LMFDB - Embedded newform 216.8.i.a.145.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,8,Mod(73,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("216.73");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	216.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$67.4751655046$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} + 332929 x^{18} - 2996076 x^{17} + 44578211685 x^{16} - 356557783716 x^{15} + \cdots + 79\!\cdots\!67 $ x^20 - 10x^19 + 332929x^18 - 2996076x^17 + 44578211685x^16 - 356557783716x^15 + 3097483289700118x^14 - 21676143504368032x^13 + 119894285805340530265x^12 - 719083941188982737718x^11 + 2563232515499525310010527x^10 - 12809571491275660354252428x^9 + 27741346343998271210179496359x^8 - 110888535856083235142883207868x^7 + 121645138768923372291810301132558x^6 - 364547360223565095382569624026928x^5 + 164798224123857530371927578258953667x^4 - 328988998658352429191508377804789922x^3 + 18304547169687091044739805564504070921x^2 - 18140113409773705053928159582048276332*x + 79919855971481307718190429644423027167
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{48}\cdot 3^{45} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			145.3
Root			$0.500000 - 157.828i$ of defining polynomial
Character	$\chi$	$=$	216.145
Dual form			216.8.i.a.73.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-130.433 + 225.916i) q^{5} +(-113.976 - 197.412i) q^{7} +O(q^{10})$	$q+(-130.433 + 225.916i) q^{5} +(-113.976 - 197.412i) q^{7} +(1739.18 + 3012.35i) q^{11} +(728.416 - 1261.65i) q^{13} +14668.5 q^{17} +6093.25 q^{19} +(1634.92 - 2831.77i) q^{23} +(5037.07 + 8724.45i) q^{25} +(55111.8 + 95456.5i) q^{29} +(16935.7 - 29333.6i) q^{31} +59464.7 q^{35} -105397. q^{37} +(-337216. + 584076. i) q^{41} +(-321128. - 556210. i) q^{43} +(122297. + 211825. i) q^{47} +(385791. - 668209. i) q^{49} -1.83400e6 q^{53} -907386. q^{55} +(1.06305e6 - 1.84126e6i) q^{59} +(833548. + 1.44375e6i) q^{61} +(190019. + 329122. i) q^{65} +(-2.39974e6 + 4.15648e6i) q^{67} -2.50821e6 q^{71} -44607.0 q^{73} +(396449. - 686670. i) q^{77} +(3.33611e6 + 5.77831e6i) q^{79} +(-991001. - 1.71646e6i) q^{83} +(-1.91325e6 + 3.31384e6i) q^{85} -8.71573e6 q^{89} -332087. q^{91} +(-794760. + 1.37656e6i) q^{95} +(429636. + 744151. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 125 q^{5} + 1245 q^{7}+O(q^{10}) $ 20 * q + 125 * q^5 + 1245 * q^7	$ 20 q + 125 q^{5} + 1245 q^{7} - 6106 q^{11} + 4937 q^{13} - 48722 q^{17} - 26882 q^{19} - 19387 q^{23} - 218957 q^{25} + 46791 q^{29} - 185039 q^{31} - 83094 q^{35} + 108420 q^{37} + 638112 q^{41} + 892628 q^{43} + 230883 q^{47} - 1034741 q^{49} + 2872940 q^{53} + 1089998 q^{55} - 2172454 q^{59} - 1878325 q^{61} - 1239133 q^{65} + 531496 q^{67} - 3723056 q^{71} - 1804522 q^{73} - 6276543 q^{77} + 3607847 q^{79} - 10794491 q^{83} + 3597658 q^{85} - 32214888 q^{89} - 16117530 q^{91} + 756868 q^{95} + 17951260 q^{97}+O(q^{100}) $ 20 * q + 125 * q^5 + 1245 * q^7 - 6106 * q^11 + 4937 * q^13 - 48722 * q^17 - 26882 * q^19 - 19387 * q^23 - 218957 * q^25 + 46791 * q^29 - 185039 * q^31 - 83094 * q^35 + 108420 * q^37 + 638112 * q^41 + 892628 * q^43 + 230883 * q^47 - 1034741 * q^49 + 2872940 * q^53 + 1089998 * q^55 - 2172454 * q^59 - 1878325 * q^61 - 1239133 * q^65 + 531496 * q^67 - 3723056 * q^71 - 1804522 * q^73 - 6276543 * q^77 + 3607847 * q^79 - 10794491 * q^83 + 3597658 * q^85 - 32214888 * q^89 - 16117530 * q^91 + 756868 * q^95 + 17951260 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−130.433	+	225.916i	−0.466651	+	0.808263i	−0.999274	−	0.0380896i	$-0.987873\pi$
$5$	−130.433	+	225.916i	−0.466651	+	0.808263i	0.532624	+	0.846352i	$0.321206\pi$
$6$		0			0
$7$	−113.976	−	197.412i	−0.125594	−	0.217535i	0.796371	−	0.604808i	$-0.206750\pi$
$7$	−113.976	−	197.412i	−0.125594	−	0.217535i	−0.921965	+	0.387273i	$0.873417\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1739.18	+	3012.35i	0.393977	+	0.682388i	0.992970	−	0.118366i	$-0.0377656\pi$
$11$	1739.18	+	3012.35i	0.393977	+	0.682388i	−0.598993	+	0.800754i	$0.704432\pi$
$12$		0			0
$13$	728.416	−	1261.65i	0.0919555	−	0.159272i	−0.816378	−	0.577517i	$-0.804022\pi$
$13$	728.416	−	1261.65i	0.0919555	−	0.159272i	0.908334	+	0.418246i	$0.137355\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14668.5			0.724124			0.362062	−	0.932154i	$-0.382073\pi$
$17$	14668.5			0.724124			0.362062	+	0.932154i	$0.382073\pi$
$18$		0			0
$19$	6093.25			0.203803			0.101902	−	0.994794i	$-0.467507\pi$
$19$	6093.25			0.203803			0.101902	+	0.994794i	$0.467507\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1634.92	−	2831.77i	0.0280188	−	0.0485299i	−0.851676	−	0.524069i	$-0.824414\pi$
$23$	1634.92	−	2831.77i	0.0280188	−	0.0485299i	0.879695	+	0.475539i	$0.157747\pi$
$24$		0			0
$25$	5037.07	+	8724.45i	0.0644744	+	0.111673i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	55111.8	+	95456.5i	0.419616	+	0.726796i	0.995901	−	0.0904529i	$-0.0288315\pi$
$29$	55111.8	+	95456.5i	0.419616	+	0.726796i	−0.576285	+	0.817249i	$0.695498\pi$
$30$		0			0
$31$	16935.7	−	29333.6i	0.102103	−	0.176848i	−0.810448	−	0.585811i	$-0.800776\pi$
$31$	16935.7	−	29333.6i	0.102103	−	0.176848i	0.912551	+	0.408963i	$0.134110\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	59464.7			0.234434
$36$		0			0
$37$	−105397.			−0.342075			−0.171037	−	0.985265i	$-0.554712\pi$
$37$	−105397.			−0.342075			−0.171037	+	0.985265i	$0.554712\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−337216.	+	584076.i	−0.764127	+	1.32351i	0.176581	+	0.984286i	$0.443496\pi$
$41$	−337216.	+	584076.i	−0.764127	+	1.32351i	−0.940707	+	0.339220i	$0.889837\pi$
$42$		0			0
$43$	−321128.	−	556210.i	−0.615940	−	1.06684i	−0.990219	−	0.139524i	$-0.955443\pi$
$43$	−321128.	−	556210.i	−0.615940	−	1.06684i	0.374278	−	0.927316i	$-0.377890\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	122297.	+	211825.i	0.171820	+	0.297601i	0.939056	−	0.343764i	$-0.111702\pi$
$47$	122297.	+	211825.i	0.171820	+	0.297601i	−0.767236	+	0.641365i	$0.778369\pi$
$48$		0			0
$49$	385791.	−	668209.i	0.468452	−	0.811383i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.83400e6			−1.69213			−0.846066	−	0.533078i	$-0.821035\pi$
$53$	−1.83400e6			−1.69213			−0.846066	+	0.533078i	$0.821035\pi$
$54$		0			0
$55$	−907386.			−0.735399
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.06305e6	−	1.84126e6i	0.673866	−	1.16717i	−0.302933	−	0.953012i	$-0.597966\pi$
$59$	1.06305e6	−	1.84126e6i	0.673866	−	1.16717i	0.976799	−	0.214158i	$-0.0687008\pi$
$60$		0			0
$61$	833548.	+	1.44375e6i	0.470193	+	0.814398i	0.999419	−	0.0340827i	$-0.0108510\pi$
$61$	833548.	+	1.44375e6i	0.470193	+	0.814398i	−0.529226	+	0.848481i	$0.677518\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	190019.	+	329122.i	0.0858222	+	0.148648i
$66$		0			0
$67$	−2.39974e6	+	4.15648e6i	−0.974772	+	1.68836i	−0.294088	+	0.955778i	$0.595016\pi$
$67$	−2.39974e6	+	4.15648e6i	−0.974772	+	1.68836i	−0.680684	+	0.732577i	$0.738317\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.50821e6			−0.831688			−0.415844	−	0.909436i	$-0.636514\pi$
$71$	−2.50821e6			−0.831688			−0.415844	+	0.909436i	$0.636514\pi$
$72$		0			0
$73$	−44607.0			−0.0134206			−0.00671032	−	0.999977i	$-0.502136\pi$
$73$	−44607.0			−0.0134206			−0.00671032	+	0.999977i	$0.502136\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	396449.	−	686670.i	0.0989624	−	0.171408i
$78$		0			0
$79$	3.33611e6	+	5.77831e6i	0.761281	+	1.31858i	0.942190	+	0.335078i	$0.108763\pi$
$79$	3.33611e6	+	5.77831e6i	0.761281	+	1.31858i	−0.180909	+	0.983500i	$0.557904\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−991001.	−	1.71646e6i	−0.190239	−	0.329504i	0.755090	−	0.655621i	$-0.227593\pi$
$83$	−991001.	−	1.71646e6i	−0.190239	−	0.329504i	−0.945330	+	0.326117i	$0.894260\pi$
$84$		0			0
$85$	−1.91325e6	+	3.31384e6i	−0.337913	+	0.585283i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.71573e6			−1.31051			−0.655253	−	0.755410i	$-0.727438\pi$
$89$	−8.71573e6			−1.31051			−0.655253	+	0.755410i	$0.727438\pi$
$90$		0			0
$91$	−332087.			−0.0461963
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−794760.	+	1.37656e6i	−0.0951050	+	0.164727i
$96$		0			0
$97$	429636.	+	744151.i	0.0477969	+	0.0827866i	0.888934	−	0.458035i	$-0.151447\pi$
$97$	429636.	+	744151.i	0.0477969	+	0.0827866i	−0.841137	+	0.540822i	$0.818113\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.89060e6	−	1.19349e7i	−0.665476	−	1.15264i	−0.979156	−	0.203110i	$-0.934895\pi$
$101$	−6.89060e6	−	1.19349e7i	−0.665476	−	1.15264i	0.313680	−	0.949529i	$-0.398438\pi$
$102$		0			0
$103$	−9.16586e6	+	1.58757e7i	−0.826500	+	1.43154i	0.0742670	+	0.997238i	$0.476338\pi$
$103$	−9.16586e6	+	1.58757e7i	−0.826500	+	1.43154i	−0.900767	+	0.434302i	$0.856995\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.50029e6			−0.512967			−0.256483	−	0.966549i	$-0.582564\pi$
$107$	−6.50029e6			−0.512967			−0.256483	+	0.966549i	$0.582564\pi$
$108$		0			0
$109$	−1.77585e7			−1.31345			−0.656725	−	0.754130i	$-0.728059\pi$
$109$	−1.77585e7			−1.31345			−0.656725	+	0.754130i	$0.728059\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.13471e7	+	1.96538e7i	−0.739793	+	1.28136i	0.212795	+	0.977097i	$0.431743\pi$
$113$	−1.13471e7	+	1.96538e7i	−0.739793	+	1.28136i	−0.952588	+	0.304263i	$0.901590\pi$
$114$		0			0
$115$	426495.	+	738710.i	0.0261500	+	0.0452931i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.67185e6	−	2.89572e6i	−0.0909457	−	0.157523i
$120$		0			0
$121$	3.69407e6	−	6.39831e6i	0.189564	−	0.328335i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.30081e7			−1.05365
$126$		0			0
$127$	−5.40018e6			−0.233935			−0.116968	−	0.993136i	$-0.537317\pi$
$127$	−5.40018e6			−0.233935			−0.116968	+	0.993136i	$0.537317\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.23198e7	+	2.13386e7i	−0.478802	+	0.829309i	−0.999705	−	0.0243073i	$-0.992262\pi$
$131$	−1.23198e7	+	2.13386e7i	−0.478802	+	0.829309i	0.520903	+	0.853616i	$0.325595\pi$
$132$		0			0
$133$	−694482.	−	1.20288e6i	−0.0255965	−	0.0443344i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.90562e6	+	3.30063e6i	0.0633161	+	0.109667i	0.895946	−	0.444163i	$-0.146499\pi$
$137$	1.90562e6	+	3.30063e6i	0.0633161	+	0.109667i	−0.832630	+	0.553830i	$0.813166\pi$
$138$		0			0
$139$	−1.09837e6	+	1.90243e6i	−0.0346894	+	0.0600837i	−0.882849	−	0.469657i	$-0.844378\pi$
$139$	−1.09837e6	+	1.90243e6i	−0.0346894	+	0.0600837i	0.848160	+	0.529741i	$0.177711\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.06740e6			0.144914
$144$		0			0
$145$	−2.87536e7			−0.783256
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.59906e6	+	7.96580e6i	−0.113898	+	0.197278i	−0.917339	−	0.398107i	$-0.869667\pi$
$149$	−4.59906e6	+	7.96580e6i	−0.113898	+	0.197278i	0.803441	+	0.595385i	$0.203000\pi$
$150$		0			0
$151$	1.60653e7	+	2.78258e7i	0.379724	+	0.657702i	0.991022	−	0.133699i	$-0.0426856\pi$
$151$	1.60653e7	+	2.78258e7i	0.379724	+	0.657702i	−0.611298	+	0.791401i	$0.709352\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.41795e6	+	7.65212e6i	0.0952928	+	0.165052i
$156$		0			0
$157$	−1.70668e7	+	2.95606e7i	−0.351969	+	0.609628i	−0.986594	−	0.163192i	$-0.947821\pi$
$157$	−1.70668e7	+	2.95606e7i	−0.351969	+	0.609628i	0.634625	+	0.772820i	$0.281154\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−745365.			−0.0140760
$162$		0			0
$163$	3.34913e7			0.605726			0.302863	−	0.953034i	$-0.402058\pi$
$163$	3.34913e7			0.605726			0.302863	+	0.953034i	$0.402058\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.22633e7	−	2.12406e7i	0.203750	−	0.352906i	−0.745984	−	0.665964i	$-0.768020\pi$
$167$	1.22633e7	−	2.12406e7i	0.203750	−	0.352906i	0.949734	+	0.313059i	$0.101354\pi$
$168$		0			0
$169$	3.03131e7	+	5.25038e7i	0.483088	+	0.836734i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.74881e7	−	6.49313e7i	−0.550468	−	0.953439i	−0.998241	−	0.0592916i	$-0.981116\pi$
$173$	−3.74881e7	−	6.49313e7i	−0.550468	−	0.953439i	0.447772	−	0.894148i	$-0.352218\pi$
$174$		0			0
$175$	1.14821e6	−	1.98875e6i	0.0161952	−	0.0280509i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.21187e7			0.679216			0.339608	−	0.940567i	$-0.389706\pi$
$179$	5.21187e7			0.679216			0.339608	+	0.940567i	$0.389706\pi$
$180$		0			0
$181$	−1.02750e8			−1.28797			−0.643987	−	0.765037i	$-0.722721\pi$
$181$	−1.02750e8			−1.28797			−0.643987	+	0.765037i	$0.722721\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.37472e7	−	2.38108e7i	0.159629	−	0.276486i
$186$		0			0
$187$	2.55111e7	+	4.41866e7i	0.285288	+	0.494134i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.65299e7	−	2.86306e7i	−0.171654	−	0.297313i	0.767344	−	0.641235i	$-0.221578\pi$
$191$	−1.65299e7	−	2.86306e7i	−0.171654	−	0.297313i	−0.938998	+	0.343922i	$0.888244\pi$
$192$		0			0
$193$	6.39107e7	−	1.10697e8i	0.639916	−	1.10837i	−0.345535	−	0.938406i	$-0.612303\pi$
$193$	6.39107e7	−	1.10697e8i	0.639916	−	1.10837i	0.985451	−	0.169961i	$-0.0543641\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.62127e7			−0.803414			−0.401707	−	0.915768i	$-0.631583\pi$
$197$	−8.62127e7			−0.803414			−0.401707	+	0.915768i	$0.631583\pi$
$198$		0			0
$199$	6.09247e7			0.548034			0.274017	−	0.961725i	$-0.411647\pi$
$199$	6.09247e7			0.548034			0.274017	+	0.961725i	$0.411647\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.25628e7	−	2.17594e7i	0.105402	−	0.182562i
$204$		0			0
$205$	−8.79682e7	−	1.52365e8i	−0.713160	−	1.23523i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.05973e7	+	1.83550e7i	0.0802939	+	0.139073i
$210$		0			0
$211$	3.02954e7	−	5.24731e7i	0.222018	−	0.384546i	−0.733403	−	0.679794i	$-0.762069\pi$
$211$	3.02954e7	−	5.24731e7i	0.222018	−	0.384546i	0.955421	+	0.295248i	$0.0954024\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.67543e8			1.14972
$216$		0			0
$217$	−7.72105e6			−0.0512941
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.06847e7	−	1.85065e7i	0.0665872	−	0.115332i
$222$		0			0
$223$	1.27224e8	+	2.20358e8i	0.768247	+	1.33064i	0.938513	+	0.345245i	$0.112204\pi$
$223$	1.27224e8	+	2.20358e8i	0.768247	+	1.33064i	−0.170265	+	0.985398i	$0.554462\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.45546e8	+	2.52094e8i	0.825869	+	1.43045i	0.901254	+	0.433292i	$0.142648\pi$
$227$	1.45546e8	+	2.52094e8i	0.825869	+	1.43045i	−0.0753852	+	0.997154i	$0.524019\pi$
$228$		0			0
$229$	1.77674e8	−	3.07740e8i	0.977686	−	1.69340i	0.306916	−	0.951737i	$-0.400703\pi$
$229$	1.77674e8	−	3.07740e8i	0.977686	−	1.69340i	0.670770	−	0.741665i	$-0.265964\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.27669e7			−0.221494			−0.110747	−	0.993849i	$-0.535324\pi$
$233$	−4.27669e7			−0.221494			−0.110747	+	0.993849i	$0.535324\pi$
$234$		0			0
$235$	−6.38063e7			−0.320720
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.46647e8	−	2.54000e8i	0.694834	−	1.20349i	−0.275403	−	0.961329i	$-0.588811\pi$
$239$	1.46647e8	−	2.54000e8i	0.694834	−	1.20349i	0.970237	−	0.242158i	$-0.0778553\pi$
$240$		0			0
$241$	−6.79400e7	−	1.17675e8i	−0.312655	−	0.541535i	0.666281	−	0.745701i	$-0.267885\pi$
$241$	−6.79400e7	−	1.17675e8i	−0.312655	−	0.541535i	−0.978936	+	0.204166i	$0.934552\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.00640e8	+	1.74313e8i	0.437207	+	0.757265i
$246$		0			0
$247$	4.43842e6	−	7.68758e6i	0.0187409	−	0.0324601i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.87241e8			1.94485			0.972423	−	0.233223i	$-0.0749271\pi$
$251$	4.87241e8			1.94485			0.972423	+	0.233223i	$0.0749271\pi$
$252$		0			0
$253$	1.13737e7			0.0441550
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.70685e6	−	2.95635e6i	0.00627235	−	0.0108640i	−0.862872	−	0.505422i	$-0.831337\pi$
$257$	1.70685e6	−	2.95635e6i	0.00627235	−	0.0108640i	0.869145	+	0.494558i	$0.164670\pi$
$258$		0			0
$259$	1.20127e7	+	2.08065e7i	0.0429625	+	0.0744133i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.27064e7	+	1.08611e8i	0.212553	+	0.368152i	0.952513	−	0.304499i	$-0.0984889\pi$
$263$	6.27064e7	+	1.08611e8i	0.212553	+	0.368152i	−0.739960	+	0.672651i	$0.765156\pi$
$264$		0			0
$265$	2.39214e8	−	4.14331e8i	0.789635	−	1.36769i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.03692e8			−0.324797			−0.162399	−	0.986725i	$-0.551923\pi$
$269$	−1.03692e8			−0.324797			−0.162399	+	0.986725i	$0.551923\pi$
$270$		0			0
$271$	646129.			0.00197209			0.000986045	−	1.00000i	$-0.499686\pi$
$271$	646129.			0.00197209			0.000986045		1.00000i	$0.499686\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.75208e7	+	3.03469e7i	−0.0508029	+	0.0879932i
$276$		0			0
$277$	−1.33457e8	−	2.31155e8i	−0.377280	−	0.653468i	0.613385	−	0.789784i	$-0.289807\pi$
$277$	−1.33457e8	−	2.31155e8i	−0.377280	−	0.653468i	−0.990665	+	0.136316i	$0.956474\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.09814e7	+	1.05623e8i	0.163955	+	0.283979i	0.936284	−	0.351244i	$-0.114241\pi$
$281$	6.09814e7	+	1.05623e8i	0.163955	+	0.283979i	−0.772328	+	0.635223i	$0.780908\pi$
$282$		0			0
$283$	3.76682e8	−	6.52432e8i	0.987921	−	1.71113i	0.359760	−	0.933045i	$-0.382858\pi$
$283$	3.76682e8	−	6.52432e8i	0.987921	−	1.71113i	0.628160	−	0.778084i	$-0.283808\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.53738e8			0.383879
$288$		0			0
$289$	−1.95175e8			−0.475644
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.43334e8	+	2.48262e8i	−0.332899	+	0.576598i	−0.983079	−	0.183182i	$-0.941360\pi$
$293$	−1.43334e8	+	2.48262e8i	−0.332899	+	0.576598i	0.650180	+	0.759780i	$0.274694\pi$
$294$		0			0
$295$	2.77314e8	+	4.80323e8i	0.628920	+	1.08932i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.38181e6	−	4.12541e6i	−0.00515296	−	0.00892519i
$300$		0			0
$301$	−7.32016e7	+	1.26789e8i	−0.154717	+	0.267978i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.34888e8			−0.877663
$306$		0			0
$307$	−7.68525e8			−1.51591			−0.757955	−	0.652307i	$-0.773801\pi$
$307$	−7.68525e8			−1.51591			−0.757955	+	0.652307i	$0.773801\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.80080e8	−	3.11908e8i	0.339473	−	0.587984i	−0.644861	−	0.764300i	$-0.723085\pi$
$311$	1.80080e8	−	3.11908e8i	0.339473	−	0.587984i	0.984334	+	0.176316i	$0.0564180\pi$
$312$		0			0
$313$	1.87316e8	+	3.24441e8i	0.345279	+	0.598041i	0.985404	−	0.170229i	$-0.0544509\pi$
$313$	1.87316e8	+	3.24441e8i	0.345279	+	0.598041i	−0.640125	+	0.768271i	$0.721118\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.50103e8	−	4.33190e8i	−0.440972	−	0.763786i	0.556790	−	0.830653i	$-0.312033\pi$
$317$	−2.50103e8	−	4.33190e8i	−0.440972	−	0.763786i	−0.997762	+	0.0668677i	$0.978699\pi$
$318$		0			0
$319$	−1.91699e8	+	3.32033e8i	−0.330638	+	0.572682i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.93786e7			0.147579
$324$		0			0
$325$	1.46763e7			0.0237151
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.78778e7	−	4.82858e7i	0.0431592	−	0.0747538i
$330$		0			0
$331$	3.20893e8	+	5.55803e8i	0.486365	+	0.842408i	0.999877	−	0.0156740i	$-0.00498938\pi$
$331$	3.20893e8	+	5.55803e8i	0.486365	+	0.842408i	−0.513513	+	0.858082i	$0.671656\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.26011e8	−	1.08428e9i	−0.909756	−	1.57574i
$336$		0			0
$337$	−2.27102e7	+	3.93352e7i	−0.0323233	+	0.0559856i	−0.881735	−	0.471746i	$-0.843624\pi$
$337$	−2.27102e7	+	3.93352e7i	−0.0323233	+	0.0559856i	0.849411	+	0.527732i	$0.176957\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.17817e8			0.160905
$342$		0			0
$343$	−3.63611e8			−0.486527
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.91787e8	+	6.78596e8i	−0.503381	+	0.871882i	0.496611	+	0.867973i	$0.334578\pi$
$347$	−3.91787e8	+	6.78596e8i	−0.503381	+	0.871882i	−0.999992	+	0.00390896i	$0.998756\pi$
$348$		0			0
$349$	5.56203e8	+	9.63372e8i	0.700397	+	1.21312i	0.968327	+	0.249685i	$0.0803271\pi$
$349$	5.56203e8	+	9.63372e8i	0.700397	+	1.21312i	−0.267930	+	0.963438i	$0.586340\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.35233e8	−	4.07435e8i	−0.284634	−	0.493000i	0.687886	−	0.725818i	$-0.258539\pi$
$353$	−2.35233e8	−	4.07435e8i	−0.284634	−	0.493000i	−0.972520	+	0.232818i	$0.925205\pi$
$354$		0			0
$355$	3.27153e8	−	5.66646e8i	0.388108	−	0.672222i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.25387e8			−0.143028			−0.0715139	−	0.997440i	$-0.522783\pi$
$359$	−1.25387e8			−0.143028			−0.0715139	+	0.997440i	$0.522783\pi$
$360$		0			0
$361$	−8.56744e8			−0.958464
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.81822e6	−	1.00775e7i	0.00626275	−	0.0108474i
$366$		0			0
$367$	4.55628e8	+	7.89171e8i	0.481149	+	0.833374i	0.999766	−	0.0216323i	$-0.00688631\pi$
$367$	4.55628e8	+	7.89171e8i	0.481149	+	0.833374i	−0.518617	+	0.855007i	$0.673553\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.09032e8	+	3.62053e8i	0.212522	+	0.368099i
$372$		0			0
$373$	−3.97568e8	+	6.88607e8i	−0.396671	+	0.687054i	−0.993313	−	0.115454i	$-0.963168\pi$
$373$	−3.97568e8	+	6.88607e8i	−0.396671	+	0.687054i	0.596642	+	0.802507i	$0.296501\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.60577e8			0.154344
$378$		0			0
$379$	−3.60938e8			−0.340561			−0.170280	−	0.985396i	$-0.554467\pi$
$379$	−3.60938e8			−0.340561			−0.170280	+	0.985396i	$0.554467\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.56811e8	+	1.31083e9i	−0.688322	+	1.19221i	0.284059	+	0.958807i	$0.408319\pi$
$383$	−7.56811e8	+	1.31083e9i	−0.688322	+	1.19221i	−0.972380	+	0.233401i	$0.925014\pi$
$384$		0			0
$385$	1.03420e8	+	1.79129e8i	0.0923617	+	0.159975i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.99181e8	+	6.91402e8i	0.343832	+	0.595535i	0.985141	−	0.171748i	$-0.0549416\pi$
$389$	3.99181e8	+	6.91402e8i	0.343832	+	0.595535i	−0.641309	+	0.767283i	$0.721608\pi$
$390$		0			0
$391$	2.39818e7	−	4.15376e7i	0.0202891	−	0.0351417i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.74055e9			−1.42101
$396$		0			0
$397$	−1.92368e9			−1.54300			−0.771500	−	0.636230i	$-0.780493\pi$
$397$	−1.92368e9			−1.54300			−0.771500	+	0.636230i	$0.780493\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.17486e8	−	1.24272e9i	0.555659	−	0.962430i	−0.442193	−	0.896920i	$-0.645799\pi$
$401$	7.17486e8	−	1.24272e9i	0.555659	−	0.962430i	0.997852	−	0.0655096i	$-0.0208673\pi$
$402$		0			0
$403$	−2.46725e7	−	4.27341e7i	−0.0187779	−	0.0325242i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.83304e8	−	3.17492e8i	−0.134770	−	0.233428i
$408$		0			0
$409$	−1.61149e8	+	2.79119e8i	−0.116465	+	0.201724i	−0.918365	−	0.395735i	$-0.870490\pi$
$409$	−1.61149e8	+	2.79119e8i	−0.116465	+	0.201724i	0.801899	+	0.597459i	$0.203823\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.84649e8			−0.338534
$414$		0			0
$415$	5.17036e8			0.355101
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.84483e8	+	1.53197e9i	−0.587408	+	1.01742i	0.407162	+	0.913356i	$0.366518\pi$
$419$	−8.84483e8	+	1.53197e9i	−0.587408	+	1.01742i	−0.994570	+	0.104065i	$0.966815\pi$
$420$		0			0
$421$	6.91884e8	+	1.19838e9i	0.451904	+	0.782720i	0.998504	−	0.0546732i	$-0.0174117\pi$
$421$	6.91884e8	+	1.19838e9i	0.451904	+	0.782720i	−0.546601	+	0.837393i	$0.684078\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.38860e7	+	1.27974e8i	0.0466875	+	0.0808651i
$426$		0			0
$427$	1.90008e8	−	3.29104e8i	0.118107	−	0.204567i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.07339e9			−1.24741			−0.623706	−	0.781659i	$-0.714374\pi$
$431$	−2.07339e9			−1.24741			−0.623706	+	0.781659i	$0.714374\pi$
$432$		0			0
$433$	−5.41827e8			−0.320740			−0.160370	−	0.987057i	$-0.551269\pi$
$433$	−5.41827e8			−0.320740			−0.160370	+	0.987057i	$0.551269\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.96198e6	−	1.72547e7i	0.00571032	−	0.00989057i
$438$		0			0
$439$	−1.60930e9	−	2.78739e9i	−0.907845	−	1.57243i	−0.817052	−	0.576565i	$-0.804393\pi$
$439$	−1.60930e9	−	2.78739e9i	−0.907845	−	1.57243i	−0.0907937	−	0.995870i	$-0.528940\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.74920e8	−	8.22585e8i	−0.259542	−	0.449540i	0.706577	−	0.707636i	$-0.250238\pi$
$443$	−4.74920e8	−	8.22585e8i	−0.259542	−	0.449540i	−0.966119	+	0.258096i	$0.916905\pi$
$444$		0			0
$445$	1.13682e9	−	1.96903e9i	0.611548	−	1.05923i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.28070e9			−0.667704			−0.333852	−	0.942626i	$-0.608349\pi$
$449$	−1.28070e9			−0.667704			−0.333852	+	0.942626i	$0.608349\pi$
$450$		0			0
$451$	−2.34593e9			−1.20419
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.33150e7	−	7.50238e7i	0.0215575	−	0.0373387i
$456$		0			0
$457$	1.32019e9	+	2.28663e9i	0.647036	+	1.12070i	0.983827	+	0.179120i	$0.0573251\pi$
$457$	1.32019e9	+	2.28663e9i	0.647036	+	1.12070i	−0.336791	+	0.941579i	$0.609342\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.79654e9	+	3.11170e9i	0.854052	+	1.47926i	0.877521	+	0.479538i	$0.159196\pi$
$461$	1.79654e9	+	3.11170e9i	0.854052	+	1.47926i	−0.0234690	+	0.999725i	$0.507471\pi$
$462$		0			0
$463$	−7.70565e8	+	1.33466e9i	−0.360808	+	0.624937i	−0.988094	−	0.153851i	$-0.950832\pi$
$463$	−7.70565e8	+	1.33466e9i	−0.360808	+	0.624937i	0.627286	+	0.778789i	$0.284166\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.25901e8			0.420684			0.210342	−	0.977628i	$-0.432542\pi$
$467$	9.25901e8			0.420684			0.210342	+	0.977628i	$0.432542\pi$
$468$		0			0
$469$	1.09405e9			0.489702
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.11700e9	−	1.93470e9i	0.485333	−	0.840621i
$474$		0			0
$475$	3.06921e7	+	5.31603e7i	0.0131401	+	0.0227593i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.54973e8	−	9.61242e8i	−0.230727	−	0.399630i	0.727295	−	0.686325i	$-0.240777\pi$
$479$	−5.54973e8	−	9.61242e8i	−0.230727	−	0.399630i	−0.958022	+	0.286694i	$0.907444\pi$
$480$		0			0
$481$	−7.67727e7	+	1.32974e8i	−0.0314557	+	0.0544828i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.24154e8			−0.0892178
$486$		0			0
$487$	−5.53047e8			−0.216975			−0.108488	−	0.994098i	$-0.534601\pi$
$487$	−5.53047e8			−0.216975			−0.108488	+	0.994098i	$0.534601\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.16524e8	−	1.41426e9i	0.311303	−	0.539193i	−0.667341	−	0.744752i	$-0.732568\pi$
$491$	8.16524e8	−	1.41426e9i	0.311303	−	0.539193i	0.978645	+	0.205559i	$0.0659011\pi$
$492$		0			0
$493$	8.08405e8	+	1.40020e9i	0.303854	+	0.526291i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.85875e8	+	4.95151e8i	0.104455	+	0.180921i
$498$		0			0
$499$	6.75301e8	−	1.16966e9i	0.243302	−	0.421411i	−0.718351	−	0.695681i	$-0.755103\pi$
$499$	6.75301e8	−	1.16966e9i	0.243302	−	0.421411i	0.961653	+	0.274270i	$0.0884361\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.86860e9			1.35540			0.677698	−	0.735341i	$-0.262978\pi$
$503$	3.86860e9			1.35540			0.677698	+	0.735341i	$0.262978\pi$
$504$		0			0
$505$	3.59504e9			1.24218
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.31621e8	+	5.74384e8i	−0.111463	+	0.193059i	−0.916360	−	0.400355i	$-0.868887\pi$
$509$	−3.31621e8	+	5.74384e8i	−0.111463	+	0.193059i	0.804897	+	0.593414i	$0.202220\pi$
$510$		0			0
$511$	5.08412e6	+	8.80595e6i	0.00168555	+	0.00291946i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.39106e9	−	4.14143e9i	−0.771374	−	1.33606i
$516$		0			0
$517$	−4.25394e8	+	7.36805e8i	−0.135386	+	0.234496i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.66784e9			0.826471			0.413235	−	0.910624i	$-0.364399\pi$
$521$	2.66784e9			0.826471			0.413235	+	0.910624i	$0.364399\pi$
$522$		0			0
$523$	6.61212e8			0.202109			0.101054	−	0.994881i	$-0.467778\pi$
$523$	6.61212e8			0.202109			0.101054	+	0.994881i	$0.467778\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.48421e8	−	4.30278e8i	0.0739352	−	0.128060i
$528$		0			0
$529$	1.69707e9	+	2.93941e9i	0.498430	+	0.863306i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.91268e8	+	8.50901e8i	0.140531	+	0.243407i
$534$		0			0
$535$	8.47851e8	−	1.46852e9i	0.239376	−	0.414612i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.68384e9			0.738238
$540$		0			0
$541$	−9.78165e8			−0.265596			−0.132798	−	0.991143i	$-0.542396\pi$
$541$	−9.78165e8			−0.265596			−0.132798	+	0.991143i	$0.542396\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.31629e9	−	4.01194e9i	0.612923	−	1.06161i
$546$		0			0
$547$	−7.23875e8	−	1.25379e9i	−0.189107	−	0.327543i	0.755846	−	0.654750i	$-0.227226\pi$
$547$	−7.23875e8	−	1.25379e9i	−0.189107	−	0.327543i	−0.944953	+	0.327207i	$0.893893\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.35810e8	+	5.81640e8i	0.0855192	+	0.148124i
$552$		0			0
$553$	7.60470e8	−	1.31717e9i	0.191225	−	0.331211i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.96274e8			−0.0726442			−0.0363221	−	0.999340i	$-0.511564\pi$
$557$	−2.96274e8			−0.0726442			−0.0363221	+	0.999340i	$0.511564\pi$
$558$		0			0
$559$	−9.35660e8			−0.226557
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8.39942e8	+	1.45482e9i	−0.198367	+	0.343582i	−0.947999	−	0.318273i	$-0.896897\pi$
$563$	−8.39942e8	+	1.45482e9i	−0.198367	+	0.343582i	0.749632	+	0.661855i	$0.230230\pi$
$564$		0			0
$565$	−2.96007e9	−	5.12699e9i	−0.690450	−	1.19589i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.40822e9	+	4.17116e9i	0.548029	+	0.949214i	0.998410	+	0.0563774i	$0.0179550\pi$
$569$	2.40822e9	+	4.17116e9i	0.548029	+	0.949214i	−0.450381	+	0.892837i	$0.648712\pi$
$570$		0			0
$571$	2.15364e9	−	3.73021e9i	0.484112	−	0.838507i	−0.515721	−	0.856757i	$-0.672476\pi$
$571$	2.15364e9	−	3.73021e9i	0.484112	−	0.838507i	0.999833	+	0.0182492i	$0.00580922\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.29408e7			0.00722598
$576$		0			0
$577$	8.72577e9			1.89099			0.945494	−	0.325641i	$-0.105580\pi$
$577$	8.72577e9			1.89099			0.945494	+	0.325641i	$0.105580\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.25900e8	+	3.91270e8i	−0.0477859	+	0.0827676i
$582$		0			0
$583$	−3.18967e9	−	5.52467e9i	−0.666662	−	1.15469i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.77807e8	+	8.27587e8i	0.0975034	+	0.168881i	0.910651	−	0.413177i	$-0.135581\pi$
$587$	4.77807e8	+	8.27587e8i	0.0975034	+	0.168881i	−0.813147	+	0.582058i	$0.802248\pi$
$588$		0			0
$589$	1.03194e8	−	1.78737e8i	0.0208089	−	0.0360421i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4.93145e9			0.971144			0.485572	−	0.874197i	$-0.338611\pi$
$593$	4.93145e9			0.971144			0.485572	+	0.874197i	$0.338611\pi$
$594$		0			0
$595$	8.72255e8			0.169759
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.83741e9	+	8.37864e9i	−0.919643	+	1.59287i	−0.119685	+	0.992812i	$0.538188\pi$
$599$	−4.83741e9	+	8.37864e9i	−0.919643	+	1.59287i	−0.799958	+	0.600056i	$0.795145\pi$
$600$		0			0
$601$	−4.57881e9	−	7.93074e9i	−0.860384	−	1.49023i	−0.871559	−	0.490291i	$-0.836890\pi$
$601$	−4.57881e9	−	7.93074e9i	−0.860384	−	1.49023i	0.0111747	−	0.999938i	$-0.496443\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.63655e8	+	1.66910e9i	0.176920	+	0.306435i
$606$		0			0
$607$	1.84770e9	−	3.20031e9i	0.335329	−	0.580807i	−0.648219	−	0.761454i	$-0.724486\pi$
$607$	1.84770e9	−	3.20031e9i	0.335329	−	0.580807i	0.983548	+	0.180647i	$0.0578191\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.56333e8			0.0631992
$612$		0			0
$613$	−8.74623e9			−1.53359			−0.766795	−	0.641892i	$-0.778150\pi$
$613$	−8.74623e9			−1.53359			−0.766795	+	0.641892i	$0.778150\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.89438e9	−	5.01321e9i	0.496087	−	0.859247i	−0.503903	−	0.863760i	$-0.668103\pi$
$617$	2.89438e9	−	5.01321e9i	0.496087	−	0.859247i	0.999990	+	0.00451290i	$0.00143651\pi$
$618$		0			0
$619$	8.89104e8	+	1.53997e9i	0.150673	+	0.260973i	0.931475	−	0.363805i	$-0.118523\pi$
$619$	8.89104e8	+	1.53997e9i	0.150673	+	0.260973i	−0.780802	+	0.624778i	$0.785189\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.93381e8	+	1.72059e9i	0.164592	+	0.285081i
$624$		0			0
$625$	2.60749e9	−	4.51631e9i	0.427212	−	0.739952i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.54601e9			−0.247705
$630$		0			0
$631$	1.06767e10			1.69174			0.845872	−	0.533386i	$-0.179081\pi$
$631$	1.06767e10			1.69174			0.845872	+	0.533386i	$0.179081\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.04361e8	−	1.21999e9i	0.109166	−	0.189081i
$636$		0			0
$637$	−5.62032e8	−	9.73468e8i	−0.0861536	−	0.149222i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.16269e8	−	2.01384e8i	−0.0174366	−	0.0302011i	0.857175	−	0.515025i	$-0.172217\pi$
$641$	−1.16269e8	−	2.01384e8i	−0.0174366	−	0.0302011i	−0.874612	+	0.484823i	$0.838884\pi$
$642$		0			0
$643$	−5.54304e9	+	9.60083e9i	−0.822261	+	1.42420i	0.0817332	+	0.996654i	$0.473954\pi$
$643$	−5.54304e9	+	9.60083e9i	−0.822261	+	1.42420i	−0.903995	+	0.427544i	$0.859379\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6.85588e9			0.995173			0.497586	−	0.867414i	$-0.334220\pi$
$647$	6.85588e9			0.995173			0.497586	+	0.867414i	$0.334220\pi$
$648$		0			0
$649$	7.39539e9			1.06195
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.89089e9	−	3.27512e9i	0.265748	−	0.460289i	−0.702011	−	0.712166i	$-0.747714\pi$
$653$	1.89089e9	−	3.27512e9i	0.265748	−	0.460289i	0.967759	+	0.251877i	$0.0810477\pi$
$654$		0			0
$655$	−3.21382e9	−	5.56650e9i	−0.446866	−	0.773995i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.10751e9	−	5.38237e9i	−0.422974	−	0.732613i	0.573255	−	0.819377i	$-0.305681\pi$
$659$	−3.10751e9	−	5.38237e9i	−0.422974	−	0.732613i	−0.996229	+	0.0867644i	$0.972347\pi$
$660$		0			0
$661$	−5.38943e9	+	9.33477e9i	−0.725835	+	1.25718i	0.232795	+	0.972526i	$0.425213\pi$
$661$	−5.38943e9	+	9.33477e9i	−0.725835	+	1.25718i	−0.958630	+	0.284657i	$0.908120\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.62333e8			0.0477785
$666$		0			0
$667$	3.60414e8			0.0470285
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.89939e9	+	5.02188e9i	−0.370491	+	0.641709i
$672$		0			0
$673$	3.13313e9	+	5.42673e9i	0.396210	+	0.686255i	0.993255	−	0.115952i	$-0.0369920\pi$
$673$	3.13313e9	+	5.42673e9i	0.396210	+	0.686255i	−0.597045	+	0.802208i	$0.703659\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.80660e8	−	4.86117e8i	−0.0347632	−	0.0602117i	0.848120	−	0.529804i	$-0.177734\pi$
$677$	−2.80660e8	−	4.86117e8i	−0.0347632	−	0.0602117i	−0.882884	+	0.469592i	$0.844401\pi$
$678$		0			0
$679$	9.79361e7	−	1.69630e8i	0.0120060	−	0.0207950i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.13589e9			0.977086			0.488543	−	0.872540i	$-0.337529\pi$
$683$	8.13589e9			0.977086			0.488543	+	0.872540i	$0.337529\pi$
$684$		0			0
$685$	−9.94220e8			−0.118186
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.33592e9	+	2.31388e9i	−0.155601	+	0.269509i
$690$		0			0
$691$	6.33201e9	+	1.09674e10i	0.730077	+	1.26453i	0.956850	+	0.290583i	$0.0938491\pi$
$691$	6.33201e9	+	1.09674e10i	0.730077	+	1.26453i	−0.226773	+	0.973948i	$0.572818\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.86527e8	−	4.96279e8i	−0.0323756	−	0.0560762i
$696$		0			0
$697$	−4.94644e9	+	8.56749e9i	−0.553323	+	0.958383i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.77555e10			−1.94680			−0.973398	−	0.229123i	$-0.926414\pi$
$701$	−1.77555e10			−1.94680			−0.973398	+	0.229123i	$0.926414\pi$
$702$		0			0
$703$	−6.42209e8			−0.0697160
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.57072e9	+	2.72057e9i	−0.167160	+	0.289529i
$708$		0			0
$709$	−6.35607e9	−	1.10090e10i	−0.669772	−	1.16008i	−0.977968	−	0.208756i	$-0.933059\pi$
$709$	−6.35607e9	−	1.10090e10i	−0.669772	−	1.16008i	0.308196	−	0.951323i	$-0.400275\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.53772e7	−	9.59161e7i	−0.00572160	−	0.00991010i
$714$		0			0
$715$	−6.60955e8	+	1.14481e9i	−0.0676240	+	0.117128i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.31278e9			−0.432719			−0.216359	−	0.976314i	$-0.569418\pi$
$719$	−4.31278e9			−0.432719			−0.216359	+	0.976314i	$0.569418\pi$
$720$		0			0
$721$	4.17874e9			0.415214
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.55204e8	+	9.61641e8i	−0.0541090	+	0.0937195i
$726$		0			0
$727$	−5.54202e8	−	9.59905e8i	−0.0534930	−	0.0926527i	0.838039	−	0.545610i	$-0.183702\pi$
$727$	−5.54202e8	−	9.59905e8i	−0.0534930	−	0.0926527i	−0.891532	+	0.452958i	$0.850369\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.71045e9	−	8.15874e9i	−0.446017	−	0.772525i
$732$		0			0
$733$	−9.67551e9	+	1.67585e10i	−0.907424	+	1.57170i	−0.0897933	+	0.995960i	$0.528621\pi$
$733$	−9.67551e9	+	1.67585e10i	−0.907424	+	1.57170i	−0.817630	+	0.575743i	$0.804713\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.66944e10			−1.53615
$738$		0			0
$739$	7.14138e9			0.650918			0.325459	−	0.945556i	$-0.394481\pi$
$739$	7.14138e9			0.650918			0.325459	+	0.945556i	$0.394481\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.61259e9	−	1.14533e10i	0.591440	−	1.02440i	−0.402599	−	0.915376i	$-0.631893\pi$
$743$	6.61259e9	−	1.14533e10i	0.591440	−	1.02440i	0.994039	−	0.109027i	$-0.0347736\pi$
$744$		0			0
$745$	−1.19974e9	−	2.07800e9i	−0.106301	−	0.184119i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.40875e8	+	1.28323e9i	0.0644256	+	0.111588i
$750$		0			0
$751$	−8.10054e9	+	1.40306e10i	−0.697870	+	1.20875i	0.271334	+	0.962485i	$0.412535\pi$
$751$	−8.10054e9	+	1.40306e10i	−0.697870	+	1.20875i	−0.969204	+	0.246261i	$0.920798\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.38174e9			−0.708794
$756$		0			0
$757$	−8.61723e9			−0.721991			−0.360996	−	0.932567i	$-0.617563\pi$
$757$	−8.61723e9			−0.721991			−0.360996	+	0.932567i	$0.617563\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.22164e9	−	1.25083e10i	0.594005	−	1.02885i	−0.399682	−	0.916654i	$-0.630879\pi$
$761$	7.22164e9	−	1.25083e10i	0.594005	−	1.02885i	0.993687	−	0.112192i	$-0.0357872\pi$
$762$		0			0
$763$	2.02404e9	+	3.50574e9i	0.164962	+	0.285722i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.54869e9	−	2.68241e9i	−0.123931	−	0.214656i
$768$		0			0
$769$	4.58904e9	−	7.94844e9i	0.363898	−	0.630289i	−0.624701	−	0.780864i	$-0.714779\pi$
$769$	4.58904e9	−	7.94844e9i	0.363898	−	0.630289i	0.988599	+	0.150575i	$0.0481124\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.82102e10			−1.41804			−0.709018	−	0.705191i	$-0.750861\pi$
$773$	−1.82102e10			−1.41804			−0.709018	+	0.705191i	$0.750861\pi$
$774$		0			0
$775$	3.41226e8			0.0263321
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.05475e9	+	3.55892e9i	−0.155732	+	0.269735i
$780$		0			0
$781$	−4.36225e9	−	7.55563e9i	−0.327666	−	0.567534i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.45215e9	−	7.71135e9i	−0.328493	−	0.568967i
$786$		0			0
$787$	−9.07135e9	+	1.57120e10i	−0.663377	+	1.14900i	0.316345	+	0.948644i	$0.397544\pi$
$787$	−9.07135e9	+	1.57120e10i	−0.663377	+	1.14900i	−0.979723	+	0.200359i	$0.935789\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.17317e9			0.371655
$792$		0			0
$793$	2.42868e9			0.172947
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.31147e10	+	2.27154e10i	−0.917604	+	1.58934i	−0.114560	+	0.993416i	$0.536546\pi$
$797$	−1.31147e10	+	2.27154e10i	−0.917604	+	1.58934i	−0.803044	+	0.595920i	$0.796788\pi$
$798$		0			0
$799$	1.79391e9	+	3.10714e9i	0.124419	+	0.215500i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.75798e7	−	1.34372e8i	−0.00528743	−	0.00915809i
$804$		0			0
$805$	9.72200e7	−	1.68390e8i	0.00656856	−	0.0113771i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.37862e10			−0.915427			−0.457714	−	0.889100i	$-0.651332\pi$
$809$	−1.37862e10			−0.915427			−0.457714	+	0.889100i	$0.651332\pi$
$810$		0			0
$811$	−5.57965e9			−0.367311			−0.183655	−	0.982991i	$-0.558793\pi$
$811$	−5.57965e9			−0.367311			−0.183655	+	0.982991i	$0.558793\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.36837e9	+	7.56624e9i	−0.282662	+	0.489585i
$816$		0			0
$817$	−1.95671e9	−	3.38913e9i	−0.125531	−	0.217426i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.35629e10	+	2.34916e10i	0.855363	+	1.48153i	0.876308	+	0.481751i	$0.159999\pi$
$821$	1.35629e10	+	2.34916e10i	0.855363	+	1.48153i	−0.0209451	+	0.999781i	$0.506668\pi$
$822$		0			0
$823$	−7.67260e9	+	1.32893e10i	−0.479781	+	0.831005i	−0.999731	−	0.0231913i	$-0.992617\pi$
$823$	−7.67260e9	+	1.32893e10i	−0.479781	+	0.831005i	0.519950	+	0.854197i	$0.325951\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.26812e10			1.39443			0.697215	−	0.716862i	$-0.254422\pi$
$827$	2.26812e10			1.39443			0.697215	+	0.716862i	$0.254422\pi$
$828$		0			0
$829$	5.58425e9			0.340427			0.170214	−	0.985407i	$-0.445554\pi$
$829$	5.58425e9			0.340427			0.170214	+	0.985407i	$0.445554\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.65895e9	−	9.80159e9i	0.339218	−	0.587542i
$834$		0			0
$835$	3.19906e9	+	5.54094e9i	0.190160	+	0.329367i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.62289e9	+	8.00708e9i	0.270238	+	0.468066i	0.968923	−	0.247364i	$-0.0795642\pi$
$839$	4.62289e9	+	8.00708e9i	0.270238	+	0.468066i	−0.698685	+	0.715430i	$0.746231\pi$
$840$		0			0
$841$	2.55031e9	−	4.41727e9i	0.147845	−	0.256075i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.58153e10			−0.901734
$846$		0			0
$847$	−1.68413e9			−0.0952324
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.72315e8	+	2.98459e8i	−0.00958451	+	0.0166009i
$852$		0			0
$853$	−1.14801e10	−	1.98840e10i	−0.633319	−	1.09694i	−0.986869	−	0.161525i	$-0.948359\pi$
$853$	−1.14801e10	−	1.98840e10i	−0.633319	−	1.09694i	0.353550	−	0.935416i	$-0.384975\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.70534e9	−	1.33460e10i	−0.418176	−	0.724302i	0.577580	−	0.816334i	$-0.303997\pi$
$857$	−7.70534e9	−	1.33460e10i	−0.418176	−	0.724302i	−0.995756	+	0.0920321i	$0.970664\pi$
$858$		0			0
$859$	5.41342e9	−	9.37631e9i	0.291404	−	0.504726i	−0.682738	−	0.730663i	$-0.739211\pi$
$859$	5.41342e9	−	9.37631e9i	0.291404	−	0.504726i	0.974142	+	0.225937i	$0.0725443\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.73971e10			1.45100			0.725498	−	0.688224i	$-0.241610\pi$
$863$	2.73971e10			1.45100			0.725498	+	0.688224i	$0.241610\pi$
$864$		0			0
$865$	1.95587e10			1.02751
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.16042e10	+	2.00991e10i	−0.599855	+	1.03898i
$870$		0			0
$871$	3.49603e9	+	6.05529e9i	0.179271	+	0.310507i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.62237e9	+	4.54207e9i	0.132332	+	0.229206i
$876$		0			0
$877$	7.85940e9	−	1.36129e10i	0.393451	−	0.681477i	−0.599451	−	0.800411i	$-0.704614\pi$
$877$	7.85940e9	−	1.36129e10i	0.393451	−	0.681477i	0.992902	+	0.118934i	$0.0379478\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.10186e10			0.542887			0.271443	−	0.962454i	$-0.412499\pi$
$881$	1.10186e10			0.542887			0.271443	+	0.962454i	$0.412499\pi$
$882$		0			0
$883$	4.13237e9			0.201993			0.100997	−	0.994887i	$-0.467797\pi$
$883$	4.13237e9			0.201993			0.100997	+	0.994887i	$0.467797\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.99860e9	+	6.92577e9i	−0.192387	+	0.333223i	−0.946041	−	0.324048i	$-0.894956\pi$
$887$	−3.99860e9	+	6.92577e9i	−0.192387	+	0.333223i	0.753654	+	0.657271i	$0.228289\pi$
$888$		0			0
$889$	6.15489e8	+	1.06606e9i	0.0293808	+	0.0508891i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.45187e8	+	1.29070e9i	0.0350175	+	0.0606521i
$894$		0			0
$895$	−6.79799e9	+	1.17745e10i	−0.316957	+	0.548985i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.73344e9			0.171376
$900$		0			0
$901$	−2.69020e10			−1.22531
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.34020e10	−	2.32129e10i	0.601034	−	1.04102i
$906$		0			0
$907$	−1.11154e10	−	1.92524e10i	−0.494650	−	0.856760i	0.505331	−	0.862926i	$-0.331371\pi$
$907$	−1.11154e10	−	1.92524e10i	−0.494650	−	0.856760i	−0.999981	+	0.00616614i	$0.998037\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.92365e9	−	1.37242e10i	−0.347225	−	0.601411i	0.638530	−	0.769597i	$-0.279543\pi$
$911$	−7.92365e9	−	1.37242e10i	−0.347225	−	0.601411i	−0.985755	+	0.168185i	$0.946209\pi$
$912$		0			0
$913$	3.44706e9	−	5.97049e9i	0.149900	−	0.259634i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.61665e9			0.240538
$918$		0			0
$919$	3.73387e10			1.58692			0.793459	−	0.608624i	$-0.208278\pi$
$919$	3.73387e10			1.58692			0.793459	+	0.608624i	$0.208278\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.82702e9	+	3.16450e9i	−0.0764783	+	0.132464i
$924$		0			0
$925$	−5.30890e8	−	9.19528e8i	−0.0220551	−	0.0382005i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.34625e9	−	1.44561e10i	−0.341536	−	0.591558i	0.643182	−	0.765713i	$-0.277614\pi$
$929$	−8.34625e9	−	1.44561e10i	−0.341536	−	0.591558i	−0.984718	+	0.174156i	$0.944280\pi$
$930$		0			0
$931$	2.35072e9	−	4.07157e9i	0.0954722	−	0.165363i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.33100e10			−0.532520
$936$		0			0
$937$	−1.74317e10			−0.692232			−0.346116	−	0.938192i	$-0.612500\pi$
$937$	−1.74317e10			−0.692232			−0.346116	+	0.938192i	$0.612500\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−9.46758e8	+	1.63983e9i	−0.0370404	+	0.0641558i	−0.883951	−	0.467579i	$-0.845126\pi$
$941$	−9.46758e8	+	1.63983e9i	−0.0370404	+	0.0641558i	0.846911	+	0.531735i	$0.178460\pi$
$942$		0			0
$943$	1.10264e9	+	1.90984e9i	0.0428198	+	0.0741660i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.24633e9	−	1.60151e10i	−0.353789	−	0.612781i	0.633121	−	0.774053i	$-0.281774\pi$
$947$	−9.24633e9	−	1.60151e10i	−0.353789	−	0.612781i	−0.986910	+	0.161272i	$0.948440\pi$
$948$		0			0
$949$	−3.24925e7	+	5.62786e7i	−0.00123410	+	0.00213753i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.23791e10			1.58609			0.793044	−	0.609165i	$-0.208495\pi$
$953$	4.23791e10			1.58609			0.793044	+	0.609165i	$0.208495\pi$
$954$		0			0
$955$	8.62416e9			0.320409
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.34388e8	−	7.52382e8i	0.0159042	−	0.0275469i
$960$		0			0
$961$	1.31827e10	+	2.28331e10i	0.479150	+	0.829912i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.66721e10	+	2.88769e10i	0.597234	+	1.03444i
$966$		0			0
$967$	7.81860e9	−	1.35422e10i	0.278059	−	0.481612i	−0.692844	−	0.721088i	$-0.743642\pi$
$967$	7.81860e9	−	1.35422e10i	0.278059	−	0.481612i	0.970902	+	0.239476i	$0.0769757\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.17665e10			0.762994			0.381497	−	0.924370i	$-0.375409\pi$
$971$	2.17665e10			0.762994			0.381497	+	0.924370i	$0.375409\pi$
$972$		0			0
$973$	5.00749e8			0.0174271
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.83796e10	−	3.18345e10i	0.630530	−	1.09211i	−0.356913	−	0.934137i	$-0.616171\pi$
$977$	1.83796e10	−	3.18345e10i	0.630530	−	1.09211i	0.987443	−	0.157973i	$-0.0504958\pi$
$978$		0			0
$979$	−1.51583e10	−	2.62549e10i	−0.516309	−	0.894274i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.00370e10	−	1.73845e10i	−0.337027	−	0.583748i	0.646845	−	0.762622i	$-0.276088\pi$
$983$	−1.00370e10	−	1.73845e10i	−0.337027	−	0.583748i	−0.983872	+	0.178873i	$0.942755\pi$
$984$		0			0
$985$	1.12450e10	−	1.94769e10i	0.374914	−	0.649370i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.10008e9			−0.0690316
$990$		0			0
$991$	−1.43677e10			−0.468955			−0.234477	−	0.972122i	$-0.575338\pi$
$991$	−1.43677e10			−0.468955			−0.234477	+	0.972122i	$0.575338\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.94658e9	+	1.37639e10i	−0.255741	+	0.442956i
$996$		0			0
$997$	3.87523e9	+	6.71210e9i	0.123841	+	0.214499i	0.921279	−	0.388902i	$-0.127145\pi$
$997$	3.87523e9	+	6.71210e9i	0.123841	+	0.214499i	−0.797438	+	0.603400i	$0.793812\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.8.i.a.145.3		20
3.2	odd	2		72.8.i.a.49.10	yes	20
4.3	odd	2		432.8.i.e.145.3		20
9.2	odd	6		72.8.i.a.25.10	✓	20
9.7	even	3	inner	216.8.i.a.73.3		20
12.11	even	2		144.8.i.e.49.1		20
36.7	odd	6		432.8.i.e.289.3		20
36.11	even	6		144.8.i.e.97.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.i.a.25.10	✓	20	9.2	odd	6
72.8.i.a.49.10	yes	20	3.2	odd	2
144.8.i.e.49.1		20	12.11	even	2
144.8.i.e.97.1		20	36.11	even	6
216.8.i.a.73.3		20	9.7	even	3	inner
216.8.i.a.145.3		20	1.1	even	1	trivial
432.8.i.e.145.3		20	4.3	odd	2
432.8.i.e.289.3		20	36.7	odd	6

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	216.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-130.433 + 225.916i) q^{5} +(-113.976 - 197.412i) q^{7} +O(q^{10})\)	\(q+(-130.433 + 225.916i) q^{5} +(-113.976 - 197.412i) q^{7} +(1739.18 + 3012.35i) q^{11} +(728.416 - 1261.65i) q^{13} +14668.5 q^{17} +6093.25 q^{19} +(1634.92 - 2831.77i) q^{23} +(5037.07 + 8724.45i) q^{25} +(55111.8 + 95456.5i) q^{29} +(16935.7 - 29333.6i) q^{31} +59464.7 q^{35} -105397. q^{37} +(-337216. + 584076. i) q^{41} +(-321128. - 556210. i) q^{43} +(122297. + 211825. i) q^{47} +(385791. - 668209. i) q^{49} -1.83400e6 q^{53} -907386. q^{55} +(1.06305e6 - 1.84126e6i) q^{59} +(833548. + 1.44375e6i) q^{61} +(190019. + 329122. i) q^{65} +(-2.39974e6 + 4.15648e6i) q^{67} -2.50821e6 q^{71} -44607.0 q^{73} +(396449. - 686670. i) q^{77} +(3.33611e6 + 5.77831e6i) q^{79} +(-991001. - 1.71646e6i) q^{83} +(-1.91325e6 + 3.31384e6i) q^{85} -8.71573e6 q^{89} -332087. q^{91} +(-794760. + 1.37656e6i) q^{95} +(429636. + 744151. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 125 q^{5} + 1245 q^{7}+O(q^{10}) \) 20 * q + 125 * q^5 + 1245 * q^7	\( 20 q + 125 q^{5} + 1245 q^{7} - 6106 q^{11} + 4937 q^{13} - 48722 q^{17} - 26882 q^{19} - 19387 q^{23} - 218957 q^{25} + 46791 q^{29} - 185039 q^{31} - 83094 q^{35} + 108420 q^{37} + 638112 q^{41} + 892628 q^{43} + 230883 q^{47} - 1034741 q^{49} + 2872940 q^{53} + 1089998 q^{55} - 2172454 q^{59} - 1878325 q^{61} - 1239133 q^{65} + 531496 q^{67} - 3723056 q^{71} - 1804522 q^{73} - 6276543 q^{77} + 3607847 q^{79} - 10794491 q^{83} + 3597658 q^{85} - 32214888 q^{89} - 16117530 q^{91} + 756868 q^{95} + 17951260 q^{97}+O(q^{100}) \) 20 * q + 125 * q^5 + 1245 * q^7 - 6106 * q^11 + 4937 * q^13 - 48722 * q^17 - 26882 * q^19 - 19387 * q^23 - 218957 * q^25 + 46791 * q^29 - 185039 * q^31 - 83094 * q^35 + 108420 * q^37 + 638112 * q^41 + 892628 * q^43 + 230883 * q^47 - 1034741 * q^49 + 2872940 * q^53 + 1089998 * q^55 - 2172454 * q^59 - 1878325 * q^61 - 1239133 * q^65 + 531496 * q^67 - 3723056 * q^71 - 1804522 * q^73 - 6276543 * q^77 + 3607847 * q^79 - 10794491 * q^83 + 3597658 * q^85 - 32214888 * q^89 - 16117530 * q^91 + 756868 * q^95 + 17951260 * q^97

Introduction

L-functions

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