LMFDB - Embedded newform 84.9.m.a.61.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,9,Mod(61,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("84.61");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.m (of order $6$, 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:	$34.2198032451$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} + \cdots + 15\!\cdots\!00 $ x^10 - 38255x^8 + 1483053595x^6 - 139470625170x^5 + 5194605060018x^4 - 71600654137860x^3 + 119846615988780x^2 + 4263507454127400*x + 15758720495290800
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{8}\cdot 7^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			61.1
Root			$-3.70642 + 2.13991i$ of defining polynomial
Character	$\chi$	$=$	84.61
Dual form			84.9.m.a.73.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 + 23.3827i) q^{3} +(-489.814 - 282.795i) q^{5} +(-1456.69 - 1908.63i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})$	\(q+(-40.5000 + 23.3827i) q^{3} +(-489.814 - 282.795i) q^{5} +(-1456.69 - 1908.63i) q^{7} +(1093.50 - 1894.00i) q^{9} +(-4106.46 - 7112.59i) q^{11} -13651.5i q^{13} +26450.0 q^{15} +(6393.53 - 3691.30i) q^{17} +(26652.7 + 15387.9i) q^{19} +(103625. + 43238.0i) q^{21} +(-57456.4 + 99517.3i) q^{23} +(-35367.0 - 61257.5i) q^{25} +102276. i q^{27} -124123. q^{29} +(-331222. + 191231. i) q^{31} +(332623. + 192040. i) q^{33} +(173760. + 1.34682e6i) q^{35} +(-678319. + 1.17488e6i) q^{37} +(319210. + 552887. i) q^{39} +2.68579e6i q^{41} -1.62184e6 q^{43} +(-1.07122e6 + 618472. i) q^{45} +(3.06709e6 + 1.77079e6i) q^{47} +(-1.52090e6 + 5.56056e6i) q^{49} +(-172625. + 298996. i) q^{51} +(4.54092e6 + 7.86511e6i) q^{53} +4.64513e6i q^{55} -1.43925e6 q^{57} +(197222. - 113866. i) q^{59} +(1.18352e7 + 6.83305e6i) q^{61} +(-5.20782e6 + 671889. i) q^{63} +(-3.86058e6 + 6.68672e6i) q^{65} +(4.13814e6 + 7.16746e6i) q^{67} -5.37394e6i q^{69} -2.81148e7 q^{71} +(1.49937e7 - 8.65664e6i) q^{73} +(2.86473e6 + 1.65395e6i) q^{75} +(-7.59343e6 + 1.81985e7i) q^{77} +(-1.26003e7 + 2.18243e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} +1.68155e7i q^{83} -4.17552e6 q^{85} +(5.02699e6 - 2.90234e6i) q^{87} +(6.75025e7 + 3.89726e7i) q^{89} +(-2.60557e7 + 1.98861e7i) q^{91} +(8.94299e6 - 1.54897e7i) q^{93} +(-8.70325e6 - 1.50745e7i) q^{95} -1.50031e8i q^{97} -1.79616e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10}) $ 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9	$ 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9} - 879 q^{11} - 75006 q^{15} - 13674 q^{17} - 29268 q^{19} - 42363 q^{21} + 312732 q^{23} - 22052 q^{25} - 289794 q^{29} + 242787 q^{31} + 71199 q^{33} + 1209372 q^{35} + 1913308 q^{37} - 1232334 q^{39} - 861848 q^{43} + 3037743 q^{45} - 305448 q^{47} + 9821659 q^{49} + 369198 q^{51} - 10663233 q^{53} + 1580472 q^{57} + 18410871 q^{59} - 13937808 q^{61} + 769824 q^{63} - 14966808 q^{65} - 20722822 q^{67} + 113032584 q^{71} + 43436322 q^{73} + 1786212 q^{75} - 98823405 q^{77} - 42189637 q^{79} - 23914845 q^{81} + 142602108 q^{85} + 11736657 q^{87} + 67171914 q^{89} - 246091266 q^{91} - 6555249 q^{93} - 140649894 q^{95} - 3844746 q^{99}+O(q^{100}) $ 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9 - 879 * q^11 - 75006 * q^15 - 13674 * q^17 - 29268 * q^19 - 42363 * q^21 + 312732 * q^23 - 22052 * q^25 - 289794 * q^29 + 242787 * q^31 + 71199 * q^33 + 1209372 * q^35 + 1913308 * q^37 - 1232334 * q^39 - 861848 * q^43 + 3037743 * q^45 - 305448 * q^47 + 9821659 * q^49 + 369198 * q^51 - 10663233 * q^53 + 1580472 * q^57 + 18410871 * q^59 - 13937808 * q^61 + 769824 * q^63 - 14966808 * q^65 - 20722822 * q^67 + 113032584 * q^71 + 43436322 * q^73 + 1786212 * q^75 - 98823405 * q^77 - 42189637 * q^79 - 23914845 * q^81 + 142602108 * q^85 + 11736657 * q^87 + 67171914 * q^89 - 246091266 * q^91 - 6555249 * q^93 - 140649894 * q^95 - 3844746 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$	−40.5000	+	23.3827i	−0.500000	+	0.288675i
$3$	−40.5000	+	23.3827i	−0.500000	+	0.288675i
$4$		0			0
$5$	−489.814	−	282.795i	−0.783703	−	0.452471i	0.0540379	−	0.998539i	$-0.482791\pi$
$5$	−489.814	−	282.795i	−0.783703	−	0.452471i	−0.837741	+	0.546068i	$0.816124\pi$
$6$		0			0
$7$	−1456.69	−	1908.63i	−0.606702	−	0.794929i
$7$	−1456.69	−	1908.63i	−0.606702	−	0.794929i
$8$		0			0
$9$	1093.50	−	1894.00i	0.166667	−	0.288675i
$10$		0			0
$11$	−4106.46	−	7112.59i	−0.280476	−	0.485799i	0.691026	−	0.722830i	$-0.257159\pi$
$11$	−4106.46	−	7112.59i	−0.280476	−	0.485799i	−0.971502	+	0.237031i	$0.923826\pi$
$12$		0			0
$13$		−	13651.5i		−	0.477978i	−0.971022	−	0.238989i	$-0.923184\pi$
$13$		−	13651.5i		−	0.477978i	0.971022	−	0.238989i	$-0.0768160\pi$
$14$		0			0
$15$	26450.0			0.522469
$16$		0			0
$17$	6393.53	−	3691.30i	0.0765499	−	0.0441961i	−0.461236	−	0.887277i	$-0.652594\pi$
$17$	6393.53	−	3691.30i	0.0765499	−	0.0441961i	0.537786	+	0.843081i	$0.319261\pi$
$18$		0			0
$19$	26652.7	+	15387.9i	0.204516	+	0.118077i	0.598760	−	0.800928i	$-0.295660\pi$
$19$	26652.7	+	15387.9i	0.204516	+	0.118077i	−0.394244	+	0.919006i	$0.628994\pi$
$20$		0			0
$21$	103625.	+	43238.0i	0.532827	+	0.222325i
$22$		0			0
$23$	−57456.4	+	99517.3i	−0.205318	+	0.355621i	−0.950234	−	0.311537i	$-0.899156\pi$
$23$	−57456.4	+	99517.3i	−0.205318	+	0.355621i	0.744916	+	0.667158i	$0.232489\pi$
$24$		0			0
$25$	−35367.0	−	61257.5i	−0.0905395	−	0.156819i
$26$		0			0
$27$			102276.i			0.192450i
$28$		0			0
$29$	−124123.			−0.175494			−0.0877468	−	0.996143i	$-0.527967\pi$
$29$	−124123.			−0.175494			−0.0877468	+	0.996143i	$0.527967\pi$
$30$		0			0
$31$	−331222.	+	191231.i	−0.358651	+	0.207067i	−0.668489	−	0.743722i	$-0.733059\pi$
$31$	−331222.	+	191231.i	−0.358651	+	0.207067i	0.309838	+	0.950789i	$0.399725\pi$
$32$		0			0
$33$	332623.	+	192040.i	0.280476	+	0.161933i
$34$		0			0
$35$	173760.	+	1.34682e6i	0.115792	+	0.897504i
$36$		0			0
$37$	−678319.	+	1.17488e6i	−0.361932	+	0.626885i	−0.988279	−	0.152660i	$-0.951216\pi$
$37$	−678319.	+	1.17488e6i	−0.361932	+	0.626885i	0.626347	+	0.779545i	$0.284549\pi$
$38$		0			0
$39$	319210.	+	552887.i	0.137980	+	0.238989i
$40$		0			0
$41$			2.68579e6i			0.950467i	0.879860	+	0.475233i	$0.157636\pi$
$41$			2.68579e6i			0.950467i	−0.879860	+	0.475233i	$0.842364\pi$
$42$		0			0
$43$	−1.62184e6			−0.474389			−0.237195	−	0.971462i	$-0.576228\pi$
$43$	−1.62184e6			−0.474389			−0.237195	+	0.971462i	$0.576228\pi$
$44$		0			0
$45$	−1.07122e6	+	618472.i	−0.261234	+	0.150824i
$46$		0			0
$47$	3.06709e6	+	1.77079e6i	0.628544	+	0.362890i	0.780188	−	0.625545i	$-0.215123\pi$
$47$	3.06709e6	+	1.77079e6i	0.628544	+	0.362890i	−0.151644	+	0.988435i	$0.548457\pi$
$48$		0			0
$49$	−1.52090e6	+	5.56056e6i	−0.263825	+	0.964571i
$50$		0			0
$51$	−172625.	+	298996.i	−0.0255166	+	0.0441961i
$52$		0			0
$53$	4.54092e6	+	7.86511e6i	0.575494	+	0.996785i	0.995988	+	0.0894893i	$0.0285235\pi$
$53$	4.54092e6	+	7.86511e6i	0.575494	+	0.996785i	−0.420494	+	0.907295i	$0.638143\pi$
$54$		0			0
$55$			4.64513e6i			0.507630i
$56$		0			0
$57$	−1.43925e6			−0.136344
$58$		0			0
$59$	197222.	−	113866.i	0.0162760	−	0.00939696i	−0.491840	−	0.870686i	$-0.663675\pi$
$59$	197222.	−	113866.i	0.0162760	−	0.00939696i	0.508116	+	0.861289i	$0.330342\pi$
$60$		0			0
$61$	1.18352e7	+	6.83305e6i	0.854783	+	0.493509i	0.862262	−	0.506463i	$-0.169047\pi$
$61$	1.18352e7	+	6.83305e6i	0.854783	+	0.493509i	−0.00747910	+	0.999972i	$0.502381\pi$
$62$		0			0
$63$	−5.20782e6	+	671889.i	−0.330593	+	0.0426516i
$64$		0			0
$65$	−3.86058e6	+	6.68672e6i	−0.216271	+	0.374593i
$66$		0			0
$67$	4.13814e6	+	7.16746e6i	0.205355	+	0.355686i	0.950246	−	0.311501i	$-0.100832\pi$
$67$	4.13814e6	+	7.16746e6i	0.205355	+	0.355686i	−0.744891	+	0.667187i	$0.767498\pi$
$68$		0			0
$69$		−	5.37394e6i		−	0.237081i
$70$		0			0
$71$	−2.81148e7			−1.10637			−0.553187	−	0.833057i	$-0.686588\pi$
$71$	−2.81148e7			−1.10637			−0.553187	+	0.833057i	$0.686588\pi$
$72$		0			0
$73$	1.49937e7	−	8.65664e6i	0.527981	−	0.304830i	−0.212213	−	0.977223i	$-0.568067\pi$
$73$	1.49937e7	−	8.65664e6i	0.527981	−	0.304830i	0.740194	+	0.672393i	$0.234734\pi$
$74$		0			0
$75$	2.86473e6	+	1.65395e6i	0.0905395	+	0.0522730i
$76$		0			0
$77$	−7.59343e6	+	1.81985e7i	−0.216011	+	0.517694i
$78$		0			0
$79$	−1.26003e7	+	2.18243e7i	−0.323498	+	0.560315i	−0.981207	−	0.192957i	$-0.938192\pi$
$79$	−1.26003e7	+	2.18243e7i	−0.323498	+	0.560315i	0.657709	+	0.753272i	$0.271526\pi$
$80$		0			0
$81$	−2.39148e6	−	4.14217e6i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			1.68155e7i			0.354321i	0.984182	+	0.177160i	$0.0566911\pi$
$83$			1.68155e7i			0.354321i	−0.984182	+	0.177160i	$0.943309\pi$
$84$		0			0
$85$	−4.17552e6			−0.0799899
$86$		0			0
$87$	5.02699e6	−	2.90234e6i	0.0877468	−	0.0506606i
$88$		0			0
$89$	6.75025e7	+	3.89726e7i	1.07587	+	0.621154i	0.929779	−	0.368117i	$-0.119997\pi$
$89$	6.75025e7	+	3.89726e7i	1.07587	+	0.621154i	0.146091	+	0.989271i	$0.453331\pi$
$90$		0			0
$91$	−2.60557e7	+	1.98861e7i	−0.379959	+	0.289990i
$92$		0			0
$93$	8.94299e6	−	1.54897e7i	0.119550	−	0.207067i
$94$		0			0
$95$	−8.70325e6	−	1.50745e7i	−0.106853	−	0.185075i
$96$		0			0
$97$		−	1.50031e8i		−	1.69470i	−0.531033	−	0.847351i	$-0.678196\pi$
$97$		−	1.50031e8i		−	1.69470i	0.531033	−	0.847351i	$-0.321804\pi$
$98$		0			0
$99$	−1.79616e7			−0.186984
$100$		0			0
$101$	9.51524e7	−	5.49363e7i	0.914396	−	0.527927i	0.0325532	−	0.999470i	$-0.489636\pi$
$101$	9.51524e7	−	5.49363e7i	0.914396	−	0.527927i	0.881843	+	0.471543i	$0.156303\pi$
$102$		0			0
$103$	−8.77965e7	−	5.06893e7i	−0.780060	−	0.450368i	0.0563912	−	0.998409i	$-0.482041\pi$
$103$	−8.77965e7	−	5.06893e7i	−0.780060	−	0.450368i	−0.836452	+	0.548041i	$0.815374\pi$
$104$		0			0
$105$	−3.85295e7	−	5.04831e7i	−0.316983	−	0.415326i
$106$		0			0
$107$	3.45484e7	−	5.98396e7i	0.263568	−	0.456513i	−0.703619	−	0.710577i	$-0.748434\pi$
$107$	3.45484e7	−	5.98396e7i	0.263568	−	0.456513i	0.967188	+	0.254064i	$0.0817673\pi$
$108$		0			0
$109$	−1.38845e8	−	2.40487e8i	−0.983615	−	1.70367i	−0.647936	−	0.761695i	$-0.724368\pi$
$109$	−1.38845e8	−	2.40487e8i	−0.983615	−	1.70367i	−0.335679	−	0.941977i	$-0.608966\pi$
$110$		0			0
$111$		−	6.34437e7i		−	0.417923i
$112$		0			0
$113$	1.03160e8			0.632700			0.316350	−	0.948642i	$-0.397543\pi$
$113$	1.03160e8			0.632700			0.316350	+	0.948642i	$0.397543\pi$
$114$		0			0
$115$	5.62859e7	−	3.24967e7i	0.321817	−	0.185801i
$116$		0			0
$117$	−2.58560e7	−	1.49280e7i	−0.137980	−	0.0796630i
$118$		0			0
$119$	−1.63587e7	−	6.82575e6i	−0.0815758	−	0.0340379i
$120$		0			0
$121$	7.34535e7	−	1.27225e8i	0.342666	−	0.593515i
$122$		0			0
$123$	−6.28010e7	−	1.08775e8i	−0.274376	−	0.475233i
$124$		0			0
$125$			2.60940e8i			1.06881i
$126$		0			0
$127$	3.26999e7			0.125699			0.0628495	−	0.998023i	$-0.479981\pi$
$127$	3.26999e7			0.125699			0.0628495	+	0.998023i	$0.479981\pi$
$128$		0			0
$129$	6.56846e7	−	3.79230e7i	0.237195	−	0.136944i
$130$		0			0
$131$	1.25512e8	+	7.24644e7i	0.426187	+	0.246059i	0.697721	−	0.716370i	$-0.254198\pi$
$131$	1.25512e8	+	7.24644e7i	0.426187	+	0.246059i	−0.271534	+	0.962429i	$0.587531\pi$
$132$		0			0
$133$	−9.45495e6	−	7.32855e7i	−0.0302171	−	0.234213i
$134$		0			0
$135$	2.89231e7	−	5.00962e7i	0.0870781	−	0.150824i
$136$		0			0
$137$	6.32330e7	+	1.09523e8i	0.179499	+	0.310901i	0.941709	−	0.336428i	$-0.109219\pi$
$137$	6.32330e7	+	1.09523e8i	0.179499	+	0.310901i	−0.762210	+	0.647330i	$0.775886\pi$
$138$		0			0
$139$			2.58673e8i			0.692933i	0.938062	+	0.346467i	$0.112619\pi$
$139$			2.58673e8i			0.692933i	−0.938062	+	0.346467i	$0.887381\pi$
$140$		0			0
$141$	−1.65623e8			−0.419029
$142$		0			0
$143$	−9.70978e7	+	5.60594e7i	−0.232202	+	0.134062i
$144$		0			0
$145$	6.07974e7	+	3.51014e7i	0.137535	+	0.0794058i
$146$		0			0
$147$	−6.84243e7	−	2.60765e8i	−0.146535	−	0.558445i
$148$		0			0
$149$	−3.65743e8	+	6.33485e8i	−0.742046	+	1.28526i	0.209516	+	0.977805i	$0.432811\pi$
$149$	−3.65743e8	+	6.33485e8i	−0.742046	+	1.28526i	−0.951562	+	0.307456i	$0.900522\pi$
$150$		0			0
$151$	−2.36502e8	−	4.09634e8i	−0.454912	−	0.787930i	0.543771	−	0.839233i	$-0.316996\pi$
$151$	−2.36502e8	−	4.09634e8i	−0.454912	−	0.787930i	−0.998683	+	0.0513032i	$0.983663\pi$
$152$		0			0
$153$		−	1.61458e7i		−	0.0294641i
$154$		0			0
$155$	2.16316e8			0.374768
$156$		0			0
$157$	−5.90781e7	+	3.41087e7i	−0.0972361	+	0.0561393i	−0.547830	−	0.836590i	$-0.684546\pi$
$157$	−5.90781e7	+	3.41087e7i	−0.0972361	+	0.0561393i	0.450593	+	0.892729i	$0.351212\pi$
$158$		0			0
$159$	−3.67815e8	−	2.12358e8i	−0.575494	−	0.332262i
$160$		0			0
$161$	2.73638e8	−	3.53034e7i	0.407260	−	0.0525428i
$162$		0			0
$163$	−2.68505e8	+	4.65065e8i	−0.380367	+	0.658814i	−0.991115	−	0.133011i	$-0.957536\pi$
$163$	−2.68505e8	+	4.65065e8i	−0.380367	+	0.658814i	0.610748	+	0.791825i	$0.290869\pi$
$164$		0			0
$165$	−1.08616e8	−	1.88128e8i	−0.146540	−	0.253815i
$166$		0			0
$167$			1.30300e9i			1.67524i	0.546253	+	0.837620i	$0.316054\pi$
$167$			1.30300e9i			1.67524i	−0.546253	+	0.837620i	$0.683946\pi$
$168$		0			0
$169$	6.29366e8			0.771537
$170$		0			0
$171$	5.82895e7	−	3.36534e7i	0.0681719	−	0.0393591i
$172$		0			0
$173$	1.04566e9	+	6.03713e8i	1.16737	+	0.673979i	0.953058	−	0.302787i	$-0.0979171\pi$
$173$	1.04566e9	+	6.03713e8i	1.16737	+	0.673979i	0.214308	+	0.976766i	$0.431250\pi$
$174$		0			0
$175$	−6.53987e7	+	1.56736e8i	−0.0697296	+	0.167115i
$176$		0			0
$177$	−5.32500e6	+	9.22317e6i	−0.00542534	+	0.00939696i
$178$		0			0
$179$	8.89506e8	+	1.54067e9i	0.866437	+	1.50071i	0.865614	+	0.500713i	$0.166929\pi$
$179$	8.89506e8	+	1.54067e9i	0.866437	+	1.50071i	0.000823108		1.00000i	$0.499738\pi$
$180$		0			0
$181$			9.12768e7i			0.0850445i	0.999096	+	0.0425222i	$0.0135393\pi$
$181$			9.12768e7i			0.0850445i	−0.999096	+	0.0425222i	$0.986461\pi$
$182$		0			0
$183$	−6.39100e8			−0.569855
$184$		0			0
$185$	6.64501e8	−	3.83650e8i	0.567294	−	0.327528i
$186$		0			0
$187$	−5.25095e7	−	3.03164e7i	−0.0429409	−	0.0247919i
$188$		0			0
$189$	1.95206e8	−	1.48984e8i	0.152984	−	0.116760i
$190$		0			0
$191$	−1.05197e9	+	1.82206e9i	−0.790438	+	1.36908i	0.135258	+	0.990810i	$0.456814\pi$
$191$	−1.05197e9	+	1.82206e9i	−0.790438	+	1.36908i	−0.925696	+	0.378269i	$0.876520\pi$
$192$		0			0
$193$	−9.00818e8	−	1.56026e9i	−0.649244	−	1.12452i	−0.983304	−	0.181971i	$-0.941752\pi$
$193$	−9.00818e8	−	1.56026e9i	−0.649244	−	1.12452i	0.334060	−	0.942552i	$-0.391581\pi$
$194$		0			0
$195$		−	3.61083e8i		−	0.249729i
$196$		0			0
$197$	1.10059e9			0.730736			0.365368	−	0.930863i	$-0.380943\pi$
$197$	1.10059e9			0.730736			0.365368	+	0.930863i	$0.380943\pi$
$198$		0			0
$199$	−2.64767e9	+	1.52864e9i	−1.68831	+	0.974746i	−0.732498	+	0.680769i	$0.761646\pi$
$199$	−2.64767e9	+	1.52864e9i	−1.68831	+	0.974746i	−0.955812	+	0.293977i	$0.905021\pi$
$200$		0			0
$201$	−3.35189e8	−	1.93522e8i	−0.205355	−	0.118562i
$202$		0			0
$203$	1.80809e8	+	2.36905e8i	0.106472	+	0.139505i
$204$		0			0
$205$	7.59527e8	−	1.31554e9i	0.430059	−	0.744884i
$206$		0			0
$207$	1.25657e8	+	2.17644e8i	0.0684393	+	0.118540i
$208$		0			0
$209$		−	2.52760e8i		−	0.132472i
$210$		0			0
$211$	−1.51847e9			−0.766083			−0.383041	−	0.923731i	$-0.625123\pi$
$211$	−1.51847e9			−0.766083			−0.383041	+	0.923731i	$0.625123\pi$
$212$		0			0
$213$	1.13865e9	−	6.57400e8i	0.553187	−	0.319382i
$214$		0			0
$215$	7.94402e8	+	4.58648e8i	0.371780	+	0.214647i
$216$		0			0
$217$	8.47476e8	+	3.53614e8i	0.382198	+	0.159474i
$218$		0			0
$219$	−4.04831e8	+	7.01188e8i	−0.175994	+	0.304830i
$220$		0			0
$221$	−5.03920e7	−	8.72815e7i	−0.0211248	−	0.0365892i
$222$		0			0
$223$		−	3.17111e9i		−	1.28231i	−0.767413	−	0.641153i	$-0.778456\pi$
$223$		−	3.17111e9i		−	1.28231i	0.767413	−	0.641153i	$-0.221544\pi$
$224$		0			0
$225$	−1.54695e8			−0.0603597
$226$		0			0
$227$	−1.03782e9	+	5.99185e8i	−0.390857	+	0.225662i	−0.682532	−	0.730856i	$-0.739121\pi$
$227$	−1.03782e9	+	5.99185e8i	−0.390857	+	0.225662i	0.291674	+	0.956518i	$0.405788\pi$
$228$		0			0
$229$	3.28021e9	+	1.89383e9i	1.19278	+	0.688651i	0.958936	−	0.283624i	$-0.0915368\pi$
$229$	3.28021e9	+	1.89383e9i	1.19278	+	0.688651i	0.233842	+	0.972275i	$0.424870\pi$
$230$		0			0
$231$	−1.17997e8	−	9.14595e8i	−0.0414402	−	0.321204i
$232$		0			0
$233$	4.07388e8	−	7.05617e8i	0.138224	−	0.239412i	−0.788600	−	0.614906i	$-0.789194\pi$
$233$	4.07388e8	−	7.05617e8i	0.138224	−	0.239412i	0.926825	+	0.375495i	$0.122527\pi$
$234$		0			0
$235$	−1.00154e9	−	1.73471e9i	−0.328394	−	0.568796i
$236$		0			0
$237$		−	1.17851e9i		−	0.373543i
$238$		0			0
$239$	−5.21004e9			−1.59679			−0.798397	−	0.602131i	$-0.794318\pi$
$239$	−5.21004e9			−1.59679			−0.798397	+	0.602131i	$0.794318\pi$
$240$		0			0
$241$	7.50376e8	−	4.33230e8i	0.222439	−	0.128425i	−0.384640	−	0.923067i	$-0.625674\pi$
$241$	7.50376e8	−	4.33230e8i	0.222439	−	0.128425i	0.607079	+	0.794641i	$0.292341\pi$
$242$		0			0
$243$	1.93710e8	+	1.11839e8i	0.0555556	+	0.0320750i
$244$		0			0
$245$	2.31745e9	−	2.29354e9i	0.643201	−	0.636564i
$246$		0			0
$247$	2.10069e8	−	3.63850e8i	0.0564384	−	0.0977541i
$248$		0			0
$249$	−3.93191e8	−	6.81027e8i	−0.102284	−	0.177160i
$250$		0			0
$251$		−	6.32101e9i		−	1.59255i	−0.604938	−	0.796273i	$-0.706802\pi$
$251$		−	6.32101e9i		−	1.59255i	0.604938	−	0.796273i	$-0.293198\pi$
$252$		0			0
$253$	9.43768e8			0.230347
$254$		0			0
$255$	1.69109e8	−	9.76349e7i	0.0399949	−	0.0230911i
$256$		0			0
$257$	2.87584e8	+	1.66037e8i	0.0659222	+	0.0380602i	0.532599	−	0.846368i	$-0.321215\pi$
$257$	2.87584e8	+	1.66037e8i	0.0659222	+	0.0380602i	−0.466677	+	0.884428i	$0.654549\pi$
$258$		0			0
$259$	3.23051e9	−	4.16785e8i	0.717914	−	0.0926218i
$260$		0			0
$261$	−1.35729e8	+	2.35089e8i	−0.0292489	+	0.0506606i
$262$		0			0
$263$	−1.41286e9	−	2.44714e9i	−0.295308	−	0.511488i	0.679749	−	0.733445i	$-0.262089\pi$
$263$	−1.41286e9	−	2.44714e9i	−0.295308	−	0.511488i	−0.975056	+	0.221957i	$0.928755\pi$
$264$		0			0
$265$		−	5.13659e9i		−	1.04158i
$266$		0			0
$267$	−3.64514e9			−0.717247
$268$		0			0
$269$	4.64753e9	−	2.68326e9i	0.887592	−	0.512452i	0.0144381	−	0.999896i	$-0.495404\pi$
$269$	4.64753e9	−	2.68326e9i	0.887592	−	0.512452i	0.873154	+	0.487444i	$0.162071\pi$
$270$		0			0
$271$	2.31806e9	+	1.33833e9i	0.429782	+	0.248135i	0.699254	−	0.714874i	$-0.253516\pi$
$271$	2.31806e9	+	1.33833e9i	0.429782	+	0.248135i	−0.269472	+	0.963008i	$0.586849\pi$
$272$		0			0
$273$	5.90265e8	−	1.41464e9i	0.106266	−	0.254680i
$274$		0			0
$275$	−2.90466e8	+	5.03102e8i	−0.0507884	+	0.0879681i
$276$		0			0
$277$	−1.44245e9	−	2.49840e9i	−0.245009	−	0.424368i	0.717125	−	0.696944i	$-0.245457\pi$
$277$	−1.44245e9	−	2.49840e9i	−0.245009	−	0.424368i	−0.962134	+	0.272577i	$0.912124\pi$
$278$		0			0
$279$			8.36444e8i			0.138045i
$280$		0			0
$281$	−5.10904e9			−0.819434			−0.409717	−	0.912213i	$-0.634373\pi$
$281$	−5.10904e9			−0.819434			−0.409717	+	0.912213i	$0.634373\pi$
$282$		0			0
$283$	−6.48419e9	+	3.74365e9i	−1.01090	+	0.583646i	−0.911456	−	0.411397i	$-0.865041\pi$
$283$	−6.48419e9	+	3.74365e9i	−1.01090	+	0.583646i	−0.0994476	+	0.995043i	$0.531708\pi$
$284$		0			0
$285$	7.04964e8	+	4.07011e8i	0.106853	+	0.0616917i
$286$		0			0
$287$	5.12617e9	−	3.91237e9i	0.755554	−	0.576650i
$288$		0			0
$289$	−3.46063e9	+	5.99398e9i	−0.496093	+	0.859259i
$290$		0			0
$291$	3.50812e9	+	6.07624e9i	0.489218	+	0.847351i
$292$		0			0
$293$			9.75896e8i			0.132414i	0.997806	+	0.0662069i	$0.0210897\pi$
$293$			9.75896e8i			0.132414i	−0.997806	+	0.0662069i	$0.978910\pi$
$294$		0			0
$295$	−1.28803e8			−0.0170074
$296$		0			0
$297$	7.27446e8	−	4.19991e8i	0.0934921	−	0.0539777i
$298$		0			0
$299$	1.35856e9	+	7.84368e8i	0.169979	+	0.0981375i
$300$		0			0
$301$	2.36252e9	+	3.09549e9i	0.287813	+	0.377106i
$302$		0			0
$303$	−2.56912e9	+	4.44984e9i	−0.304799	+	0.527927i
$304$		0			0
$305$	−3.86470e9	−	6.69385e9i	−0.446597	−	0.773529i
$306$		0			0
$307$			5.89263e9i			0.663370i	0.943390	+	0.331685i	$0.107617\pi$
$307$			5.89263e9i			0.663370i	−0.943390	+	0.331685i	$0.892383\pi$
$308$		0			0
$309$	4.74101e9			0.520040
$310$		0			0
$311$	−1.04962e10	+	6.05996e9i	−1.12199	+	0.647781i	−0.941908	−	0.335870i	$-0.890970\pi$
$311$	−1.04962e10	+	6.05996e9i	−1.12199	+	0.647781i	−0.180082	+	0.983652i	$0.557636\pi$
$312$		0			0
$313$	7.55111e9	+	4.35963e9i	0.786744	+	0.454227i	0.838815	−	0.544417i	$-0.183249\pi$
$313$	7.55111e9	+	4.35963e9i	0.786744	+	0.454227i	−0.0520711	+	0.998643i	$0.516582\pi$
$314$		0			0
$315$	2.74087e9	+	1.14364e9i	0.278386	+	0.116158i
$316$		0			0
$317$	−6.70410e9	+	1.16118e10i	−0.663901	+	1.14991i	0.315681	+	0.948866i	$0.397767\pi$
$317$	−6.70410e9	+	1.16118e10i	−0.663901	+	1.14991i	−0.979582	+	0.201045i	$0.935566\pi$
$318$		0			0
$319$	5.09707e8	+	8.82838e8i	0.0492218	+	0.0852547i
$320$		0			0
$321$			3.23134e9i			0.304342i
$322$		0			0
$323$	2.27206e8			0.0208742
$324$		0			0
$325$	−8.36258e8	+	4.82814e8i	−0.0749561	+	0.0432759i
$326$		0			0
$327$	1.12465e10	+	6.49315e9i	0.983615	+	0.567890i
$328$		0			0
$329$	−1.08804e9	−	8.43342e9i	−0.0928670	−	0.719814i
$330$		0			0
$331$	−8.77673e9	+	1.52017e10i	−0.731174	+	1.26643i	0.225208	+	0.974311i	$0.427694\pi$
$331$	−8.77673e9	+	1.52017e10i	−0.731174	+	1.26643i	−0.956382	+	0.292120i	$0.905639\pi$
$332$		0			0
$333$	1.48348e9	+	2.56947e9i	0.120644	+	0.208962i
$334$		0			0
$335$		−	4.68097e9i		−	0.371669i
$336$		0			0
$337$	3.68707e9			0.285865			0.142933	−	0.989732i	$-0.454347\pi$
$337$	3.68707e9			0.285865			0.142933	+	0.989732i	$0.454347\pi$
$338$		0			0
$339$	−4.17799e9	+	2.41216e9i	−0.316350	+	0.182645i
$340$		0			0
$341$	2.72029e9	+	1.57056e9i	0.201186	+	0.116155i
$342$		0			0
$343$	1.28285e10	−	5.19719e9i	0.926829	−	0.375485i
$344$		0			0
$345$	−1.51972e9	+	2.63223e9i	−0.107272	+	0.185801i
$346$		0			0
$347$	−8.27333e9	−	1.43298e10i	−0.570640	−	0.988378i	−0.996500	−	0.0835889i	$-0.973362\pi$
$347$	−8.27333e9	−	1.43298e10i	−0.570640	−	0.988378i	0.425860	−	0.904789i	$-0.359972\pi$
$348$		0			0
$349$			8.55255e9i			0.576493i	0.957556	+	0.288246i	$0.0930722\pi$
$349$			8.55255e9i			0.576493i	−0.957556	+	0.288246i	$0.906928\pi$
$350$		0			0
$351$	1.39622e9			0.0919870
$352$		0			0
$353$	−5.79315e9	+	3.34468e9i	−0.373092	+	0.215405i	−0.674809	−	0.737993i	$-0.735774\pi$
$353$	−5.79315e9	+	3.34468e9i	−0.373092	+	0.215405i	0.301716	+	0.953398i	$0.402440\pi$
$354$		0			0
$355$	1.37710e10	+	7.95071e9i	0.867068	+	0.500602i
$356$		0			0
$357$	8.22132e8	−	1.06068e8i	0.0506138	−	0.00652995i
$358$		0			0
$359$	2.06593e9	−	3.57829e9i	0.124376	−	0.215426i	−0.797113	−	0.603830i	$-0.793640\pi$
$359$	2.06593e9	−	3.57829e9i	0.124376	−	0.215426i	0.921489	+	0.388405i	$0.126974\pi$
$360$		0			0
$361$	−8.01820e9	−	1.38879e10i	−0.472116	−	0.817728i
$362$		0			0
$363$			6.87016e9i			0.395677i
$364$		0			0
$365$	−9.79220e9			−0.551707
$366$		0			0
$367$	−1.18309e9	+	6.83059e8i	−0.0652161	+	0.0376525i	−0.532253	−	0.846585i	$-0.678655\pi$
$367$	−1.18309e9	+	6.83059e8i	−0.0652161	+	0.0376525i	0.467037	+	0.884238i	$0.345321\pi$
$368$		0			0
$369$	5.08688e9	+	2.93691e9i	0.274376	+	0.158411i
$370$		0			0
$371$	8.39682e9	−	2.01240e10i	0.443220	−	1.06223i
$372$		0			0
$373$	−1.47249e10	+	2.55043e10i	−0.760707	+	1.31758i	0.181780	+	0.983339i	$0.441814\pi$
$373$	−1.47249e10	+	2.55043e10i	−0.760707	+	1.31758i	−0.942487	+	0.334244i	$0.891519\pi$
$374$		0			0
$375$	−6.10147e9	−	1.05681e10i	−0.308538	−	0.534404i
$376$		0			0
$377$			1.69447e9i			0.0838821i
$378$		0			0
$379$	−3.21245e8			−0.0155697			−0.00778485	−	0.999970i	$-0.502478\pi$
$379$	−3.21245e8			−0.0155697			−0.00778485	+	0.999970i	$0.502478\pi$
$380$		0			0
$381$	−1.32435e9	+	7.64612e8i	−0.0628495	+	0.0362862i
$382$		0			0
$383$	1.17225e10	+	6.76799e9i	0.544785	+	0.314532i	0.747016	−	0.664806i	$-0.231486\pi$
$383$	1.17225e10	+	6.76799e9i	0.544785	+	0.314532i	−0.202231	+	0.979338i	$0.564819\pi$
$384$		0			0
$385$	8.86582e9	−	6.76653e9i	0.403530	−	0.307980i
$386$		0			0
$387$	−1.77348e9	+	3.07177e9i	−0.0790649	+	0.136944i
$388$		0			0
$389$	−1.51250e10	−	2.61972e10i	−0.660535	−	1.14408i	−0.980475	−	0.196643i	$-0.936996\pi$
$389$	−1.51250e10	−	2.61972e10i	−0.660535	−	1.14408i	0.319940	−	0.947438i	$-0.396337\pi$
$390$		0			0
$391$			8.48356e8i			0.0362970i
$392$		0			0
$393$	−6.77765e9			−0.284125
$394$		0			0
$395$	1.23436e10	−	7.12657e9i	0.507053	−	0.292747i
$396$		0			0
$397$	1.60828e10	+	9.28540e9i	0.647440	+	0.373799i	0.787475	−	0.616347i	$-0.211388\pi$
$397$	1.60828e10	+	9.28540e9i	0.647440	+	0.373799i	−0.140035	+	0.990147i	$0.544721\pi$
$398$		0			0
$399$	2.09654e9	+	2.74698e9i	0.0827201	+	0.108384i
$400$		0			0
$401$	−1.89713e10	+	3.28592e10i	−0.733701	+	1.27081i	0.221589	+	0.975140i	$0.428876\pi$
$401$	−1.89713e10	+	3.28592e10i	−0.733701	+	1.27081i	−0.955291	+	0.295668i	$0.904458\pi$
$402$		0			0
$403$	2.61060e9	+	4.52169e9i	0.0989737	+	0.171427i
$404$		0			0
$405$			2.70519e9i			0.100549i
$406$		0			0
$407$	1.11419e10			0.406054
$408$		0			0
$409$	2.67249e10	−	1.54296e10i	0.955042	−	0.551394i	0.0603984	−	0.998174i	$-0.480763\pi$
$409$	2.67249e10	−	1.54296e10i	0.955042	−	0.551394i	0.894644	+	0.446781i	$0.147430\pi$
$410$		0			0
$411$	−5.12188e9	−	2.95712e9i	−0.179499	−	0.103634i
$412$		0			0
$413$	−5.04620e8	−	2.10555e8i	−0.0173446	−	0.00723712i
$414$		0			0
$415$	4.75532e9	−	8.23646e9i	0.160320	−	0.277682i
$416$		0			0
$417$	−6.04846e9	−	1.04762e10i	−0.200033	−	0.346467i
$418$		0			0
$419$		−	3.55878e10i		−	1.15464i	−0.816519	−	0.577318i	$-0.804099\pi$
$419$		−	3.55878e10i		−	1.15464i	0.816519	−	0.577318i	$-0.195901\pi$
$420$		0			0
$421$	−4.75589e10			−1.51392			−0.756961	−	0.653460i	$-0.773317\pi$
$421$	−4.75589e10			−1.51392			−0.756961	+	0.653460i	$0.773317\pi$
$422$		0			0
$423$	6.70773e9	−	3.87271e9i	0.209515	−	0.120963i
$424$		0			0
$425$	−4.52240e8	−	2.61101e8i	−0.0138616	−	0.00800299i
$426$		0			0
$427$	−4.19849e9	−	3.25426e10i	−0.126294	−	0.978905i
$428$		0			0
$429$	2.62164e9	−	4.54081e9i	0.0774005	−	0.134062i
$430$		0			0
$431$	−2.46357e10	−	4.26703e10i	−0.713930	−	1.23656i	−0.963371	−	0.268174i	$-0.913580\pi$
$431$	−2.46357e10	−	4.26703e10i	−0.713930	−	1.23656i	0.249440	−	0.968390i	$-0.419753\pi$
$432$		0			0
$433$		−	3.81881e10i		−	1.08637i	−0.839614	−	0.543184i	$-0.817219\pi$
$433$		−	3.81881e10i		−	1.08637i	0.839614	−	0.543184i	$-0.182781\pi$
$434$		0			0
$435$	−3.28306e9			−0.0916899
$436$		0			0
$437$	−3.06273e9	+	1.76827e9i	−0.0839815	+	0.0484867i
$438$		0			0
$439$	1.68590e10	+	9.73352e9i	0.453913	+	0.262067i	0.709481	−	0.704724i	$-0.248929\pi$
$439$	1.68590e10	+	9.73352e9i	0.453913	+	0.262067i	−0.255568	+	0.966791i	$0.582263\pi$
$440$		0			0
$441$	8.86858e9	+	8.96105e9i	0.234477	+	0.236922i
$442$		0			0
$443$	7.42727e9	−	1.28644e10i	0.192848	−	0.334022i	−0.753345	−	0.657625i	$-0.771561\pi$
$443$	7.42727e9	−	1.28644e10i	0.192848	−	0.334022i	0.946193	+	0.323603i	$0.104894\pi$
$444$		0			0
$445$	−2.20425e10	−	3.81787e10i	−0.562109	−	0.973601i
$446$		0			0
$447$		−	3.42082e10i		−	0.856841i
$448$		0			0
$449$	−1.59088e10			−0.391429			−0.195714	−	0.980661i	$-0.562703\pi$
$449$	−1.59088e10			−0.391429			−0.195714	+	0.980661i	$0.562703\pi$
$450$		0			0
$451$	1.91029e10	−	1.10291e10i	0.461736	−	0.266584i
$452$		0			0
$453$	1.91567e10	+	1.10601e10i	0.454912	+	0.262643i
$454$		0			0
$455$	1.83861e10	−	2.37209e9i	0.428987	−	0.0553459i
$456$		0			0
$457$	−6.88140e9	+	1.19189e10i	−0.157765	+	0.273258i	−0.934063	−	0.357109i	$-0.883762\pi$
$457$	−6.88140e9	+	1.19189e10i	−0.157765	+	0.273258i	0.776297	+	0.630367i	$0.217096\pi$
$458$		0			0
$459$	3.77531e8	+	6.53903e8i	0.00850555	+	0.0147320i
$460$		0			0
$461$			2.30707e10i			0.510806i	0.966835	+	0.255403i	$0.0822082\pi$
$461$			2.30707e10i			0.510806i	−0.966835	+	0.255403i	$0.917792\pi$
$462$		0			0
$463$	−4.27958e10			−0.931273			−0.465636	−	0.884976i	$-0.654175\pi$
$463$	−4.27958e10			−0.931273			−0.465636	+	0.884976i	$0.654175\pi$
$464$		0			0
$465$	−8.76081e9	+	5.05806e9i	−0.187384	+	0.108186i
$466$		0			0
$467$	8.02470e10	+	4.63306e10i	1.68718	+	0.974093i	0.956663	+	0.291198i	$0.0940538\pi$
$467$	8.02470e10	+	4.63306e10i	1.68718	+	0.974093i	0.730516	+	0.682895i	$0.239280\pi$
$468$		0			0
$469$	7.65201e9	−	1.83389e10i	0.158156	−	0.379038i
$470$		0			0
$471$	1.59511e9	−	2.76281e9i	0.0324120	−	0.0561393i
$472$		0			0
$473$	6.66002e9	+	1.15355e10i	0.133055	+	0.230458i
$474$		0			0
$475$		−	2.17690e9i		−	0.0427626i
$476$		0			0
$477$	1.98620e10			0.383663
$478$		0			0
$479$	−5.36846e9	+	3.09948e9i	−0.101978	+	0.0588772i	−0.550122	−	0.835084i	$-0.685419\pi$
$479$	−5.36846e9	+	3.09948e9i	−0.101978	+	0.0588772i	0.448143	+	0.893962i	$0.352085\pi$
$480$		0			0
$481$	1.60390e10	+	9.26009e9i	0.299637	+	0.172996i
$482$		0			0
$483$	−1.02568e10	+	7.82817e9i	−0.188462	+	0.143837i
$484$		0			0
$485$	−4.24279e10	+	7.34872e10i	−0.766804	+	1.32814i
$486$		0			0
$487$	1.27148e9	+	2.20226e9i	0.0226044	+	0.0391519i	0.877106	−	0.480296i	$-0.159471\pi$
$487$	1.27148e9	+	2.20226e9i	0.0226044	+	0.0391519i	−0.854502	+	0.519448i	$0.826138\pi$
$488$		0			0
$489$		−	2.51135e10i		−	0.439210i
$490$		0			0
$491$	4.42855e10			0.761966			0.380983	−	0.924582i	$-0.375586\pi$
$491$	4.42855e10			0.761966			0.380983	+	0.924582i	$0.375586\pi$
$492$		0			0
$493$	−7.93585e8	+	4.58177e8i	−0.0134340	+	0.00775613i
$494$		0			0
$495$	8.79787e9	+	5.07945e9i	0.146540	+	0.0846050i
$496$		0			0
$497$	4.09546e10	+	5.36606e10i	0.671239	+	0.879489i
$498$		0			0
$499$	−1.84607e10	+	3.19749e10i	−0.297746	+	0.515711i	−0.975620	−	0.219467i	$-0.929568\pi$
$499$	−1.84607e10	+	3.19749e10i	−0.297746	+	0.515711i	0.677874	+	0.735178i	$0.262902\pi$
$500$		0			0
$501$	−3.04675e10	−	5.27713e10i	−0.483600	−	0.837620i
$502$		0			0
$503$			1.57937e10i			0.246724i	0.992362	+	0.123362i	$0.0393677\pi$
$503$			1.57937e10i			0.246724i	−0.992362	+	0.123362i	$0.960632\pi$
$504$		0			0
$505$	−6.21427e10			−0.955487
$506$		0			0
$507$	−2.54893e10	+	1.47163e10i	−0.385768	+	0.222723i
$508$		0			0
$509$	3.02409e10	+	1.74596e10i	0.450530	+	0.260114i	0.708054	−	0.706158i	$-0.249573\pi$
$509$	3.02409e10	+	1.74596e10i	0.450530	+	0.260114i	−0.257524	+	0.966272i	$0.582907\pi$
$510$		0			0
$511$	−3.83635e10	−	1.60074e10i	−0.562646	−	0.234767i
$512$		0			0
$513$	−1.57382e9	+	2.72593e9i	−0.0227240	+	0.0393591i
$514$		0			0
$515$	2.86693e10	+	4.96567e10i	0.407557	+	0.705910i
$516$		0			0
$517$		−	2.90866e10i		−	0.407128i
$518$		0			0
$519$	−5.64658e10			−0.778244
$520$		0			0
$521$	−1.03962e11	+	6.00222e10i	−1.41098	+	0.814631i	−0.995481	−	0.0949612i	$-0.969727\pi$
$521$	−1.03962e11	+	6.00222e10i	−1.41098	+	0.814631i	−0.415502	+	0.909592i	$0.636394\pi$
$522$		0			0
$523$	4.75841e10	+	2.74727e10i	0.635997	+	0.367193i	0.783071	−	0.621933i	$-0.213652\pi$
$523$	4.75841e10	+	2.74727e10i	0.635997	+	0.367193i	−0.147074	+	0.989125i	$0.546986\pi$
$524$		0			0
$525$	−1.01625e9	−	7.87699e9i	−0.0133772	−	0.103687i
$526$		0			0
$527$	−1.41178e9	+	2.44528e9i	−0.0183031	+	0.0317020i
$528$		0			0
$529$	3.25530e10	+	5.63835e10i	0.415689	+	0.719995i
$530$		0			0
$531$		−	4.98051e8i		−	0.00626464i
$532$		0			0
$533$	3.66652e10			0.454303
$534$		0			0
$535$	−3.38446e10	+	1.95402e10i	−0.413118	+	0.238514i
$536$		0			0
$537$	−7.20500e10	−	4.15981e10i	−0.866437	−	0.500237i
$538$		0			0
$539$	4.57955e10	−	1.20166e10i	0.542585	−	0.142373i
$540$		0			0
$541$	−2.78606e10	+	4.82561e10i	−0.325239	+	0.563330i	−0.981561	−	0.191151i	$-0.938778\pi$
$541$	−2.78606e10	+	4.82561e10i	−0.325239	+	0.563330i	0.656322	+	0.754481i	$0.272111\pi$
$542$		0			0
$543$	−2.13430e9	−	3.69671e9i	−0.0245502	−	0.0425222i
$544$		0			0
$545$			1.57059e11i			1.78023i
$546$		0			0
$547$	5.94969e10			0.664577			0.332288	−	0.943178i	$-0.392179\pi$
$547$	5.94969e10			0.664577			0.332288	+	0.943178i	$0.392179\pi$
$548$		0			0
$549$	2.58835e10	−	1.49439e10i	0.284928	−	0.164503i
$550$		0			0
$551$	−3.30822e9	−	1.91000e9i	−0.0358912	−	0.0207218i
$552$		0			0
$553$	6.00091e10	−	7.74209e9i	0.641678	−	0.0827862i
$554$		0			0
$555$	−1.79415e10	+	3.10756e10i	−0.189098	+	0.327528i
$556$		0			0
$557$	4.87231e10	+	8.43909e10i	0.506191	+	0.876748i	0.999974	+	0.00716351i	$0.00228023\pi$
$557$	4.87231e10	+	8.43909e10i	0.506191	+	0.876748i	−0.493783	+	0.869585i	$0.664386\pi$
$558$		0			0
$559$			2.21406e10i			0.226748i
$560$		0			0
$561$	2.83551e9			0.0286273
$562$		0			0
$563$	6.40959e10	−	3.70058e10i	0.637965	−	0.368329i	−0.145865	−	0.989304i	$-0.546597\pi$
$563$	6.40959e10	−	3.70058e10i	0.637965	−	0.368329i	0.783830	+	0.620975i	$0.213263\pi$
$564$		0			0
$565$	−5.05293e10	−	2.91731e10i	−0.495849	−	0.286279i
$566$		0			0
$567$	−4.42220e9	+	1.05983e10i	−0.0427864	+	0.102543i
$568$		0			0
$569$	−2.27452e10	+	3.93958e10i	−0.216990	+	0.375838i	−0.953886	−	0.300168i	$-0.902957\pi$
$569$	−2.27452e10	+	3.93958e10i	−0.216990	+	0.375838i	0.736896	+	0.676006i	$0.236291\pi$
$570$		0			0
$571$	2.29721e10	+	3.97888e10i	0.216101	+	0.374297i	0.953613	−	0.301037i	$-0.0973327\pi$
$571$	2.29721e10	+	3.97888e10i	0.216101	+	0.374297i	−0.737512	+	0.675334i	$0.763999\pi$
$572$		0			0
$573$		−	9.83911e10i		−	0.912719i
$574$		0			0
$575$	8.12824e9			0.0743575
$576$		0			0
$577$	1.95631e10	−	1.12948e10i	0.176496	−	0.101900i	−0.409149	−	0.912467i	$-0.634174\pi$
$577$	1.95631e10	−	1.12948e10i	0.176496	−	0.101900i	0.585645	+	0.810567i	$0.300841\pi$
$578$		0			0
$579$	7.29663e10	+	4.21271e10i	0.649244	+	0.374841i
$580$		0			0
$581$	3.20944e10	−	2.44950e10i	0.281660	−	0.214967i
$582$		0			0
$583$	3.72942e10	−	6.45954e10i	0.322825	−	0.559149i
$584$		0			0
$585$	8.44309e9	+	1.46239e10i	0.0720905	+	0.124864i
$586$		0			0
$587$			5.75424e10i			0.484658i	0.970194	+	0.242329i	$0.0779113\pi$
$587$			5.75424e10i			0.484658i	−0.970194	+	0.242329i	$0.922089\pi$
$588$		0			0
$589$	−1.17706e10			−0.0977997
$590$		0			0
$591$	−4.45739e10	+	2.57347e10i	−0.365368	+	0.210945i
$592$		0			0
$593$	−1.27721e11	−	7.37397e10i	−1.03286	−	0.596325i	−0.115061	−	0.993358i	$-0.536706\pi$
$593$	−1.27721e11	−	7.37397e10i	−1.03286	−	0.596325i	−0.917804	+	0.397034i	$0.870040\pi$
$594$		0			0
$595$	6.08245e9	+	7.96951e9i	0.0485300	+	0.0635863i
$596$		0			0
$597$	7.14872e10	−	1.23819e11i	0.562770	−	0.974746i
$598$		0			0
$599$	−5.35270e10	−	9.27114e10i	−0.415782	−	0.720155i	0.579728	−	0.814810i	$-0.303159\pi$
$599$	−5.35270e10	−	9.27114e10i	−0.415782	−	0.720155i	−0.995510	+	0.0946547i	$0.969825\pi$
$600$		0			0
$601$		−	1.21889e11i		−	0.934257i	−0.884189	−	0.467129i	$-0.845289\pi$
$601$		−	1.21889e11i		−	0.934257i	0.884189	−	0.467129i	$-0.154711\pi$
$602$		0			0
$603$	1.81002e10			0.136903
$604$		0			0
$605$	−7.19572e10	+	4.15445e10i	−0.537097	+	0.310093i
$606$		0			0
$607$	−1.76210e11	−	1.01735e11i	−1.29800	−	0.749402i	−0.317943	−	0.948110i	$-0.602992\pi$
$607$	−1.76210e11	−	1.01735e11i	−1.29800	−	0.749402i	−0.980059	+	0.198708i	$0.936325\pi$
$608$		0			0
$609$	−1.28622e10	−	5.36684e9i	−0.0935078	−	0.0390166i
$610$		0			0
$611$	2.41740e10	−	4.18705e10i	0.173453	−	0.300430i
$612$		0			0
$613$	−3.36957e10	−	5.83627e10i	−0.238634	−	0.413327i	0.721688	−	0.692218i	$-0.243366\pi$
$613$	−3.36957e10	−	5.83627e10i	−0.238634	−	0.413327i	−0.960323	+	0.278891i	$0.910033\pi$
$614$		0			0
$615$			7.10392e10i			0.496589i
$616$		0			0
$617$	1.24394e11			0.858338			0.429169	−	0.903224i	$-0.358807\pi$
$617$	1.24394e11			0.858338			0.429169	+	0.903224i	$0.358807\pi$
$618$		0			0
$619$	1.35672e9	−	7.83300e8i	0.00924115	−	0.00533538i	−0.495372	−	0.868681i	$-0.664968\pi$
$619$	1.35672e9	−	7.83300e8i	0.00924115	−	0.00533538i	0.504613	+	0.863345i	$0.331635\pi$
$620$		0			0
$621$	−1.01782e10	−	5.87640e9i	−0.0684393	−	0.0395134i
$622$		0			0
$623$	−2.39463e10	−	1.85608e11i	−0.158959	−	1.23210i
$624$		0			0
$625$	5.99771e10	−	1.03883e11i	0.393066	−	0.680810i
$626$		0			0
$627$	5.91020e9	+	1.02368e10i	0.0382412	+	0.0662358i
$628$		0			0
$629$			1.00155e10i			0.0639840i
$630$		0			0
$631$	−1.06449e11			−0.671466			−0.335733	−	0.941957i	$-0.608984\pi$
$631$	−1.06449e11			−0.671466			−0.335733	+	0.941957i	$0.608984\pi$
$632$		0			0
$633$	6.14979e10	−	3.55058e10i	0.383041	−	0.221149i
$634$		0			0
$635$	−1.60169e10	−	9.24736e9i	−0.0985107	−	0.0568752i
$636$		0			0
$637$	7.59101e10	+	2.07626e10i	0.461044	+	0.126103i
$638$		0			0
$639$	−3.07435e10	+	5.32494e10i	−0.184396	+	0.319382i
$640$		0			0
$641$	3.65101e10	+	6.32373e10i	0.216262	+	0.374577i	0.953662	−	0.300879i	$-0.0972801\pi$
$641$	3.65101e10	+	6.32373e10i	0.216262	+	0.374577i	−0.737400	+	0.675456i	$0.763947\pi$
$642$		0			0
$643$			1.01096e10i			0.0591412i	0.999563	+	0.0295706i	$0.00941398\pi$
$643$			1.01096e10i			0.0591412i	−0.999563	+	0.0295706i	$0.990586\pi$
$644$		0			0
$645$	−4.28977e10			−0.247854
$646$		0			0
$647$	−2.37711e10	+	1.37242e10i	−0.135654	+	0.0783197i	−0.566291	−	0.824205i	$-0.691622\pi$
$647$	−2.37711e10	+	1.37242e10i	−0.135654	+	0.0783197i	0.430637	+	0.902525i	$0.358289\pi$
$648$		0			0
$649$	−1.61977e9	−	9.35174e8i	−0.00913007	−	0.00527125i
$650$		0			0
$651$	−4.25912e10	+	5.49492e9i	−0.237135	+	0.0305941i
$652$		0			0
$653$	9.08593e10	−	1.57373e11i	0.499709	−	0.865521i	−0.500291	−	0.865857i	$-0.666774\pi$
$653$	9.08593e10	−	1.57373e11i	0.499709	−	0.865521i	1.00000		0.000336511i	$0.000107115\pi$
$654$		0			0
$655$	−4.09851e10	−	7.09882e10i	−0.222669	−	0.385675i
$656$		0			0
$657$		−	3.78641e10i		−	0.203220i
$658$		0			0
$659$	1.63076e11			0.864668			0.432334	−	0.901714i	$-0.357690\pi$
$659$	1.63076e11			0.864668			0.432334	+	0.901714i	$0.357690\pi$
$660$		0			0
$661$	−3.66401e10	+	2.11542e10i	−0.191934	+	0.110813i	−0.592887	−	0.805285i	$-0.702012\pi$
$661$	−3.66401e10	+	2.11542e10i	−0.191934	+	0.110813i	0.400954	+	0.916098i	$0.368679\pi$
$662$		0			0
$663$	4.08175e9	+	2.35660e9i	0.0211248	+	0.0121964i
$664$		0			0
$665$	−1.60936e10	+	3.85701e10i	−0.0822936	+	0.197226i
$666$		0			0
$667$	7.13167e9	−	1.23524e10i	0.0360320	−	0.0624092i
$668$		0			0
$669$	7.41491e10	+	1.28430e11i	0.370170	+	0.641153i
$670$		0			0
$671$		−	1.12238e11i		−	0.553670i
$672$		0			0
$673$	2.61487e11			1.27465			0.637324	−	0.770596i	$-0.280041\pi$
$673$	2.61487e11			1.27465			0.637324	+	0.770596i	$0.280041\pi$
$674$		0			0
$675$	6.26516e9	−	3.61719e9i	0.0301798	−	0.0174243i
$676$		0			0
$677$	−2.24856e11	−	1.29821e11i	−1.07041	−	0.618000i	−0.142115	−	0.989850i	$-0.545390\pi$
$677$	−2.24856e11	−	1.29821e11i	−1.07041	−	0.618000i	−0.928293	+	0.371850i	$0.878724\pi$
$678$		0			0
$679$	−2.86352e11	+	2.18549e11i	−1.34717	+	1.02818i
$680$		0			0
$681$	2.80211e10	−	4.85340e10i	0.130286	−	0.225662i
$682$		0			0
$683$	−2.05498e10	−	3.55933e10i	−0.0944333	−	0.163563i	0.814939	−	0.579547i	$-0.196771\pi$
$683$	−2.05498e10	−	3.55933e10i	−0.0944333	−	0.163563i	−0.909372	+	0.415984i	$0.863437\pi$
$684$		0			0
$685$		−	7.15278e10i		−	0.324872i
$686$		0			0
$687$	−1.77131e11			−0.795185
$688$		0			0
$689$	1.07371e11	−	6.19906e10i	0.476441	−	0.275074i
$690$		0			0
$691$	7.68717e10	+	4.43819e10i	0.337174	+	0.194668i	0.659022	−	0.752124i	$-0.270971\pi$
$691$	7.68717e10	+	4.43819e10i	0.337174	+	0.194668i	−0.321848	+	0.946791i	$0.604304\pi$
$692$		0			0
$693$	2.61646e10	+	3.42820e10i	0.113444	+	0.148639i
$694$		0			0
$695$	7.31512e10	−	1.26702e11i	0.313532	−	0.543054i
$696$		0			0
$697$	9.91408e9	+	1.71717e10i	0.0420070	+	0.0727582i
$698$		0			0
$699$			3.81033e10i			0.159608i
$700$		0			0
$701$	−5.41049e10			−0.224060			−0.112030	−	0.993705i	$-0.535735\pi$
$701$	−5.41049e10			−0.224060			−0.112030	+	0.993705i	$0.535735\pi$
$702$		0			0
$703$	−3.61581e10	+	2.08759e10i	−0.148042	+	0.0854719i
$704$		0			0
$705$	8.11245e10	+	4.68373e10i	0.328394	+	0.189599i
$706$		0			0
$707$	−2.43461e11	−	1.01585e11i	−0.974431	−	0.406586i
$708$		0			0
$709$	1.63603e11	−	2.83368e11i	0.647449	−	1.12141i	−0.336281	−	0.941762i	$-0.609169\pi$
$709$	1.63603e11	−	2.83368e11i	0.647449	−	1.12141i	0.983730	−	0.179653i	$-0.0574973\pi$
$710$		0			0
$711$	2.75568e10	+	4.77298e10i	0.107833	+	0.186772i
$712$		0			0
$713$		−	4.39498e10i		−	0.170058i
$714$		0			0
$715$	6.34132e10			0.242636
$716$		0			0
$717$	2.11006e11	−	1.21825e11i	0.798397	−	0.460955i
$718$		0			0
$719$	−4.11971e11	−	2.37851e11i	−1.54153	−	0.890000i	−0.998743	−	0.0501272i	$-0.984037\pi$
$719$	−4.11971e11	−	2.37851e11i	−1.54153	−	0.890000i	−0.542783	−	0.839873i	$-0.682629\pi$
$720$		0			0
$721$	3.11455e10	+	2.41409e11i	0.115253	+	0.893332i
$722$		0			0
$723$	−2.02602e10	+	3.50916e10i	−0.0741463	+	0.128425i
$724$		0			0
$725$	4.38987e9	+	7.60347e9i	0.0158891	+	0.0275207i
$726$		0			0
$727$			1.61243e11i			0.577222i	0.957446	+	0.288611i	$0.0931934\pi$
$727$			1.61243e11i			0.577222i	−0.957446	+	0.288611i	$0.906807\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$	−1.03693e10	+	5.98671e9i	−0.0363145	+	0.0209662i
$732$		0			0
$733$	2.16374e11	+	1.24924e11i	0.749530	+	0.432741i	0.825524	−	0.564367i	$-0.190880\pi$
$733$	2.16374e11	+	1.24924e11i	0.749530	+	0.432741i	−0.0759941	+	0.997108i	$0.524213\pi$
$734$		0			0
$735$	−4.02278e10	+	1.47077e11i	−0.137840	+	0.503958i
$736$		0			0
$737$	3.39862e10	−	5.88657e10i	0.115195	−	0.199523i
$738$		0			0
$739$	2.29888e11	+	3.98177e11i	0.770793	+	1.33505i	0.937129	+	0.348984i	$0.113473\pi$
$739$	2.29888e11	+	3.98177e11i	0.770793	+	1.33505i	−0.166335	+	0.986069i	$0.553193\pi$
$740$		0			0
$741$			1.96479e10i			0.0651694i
$742$		0			0
$743$	2.61501e10			0.0858060			0.0429030	−	0.999079i	$-0.486339\pi$
$743$	2.61501e10			0.0858060			0.0429030	+	0.999079i	$0.486339\pi$
$744$		0			0
$745$	3.58292e11	−	2.06860e11i	1.16309	−	0.671509i
$746$		0			0
$747$	3.18485e10	+	1.83877e10i	0.102284	+	0.0590535i
$748$		0			0
$749$	−1.64538e11	+	2.12279e10i	−0.522803	+	0.0674496i
$750$		0			0
$751$	−7.92470e10	+	1.37260e11i	−0.249128	+	0.431503i	−0.963284	−	0.268484i	$-0.913477\pi$
$751$	−7.92470e10	+	1.37260e11i	−0.249128	+	0.431503i	0.714156	+	0.699987i	$0.246811\pi$
$752$		0			0
$753$	1.47802e11	+	2.56001e11i	0.459728	+	0.796273i
$754$		0			0
$755$			2.67526e11i			0.823338i
$756$		0			0
$757$	−4.14772e10			−0.126307			−0.0631533	−	0.998004i	$-0.520116\pi$
$757$	−4.14772e10			−0.126307			−0.0631533	+	0.998004i	$0.520116\pi$
$758$		0			0
$759$	−3.82226e10	+	2.20678e10i	−0.115174	+	0.0664955i
$760$		0			0
$761$	−3.23340e11	−	1.86680e11i	−0.964096	−	0.556621i	−0.0666649	−	0.997775i	$-0.521236\pi$
$761$	−3.23340e11	−	1.86680e11i	−0.964096	−	0.556621i	−0.897431	+	0.441154i	$0.854569\pi$
$762$		0			0
$763$	−2.56745e11	+	6.15319e11i	−0.757537	+	1.81553i
$764$		0			0
$765$	−4.56593e9	+	7.90843e9i	−0.0133316	+	0.0230911i
$766$		0			0
$767$	−1.55445e9	−	2.69239e9i	−0.00449154	−	0.00777958i
$768$		0			0
$769$			9.91215e10i			0.283441i	0.989907	+	0.141721i	$0.0452634\pi$
$769$			9.91215e10i			0.283441i	−0.989907	+	0.141721i	$0.954737\pi$
$770$		0			0
$771$	−1.55295e10			−0.0439482
$772$		0			0
$773$	3.49902e10	−	2.02016e10i	0.0980004	−	0.0565806i	−0.450199	−	0.892928i	$-0.648647\pi$
$773$	3.49902e10	−	2.02016e10i	0.0980004	−	0.0565806i	0.548199	+	0.836348i	$0.315314\pi$
$774$		0			0
$775$	2.34286e10	+	1.35265e10i	0.0649442	+	0.0374956i
$776$		0			0
$777$	−1.21090e11	+	9.24179e10i	−0.332219	+	0.253555i
$778$		0			0
$779$	−4.13288e10	+	7.15836e10i	−0.112229	+	0.194386i
$780$		0			0
$781$	1.15452e11	+	1.99969e11i	0.310312	+	0.537475i
$782$		0			0
$783$		−	1.26948e10i		−	0.0337737i
$784$		0			0
$785$	3.85831e10			0.101606
$786$		0			0
$787$	−3.27582e11	+	1.89129e11i	−0.853927	+	0.493015i	−0.861974	−	0.506953i	$-0.830772\pi$
$787$	−3.27582e11	+	1.89129e11i	−0.853927	+	0.493015i	0.00804708	+	0.999968i	$0.497439\pi$
$788$		0			0
$789$	1.14441e11	+	6.60727e10i	0.295308	+	0.170496i
$790$		0			0
$791$	−1.50273e11	−	1.96894e11i	−0.383861	−	0.502952i
$792$		0			0
$793$	9.32816e10	−	1.61568e11i	0.235887	−	0.408567i
$794$		0			0
$795$	1.20107e11	+	2.08032e11i	0.300678	+	0.520789i
$796$		0			0
$797$			7.62464e11i			1.88967i	0.327546	+	0.944835i	$0.393778\pi$
$797$			7.62464e11i			1.88967i	−0.327546	+	0.944835i	$0.606222\pi$
$798$		0			0
$799$	2.61460e10			0.0641533
$800$		0			0
$801$	1.47628e11	−	8.52331e10i	0.358623	−	0.207051i
$802$		0			0
$803$	−1.23142e11	−	7.10962e10i	−0.296173	−	0.170995i
$804$		0			0
$805$	−1.44015e11	−	6.00911e10i	−0.342945	−	0.143096i
$806$		0			0
$807$	−1.25483e11	+	2.17344e11i	−0.295864	+	0.512452i
$808$		0			0
$809$	6.60098e9	+	1.14332e10i	0.0154104	+	0.0266916i	0.873628	−	0.486595i	$-0.161761\pi$
$809$	6.60098e9	+	1.14332e10i	0.0154104	+	0.0266916i	−0.858217	+	0.513286i	$0.828428\pi$
$810$		0			0
$811$			1.57984e11i			0.365198i	0.983187	+	0.182599i	$0.0584510\pi$
$811$			1.57984e11i			0.365198i	−0.983187	+	0.182599i	$0.941549\pi$
$812$		0			0
$813$	−1.25175e11			−0.286521
$814$		0			0
$815$	2.63036e11	−	1.51864e11i	0.596189	−	0.344210i
$816$		0			0
$817$	−4.32265e10	−	2.49568e10i	−0.0970201	−	0.0560146i
$818$		0			0
$819$	9.17231e9	+	7.10948e10i	0.0203865	+	0.158016i
$820$		0			0
$821$	4.59911e10	−	7.96590e10i	0.101228	−	0.175332i	−0.810963	−	0.585098i	$-0.801056\pi$
$821$	4.59911e10	−	7.96590e10i	0.101228	−	0.175332i	0.912191	+	0.409765i	$0.134389\pi$
$822$		0			0
$823$	2.71088e10	+	4.69538e10i	0.0590895	+	0.102346i	0.894057	−	0.447953i	$-0.147847\pi$
$823$	2.71088e10	+	4.69538e10i	0.0590895	+	0.102346i	−0.834967	+	0.550299i	$0.814514\pi$
$824$		0			0
$825$		−	2.71675e10i		−	0.0586454i
$826$		0			0
$827$	7.45511e11			1.59379			0.796897	−	0.604115i	$-0.206473\pi$
$827$	7.45511e11			1.59379			0.796897	+	0.604115i	$0.206473\pi$
$828$		0			0
$829$	−1.02171e11	+	5.89884e10i	−0.216326	+	0.124896i	−0.604248	−	0.796796i	$-0.706526\pi$
$829$	−1.02171e11	+	5.89884e10i	−0.216326	+	0.124896i	0.387922	+	0.921692i	$0.373193\pi$
$830$		0			0
$831$	1.16838e11	+	6.74567e10i	0.245009	+	0.141456i
$832$		0			0
$833$	1.08018e10	+	4.11657e10i	0.0224345	+	0.0854979i
$834$		0			0
$835$	3.68480e11	−	6.38226e11i	0.757998	−	1.31289i
$836$		0			0
$837$	−1.95583e10	−	3.38760e10i	−0.0398501	−	0.0690224i
$838$		0			0
$839$		−	2.34278e11i		−	0.472807i	−0.971655	−	0.236403i	$-0.924031\pi$
$839$		−	2.34278e11i		−	0.472807i	0.971655	−	0.236403i	$-0.0759687\pi$
$840$		0			0
$841$	−4.84840e11			−0.969202
$842$		0			0
$843$	2.06916e11	−	1.19463e11i	0.409717	−	0.236550i
$844$		0			0
$845$	−3.08273e11	−	1.77981e11i	−0.604656	−	0.349098i
$846$		0			0
$847$	−3.49824e11	+	4.51327e10i	−0.679698	+	0.0876915i
$848$		0			0
$849$	1.75073e11	−	3.03235e11i	0.336968	−	0.583646i
$850$		0			0
$851$	−7.79475e10	−	1.35009e11i	−0.148622	−	0.257421i
$852$		0			0
$853$			4.56237e10i			0.0861776i	0.999071	+	0.0430888i	$0.0137198\pi$
$853$			4.56237e10i			0.0861776i	−0.999071	+	0.0430888i	$0.986280\pi$
$854$		0			0
$855$	−3.80680e10			−0.0712354
$856$		0			0
$857$	−2.11961e11	+	1.22376e11i	−0.392945	+	0.226867i	−0.683436	−	0.730011i	$-0.739515\pi$
$857$	−2.11961e11	+	1.22376e11i	−0.392945	+	0.226867i	0.290490	+	0.956878i	$0.406182\pi$
$858$		0			0
$859$	−3.62298e11	−	2.09173e11i	−0.665416	−	0.384178i	0.128922	−	0.991655i	$-0.458848\pi$
$859$	−3.62298e11	−	2.09173e11i	−0.665416	−	0.384178i	−0.794337	+	0.607477i	$0.792182\pi$
$860$		0			0
$861$	−1.16128e11	+	2.78315e11i	−0.211312	+	0.506435i
$862$		0			0
$863$	3.32103e11	−	5.75219e11i	0.598728	−	1.03703i	−0.394281	−	0.918990i	$-0.629006\pi$
$863$	3.32103e11	−	5.75219e11i	0.598728	−	1.03703i	0.993009	−	0.118037i	$-0.0376603\pi$
$864$		0			0
$865$	−3.41454e11	−	5.91415e11i	−0.609912	−	1.05640i
$866$		0			0
$867$		−	3.23675e11i		−	0.572839i
$868$		0			0
$869$	2.06970e11			0.362934
$870$		0			0
$871$	9.78469e10	−	5.64919e10i	0.170010	−	0.0981553i
$872$		0			0
$873$	−2.84158e11	−	1.64059e11i	−0.489218	−	0.282450i
$874$		0			0
$875$	4.98036e11	−	3.80109e11i	0.849627	−	0.648448i
$876$		0			0
$877$	1.19137e11	−	2.06352e11i	0.201395	−	0.348827i	−0.747583	−	0.664168i	$-0.768786\pi$
$877$	1.19137e11	−	2.06352e11i	0.201395	−	0.348827i	0.948978	+	0.315342i	$0.102119\pi$
$878$		0			0
$879$	−2.28191e10	−	3.95238e10i	−0.0382245	−	0.0662069i
$880$		0			0
$881$			8.99876e11i			1.49375i	0.664962	+	0.746877i	$0.268448\pi$
$881$			8.99876e11i			1.49375i	−0.664962	+	0.746877i	$0.731552\pi$
$882$		0			0
$883$	9.52626e11			1.56704			0.783519	−	0.621368i	$-0.213423\pi$
$883$	9.52626e11			1.56704			0.783519	+	0.621368i	$0.213423\pi$
$884$		0			0
$885$	5.21653e9	−	3.01176e9i	0.00850371	−	0.00490962i
$886$		0			0
$887$	−1.03165e12	−	5.95622e11i	−1.66662	−	0.962225i	−0.969439	−	0.245331i	$-0.921103\pi$
$887$	−1.03165e12	−	5.95622e11i	−1.66662	−	0.962225i	−0.697182	−	0.716894i	$-0.745563\pi$
$888$		0			0
$889$	−4.76337e10	−	6.24119e10i	−0.0762619	−	0.0999218i
$890$		0			0
$891$	−1.96410e10	+	3.40193e10i	−0.0311640	+	0.0539777i
$892$		0			0
$893$	5.44975e10	+	9.43925e10i	0.0856981	+	0.148433i
$894$		0			0
$895$		−	1.00619e12i		−	1.56815i
$896$		0			0
$897$	−7.33625e10			−0.113319
$898$		0			0
$899$	4.11123e10	−	2.37362e10i	0.0629409	−	0.0363390i
$900$		0			0
$901$	5.80650e10	+	3.35239e10i	0.0881080	+	0.0508692i
$902$		0			0
$903$	−1.68063e11	−	7.01251e10i	−0.252768	−	0.105469i
$904$		0			0
$905$	2.58126e10	−	4.47087e10i	0.0384802	−	0.0666496i
$906$		0			0
$907$	5.10644e11	+	8.84462e11i	0.754552	+	1.30692i	0.945597	+	0.325342i	$0.105479\pi$
$907$	5.10644e11	+	8.84462e11i	0.754552	+	1.30692i	−0.191044	+	0.981581i	$0.561187\pi$
$908$		0			0
$909$		−	2.40291e11i		−	0.351951i
$910$		0			0
$911$	6.98995e11			1.01485			0.507424	−	0.861697i	$-0.330598\pi$
$911$	6.98995e11			1.01485			0.507424	+	0.861697i	$0.330598\pi$
$912$		0			0
$913$	1.19602e11	−	6.90520e10i	0.172129	−	0.0993786i
$914$		0			0
$915$	3.13040e11	+	1.80734e11i	0.446597	+	0.257843i
$916$		0			0
$917$	−4.45249e10	−	3.45114e11i	−0.0629689	−	0.488073i
$918$		0			0
$919$	1.30804e11	−	2.26560e11i	0.183384	−	0.317630i	−0.759647	−	0.650336i	$-0.774628\pi$
$919$	1.30804e11	−	2.26560e11i	0.183384	−	0.317630i	0.943031	+	0.332706i	$0.107962\pi$
$920$		0			0
$921$	−1.37786e11	−	2.38652e11i	−0.191498	−	0.331685i
$922$		0			0
$923$			3.83810e11i			0.528822i
$924$		0			0
$925$	9.59604e10			0.131077
$926$		0			0
$927$	−1.92011e11	+	1.10858e11i	−0.260020	+	0.150123i
$928$		0			0
$929$	−6.71373e11	−	3.87617e11i	−0.901366	−	0.520404i	−0.0237226	−	0.999719i	$-0.507552\pi$
$929$	−6.71373e11	−	3.87617e11i	−0.901366	−	0.520404i	−0.877643	+	0.479315i	$0.840885\pi$
$930$		0			0
$931$	−1.26102e11	+	1.24800e11i	−0.167850	+	0.166118i
$932$		0			0
$933$	2.83396e11	−	4.90857e11i	0.373997	−	0.647781i
$934$		0			0
$935$	1.71466e10	+	2.96988e10i	0.0224353	+	0.0388590i
$936$		0			0
$937$		−	1.10001e11i		−	0.142705i	−0.997451	−	0.0713525i	$-0.977268\pi$
$937$		−	1.10001e11i		−	0.142705i	0.997451	−	0.0713525i	$-0.0227315\pi$
$938$		0			0
$939$	−4.07760e11			−0.524496
$940$		0			0
$941$	−8.90460e11	+	5.14107e11i	−1.13568	+	0.655685i	−0.945357	−	0.326037i	$-0.894287\pi$
$941$	−8.90460e11	+	5.14107e11i	−1.13568	+	0.655685i	−0.190323	+	0.981722i	$0.560953\pi$
$942$		0			0
$943$	−2.67283e11	−	1.54316e11i	−0.338006	−	0.195148i
$944$		0			0
$945$	−1.37747e11	+	1.77714e10i	−0.172725	+	0.0222841i
$946$		0			0
$947$	−2.70686e11	+	4.68842e11i	−0.336563	+	0.582944i	−0.983784	−	0.179359i	$-0.942598\pi$
$947$	−2.70686e11	+	4.68842e11i	−0.336563	+	0.582944i	0.647221	+	0.762302i	$0.275931\pi$
$948$		0			0
$949$	−1.18176e11	−	2.04688e11i	−0.145702	−	0.252364i
$950$		0			0
$951$		−	6.27040e11i		−	0.766607i
$952$		0			0
$953$	1.03712e12			1.25736			0.628679	−	0.777665i	$-0.283596\pi$
$953$	1.03712e12			1.25736			0.628679	+	0.777665i	$0.283596\pi$
$954$		0			0
$955$	1.03054e12	−	5.94980e11i	1.23894	−	0.715301i
$956$		0			0
$957$	−4.12862e10	−	2.38366e10i	−0.0492218	−	0.0284182i
$958$		0			0
$959$	1.16927e11	−	2.80229e11i	0.138242	−	0.331313i
$960$		0			0
$961$	−3.53307e11	+	6.11946e11i	−0.414246	+	0.717496i
$962$		0			0
$963$	−7.55574e10	−	1.30869e11i	−0.0878561	−	0.152171i
$964$		0			0
$965$			1.01899e12i			1.17506i
$966$		0			0
$967$	−1.59184e12			−1.82051			−0.910255	−	0.414048i	$-0.864115\pi$
$967$	−1.59184e12			−1.82051			−0.910255	+	0.414048i	$0.864115\pi$
$968$		0			0
$969$	−9.20186e9	+	5.31269e9i	−0.0104371	+	0.00602587i
$970$		0			0
$971$	4.29931e11	+	2.48221e11i	0.483639	+	0.279229i	0.721932	−	0.691964i	$-0.243254\pi$
$971$	4.29931e11	+	2.48221e11i	0.483639	+	0.279229i	−0.238293	+	0.971193i	$0.576588\pi$
$972$		0			0
$973$	4.93709e11	−	3.76806e11i	0.550833	−	0.420404i
$974$		0			0
$975$	2.25790e10	−	3.91079e10i	0.0249854	−	0.0432759i
$976$		0			0
$977$	9.39687e9	+	1.62759e10i	0.0103135	+	0.0178635i	0.871136	−	0.491042i	$-0.163384\pi$
$977$	9.39687e9	+	1.62759e10i	0.0103135	+	0.0178635i	−0.860823	+	0.508905i	$0.830050\pi$
$978$		0			0
$979$		−	6.40157e11i		−	0.696876i
$980$		0			0
$981$	−6.07309e11			−0.655743
$982$		0			0
$983$	−8.20014e11	+	4.73435e11i	−0.878228	+	0.507045i	−0.870074	−	0.492922i	$-0.835929\pi$
$983$	−8.20014e11	+	4.73435e11i	−0.878228	+	0.507045i	−0.00815427	+	0.999967i	$0.502596\pi$
$984$		0			0
$985$	−5.39085e11	−	3.11241e11i	−0.572680	−	0.330637i
$986$		0			0
$987$	2.41262e11	+	3.16112e11i	0.254226	+	0.333098i
$988$		0			0
$989$	9.31852e10	−	1.61401e11i	0.0974006	−	0.168703i
$990$		0			0
$991$	2.45053e11	+	4.24444e11i	0.254077	+	0.440074i	0.964644	−	0.263555i	$-0.0848950\pi$
$991$	2.45053e11	+	4.24444e11i	0.254077	+	0.440074i	−0.710567	+	0.703629i	$0.751562\pi$
$992$		0			0
$993$		−	8.20894e11i		−	0.844287i
$994$		0			0
$995$	1.72916e12			1.76418
$996$		0			0
$997$	1.52518e11	−	8.80561e10i	0.154362	−	0.0891208i	−0.420830	−	0.907140i	$-0.638261\pi$
$997$	1.52518e11	−	8.80561e10i	0.154362	−	0.0891208i	0.575191	+	0.818019i	$0.304928\pi$
$998$		0			0
$999$	−1.20162e11	−	6.93756e10i	−0.120644	−	0.0696538i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.m.a.61.1	✓	10
3.2	odd	2		252.9.z.b.145.5		10
7.3	odd	6	inner	84.9.m.a.73.1	yes	10
21.17	even	6		252.9.z.b.73.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.m.a.61.1	✓	10	1.1	even	1	trivial
84.9.m.a.73.1	yes	10	7.3	odd	6	inner
252.9.z.b.73.5		10	21.17	even	6
252.9.z.b.145.5		10	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.m (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-40.5000 + 23.3827i) q^{3} +(-489.814 - 282.795i) q^{5} +(-1456.69 - 1908.63i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})\)	\(q+(-40.5000 + 23.3827i) q^{3} +(-489.814 - 282.795i) q^{5} +(-1456.69 - 1908.63i) q^{7} +(1093.50 - 1894.00i) q^{9} +(-4106.46 - 7112.59i) q^{11} -13651.5i q^{13} +26450.0 q^{15} +(6393.53 - 3691.30i) q^{17} +(26652.7 + 15387.9i) q^{19} +(103625. + 43238.0i) q^{21} +(-57456.4 + 99517.3i) q^{23} +(-35367.0 - 61257.5i) q^{25} +102276. i q^{27} -124123. q^{29} +(-331222. + 191231. i) q^{31} +(332623. + 192040. i) q^{33} +(173760. + 1.34682e6i) q^{35} +(-678319. + 1.17488e6i) q^{37} +(319210. + 552887. i) q^{39} +2.68579e6i q^{41} -1.62184e6 q^{43} +(-1.07122e6 + 618472. i) q^{45} +(3.06709e6 + 1.77079e6i) q^{47} +(-1.52090e6 + 5.56056e6i) q^{49} +(-172625. + 298996. i) q^{51} +(4.54092e6 + 7.86511e6i) q^{53} +4.64513e6i q^{55} -1.43925e6 q^{57} +(197222. - 113866. i) q^{59} +(1.18352e7 + 6.83305e6i) q^{61} +(-5.20782e6 + 671889. i) q^{63} +(-3.86058e6 + 6.68672e6i) q^{65} +(4.13814e6 + 7.16746e6i) q^{67} -5.37394e6i q^{69} -2.81148e7 q^{71} +(1.49937e7 - 8.65664e6i) q^{73} +(2.86473e6 + 1.65395e6i) q^{75} +(-7.59343e6 + 1.81985e7i) q^{77} +(-1.26003e7 + 2.18243e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} +1.68155e7i q^{83} -4.17552e6 q^{85} +(5.02699e6 - 2.90234e6i) q^{87} +(6.75025e7 + 3.89726e7i) q^{89} +(-2.60557e7 + 1.98861e7i) q^{91} +(8.94299e6 - 1.54897e7i) q^{93} +(-8.70325e6 - 1.50745e7i) q^{95} -1.50031e8i q^{97} -1.79616e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10}) \) 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9	\( 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9} - 879 q^{11} - 75006 q^{15} - 13674 q^{17} - 29268 q^{19} - 42363 q^{21} + 312732 q^{23} - 22052 q^{25} - 289794 q^{29} + 242787 q^{31} + 71199 q^{33} + 1209372 q^{35} + 1913308 q^{37} - 1232334 q^{39} - 861848 q^{43} + 3037743 q^{45} - 305448 q^{47} + 9821659 q^{49} + 369198 q^{51} - 10663233 q^{53} + 1580472 q^{57} + 18410871 q^{59} - 13937808 q^{61} + 769824 q^{63} - 14966808 q^{65} - 20722822 q^{67} + 113032584 q^{71} + 43436322 q^{73} + 1786212 q^{75} - 98823405 q^{77} - 42189637 q^{79} - 23914845 q^{81} + 142602108 q^{85} + 11736657 q^{87} + 67171914 q^{89} - 246091266 q^{91} - 6555249 q^{93} - 140649894 q^{95} - 3844746 q^{99}+O(q^{100}) \) 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9 - 879 * q^11 - 75006 * q^15 - 13674 * q^17 - 29268 * q^19 - 42363 * q^21 + 312732 * q^23 - 22052 * q^25 - 289794 * q^29 + 242787 * q^31 + 71199 * q^33 + 1209372 * q^35 + 1913308 * q^37 - 1232334 * q^39 - 861848 * q^43 + 3037743 * q^45 - 305448 * q^47 + 9821659 * q^49 + 369198 * q^51 - 10663233 * q^53 + 1580472 * q^57 + 18410871 * q^59 - 13937808 * q^61 + 769824 * q^63 - 14966808 * q^65 - 20722822 * q^67 + 113032584 * q^71 + 43436322 * q^73 + 1786212 * q^75 - 98823405 * q^77 - 42189637 * q^79 - 23914845 * q^81 + 142602108 * q^85 + 11736657 * q^87 + 67171914 * q^89 - 246091266 * q^91 - 6555249 * q^93 - 140649894 * q^95 - 3844746 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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