LMFDB - Embedded newform 72.8.f.a.35.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(35,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("72.35");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$22.4917218349$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			35.27
Character	$\chi$	$=$	72.35
Dual form			72.8.f.a.35.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(11.3087 - 0.338231i) q^{2} +(127.771 - 7.64988i) q^{4} -93.0830 q^{5} +1025.06i q^{7} +(1442.33 - 129.726i) q^{8} +O(q^{10})$	\(q+(11.3087 - 0.338231i) q^{2} +(127.771 - 7.64988i) q^{4} -93.0830 q^{5} +1025.06i q^{7} +(1442.33 - 129.726i) q^{8} +(-1052.64 + 31.4836i) q^{10} +1545.61i q^{11} +5677.41i q^{13} +(346.709 + 11592.1i) q^{14} +(16267.0 - 1954.87i) q^{16} +15703.6i q^{17} +33939.0 q^{19} +(-11893.3 + 712.074i) q^{20} +(522.773 + 17478.7i) q^{22} +30068.2 q^{23} -69460.6 q^{25} +(1920.28 + 64203.9i) q^{26} +(7841.62 + 130974. i) q^{28} +99025.6 q^{29} +152785. i q^{31} +(183296. - 27608.9i) q^{32} +(5311.46 + 177587. i) q^{34} -95416.0i q^{35} +336316. i q^{37} +(383805. - 11479.2i) q^{38} +(-134257. + 12075.3i) q^{40} -663449. i q^{41} -707879. q^{43} +(11823.7 + 197484. i) q^{44} +(340031. - 10170.0i) q^{46} +402012. q^{47} -227213. q^{49} +(-785505. + 23493.7i) q^{50} +(43431.5 + 725410. i) q^{52} +662469. q^{53} -143870. i q^{55} +(132977. + 1.47848e6i) q^{56} +(1.11985e6 - 33493.6i) q^{58} -1.74805e6i q^{59} +321803. i q^{61} +(51676.6 + 1.72779e6i) q^{62} +(2.06349e6 - 374216. i) q^{64} -528470. i q^{65} -464913. q^{67} +(120131. + 2.00647e6i) q^{68} +(-32272.7 - 1.07903e6i) q^{70} -3.53880e6 q^{71} -3.41741e6 q^{73} +(113753. + 3.80328e6i) q^{74} +(4.33643e6 - 259630. i) q^{76} -1.58435e6 q^{77} -7.60430e6i q^{79} +(-1.51418e6 + 181965. i) q^{80} +(-224399. - 7.50271e6i) q^{82} -8.11070e6i q^{83} -1.46174e6i q^{85} +(-8.00515e6 + 239427. i) q^{86} +(200506. + 2.22928e6i) q^{88} -1.00923e7i q^{89} -5.81971e6 q^{91} +(3.84185e6 - 230018. i) q^{92} +(4.54621e6 - 135973. i) q^{94} -3.15915e6 q^{95} -5.88229e6 q^{97} +(-2.56947e6 + 76850.4i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 52 q^{4}+O(q^{10}) $ 28 * q + 52 * q^4	$ 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	11.3087	−	0.338231i	0.999553	−	0.0298957i
$2$	11.3087	−	0.338231i	0.999553	−	0.0298957i
$3$		0			0
$3$		0			0
$4$	127.771	−	7.64988i	0.998212	−	0.0597647i
$5$	−93.0830			−0.333024			−0.166512	−	0.986039i	$-0.553250\pi$
$5$	−93.0830			−0.333024			−0.166512	+	0.986039i	$0.553250\pi$
$6$		0			0
$7$			1025.06i			1.12956i	0.825243	+	0.564778i	$0.191038\pi$
$7$			1025.06i			1.12956i	−0.825243	+	0.564778i	$0.808962\pi$
$8$	1442.33	−	129.726i	0.995980	−	0.0895803i
$9$		0			0
$10$	−1052.64	+	31.4836i	−0.332875	+	0.00995598i
$11$			1545.61i			0.350126i	0.984557	+	0.175063i	$0.0560130\pi$
$11$			1545.61i			0.350126i	−0.984557	+	0.175063i	$0.943987\pi$
$12$		0			0
$13$			5677.41i			0.716719i	0.933584	+	0.358359i	$0.116664\pi$
$13$			5677.41i			0.716719i	−0.933584	+	0.358359i	$0.883336\pi$
$14$	346.709	+	11592.1i	0.0337689	+	1.12905i
$15$		0			0
$16$	16267.0	−	1954.87i	0.992856	−	0.119316i
$17$			15703.6i			0.775227i	0.921822	+	0.387613i	$0.126700\pi$
$17$			15703.6i			0.775227i	−0.921822	+	0.387613i	$0.873300\pi$
$18$		0			0
$19$	33939.0			1.13517			0.567586	−	0.823314i	$-0.307877\pi$
$19$	33939.0			1.13517			0.567586	+	0.823314i	$0.307877\pi$
$20$	−11893.3	+	712.074i	−0.332428	+	0.0199031i
$21$		0			0
$22$	522.773	+	17478.7i	0.0104673	+	0.349970i
$23$	30068.2			0.515300			0.257650	−	0.966238i	$-0.417052\pi$
$23$	30068.2			0.515300			0.257650	+	0.966238i	$0.417052\pi$
$24$		0			0
$25$	−69460.6			−0.889095
$26$	1920.28	+	64203.9i	0.0214268	+	0.716398i
$27$		0			0
$28$	7841.62	+	130974.i	0.0675076	+	1.12754i
$29$	99025.6			0.753971			0.376985	−	0.926219i	$-0.376961\pi$
$29$	99025.6			0.753971			0.376985	+	0.926219i	$0.376961\pi$
$30$		0			0
$31$			152785.i			0.921115i	0.887630	+	0.460558i	$0.152351\pi$
$31$			152785.i			0.921115i	−0.887630	+	0.460558i	$0.847649\pi$
$32$	183296.	−	27608.9i	0.988846	−	0.148945i
$33$		0			0
$34$	5311.46	+	177587.i	0.0231760	+	0.774880i
$35$		−	95416.0i		−	0.376169i
$36$		0			0
$37$			336316.i			1.09154i	0.837933	+	0.545772i	$0.183764\pi$
$37$			336316.i			1.09154i	−0.837933	+	0.545772i	$0.816236\pi$
$38$	383805.	−	11479.2i	1.13467	−	0.0339368i
$39$		0			0
$40$	−134257.	+	12075.3i	−0.331685	+	0.0298324i
$41$		−	663449.i		−	1.50336i	−0.659527	−	0.751681i	$-0.729243\pi$
$41$		−	663449.i		−	1.50336i	0.659527	−	0.751681i	$-0.270757\pi$
$42$		0			0
$43$	−707879.			−1.35775			−0.678874	−	0.734255i	$-0.737532\pi$
$43$	−707879.			−1.35775			−0.678874	+	0.734255i	$0.737532\pi$
$44$	11823.7	+	197484.i	0.0209252	+	0.349500i
$45$		0			0
$46$	340031.	−	10170.0i	0.515069	−	0.0154052i
$47$	402012.			0.564802			0.282401	−	0.959296i	$-0.408869\pi$
$47$	402012.			0.564802			0.282401	+	0.959296i	$0.408869\pi$
$48$		0			0
$49$	−227213.			−0.275896
$50$	−785505.	+	23493.7i	−0.888698	+	0.0265801i
$51$		0			0
$52$	43431.5	+	725410.i	0.0428345	+	0.715438i
$53$	662469.			0.611223			0.305612	−	0.952156i	$-0.401139\pi$
$53$	662469.			0.611223			0.305612	+	0.952156i	$0.401139\pi$
$54$		0			0
$55$		−	143870.i		−	0.116600i
$56$	132977.	+	1.47848e6i	0.101186	+	1.12501i
$57$		0			0
$58$	1.11985e6	−	33493.6i	0.753634	−	0.0225405i
$59$		−	1.74805e6i		−	1.10808i	−0.832489	−	0.554041i	$-0.813085\pi$
$59$		−	1.74805e6i		−	1.10808i	0.832489	−	0.554041i	$-0.186915\pi$
$60$		0			0
$61$			321803.i			0.181525i	0.995873	+	0.0907624i	$0.0289304\pi$
$61$			321803.i			0.181525i	−0.995873	+	0.0907624i	$0.971070\pi$
$62$	51676.6	+	1.72779e6i	0.0275374	+	0.920704i
$63$		0			0
$64$	2.06349e6	−	374216.i	0.983951	−	0.178440i
$65$		−	528470.i		−	0.238684i
$66$		0			0
$67$	−464913.			−0.188847			−0.0944234	−	0.995532i	$-0.530101\pi$
$67$	−464913.			−0.188847			−0.0944234	+	0.995532i	$0.530101\pi$
$68$	120131.	+	2.00647e6i	0.0463312	+	0.773841i
$69$		0			0
$70$	−32272.7	−	1.07903e6i	−0.0112458	−	0.376001i
$71$	−3.53880e6			−1.17342			−0.586708	−	0.809799i	$-0.699576\pi$
$71$	−3.53880e6			−1.17342			−0.586708	+	0.809799i	$0.699576\pi$
$72$		0			0
$73$	−3.41741e6			−1.02817			−0.514087	−	0.857738i	$-0.671869\pi$
$73$	−3.41741e6			−1.02817			−0.514087	+	0.857738i	$0.671869\pi$
$74$	113753.	+	3.80328e6i	0.0326325	+	1.09106i
$75$		0			0
$76$	4.33643e6	−	259630.i	1.13314	−	0.0678433i
$77$	−1.58435e6			−0.395487
$78$		0			0
$79$		−	7.60430e6i		−	1.73526i	−0.497210	−	0.867630i	$-0.665642\pi$
$79$		−	7.60430e6i		−	1.73526i	0.497210	−	0.867630i	$-0.334358\pi$
$80$	−1.51418e6	+	181965.i	−0.330645	+	0.0397350i
$81$		0			0
$82$	−224399.	−	7.50271e6i	−0.0449441	−	1.50269i
$83$		−	8.11070e6i		−	1.55699i	−0.627653	−	0.778494i	$-0.715984\pi$
$83$		−	8.11070e6i		−	1.55699i	0.627653	−	0.778494i	$-0.284016\pi$
$84$		0			0
$85$		−	1.46174e6i		−	0.258169i
$86$	−8.00515e6	+	239427.i	−1.35714	+	0.0405908i
$87$		0			0
$88$	200506.	+	2.22928e6i	0.0313644	+	0.348719i
$89$		−	1.00923e7i		−	1.51749i	−0.651387	−	0.758746i	$-0.725813\pi$
$89$		−	1.00923e7i		−	1.51749i	0.651387	−	0.758746i	$-0.274187\pi$
$90$		0			0
$91$	−5.81971e6			−0.809574
$92$	3.84185e6	−	230018.i	0.514379	−	0.0307967i
$93$		0			0
$94$	4.54621e6	−	135973.i	0.564550	−	0.0168852i
$95$	−3.15915e6			−0.378039
$96$		0			0
$97$	−5.88229e6			−0.654403			−0.327201	−	0.944955i	$-0.606106\pi$
$97$	−5.88229e6			−0.654403			−0.327201	+	0.944955i	$0.606106\pi$
$98$	−2.56947e6	+	76850.4i	−0.275773	+	0.00824812i
$99$		0			0
$100$	−8.87506e6	+	531365.i	−0.887506	+	0.0531365i
$101$	1.32223e7			1.27697			0.638486	−	0.769633i	$-0.279561\pi$
$101$	1.32223e7			1.27697			0.638486	+	0.769633i	$0.279561\pi$
$102$		0			0
$103$		−	8.86665e6i		−	0.799520i	−0.916620	−	0.399760i	$-0.869093\pi$
$103$		−	8.86665e6i		−	0.799520i	0.916620	−	0.399760i	$-0.130907\pi$
$104$	736508.	+	8.18872e6i	0.0642038	+	0.713837i
$105$		0			0
$106$	7.49163e6	−	224068.i	0.610950	−	0.0182729i
$107$			4.71318e6i			0.371938i	0.982556	+	0.185969i	$0.0595424\pi$
$107$			4.71318e6i			0.371938i	−0.982556	+	0.185969i	$0.940458\pi$
$108$		0			0
$109$			2.63965e7i			1.95233i	0.217027	+	0.976166i	$0.430364\pi$
$109$			2.63965e7i			1.95233i	−0.217027	+	0.976166i	$0.569636\pi$
$110$	−48661.2	−	1.62697e6i	−0.00348585	−	0.116548i
$111$		0			0
$112$	2.00387e6	+	1.66747e7i	0.134774	+	1.12149i
$113$		−	3.41021e6i		−	0.222334i	−0.993802	−	0.111167i	$-0.964541\pi$
$113$		−	3.41021e6i		−	0.222334i	0.993802	−	0.111167i	$-0.0354589\pi$
$114$		0			0
$115$	−2.79884e6			−0.171607
$116$	1.26526e7	−	757534.i	0.752623	−	0.0450608i
$117$		0			0
$118$	−591246.	−	1.97681e7i	−0.0331269	−	1.10759i
$119$	−1.60972e7			−0.875662
$120$		0			0
$121$	1.70983e7			0.877412
$122$	108844.	+	3.63916e6i	0.00542681	+	0.181444i
$123$		0			0
$124$	1.16879e6	+	1.95215e7i	0.0550502	+	0.919469i
$125$	1.37377e7			0.629114
$126$		0			0
$127$		−	9.87499e6i		−	0.427783i	−0.976857	−	0.213892i	$-0.931386\pi$
$127$		−	9.87499e6i		−	0.427783i	0.976857	−	0.213892i	$-0.0686139\pi$
$128$	2.32088e7	−	4.92982e6i	0.978176	−	0.207776i
$129$		0			0
$130$	−178745.	−	5.97629e6i	−0.00713564	−	0.238578i
$131$			2.16683e7i			0.842124i	0.907032	+	0.421062i	$0.138342\pi$
$131$			2.16683e7i			0.842124i	−0.907032	+	0.421062i	$0.861658\pi$
$132$		0			0
$133$			3.47897e7i			1.28224i
$134$	−5.25754e6	+	157248.i	−0.188762	+	0.00564571i
$135$		0			0
$136$	2.03717e6	+	2.26499e7i	0.0694450	+	0.772110i
$137$			5.12379e7i			1.70243i	0.524818	+	0.851214i	$0.324133\pi$
$137$			5.12379e7i			1.70243i	−0.524818	+	0.851214i	$0.675867\pi$
$138$		0			0
$139$	3.92028e7			1.23813			0.619063	−	0.785341i	$-0.287512\pi$
$139$	3.92028e7			1.23813			0.619063	+	0.785341i	$0.287512\pi$
$140$	−729921.	−	1.21914e7i	−0.0224816	−	0.375497i
$141$		0			0
$142$	−4.00191e7	+	1.19693e6i	−1.17289	+	0.0350801i
$143$	−8.77505e6			−0.250942
$144$		0			0
$145$	−9.21760e6			−0.251090
$146$	−3.86463e7	+	1.15587e6i	−1.02771	+	0.0307380i
$147$		0			0
$148$	2.57278e6	+	4.29715e7i	0.0652359	+	1.08959i
$149$	7.23566e7			1.79195			0.895975	−	0.444104i	$-0.146478\pi$
$149$	7.23566e7			1.79195			0.895975	+	0.444104i	$0.146478\pi$
$150$		0			0
$151$		−	5.61796e7i		−	1.32788i	−0.747786	−	0.663940i	$-0.768883\pi$
$151$		−	5.61796e7i		−	1.32788i	0.747786	−	0.663940i	$-0.231117\pi$
$152$	4.89514e7	−	4.40278e6i	1.13061	−	0.101689i
$153$		0			0
$154$	−1.79168e7	+	535875.i	−0.395310	+	0.0118234i
$155$		−	1.42217e7i		−	0.306753i
$156$		0			0
$157$			8.29634e7i			1.71095i	0.517843	+	0.855476i	$0.326735\pi$
$157$			8.29634e7i			1.71095i	−0.517843	+	0.855476i	$0.673265\pi$
$158$	−2.57201e6	−	8.59944e7i	−0.0518768	−	1.73448i
$159$		0			0
$160$	−1.70618e7	+	2.56992e6i	−0.329309	+	0.0496021i
$161$			3.08218e7i			0.582060i
$162$		0			0
$163$	1.32708e7			0.240017			0.120008	−	0.992773i	$-0.461708\pi$
$163$	1.32708e7			0.240017			0.120008	+	0.992773i	$0.461708\pi$
$164$	−5.07530e6	−	8.47696e7i	−0.0898480	−	1.50068i
$165$		0			0
$166$	−2.74329e6	−	9.17211e7i	−0.0465472	−	1.55629i
$167$	−6.62945e7			−1.10146			−0.550732	−	0.834682i	$-0.685651\pi$
$167$	−6.62945e7			−1.10146			−0.550732	+	0.834682i	$0.685651\pi$
$168$		0			0
$169$	3.05155e7			0.486314
$170$	−494406.	−	1.65303e7i	−0.00771814	−	0.258054i
$171$		0			0
$172$	−9.04465e7	+	5.41519e6i	−1.35532	+	0.0811454i
$173$	−5.99909e7			−0.880895			−0.440448	−	0.897778i	$-0.645180\pi$
$173$	−5.99909e7			−0.880895			−0.440448	+	0.897778i	$0.645180\pi$
$174$		0			0
$175$		−	7.12015e7i		−	1.00428i
$176$	3.02146e6	+	2.51423e7i	0.0417756	+	0.347625i
$177$		0			0
$178$	−3.41354e6	−	1.14131e8i	−0.0453665	−	1.51681i
$179$			8.01095e7i			1.04399i	0.852947	+	0.521997i	$0.174813\pi$
$179$			8.01095e7i			1.04399i	−0.852947	+	0.521997i	$0.825187\pi$
$180$		0			0
$181$		−	3.83641e7i		−	0.480894i	−0.970662	−	0.240447i	$-0.922706\pi$
$181$		−	3.83641e7i		−	0.480894i	0.970662	−	0.240447i	$-0.0772941\pi$
$182$	−6.58131e7	+	1.96841e6i	−0.809212	+	0.0242028i
$183$		0			0
$184$	4.33683e7	−	3.90063e6i	0.513228	−	0.0461607i
$185$		−	3.13053e7i		−	0.363510i
$186$		0			0
$187$	−2.42716e7			−0.271427
$188$	5.13655e7	−	3.07534e6i	0.563792	−	0.0337552i
$189$		0			0
$190$	−3.57257e7	+	1.06852e6i	−0.377871	+	0.0113018i
$191$	−1.26246e8			−1.31100			−0.655499	−	0.755196i	$-0.727542\pi$
$191$	−1.26246e8			−1.31100			−0.655499	+	0.755196i	$0.727542\pi$
$192$		0			0
$193$	1.20271e8			1.20423			0.602115	−	0.798409i	$-0.294325\pi$
$193$	1.20271e8			1.20423			0.602115	+	0.798409i	$0.294325\pi$
$194$	−6.65207e7	+	1.98957e6i	−0.654110	+	0.0195638i
$195$		0			0
$196$	−2.90312e7	+	1.73815e6i	−0.275403	+	0.0164889i
$197$	1.23903e8			1.15465			0.577326	−	0.816514i	$-0.304096\pi$
$197$	1.23903e8			1.15465			0.577326	+	0.816514i	$0.304096\pi$
$198$		0			0
$199$		−	1.68027e8i		−	1.51145i	−0.654888	−	0.755726i	$-0.727284\pi$
$199$		−	1.68027e8i		−	1.51145i	0.654888	−	0.755726i	$-0.272716\pi$
$200$	−1.00185e8	+	9.01085e6i	−0.885521	+	0.0796454i
$201$		0			0
$202$	1.49526e8	−	4.47219e6i	1.27640	−	0.0381760i
$203$			1.01508e8i			0.851652i
$204$		0			0
$205$			6.17558e7i			0.500655i
$206$	−2.99898e6	−	1.00270e8i	−0.0239022	−	0.799163i
$207$		0			0
$208$	1.10986e7	+	9.23542e7i	0.0855158	+	0.711599i
$209$			5.24564e7i			0.397454i
$210$		0			0
$211$	8.24919e7			0.604537			0.302268	−	0.953223i	$-0.402256\pi$
$211$	8.24919e7			0.604537			0.302268	+	0.953223i	$0.402256\pi$
$212$	8.46444e7	−	5.06781e6i	0.610130	−	0.0365296i
$213$		0			0
$214$	1.59414e6	+	5.32997e7i	0.0111194	+	0.371772i
$215$	6.58914e7			0.452162
$216$		0			0
$217$	−1.56614e8			−1.04045
$218$	8.92812e6	+	2.98509e8i	0.0583663	+	1.95146i
$219$		0			0
$220$	−1.10059e6	−	1.83824e7i	−0.00696858	−	0.116392i
$221$	−8.91560e7			−0.555620
$222$		0			0
$223$			1.10837e8i			0.669296i	0.942343	+	0.334648i	$0.108617\pi$
$223$			1.10837e8i			0.669296i	−0.942343	+	0.334648i	$0.891383\pi$
$224$	2.83009e7	+	1.87890e8i	0.168241	+	1.11696i
$225$		0			0
$226$	−1.15344e6	−	3.85649e7i	−0.00664685	−	0.222235i
$227$		−	2.44393e8i		−	1.38675i	−0.720576	−	0.693376i	$-0.756123\pi$
$227$		−	2.44393e8i		−	1.38675i	0.720576	−	0.693376i	$-0.243877\pi$
$228$		0			0
$229$			2.49235e7i			0.137146i	0.997646	+	0.0685732i	$0.0218447\pi$
$229$			2.49235e7i			0.137146i	−0.997646	+	0.0685732i	$0.978155\pi$
$230$	−3.16511e7	+	946654.i	−0.171530	+	0.00513031i
$231$		0			0
$232$	1.42828e8	−	1.28462e7i	0.750940	−	0.0675409i
$233$		−	8.59209e7i		−	0.444993i	−0.974934	−	0.222496i	$-0.928579\pi$
$233$		−	8.59209e7i		−	0.444993i	0.974934	−	0.222496i	$-0.0714206\pi$
$234$		0			0
$235$	−3.74204e7			−0.188092
$236$	−1.33724e7	−	2.23351e8i	−0.0662242	−	1.10610i
$237$		0			0
$238$	−1.82038e8	+	5.44458e6i	−0.875271	+	0.0261785i
$239$	1.03442e8			0.490122			0.245061	−	0.969508i	$-0.421192\pi$
$239$	1.03442e8			0.490122			0.245061	+	0.969508i	$0.421192\pi$
$240$		0			0
$241$	−2.12373e8			−0.977325			−0.488662	−	0.872473i	$-0.662515\pi$
$241$	−2.12373e8			−0.977325			−0.488662	+	0.872473i	$0.662515\pi$
$242$	1.93358e8	−	5.78317e6i	0.877019	−	0.0262308i
$243$		0			0
$244$	2.46176e6	+	4.11172e7i	0.0108488	+	0.181200i
$245$	2.11496e7			0.0918801
$246$		0			0
$247$			1.92686e8i			0.813599i
$248$	1.98202e7	+	2.20366e8i	0.0825137	+	0.917412i
$249$		0			0
$250$	1.55355e8	−	4.64652e6i	0.628832	−	0.0188078i
$251$		−	2.88318e8i		−	1.15084i	−0.817859	−	0.575418i	$-0.804839\pi$
$251$		−	2.88318e8i		−	1.15084i	0.817859	−	0.575418i	$-0.195161\pi$
$252$		0			0
$253$			4.64736e7i			0.180420i
$254$	−3.34003e6	−	1.11673e8i	−0.0127889	−	0.427592i
$255$		0			0
$256$	2.60792e8	−	6.35995e7i	0.971528	−	0.236927i
$257$		−	3.06607e8i		−	1.12672i	−0.826211	−	0.563361i	$-0.809508\pi$
$257$		−	3.06607e8i		−	1.12672i	0.826211	−	0.563361i	$-0.190492\pi$
$258$		0			0
$259$	−3.44745e8			−1.23296
$260$	−4.04274e6	−	6.75233e7i	−0.0142649	−	0.238258i
$261$		0			0
$262$	7.32891e6	+	2.45040e8i	0.0251759	+	0.841747i
$263$	7.85030e7			0.266098			0.133049	−	0.991109i	$-0.457523\pi$
$263$	7.85030e7			0.266098			0.133049	+	0.991109i	$0.457523\pi$
$264$		0			0
$265$	−6.16645e7			−0.203552
$266$	1.17670e7	+	3.93424e8i	0.0383335	+	1.28167i
$267$		0			0
$268$	−5.94025e7	+	3.55653e6i	−0.188509	+	0.0112864i
$269$	2.64506e8			0.828519			0.414259	−	0.910159i	$-0.364041\pi$
$269$	2.64506e8			0.828519			0.414259	+	0.910159i	$0.364041\pi$
$270$		0			0
$271$			3.60573e8i			1.10053i	0.834991	+	0.550263i	$0.185473\pi$
$271$			3.60573e8i			1.10053i	−0.834991	+	0.550263i	$0.814527\pi$
$272$	3.06985e7	+	2.55450e8i	0.0924968	+	0.769689i
$273$		0			0
$274$	1.73302e7	+	5.79431e8i	0.0508953	+	1.70167i
$275$		−	1.07359e8i		−	0.311296i
$276$		0			0
$277$		−	2.46978e8i		−	0.698198i	−0.937086	−	0.349099i	$-0.886488\pi$
$277$		−	2.46978e8i		−	0.698198i	0.937086	−	0.349099i	$-0.113512\pi$
$278$	4.43331e8	−	1.32596e7i	1.23757	−	0.0370147i
$279$		0			0
$280$	−1.23779e7	−	1.37622e8i	−0.0336973	−	0.374657i
$281$		−	1.16347e8i		−	0.312812i	−0.987693	−	0.156406i	$-0.950009\pi$
$281$		−	1.16347e8i		−	0.312812i	0.987693	−	0.156406i	$-0.0499908\pi$
$282$		0			0
$283$	−3.05548e8			−0.801358			−0.400679	−	0.916219i	$-0.631226\pi$
$283$	−3.05548e8			−0.801358			−0.400679	+	0.916219i	$0.631226\pi$
$284$	−4.52157e8	+	2.70714e7i	−1.17132	+	0.0701289i
$285$		0			0
$286$	−9.92340e7	+	2.96800e6i	−0.250830	+	0.00750209i
$287$	6.80077e8			1.69813
$288$		0			0
$289$	1.63735e8			0.399023
$290$	−1.04239e8	+	3.11768e6i	−0.250978	+	0.00750652i
$291$		0			0
$292$	−4.36646e8	+	2.61428e7i	−1.02634	+	0.0614485i
$293$	−7.14442e8			−1.65932			−0.829661	−	0.558268i	$-0.811466\pi$
$293$	−7.14442e8			−1.65932			−0.829661	+	0.558268i	$0.811466\pi$
$294$		0			0
$295$			1.62714e8i			0.369018i
$296$	4.36290e7	+	4.85080e8i	0.0977809	+	1.08716i
$297$		0			0
$298$	8.18255e8	−	2.44733e7i	1.79115	−	0.0535716i
$299$			1.70709e8i			0.369325i
$300$		0			0
$301$		−	7.25621e8i		−	1.53365i
$302$	−1.90017e7	−	6.35315e8i	−0.0396979	−	1.32729i
$303$		0			0
$304$	5.52085e8	−	6.63464e7i	1.12706	−	0.135444i
$305$		−	2.99544e7i		−	0.0604521i
$306$		0			0
$307$	3.95766e8			0.780647			0.390323	−	0.920678i	$-0.372363\pi$
$307$	3.95766e8			0.780647			0.390323	+	0.920678i	$0.372363\pi$
$308$	−2.02434e8	+	1.21201e7i	−0.394780	+	0.0236362i
$309$		0			0
$310$	−4.81021e6	−	1.60828e8i	−0.00917061	−	0.306616i
$311$	−4.17018e8			−0.786129			−0.393064	−	0.919511i	$-0.628585\pi$
$311$	−4.17018e8			−0.786129			−0.393064	+	0.919511i	$0.628585\pi$
$312$		0			0
$313$	2.16149e8			0.398426			0.199213	−	0.979956i	$-0.436161\pi$
$313$	2.16149e8			0.398426			0.199213	+	0.979956i	$0.436161\pi$
$314$	2.80608e7	+	9.38204e8i	0.0511501	+	1.71019i
$315$		0			0
$316$	−5.81720e7	−	9.71611e8i	−0.103707	−	1.73216i
$317$	−6.69525e8			−1.18048			−0.590241	−	0.807227i	$-0.700967\pi$
$317$	−6.69525e8			−1.18048			−0.590241	+	0.807227i	$0.700967\pi$
$318$		0			0
$319$			1.53055e8i			0.263985i
$320$	−1.92076e8	+	3.48332e7i	−0.327679	+	0.0594248i
$321$		0			0
$322$	1.04249e7	+	3.48553e8i	0.0174011	+	0.581800i
$323$			5.32966e8i			0.880016i
$324$		0			0
$325$		−	3.94356e8i		−	0.637231i
$326$	1.50075e8	−	4.48862e6i	0.239910	−	0.00717548i
$327$		0			0
$328$	−8.60666e7	−	9.56913e8i	−0.134672	−	1.49732i
$329$			4.12088e8i			0.637975i
$330$		0			0
$331$	2.27775e8			0.345230			0.172615	−	0.984989i	$-0.444778\pi$
$331$	2.27775e8			0.345230			0.172615	+	0.984989i	$0.444778\pi$
$332$	−6.20459e7	−	1.03631e9i	−0.0930529	−	1.55420i
$333$		0			0
$334$	−7.49702e8	+	2.24229e7i	−1.10097	+	0.0329290i
$335$	4.32755e7			0.0628905
$336$		0			0
$337$	−3.09334e8			−0.440274			−0.220137	−	0.975469i	$-0.570650\pi$
$337$	−3.09334e8			−0.440274			−0.220137	+	0.975469i	$0.570650\pi$
$338$	3.45089e8	−	1.03213e7i	0.486097	−	0.0145387i
$339$		0			0
$340$	−1.11821e7	−	1.86768e8i	−0.0154294	−	0.257707i
$341$	−2.36145e8			−0.322507
$342$		0			0
$343$			6.11277e8i			0.817915i
$344$	−1.02100e9	+	9.18303e7i	−1.35229	+	0.121627i
$345$		0			0
$346$	−6.78417e8	+	2.02908e7i	−0.880502	+	0.0263350i
$347$		−	4.22835e8i		−	0.543272i	−0.962400	−	0.271636i	$-0.912435\pi$
$347$		−	4.22835e8i		−	0.543272i	0.962400	−	0.271636i	$-0.0875647\pi$
$348$		0			0
$349$		−	1.23470e9i		−	1.55479i	−0.629011	−	0.777396i	$-0.716540\pi$
$349$		−	1.23470e9i		−	1.55479i	0.629011	−	0.777396i	$-0.283460\pi$
$350$	−2.40826e7	−	8.05193e8i	−0.0300237	−	1.00383i
$351$		0			0
$352$	4.26726e7	+	2.83304e8i	0.0521494	+	0.346221i
$353$			1.17813e8i			0.142554i	0.997457	+	0.0712772i	$0.0227075\pi$
$353$			1.17813e8i			0.142554i	−0.997457	+	0.0712772i	$0.977292\pi$
$354$		0			0
$355$	3.29402e8			0.390775
$356$	−7.72051e7	−	1.28951e9i	−0.0906924	−	1.51478i
$357$		0			0
$358$	2.70955e7	+	9.05930e8i	0.0312110	+	1.04353i
$359$	1.21083e9			1.38119			0.690595	−	0.723242i	$-0.257349\pi$
$359$	1.21083e9			1.38119			0.690595	+	0.723242i	$0.257349\pi$
$360$		0			0
$361$	2.57987e8			0.288617
$362$	−1.29759e7	−	4.33846e8i	−0.0143767	−	0.480679i
$363$		0			0
$364$	−7.43591e8	+	4.45201e7i	−0.808127	+	0.0483839i
$365$	3.18102e8			0.342406
$366$		0			0
$367$			2.02237e8i			0.213565i	0.994282	+	0.106782i	$0.0340548\pi$
$367$			2.02237e8i			0.213565i	−0.994282	+	0.106782i	$0.965945\pi$
$368$	4.89118e8	−	5.87794e7i	0.511618	−	0.0614833i
$369$		0			0
$370$	−1.05884e7	−	3.54021e8i	−0.0108674	−	0.363348i
$371$			6.79072e8i			0.690411i
$372$		0			0
$373$			1.43859e8i			0.143534i	0.997421	+	0.0717671i	$0.0228638\pi$
$373$			1.43859e8i			0.143534i	−0.997421	+	0.0717671i	$0.977136\pi$
$374$	−2.74480e8	+	8.20943e6i	−0.271306	+	0.00811451i
$375$		0			0
$376$	5.79835e8	−	5.21514e7i	0.562531	−	0.0505951i
$377$			5.62209e8i			0.540385i
$378$		0			0
$379$	8.77228e8			0.827705			0.413852	−	0.910344i	$-0.364183\pi$
$379$	8.77228e8			0.827705			0.413852	+	0.910344i	$0.364183\pi$
$380$	−4.03648e8	+	2.41671e7i	−0.377364	+	0.0225934i
$381$		0			0
$382$	−1.42768e9	+	4.27005e7i	−1.31041	+	0.0391932i
$383$	−2.04366e8			−0.185871			−0.0929356	−	0.995672i	$-0.529625\pi$
$383$	−2.04366e8			−0.185871			−0.0929356	+	0.995672i	$0.529625\pi$
$384$		0			0
$385$	1.47476e8			0.131707
$386$	1.36010e9	−	4.06793e7i	1.20369	−	0.0360013i
$387$		0			0
$388$	−7.51587e8	+	4.49988e7i	−0.653233	+	0.0391102i
$389$	−1.58576e9			−1.36589			−0.682943	−	0.730472i	$-0.739300\pi$
$389$	−1.58576e9			−1.36589			−0.682943	+	0.730472i	$0.739300\pi$
$390$		0			0
$391$			4.72180e8i			0.399474i
$392$	−3.27716e8	+	2.94754e7i	−0.274787	+	0.0247149i
$393$		0			0
$394$	1.40118e9	−	4.19080e7i	1.15414	−	0.0345191i
$395$			7.07831e8i			0.577883i
$396$		0			0
$397$			1.83002e8i			0.146787i	0.997303	+	0.0733935i	$0.0233829\pi$
$397$			1.83002e8i			0.146787i	−0.997303	+	0.0733935i	$0.976617\pi$
$398$	−5.68321e7	−	1.90016e9i	−0.0451859	−	1.51078i
$399$		0			0
$400$	−1.12991e9	+	1.35786e8i	−0.882744	+	0.106083i
$401$			1.57329e9i			1.21844i	0.793001	+	0.609220i	$0.208517\pi$
$401$			1.57329e9i			1.21844i	−0.793001	+	0.609220i	$0.791483\pi$
$402$		0			0
$403$	−8.67422e8			−0.660181
$404$	1.68943e9	−	1.01149e8i	1.27469	−	0.0763178i
$405$		0			0
$406$	3.43330e7	+	1.14791e9i	0.0254607	+	0.851271i
$407$	−5.19813e8			−0.382179
$408$		0			0
$409$	1.66645e9			1.20437			0.602187	−	0.798355i	$-0.294296\pi$
$409$	1.66645e9			1.20437			0.602187	+	0.798355i	$0.294296\pi$
$410$	2.08877e7	+	6.98374e8i	0.0149675	+	0.500432i
$411$		0			0
$412$	−6.78288e7	−	1.13290e9i	−0.0477831	−	0.798091i
$413$	1.79186e9			1.25164
$414$		0			0
$415$			7.54968e8i			0.518514i
$416$	1.56747e8	+	1.04065e9i	0.106751	+	0.708724i
$417$		0			0
$418$	1.77424e7	+	5.93211e8i	0.0118822	+	0.397276i
$419$			8.92118e8i			0.592479i	0.955114	+	0.296240i	$0.0957327\pi$
$419$			8.92118e8i			0.592479i	−0.955114	+	0.296240i	$0.904267\pi$
$420$		0			0
$421$			1.98849e9i			1.29878i	0.760455	+	0.649391i	$0.224976\pi$
$421$			1.98849e9i			1.29878i	−0.760455	+	0.649391i	$0.775024\pi$
$422$	9.32872e8	−	2.79013e7i	0.604267	−	0.0180731i
$423$		0			0
$424$	9.55500e8	−	8.59394e7i	0.608766	−	0.0547535i
$425$		−	1.09078e9i		−	0.689250i
$426$		0			0
$427$	−3.29869e8			−0.205042
$428$	3.60552e7	+	6.02208e8i	0.0222288	+	0.371273i
$429$		0			0
$430$	7.45143e8	−	2.22865e7i	0.451960	−	0.0135177i
$431$	9.29190e8			0.559029			0.279514	−	0.960142i	$-0.409827\pi$
$431$	9.29190e8			0.559029			0.279514	+	0.960142i	$0.409827\pi$
$432$		0			0
$433$	1.65923e9			0.982197			0.491099	−	0.871104i	$-0.336595\pi$
$433$	1.65923e9			0.982197			0.491099	+	0.871104i	$0.336595\pi$
$434$	−1.77109e9	+	5.29718e7i	−1.03999	+	0.0311050i
$435$		0			0
$436$	2.01930e8	+	3.37271e9i	0.116680	+	1.94884i
$437$	1.02049e9			0.584954
$438$		0			0
$439$		−	9.17987e7i		−	0.0517858i	−0.999665	−	0.0258929i	$-0.991757\pi$
$439$		−	9.17987e7i		−	0.0517858i	0.999665	−	0.0258929i	$-0.00824288\pi$
$440$	−1.86636e7	−	2.07508e8i	−0.0104451	−	0.116132i
$441$		0			0
$442$	−1.00823e9	+	3.01553e7i	−0.555371	+	0.0166106i
$443$			7.47135e8i			0.408306i	0.978939	+	0.204153i	$0.0654441\pi$
$443$			7.47135e8i			0.408306i	−0.978939	+	0.204153i	$0.934556\pi$
$444$		0			0
$445$			9.39424e8i			0.505361i
$446$	3.74886e7	+	1.25342e9i	0.0200091	+	0.668996i
$447$		0			0
$448$	3.83595e8	+	2.11521e9i	0.201558	+	1.11143i
$449$			3.67451e8i			0.191574i	0.995402	+	0.0957871i	$0.0305368\pi$
$449$			3.67451e8i			0.191574i	−0.995402	+	0.0957871i	$0.969463\pi$
$450$		0			0
$451$	1.02543e9			0.526367
$452$	−2.60877e7	−	4.35727e8i	−0.0132877	−	0.221937i
$453$		0			0
$454$	−8.26615e7	−	2.76376e9i	−0.0414579	−	1.38613i
$455$	5.41716e8			0.269607
$456$		0			0
$457$	−3.88980e9			−1.90643			−0.953215	−	0.302292i	$-0.902248\pi$
$457$	−3.88980e9			−1.90643			−0.953215	+	0.302292i	$0.902248\pi$
$458$	8.42990e6	+	2.81851e8i	0.00410009	+	0.137085i
$459$		0			0
$460$	−3.57611e8	+	2.14108e7i	−0.171300	+	0.0102560i
$461$	7.45672e8			0.354482			0.177241	−	0.984167i	$-0.443283\pi$
$461$	7.45672e8			0.354482			0.177241	+	0.984167i	$0.443283\pi$
$462$		0			0
$463$		−	1.76294e9i		−	0.825473i	−0.910850	−	0.412737i	$-0.864573\pi$
$463$		−	1.76294e9i		−	0.825473i	0.910850	−	0.412737i	$-0.135427\pi$
$464$	1.61085e9	−	1.93582e8i	0.748585	−	0.0899606i
$465$		0			0
$466$	−2.90611e7	−	9.71649e8i	−0.0133034	−	0.444794i
$467$		−	2.95161e9i		−	1.34106i	−0.741881	−	0.670531i	$-0.766066\pi$
$467$		−	2.95161e9i		−	1.34106i	0.741881	−	0.670531i	$-0.233934\pi$
$468$		0			0
$469$		−	4.76565e8i		−	0.213313i
$470$	−4.23175e8	+	1.26568e7i	−0.188008	+	0.00562316i
$471$		0			0
$472$	−2.26768e8	−	2.52127e9i	−0.0992623	−	1.10363i
$473$		−	1.09410e9i		−	0.475383i
$474$		0			0
$475$	−2.35742e9			−1.00928
$476$	−2.05676e9	+	1.23142e8i	−0.874097	+	0.0523337i
$477$		0			0
$478$	1.16979e9	−	3.49873e7i	0.489903	−	0.0146526i
$479$	−4.12030e9			−1.71299			−0.856495	−	0.516156i	$-0.827363\pi$
$479$	−4.12030e9			−1.71299			−0.856495	+	0.516156i	$0.827363\pi$
$480$		0			0
$481$	−1.90941e9			−0.782331
$482$	−2.40165e9	+	7.18311e7i	−0.976888	+	0.0292178i
$483$		0			0
$484$	2.18467e9	−	1.30800e8i	0.875843	−	0.0524382i
$485$	5.47541e8			0.217932
$486$		0			0
$487$		−	4.42855e9i		−	1.73744i	−0.495304	−	0.868720i	$-0.664943\pi$
$487$		−	4.42855e9i		−	1.73744i	0.495304	−	0.868720i	$-0.335057\pi$
$488$	4.17463e7	+	4.64147e8i	0.0162610	+	0.180795i
$489$		0			0
$490$	2.39174e8	−	7.15347e6i	0.0918390	−	0.00274682i
$491$		−	4.43647e9i		−	1.69142i	−0.533641	−	0.845711i	$-0.679177\pi$
$491$		−	4.43647e9i		−	1.69142i	0.533641	−	0.845711i	$-0.320823\pi$
$492$		0			0
$493$			1.55506e9i			0.584498i
$494$	6.51724e7	+	2.17902e9i	0.0243231	+	0.813236i
$495$		0			0
$496$	2.98674e8	+	2.48534e9i	0.109904	+	0.914535i
$497$		−	3.62750e9i		−	1.32544i
$498$		0			0
$499$	3.41428e9			1.23012			0.615060	−	0.788481i	$-0.289132\pi$
$499$	3.41428e9			1.23012			0.615060	+	0.788481i	$0.289132\pi$
$500$	1.75528e9	−	1.05092e8i	0.627989	−	0.0375988i
$501$		0			0
$502$	−9.75182e7	−	3.26049e9i	−0.0344051	−	1.15032i
$503$	−2.56937e9			−0.900201			−0.450100	−	0.892978i	$-0.648612\pi$
$503$	−2.56937e9			−0.900201			−0.450100	+	0.892978i	$0.648612\pi$
$504$		0			0
$505$	−1.23077e9			−0.425262
$506$	1.57188e7	+	5.25554e8i	0.00539378	+	0.180339i
$507$		0			0
$508$	−7.55425e7	−	1.26174e9i	−0.0255663	−	0.427019i
$509$	−1.06039e9			−0.356414			−0.178207	−	0.983993i	$-0.557030\pi$
$509$	−1.06039e9			−0.356414			−0.178207	+	0.983993i	$0.557030\pi$
$510$		0			0
$511$		−	3.50306e9i		−	1.16138i
$512$	2.92770e9	−	8.07433e8i	0.964010	−	0.265865i
$513$		0			0
$514$	−1.03704e8	−	3.46731e9i	−0.0336841	−	1.12622i
$515$			8.25334e8i			0.266259i
$516$		0			0
$517$			6.21352e8i			0.197752i
$518$	−3.89861e9	+	1.16604e8i	−1.23241	+	0.0368602i
$519$		0			0
$520$	−6.85564e7	−	7.62230e8i	−0.0213814	−	0.237725i
$521$		−	4.18471e8i		−	0.129638i	−0.997897	−	0.0648191i	$-0.979353\pi$
$521$		−	4.18471e8i		−	0.129638i	0.997897	−	0.0648191i	$-0.0206470\pi$
$522$		0			0
$523$	−3.84414e9			−1.17502			−0.587508	−	0.809218i	$-0.699891\pi$
$523$	−3.84414e9			−1.17502			−0.587508	+	0.809218i	$0.699891\pi$
$524$	1.65760e8	+	2.76859e9i	0.0503293	+	0.840618i
$525$		0			0
$526$	8.87763e8	−	2.65522e7i	0.265979	−	0.00795518i
$527$	−2.39927e9			−0.714073
$528$		0			0
$529$	−2.50073e9			−0.734466
$530$	−6.97343e8	+	2.08569e7i	−0.203461	+	0.00608533i
$531$		0			0
$532$	2.66137e8	+	4.44512e9i	0.0766327	+	1.27995i
$533$	3.76667e9			1.07749
$534$		0			0
$535$		−	4.38717e8i		−	0.123864i
$536$	−6.70559e8	+	6.03113e7i	−0.188088	+	0.0169169i
$537$		0			0
$538$	2.99120e9	−	8.94642e7i	0.828148	−	0.0247692i
$539$		−	3.51181e8i		−	0.0965986i
$540$		0			0
$541$			3.23668e9i			0.878838i	0.898282	+	0.439419i	$0.144816\pi$
$541$			3.23668e9i			0.878838i	−0.898282	+	0.439419i	$0.855184\pi$
$542$	1.21957e8	+	4.07759e9i	0.0329010	+	1.10003i
$543$		0			0
$544$	4.33560e8	+	2.87842e9i	0.115466	+	0.766580i
$545$		−	2.45706e9i		−	0.650173i
$546$		0			0
$547$	−4.94425e9			−1.29165			−0.645824	−	0.763486i	$-0.723486\pi$
$547$	−4.94425e9			−1.29165			−0.645824	+	0.763486i	$0.723486\pi$
$548$	3.91963e8	+	6.54672e9i	0.101745	+	1.69939i
$549$		0			0
$550$	−3.63121e7	−	1.21408e9i	−0.00930640	−	0.311156i
$551$	3.36083e9			0.855887
$552$		0			0
$553$	7.79490e9			1.96007
$554$	−8.35356e7	−	2.79299e9i	−0.0208731	−	0.697886i
$555$		0			0
$556$	5.00899e9	−	2.99897e8i	1.23591	−	0.0739963i
$557$	−1.48349e9			−0.363740			−0.181870	−	0.983323i	$-0.558215\pi$
$557$	−1.48349e9			−0.363740			−0.181870	+	0.983323i	$0.558215\pi$
$558$		0			0
$559$		−	4.01892e9i		−	0.973123i
$560$	−1.86526e8	−	1.55213e9i	−0.0448829	−	0.373482i
$561$		0			0
$562$	−3.93522e7	−	1.31573e9i	−0.00935174	−	0.312672i
$563$			4.65548e9i			1.09947i	0.835338	+	0.549737i	$0.185272\pi$
$563$			4.65548e9i			1.09947i	−0.835338	+	0.549737i	$0.814728\pi$
$564$		0			0
$565$			3.17433e8i			0.0740426i
$566$	−3.45533e9	+	1.03346e8i	−0.801000	+	0.0239572i
$567$		0			0
$568$	−5.10413e9	+	4.59075e8i	−1.16870	+	0.105115i
$569$			4.03603e9i			0.918462i	0.888317	+	0.459231i	$0.151875\pi$
$569$			4.03603e9i			0.918462i	−0.888317	+	0.459231i	$0.848125\pi$
$570$		0			0
$571$	4.33338e9			0.974094			0.487047	−	0.873376i	$-0.338074\pi$
$571$	4.33338e9			0.974094			0.487047	+	0.873376i	$0.338074\pi$
$572$	−1.12120e9	+	6.71281e7i	−0.250493	+	0.0149975i
$573$		0			0
$574$	7.69075e9	−	2.30023e8i	1.69737	−	0.0507669i
$575$	−2.08855e9			−0.458150
$576$		0			0
$577$	1.65483e9			0.358623			0.179311	−	0.983792i	$-0.442613\pi$
$577$	1.65483e9			0.358623			0.179311	+	0.983792i	$0.442613\pi$
$578$	1.85162e9	−	5.53802e7i	0.398845	−	0.0119291i
$579$		0			0
$580$	−1.17774e9	+	7.05135e7i	−0.250641	+	0.0150063i
$581$	8.31398e9			1.75870
$582$		0			0
$583$			1.02392e9i			0.214005i
$584$	−4.92904e9	+	4.43327e8i	−1.02404	+	0.0921041i
$585$		0			0
$586$	−8.07938e9	+	2.41647e8i	−1.65858	+	0.0496066i
$587$		−	5.30362e9i		−	1.08228i	−0.840933	−	0.541140i	$-0.817993\pi$
$587$		−	5.30362e9i		−	1.08228i	0.840933	−	0.541140i	$-0.182007\pi$
$588$		0			0
$589$			5.18537e9i			1.04563i
$590$	5.50349e7	+	1.84007e9i	0.0110320	+	0.368853i
$591$		0			0
$592$	6.57454e8	+	5.47084e9i	0.130238	+	1.08375i
$593$			7.54656e7i			0.0148613i	0.999972	+	0.00743066i	$0.00236527\pi$
$593$			7.54656e7i			0.0148613i	−0.999972	+	0.00743066i	$0.997635\pi$
$594$		0			0
$595$	1.49838e9			0.291616
$596$	9.24508e9	−	5.53519e8i	1.78875	−	0.107095i
$597$		0			0
$598$	5.77393e7	+	1.93049e9i	0.0110412	+	0.369160i
$599$	−7.13988e9			−1.35737			−0.678683	−	0.734431i	$-0.737449\pi$
$599$	−7.13988e9			−1.35737			−0.678683	+	0.734431i	$0.737449\pi$
$600$		0			0
$601$	−9.21234e9			−1.73105			−0.865524	−	0.500867i	$-0.833015\pi$
$601$	−9.21234e9			−1.73105			−0.865524	+	0.500867i	$0.833015\pi$
$602$	−2.45428e8	−	8.20579e9i	−0.0458496	−	1.53297i
$603$		0			0
$604$	−4.29767e8	−	7.17813e9i	−0.0793604	−	1.32551i
$605$	−1.59156e9			−0.292199
$606$		0			0
$607$			2.53391e9i			0.459866i	0.973206	+	0.229933i	$0.0738507\pi$
$607$			2.53391e9i			0.459866i	−0.973206	+	0.229933i	$0.926149\pi$
$608$	6.22090e9	−	9.37020e8i	1.12251	−	0.169078i
$609$		0			0
$610$	−1.01315e7	−	3.38744e8i	−0.00180726	−	0.0604251i
$611$			2.28239e9i			0.404804i
$612$		0			0
$613$			2.83258e9i			0.496674i	0.968674	+	0.248337i	$0.0798840\pi$
$613$			2.83258e9i			0.496674i	−0.968674	+	0.248337i	$0.920116\pi$
$614$	4.47558e9	−	1.33861e8i	0.780298	−	0.0233380i
$615$		0			0
$616$	−2.28515e9	+	2.05531e8i	−0.393897	+	0.0354278i
$617$			1.09456e9i			0.187604i	0.995591	+	0.0938018i	$0.0299020\pi$
$617$			1.09456e9i			0.187604i	−0.995591	+	0.0938018i	$0.970098\pi$
$618$		0			0
$619$	−3.22342e9			−0.546260			−0.273130	−	0.961977i	$-0.588059\pi$
$619$	−3.22342e9			−0.546260			−0.273130	+	0.961977i	$0.588059\pi$
$620$	−1.08794e8	−	1.81712e9i	−0.0183330	−	0.306205i
$621$		0			0
$622$	−4.71591e9	+	1.41049e8i	−0.785777	+	0.0235019i
$623$	1.03453e10			1.71409
$624$		0			0
$625$	4.14786e9			0.679585
$626$	2.44435e9	−	7.31084e7i	0.398248	−	0.0119112i
$627$		0			0
$628$	6.34660e8	+	1.06003e10i	0.102255	+	1.70789i
$629$	−5.28138e9			−0.846195
$630$		0			0
$631$			8.31648e9i			1.31776i	0.752248	+	0.658880i	$0.228970\pi$
$631$			8.31648e9i			1.31776i	−0.752248	+	0.658880i	$0.771030\pi$
$632$	−9.86476e8	−	1.09679e10i	−0.155445	−	1.72828i
$633$		0			0
$634$	−7.57143e9	+	2.26454e8i	−1.17995	+	0.0352914i
$635$			9.19193e8i			0.142462i
$636$		0			0
$637$		−	1.28998e9i		−	0.197740i
$638$	5.17679e7	+	1.73084e9i	0.00789202	+	0.263867i
$639$		0			0
$640$	−2.16034e9	+	4.58882e8i	−0.325756	+	0.0691945i
$641$			8.71195e9i			1.30651i	0.757138	+	0.653255i	$0.226597\pi$
$641$			8.71195e9i			1.30651i	−0.757138	+	0.653255i	$0.773403\pi$
$642$		0			0
$643$	9.96578e9			1.47834			0.739168	−	0.673522i	$-0.235219\pi$
$643$	9.96578e9			1.47834			0.739168	+	0.673522i	$0.235219\pi$
$644$	2.35783e8	+	3.93814e9i	0.0347866	+	0.581019i
$645$		0			0
$646$	1.80266e8	+	6.02713e9i	0.0263087	+	0.879623i
$647$	4.53985e9			0.658987			0.329493	−	0.944158i	$-0.393122\pi$
$647$	4.53985e9			0.658987			0.329493	+	0.944158i	$0.393122\pi$
$648$		0			0
$649$	2.70180e9			0.387969
$650$	−1.33384e8	−	4.45964e9i	−0.0190505	−	0.636946i
$651$		0			0
$652$	1.69563e9	−	1.01520e8i	0.239588	−	0.0143445i
$653$	2.90772e9			0.408655			0.204328	−	0.978903i	$-0.434499\pi$
$653$	2.90772e9			0.408655			0.204328	+	0.978903i	$0.434499\pi$
$654$		0			0
$655$		−	2.01695e9i		−	0.280447i
$656$	−1.29695e9	−	1.07923e10i	−0.179375	−	1.49262i
$657$		0			0
$658$	1.39381e8	+	4.66016e9i	0.0190727	+	0.637690i
$659$			3.96130e9i			0.539186i	0.962974	+	0.269593i	$0.0868892\pi$
$659$			3.96130e9i			0.539186i	−0.962974	+	0.269593i	$0.913111\pi$
$660$		0			0
$661$		−	4.86484e9i		−	0.655184i	−0.944819	−	0.327592i	$-0.893763\pi$
$661$		−	4.86484e9i		−	0.655184i	0.944819	−	0.327592i	$-0.106237\pi$
$662$	2.57583e9	−	7.70408e7i	0.345076	−	0.0103209i
$663$		0			0
$664$	−1.05217e9	−	1.16983e10i	−0.139475	−	1.55073i
$665$		−	3.23833e9i		−	0.427017i
$666$		0			0
$667$	2.97752e9			0.388521
$668$	−8.47053e9	+	5.07145e8i	−1.09949	+	0.0658286i
$669$		0			0
$670$	4.89387e8	−	1.46371e7i	0.0628624	−	0.00188016i
$671$	−4.97381e8			−0.0635566
$672$		0			0
$673$	−2.43436e9			−0.307845			−0.153923	−	0.988083i	$-0.549191\pi$
$673$	−2.43436e9			−0.307845			−0.153923	+	0.988083i	$0.549191\pi$
$674$	−3.49815e9	+	1.04626e8i	−0.440077	+	0.0131623i
$675$		0			0
$676$	3.89900e9	−	2.33440e8i	0.485445	−	0.0290644i
$677$	1.30568e9			0.161725			0.0808624	−	0.996725i	$-0.474233\pi$
$677$	1.30568e9			0.161725			0.0808624	+	0.996725i	$0.474233\pi$
$678$		0			0
$679$		−	6.02972e9i		−	0.739184i
$680$	−1.89626e8	−	2.10832e9i	−0.0231268	−	0.257131i
$681$		0			0
$682$	−2.67048e9	+	7.98717e7i	−0.322363	+	0.00964157i
$683$			7.27394e9i			0.873569i	0.899566	+	0.436785i	$0.143883\pi$
$683$			7.27394e9i			0.873569i	−0.899566	+	0.436785i	$0.856117\pi$
$684$		0			0
$685$		−	4.76937e9i		−	0.566949i
$686$	2.06753e8	+	6.91271e9i	0.0244522	+	0.817550i
$687$		0			0
$688$	−1.15150e10	+	1.38381e9i	−1.34805	+	0.162001i
$689$			3.76111e9i			0.438075i
$690$		0			0
$691$	−2.01649e9			−0.232500			−0.116250	−	0.993220i	$-0.537087\pi$
$691$	−2.01649e9			−0.232500			−0.116250	+	0.993220i	$0.537087\pi$
$692$	−7.66511e9	+	4.58924e8i	−0.879321	+	0.0526464i
$693$		0			0
$694$	−1.43016e8	−	4.78169e9i	−0.0162415	−	0.543030i
$695$	−3.64911e9			−0.412326
$696$		0			0
$697$	1.04185e10			1.16545
$698$	−4.17614e8	−	1.39628e10i	−0.0464816	−	1.55410i
$699$		0			0
$700$	−5.44683e8	−	9.09750e9i	−0.0600206	−	1.00249i
$701$	−1.51344e10			−1.65940			−0.829700	−	0.558210i	$-0.811488\pi$
$701$	−1.51344e10			−1.65940			−0.829700	+	0.558210i	$0.811488\pi$
$702$		0			0
$703$			1.14142e10i			1.23909i
$704$	5.78391e8	+	3.18935e9i	0.0624766	+	0.344507i
$705$		0			0
$706$	3.98480e7	+	1.33230e9i	0.00426177	+	0.142491i
$707$			1.35537e10i			1.44241i
$708$		0			0
$709$		−	1.47287e10i		−	1.55204i	−0.630708	−	0.776020i	$-0.717235\pi$
$709$		−	1.47287e10i		−	1.55204i	0.630708	−	0.776020i	$-0.282765\pi$
$710$	3.72509e9	−	1.11414e8i	0.390601	−	0.0116825i
$711$		0			0
$712$	−1.30924e9	−	1.45565e10i	−0.135937	−	1.51139i
$713$			4.59396e9i			0.474650i
$714$		0			0
$715$	8.16808e8			0.0835697
$716$	6.12828e8	+	1.02357e10i	0.0623940	+	1.04213i
$717$		0			0
$718$	1.36929e10	−	4.09541e8i	1.38057	−	0.0412916i
$719$	−2.99832e9			−0.300834			−0.150417	−	0.988623i	$-0.548062\pi$
$719$	−2.99832e9			−0.300834			−0.150417	+	0.988623i	$0.548062\pi$
$720$		0			0
$721$	9.08888e9			0.903103
$722$	2.91748e9	−	8.72592e7i	0.288488	−	0.00862842i
$723$		0			0
$724$	−2.93481e8	−	4.90182e9i	−0.0287405	−	0.480035i
$725$	−6.87837e9			−0.670352
$726$		0			0
$727$			9.00906e9i			0.869579i	0.900532	+	0.434789i	$0.143177\pi$
$727$			9.00906e9i			0.869579i	−0.900532	+	0.434789i	$0.856823\pi$
$728$	−8.39396e9	+	7.54968e8i	−0.806319	+	0.0725218i
$729$		0			0
$730$	3.59731e9	−	1.07592e8i	0.342253	−	0.0102365i
$731$		−	1.11163e10i		−	1.05256i
$732$		0			0
$733$		−	8.89008e9i		−	0.833761i	−0.908961	−	0.416880i	$-0.863123\pi$
$733$		−	8.89008e9i		−	0.833761i	0.908961	−	0.416880i	$-0.136877\pi$
$734$	6.84029e7	+	2.28703e9i	0.00638467	+	0.213469i
$735$		0			0
$736$	5.51138e9	−	8.30150e8i	0.509552	−	0.0767511i
$737$		−	7.18573e8i		−	0.0661202i
$738$		0			0
$739$	8.53349e9			0.777806			0.388903	−	0.921279i	$-0.372854\pi$
$739$	8.53349e9			0.777806			0.388903	+	0.921279i	$0.372854\pi$
$740$	−2.39482e8	−	3.99992e9i	−0.0217251	−	0.362861i
$741$		0			0
$742$	2.29684e8	+	7.67939e9i	0.0206403	+	0.690102i
$743$	1.12936e10			1.01011			0.505057	−	0.863086i	$-0.331472\pi$
$743$	1.12936e10			1.01011			0.505057	+	0.863086i	$0.331472\pi$
$744$		0			0
$745$	−6.73516e9			−0.596762
$746$	4.86576e7	+	1.62685e9i	0.00429106	+	0.143470i
$747$		0			0
$748$	−3.10122e9	+	1.85675e8i	−0.270942	+	0.0162218i
$749$	−4.83131e9			−0.420125
$750$		0			0
$751$			6.47122e9i			0.557502i	0.960363	+	0.278751i	$0.0899204\pi$
$751$			6.47122e9i			0.557502i	−0.960363	+	0.278751i	$0.910080\pi$
$752$	6.53951e9	−	7.85880e8i	0.560767	−	0.0673898i
$753$		0			0
$754$	1.90157e8	+	6.35783e9i	0.0161552	+	0.540143i
$755$			5.22936e9i			0.442216i
$756$		0			0
$757$		−	4.70372e9i		−	0.394100i	−0.980393	−	0.197050i	$-0.936864\pi$
$757$		−	4.70372e9i		−	0.394100i	0.980393	−	0.197050i	$-0.0631361\pi$
$758$	9.92027e9	−	2.96706e8i	0.827335	−	0.0247448i
$759$		0			0
$760$	−4.55654e9	+	4.09824e8i	−0.376520	+	0.0338649i
$761$		−	9.44372e9i		−	0.776778i	−0.921496	−	0.388389i	$-0.873032\pi$
$761$		−	9.44372e9i		−	0.776778i	0.921496	−	0.388389i	$-0.126968\pi$
$762$		0			0
$763$	−2.70581e10			−2.20527
$764$	−1.61306e10	+	9.65770e8i	−1.30865	+	0.0783514i
$765$		0			0
$766$	−2.31110e9	+	6.91228e7i	−0.185788	+	0.00555675i
$767$	9.92441e9			0.794183
$768$		0			0
$769$	−5.19374e9			−0.411849			−0.205925	−	0.978568i	$-0.566020\pi$
$769$	−5.19374e9			−0.411849			−0.205925	+	0.978568i	$0.566020\pi$
$770$	1.66775e9	−	4.98809e7i	0.131648	−	0.00393746i
$771$		0			0
$772$	1.53671e10	−	9.20057e8i	1.20208	−	0.0719704i
$773$	−1.43367e10			−1.11640			−0.558202	−	0.829705i	$-0.688509\pi$
$773$	−1.43367e10			−1.11640			−0.558202	+	0.829705i	$0.688509\pi$
$774$		0			0
$775$		−	1.06125e10i		−	0.818959i
$776$	−8.48421e9	+	7.63086e8i	−0.651772	+	0.0586216i
$777$		0			0
$778$	−1.79328e10	+	5.36355e8i	−1.36528	+	0.0408341i
$779$		−	2.25168e10i		−	1.70658i
$780$		0			0
$781$		−	5.46960e9i		−	0.410844i
$782$	1.59706e8	+	5.33972e9i	0.0119426	+	0.399296i
$783$		0			0
$784$	−3.69606e9	+	4.44171e8i	−0.273926	+	0.0329188i
$785$		−	7.72248e9i		−	0.569788i
$786$		0			0
$787$	6.32430e9			0.462489			0.231244	−	0.972896i	$-0.425720\pi$
$787$	6.32430e9			0.462489			0.231244	+	0.972896i	$0.425720\pi$
$788$	1.58313e10	−	9.47846e8i	1.15259	−	0.0690074i
$789$		0			0
$790$	2.39411e8	+	8.00462e9i	0.0172762	+	0.577625i
$791$	3.49568e9			0.251139
$792$		0			0
$793$	−1.82701e9			−0.130102
$794$	6.18968e7	+	2.06950e9i	0.00438830	+	0.146721i
$795$		0			0
$796$	−1.28539e9	−	2.14691e10i	−0.0903315	−	1.50875i
$797$	1.99261e10			1.39418			0.697091	−	0.716983i	$-0.254477\pi$
$797$	1.99261e10			1.39418			0.697091	+	0.716983i	$0.254477\pi$
$798$		0			0
$799$			6.31304e9i			0.437850i
$800$	−1.27319e10	+	1.91773e9i	−0.879178	+	0.132426i
$801$		0			0
$802$	5.32137e8	+	1.77918e10i	0.0364261	+	1.21790i
$803$		−	5.28197e9i		−	0.359991i
$804$		0			0
$805$		−	2.86899e9i		−	0.193840i
$806$	−9.80937e9	+	2.93389e8i	−0.659885	+	0.0197366i
$807$		0			0
$808$	1.90709e10	−	1.71527e9i	1.27184	−	0.114391i
$809$		−	4.18784e9i		−	0.278080i	−0.990287	−	0.139040i	$-0.955598\pi$
$809$		−	4.18784e9i		−	0.278080i	0.990287	−	0.139040i	$-0.0444017\pi$
$810$		0			0
$811$	−3.77264e9			−0.248355			−0.124177	−	0.992260i	$-0.539629\pi$
$811$	−3.77264e9			−0.248355			−0.124177	+	0.992260i	$0.539629\pi$
$812$	7.76521e8	+	1.29697e10i	0.0508987	+	0.850130i
$813$		0			0
$814$	−5.87838e9	+	1.75817e8i	−0.382008	+	0.0114255i
$815$	−1.23529e9			−0.0799314
$816$		0			0
$817$	−2.40247e10			−1.54128
$818$	1.88453e10	−	5.63646e8i	1.20384	−	0.0360056i
$819$		0			0
$820$	4.72424e8	+	7.89061e9i	0.0299215	+	0.499761i
$821$	−8.35665e9			−0.527025			−0.263512	−	0.964656i	$-0.584881\pi$
$821$	−8.35665e9			−0.527025			−0.263512	+	0.964656i	$0.584881\pi$
$822$		0			0
$823$			2.54233e10i			1.58977i	0.606763	+	0.794883i	$0.292468\pi$
$823$			2.54233e10i			1.58977i	−0.606763	+	0.794883i	$0.707532\pi$
$824$	−1.15024e9	−	1.27887e10i	−0.0716212	−	0.796306i
$825$		0			0
$826$	2.02636e10	−	6.06064e8i	1.25108	−	0.0374187i
$827$		−	2.71179e10i		−	1.66720i	−0.552371	−	0.833599i	$-0.686277\pi$
$827$		−	2.71179e10i		−	1.66720i	0.552371	−	0.833599i	$-0.313723\pi$
$828$		0			0
$829$		−	1.95618e10i		−	1.19252i	−0.802790	−	0.596262i	$-0.796652\pi$
$829$		−	1.95618e10i		−	1.19252i	0.802790	−	0.596262i	$-0.203348\pi$
$830$	2.55354e8	+	8.53767e9i	0.0155013	+	0.518282i
$831$		0			0
$832$	2.12458e9	+	1.17153e10i	0.127891	+	0.705216i
$833$		−	3.56806e9i		−	0.213882i
$834$		0			0
$835$	6.17089e9			0.366813
$836$	4.01285e8	+	6.70242e9i	0.0237537	+	0.396743i
$837$		0			0
$838$	3.01742e8	+	1.00886e10i	0.0177126	+	0.592214i
$839$	1.86844e10			1.09223			0.546113	−	0.837711i	$-0.316107\pi$
$839$	1.86844e10			1.09223			0.546113	+	0.837711i	$0.316107\pi$
$840$		0			0
$841$	−7.44381e9			−0.431528
$842$	6.72570e8	+	2.24871e10i	0.0388280	+	1.29820i
$843$		0			0
$844$	1.05401e10	−	6.31053e8i	0.603456	−	0.0361300i
$845$	−2.84047e9			−0.161954
$846$		0			0
$847$			1.75268e10i			0.991085i
$848$	1.07763e10	−	1.29504e9i	0.606857	−	0.0729285i
$849$		0			0
$850$	−3.68937e8	−	1.23353e10i	−0.0206056	−	0.688942i
$851$			1.01124e10i			0.562473i
$852$		0			0
$853$			3.55557e10i			1.96150i	0.195273	+	0.980749i	$0.437441\pi$
$853$			3.55557e10i			1.96150i	−0.195273	+	0.980749i	$0.562559\pi$
$854$	−3.73037e9	+	1.11572e8i	−0.204951	+	0.00612989i
$855$		0			0
$856$	6.11422e8	+	6.79797e9i	0.0333183	+	0.370443i
$857$		−	4.79883e6i		−	0.000260437i	−1.00000		0.000130218i	$-0.999959\pi$
$857$		−	4.79883e6i		−	0.000260437i	1.00000		0.000130218i	$-4.14498e-5\pi$
$858$		0			0
$859$	−1.84728e10			−0.994388			−0.497194	−	0.867639i	$-0.665636\pi$
$859$	−1.84728e10			−0.994388			−0.497194	+	0.867639i	$0.665636\pi$
$860$	8.41903e9	−	5.04062e8i	0.451354	−	0.0270233i
$861$		0			0
$862$	1.05079e10	−	3.14281e8i	0.558779	−	0.0167126i
$863$	6.22171e9			0.329513			0.164756	−	0.986334i	$-0.447316\pi$
$863$	6.22171e9			0.329513			0.164756	+	0.986334i	$0.447316\pi$
$864$		0			0
$865$	5.58414e9			0.293359
$866$	1.87636e10	−	5.61203e8i	0.981758	−	0.0293635i
$867$		0			0
$868$	−2.00108e10	+	1.19808e9i	−1.03859	+	0.0621823i
$869$	1.17533e10			0.607560
$870$		0			0
$871$		−	2.63950e9i		−	0.135350i
$872$	3.42431e9	+	3.80725e10i	0.174890	+	1.94448i
$873$		0			0
$874$	1.15403e10	−	3.45160e8i	0.584693	−	0.0174876i
$875$			1.40820e10i			0.710619i
$876$		0			0
$877$			1.16634e10i			0.583884i	0.956436	+	0.291942i	$0.0943014\pi$
$877$			1.16634e10i			0.583884i	−0.956436	+	0.291942i	$0.905699\pi$
$878$	−3.10492e7	−	1.03812e9i	−0.00154817	−	0.0517626i
$879$		0			0
$880$	−2.81246e8	−	2.34032e9i	−0.0139123	−	0.115767i
$881$		−	9.09495e9i		−	0.448110i	−0.974577	−	0.224055i	$-0.928070\pi$
$881$		−	9.09495e9i		−	0.448110i	0.974577	−	0.224055i	$-0.0719295\pi$
$882$		0			0
$883$	1.53416e10			0.749907			0.374954	−	0.927044i	$-0.377659\pi$
$883$	1.53416e10			0.749907			0.374954	+	0.927044i	$0.377659\pi$
$884$	−1.13916e10	+	6.82033e8i	−0.554626	+	0.0332064i
$885$		0			0
$886$	2.52705e8	+	8.44909e9i	0.0122066	+	0.408124i
$887$	−2.84449e10			−1.36858			−0.684291	−	0.729209i	$-0.739888\pi$
$887$	−2.84449e10			−1.36858			−0.684291	+	0.729209i	$0.739888\pi$
$888$		0			0
$889$	1.01225e10			0.483205
$890$	3.17742e8	+	1.06236e10i	0.0151081	+	0.505135i
$891$		0			0
$892$	8.47890e8	+	1.41618e10i	0.0400002	+	0.668099i
$893$	1.36439e10			0.641148
$894$		0			0
$895$		−	7.45683e9i		−	0.347675i
$896$	5.05338e9	+	2.37905e10i	0.234695	+	1.10490i
$897$		0			0
$898$	1.24283e8	+	4.15537e9i	0.00572725	+	0.191489i
$899$			1.51296e10i			0.694494i
$900$		0			0
$901$			1.04032e10i			0.473837i
$902$	1.15962e10	−	3.46833e8i	0.526131	−	0.0157361i
$903$		0			0
$904$	−4.42393e8	−	4.91866e9i	−0.0199168	−	0.221441i
$905$			3.57104e9i			0.160149i
$906$		0			0
$907$	−1.63965e10			−0.729666			−0.364833	−	0.931073i	$-0.618874\pi$
$907$	−1.63965e10			−0.729666			−0.364833	+	0.931073i	$0.618874\pi$
$908$	−1.86958e9	−	3.12264e10i	−0.0828788	−	1.38427i
$909$		0			0
$910$	6.12608e9	−	1.83225e8i	0.269487	−	0.00806010i
$911$	1.21999e10			0.534617			0.267309	−	0.963611i	$-0.413866\pi$
$911$	1.21999e10			0.534617			0.267309	+	0.963611i	$0.413866\pi$
$912$		0			0
$913$	1.25360e10			0.545142
$914$	−4.39884e10	+	1.31565e9i	−1.90558	+	0.0569941i
$915$		0			0
$916$	1.90662e8	+	3.18450e9i	0.00819651	+	0.136901i
$917$	−2.22114e10			−0.951226
$918$		0			0
$919$		−	3.87196e10i		−	1.64561i	−0.568326	−	0.822803i	$-0.692409\pi$
$919$		−	3.87196e10i		−	1.64561i	0.568326	−	0.822803i	$-0.307591\pi$
$920$	−4.03685e9	+	3.63082e8i	−0.170917	+	0.0153726i
$921$		0			0
$922$	8.43255e9	−	2.52210e8i	0.354324	−	0.0105975i
$923$		−	2.00912e10i		−	0.841009i
$924$		0			0
$925$		−	2.33607e10i		−	0.970487i
$926$	−5.96280e8	−	1.99364e10i	−0.0246781	−	0.825105i
$927$		0			0
$928$	1.81510e10	−	2.73399e9i	0.745561	−	0.112300i
$929$		−	3.34318e10i		−	1.36806i	−0.729455	−	0.684029i	$-0.760226\pi$
$929$		−	3.34318e10i		−	1.36806i	0.729455	−	0.684029i	$-0.239774\pi$
$930$		0			0
$931$	−7.71138e9			−0.313190
$932$	−6.57284e8	−	1.09782e10i	−0.0265949	−	0.444197i
$933$		0			0
$934$	−9.98325e8	−	3.33787e10i	−0.0400920	−	1.34046i
$935$	2.25928e9			0.0903917
$936$		0			0
$937$	−1.13884e8			−0.00452247			−0.00226123	−	0.999997i	$-0.500720\pi$
$937$	−1.13884e8			−0.00452247			−0.00226123	+	0.999997i	$0.500720\pi$
$938$	−1.61189e8	−	5.38931e9i	−0.00637715	−	0.213218i
$939$		0			0
$940$	−4.78126e9	+	2.86262e8i	−0.187756	+	0.0112413i
$941$	3.30760e10			1.29404			0.647022	−	0.762472i	$-0.276014\pi$
$941$	3.30760e10			1.29404			0.647022	+	0.762472i	$0.276014\pi$
$942$		0			0
$943$		−	1.99487e10i		−	0.774682i
$944$	−3.41721e9	−	2.84355e10i	−0.132212	−	1.10017i
$945$		0			0
$946$	−3.70060e8	−	1.23728e10i	−0.0142119	−	0.475171i
$947$			4.30694e10i			1.64795i	0.566627	+	0.823974i	$0.308248\pi$
$947$			4.30694e10i			1.64795i	−0.566627	+	0.823974i	$0.691752\pi$
$948$		0			0
$949$		−	1.94020e10i		−	0.736912i
$950$	−2.66593e10	+	7.97355e8i	−1.00883	+	0.0301730i
$951$		0			0
$952$	−2.32175e10	+	2.08823e9i	−0.872142	+	0.0784420i
$953$			2.45613e10i			0.919233i	0.888118	+	0.459616i	$0.152013\pi$
$953$			2.45613e10i			0.919233i	−0.888118	+	0.459616i	$0.847987\pi$
$954$		0			0
$955$	1.17514e10			0.436594
$956$	1.32169e10	−	7.91319e8i	0.489246	−	0.0292920i
$957$		0			0
$958$	−4.65951e10	+	1.39362e9i	−1.71222	+	0.0512110i
$959$	−5.25221e10			−1.92299
$960$		0			0
$961$	4.16944e9			0.151546
$962$	−2.15928e10	+	6.45821e8i	−0.781981	+	0.0233883i
$963$		0			0
$964$	−2.71351e10	+	1.62463e9i	−0.975578	+	0.0584095i
$965$	−1.11952e10			−0.401037
$966$		0			0
$967$		−	2.28479e10i		−	0.812556i	−0.913750	−	0.406278i	$-0.866826\pi$
$967$		−	2.28479e10i		−	0.812556i	0.913750	−	0.406278i	$-0.133174\pi$
$968$	2.46614e10	−	2.21809e9i	0.873884	−	0.0785988i
$969$		0			0
$970$	6.19195e9	−	1.85195e8i	0.217834	−	0.00651522i
$971$		−	4.59295e10i		−	1.60999i	−0.593279	−	0.804997i	$-0.702167\pi$
$971$		−	4.59295e10i		−	1.60999i	0.593279	−	0.804997i	$-0.297833\pi$
$972$		0			0
$973$			4.01854e10i			1.39853i
$974$	−1.49787e9	−	5.00809e10i	−0.0519420	−	1.73666i
$975$		0			0
$976$	6.29083e8	+	5.23476e9i	0.0216588	+	0.180228i
$977$			4.53672e10i			1.55636i	0.628038	+	0.778182i	$0.283858\pi$
$977$			4.53672e10i			1.55636i	−0.628038	+	0.778182i	$0.716142\pi$
$978$		0			0
$979$	1.55988e10			0.531314
$980$	2.70231e9	−	1.61792e8i	0.0917158	−	0.00549119i
$981$		0			0
$982$	−1.50055e9	−	5.01705e10i	−0.0505663	−	1.69067i
$983$	−2.77122e10			−0.930536			−0.465268	−	0.885170i	$-0.654042\pi$
$983$	−2.77122e10			−0.930536			−0.465268	+	0.885170i	$0.654042\pi$
$984$		0			0
$985$	−1.15333e10			−0.384527
$986$	5.25971e8	+	1.75856e10i	0.0174740	+	0.584237i
$987$		0			0
$988$	1.47402e9	+	2.46197e10i	0.0486245	+	0.812145i
$989$	−2.12846e10			−0.699647
$990$		0			0
$991$		−	7.33226e9i		−	0.239321i	−0.992815	−	0.119660i	$-0.961819\pi$
$991$		−	7.33226e9i		−	0.239321i	0.992815	−	0.119660i	$-0.0381806\pi$
$992$	4.21822e9	+	2.80049e10i	0.137195	+	0.910841i
$993$		0			0
$994$	−1.22693e9	−	4.10221e10i	−0.0396249	−	1.32485i
$995$			1.56405e10i			0.503350i
$996$		0			0
$997$			2.27024e10i			0.725502i	0.931886	+	0.362751i	$0.118162\pi$
$997$			2.27024e10i			0.725502i	−0.931886	+	0.362751i	$0.881838\pi$
$998$	3.86109e10	−	1.15482e9i	1.22957	−	0.0367753i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.f.a.35.27	yes	28
3.2	odd	2	inner	72.8.f.a.35.2	yes	28
4.3	odd	2		288.8.f.a.143.13		28
8.3	odd	2	inner	72.8.f.a.35.1	✓	28
8.5	even	2		288.8.f.a.143.16		28
12.11	even	2		288.8.f.a.143.15		28
24.5	odd	2		288.8.f.a.143.14		28
24.11	even	2	inner	72.8.f.a.35.28	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.1	✓	28	8.3	odd	2	inner
72.8.f.a.35.2	yes	28	3.2	odd	2	inner
72.8.f.a.35.27	yes	28	1.1	even	1	trivial
72.8.f.a.35.28	yes	28	24.11	even	2	inner
288.8.f.a.143.13		28	4.3	odd	2
288.8.f.a.143.14		28	24.5	odd	2
288.8.f.a.143.15		28	12.11	even	2
288.8.f.a.143.16		28	8.5	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(11.3087 - 0.338231i) q^{2} +(127.771 - 7.64988i) q^{4} -93.0830 q^{5} +1025.06i q^{7} +(1442.33 - 129.726i) q^{8} +O(q^{10})\)	\(q+(11.3087 - 0.338231i) q^{2} +(127.771 - 7.64988i) q^{4} -93.0830 q^{5} +1025.06i q^{7} +(1442.33 - 129.726i) q^{8} +(-1052.64 + 31.4836i) q^{10} +1545.61i q^{11} +5677.41i q^{13} +(346.709 + 11592.1i) q^{14} +(16267.0 - 1954.87i) q^{16} +15703.6i q^{17} +33939.0 q^{19} +(-11893.3 + 712.074i) q^{20} +(522.773 + 17478.7i) q^{22} +30068.2 q^{23} -69460.6 q^{25} +(1920.28 + 64203.9i) q^{26} +(7841.62 + 130974. i) q^{28} +99025.6 q^{29} +152785. i q^{31} +(183296. - 27608.9i) q^{32} +(5311.46 + 177587. i) q^{34} -95416.0i q^{35} +336316. i q^{37} +(383805. - 11479.2i) q^{38} +(-134257. + 12075.3i) q^{40} -663449. i q^{41} -707879. q^{43} +(11823.7 + 197484. i) q^{44} +(340031. - 10170.0i) q^{46} +402012. q^{47} -227213. q^{49} +(-785505. + 23493.7i) q^{50} +(43431.5 + 725410. i) q^{52} +662469. q^{53} -143870. i q^{55} +(132977. + 1.47848e6i) q^{56} +(1.11985e6 - 33493.6i) q^{58} -1.74805e6i q^{59} +321803. i q^{61} +(51676.6 + 1.72779e6i) q^{62} +(2.06349e6 - 374216. i) q^{64} -528470. i q^{65} -464913. q^{67} +(120131. + 2.00647e6i) q^{68} +(-32272.7 - 1.07903e6i) q^{70} -3.53880e6 q^{71} -3.41741e6 q^{73} +(113753. + 3.80328e6i) q^{74} +(4.33643e6 - 259630. i) q^{76} -1.58435e6 q^{77} -7.60430e6i q^{79} +(-1.51418e6 + 181965. i) q^{80} +(-224399. - 7.50271e6i) q^{82} -8.11070e6i q^{83} -1.46174e6i q^{85} +(-8.00515e6 + 239427. i) q^{86} +(200506. + 2.22928e6i) q^{88} -1.00923e7i q^{89} -5.81971e6 q^{91} +(3.84185e6 - 230018. i) q^{92} +(4.54621e6 - 135973. i) q^{94} -3.15915e6 q^{95} -5.88229e6 q^{97} +(-2.56947e6 + 76850.4i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 52 q^{4}+O(q^{10}) \) 28 * q + 52 * q^4	\( 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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