LMFDB - Embedded newform 168.10.q.a.25.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,10,Mod(25,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("168.25");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	168.q (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$86.5260204755$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 18660372 x^{14} - 3458782984 x^{13} + 143123973101310 x^{12} + \cdots + 50\!\cdots\!97 $ x^16 - 18660372x^14 - 3458782984x^13 + 143123973101310x^12 + 48872053104519072x^11 - 578734520240854872804x^10 - 270042331189660586117064x^9 + 1317886932354758017843044627x^8 + 736352464217070243815722699808x^7 - 1673655580870016843616444263455812x^6 - 1020366474648370188416591376208359000x^5 + 1103325097889381327876435664684216492298x^4 + 649904594580478032138799228161609255854736x^3 - 331733869309416601638773034133657913991577560x^2 - 128648437449643500784782173321364984397814457552x + 50061741614807564009961574976312432423689032591597
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{38}\cdot 3^{5}\cdot 5^{2}\cdot 7^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.8
Root			$2262.65 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	168.25
Dual form			168.10.q.a.121.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 + 70.1481i) q^{3} +(1119.08 + 1938.30i) q^{5} +(-5033.95 + 3874.66i) q^{7} +(-3280.50 - 5681.99i) q^{9} +O(q^{10})$	\(q+(-40.5000 + 70.1481i) q^{3} +(1119.08 + 1938.30i) q^{5} +(-5033.95 + 3874.66i) q^{7} +(-3280.50 - 5681.99i) q^{9} +(21727.9 - 37633.8i) q^{11} +76526.3 q^{13} -181290. q^{15} +(-178621. + 309380. i) q^{17} +(342047. + 592443. i) q^{19} +(-67925.2 - 510045. i) q^{21} +(1.28747e6 + 2.22997e6i) q^{23} +(-1.52810e6 + 2.64675e6i) q^{25} +531441. q^{27} -1.81017e6 q^{29} +(-2.13440e6 + 3.69689e6i) q^{31} +(1.75996e6 + 3.04834e6i) q^{33} +(-1.31436e7 - 5.42124e6i) q^{35} +(-3.94948e6 - 6.84070e6i) q^{37} +(-3.09932e6 + 5.36817e6i) q^{39} -1.98724e6 q^{41} +3.62794e7 q^{43} +(7.34226e6 - 1.27172e7i) q^{45} +(2.89343e7 + 5.01156e7i) q^{47} +(1.03276e7 - 3.90097e7i) q^{49} +(-1.44683e7 - 2.50598e7i) q^{51} +(-2.54809e7 + 4.41342e7i) q^{53} +9.72608e7 q^{55} -5.54116e7 q^{57} +(2.82882e7 - 4.89966e7i) q^{59} +(-8.27112e7 - 1.43260e8i) q^{61} +(3.85297e7 + 1.58920e7i) q^{63} +(8.56388e7 + 1.48331e8i) q^{65} +(-1.33171e8 + 2.30659e8i) q^{67} -2.08571e8 q^{69} -2.08661e7 q^{71} +(9.31603e7 - 1.61358e8i) q^{73} +(-1.23776e8 - 2.14387e8i) q^{75} +(3.64413e7 + 2.73635e8i) q^{77} +(-2.08759e8 - 3.61582e8i) q^{79} +(-2.15234e7 + 3.72796e7i) q^{81} -3.93607e8 q^{83} -7.99561e8 q^{85} +(7.33119e7 - 1.26980e8i) q^{87} +(1.43611e8 + 2.48741e8i) q^{89} +(-3.85229e8 + 2.96514e8i) q^{91} +(-1.72887e8 - 2.99448e8i) q^{93} +(-7.65554e8 + 1.32598e9i) q^{95} -3.73818e7 q^{97} -2.85114e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) $ 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9	$ 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9} + 32460 q^{11} + 119048 q^{13} + 31752 q^{15} + 208352 q^{17} + 914588 q^{19} - 428652 q^{21} + 460920 q^{23} - 3040180 q^{25} + 8503056 q^{27} - 16376136 q^{29} - 944064 q^{31} + 2629260 q^{33} - 15546664 q^{35} - 9826516 q^{37} - 4821444 q^{39} + 11449216 q^{41} - 6933624 q^{43} - 1285956 q^{45} + 26549360 q^{47} + 83657504 q^{49} + 16876512 q^{51} - 15354476 q^{53} + 134121944 q^{55} - 148163256 q^{57} + 18404996 q^{59} - 260632792 q^{61} + 35823060 q^{63} + 191461840 q^{65} + 53879788 q^{67} - 74669040 q^{69} - 164207456 q^{71} + 248475540 q^{73} - 246254580 q^{75} + 670121788 q^{77} + 16631256 q^{79} - 344373768 q^{81} - 1138943272 q^{83} - 1690136272 q^{85} + 663233508 q^{87} + 236796360 q^{89} - 1455575212 q^{91} - 76469184 q^{93} + 182450488 q^{95} + 1339799464 q^{97} - 425940120 q^{99}+O(q^{100}) $ 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9 + 32460 * q^11 + 119048 * q^13 + 31752 * q^15 + 208352 * q^17 + 914588 * q^19 - 428652 * q^21 + 460920 * q^23 - 3040180 * q^25 + 8503056 * q^27 - 16376136 * q^29 - 944064 * q^31 + 2629260 * q^33 - 15546664 * q^35 - 9826516 * q^37 - 4821444 * q^39 + 11449216 * q^41 - 6933624 * q^43 - 1285956 * q^45 + 26549360 * q^47 + 83657504 * q^49 + 16876512 * q^51 - 15354476 * q^53 + 134121944 * q^55 - 148163256 * q^57 + 18404996 * q^59 - 260632792 * q^61 + 35823060 * q^63 + 191461840 * q^65 + 53879788 * q^67 - 74669040 * q^69 - 164207456 * q^71 + 248475540 * q^73 - 246254580 * q^75 + 670121788 * q^77 + 16631256 * q^79 - 344373768 * q^81 - 1138943272 * q^83 - 1690136272 * q^85 + 663233508 * q^87 + 236796360 * q^89 - 1455575212 * q^91 - 76469184 * q^93 + 182450488 * q^95 + 1339799464 * q^97 - 425940120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−40.5000	+	70.1481i	−0.288675	+	0.500000i
$3$	−40.5000	+	70.1481i	−0.288675	+	0.500000i
$4$		0			0
$5$	1119.08	+	1938.30i	0.800746	+	1.38693i	0.919126	+	0.393965i	$0.128897\pi$
$5$	1119.08	+	1938.30i	0.800746	+	1.38693i	−0.118379	+	0.992968i	$0.537770\pi$
$6$		0			0
$7$	−5033.95	+	3874.66i	−0.792442	+	0.609948i
$7$	−5033.95	+	3874.66i	−0.792442	+	0.609948i
$8$		0			0
$9$	−3280.50	−	5681.99i	−0.166667	−	0.288675i
$10$		0			0
$11$	21727.9	−	37633.8i	0.447456	−	0.775017i	−0.550763	−	0.834662i	$-0.685663\pi$
$11$	21727.9	−	37633.8i	0.447456	−	0.775017i	0.998220	+	0.0596442i	$0.0189966\pi$
$12$		0			0
$13$	76526.3			0.743132			0.371566	−	0.928407i	$-0.378821\pi$
$13$	76526.3			0.743132			0.371566	+	0.928407i	$0.378821\pi$
$14$		0			0
$15$	−181290.			−0.924622
$16$		0			0
$17$	−178621.	+	309380.i	−0.518695	+	0.898405i	0.481069	+	0.876682i	$0.340248\pi$
$17$	−178621.	+	309380.i	−0.518695	+	0.898405i	−0.999764	+	0.0217229i	$0.993085\pi$
$18$		0			0
$19$	342047.	+	592443.i	0.602136	+	1.04293i	0.992497	+	0.122268i	$0.0390169\pi$
$19$	342047.	+	592443.i	0.602136	+	1.04293i	−0.390361	+	0.920662i	$0.627650\pi$
$20$		0			0
$21$	−67925.2	−	510045.i	−0.0762157	−	0.572298i
$22$		0			0
$23$	1.28747e6	+	2.22997e6i	0.959319	+	1.66159i	0.724160	+	0.689632i	$0.242228\pi$
$23$	1.28747e6	+	2.22997e6i	0.959319	+	1.66159i	0.235158	+	0.971957i	$0.424439\pi$
$24$		0			0
$25$	−1.52810e6	+	2.64675e6i	−0.782389	+	1.35514i
$26$		0			0
$27$	531441.			0.192450
$28$		0			0
$29$	−1.81017e6			−0.475257			−0.237629	−	0.971356i	$-0.576370\pi$
$29$	−1.81017e6			−0.475257			−0.237629	+	0.971356i	$0.576370\pi$
$30$		0			0
$31$	−2.13440e6	+	3.69689e6i	−0.415096	+	0.718967i	−0.995439	−	0.0954051i	$-0.969585\pi$
$31$	−2.13440e6	+	3.69689e6i	−0.415096	+	0.718967i	0.580342	+	0.814373i	$0.302919\pi$
$32$		0			0
$33$	1.75996e6	+	3.04834e6i	0.258339	+	0.447456i
$34$		0			0
$35$	−1.31436e7	−	5.42124e6i	−1.48050	−	0.610650i
$36$		0			0
$37$	−3.94948e6	−	6.84070e6i	−0.346443	−	0.600057i	0.639172	−	0.769064i	$-0.279277\pi$
$37$	−3.94948e6	−	6.84070e6i	−0.346443	−	0.600057i	−0.985615	+	0.169007i	$0.945944\pi$
$38$		0			0
$39$	−3.09932e6	+	5.36817e6i	−0.214524	+	0.371566i
$40$		0			0
$41$	−1.98724e6			−0.109831			−0.0549153	−	0.998491i	$-0.517489\pi$
$41$	−1.98724e6			−0.109831			−0.0549153	+	0.998491i	$0.517489\pi$
$42$		0			0
$43$	3.62794e7			1.61827			0.809137	−	0.587620i	$-0.199935\pi$
$43$	3.62794e7			1.61827			0.809137	+	0.587620i	$0.199935\pi$
$44$		0			0
$45$	7.34226e6	−	1.27172e7i	0.266915	−	0.462311i
$46$		0			0
$47$	2.89343e7	+	5.01156e7i	0.864912	+	1.49807i	0.867134	+	0.498074i	$0.165959\pi$
$47$	2.89343e7	+	5.01156e7i	0.864912	+	1.49807i	−0.00222240	+	0.999998i	$0.500707\pi$
$48$		0			0
$49$	1.03276e7	−	3.90097e7i	0.255928	−	0.966696i
$50$		0			0
$51$	−1.44683e7	−	2.50598e7i	−0.299468	−	0.518695i
$52$		0			0
$53$	−2.54809e7	+	4.41342e7i	−0.443582	+	0.768306i	−0.997952	−	0.0639639i	$-0.979626\pi$
$53$	−2.54809e7	+	4.41342e7i	−0.443582	+	0.768306i	0.554370	+	0.832270i	$0.312959\pi$
$54$		0			0
$55$	9.72608e7			1.43320
$56$		0			0
$57$	−5.54116e7			−0.695287
$58$		0			0
$59$	2.82882e7	−	4.89966e7i	0.303928	−	0.526419i	−0.673094	−	0.739557i	$-0.735035\pi$
$59$	2.82882e7	−	4.89966e7i	0.303928	−	0.526419i	0.977022	+	0.213138i	$0.0683683\pi$
$60$		0			0
$61$	−8.27112e7	−	1.43260e8i	−0.764857	−	1.32477i	−0.940322	−	0.340286i	$-0.889476\pi$
$61$	−8.27112e7	−	1.43260e8i	−0.764857	−	1.32477i	0.175465	−	0.984486i	$-0.443857\pi$
$62$		0			0
$63$	3.85297e7	+	1.58920e7i	0.308150	+	0.127100i
$64$		0			0
$65$	8.56388e7	+	1.48331e8i	0.595060	+	1.03067i
$66$		0			0
$67$	−1.33171e8	+	2.30659e8i	−0.807372	+	1.39841i	0.107306	+	0.994226i	$0.465778\pi$
$67$	−1.33171e8	+	2.30659e8i	−0.807372	+	1.39841i	−0.914678	+	0.404184i	$0.867556\pi$
$68$		0			0
$69$	−2.08571e8			−1.10773
$70$		0			0
$71$	−2.08661e7			−0.0974492			−0.0487246	−	0.998812i	$-0.515516\pi$
$71$	−2.08661e7			−0.0974492			−0.0487246	+	0.998812i	$0.515516\pi$
$72$		0			0
$73$	9.31603e7	−	1.61358e8i	0.383953	−	0.665026i	−0.607670	−	0.794189i	$-0.707896\pi$
$73$	9.31603e7	−	1.61358e8i	0.383953	−	0.665026i	0.991623	+	0.129163i	$0.0412292\pi$
$74$		0			0
$75$	−1.23776e8	−	2.14387e8i	−0.451712	−	0.782389i
$76$		0			0
$77$	3.64413e7	+	2.73635e8i	0.118137	+	0.887081i
$78$		0			0
$79$	−2.08759e8	−	3.61582e8i	−0.603009	−	1.04444i	−0.992363	−	0.123354i	$-0.960635\pi$
$79$	−2.08759e8	−	3.61582e8i	−0.603009	−	1.04444i	0.389354	−	0.921088i	$-0.372698\pi$
$80$		0			0
$81$	−2.15234e7	+	3.72796e7i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−3.93607e8			−0.910357			−0.455178	−	0.890400i	$-0.650425\pi$
$83$	−3.93607e8			−0.910357			−0.455178	+	0.890400i	$0.650425\pi$
$84$		0			0
$85$	−7.99561e8			−1.66137
$86$		0			0
$87$	7.33119e7	−	1.26980e8i	0.137195	−	0.237629i
$88$		0			0
$89$	1.43611e8	+	2.48741e8i	0.242623	+	0.420236i	0.961461	−	0.274943i	$-0.0886588\pi$
$89$	1.43611e8	+	2.48741e8i	0.242623	+	0.420236i	−0.718838	+	0.695178i	$0.755325\pi$
$90$		0			0
$91$	−3.85229e8	+	2.96514e8i	−0.588888	+	0.453271i
$92$		0			0
$93$	−1.72887e8	−	2.99448e8i	−0.239656	−	0.415096i
$94$		0			0
$95$	−7.65554e8	+	1.32598e9i	−0.964316	+	1.67024i
$96$		0			0
$97$	−3.73818e7			−0.0428733			−0.0214367	−	0.999770i	$-0.506824\pi$
$97$	−3.73818e7			−0.0428733			−0.0214367	+	0.999770i	$0.506824\pi$
$98$		0			0
$99$	−2.85114e8			−0.298304
$100$		0			0
$101$	6.71525e8	−	1.16311e9i	0.642119	−	1.11218i	−0.342840	−	0.939394i	$-0.611389\pi$
$101$	6.71525e8	−	1.16311e9i	0.642119	−	1.11218i	0.984959	−	0.172789i	$-0.0552780\pi$
$102$		0			0
$103$	−7.95557e8	−	1.37795e9i	−0.696472	−	1.20633i	−0.969682	−	0.244371i	$-0.921419\pi$
$103$	−7.95557e8	−	1.37795e9i	−0.696472	−	1.20633i	0.273210	−	0.961955i	$-0.411915\pi$
$104$		0			0
$105$	9.12606e8	−	7.02439e8i	0.732709	−	0.563971i
$106$		0			0
$107$	−1.08097e8	−	1.87229e8i	−0.0797233	−	0.138085i	0.823407	−	0.567451i	$-0.192070\pi$
$107$	−1.08097e8	−	1.87229e8i	−0.0797233	−	0.138085i	−0.903131	+	0.429366i	$0.858737\pi$
$108$		0			0
$109$	1.23601e9	−	2.14084e9i	0.838695	−	1.45266i	−0.0522913	−	0.998632i	$-0.516652\pi$
$109$	1.23601e9	−	2.14084e9i	0.838695	−	1.45266i	0.890986	−	0.454030i	$-0.150014\pi$
$110$		0			0
$111$	6.39815e8			0.400038
$112$		0			0
$113$	−6.59446e8			−0.380475			−0.190238	−	0.981738i	$-0.560926\pi$
$113$	−6.59446e8			−0.380475			−0.190238	+	0.981738i	$0.560926\pi$
$114$		0			0
$115$	−2.88156e9	+	4.99101e9i	−1.53634	+	2.66102i
$116$		0			0
$117$	−2.51045e8	−	4.34822e8i	−0.123855	−	0.214524i
$118$		0			0
$119$	−2.99577e8	−	2.24950e9i	−0.136945	−	1.02831i
$120$		0			0
$121$	2.34770e8	+	4.06634e8i	0.0995654	+	0.172452i
$122$		0			0
$123$	8.04832e7	−	1.39401e8i	0.0317053	−	0.0549153i
$124$		0			0
$125$	−2.46887e9			−0.904487
$126$		0			0
$127$	−9.28749e8			−0.316798			−0.158399	−	0.987375i	$-0.550633\pi$
$127$	−9.28749e8			−0.316798			−0.158399	+	0.987375i	$0.550633\pi$
$128$		0			0
$129$	−1.46932e9	+	2.54493e9i	−0.467155	+	0.809137i
$130$		0			0
$131$	−1.57909e9	−	2.73506e9i	−0.468474	−	0.811420i	0.530877	−	0.847449i	$-0.321863\pi$
$131$	−1.57909e9	−	2.73506e9i	−0.468474	−	0.811420i	−0.999351	+	0.0360287i	$0.988529\pi$
$132$		0			0
$133$	−4.01736e9	−	1.65701e9i	−1.11329	−	0.459190i
$134$		0			0
$135$	5.94723e8	+	1.03009e9i	0.154104	+	0.266915i
$136$		0			0
$137$	−3.40665e9	+	5.90050e9i	−0.826201	+	1.43102i	0.0747975	+	0.997199i	$0.476169\pi$
$137$	−3.40665e9	+	5.90050e9i	−0.826201	+	1.43102i	−0.900998	+	0.433823i	$0.857164\pi$
$138$		0			0
$139$	2.40441e8			0.0546314			0.0273157	−	0.999627i	$-0.491304\pi$
$139$	2.40441e8			0.0546314			0.0273157	+	0.999627i	$0.491304\pi$
$140$		0			0
$141$	−4.68735e9			−0.998714
$142$		0			0
$143$	1.66276e9	−	2.87998e9i	0.332519	−	0.575940i
$144$		0			0
$145$	−2.02572e9	−	3.50865e9i	−0.380560	−	0.659150i
$146$		0			0
$147$	2.31818e9	+	2.30435e9i	0.409468	+	0.407025i
$148$		0			0
$149$	1.22010e9	+	2.11328e9i	0.202795	+	0.351252i	0.949428	−	0.313985i	$-0.101664\pi$
$149$	1.22010e9	+	2.11328e9i	0.202795	+	0.351252i	−0.746633	+	0.665236i	$0.768331\pi$
$150$		0			0
$151$	−5.10660e9	+	8.84490e9i	−0.799348	+	1.38451i	0.120693	+	0.992690i	$0.461488\pi$
$151$	−5.10660e9	+	8.84490e9i	−0.799348	+	1.38451i	−0.920041	+	0.391821i	$0.871845\pi$
$152$		0			0
$153$	2.34386e9			0.345796
$154$		0			0
$155$	−9.55424e9			−1.32955
$156$		0			0
$157$	−1.81633e9	+	3.14598e9i	−0.238587	+	0.413245i	−0.960309	−	0.278938i	$-0.910018\pi$
$157$	−1.81633e9	+	3.14598e9i	−0.238587	+	0.413245i	0.721722	+	0.692183i	$0.243351\pi$
$158$		0			0
$159$	−2.06395e9	−	3.57487e9i	−0.256102	−	0.443582i
$160$		0			0
$161$	−1.51214e10	−	6.23702e9i	−1.77369	−	0.731578i
$162$		0			0
$163$	6.75643e9	+	1.17025e10i	0.749675	+	1.29848i	0.947978	+	0.318335i	$0.103124\pi$
$163$	6.75643e9	+	1.17025e10i	0.749675	+	1.29848i	−0.198303	+	0.980141i	$0.563543\pi$
$164$		0			0
$165$	−3.93906e9	+	6.82266e9i	−0.413728	+	0.716598i
$166$		0			0
$167$	1.46629e10			1.45880			0.729402	−	0.684086i	$-0.239799\pi$
$167$	1.46629e10			1.45880			0.729402	+	0.684086i	$0.239799\pi$
$168$		0			0
$169$	−4.74822e9			−0.447756
$170$		0			0
$171$	2.24417e9	−	3.88702e9i	0.200712	−	0.347643i
$172$		0			0
$173$	−9.69379e9	−	1.67901e10i	−0.822785	−	1.42511i	−0.903601	−	0.428376i	$-0.859086\pi$
$173$	−9.69379e9	−	1.67901e10i	−0.822785	−	1.42511i	0.0808156	−	0.996729i	$-0.474248\pi$
$174$		0			0
$175$	−2.56288e9	−	1.92445e10i	−0.206565	−	1.55108i
$176$		0			0
$177$	2.29134e9	+	3.96872e9i	0.175473	+	0.303928i
$178$		0			0
$179$	−3.10462e9	+	5.37735e9i	−0.226032	+	0.391498i	−0.956628	−	0.291311i	$-0.905909\pi$
$179$	−3.10462e9	+	5.37735e9i	−0.226032	+	0.391498i	0.730597	+	0.682809i	$0.239242\pi$
$180$		0			0
$181$	7.36953e9			0.510372			0.255186	−	0.966892i	$-0.417863\pi$
$181$	7.36953e9			0.510372			0.255186	+	0.966892i	$0.417863\pi$
$182$		0			0
$183$	1.33992e10			0.883181
$184$		0			0
$185$	8.83954e9	−	1.53105e10i	0.554826	−	0.960987i
$186$		0			0
$187$	7.76211e9	+	1.34444e10i	0.464186	+	0.803995i
$188$		0			0
$189$	−2.67524e9	+	2.05915e9i	−0.152505	+	0.117384i
$190$		0			0
$191$	−1.46033e10	−	2.52937e10i	−0.793965	−	1.37519i	−0.923494	−	0.383612i	$-0.874680\pi$
$191$	−1.46033e10	−	2.52937e10i	−0.793965	−	1.37519i	0.129529	−	0.991576i	$-0.458653\pi$
$192$		0			0
$193$	7.02494e9	−	1.21676e10i	0.364447	−	0.631241i	−0.624240	−	0.781233i	$-0.714591\pi$
$193$	7.02494e9	−	1.21676e10i	0.364447	−	0.631241i	0.988687	+	0.149991i	$0.0479246\pi$
$194$		0			0
$195$	−1.38735e10			−0.687116
$196$		0			0
$197$	−1.95131e10			−0.923056			−0.461528	−	0.887126i	$-0.652699\pi$
$197$	−1.95131e10			−0.923056			−0.461528	+	0.887126i	$0.652699\pi$
$198$		0			0
$199$	−3.54871e9	+	6.14655e9i	−0.160410	+	0.277838i	−0.935016	−	0.354606i	$-0.884615\pi$
$199$	−3.54871e9	+	6.14655e9i	−0.160410	+	0.277838i	0.774606	+	0.632444i	$0.217948\pi$
$200$		0			0
$201$	−1.07869e10	−	1.86834e10i	−0.466137	−	0.807372i
$202$		0			0
$203$	9.11230e9	−	7.01380e9i	0.376613	−	0.289882i
$204$		0			0
$205$	−2.22387e9	−	3.85186e9i	−0.0879464	−	0.152328i
$206$		0			0
$207$	8.44711e9	−	1.46308e10i	0.319773	−	0.553863i
$208$		0			0
$209$	2.97279e10			1.07772
$210$		0			0
$211$	−1.57776e10			−0.547986			−0.273993	−	0.961732i	$-0.588345\pi$
$211$	−1.57776e10			−0.547986			−0.273993	+	0.961732i	$0.588345\pi$
$212$		0			0
$213$	8.45076e8	−	1.46371e9i	0.0281311	−	0.0487246i
$214$		0			0
$215$	4.05994e10	+	7.03203e10i	1.29583	+	2.24444i
$216$		0			0
$217$	−3.57975e9	−	2.68800e10i	−0.109593	−	0.822927i
$218$		0			0
$219$	7.54598e9	+	1.30700e10i	0.221675	+	0.383953i
$220$		0			0
$221$	−1.36692e10	+	2.36757e10i	−0.385458	+	0.667633i
$222$		0			0
$223$	1.25257e10			0.339179			0.169590	−	0.985515i	$-0.445756\pi$
$223$	1.25257e10			0.339179			0.169590	+	0.985515i	$0.445756\pi$
$224$		0			0
$225$	2.00518e10			0.521593
$226$		0			0
$227$	−2.16148e10	+	3.74380e10i	−0.540301	+	0.935828i	0.458586	+	0.888650i	$0.348356\pi$
$227$	−2.16148e10	+	3.74380e10i	−0.540301	+	0.935828i	−0.998886	+	0.0471782i	$0.984977\pi$
$228$		0			0
$229$	−1.51365e10	−	2.62173e10i	−0.363720	−	0.629981i	0.624850	−	0.780745i	$-0.285160\pi$
$229$	−1.51365e10	−	2.62173e10i	−0.363720	−	0.629981i	−0.988570	+	0.150764i	$0.951827\pi$
$230$		0			0
$231$	−2.06708e10	−	8.52593e9i	−0.477644	−	0.197010i
$232$		0			0
$233$	3.00003e10	+	5.19620e10i	0.666844	+	1.15501i	0.978782	+	0.204905i	$0.0656884\pi$
$233$	3.00003e10	+	5.19620e10i	0.666844	+	1.15501i	−0.311938	+	0.950102i	$0.600978\pi$
$234$		0			0
$235$	−6.47593e10	+	1.12166e11i	−1.38515	+	2.39915i
$236$		0			0
$237$	3.38190e10			0.696295
$238$		0			0
$239$	6.01972e10			1.19340			0.596699	−	0.802465i	$-0.296478\pi$
$239$	6.01972e10			1.19340			0.596699	+	0.802465i	$0.296478\pi$
$240$		0			0
$241$	−1.42616e10	+	2.47018e10i	−0.272327	+	0.471685i	−0.969457	−	0.245260i	$-0.921127\pi$
$241$	−1.42616e10	+	2.47018e10i	−0.272327	+	0.471685i	0.697130	+	0.716945i	$0.254460\pi$
$242$		0			0
$243$	−1.74339e9	−	3.01964e9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	8.71697e10	−	2.36369e10i	1.54568	−	0.419124i
$246$		0			0
$247$	2.61756e10	+	4.53375e10i	0.447466	+	0.775034i
$248$		0			0
$249$	1.59411e10	−	2.76108e10i	0.262797	−	0.455178i
$250$		0			0
$251$	7.88463e10			1.25386			0.626931	−	0.779075i	$-0.284311\pi$
$251$	7.88463e10			1.25386			0.626931	+	0.779075i	$0.284311\pi$
$252$		0			0
$253$	1.11896e11			1.71701
$254$		0			0
$255$	3.23822e10	−	5.60877e10i	0.479596	−	0.830685i
$256$		0			0
$257$	−4.93329e10	−	8.54471e10i	−0.705403	−	1.22179i	−0.966546	−	0.256494i	$-0.917433\pi$
$257$	−4.93329e10	−	8.54471e10i	−0.705403	−	1.22179i	0.261142	−	0.965300i	$-0.415901\pi$
$258$		0			0
$259$	4.63868e10	+	1.91328e10i	0.640539	+	0.264198i
$260$		0			0
$261$	5.93827e9	+	1.02854e10i	0.0792095	+	0.137195i
$262$		0			0
$263$	1.28194e10	−	2.22038e10i	0.165221	−	0.286171i	−0.771513	−	0.636214i	$-0.780500\pi$
$263$	1.28194e10	−	2.22038e10i	0.165221	−	0.286171i	0.936734	+	0.350043i	$0.113833\pi$
$264$		0			0
$265$	−1.14060e11			−1.42079
$266$		0			0
$267$	−2.32650e10			−0.280157
$268$		0			0
$269$	5.66801e10	−	9.81728e10i	0.660002	−	1.14316i	−0.320612	−	0.947211i	$-0.603889\pi$
$269$	5.66801e10	−	9.81728e10i	0.660002	−	1.14316i	0.980614	−	0.195947i	$-0.0627780\pi$
$270$		0			0
$271$	8.43310e9	+	1.46066e10i	0.0949785	+	0.164508i	0.909600	−	0.415486i	$-0.136388\pi$
$271$	8.43310e9	+	1.46066e10i	0.0949785	+	0.164508i	−0.814621	+	0.579993i	$0.803055\pi$
$272$		0			0
$273$	−5.19807e9	−	3.90319e10i	−0.0566383	−	0.425292i
$274$		0			0
$275$	6.64050e10	+	1.15017e11i	0.700170	+	1.21273i
$276$		0			0
$277$	−1.45015e10	+	2.51174e10i	−0.147998	+	0.256340i	−0.930487	−	0.366324i	$-0.880616\pi$
$277$	−1.45015e10	+	2.51174e10i	−0.147998	+	0.256340i	0.782490	+	0.622664i	$0.213949\pi$
$278$		0			0
$279$	2.80076e10			0.276731
$280$		0			0
$281$	−6.02266e10			−0.576249			−0.288124	−	0.957593i	$-0.593032\pi$
$281$	−6.02266e10			−0.576249			−0.288124	+	0.957593i	$0.593032\pi$
$282$		0			0
$283$	8.33372e10	−	1.44344e11i	0.772324	−	1.33771i	−0.163962	−	0.986467i	$-0.552427\pi$
$283$	8.33372e10	−	1.44344e11i	0.772324	−	1.33771i	0.936286	−	0.351238i	$-0.114239\pi$
$284$		0			0
$285$	−6.20099e10	−	1.07404e11i	−0.556748	−	0.964316i
$286$		0			0
$287$	1.00037e10	−	7.69989e9i	0.0870343	−	0.0669909i
$288$		0			0
$289$	−4.51679e9	−	7.82331e9i	−0.0380881	−	0.0659706i
$290$		0			0
$291$	1.51396e9	−	2.62226e9i	0.0123765	−	0.0214367i
$292$		0			0
$293$	7.10260e9			0.0563006			0.0281503	−	0.999604i	$-0.491038\pi$
$293$	7.10260e9			0.0563006			0.0281503	+	0.999604i	$0.491038\pi$
$294$		0			0
$295$	1.26627e11			0.973478
$296$		0			0
$297$	1.15471e10	−	2.00002e10i	0.0861130	−	0.149152i
$298$		0			0
$299$	9.85256e10	+	1.70651e11i	0.712900	+	1.23478i
$300$		0			0
$301$	−1.82629e11	+	1.40570e11i	−1.28239	+	0.987063i
$302$		0			0
$303$	5.43935e10	+	9.42123e10i	0.370728	+	0.642119i
$304$		0			0
$305$	1.85120e11	−	3.20638e11i	1.22491	−	2.12161i
$306$		0			0
$307$	4.39573e10			0.282428			0.141214	−	0.989979i	$-0.454899\pi$
$307$	4.39573e10			0.282428			0.141214	+	0.989979i	$0.454899\pi$
$308$		0			0
$309$	1.28880e11			0.804217
$310$		0			0
$311$	7.48583e10	−	1.29658e11i	0.453751	−	0.785921i	−0.544864	−	0.838524i	$-0.683419\pi$
$311$	7.48583e10	−	1.29658e11i	0.453751	−	0.785921i	0.998615	+	0.0526039i	$0.0167521\pi$
$312$		0			0
$313$	1.97266e10	+	3.41675e10i	0.116172	+	0.201217i	0.918248	−	0.396006i	$-0.129604\pi$
$313$	1.97266e10	+	3.41675e10i	0.116172	+	0.201217i	−0.802075	+	0.597223i	$0.796271\pi$
$314$		0			0
$315$	1.23142e10	+	9.24663e10i	0.0704707	+	0.529159i
$316$		0			0
$317$	−4.36470e10	−	7.55987e10i	−0.242766	−	0.420482i	0.718735	−	0.695284i	$-0.244721\pi$
$317$	−4.36470e10	−	7.55987e10i	−0.242766	−	0.420482i	−0.961501	+	0.274801i	$0.911388\pi$
$318$		0			0
$319$	−3.93312e10	+	6.81237e10i	−0.212657	+	0.368332i
$320$		0			0
$321$	1.75117e10			0.0920566
$322$		0			0
$323$	−2.44387e11			−1.24930
$324$		0			0
$325$	−1.16940e11	+	2.02546e11i	−0.581418	+	1.00705i
$326$		0			0
$327$	1.00117e11	+	1.73408e11i	0.484221	+	0.838695i
$328$		0			0
$329$	−3.39834e11	−	1.40169e11i	−1.59914	−	0.659583i
$330$		0			0
$331$	1.35652e11	+	2.34957e11i	0.621157	+	1.07588i	0.989271	+	0.146095i	$0.0466705\pi$
$331$	1.35652e11	+	2.34957e11i	0.621157	+	1.07588i	−0.368113	+	0.929781i	$0.619996\pi$
$332$		0			0
$333$	−2.59125e10	+	4.48818e10i	−0.115481	+	0.200019i
$334$		0			0
$335$	−5.96115e11			−2.58600
$336$		0			0
$337$	9.46122e10			0.399588			0.199794	−	0.979838i	$-0.435973\pi$
$337$	9.46122e10			0.399588			0.199794	+	0.979838i	$0.435973\pi$
$338$		0			0
$339$	2.67076e10	−	4.62589e10i	0.109834	−	0.190238i
$340$		0			0
$341$	9.27522e10	+	1.60651e11i	0.371475	+	0.643413i
$342$		0			0
$343$	9.91607e10	+	2.36388e11i	0.386826	+	0.922153i
$344$		0			0
$345$	−2.33407e11	−	4.04272e11i	−0.887007	−	1.53634i
$346$		0			0
$347$	1.35367e11	−	2.34463e11i	0.501224	−	0.868145i	−0.498775	−	0.866731i	$-0.666217\pi$
$347$	1.35367e11	−	2.34463e11i	0.501224	−	0.868145i	0.999999	−	0.00141370i	$-0.000449996\pi$
$348$		0			0
$349$	−3.42609e11			−1.23619			−0.618094	−	0.786105i	$-0.712095\pi$
$349$	−3.42609e11			−1.23619			−0.618094	+	0.786105i	$0.712095\pi$
$350$		0			0
$351$	4.06692e10			0.143016
$352$		0			0
$353$	4.27699e10	−	7.40796e10i	0.146606	−	0.253929i	−0.783365	−	0.621562i	$-0.786498\pi$
$353$	4.27699e10	−	7.40796e10i	0.146606	−	0.253929i	0.929971	+	0.367633i	$0.119832\pi$
$354$		0			0
$355$	−2.33507e10	−	4.04447e10i	−0.0780320	−	0.135155i
$356$		0			0
$357$	1.69931e11	+	7.00899e10i	0.553688	+	0.228375i
$358$		0			0
$359$	−1.79418e11	−	3.10761e11i	−0.570087	−	0.987420i	−0.996556	−	0.0829179i	$-0.973576\pi$
$359$	−1.79418e11	−	3.10761e11i	−0.570087	−	0.987420i	0.426469	−	0.904502i	$-0.359757\pi$
$360$		0			0
$361$	−7.26485e10	+	1.25831e11i	−0.225136	+	0.389947i
$362$		0			0
$363$	−3.80327e10			−0.114968
$364$		0			0
$365$	4.17014e11			1.22980
$366$		0			0
$367$	−2.31402e11	+	4.00801e11i	−0.665841	+	1.15327i	0.313215	+	0.949682i	$0.398594\pi$
$367$	−2.31402e11	+	4.00801e11i	−0.665841	+	1.15327i	−0.979056	+	0.203589i	$0.934739\pi$
$368$		0			0
$369$	6.51914e9	+	1.12915e10i	0.0183051	+	0.0317053i
$370$		0			0
$371$	−4.27357e10	−	3.20899e11i	−0.117114	−	0.879399i
$372$		0			0
$373$	4.48202e10	+	7.76308e10i	0.119890	+	0.207656i	0.919724	−	0.392566i	$-0.128412\pi$
$373$	4.48202e10	+	7.76308e10i	0.119890	+	0.207656i	−0.799834	+	0.600222i	$0.795079\pi$
$374$		0			0
$375$	9.99891e10	−	1.73186e11i	0.261103	−	0.452244i
$376$		0			0
$377$	−1.38526e11			−0.353179
$378$		0			0
$379$	−5.23217e11			−1.30258			−0.651291	−	0.758828i	$-0.725772\pi$
$379$	−5.23217e11			−1.30258			−0.651291	+	0.758828i	$0.725772\pi$
$380$		0			0
$381$	3.76144e10	−	6.51500e10i	0.0914516	−	0.158399i
$382$		0			0
$383$	3.88836e11	+	6.73485e11i	0.923363	+	1.59931i	0.794173	+	0.607692i	$0.207905\pi$
$383$	3.88836e11	+	6.73485e11i	0.923363	+	1.59931i	0.129190	+	0.991620i	$0.458762\pi$
$384$		0			0
$385$	−4.89605e11	+	3.76853e11i	−1.13572	+	0.874175i
$386$		0			0
$387$	−1.19015e11	−	2.06139e11i	−0.269712	−	0.467155i
$388$		0			0
$389$	2.92412e11	−	5.06472e11i	0.647473	−	1.12146i	−0.336251	−	0.941772i	$-0.609159\pi$
$389$	2.92412e11	−	5.06472e11i	0.647473	−	1.12146i	0.983724	−	0.179684i	$-0.0575077\pi$
$390$		0			0
$391$	−9.19878e11			−1.99037
$392$		0			0
$393$	2.55812e11			0.540947
$394$		0			0
$395$	4.67235e11	−	8.09275e11i	0.965715	−	1.67267i
$396$		0			0
$397$	−8.41067e10	−	1.45677e11i	−0.169931	−	0.294330i	0.768464	−	0.639893i	$-0.221021\pi$
$397$	−8.41067e10	−	1.45677e11i	−0.169931	−	0.294330i	−0.938396	+	0.345563i	$0.887688\pi$
$398$		0			0
$399$	2.78939e11	−	2.14701e11i	0.550974	−	0.424089i
$400$		0			0
$401$	4.06255e11	+	7.03654e11i	0.784601	+	1.35897i	0.929237	+	0.369484i	$0.120466\pi$
$401$	4.06255e11	+	7.03654e11i	0.784601	+	1.35897i	−0.144636	+	0.989485i	$0.546201\pi$
$402$		0			0
$403$	−1.63338e11	+	2.82910e11i	−0.308471	+	0.534287i
$404$		0			0
$405$	−9.63452e10			−0.177944
$406$		0			0
$407$	−3.43256e11			−0.620073
$408$		0			0
$409$	−6.75744e10	+	1.17042e11i	−0.119406	+	0.206818i	−0.919533	−	0.393014i	$-0.871432\pi$
$409$	−6.75744e10	+	1.17042e11i	−0.119406	+	0.206818i	0.800126	+	0.599832i	$0.204766\pi$
$410$		0			0
$411$	−2.75939e11	−	4.77940e11i	−0.477007	−	0.826201i
$412$		0			0
$413$	4.74440e10	+	3.56253e11i	0.0802428	+	0.602537i
$414$		0			0
$415$	−4.40477e11	−	7.62928e11i	−0.728965	−	1.26260i
$416$		0			0
$417$	−9.73787e9	+	1.68665e10i	−0.0157707	+	0.0273157i
$418$		0			0
$419$	7.86856e10			0.124719			0.0623594	−	0.998054i	$-0.480138\pi$
$419$	7.86856e10			0.124719			0.0623594	+	0.998054i	$0.480138\pi$
$420$		0			0
$421$	−1.03537e12			−1.60630			−0.803152	−	0.595775i	$-0.796845\pi$
$421$	−1.03537e12			−1.60630			−0.803152	+	0.595775i	$0.796845\pi$
$422$		0			0
$423$	1.89838e11	−	3.28808e11i	0.288304	−	0.499357i
$424$		0			0
$425$	−5.45902e11	−	9.45530e11i	−0.811642	−	1.40580i
$426$		0			0
$427$	9.71448e11	+	4.00685e11i	1.41415	+	0.583281i
$428$		0			0
$429$	1.34683e11	+	2.33278e11i	0.191980	+	0.332519i
$430$		0			0
$431$	−2.55714e11	+	4.42909e11i	−0.356949	+	0.618254i	−0.987449	−	0.157935i	$-0.949516\pi$
$431$	−2.55714e11	+	4.42909e11i	−0.356949	+	0.618254i	0.630501	+	0.776189i	$0.282850\pi$
$432$		0			0
$433$	−4.06519e11			−0.555757			−0.277879	−	0.960616i	$-0.589631\pi$
$433$	−4.06519e11			−0.555757			−0.277879	+	0.960616i	$0.589631\pi$
$434$		0			0
$435$	3.28167e11			0.439433
$436$		0			0
$437$	−8.80753e11	+	1.52551e12i	−1.15528	+	2.00101i
$438$		0			0
$439$	5.98869e11	+	1.03727e12i	0.769558	+	1.33291i	0.937803	+	0.347169i	$0.112857\pi$
$439$	5.98869e11	+	1.03727e12i	0.769558	+	1.33291i	−0.168244	+	0.985745i	$0.553810\pi$
$440$		0			0
$441$	−2.55532e11	+	6.92899e10i	−0.321716	+	0.0872361i
$442$		0			0
$443$	2.59781e10	+	4.49954e10i	0.0320472	+	0.0555074i	0.881604	−	0.471989i	$-0.156464\pi$
$443$	2.59781e10	+	4.49954e10i	0.0320472	+	0.0555074i	−0.849557	+	0.527497i	$0.823131\pi$
$444$		0			0
$445$	−3.21423e11	+	5.56721e11i	−0.388559	+	0.673004i
$446$		0			0
$447$	−1.97656e11			−0.234168
$448$		0			0
$449$	−1.12713e11			−0.130877			−0.0654387	−	0.997857i	$-0.520845\pi$
$449$	−1.12713e11			−0.130877			−0.0654387	+	0.997857i	$0.520845\pi$
$450$		0			0
$451$	−4.31786e10	+	7.47875e10i	−0.0491444	+	0.0851206i
$452$		0			0
$453$	−4.13635e11	−	7.16437e11i	−0.461504	−	0.799348i
$454$		0			0
$455$	−1.00583e12	−	4.14868e11i	−1.10021	−	0.453793i
$456$		0			0
$457$	8.23390e11	+	1.42615e12i	0.883045	+	1.52948i	0.847938	+	0.530095i	$0.177844\pi$
$457$	8.23390e11	+	1.42615e12i	0.883045	+	1.52948i	0.0351071	+	0.999384i	$0.488823\pi$
$458$		0			0
$459$	−9.49264e10	+	1.64417e11i	−0.0998228	+	0.172898i
$460$		0			0
$461$	5.84023e11			0.602248			0.301124	−	0.953585i	$-0.402638\pi$
$461$	5.84023e11			0.602248			0.301124	+	0.953585i	$0.402638\pi$
$462$		0			0
$463$	−1.20511e12			−1.21874			−0.609369	−	0.792886i	$-0.708577\pi$
$463$	−1.20511e12			−1.21874			−0.609369	+	0.792886i	$0.708577\pi$
$464$		0			0
$465$	3.86947e11	−	6.70211e11i	0.383807	−	0.664773i
$466$		0			0
$467$	−1.41545e11	−	2.45162e11i	−0.137711	−	0.238522i	0.788919	−	0.614497i	$-0.210641\pi$
$467$	−1.41545e11	−	2.45162e11i	−0.137711	−	0.238522i	−0.926630	+	0.375975i	$0.877308\pi$
$468$		0			0
$469$	−2.23350e11	−	1.67712e12i	−0.213161	−	1.60061i
$470$		0			0
$471$	−1.47123e11	−	2.54824e11i	−0.137748	−	0.238587i
$472$		0			0
$473$	7.88275e11	−	1.36533e12i	0.724107	−	1.25419i
$474$		0			0
$475$	−2.09073e12			−1.88442
$476$		0			0
$477$	3.34361e11			0.295721
$478$		0			0
$479$	−5.65778e11	+	9.79955e11i	−0.491062	+	0.850544i	−0.999947	−	0.0102907i	$-0.996724\pi$
$479$	−5.65778e11	+	9.79955e11i	−0.491062	+	0.850544i	0.508886	+	0.860834i	$0.330058\pi$
$480$		0			0
$481$	−3.02239e11	−	5.23493e11i	−0.257453	−	0.445921i
$482$		0			0
$483$	1.04993e12	−	8.08141e11i	0.877808	−	0.675655i
$484$		0			0
$485$	−4.18331e10	−	7.24571e10i	−0.0343307	−	0.0594625i
$486$		0			0
$487$	3.30522e11	−	5.72480e11i	0.266268	−	0.461190i	−0.701627	−	0.712545i	$-0.747543\pi$
$487$	3.30522e11	−	5.72480e11i	0.266268	−	0.461190i	0.967895	+	0.251354i	$0.0808759\pi$
$488$		0			0
$489$	−1.09454e12			−0.865650
$490$		0			0
$491$	1.32432e12			1.02831			0.514157	−	0.857696i	$-0.328105\pi$
$491$	1.32432e12			1.02831			0.514157	+	0.857696i	$0.328105\pi$
$492$		0			0
$493$	3.23334e11	−	5.60031e11i	0.246513	−	0.426973i
$494$		0			0
$495$	−3.19064e11	−	5.52635e11i	−0.238866	−	0.413728i
$496$		0			0
$497$	1.05039e11	−	8.08490e10i	0.0772228	−	0.0594389i
$498$		0			0
$499$	2.08953e11	+	3.61916e11i	0.150867	+	0.261310i	0.931547	−	0.363622i	$-0.118460\pi$
$499$	2.08953e11	+	3.61916e11i	0.150867	+	0.261310i	−0.780679	+	0.624932i	$0.785127\pi$
$500$		0			0
$501$	−5.93849e11	+	1.02858e12i	−0.421120	+	0.729402i
$502$		0			0
$503$	1.60497e12			1.11792			0.558960	−	0.829195i	$-0.311201\pi$
$503$	1.60497e12			1.11792			0.558960	+	0.829195i	$0.311201\pi$
$504$		0			0
$505$	3.00595e12			2.05670
$506$		0			0
$507$	1.92303e11	−	3.33079e11i	0.129256	−	0.223878i
$508$		0			0
$509$	7.99426e11	+	1.38465e12i	0.527896	+	0.914342i	0.999471	+	0.0325164i	$0.0103521\pi$
$509$	7.99426e11	+	1.38465e12i	0.527896	+	0.914342i	−0.471576	+	0.881826i	$0.656315\pi$
$510$		0			0
$511$	1.56245e11	+	1.17323e12i	0.101371	+	0.761185i
$512$		0			0
$513$	1.81778e11	+	3.14848e11i	0.115881	+	0.200712i
$514$		0			0
$515$	1.78058e12	−	3.08405e12i	1.11540	−	1.93192i
$516$		0			0
$517$	2.51472e12			1.54804
$518$		0			0
$519$	1.57039e12			0.950070
$520$		0			0
$521$	−1.18219e11	+	2.04761e11i	−0.0702936	+	0.121752i	−0.899030	−	0.437887i	$-0.855727\pi$
$521$	−1.18219e11	+	2.04761e11i	−0.0702936	+	0.121752i	0.828736	+	0.559639i	$0.189060\pi$
$522$		0			0
$523$	−4.44068e11	−	7.69148e11i	−0.259532	−	0.449523i	0.706584	−	0.707629i	$-0.250235\pi$
$523$	−4.44068e11	−	7.69148e11i	−0.259532	−	0.449523i	−0.966117	+	0.258106i	$0.916902\pi$
$524$		0			0
$525$	1.45376e12	+	5.99621e11i	0.835172	+	0.344477i
$526$		0			0
$527$	−7.62497e11	−	1.32068e12i	−0.430616	−	0.745849i
$528$		0			0
$529$	−2.41460e12	+	4.18221e12i	−1.34058	+	2.32196i
$530$		0			0
$531$	−3.71198e11			−0.202619
$532$		0			0
$533$	−1.52076e11			−0.0816185
$534$		0			0
$535$	2.41937e11	−	4.19047e11i	0.127676	−	0.221142i
$536$		0			0
$537$	−2.51474e11	−	4.35566e11i	−0.130499	−	0.226032i
$538$		0			0
$539$	−1.24369e12	−	1.23627e12i	−0.634690	−	0.630903i
$540$		0			0
$541$	8.61983e11	+	1.49300e12i	0.432624	+	0.749327i	0.997098	−	0.0761235i	$-0.0242543\pi$
$541$	8.61983e11	+	1.49300e12i	0.432624	+	0.749327i	−0.564474	+	0.825451i	$0.690921\pi$
$542$		0			0
$543$	−2.98466e11	+	5.16959e11i	−0.147332	+	0.255186i
$544$		0			0
$545$	5.53278e12			2.68633
$546$		0			0
$547$	4.82103e10			0.0230248			0.0115124	−	0.999934i	$-0.496335\pi$
$547$	4.82103e10			0.0230248			0.0115124	+	0.999934i	$0.496335\pi$
$548$		0			0
$549$	−5.42668e11	+	9.39929e11i	−0.254952	+	0.441590i
$550$		0			0
$551$	−6.19164e11	−	1.07242e12i	−0.286169	−	0.495660i
$552$		0			0
$553$	2.45189e12	+	1.01131e12i	1.11490	+	0.459856i
$554$		0			0
$555$	7.16003e11	+	1.24015e12i	0.320329	+	0.554826i
$556$		0			0
$557$	5.83425e11	−	1.01052e12i	0.256824	−	0.444833i	−0.708565	−	0.705646i	$-0.750657\pi$
$557$	5.83425e11	−	1.01052e12i	0.256824	−	0.444833i	0.965389	+	0.260813i	$0.0839905\pi$
$558$		0			0
$559$	2.77633e12			1.20259
$560$		0			0
$561$	−1.25746e12			−0.535996
$562$		0			0
$563$	−9.06069e11	+	1.56936e12i	−0.380079	+	0.658315i	−0.991073	−	0.133319i	$-0.957436\pi$
$563$	−9.06069e11	+	1.56936e12i	−0.380079	+	0.658315i	0.610995	+	0.791635i	$0.290770\pi$
$564$		0			0
$565$	−7.37971e11	−	1.27820e12i	−0.304664	−	0.527693i
$566$		0			0
$567$	−3.60982e10	−	2.71059e11i	−0.0146677	−	0.110139i
$568$		0			0
$569$	1.02107e12	+	1.76855e12i	0.408367	+	0.707313i	0.994707	−	0.102752i	$-0.0327649\pi$
$569$	1.02107e12	+	1.76855e12i	0.408367	+	0.707313i	−0.586340	+	0.810065i	$0.699432\pi$
$570$		0			0
$571$	−9.68802e11	+	1.67801e12i	−0.381393	+	0.660592i	−0.991262	−	0.131911i	$-0.957889\pi$
$571$	−9.68802e11	+	1.67801e12i	−0.381393	+	0.660592i	0.609869	+	0.792502i	$0.291222\pi$
$572$		0			0
$573$	2.36574e12			0.916792
$574$		0			0
$575$	−7.86957e12			−3.00224
$576$		0			0
$577$	−1.60072e12	+	2.77252e12i	−0.601206	+	1.04132i	0.391432	+	0.920207i	$0.371980\pi$
$577$	−1.60072e12	+	2.77252e12i	−0.601206	+	1.04132i	−0.992639	+	0.121113i	$0.961354\pi$
$578$		0			0
$579$	5.69020e11	+	9.85572e11i	0.210414	+	0.364447i
$580$		0			0
$581$	1.98140e12	−	1.52510e12i	0.721405	−	0.555270i
$582$		0			0
$583$	1.10729e12	+	1.91789e12i	0.396967	+	0.687567i
$584$		0			0
$585$	5.61876e11	−	9.73198e11i	0.198353	−	0.343558i
$586$		0			0
$587$	−9.90821e11			−0.344448			−0.172224	−	0.985058i	$-0.555095\pi$
$587$	−9.90821e11			−0.344448			−0.172224	+	0.985058i	$0.555095\pi$
$588$		0			0
$589$	−2.92026e12			−0.999777
$590$		0			0
$591$	7.90280e11	−	1.36881e12i	0.266463	−	0.461528i
$592$		0			0
$593$	1.51278e12	+	2.62021e12i	0.502376	+	0.870140i	0.999996	+	0.00274531i	$0.000873861\pi$
$593$	1.51278e12	+	2.62021e12i	0.502376	+	0.870140i	−0.497621	+	0.867395i	$0.665793\pi$
$594$		0			0
$595$	4.02495e12	−	3.09803e12i	1.31654	−	1.01335i
$596$		0			0
$597$	−2.87446e11	−	4.97870e11i	−0.0926128	−	0.160410i
$598$		0			0
$599$	2.49941e12	−	4.32911e12i	0.793262	−	1.37397i	−0.130674	−	0.991425i	$-0.541714\pi$
$599$	2.49941e12	−	4.32911e12i	0.793262	−	1.37397i	0.923937	−	0.382545i	$-0.124952\pi$
$600$		0			0
$601$	−2.16732e12			−0.677622			−0.338811	−	0.940854i	$-0.610025\pi$
$601$	−2.16732e12			−0.677622			−0.338811	+	0.940854i	$0.610025\pi$
$602$		0			0
$603$	1.74747e12			0.538248
$604$		0			0
$605$	−5.25451e11	+	9.10108e11i	−0.159453	+	0.276181i
$606$		0			0
$607$	−2.30554e12	−	3.99331e12i	−0.689323	−	1.19394i	−0.972057	−	0.234745i	$-0.924575\pi$
$607$	−2.30554e12	−	3.99331e12i	−0.689323	−	1.19394i	0.282734	−	0.959198i	$-0.408759\pi$
$608$		0			0
$609$	1.22956e11	+	9.23269e11i	0.0362220	+	0.271988i
$610$		0			0
$611$	2.21423e12	+	3.83516e12i	0.642743	+	1.11326i
$612$		0			0
$613$	9.09447e11	−	1.57521e12i	0.260139	−	0.450574i	−0.706140	−	0.708072i	$-0.749565\pi$
$613$	9.09447e11	−	1.57521e12i	0.260139	−	0.450574i	0.966279	+	0.257499i	$0.0828983\pi$
$614$		0			0
$615$	3.60268e11			0.101552
$616$		0			0
$617$	−3.20005e12			−0.888943			−0.444471	−	0.895793i	$-0.646608\pi$
$617$	−3.20005e12			−0.888943			−0.444471	+	0.895793i	$0.646608\pi$
$618$		0			0
$619$	−2.01099e12	+	3.48314e12i	−0.550557	+	0.953593i	0.447678	+	0.894195i	$0.352251\pi$
$619$	−2.01099e12	+	3.48314e12i	−0.550557	+	0.953593i	−0.998234	+	0.0593974i	$0.981082\pi$
$620$		0			0
$621$	6.84216e11	+	1.18510e12i	0.184621	+	0.319773i
$622$		0			0
$623$	−1.68672e12	−	6.95707e11i	−0.448586	−	0.185025i
$624$		0			0
$625$	2.21726e11	+	3.84041e11i	0.0581242	+	0.100674i
$626$		0			0
$627$	−1.20398e12	+	2.08535e12i	−0.311111	+	0.538859i
$628$		0			0
$629$	2.82183e12			0.718793
$630$		0			0
$631$	5.66829e12			1.42338			0.711689	−	0.702495i	$-0.247931\pi$
$631$	5.66829e12			1.42338			0.711689	+	0.702495i	$0.247931\pi$
$632$		0			0
$633$	6.38992e11	−	1.10677e12i	0.158190	−	0.273993i
$634$		0			0
$635$	−1.03934e12	−	1.80019e12i	−0.253674	−	0.439377i
$636$		0			0
$637$	7.90333e11	−	2.98527e12i	0.190188	−	0.718382i
$638$		0			0
$639$	6.84511e10	+	1.18561e11i	0.0162415	+	0.0281311i
$640$		0			0
$641$	1.78359e12	−	3.08927e12i	0.417287	−	0.722762i	−0.578379	−	0.815768i	$-0.696314\pi$
$641$	1.78359e12	−	3.08927e12i	0.417287	−	0.722762i	0.995666	+	0.0930063i	$0.0296477\pi$
$642$		0			0
$643$	3.93521e12			0.907860			0.453930	−	0.891037i	$-0.350022\pi$
$643$	3.93521e12			0.907860			0.453930	+	0.891037i	$0.350022\pi$
$644$		0			0
$645$	−6.57711e12			−1.49629
$646$		0			0
$647$	1.91864e11	−	3.32318e11i	0.0430451	−	0.0745563i	−0.843700	−	0.536815i	$-0.819627\pi$
$647$	1.91864e11	−	3.32318e11i	0.0430451	−	0.0745563i	0.886745	+	0.462258i	$0.152961\pi$
$648$		0			0
$649$	−1.22929e12	−	2.12919e12i	−0.271989	−	0.471100i
$650$		0			0
$651$	2.03056e12	+	8.37529e11i	0.443100	+	0.182762i
$652$		0			0
$653$	4.15440e12	+	7.19564e12i	0.894128	+	1.54867i	0.834881	+	0.550431i	$0.185537\pi$
$653$	4.15440e12	+	7.19564e12i	0.894128	+	1.54867i	0.0592471	+	0.998243i	$0.481130\pi$
$654$		0			0
$655$	3.53424e12	−	6.12148e12i	0.750257	−	1.29948i
$656$		0			0
$657$	−1.22245e12			−0.255969
$658$		0			0
$659$	2.51653e12			0.519779			0.259889	−	0.965638i	$-0.416314\pi$
$659$	2.51653e12			0.519779			0.259889	+	0.965638i	$0.416314\pi$
$660$		0			0
$661$	−6.09178e10	+	1.05513e11i	−0.0124119	+	0.0214980i	−0.872165	−	0.489212i	$-0.837284\pi$
$661$	−6.09178e10	+	1.05513e11i	−0.0124119	+	0.0214980i	0.859753	+	0.510710i	$0.170618\pi$
$662$		0			0
$663$	−1.10720e12	−	1.91773e12i	−0.222544	−	0.385458i
$664$		0			0
$665$	−1.28396e12	−	9.64116e12i	−0.254598	−	1.91175i
$666$		0			0
$667$	−2.33055e12	−	4.03662e12i	−0.455923	−	0.789682i
$668$		0			0
$669$	−5.07290e11	+	8.78652e11i	−0.0979126	+	0.169590i
$670$		0			0
$671$	−7.18857e12			−1.36896
$672$		0			0
$673$	2.14126e11			0.0402347			0.0201174	−	0.999798i	$-0.493596\pi$
$673$	2.14126e11			0.0402347			0.0201174	+	0.999798i	$0.493596\pi$
$674$		0			0
$675$	−8.12097e11	+	1.40659e12i	−0.150571	+	0.260796i
$676$		0			0
$677$	3.55632e12	+	6.15972e12i	0.650656	+	1.12697i	0.982964	+	0.183798i	$0.0588394\pi$
$677$	3.55632e12	+	6.15972e12i	0.650656	+	1.12697i	−0.332308	+	0.943171i	$0.607827\pi$
$678$		0			0
$679$	1.88178e11	−	1.44842e11i	0.0339746	−	0.0261505i
$680$		0			0
$681$	−1.75080e12	−	3.03248e12i	−0.311943	−	0.540301i
$682$		0			0
$683$	−2.81411e12	+	4.87418e12i	−0.494821	+	0.857055i	−0.999982	−	0.00597028i	$-0.998100\pi$
$683$	−2.81411e12	+	4.87418e12i	−0.494821	+	0.857055i	0.505161	+	0.863025i	$0.331433\pi$
$684$		0			0
$685$	−1.52492e13			−2.64631
$686$		0			0
$687$	2.45212e12			0.419988
$688$		0			0
$689$	−1.94996e12	+	3.37743e12i	−0.329640	+	0.570952i
$690$		0			0
$691$	−1.11547e12	−	1.93205e12i	−0.186126	−	0.322379i	0.757830	−	0.652453i	$-0.226260\pi$
$691$	−1.11547e12	−	1.93205e12i	−0.186126	−	0.322379i	−0.943955	+	0.330073i	$0.892926\pi$
$692$		0			0
$693$	1.43525e12	−	1.10472e12i	0.236389	−	0.181950i
$694$		0			0
$695$	2.69072e11	+	4.66047e11i	0.0437459	+	0.0757701i
$696$		0			0
$697$	3.54962e11	−	6.14813e11i	0.0569685	−	0.0986723i
$698$		0			0
$699$	−4.86005e12			−0.770005
$700$		0			0
$701$	−2.16595e12			−0.338780			−0.169390	−	0.985549i	$-0.554180\pi$
$701$	−2.16595e12			−0.338780			−0.169390	+	0.985549i	$0.554180\pi$
$702$		0			0
$703$	2.70181e12	−	4.67968e12i	0.417212	−	0.722632i
$704$		0			0
$705$	−5.24550e12	−	9.08548e12i	−0.799717	−	1.38515i
$706$		0			0
$707$	1.12626e12	+	8.45699e12i	0.169532	+	1.27300i
$708$		0			0
$709$	−5.38117e12	−	9.32045e12i	−0.799776	−	1.38525i	−0.919762	−	0.392477i	$-0.871618\pi$
$709$	−5.38117e12	−	9.32045e12i	−0.799776	−	1.38525i	0.119986	−	0.992776i	$-0.461715\pi$
$710$		0			0
$711$	−1.36967e12	+	2.37234e12i	−0.201003	+	0.348148i
$712$		0			0
$713$	−1.09919e13			−1.59284
$714$		0			0
$715$	7.44301e12			1.06505
$716$		0			0
$717$	−2.43799e12	+	4.22272e12i	−0.344505	+	0.596699i
$718$		0			0
$719$	3.94236e12	+	6.82837e12i	0.550144	+	0.952877i	0.998264	+	0.0589036i	$0.0187605\pi$
$719$	3.94236e12	+	6.82837e12i	0.550144	+	0.952877i	−0.448120	+	0.893973i	$0.647906\pi$
$720$		0			0
$721$	9.34386e12	+	3.85399e12i	1.28771	+	0.531131i
$722$		0			0
$723$	−1.15519e12	−	2.00085e12i	−0.157228	−	0.272327i
$724$		0			0
$725$	2.76613e12	−	4.79107e12i	0.371836	−	0.644039i
$726$		0			0
$727$	−9.08535e12			−1.20625			−0.603124	−	0.797647i	$-0.706078\pi$
$727$	−9.08535e12			−1.20625			−0.603124	+	0.797647i	$0.706078\pi$
$728$		0			0
$729$	2.82430e11			0.0370370
$730$		0			0
$731$	−6.48025e12	+	1.12241e13i	−0.839390	+	1.45387i
$732$		0			0
$733$	−2.87475e12	−	4.97922e12i	−0.367818	−	0.637079i	0.621406	−	0.783488i	$-0.286562\pi$
$733$	−2.87475e12	−	4.97922e12i	−0.367818	−	0.637079i	−0.989224	+	0.146410i	$0.953228\pi$
$734$		0			0
$735$	−1.87230e12	+	7.07208e12i	−0.236636	+	0.893828i
$736$		0			0
$737$	5.78706e12	+	1.00235e13i	0.722528	+	1.25145i
$738$		0			0
$739$	5.68350e12	−	9.84412e12i	0.700997	−	1.21416i	−0.267120	−	0.963663i	$-0.586072\pi$
$739$	5.68350e12	−	9.84412e12i	0.700997	−	1.21416i	0.968117	−	0.250499i	$-0.0805947\pi$
$740$		0			0
$741$	−4.24045e12			−0.516690
$742$		0			0
$743$	−1.31443e13			−1.58229			−0.791146	−	0.611627i	$-0.790515\pi$
$743$	−1.31443e13			−1.58229			−0.791146	+	0.611627i	$0.790515\pi$
$744$		0			0
$745$	−2.73077e12	+	4.72984e12i	−0.324775	+	0.562527i
$746$		0			0
$747$	1.29123e12	+	2.23647e12i	0.151726	+	0.262797i
$748$		0			0
$749$	1.26960e12	+	5.23662e11i	0.147401	+	0.0607971i
$750$		0			0
$751$	5.84449e12	+	1.01230e13i	0.670451	+	1.16125i	0.977776	+	0.209650i	$0.0672325\pi$
$751$	5.84449e12	+	1.01230e13i	0.670451	+	1.16125i	−0.307326	+	0.951604i	$0.599434\pi$
$752$		0			0
$753$	−3.19328e12	+	5.53091e12i	−0.361959	+	0.626931i
$754$		0			0
$755$	−2.28587e13			−2.56030
$756$		0			0
$757$	6.61213e12			0.731830			0.365915	−	0.930648i	$-0.380756\pi$
$757$	6.61213e12			0.731830			0.365915	+	0.930648i	$0.380756\pi$
$758$		0			0
$759$	−4.53180e12	+	7.84931e12i	−0.495659	+	0.858507i
$760$		0			0
$761$	5.09213e10	+	8.81982e10i	0.00550387	+	0.00953298i	0.868764	−	0.495226i	$-0.164915\pi$
$761$	5.09213e10	+	8.81982e10i	0.00550387	+	0.00953298i	−0.863260	+	0.504759i	$0.831581\pi$
$762$		0			0
$763$	2.07300e12	+	1.55660e13i	0.221431	+	1.66271i
$764$		0			0
$765$	2.62296e12	+	4.54310e12i	0.276895	+	0.479596i
$766$		0			0
$767$	2.16479e12	−	3.74953e12i	0.225859	−	0.391199i
$768$		0			0
$769$	4.02104e12			0.414639			0.207320	−	0.978273i	$-0.433526\pi$
$769$	4.02104e12			0.414639			0.207320	+	0.978273i	$0.433526\pi$
$770$		0			0
$771$	7.99193e12			0.814530
$772$		0			0
$773$	5.21158e12	−	9.02672e12i	0.525003	−	0.909331i	−0.474573	−	0.880216i	$-0.657398\pi$
$773$	5.21158e12	−	9.02672e12i	0.525003	−	0.909331i	0.999576	−	0.0291155i	$-0.00926905\pi$
$774$		0			0
$775$	−6.52317e12	−	1.12985e13i	−0.649533	−	1.12502i
$776$		0			0
$777$	−3.22080e12	+	2.47907e12i	−0.317007	+	0.244002i
$778$		0			0
$779$	−6.79730e11	−	1.17733e12i	−0.0661329	−	0.114546i
$780$		0			0
$781$	−4.53376e11	+	7.85270e11i	−0.0436043	+	0.0755248i
$782$		0			0
$783$	−9.61999e11			−0.0914633
$784$		0			0
$785$	−8.13046e12			−0.764191
$786$		0			0
$787$	7.17289e12	−	1.24238e13i	0.666511	−	1.15443i	−0.312362	−	0.949963i	$-0.601120\pi$
$787$	7.17289e12	−	1.24238e13i	0.666511	−	1.15443i	0.978873	−	0.204468i	$-0.0655466\pi$
$788$		0			0
$789$	1.03837e12	+	1.79851e12i	0.0953905	+	0.165221i
$790$		0			0
$791$	3.31961e12	−	2.55513e12i	0.301504	−	0.232070i
$792$		0			0
$793$	−6.32959e12	−	1.09632e13i	−0.568389	−	0.984479i
$794$		0			0
$795$	4.61945e12	−	8.00112e12i	0.410145	−	0.710393i
$796$		0			0
$797$	1.05784e13			0.928659			0.464329	−	0.885663i	$-0.346295\pi$
$797$	1.05784e13			0.928659			0.464329	+	0.885663i	$0.346295\pi$
$798$		0			0
$799$	−2.06730e13			−1.79450
$800$		0			0
$801$	9.42231e11	−	1.63199e12i	0.0808744	−	0.140079i
$802$		0			0
$803$	−4.04836e12	−	7.01196e12i	−0.343604	−	0.595140i
$804$		0			0
$805$	−4.83286e12	−	3.62896e13i	−0.405623	−	3.04579i
$806$		0			0
$807$	4.59109e12	+	7.95200e12i	0.381052	+	0.660002i
$808$		0			0
$809$	8.20653e12	−	1.42141e13i	0.673583	−	1.16668i	−0.303298	−	0.952896i	$-0.598088\pi$
$809$	8.20653e12	−	1.42141e13i	0.673583	−	1.16668i	0.976881	−	0.213784i	$-0.0685789\pi$
$810$		0			0
$811$	1.50696e13			1.22323			0.611616	−	0.791155i	$-0.290520\pi$
$811$	1.50696e13			1.22323			0.611616	+	0.791155i	$0.290520\pi$
$812$		0			0
$813$	−1.36616e12			−0.109672
$814$		0			0
$815$	−1.51219e13	+	2.61920e13i	−1.20060	+	2.07950i
$816$		0			0
$817$	1.24093e13	+	2.14935e13i	0.974421	+	1.68775i
$818$		0			0
$819$	2.94853e12	+	1.21616e12i	0.228996	+	0.0944522i
$820$		0			0
$821$	−1.01816e13	−	1.76351e13i	−0.782120	−	1.35467i	−0.930705	−	0.365771i	$-0.880805\pi$
$821$	−1.01816e13	−	1.76351e13i	−0.782120	−	1.35467i	0.148585	−	0.988900i	$-0.452528\pi$
$822$		0			0
$823$	1.65454e12	−	2.86576e12i	0.125713	−	0.217741i	−0.796299	−	0.604904i	$-0.793212\pi$
$823$	1.65454e12	−	2.86576e12i	0.125713	−	0.217741i	0.922011	+	0.387163i	$0.126545\pi$
$824$		0			0
$825$	−1.07576e13			−0.808487
$826$		0			0
$827$	1.79697e13			1.33588			0.667939	−	0.744216i	$-0.267177\pi$
$827$	1.79697e13			1.33588			0.667939	+	0.744216i	$0.267177\pi$
$828$		0			0
$829$	8.68903e12	−	1.50498e13i	0.638963	−	1.10672i	−0.346697	−	0.937977i	$-0.612697\pi$
$829$	8.68903e12	−	1.50498e13i	0.638963	−	1.10672i	0.985661	−	0.168740i	$-0.0539698\pi$
$830$		0			0
$831$	−1.17462e12	−	2.03451e12i	−0.0854465	−	0.147998i
$832$		0			0
$833$	1.02241e13	+	1.01631e13i	0.735737	+	0.731347i
$834$		0			0
$835$	1.64089e13	+	2.84211e13i	1.16813	+	2.02326i
$836$		0			0
$837$	−1.13431e12	+	1.96468e12i	−0.0798853	+	0.138365i
$838$		0			0
$839$	−2.06956e13			−1.44195			−0.720973	−	0.692963i	$-0.756305\pi$
$839$	−2.06956e13			−1.44195			−0.720973	+	0.692963i	$0.756305\pi$
$840$		0			0
$841$	−1.12304e13			−0.774131
$842$		0			0
$843$	2.43918e12	−	4.22478e12i	0.166349	−	0.288124i
$844$		0			0
$845$	−5.31363e12	−	9.20347e12i	−0.358539	−	0.621007i
$846$		0			0
$847$	−2.75739e12	−	1.13732e12i	−0.184087	−	0.0759287i
$848$		0			0
$849$	6.75031e12	+	1.16919e13i	0.445902	+	0.772324i
$850$		0			0
$851$	1.01697e13	−	1.76144e13i	0.664699	−	1.15129i
$852$		0			0
$853$	−3.03462e13			−1.96261			−0.981303	−	0.192469i	$-0.938351\pi$
$853$	−3.03462e13			−1.96261			−0.981303	+	0.192469i	$0.938351\pi$
$854$		0			0
$855$	1.00456e13			0.642878
$856$		0			0
$857$	9.81022e12	−	1.69918e13i	0.621248	−	1.07603i	−0.368006	−	0.929824i	$-0.619959\pi$
$857$	9.81022e12	−	1.69918e13i	0.621248	−	1.07603i	0.989254	−	0.146209i	$-0.0467073\pi$
$858$		0			0
$859$	4.87339e12	+	8.44096e12i	0.305395	+	0.528960i	0.977349	−	0.211633i	$-0.0678781\pi$
$859$	4.87339e12	+	8.44096e12i	0.305395	+	0.528960i	−0.671954	+	0.740593i	$0.734545\pi$
$860$		0			0
$861$	1.34984e11	+	1.01358e12i	0.00837081	+	0.0628557i
$862$		0			0
$863$	1.93961e12	+	3.35951e12i	0.119033	+	0.206171i	0.919385	−	0.393360i	$-0.128687\pi$
$863$	1.93961e12	+	3.35951e12i	0.119033	+	0.206171i	−0.800352	+	0.599531i	$0.795354\pi$
$864$		0			0
$865$	2.16962e13	−	3.75789e13i	1.31768	−	2.28230i
$866$		0			0
$867$	7.31720e11			0.0439804
$868$		0			0
$869$	−1.81436e13			−1.07928
$870$		0			0
$871$	−1.01911e13	+	1.76515e13i	−0.599984	+	1.03920i
$872$		0			0
$873$	1.22631e11	+	2.12403e11i	0.00714556	+	0.0123765i
$874$		0			0
$875$	1.24281e13	−	9.56602e12i	0.716753	−	0.551690i
$876$		0			0
$877$	1.33152e13	+	2.30626e13i	0.760063	+	1.31647i	0.942818	+	0.333309i	$0.108165\pi$
$877$	1.33152e13	+	2.30626e13i	0.760063	+	1.31647i	−0.182754	+	0.983159i	$0.558501\pi$
$878$		0			0
$879$	−2.87655e11	+	4.98233e11i	−0.0162526	+	0.0281503i
$880$		0			0
$881$	−1.31525e13			−0.735557			−0.367779	−	0.929913i	$-0.619882\pi$
$881$	−1.31525e13			−0.735557			−0.367779	+	0.929913i	$0.619882\pi$
$882$		0			0
$883$	1.96529e13			1.08794			0.543968	−	0.839106i	$-0.316921\pi$
$883$	1.96529e13			1.08794			0.543968	+	0.839106i	$0.316921\pi$
$884$		0			0
$885$	−5.12838e12	+	8.88262e12i	−0.281019	+	0.486739i
$886$		0			0
$887$	−2.48541e12	−	4.30485e12i	−0.134816	−	0.233508i	0.790711	−	0.612189i	$-0.209711\pi$
$887$	−2.48541e12	−	4.30485e12i	−0.134816	−	0.233508i	−0.925527	+	0.378681i	$0.876378\pi$
$888$		0			0
$889$	4.67527e12	−	3.59859e12i	0.251044	−	0.193230i
$890$		0			0
$891$	9.35315e11	+	1.62001e12i	0.0497174	+	0.0861130i
$892$		0			0
$893$	−1.97938e13	+	3.42838e13i	−1.04159	+	1.80409i
$894$		0			0
$895$	−1.38972e13			−0.723976
$896$		0			0
$897$	−1.59611e13			−0.823186
$898$		0			0
$899$	3.86363e12	−	6.69201e12i	0.197277	−	0.341694i
$900$		0			0
$901$	−9.10284e12	−	1.57666e13i	−0.460167	−	0.797032i
$902$		0			0
$903$	−2.46429e12	−	1.85041e13i	−0.123338	−	0.926134i
$904$		0			0
$905$	8.24708e12	+	1.42844e13i	0.408678	+	0.707851i
$906$		0			0
$907$	1.30287e13	−	2.25664e13i	0.639248	−	1.10721i	−0.346350	−	0.938105i	$-0.612579\pi$
$907$	1.30287e13	−	2.25664e13i	0.639248	−	1.10721i	0.985598	−	0.169105i	$-0.0540876\pi$
$908$		0			0
$909$	−8.81175e12			−0.428080
$910$		0			0
$911$	2.46116e13			1.18388			0.591938	−	0.805983i	$-0.298363\pi$
$911$	2.46116e13			1.18388			0.591938	+	0.805983i	$0.298363\pi$
$912$		0			0
$913$	−8.55226e12	+	1.48130e13i	−0.407345	+	0.705542i
$914$		0			0
$915$	1.49948e13	+	2.59717e13i	0.707204	+	1.22491i
$916$		0			0
$917$	1.85465e13	+	7.64971e12i	0.866162	+	0.357259i
$918$		0			0
$919$	1.71326e13	+	2.96745e13i	0.792324	+	1.37235i	0.924525	+	0.381122i	$0.124462\pi$
$919$	1.71326e13	+	2.96745e13i	0.792324	+	1.37235i	−0.132201	+	0.991223i	$0.542204\pi$
$920$		0			0
$921$	−1.78027e12	+	3.08352e12i	−0.0815300	+	0.141214i
$922$		0			0
$923$	−1.59680e12			−0.0724175
$924$		0			0
$925$	2.41408e13			1.08421
$926$		0			0
$927$	−5.21965e12	+	9.04070e12i	−0.232157	+	0.402108i
$928$		0			0
$929$	1.65657e13	+	2.86927e13i	0.729693	+	1.26387i	0.957013	+	0.290045i	$0.0936704\pi$
$929$	1.65657e13	+	2.86927e13i	0.729693	+	1.26387i	−0.227320	+	0.973820i	$0.572996\pi$
$930$		0			0
$931$	2.66435e13	−	7.22463e12i	1.16230	−	0.315168i
$932$		0			0
$933$	6.06352e12	+	1.05023e13i	0.261974	+	0.453751i
$934$		0			0
$935$	−1.73728e13	+	3.00906e13i	−0.743391	+	1.28759i
$936$		0			0
$937$	7.88241e12			0.334065			0.167032	−	0.985951i	$-0.446582\pi$
$937$	7.88241e12			0.334065			0.167032	+	0.985951i	$0.446582\pi$
$938$		0			0
$939$	−3.19571e12			−0.134144
$940$		0			0
$941$	−1.59430e12	+	2.76141e12i	−0.0662852	+	0.114809i	−0.897263	−	0.441496i	$-0.854448\pi$
$941$	−1.59430e12	+	2.76141e12i	−0.0662852	+	0.114809i	0.830978	+	0.556305i	$0.187781\pi$
$942$		0			0
$943$	−2.55852e12	−	4.43148e12i	−0.105362	−	0.182493i
$944$		0			0
$945$	−6.98506e12	−	2.88107e12i	−0.284923	−	0.117520i
$946$		0			0
$947$	1.74441e13	+	3.02140e13i	0.704812	+	1.22077i	0.966759	+	0.255688i	$0.0823020\pi$
$947$	1.74441e13	+	3.02140e13i	0.704812	+	1.22077i	−0.261947	+	0.965082i	$0.584365\pi$
$948$		0			0
$949$	7.12921e12	−	1.23482e13i	0.285327	−	0.494202i
$950$		0			0
$951$	7.07081e12			0.280322
$952$		0			0
$953$	2.45297e13			0.963328			0.481664	−	0.876356i	$-0.340033\pi$
$953$	2.45297e13			0.963328			0.481664	+	0.876356i	$0.340033\pi$
$954$		0			0
$955$	3.26845e13	−	5.66112e13i	1.27153	−	2.20235i
$956$		0			0
$957$	−3.18583e12	−	5.51802e12i	−0.122777	−	0.212657i
$958$		0			0
$959$	−5.71353e12	−	4.29024e13i	−0.218133	−	1.63794i
$960$		0			0
$961$	4.10847e12	+	7.11607e12i	0.155391	+	0.269144i
$962$		0			0
$963$	−7.09222e11	+	1.22841e12i	−0.0265744	+	0.0460283i
$964$		0			0
$965$	3.14458e13			1.16732
$966$		0			0
$967$	−1.73773e13			−0.639092			−0.319546	−	0.947571i	$-0.603530\pi$
$967$	−1.73773e13			−0.639092			−0.319546	+	0.947571i	$0.603530\pi$
$968$		0			0
$969$	9.89766e12	−	1.71433e13i	0.360642	−	0.624649i
$970$		0			0
$971$	−7.78317e12	−	1.34808e13i	−0.280977	−	0.486666i	0.690649	−	0.723190i	$-0.257325\pi$
$971$	−7.78317e12	−	1.34808e13i	−0.280977	−	0.486666i	−0.971626	+	0.236525i	$0.923992\pi$
$972$		0			0
$973$	−1.21037e12	+	9.31629e11i	−0.0432922	+	0.0333223i
$974$		0			0
$975$	−9.47215e12	−	1.64062e13i	−0.335682	−	0.581418i
$976$		0			0
$977$	2.00254e13	−	3.46850e13i	0.703163	−	1.21791i	−0.264188	−	0.964471i	$-0.585104\pi$
$977$	2.00254e13	−	3.46850e13i	0.703163	−	1.21791i	0.967351	−	0.253442i	$-0.0815628\pi$
$978$		0			0
$979$	1.24815e13			0.434253
$980$		0			0
$981$	−1.62190e13			−0.559130
$982$		0			0
$983$	−3.52517e11	+	6.10577e11i	−0.0120417	+	0.0208569i	−0.871983	−	0.489535i	$-0.837166\pi$
$983$	−3.52517e11	+	6.10577e11i	−0.0120417	+	0.0208569i	0.859942	+	0.510392i	$0.170500\pi$
$984$		0			0
$985$	−2.18366e13	−	3.78222e13i	−0.739134	−	1.28022i
$986$		0			0
$987$	2.35959e13	−	1.81619e13i	0.791423	−	0.609164i
$988$		0			0
$989$	4.67088e13	+	8.09019e13i	1.55244	+	2.68891i
$990$		0			0
$991$	−3.42963e12	+	5.94029e12i	−0.112958	+	0.195648i	−0.916961	−	0.398976i	$-0.869366\pi$
$991$	−3.42963e12	+	5.94029e12i	−0.112958	+	0.195648i	0.804004	+	0.594624i	$0.202699\pi$
$992$		0			0
$993$	−2.19757e13			−0.717251
$994$		0			0
$995$	−1.58851e13			−0.513791
$996$		0			0
$997$	−1.96931e13	+	3.41094e13i	−0.631226	+	1.09332i	0.356075	+	0.934457i	$0.384115\pi$
$997$	−1.96931e13	+	3.41094e13i	−0.631226	+	1.09332i	−0.987301	+	0.158859i	$0.949219\pi$
$998$		0			0
$999$	−2.09891e12	−	3.63543e12i	−0.0666730	−	0.115481i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.10.q.a.25.8	✓	16
7.2	even	3	inner	168.10.q.a.121.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.10.q.a.25.8	✓	16	1.1	even	1	trivial
168.10.q.a.121.8	yes	16	7.2	even	3	inner

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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	168.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-40.5000 + 70.1481i) q^{3} +(1119.08 + 1938.30i) q^{5} +(-5033.95 + 3874.66i) q^{7} +(-3280.50 - 5681.99i) q^{9} +O(q^{10})\)	\(q+(-40.5000 + 70.1481i) q^{3} +(1119.08 + 1938.30i) q^{5} +(-5033.95 + 3874.66i) q^{7} +(-3280.50 - 5681.99i) q^{9} +(21727.9 - 37633.8i) q^{11} +76526.3 q^{13} -181290. q^{15} +(-178621. + 309380. i) q^{17} +(342047. + 592443. i) q^{19} +(-67925.2 - 510045. i) q^{21} +(1.28747e6 + 2.22997e6i) q^{23} +(-1.52810e6 + 2.64675e6i) q^{25} +531441. q^{27} -1.81017e6 q^{29} +(-2.13440e6 + 3.69689e6i) q^{31} +(1.75996e6 + 3.04834e6i) q^{33} +(-1.31436e7 - 5.42124e6i) q^{35} +(-3.94948e6 - 6.84070e6i) q^{37} +(-3.09932e6 + 5.36817e6i) q^{39} -1.98724e6 q^{41} +3.62794e7 q^{43} +(7.34226e6 - 1.27172e7i) q^{45} +(2.89343e7 + 5.01156e7i) q^{47} +(1.03276e7 - 3.90097e7i) q^{49} +(-1.44683e7 - 2.50598e7i) q^{51} +(-2.54809e7 + 4.41342e7i) q^{53} +9.72608e7 q^{55} -5.54116e7 q^{57} +(2.82882e7 - 4.89966e7i) q^{59} +(-8.27112e7 - 1.43260e8i) q^{61} +(3.85297e7 + 1.58920e7i) q^{63} +(8.56388e7 + 1.48331e8i) q^{65} +(-1.33171e8 + 2.30659e8i) q^{67} -2.08571e8 q^{69} -2.08661e7 q^{71} +(9.31603e7 - 1.61358e8i) q^{73} +(-1.23776e8 - 2.14387e8i) q^{75} +(3.64413e7 + 2.73635e8i) q^{77} +(-2.08759e8 - 3.61582e8i) q^{79} +(-2.15234e7 + 3.72796e7i) q^{81} -3.93607e8 q^{83} -7.99561e8 q^{85} +(7.33119e7 - 1.26980e8i) q^{87} +(1.43611e8 + 2.48741e8i) q^{89} +(-3.85229e8 + 2.96514e8i) q^{91} +(-1.72887e8 - 2.99448e8i) q^{93} +(-7.65554e8 + 1.32598e9i) q^{95} -3.73818e7 q^{97} -2.85114e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) \) 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9} + 32460 q^{11} + 119048 q^{13} + 31752 q^{15} + 208352 q^{17} + 914588 q^{19} - 428652 q^{21} + 460920 q^{23} - 3040180 q^{25} + 8503056 q^{27} - 16376136 q^{29} - 944064 q^{31} + 2629260 q^{33} - 15546664 q^{35} - 9826516 q^{37} - 4821444 q^{39} + 11449216 q^{41} - 6933624 q^{43} - 1285956 q^{45} + 26549360 q^{47} + 83657504 q^{49} + 16876512 q^{51} - 15354476 q^{53} + 134121944 q^{55} - 148163256 q^{57} + 18404996 q^{59} - 260632792 q^{61} + 35823060 q^{63} + 191461840 q^{65} + 53879788 q^{67} - 74669040 q^{69} - 164207456 q^{71} + 248475540 q^{73} - 246254580 q^{75} + 670121788 q^{77} + 16631256 q^{79} - 344373768 q^{81} - 1138943272 q^{83} - 1690136272 q^{85} + 663233508 q^{87} + 236796360 q^{89} - 1455575212 q^{91} - 76469184 q^{93} + 182450488 q^{95} + 1339799464 q^{97} - 425940120 q^{99}+O(q^{100}) \) 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9 + 32460 * q^11 + 119048 * q^13 + 31752 * q^15 + 208352 * q^17 + 914588 * q^19 - 428652 * q^21 + 460920 * q^23 - 3040180 * q^25 + 8503056 * q^27 - 16376136 * q^29 - 944064 * q^31 + 2629260 * q^33 - 15546664 * q^35 - 9826516 * q^37 - 4821444 * q^39 + 11449216 * q^41 - 6933624 * q^43 - 1285956 * q^45 + 26549360 * q^47 + 83657504 * q^49 + 16876512 * q^51 - 15354476 * q^53 + 134121944 * q^55 - 148163256 * q^57 + 18404996 * q^59 - 260632792 * q^61 + 35823060 * q^63 + 191461840 * q^65 + 53879788 * q^67 - 74669040 * q^69 - 164207456 * q^71 + 248475540 * q^73 - 246254580 * q^75 + 670121788 * q^77 + 16631256 * q^79 - 344373768 * q^81 - 1138943272 * q^83 - 1690136272 * q^85 + 663233508 * q^87 + 236796360 * q^89 - 1455575212 * q^91 - 76469184 * q^93 + 182450488 * q^95 + 1339799464 * q^97 - 425940120 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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