3.10729e11			0.396299			0.198150	+	0.980172i	$0.436507\pi$
$942$		0			0
$943$	2.17628e11	+	2.17628e11i	0.275213	+	0.275213i
$944$		0			0
$945$	−8.09265e11	−	1.02393e12i	−1.01476	−	1.28393i
$946$		0			0
$947$	4.81109e11	−	4.81109e11i	0.598197	−	0.598197i	−0.341636	−	0.939832i	$-0.610981\pi$
$947$	4.81109e11	−	4.81109e11i	0.598197	−	0.598197i	0.939832	+	0.341636i	$0.110981\pi$
$948$		0			0
$949$		−	1.04019e12i		−	1.28247i
$950$		0			0
$951$	−1.02618e11			−0.125459
$952$		0			0
$953$	2.19961e11	+	2.19961e11i	0.266670	+	0.266670i	0.827757	−	0.561087i	$-0.189617\pi$
$953$	2.19961e11	+	2.19961e11i	0.266670	+	0.266670i	−0.561087	+	0.827757i	$0.689617\pi$
$954$		0			0
$955$	2.72656e11	−	2.15495e11i	0.327794	−	0.259074i
$956$		0			0
$957$	−1.18696e10	+	1.18696e10i	−0.0141510	+	0.0141510i
$958$		0			0
$959$			7.70338e10i			0.0910767i
$960$		0			0
$961$	−7.93283e11			−0.930110
$962$		0			0
$963$	5.84196e10	+	5.84196e10i	0.0679287	+	0.0679287i
$964$		0			0
$965$	−1.51677e11	−	1.77609e10i	−0.174909	−	0.0204812i
$966$		0			0
$967$	−6.94014e11	+	6.94014e11i	−0.793711	+	0.793711i	−0.982095	−	0.188384i	$-0.939675\pi$
$967$	−6.94014e11	+	6.94014e11i	−0.793711	+	0.793711i	0.188384	+	0.982095i	$0.439675\pi$
$968$		0			0
$969$			8.19453e11i			0.929456i
$970$		0			0
$971$	−1.29419e12			−1.45587			−0.727935	−	0.685646i	$-0.759520\pi$
$971$	−1.29419e12			−1.45587			−0.727935	+	0.685646i	$0.759520\pi$
$972$		0			0
$973$	1.05043e12	+	1.05043e12i	1.17197	+	1.17197i
$974$		0			0
$975$	−1.12584e12	−	2.67328e11i	−1.24582	−	0.295819i
$976$		0			0
$977$	2.00533e11	−	2.00533e11i	0.220093	−	0.220093i	−0.588444	−	0.808538i	$-0.700259\pi$
$977$	2.00533e11	−	2.00533e11i	0.220093	−	0.220093i	0.808538	+	0.588444i	$0.200259\pi$
$978$		0			0
$979$			3.09801e10i			0.0337250i
$980$		0			0
$981$	−8.90367e10			−0.0961375
$982$		0			0
$983$	−4.14637e11	−	4.14637e11i	−0.444072	−	0.444072i	0.449306	−	0.893378i	$-0.351671\pi$
$983$	−4.14637e11	−	4.14637e11i	−0.444072	−	0.444072i	−0.893378	+	0.449306i	$0.851671\pi$
$984$		0			0
$985$	1.42915e11	−	1.22049e12i	0.151821	−	1.29655i
$986$		0			0
$987$	6.30619e11	−	6.30619e11i	0.664505	−	0.664505i
$988$		0			0
$989$			4.18698e11i			0.437638i
$990$		0			0
$991$	6.51779e11			0.675780			0.337890	−	0.941186i	$-0.390287\pi$
$991$	6.51779e11			0.675780			0.337890	+	0.941186i	$0.390287\pi$
$992$		0			0
$993$	−4.04041e11	−	4.04041e11i	−0.415555	−	0.415555i
$994$		0			0
$995$	6.97667e10	+	8.82725e10i	0.0711796	+	0.0900603i
$996$		0			0
$997$	−6.71342e11	+	6.71342e11i	−0.679459	+	0.679459i	−0.959878	−	0.280419i	$-0.909527\pi$
$997$	−6.71342e11	+	6.71342e11i	−0.679459	+	0.679459i	0.280419	+	0.959878i	$0.409527\pi$
$998$		0			0
$999$		−	1.59680e12i		−	1.60320i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.9.p.b.33.1		4
4.3	odd	2		10.9.c.a.3.2	✓	4
5.2	odd	4	inner	80.9.p.b.17.1		4
12.11	even	2		90.9.g.b.73.2		4
20.3	even	4		50.9.c.e.7.1		4
20.7	even	4		10.9.c.a.7.2	yes	4
20.19	odd	2		50.9.c.e.43.1		4
60.47	odd	4		90.9.g.b.37.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.9.c.a.3.2	✓	4	4.3	odd	2
10.9.c.a.7.2	yes	4	20.7	even	4
50.9.c.e.7.1		4	20.3	even	4
50.9.c.e.43.1		4	20.19	odd	2
80.9.p.b.17.1		4	5.2	odd	4	inner
80.9.p.b.33.1		4	1.1	even	1	trivial
90.9.g.b.37.2		4	60.47	odd	4
90.9.g.b.73.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	80.p (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-52.9493 - 52.9493i) q^{3} +(-490.341 + 387.544i) q^{5} +(2624.01 - 2624.01i) q^{7} -953.736i q^{9} -5372.73 q^{11} +(27972.7 + 27972.7i) q^{13} +(46483.4 + 5443.05i) q^{15} +(53444.2 - 53444.2i) q^{17} -144788. i q^{19} -277879. q^{21} +(-52148.7 - 52148.7i) q^{23} +(90244.3 - 380058. i) q^{25} +(-397900. + 397900. i) q^{27} +41723.4i q^{29} -244148. q^{31} +(284482. + 284482. i) q^{33} +(-269741. + 2.30358e6i) q^{35} +(-2.00653e6 + 2.00653e6i) q^{37} -2.96228e6i q^{39} -4.17322e6 q^{41} +(-4.01446e6 - 4.01446e6i) q^{43} +(369615. + 467656. i) q^{45} +(-2.26940e6 + 2.26940e6i) q^{47} -8.00606e6i q^{49} -5.65967e6 q^{51} +(-3.72734e6 - 3.72734e6i) q^{53} +(2.63447e6 - 2.08217e6i) q^{55} +(-7.66643e6 + 7.66643e6i) q^{57} +1.12165e7i q^{59} +2.02002e7 q^{61} +(-2.50261e6 - 2.50261e6i) q^{63} +(-2.45569e7 - 2.87552e6i) q^{65} +(1.30857e7 - 1.30857e7i) q^{67} +5.52248e6i q^{69} -2.61041e7 q^{71} +(-1.85930e7 - 1.85930e7i) q^{73} +(-2.49022e7 + 1.53454e7i) q^{75} +(-1.40981e7 + 1.40981e7i) q^{77} -3.07594e6i q^{79} +3.58796e7 q^{81} +(-2.41522e7 - 2.41522e7i) q^{83} +(-5.49393e6 + 4.69179e7i) q^{85} +(2.20923e6 - 2.20923e6i) q^{87} -5.76618e6i q^{89} +1.46802e8 q^{91} +(1.29275e7 + 1.29275e7i) q^{93} +(5.61117e7 + 7.09955e7i) q^{95} +(2.86891e7 - 2.86891e7i) q^{97} +5.12416e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 54 q^{3} + 90 q^{5} + 1186 q^{7} + 19852 q^{11} + 73704 q^{13} + 137490 q^{15} + 198944 q^{17} - 766572 q^{21} - 631334 q^{23} + 545600 q^{25} - 184680 q^{27} + 2329892 q^{31} + 1362948 q^{33} - 1403870 q^{35}+ \cdots + 139631604 q^{97}+O(q^{100}) \) 4 * q - 54 * q^3 + 90 * q^5 + 1186 * q^7 + 19852 * q^11 + 73704 * q^13 + 137490 * q^15 + 198944 * q^17 - 766572 * q^21 - 631334 * q^23 + 545600 * q^25 - 184680 * q^27 + 2329892 * q^31 + 1362948 * q^33 - 1403870 * q^35 - 976356 * q^37 - 4284652 * q^41 - 7146534 * q^43 - 2624430 * q^45 + 3300186 * q^47 - 6541788 * q^51 + 9989164 * q^53 + 21649020 * q^55 - 19136400 * q^57 + 85377108 * q^61 + 16175226 * q^63 - 33878940 * q^65 + 57195466 * q^67 - 113004028 * q^71 - 60875756 * q^73 - 22407150 * q^75 - 90339932 * q^77 + 35054064 * q^81 - 71051214 * q^83 + 8601320 * q^85 - 58152480 * q^87 + 221467572 * q^91 + 98985108 * q^93 + 159500600 * q^95 + 139631604 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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