LMFDB - Embedded newform 168.10.q.a.25.6

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			25.6
Root			$1263.53 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	168.25
Dual form			168.10.q.a.121.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 + 70.1481i) q^{3} +(619.513 + 1073.03i) q^{5} +(-1334.90 - 6210.61i) q^{7} +(-3280.50 - 5681.99i) q^{9} +O(q^{10})$	\(q+(-40.5000 + 70.1481i) q^{3} +(619.513 + 1073.03i) q^{5} +(-1334.90 - 6210.61i) q^{7} +(-3280.50 - 5681.99i) q^{9} +(-242.537 + 420.086i) q^{11} -50302.6 q^{13} -100361. q^{15} +(-49431.1 + 85617.3i) q^{17} +(135342. + 234420. i) q^{19} +(489725. + 157889. i) q^{21} +(-241669. - 418583. i) q^{23} +(208971. - 361948. i) q^{25} +531441. q^{27} +3.37032e6 q^{29} +(1.14970e6 - 1.99134e6i) q^{31} +(-19645.5 - 34027.0i) q^{33} +(5.83717e6 - 5.27993e6i) q^{35} +(4.97259e6 + 8.61278e6i) q^{37} +(2.03726e6 - 3.52863e6i) q^{39} +1.64819e7 q^{41} -5.50320e6 q^{43} +(4.06462e6 - 7.04013e6i) q^{45} +(-2.13119e7 - 3.69132e7i) q^{47} +(-3.67897e7 + 1.65810e7i) q^{49} +(-4.00392e6 - 6.93500e6i) q^{51} +(-8.72236e6 + 1.51076e7i) q^{53} -601018. q^{55} -2.19254e7 q^{57} +(1.92959e7 - 3.34215e7i) q^{59} +(4.36736e7 + 7.56449e7i) q^{61} +(-3.09095e7 + 2.79588e7i) q^{63} +(-3.11631e7 - 5.39761e7i) q^{65} +(-4.38170e7 + 7.58932e7i) q^{67} +3.91504e7 q^{69} +2.43190e8 q^{71} +(-5.75235e7 + 9.96336e7i) q^{73} +(1.69266e7 + 2.93178e7i) q^{75} +(2.93275e6 + 945530. i) q^{77} +(2.30749e8 + 3.99670e8i) q^{79} +(-2.15234e7 + 3.72796e7i) q^{81} +5.07829e8 q^{83} -1.22493e8 q^{85} +(-1.36498e8 + 2.36421e8i) q^{87} +(5.49237e7 + 9.51306e7i) q^{89} +(6.71488e7 + 3.12410e8i) q^{91} +(9.31256e7 + 1.61298e8i) q^{93} +(-1.67692e8 + 2.90452e8i) q^{95} -4.30927e8 q^{97} +3.18257e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	619.513	+	1073.03i	0.443287	+	0.767796i	0.997931	−	0.0642920i	$-0.0204789\pi$
$5$	619.513	+	1073.03i	0.443287	+	0.767796i	−0.554644	+	0.832088i	$0.687146\pi$
$6$		0			0
$7$	−1334.90	−	6210.61i	−0.210139	−	0.977672i
$7$	−1334.90	−	6210.61i	−0.210139	−	0.977672i
$8$		0			0
$9$	−3280.50	−	5681.99i	−0.166667	−	0.288675i
$10$		0			0
$11$	−242.537	+	420.086i	−0.00499471	+	0.00865109i	−0.868512	−	0.495668i	$-0.834923\pi$
$11$	−242.537	+	420.086i	−0.00499471	+	0.00865109i	0.863517	+	0.504319i	$0.168257\pi$
$12$		0			0
$13$	−50302.6			−0.488479			−0.244239	−	0.969715i	$-0.578538\pi$
$13$	−50302.6			−0.488479			−0.244239	+	0.969715i	$0.578538\pi$
$14$		0			0
$15$	−100361.			−0.511864
$16$		0			0
$17$	−49431.1	+	85617.3i	−0.143543	+	0.248623i	−0.928828	−	0.370511i	$-0.879183\pi$
$17$	−49431.1	+	85617.3i	−0.143543	+	0.248623i	0.785286	+	0.619134i	$0.212516\pi$
$18$		0			0
$19$	135342.	+	234420.i	0.238255	+	0.412670i	0.960214	−	0.279266i	$-0.0900912\pi$
$19$	135342.	+	234420.i	0.238255	+	0.412670i	−0.721959	+	0.691936i	$0.756758\pi$
$20$		0			0
$21$	489725.	+	157889.i	0.549498	+	0.177160i
$22$		0			0
$23$	−241669.	−	418583.i	−0.180072	−	0.311894i	0.761833	−	0.647774i	$-0.224300\pi$
$23$	−241669.	−	418583.i	−0.180072	−	0.311894i	−0.941905	+	0.335880i	$0.890966\pi$
$24$		0			0
$25$	208971.	−	361948.i	0.106993	−	0.185317i
$26$		0			0
$27$	531441.			0.192450
$28$		0			0
$29$	3.37032e6			0.884871			0.442436	−	0.896800i	$-0.354114\pi$
$29$	3.37032e6			0.884871			0.442436	+	0.896800i	$0.354114\pi$
$30$		0			0
$31$	1.14970e6	−	1.99134e6i	0.223592	−	0.387273i	−0.732304	−	0.680978i	$-0.761555\pi$
$31$	1.14970e6	−	1.99134e6i	0.223592	−	0.387273i	0.955896	+	0.293705i	$0.0948883\pi$
$32$		0			0
$33$	−19645.5	−	34027.0i	−0.00288370	−	0.00499471i
$34$		0			0
$35$	5.83717e6	−	5.27993e6i	0.657500	−	0.594733i
$36$		0			0
$37$	4.97259e6	+	8.61278e6i	0.436189	+	0.755502i	0.997392	−	0.0721763i	$-0.0229944\pi$
$37$	4.97259e6	+	8.61278e6i	0.436189	+	0.755502i	−0.561202	+	0.827679i	$0.689661\pi$
$38$		0			0
$39$	2.03726e6	−	3.52863e6i	0.141012	−	0.244239i
$40$		0			0
$41$	1.64819e7			0.910921			0.455461	−	0.890256i	$-0.349475\pi$
$41$	1.64819e7			0.910921			0.455461	+	0.890256i	$0.349475\pi$
$42$		0			0
$43$	−5.50320e6			−0.245475			−0.122738	−	0.992439i	$-0.539167\pi$
$43$	−5.50320e6			−0.245475			−0.122738	+	0.992439i	$0.539167\pi$
$44$		0			0
$45$	4.06462e6	−	7.04013e6i	0.147762	−	0.255932i
$46$		0			0
$47$	−2.13119e7	−	3.69132e7i	−0.637061	−	1.10342i	−0.986074	−	0.166304i	$-0.946817\pi$
$47$	−2.13119e7	−	3.69132e7i	−0.637061	−	1.10342i	0.349013	−	0.937118i	$-0.386517\pi$
$48$		0			0
$49$	−3.67897e7	+	1.65810e7i	−0.911683	+	0.410893i
$50$		0			0
$51$	−4.00392e6	−	6.93500e6i	−0.0828743	−	0.143543i
$52$		0			0
$53$	−8.72236e6	+	1.51076e7i	−0.151842	+	0.262998i	−0.931905	−	0.362703i	$-0.881854\pi$
$53$	−8.72236e6	+	1.51076e7i	−0.151842	+	0.262998i	0.780063	+	0.625702i	$0.215187\pi$
$54$		0			0
$55$	−601018.			−0.00885636
$56$		0			0
$57$	−2.19254e7			−0.275113
$58$		0			0
$59$	1.92959e7	−	3.34215e7i	0.207315	−	0.359080i	−0.743553	−	0.668677i	$-0.766861\pi$
$59$	1.92959e7	−	3.34215e7i	0.207315	−	0.359080i	0.950868	+	0.309597i	$0.100194\pi$
$60$		0			0
$61$	4.36736e7	+	7.56449e7i	0.403864	+	0.699513i	0.994188	−	0.107653i	$-0.0343337\pi$
$61$	4.36736e7	+	7.56449e7i	0.403864	+	0.699513i	−0.590325	+	0.807166i	$0.701000\pi$
$62$		0			0
$63$	−3.09095e7	+	2.79588e7i	−0.247206	+	0.223607i
$64$		0			0
$65$	−3.11631e7	−	5.39761e7i	−0.216536	−	0.375052i
$66$		0			0
$67$	−4.38170e7	+	7.58932e7i	−0.265647	+	0.460115i	−0.967733	−	0.251978i	$-0.918919\pi$
$67$	−4.38170e7	+	7.58932e7i	−0.265647	+	0.460115i	0.702086	+	0.712093i	$0.252252\pi$
$68$		0			0
$69$	3.91504e7			0.207929
$70$		0			0
$71$	2.43190e8			1.13575			0.567875	−	0.823115i	$-0.307766\pi$
$71$	2.43190e8			1.13575			0.567875	+	0.823115i	$0.307766\pi$
$72$		0			0
$73$	−5.75235e7	+	9.96336e7i	−0.237079	+	0.410632i	−0.959875	−	0.280429i	$-0.909523\pi$
$73$	−5.75235e7	+	9.96336e7i	−0.237079	+	0.410632i	0.722796	+	0.691061i	$0.242857\pi$
$74$		0			0
$75$	1.69266e7	+	2.93178e7i	0.0617725	+	0.106993i
$76$		0			0
$77$	2.93275e6	+	945530.i	0.00950751	+	0.00306526i
$78$		0			0
$79$	2.30749e8	+	3.99670e8i	0.666528	+	1.15446i	0.978869	+	0.204491i	$0.0655538\pi$
$79$	2.30749e8	+	3.99670e8i	0.666528	+	1.15446i	−0.312340	+	0.949970i	$0.601113\pi$
$80$		0			0
$81$	−2.15234e7	+	3.72796e7i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	5.07829e8			1.17454			0.587268	−	0.809393i	$-0.300204\pi$
$83$	5.07829e8			1.17454			0.587268	+	0.809393i	$0.300204\pi$
$84$		0			0
$85$	−1.22493e8			−0.254522
$86$		0			0
$87$	−1.36498e8	+	2.36421e8i	−0.255440	+	0.442436i
$88$		0			0
$89$	5.49237e7	+	9.51306e7i	0.0927907	+	0.160718i	0.908684	−	0.417484i	$-0.137088\pi$
$89$	5.49237e7	+	9.51306e7i	0.0927907	+	0.160718i	−0.815894	+	0.578202i	$0.803755\pi$
$90$		0			0
$91$	6.71488e7	+	3.12410e8i	0.102648	+	0.477572i
$92$		0			0
$93$	9.31256e7	+	1.61298e8i	0.129091	+	0.223592i
$94$		0			0
$95$	−1.67692e8	+	2.90452e8i	−0.211231	+	0.365863i
$96$		0			0
$97$	−4.30927e8			−0.494232			−0.247116	−	0.968986i	$-0.579483\pi$
$97$	−4.30927e8			−0.494232			−0.247116	+	0.968986i	$0.579483\pi$
$98$		0			0
$99$	3.18257e6			0.00332981
$100$		0			0
$101$	−4.79527e8	+	8.30565e8i	−0.458529	+	0.794196i	−0.998883	−	0.0472419i	$-0.984957\pi$
$101$	−4.79527e8	+	8.30565e8i	−0.458529	+	0.794196i	0.540354	+	0.841438i	$0.318290\pi$
$102$		0			0
$103$	2.06517e8	+	3.57698e8i	0.180796	+	0.313147i	0.942152	−	0.335187i	$-0.108799\pi$
$103$	2.06517e8	+	3.57698e8i	0.180796	+	0.313147i	−0.761356	+	0.648334i	$0.775466\pi$
$104$		0			0
$105$	1.33972e8	+	6.23303e8i	0.107562	+	0.500435i
$106$		0			0
$107$	6.45695e8	+	1.11838e9i	0.476212	+	0.824823i	0.999629	−	0.0272537i	$-0.00867618\pi$
$107$	6.45695e8	+	1.11838e9i	0.476212	+	0.824823i	−0.523417	+	0.852077i	$0.675343\pi$
$108$		0			0
$109$	−5.88287e7	+	1.01894e8i	−0.0399181	+	0.0691401i	−0.885294	−	0.465031i	$-0.846043\pi$
$109$	−5.88287e7	+	1.01894e8i	−0.0399181	+	0.0691401i	0.845376	+	0.534172i	$0.179376\pi$
$110$		0			0
$111$	−8.05560e8			−0.503668
$112$		0			0
$113$	−1.48912e9			−0.859164			−0.429582	−	0.903028i	$-0.641339\pi$
$113$	−1.48912e9			−0.859164			−0.429582	+	0.903028i	$0.641339\pi$
$114$		0			0
$115$	2.99434e8	−	5.18635e8i	0.159647	−	0.276517i
$116$		0			0
$117$	1.65018e8	+	2.85819e8i	0.0814131	+	0.141012i
$118$		0			0
$119$	5.97721e8	+	1.92707e8i	0.273235	+	0.0880921i
$120$		0			0
$121$	1.17886e9	+	2.04184e9i	0.499950	+	0.865939i
$122$		0			0
$123$	−6.67518e8	+	1.15618e9i	−0.262960	+	0.455461i
$124$		0			0
$125$	2.93781e9			1.07629
$126$		0			0
$127$	6.27563e8			0.214062			0.107031	−	0.994256i	$-0.465866\pi$
$127$	6.27563e8			0.214062			0.107031	+	0.994256i	$0.465866\pi$
$128$		0			0
$129$	2.22880e8	−	3.86039e8i	0.0708626	−	0.122738i
$130$		0			0
$131$	8.15939e8	+	1.41325e9i	0.242068	+	0.419273i	0.961303	−	0.275493i	$-0.0888411\pi$
$131$	8.15939e8	+	1.41325e9i	0.242068	+	0.419273i	−0.719235	+	0.694766i	$0.755508\pi$
$132$		0			0
$133$	1.27522e9	−	1.15348e9i	0.353389	−	0.319653i
$134$		0			0
$135$	3.29234e8	+	5.70251e8i	0.0853106	+	0.147762i
$136$		0			0
$137$	−2.60530e9	+	4.51250e9i	−0.631851	+	1.09440i	0.355322	+	0.934744i	$0.384371\pi$
$137$	−2.60530e9	+	4.51250e9i	−0.631851	+	1.09440i	−0.987173	+	0.159654i	$0.948962\pi$
$138$		0			0
$139$	7.58741e9			1.72396			0.861980	−	0.506943i	$-0.169224\pi$
$139$	7.58741e9			1.72396			0.861980	+	0.506943i	$0.169224\pi$
$140$		0			0
$141$	3.45252e9			0.735615
$142$		0			0
$143$	1.22002e7	−	2.11314e7i	0.00243981	−	0.00422587i
$144$		0			0
$145$	2.08796e9	+	3.61645e9i	0.392252	+	0.679401i
$146$		0			0
$147$	3.26856e8	−	3.25226e9i	0.0577336	−	0.574456i
$148$		0			0
$149$	3.48017e9	+	6.02784e9i	0.578446	+	1.00190i	0.995658	+	0.0930886i	$0.0296740\pi$
$149$	3.48017e9	+	6.02784e9i	0.578446	+	1.00190i	−0.417212	+	0.908809i	$0.636993\pi$
$150$		0			0
$151$	−1.27175e9	+	2.20273e9i	−0.199069	+	0.344798i	−0.948227	−	0.317594i	$-0.897125\pi$
$151$	−1.27175e9	+	2.20273e9i	−0.199069	+	0.344798i	0.749158	+	0.662392i	$0.230459\pi$
$152$		0			0
$153$	6.48636e8			0.0956950
$154$		0			0
$155$	2.84901e9			0.396462
$156$		0			0
$157$	−3.48898e9	+	6.04309e9i	−0.458300	+	0.793800i	−0.998871	−	0.0474988i	$-0.984875\pi$
$157$	−3.48898e9	+	6.04309e9i	−0.458300	+	0.793800i	0.540571	+	0.841299i	$0.318208\pi$
$158$		0			0
$159$	−7.06511e8	−	1.22371e9i	−0.0876661	−	0.151842i
$160$		0			0
$161$	−2.27705e9	+	2.05968e9i	−0.267090	+	0.241592i
$162$		0			0
$163$	4.12936e9	+	7.15226e9i	0.458183	+	0.793596i	0.998865	−	0.0476311i	$-0.0151672\pi$
$163$	4.12936e9	+	7.15226e9i	0.458183	+	0.793596i	−0.540682	+	0.841227i	$0.681834\pi$
$164$		0			0
$165$	2.43412e7	−	4.21603e7i	0.00255661	−	0.00442818i
$166$		0			0
$167$	−3.99658e9			−0.397617			−0.198808	−	0.980038i	$-0.563707\pi$
$167$	−3.99658e9			−0.397617			−0.198808	+	0.980038i	$0.563707\pi$
$168$		0			0
$169$	−8.07415e9			−0.761389
$170$		0			0
$171$	8.87981e8	−	1.53803e9i	0.0794184	−	0.137557i
$172$		0			0
$173$	5.90438e8	+	1.02267e9i	0.0501149	+	0.0868015i	0.889995	−	0.455971i	$-0.150708\pi$
$173$	5.90438e8	+	1.02267e9i	0.0501149	+	0.0868015i	−0.839880	+	0.542772i	$0.817375\pi$
$174$		0			0
$175$	−2.52687e9	−	8.14673e8i	−0.203663	−	0.0656617i
$176$		0			0
$177$	1.56297e9	+	2.70714e9i	0.119693	+	0.207315i
$178$		0			0
$179$	−6.41465e9	+	1.11105e10i	−0.467019	+	0.808900i	−0.999290	−	0.0376738i	$-0.988005\pi$
$179$	−6.41465e9	+	1.11105e10i	−0.467019	+	0.808900i	0.532271	+	0.846574i	$0.321339\pi$
$180$		0			0
$181$	5.19082e9			0.359486			0.179743	−	0.983714i	$-0.442473\pi$
$181$	5.19082e9			0.359486			0.179743	+	0.983714i	$0.442473\pi$
$182$		0			0
$183$	−7.07513e9			−0.466342
$184$		0			0
$185$	−6.16117e9	+	1.06715e10i	−0.386714	+	0.669809i
$186$		0			0
$187$	−2.39777e7	−	4.15307e7i	−0.00143391	−	0.00248360i
$188$		0			0
$189$	−7.09418e8	−	3.30057e9i	−0.0404412	−	0.188153i
$190$		0			0
$191$	−6.57224e9	−	1.13835e10i	−0.357325	−	0.618905i	0.630188	−	0.776443i	$-0.282978\pi$
$191$	−6.57224e9	−	1.13835e10i	−0.357325	−	0.618905i	−0.987513	+	0.157538i	$0.949644\pi$
$192$		0			0
$193$	−9.75794e9	+	1.69012e10i	−0.506233	+	0.876821i	0.493741	+	0.869609i	$0.335629\pi$
$193$	−9.75794e9	+	1.69012e10i	−0.506233	+	0.876821i	−0.999974	+	0.00721205i	$0.997704\pi$
$194$		0			0
$195$	5.04842e9			0.250035
$196$		0			0
$197$	8.62196e9			0.407857			0.203929	−	0.978986i	$-0.434629\pi$
$197$	8.62196e9			0.407857			0.203929	+	0.978986i	$0.434629\pi$
$198$		0			0
$199$	1.13576e10	−	1.96719e10i	0.513389	−	0.889216i	−0.486491	−	0.873686i	$-0.661723\pi$
$199$	1.13576e10	−	1.96719e10i	0.513389	−	0.889216i	0.999879	−	0.0155297i	$-0.00494347\pi$
$200$		0			0
$201$	−3.54917e9	−	6.14735e9i	−0.153372	−	0.265647i
$202$		0			0
$203$	−4.49903e9	−	2.09317e10i	−0.185946	−	0.865114i
$204$		0			0
$205$	1.02108e10	+	1.76856e10i	0.403800	+	0.699401i
$206$		0			0
$207$	−1.58559e9	+	2.74633e9i	−0.0600240	+	0.103965i
$208$		0			0
$209$	−1.31302e8			−0.00476006
$210$		0			0
$211$	2.34313e10			0.813813			0.406906	−	0.913470i	$-0.366607\pi$
$211$	2.34313e10			0.813813			0.406906	+	0.913470i	$0.366607\pi$
$212$		0			0
$213$	−9.84919e9	+	1.70593e10i	−0.327863	+	0.567875i
$214$		0			0
$215$	−3.40930e9	−	5.90509e9i	−0.108816	−	0.188475i
$216$		0			0
$217$	−1.39021e10	−	4.48210e9i	−0.425611	−	0.137219i
$218$		0			0
$219$	−4.65940e9	−	8.07032e9i	−0.136877	−	0.237079i
$220$		0			0
$221$	2.48652e9	−	4.30677e9i	0.0701174	−	0.121447i
$222$		0			0
$223$	2.38366e10			0.645466			0.322733	−	0.946490i	$-0.395398\pi$
$223$	2.38366e10			0.645466			0.322733	+	0.946490i	$0.395398\pi$
$224$		0			0
$225$	−2.74212e9			−0.0713287
$226$		0			0
$227$	1.97436e10	−	3.41969e10i	0.493525	−	0.854811i	−0.506447	−	0.862271i	$-0.669041\pi$
$227$	1.97436e10	−	3.41969e10i	0.493525	−	0.854811i	0.999972	+	0.00746049i	$0.00237477\pi$
$228$		0			0
$229$	−2.28526e10	−	3.95819e10i	−0.549132	−	0.951124i	−0.998334	−	0.0576944i	$-0.981625\pi$
$229$	−2.28526e10	−	3.95819e10i	−0.549132	−	0.951124i	0.449202	−	0.893430i	$-0.351708\pi$
$230$		0			0
$231$	−1.85103e8	+	1.67433e8i	−0.00427721	+	0.00386889i
$232$		0			0
$233$	−7.38127e8	−	1.27847e9i	−0.0164070	−	0.0284178i	0.857705	−	0.514142i	$-0.171889\pi$
$233$	−7.38127e8	−	1.27847e9i	−0.0164070	−	0.0284178i	−0.874112	+	0.485724i	$0.838556\pi$
$234$		0			0
$235$	2.64059e10	−	4.57364e10i	0.564802	−	0.978266i
$236$		0			0
$237$	−3.73814e10			−0.769641
$238$		0			0
$239$	−1.54769e10			−0.306827			−0.153414	−	0.988162i	$-0.549027\pi$
$239$	−1.54769e10			−0.306827			−0.153414	+	0.988162i	$0.549027\pi$
$240$		0			0
$241$	1.19199e10	−	2.06459e10i	0.227613	−	0.394237i	−0.729487	−	0.683994i	$-0.760241\pi$
$241$	1.19199e10	−	2.06459e10i	0.227613	−	0.394237i	0.957100	+	0.289757i	$0.0935746\pi$
$242$		0			0
$243$	−1.74339e9	−	3.01964e9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	−4.05836e10	−	2.92042e10i	−0.719620	−	0.517843i
$246$		0			0
$247$	−6.80807e9	−	1.17919e10i	−0.116383	−	0.201580i
$248$		0			0
$249$	−2.05671e10	+	3.56232e10i	−0.339059	+	0.587268i
$250$		0			0
$251$	1.54069e10			0.245011			0.122505	−	0.992468i	$-0.460907\pi$
$251$	1.54069e10			0.245011			0.122505	+	0.992468i	$0.460907\pi$
$252$		0			0
$253$	2.34455e8			0.00359763
$254$		0			0
$255$	4.96096e9	−	8.59264e9i	0.0734742	−	0.127261i
$256$		0			0
$257$	−3.22043e10	−	5.57795e10i	−0.460484	−	0.797582i	0.538501	−	0.842625i	$-0.318991\pi$
$257$	−3.22043e10	−	5.57795e10i	−0.460484	−	0.797582i	−0.998985	+	0.0450426i	$0.985658\pi$
$258$		0			0
$259$	4.68527e10	−	4.23800e10i	0.646973	−	0.585210i
$260$		0			0
$261$	−1.10563e10	−	1.91501e10i	−0.147479	−	0.255440i
$262$		0			0
$263$	4.34891e10	−	7.53253e10i	0.560505	−	0.970823i	−0.436947	−	0.899487i	$-0.643940\pi$
$263$	4.34891e10	−	7.53253e10i	0.560505	−	0.970823i	0.997452	−	0.0713359i	$-0.0227262\pi$
$264$		0			0
$265$	−2.16144e10			−0.269239
$266$		0			0
$267$	−8.89764e9			−0.107146
$268$		0			0
$269$	4.07094e9	−	7.05108e9i	0.0474034	−	0.0821052i	−0.841350	−	0.540490i	$-0.818239\pi$
$269$	4.07094e9	−	7.05108e9i	0.0474034	−	0.0821052i	0.888754	+	0.458385i	$0.151572\pi$
$270$		0			0
$271$	−1.64573e10	−	2.85049e10i	−0.185352	−	0.321039i	0.758343	−	0.651856i	$-0.226009\pi$
$271$	−1.64573e10	−	2.85049e10i	−0.185352	−	0.321039i	−0.943695	+	0.330817i	$0.892676\pi$
$272$		0			0
$273$	−2.46345e10	−	7.94225e9i	−0.268418	−	0.0865389i
$274$		0			0
$275$	1.01366e8	+	1.75571e8i	0.00106880	+	0.00185121i
$276$		0			0
$277$	−1.77813e10	+	3.07981e10i	−0.181470	+	0.314315i	−0.942381	−	0.334541i	$-0.891419\pi$
$277$	−1.77813e10	+	3.07981e10i	−0.181470	+	0.314315i	0.760912	+	0.648856i	$0.224752\pi$
$278$		0			0
$279$	−1.50863e10			−0.149061
$280$		0			0
$281$	−9.72117e10			−0.930123			−0.465061	−	0.885278i	$-0.653968\pi$
$281$	−9.72117e10			−0.930123			−0.465061	+	0.885278i	$0.653968\pi$
$282$		0			0
$283$	7.39125e10	−	1.28020e11i	0.684982	−	1.18642i	−0.288461	−	0.957492i	$-0.593143\pi$
$283$	7.39125e10	−	1.28020e11i	0.684982	−	1.18642i	0.973443	−	0.228932i	$-0.0735232\pi$
$284$		0			0
$285$	−1.35831e10	−	2.35266e10i	−0.121954	−	0.211231i
$286$		0			0
$287$	−2.20017e10	−	1.02363e11i	−0.191420	−	0.890582i
$288$		0			0
$289$	5.44071e10	+	9.42358e10i	0.458791	+	0.794649i
$290$		0			0
$291$	1.74526e10	−	3.02287e10i	0.142673	−	0.247116i
$292$		0			0
$293$	−1.75657e11			−1.39239			−0.696194	−	0.717854i	$-0.745125\pi$
$293$	−1.75657e11			−1.39239			−0.696194	+	0.717854i	$0.745125\pi$
$294$		0			0
$295$	4.78162e10			0.367601
$296$		0			0
$297$	−1.28894e8	+	2.23251e8i	−0.000961233	+	0.00166490i
$298$		0			0
$299$	1.21566e10	+	2.10558e10i	0.0879613	+	0.152353i
$300$		0			0
$301$	7.34620e9	+	3.41782e10i	0.0515838	+	0.239994i
$302$		0			0
$303$	−3.88417e10	−	6.72758e10i	−0.264732	−	0.458529i
$304$		0			0
$305$	−5.41127e10	+	9.37259e10i	−0.358055	+	0.620170i
$306$		0			0
$307$	−5.99877e10			−0.385425			−0.192712	−	0.981255i	$-0.561728\pi$
$307$	−5.99877e10			−0.385425			−0.192712	+	0.981255i	$0.561728\pi$
$308$		0			0
$309$	−3.34557e10			−0.208765
$310$		0			0
$311$	5.43375e10	−	9.41152e10i	0.329365	−	0.570477i	−0.653021	−	0.757340i	$-0.726499\pi$
$311$	5.43375e10	−	9.41152e10i	0.329365	−	0.570477i	0.982386	+	0.186863i	$0.0598320\pi$
$312$		0			0
$313$	9.52628e10	+	1.65000e11i	0.561014	+	0.971705i	0.997408	+	0.0719494i	$0.0229220\pi$
$313$	9.52628e10	+	1.65000e11i	0.561014	+	0.971705i	−0.436394	+	0.899756i	$0.643745\pi$
$314$		0			0
$315$	−4.91493e10	−	1.58459e10i	−0.281268	−	0.0906818i
$316$		0			0
$317$	2.81816e10	+	4.88119e10i	0.156747	+	0.271493i	0.933694	−	0.358073i	$-0.116566\pi$
$317$	2.81816e10	+	4.88119e10i	0.156747	+	0.271493i	−0.776947	+	0.629566i	$0.783233\pi$
$318$		0			0
$319$	−8.17426e8	+	1.41582e9i	−0.00441968	+	0.00765511i
$320$		0			0
$321$	−1.04603e11			−0.549882
$322$		0			0
$323$	−2.67605e10			−0.136799
$324$		0			0
$325$	−1.05118e10	+	1.82069e10i	−0.0522638	+	0.0905236i
$326$		0			0
$327$	−4.76512e9	−	8.25343e9i	−0.0230467	−	0.0399181i
$328$		0			0
$329$	−2.00805e11	+	1.81635e11i	−0.944913	+	0.854708i
$330$		0			0
$331$	−1.97072e10	−	3.41339e10i	−0.0902399	−	0.156300i	0.817372	−	0.576110i	$-0.195430\pi$
$331$	−1.97072e10	−	3.41339e10i	−0.0902399	−	0.156300i	−0.907612	+	0.419810i	$0.862097\pi$
$332$		0			0
$333$	3.26252e10	−	5.65085e10i	0.145396	−	0.251834i
$334$		0			0
$335$	−1.08581e11			−0.471032
$336$		0			0
$337$	4.14853e11			1.75210			0.876051	−	0.482218i	$-0.160169\pi$
$337$	4.14853e11			1.75210			0.876051	+	0.482218i	$0.160169\pi$
$338$		0			0
$339$	6.03092e10	−	1.04459e11i	0.248019	−	0.429582i
$340$		0			0
$341$	5.57688e8	+	9.65945e8i	0.00223356	+	0.00386863i
$342$		0			0
$343$	1.52089e11	+	2.06353e11i	0.593299	+	0.804982i
$344$		0			0
$345$	2.42542e10	+	4.20095e10i	0.0921723	+	0.159647i
$346$		0			0
$347$	8.15578e9	−	1.41262e10i	0.0301983	−	0.0523051i	−0.850531	−	0.525924i	$-0.823719\pi$
$347$	8.15578e9	−	1.41262e10i	0.0301983	−	0.0523051i	0.880730	+	0.473619i	$0.157053\pi$
$348$		0			0
$349$	2.48881e11			0.898002			0.449001	−	0.893531i	$-0.351780\pi$
$349$	2.48881e11			0.898002			0.449001	+	0.893531i	$0.351780\pi$
$350$		0			0
$351$	−2.67329e10			−0.0940077
$352$		0			0
$353$	−6.88198e10	+	1.19199e11i	−0.235900	+	0.408590i	−0.959534	−	0.281594i	$-0.909137\pi$
$353$	−6.88198e10	+	1.19199e11i	−0.235900	+	0.408590i	0.723634	+	0.690184i	$0.242470\pi$
$354$		0			0
$355$	1.50659e11	+	2.60949e11i	0.503463	+	0.872024i
$356$		0			0
$357$	−3.77257e10	+	3.41243e10i	−0.122922	+	0.111188i
$358$		0			0
$359$	4.24601e10	+	7.35431e10i	0.134914	+	0.233678i	0.925565	−	0.378590i	$-0.123591\pi$
$359$	4.24601e10	+	7.35431e10i	0.134914	+	0.233678i	−0.790651	+	0.612267i	$0.790258\pi$
$360$		0			0
$361$	1.24709e11	−	2.16002e11i	0.386469	−	0.669384i
$362$		0			0
$363$	−1.90975e11			−0.577293
$364$		0			0
$365$	−1.42546e11			−0.420375
$366$		0			0
$367$	2.26393e11	−	3.92123e11i	0.651426	−	1.12830i	−0.331352	−	0.943507i	$-0.607505\pi$
$367$	2.26393e11	−	3.92123e11i	0.651426	−	1.12830i	0.982777	−	0.184795i	$-0.0591621\pi$
$368$		0			0
$369$	−5.40690e10	−	9.36502e10i	−0.151820	−	0.262960i
$370$		0			0
$371$	1.05471e11	+	3.40041e10i	0.289034	+	0.0931856i
$372$		0			0
$373$	2.02282e11	+	3.50363e11i	0.541087	+	0.937191i	0.998842	+	0.0481123i	$0.0153205\pi$
$373$	2.02282e11	+	3.50363e11i	0.541087	+	0.937191i	−0.457755	+	0.889079i	$0.651346\pi$
$374$		0			0
$375$	−1.18981e11	+	2.06082e11i	−0.310698	+	0.538144i
$376$		0			0
$377$	−1.69536e11			−0.432241
$378$		0			0
$379$	2.61857e11			0.651911			0.325955	−	0.945385i	$-0.394314\pi$
$379$	2.61857e11			0.651911			0.325955	+	0.945385i	$0.394314\pi$
$380$		0			0
$381$	−2.54163e10	+	4.40223e10i	−0.0617945	+	0.107031i
$382$		0			0
$383$	2.84985e11	+	4.93609e11i	0.676749	+	1.17216i	0.975954	+	0.217975i	$0.0699452\pi$
$383$	2.84985e11	+	4.93609e11i	0.676749	+	1.17216i	−0.299205	+	0.954189i	$0.596722\pi$
$384$		0			0
$385$	8.02297e8	+	3.73269e9i	0.00186107	+	0.00865862i
$386$		0			0
$387$	1.80533e10	+	3.12692e10i	0.0409125	+	0.0708626i
$388$		0			0
$389$	2.03433e11	−	3.52356e11i	0.450452	−	0.780205i	−0.547962	−	0.836503i	$-0.684596\pi$
$389$	2.03433e11	−	3.52356e11i	0.450452	−	0.780205i	0.998414	+	0.0562979i	$0.0179297\pi$
$390$		0			0
$391$	4.77840e10			0.103392
$392$		0			0
$393$	−1.32182e11			−0.279516
$394$		0			0
$395$	−2.85904e11	+	4.95201e11i	−0.590927	+	1.02352i
$396$		0			0
$397$	4.12773e9	+	7.14943e9i	0.00833976	+	0.0144449i	0.870165	−	0.492760i	$-0.164012\pi$
$397$	4.12773e9	+	7.14943e9i	0.00833976	+	0.0144449i	−0.861825	+	0.507205i	$0.830679\pi$
$398$		0			0
$399$	2.92682e10	+	1.36170e11i	0.0578120	+	0.268970i
$400$		0			0
$401$	5.91429e9	+	1.02439e10i	0.0114223	+	0.0197840i	0.871680	−	0.490075i	$-0.163031\pi$
$401$	5.91429e9	+	1.02439e10i	0.0114223	+	0.0197840i	−0.860258	+	0.509859i	$0.829697\pi$
$402$		0			0
$403$	−5.78329e10	+	1.00169e11i	−0.109220	+	0.189175i
$404$		0			0
$405$	−5.33360e10			−0.0985082
$406$		0			0
$407$	−4.82414e9			−0.00871456
$408$		0			0
$409$	−1.91350e11	+	3.31427e11i	−0.338122	+	0.585644i	−0.984079	−	0.177729i	$-0.943125\pi$
$409$	−1.91350e11	+	3.31427e11i	−0.338122	+	0.585644i	0.645958	+	0.763373i	$0.276458\pi$
$410$		0			0
$411$	−2.11029e11	−	3.65513e11i	−0.364799	−	0.631851i
$412$		0			0
$413$	−2.33326e11	−	7.52251e10i	−0.394628	−	0.127229i
$414$		0			0
$415$	3.14607e11	+	5.44914e11i	0.520657	+	0.901804i
$416$		0			0
$417$	−3.07290e11	+	5.32242e11i	−0.497664	+	0.861980i
$418$		0			0
$419$	−9.52983e11			−1.51050			−0.755252	−	0.655434i	$-0.772486\pi$
$419$	−9.52983e11			−1.51050			−0.755252	+	0.655434i	$0.772486\pi$
$420$		0			0
$421$	1.14473e12			1.77597			0.887983	−	0.459876i	$-0.152106\pi$
$421$	1.14473e12			1.77597			0.887983	+	0.459876i	$0.152106\pi$
$422$		0			0
$423$	−1.39827e11	+	2.42188e11i	−0.212354	+	0.367807i
$424$		0			0
$425$	2.06593e10	+	3.57830e10i	0.0307161	+	0.0532019i
$426$		0			0
$427$	4.11501e11	−	3.72218e11i	0.599026	−	0.541841i
$428$		0			0
$429$	9.88219e8	+	1.71165e9i	0.00140862	+	0.00243981i
$430$		0			0
$431$	2.91882e11	−	5.05554e11i	0.407436	−	0.705699i	−0.587166	−	0.809467i	$-0.699756\pi$
$431$	2.91882e11	−	5.05554e11i	0.407436	−	0.705699i	0.994602	+	0.103767i	$0.0330897\pi$
$432$		0			0
$433$	1.91734e11			0.262122			0.131061	−	0.991374i	$-0.458162\pi$
$433$	1.91734e11			0.262122			0.131061	+	0.991374i	$0.458162\pi$
$434$		0			0
$435$	−3.38249e11			−0.452934
$436$		0			0
$437$	6.54161e10	−	1.13304e11i	0.0858061	−	0.148621i
$438$		0			0
$439$	5.54363e11	+	9.60185e11i	0.712368	+	1.23386i	0.963966	+	0.266025i	$0.0857104\pi$
$439$	5.54363e11	+	9.60185e11i	0.712368	+	1.23386i	−0.251599	+	0.967832i	$0.580956\pi$
$440$		0			0
$441$	2.14902e11	+	1.54645e11i	0.270562	+	0.194698i
$442$		0			0
$443$	3.11302e11	+	5.39190e11i	0.384030	+	0.665159i	0.991634	−	0.129081i	$-0.0412027\pi$
$443$	3.11302e11	+	5.39190e11i	0.384030	+	0.665159i	−0.607604	+	0.794240i	$0.707869\pi$
$444$		0			0
$445$	−6.80518e10	+	1.17869e11i	−0.0822659	+	0.142489i
$446$		0			0
$447$	−5.63788e11			−0.667932
$448$		0			0
$449$	−7.73398e9			−0.00898037			−0.00449019	−	0.999990i	$-0.501429\pi$
$449$	−7.73398e9			−0.00898037			−0.00449019	+	0.999990i	$0.501429\pi$
$450$		0			0
$451$	−3.99747e9	+	6.92383e9i	−0.00454979	+	0.00788046i
$452$		0			0
$453$	−1.03011e11	−	1.78421e11i	−0.114933	−	0.199069i
$454$		0			0
$455$	−2.93625e11	+	2.65594e11i	−0.321175	+	0.290514i
$456$		0			0
$457$	3.94156e11	+	6.82699e11i	0.422713	+	0.732160i	0.996204	−	0.0870516i	$-0.0277445\pi$
$457$	3.94156e11	+	6.82699e11i	0.422713	+	0.732160i	−0.573491	+	0.819212i	$0.694411\pi$
$458$		0			0
$459$	−2.62697e10	+	4.55005e10i	−0.0276248	+	0.0478475i
$460$		0			0
$461$	1.48096e12			1.52718			0.763589	−	0.645703i	$-0.223435\pi$
$461$	1.48096e12			1.52718			0.763589	+	0.645703i	$0.223435\pi$
$462$		0			0
$463$	5.94972e11			0.601703			0.300851	−	0.953671i	$-0.402729\pi$
$463$	5.94972e11			0.601703			0.300851	+	0.953671i	$0.402729\pi$
$464$		0			0
$465$	−1.15385e11	+	1.99853e11i	−0.114449	+	0.198231i
$466$		0			0
$467$	2.02132e11	+	3.50103e11i	0.196657	+	0.340619i	0.947442	−	0.319926i	$-0.103658\pi$
$467$	2.02132e11	+	3.50103e11i	0.196657	+	0.340619i	−0.750786	+	0.660546i	$0.770325\pi$
$468$		0			0
$469$	5.29834e11	+	1.70820e11i	0.505664	+	0.163028i
$470$		0			0
$471$	−2.82608e11	−	4.89491e11i	−0.264600	−	0.458300i
$472$		0			0
$473$	1.33473e9	−	2.31182e9i	0.00122608	−	0.00212363i
$474$		0			0
$475$	1.13130e11			0.101967
$476$		0			0
$477$	1.14455e11			0.101228
$478$		0			0
$479$	5.81955e10	−	1.00798e11i	0.0505103	−	0.0874864i	−0.839665	−	0.543105i	$-0.817249\pi$
$479$	5.81955e10	−	1.00798e11i	0.0505103	−	0.0874864i	0.890175	+	0.455619i	$0.150582\pi$
$480$		0			0
$481$	−2.50134e11	−	4.33246e11i	−0.213069	−	0.369047i
$482$		0			0
$483$	−5.22617e10	−	2.43148e11i	−0.0436940	−	0.203286i
$484$		0			0
$485$	−2.66965e11	−	4.62397e11i	−0.219087	−	0.379470i
$486$		0			0
$487$	−1.78457e11	+	3.09096e11i	−0.143765	+	0.249008i	−0.928911	−	0.370302i	$-0.879254\pi$
$487$	−1.78457e11	+	3.09096e11i	−0.143765	+	0.249008i	0.785147	+	0.619310i	$0.212588\pi$
$488$		0			0
$489$	−6.68957e11			−0.529064
$490$		0			0
$491$	−2.40265e12			−1.86563			−0.932813	−	0.360362i	$-0.882653\pi$
$491$	−2.40265e12			−1.86563			−0.932813	+	0.360362i	$0.882653\pi$
$492$		0			0
$493$	−1.66599e11	+	2.88558e11i	−0.127017	+	0.219999i
$494$		0			0
$495$	1.97164e9	+	3.41498e9i	0.00147606	+	0.00255661i
$496$		0			0
$497$	−3.24633e11	−	1.51036e12i	−0.238665	−	1.11039i
$498$		0			0
$499$	−5.01445e10	−	8.68529e10i	−0.0362052	−	0.0627093i	0.847355	−	0.531027i	$-0.178194\pi$
$499$	−5.01445e10	−	8.68529e10i	−0.0362052	−	0.0627093i	−0.883560	+	0.468318i	$0.844860\pi$
$500$		0			0
$501$	1.61862e11	−	2.80353e11i	0.114782	−	0.198808i
$502$		0			0
$503$	−7.46246e11			−0.519788			−0.259894	−	0.965637i	$-0.583688\pi$
$503$	−7.46246e11			−0.519788			−0.259894	+	0.965637i	$0.583688\pi$
$504$		0			0
$505$	−1.18829e12			−0.813040
$506$		0			0
$507$	3.27003e11	−	5.66386e11i	0.219794	−	0.380694i
$508$		0			0
$509$	−1.08713e12	−	1.88297e12i	−0.717882	−	1.24341i	−0.961837	−	0.273622i	$-0.911778\pi$
$509$	−1.08713e12	−	1.88297e12i	−0.717882	−	1.24341i	0.243956	−	0.969786i	$-0.421555\pi$
$510$		0			0
$511$	6.95573e11	+	2.24255e11i	0.451283	+	0.145495i
$512$		0			0
$513$	7.19264e10	+	1.24580e11i	0.0458522	+	0.0794184i
$514$		0			0
$515$	−2.55880e11	+	4.43197e11i	−0.160289	+	0.277628i
$516$		0			0
$517$	2.06756e10			0.0127277
$518$		0			0
$519$	−9.56509e10			−0.0578677
$520$		0			0
$521$	3.64659e11	−	6.31608e11i	0.216829	−	0.375559i	−0.737008	−	0.675884i	$-0.763762\pi$
$521$	3.64659e11	−	6.31608e11i	0.216829	−	0.375559i	0.953837	+	0.300325i	$0.0970953\pi$
$522$		0			0
$523$	−7.33806e11	−	1.27099e12i	−0.428868	−	0.742821i	0.567905	−	0.823094i	$-0.307754\pi$
$523$	−7.33806e11	−	1.27099e12i	−0.428868	−	0.742821i	−0.996773	+	0.0802731i	$0.974421\pi$
$524$		0			0
$525$	1.59486e11	−	1.44261e11i	0.0916233	−	0.0828766i
$526$		0			0
$527$	1.13662e11	+	1.96868e11i	0.0641899	+	0.111180i
$528$		0			0
$529$	7.83768e11	−	1.35753e12i	0.435148	−	0.753699i
$530$		0			0
$531$	−2.53201e11			−0.138210
$532$		0			0
$533$	−8.29084e11			−0.444965
$534$		0			0
$535$	−8.00032e11	+	1.38570e12i	−0.422197	+	0.731267i
$536$		0			0
$537$	−5.19586e11	−	8.99950e11i	−0.269633	−	0.467019i
$538$		0			0
$539$	1.95740e9	−	1.94763e10i	0.000998918	−	0.00993935i
$540$		0			0
$541$	−5.25528e11	−	9.10241e11i	−0.263760	−	0.456845i	0.703478	−	0.710717i	$-0.251629\pi$
$541$	−5.25528e11	−	9.10241e11i	−0.263760	−	0.456845i	−0.967238	+	0.253872i	$0.918296\pi$
$542$		0			0
$543$	−2.10228e11	+	3.64126e11i	−0.103775	+	0.179743i
$544$		0			0
$545$	−1.45780e11			−0.0707807
$546$		0			0
$547$	4.30455e11			0.205582			0.102791	−	0.994703i	$-0.467223\pi$
$547$	4.30455e11			0.205582			0.102791	+	0.994703i	$0.467223\pi$
$548$		0			0
$549$	2.86543e11	−	4.96306e11i	0.134621	−	0.233171i
$550$		0			0
$551$	4.56147e11	+	7.90069e11i	0.210825	+	0.365160i
$552$		0			0
$553$	2.17417e12	−	1.96661e12i	0.988620	−	0.894243i
$554$		0			0
$555$	−4.99055e11	−	8.64388e11i	−0.223270	−	0.386714i
$556$		0			0
$557$	1.49355e12	−	2.58691e12i	0.657465	−	1.13876i	−0.323805	−	0.946124i	$-0.604962\pi$
$557$	1.49355e12	−	2.58691e12i	0.657465	−	1.13876i	0.981270	−	0.192639i	$-0.0617046\pi$
$558$		0			0
$559$	2.76826e11			0.119909
$560$		0			0
$561$	3.88439e9			0.00165573
$562$		0			0
$563$	−1.61966e12	+	2.80534e12i	−0.679418	+	1.17679i	0.295739	+	0.955269i	$0.404434\pi$
$563$	−1.61966e12	+	2.80534e12i	−0.679418	+	1.17679i	−0.975156	+	0.221517i	$0.928899\pi$
$564$		0			0
$565$	−9.22527e11	−	1.59786e12i	−0.380856	−	0.659662i
$566$		0			0
$567$	2.60260e11	+	8.39088e10i	0.105751	+	0.0340945i
$568$		0			0
$569$	−1.51421e11	−	2.62270e11i	−0.0605595	−	0.104892i	0.834156	−	0.551528i	$-0.185955\pi$
$569$	−1.51421e11	−	2.62270e11i	−0.0605595	−	0.104892i	−0.894716	+	0.446636i	$0.852622\pi$
$570$		0			0
$571$	1.44695e12	−	2.50619e12i	0.569628	−	0.986624i	−0.426975	−	0.904263i	$-0.640421\pi$
$571$	1.44695e12	−	2.50619e12i	0.569628	−	0.986624i	0.996603	−	0.0823606i	$-0.0262459\pi$
$572$		0			0
$573$	1.06470e12			0.412603
$574$		0			0
$575$	−2.02007e11			−0.0770658
$576$		0			0
$577$	1.94900e12	−	3.37577e12i	0.732017	−	1.26789i	−0.224003	−	0.974588i	$-0.571913\pi$
$577$	1.94900e12	−	3.37577e12i	0.732017	−	1.26789i	0.956020	−	0.293302i	$-0.0947541\pi$
$578$		0			0
$579$	−7.90393e11	−	1.36900e12i	−0.292274	−	0.506233i
$580$		0			0
$581$	−6.77899e11	−	3.15393e12i	−0.246816	−	1.14831i
$582$		0			0
$583$	−4.23098e9	−	7.32828e9i	−0.00151682	−	0.00262720i
$584$		0			0
$585$	−2.04461e11	+	3.54137e11i	−0.0721788	+	0.125017i
$586$		0			0
$587$	2.07278e12			0.720581			0.360290	−	0.932840i	$-0.382678\pi$
$587$	2.07278e12			0.720581			0.360290	+	0.932840i	$0.382678\pi$
$588$		0			0
$589$	6.22411e11			0.213088
$590$		0			0
$591$	−3.49190e11	+	6.04814e11i	−0.117738	+	0.203929i
$592$		0			0
$593$	−2.67435e11	−	4.63211e11i	−0.0888121	−	0.153827i	0.818197	−	0.574938i	$-0.194974\pi$
$593$	−2.67435e11	−	4.63211e11i	−0.0888121	−	0.153827i	−0.907009	+	0.421111i	$0.861640\pi$
$594$		0			0
$595$	1.63515e11	+	7.60755e11i	0.0534850	+	0.248839i
$596$		0			0
$597$	9.19963e11	+	1.59342e12i	0.296405	+	0.513389i
$598$		0			0
$599$	−5.32536e10	+	9.22379e10i	−0.0169016	+	0.0292744i	−0.874353	−	0.485291i	$-0.838714\pi$
$599$	−5.32536e10	+	9.22379e10i	−0.0169016	+	0.0292744i	0.857451	+	0.514566i	$0.172047\pi$
$600$		0			0
$601$	3.48662e12			1.09011			0.545053	−	0.838401i	$-0.316509\pi$
$601$	3.48662e12			1.09011			0.545053	+	0.838401i	$0.316509\pi$
$602$		0			0
$603$	5.74966e11			0.177098
$604$		0			0
$605$	−1.46063e12	+	2.52989e12i	−0.443243	+	0.767719i
$606$		0			0
$607$	−2.83542e12	−	4.91109e12i	−0.847751	−	1.46835i	−0.883211	−	0.468976i	$-0.844623\pi$
$607$	−2.83542e12	−	4.91109e12i	−0.847751	−	1.46835i	0.0354604	−	0.999371i	$-0.488710\pi$
$608$		0			0
$609$	1.65053e12	+	5.32137e11i	0.486235	+	0.156764i
$610$		0			0
$611$	1.07204e12	+	1.85683e12i	0.311191	+	0.538998i
$612$		0			0
$613$	−2.38114e12	+	4.12426e12i	−0.681104	+	1.17971i	0.293540	+	0.955947i	$0.405167\pi$
$613$	−2.38114e12	+	4.12426e12i	−0.681104	+	1.17971i	−0.974644	+	0.223760i	$0.928167\pi$
$614$		0			0
$615$	−1.65414e12			−0.466268
$616$		0			0
$617$	−4.84504e12			−1.34590			−0.672952	−	0.739686i	$-0.734974\pi$
$617$	−4.84504e12			−1.34590			−0.672952	+	0.739686i	$0.734974\pi$
$618$		0			0
$619$	6.80325e11	−	1.17836e12i	0.186255	−	0.322604i	−0.757744	−	0.652552i	$-0.773698\pi$
$619$	6.80325e11	−	1.17836e12i	0.186255	−	0.322604i	0.943999	+	0.329949i	$0.107032\pi$
$620$		0			0
$621$	−1.28433e11	−	2.22452e11i	−0.0346549	−	0.0600240i
$622$		0			0
$623$	5.17502e11	−	4.68099e11i	0.137631	−	0.124492i
$624$		0			0
$625$	1.41186e12	+	2.44542e12i	0.370112	+	0.641053i
$626$		0			0
$627$	5.31773e9	−	9.21057e9i	0.00137411	−	0.00238003i
$628$		0			0
$629$	−9.83204e11			−0.250447
$630$		0			0
$631$	−6.43598e12			−1.61615			−0.808077	−	0.589077i	$-0.799491\pi$
$631$	−6.43598e12			−1.61615			−0.808077	+	0.589077i	$0.799491\pi$
$632$		0			0
$633$	−9.48966e11	+	1.64366e12i	−0.234928	+	0.406906i
$634$		0			0
$635$	3.88783e11	+	6.73392e11i	0.0948911	+	0.164356i
$636$		0			0
$637$	1.85062e12	−	8.34069e11i	0.445338	−	0.200713i
$638$		0			0
$639$	−7.97784e11	−	1.38180e12i	−0.189292	−	0.327863i
$640$		0			0
$641$	3.21481e12	−	5.56821e12i	0.752131	−	1.30273i	−0.194657	−	0.980871i	$-0.562359\pi$
$641$	3.21481e12	−	5.56821e12i	0.752131	−	1.30273i	0.946788	−	0.321858i	$-0.104307\pi$
$642$		0			0
$643$	−8.64189e12			−1.99370			−0.996848	−	0.0793299i	$-0.974722\pi$
$643$	−8.64189e12			−1.99370			−0.996848	+	0.0793299i	$0.974722\pi$
$644$		0			0
$645$	5.52307e11			0.125650
$646$		0			0
$647$	−2.29186e12	+	3.96962e12i	−0.514185	+	0.890595i	0.485679	+	0.874137i	$0.338572\pi$
$647$	−2.29186e12	+	3.96962e12i	−0.514185	+	0.890595i	−0.999865	+	0.0164578i	$0.994761\pi$
$648$		0			0
$649$	9.35993e9	+	1.62119e10i	0.00207096	+	0.00358701i
$650$		0			0
$651$	8.77448e11	−	7.93683e11i	0.191473	−	0.173194i
$652$		0			0
$653$	7.28728e11	+	1.26219e12i	0.156840	+	0.271655i	0.933727	−	0.357985i	$-0.116536\pi$
$653$	7.28728e11	+	1.26219e12i	0.156840	+	0.271655i	−0.776888	+	0.629639i	$0.783203\pi$
$654$		0			0
$655$	−1.01097e12	+	1.75105e12i	−0.214611	+	0.371717i
$656$		0			0
$657$	7.54823e11			0.158052
$658$		0			0
$659$	2.90180e12			0.599353			0.299677	−	0.954041i	$-0.403121\pi$
$659$	2.90180e12			0.599353			0.299677	+	0.954041i	$0.403121\pi$
$660$		0			0
$661$	−4.40714e12	+	7.63338e12i	−0.897945	+	1.55529i	−0.0678284	+	0.997697i	$0.521607\pi$
$661$	−4.40714e12	+	7.63338e12i	−0.897945	+	1.55529i	−0.830117	+	0.557590i	$0.811726\pi$
$662$		0			0
$663$	2.01408e11	+	3.48849e11i	0.0404823	+	0.0701174i
$664$		0			0
$665$	2.02773e12	+	6.53749e11i	0.402081	+	0.129632i
$666$		0			0
$667$	−8.14503e11	−	1.41076e12i	−0.159341	−	0.275986i
$668$		0			0
$669$	−9.65384e11	+	1.67209e12i	−0.186330	+	0.322733i
$670$		0			0
$671$	−4.23698e10			−0.00806873
$672$		0			0
$673$	2.32198e12			0.436305			0.218153	−	0.975915i	$-0.429997\pi$
$673$	2.32198e12			0.436305			0.218153	+	0.975915i	$0.429997\pi$
$674$		0			0
$675$	1.11056e11	−	1.92354e11i	0.0205908	−	0.0356644i
$676$		0			0
$677$	−4.93841e10	−	8.55357e10i	−0.00903520	−	0.0156494i	0.861472	−	0.507804i	$-0.169543\pi$
$677$	−4.93841e10	−	8.55357e10i	−0.00903520	−	0.0156494i	−0.870508	+	0.492155i	$0.836209\pi$
$678$		0			0
$679$	5.75243e11	+	2.67632e12i	0.103857	+	0.483197i
$680$		0			0
$681$	1.59923e12	+	2.76995e12i	0.284937	+	0.493525i
$682$		0			0
$683$	5.13429e12	−	8.89285e12i	0.902790	−	1.56368i	0.0789345	−	0.996880i	$-0.474848\pi$
$683$	5.13429e12	−	8.89285e12i	0.902790	−	1.56368i	0.823856	−	0.566799i	$-0.191818\pi$
$684$		0			0
$685$	−6.45605e12			−1.12036
$686$		0			0
$687$	3.70213e12			0.634083
$688$		0			0
$689$	4.38757e11	−	7.59950e11i	0.0741717	−	0.128469i
$690$		0			0
$691$	−1.21427e12	−	2.10317e12i	−0.202611	−	0.350932i	0.746758	−	0.665096i	$-0.231609\pi$
$691$	−1.21427e12	−	2.10317e12i	−0.202611	−	0.350932i	−0.949369	+	0.314164i	$0.898276\pi$
$692$		0			0
$693$	−4.24840e9	−	1.97657e10i	−0.000699722	−	0.00325546i
$694$		0			0
$695$	4.70050e12	+	8.14150e12i	0.764209	+	1.32365i
$696$		0			0
$697$	−8.14721e11	+	1.41114e12i	−0.130756	+	0.226476i
$698$		0			0
$699$	1.19577e11			0.0189452
$700$		0			0
$701$	−1.81015e12			−0.283129			−0.141565	−	0.989929i	$-0.545213\pi$
$701$	−1.81015e12			−0.283129			−0.141565	+	0.989929i	$0.545213\pi$
$702$		0			0
$703$	−1.34600e12	+	2.33135e12i	−0.207849	+	0.360004i
$704$		0			0
$705$	2.13888e12	+	3.70465e12i	0.326089	+	0.564802i
$706$		0			0
$707$	5.79843e12	+	1.86944e12i	0.872817	+	0.281400i
$708$		0			0
$709$	4.32511e12	+	7.49132e12i	0.642820	+	1.11340i	0.984800	+	0.173690i	$0.0555691\pi$
$709$	4.32511e12	+	7.49132e12i	0.642820	+	1.11340i	−0.341980	+	0.939707i	$0.611098\pi$
$710$		0			0
$711$	1.51395e12	−	2.62223e12i	0.222176	−	0.384820i
$712$		0			0
$713$	−1.11139e12			−0.161051
$714$		0			0
$715$	3.02328e10			0.00432614
$716$		0			0
$717$	6.26815e11	−	1.08568e12i	0.0885734	−	0.153414i
$718$		0			0
$719$	4.85178e12	+	8.40353e12i	0.677050	+	1.17269i	0.975865	+	0.218375i	$0.0700754\pi$
$719$	4.85178e12	+	8.40353e12i	0.677050	+	1.17269i	−0.298815	+	0.954311i	$0.596591\pi$
$720$		0			0
$721$	1.94584e12	−	1.76009e12i	0.268163	−	0.242563i
$722$		0			0
$723$	9.65513e11	+	1.67232e12i	0.131412	+	0.227613i
$724$		0			0
$725$	7.04299e11	−	1.21988e12i	0.0946751	−	0.163982i
$726$		0			0
$727$	−1.64579e11			−0.0218509			−0.0109254	−	0.999940i	$-0.503478\pi$
$727$	−1.64579e11			−0.0218509			−0.0109254	+	0.999940i	$0.503478\pi$
$728$		0			0
$729$	2.82430e11			0.0370370
$730$		0			0
$731$	2.72030e11	−	4.71169e11i	0.0352361	−	0.0610307i
$732$		0			0
$733$	2.98979e12	+	5.17848e12i	0.382537	+	0.662573i	0.991424	−	0.130683i	$-0.0417171\pi$
$733$	2.98979e12	+	5.17848e12i	0.382537	+	0.662573i	−0.608887	+	0.793257i	$0.708384\pi$
$734$		0			0
$735$	3.69225e12	−	1.66409e12i	0.466658	−	0.210321i
$736$		0			0
$737$	−2.12544e10	−	3.68138e10i	−0.00265366	−	0.00459628i
$738$		0			0
$739$	3.31912e12	−	5.74889e12i	0.409377	−	0.709062i	−0.585443	−	0.810714i	$-0.699079\pi$
$739$	3.31912e12	−	5.74889e12i	0.409377	−	0.709062i	0.994820	+	0.101652i	$0.0324127\pi$
$740$		0			0
$741$	1.10291e12			0.134387
$742$		0			0
$743$	1.49130e13			1.79521			0.897607	−	0.440797i	$-0.145304\pi$
$743$	1.49130e13			1.79521			0.897607	+	0.440797i	$0.145304\pi$
$744$		0			0
$745$	−4.31202e12	+	7.46864e12i	−0.512835	+	0.888257i
$746$		0			0
$747$	−1.66593e12	−	2.88548e12i	−0.195756	−	0.339059i
$748$		0			0
$749$	6.08386e12	−	5.50307e12i	0.706336	−	0.638906i
$750$		0			0
$751$	2.61444e12	+	4.52835e12i	0.299916	+	0.519469i	0.976116	−	0.217248i	$-0.0697081\pi$
$751$	2.61444e12	+	4.52835e12i	0.299916	+	0.519469i	−0.676201	+	0.736718i	$0.736375\pi$
$752$		0			0
$753$	−6.23981e11	+	1.08077e12i	−0.0707284	+	0.122505i
$754$		0			0
$755$	−3.15145e12			−0.352979
$756$		0			0
$757$	−1.47182e13			−1.62901			−0.814505	−	0.580157i	$-0.802991\pi$
$757$	−1.47182e13			−1.62901			−0.814505	+	0.580157i	$0.802991\pi$
$758$		0			0
$759$	−9.49541e9	+	1.64465e10i	−0.00103855	+	0.00179882i
$760$		0			0
$761$	1.98433e12	+	3.43697e12i	0.214478	+	0.371488i	0.953111	−	0.302621i	$-0.0978615\pi$
$761$	1.98433e12	+	3.43697e12i	0.214478	+	0.371488i	−0.738633	+	0.674108i	$0.764528\pi$
$762$		0			0
$763$	7.11355e11	+	2.29344e11i	0.0759847	+	0.0244977i
$764$		0			0
$765$	4.01838e11	+	6.96004e11i	0.0424204	+	0.0734742i
$766$		0			0
$767$	−9.70635e11	+	1.68119e12i	−0.101269	+	0.175403i
$768$		0			0
$769$	−1.24543e13			−1.28425			−0.642127	−	0.766598i	$-0.721948\pi$
$769$	−1.24543e13			−1.28425			−0.642127	+	0.766598i	$0.721948\pi$
$770$		0			0
$771$	5.21710e12			0.531722
$772$		0			0
$773$	6.10048e11	−	1.05663e12i	0.0614549	−	0.106443i	−0.833661	−	0.552276i	$-0.813759\pi$
$773$	6.10048e11	−	1.05663e12i	0.0614549	−	0.106443i	0.895116	+	0.445833i	$0.147093\pi$
$774$		0			0
$775$	−4.80507e11	−	8.32263e11i	−0.0478456	−	0.0828711i
$776$		0			0
$777$	1.07534e12	+	5.00302e12i	0.105840	+	0.492422i
$778$		0			0
$779$	2.23070e12	+	3.86369e12i	0.217032	+	0.375910i
$780$		0			0
$781$	−5.89825e10	+	1.02161e11i	−0.00567274	+	0.00982548i
$782$		0			0
$783$	1.79113e12			0.170294
$784$		0			0
$785$	−8.64587e12			−0.812635
$786$		0			0
$787$	2.78611e12	−	4.82569e12i	0.258888	−	0.448408i	−0.707056	−	0.707157i	$-0.749977\pi$
$787$	2.78611e12	−	4.82569e12i	0.258888	−	0.448408i	0.965944	+	0.258750i	$0.0833105\pi$
$788$		0			0
$789$	3.52262e12	+	6.10135e12i	0.323608	+	0.560505i
$790$		0			0
$791$	1.98782e12	+	9.24832e12i	0.180544	+	0.839980i
$792$		0			0
$793$	−2.19690e12	−	3.80514e12i	−0.197279	−	0.341697i
$794$		0			0
$795$	8.75385e11	−	1.51621e12i	0.0777225	−	0.134619i
$796$		0			0
$797$	−7.88447e12			−0.692165			−0.346083	−	0.938204i	$-0.612488\pi$
$797$	−7.88447e12			−0.692165			−0.346083	+	0.938204i	$0.612488\pi$
$798$		0			0
$799$	4.21388e12			0.365781
$800$		0			0
$801$	3.60354e11	−	6.24152e11i	0.0309302	−	0.0535728i
$802$		0			0
$803$	−2.79031e10	−	4.83296e10i	−0.00236828	−	0.00410198i
$804$		0			0
$805$	−3.62076e12	−	1.16734e12i	−0.303891	−	0.0979756i
$806$		0			0
$807$	3.29746e11	+	5.71138e11i	0.0273684	+	0.0474034i
$808$		0			0
$809$	1.47948e11	−	2.56253e11i	0.0121434	−	0.0210330i	−0.859890	−	0.510480i	$-0.829468\pi$
$809$	1.47948e11	−	2.56253e11i	0.0121434	−	0.0210330i	0.872033	+	0.489447i	$0.162801\pi$
$810$		0			0
$811$	−1.84219e13			−1.49534			−0.747670	−	0.664070i	$-0.768828\pi$
$811$	−1.84219e13			−1.49534			−0.747670	+	0.664070i	$0.768828\pi$
$812$		0			0
$813$	2.66609e12			0.214026
$814$		0			0
$815$	−5.11638e12	+	8.86183e12i	−0.406213	+	0.703582i
$816$		0			0
$817$	−7.44816e11	−	1.29006e12i	−0.0584857	−	0.101300i
$818$		0			0
$819$	1.55483e12	−	1.40640e12i	0.120755	−	0.109227i
$820$		0			0
$821$	8.32591e12	+	1.44209e13i	0.639569	+	1.10777i	0.985527	+	0.169517i	$0.0542206\pi$
$821$	8.32591e12	+	1.44209e13i	0.639569	+	1.10777i	−0.345958	+	0.938250i	$0.612446\pi$
$822$		0			0
$823$	8.84128e12	−	1.53136e13i	0.671763	−	1.16353i	−0.305641	−	0.952147i	$-0.598871\pi$
$823$	8.84128e12	−	1.53136e13i	0.671763	−	1.16353i	0.977404	−	0.211381i	$-0.0677960\pi$
$824$		0			0
$825$	−1.64213e10			−0.00123414
$826$		0			0
$827$	8.42122e11			0.0626038			0.0313019	−	0.999510i	$-0.490035\pi$
$827$	8.42122e11			0.0626038			0.0313019	+	0.999510i	$0.490035\pi$
$828$		0			0
$829$	−7.74310e12	+	1.34114e13i	−0.569403	+	0.986234i	0.427223	+	0.904147i	$0.359492\pi$
$829$	−7.74310e12	+	1.34114e13i	−0.569403	+	0.986234i	−0.996625	+	0.0820877i	$0.973841\pi$
$830$		0			0
$831$	−1.44028e12	−	2.49464e12i	−0.104772	−	0.181470i
$832$		0			0
$833$	3.98935e11	−	3.96945e12i	0.0287078	−	0.285646i
$834$		0			0
$835$	−2.47593e12	−	4.28844e12i	−0.176258	−	0.305289i
$836$		0			0
$837$	6.10997e11	−	1.05828e12i	0.0430303	−	0.0745307i
$838$		0			0
$839$	−6.76138e12			−0.471093			−0.235546	−	0.971863i	$-0.575688\pi$
$839$	−6.76138e12			−0.471093			−0.235546	+	0.971863i	$0.575688\pi$
$840$		0			0
$841$	−3.14809e12			−0.217003
$842$		0			0
$843$	3.93707e12	−	6.81921e12i	0.268503	−	0.465061i
$844$		0			0
$845$	−5.00203e12	−	8.66378e12i	−0.337514	−	0.584591i
$846$		0			0
$847$	1.11074e13	−	1.00471e13i	0.741545	−	0.670754i
$848$		0			0
$849$	5.98691e12	+	1.03696e13i	0.395474	+	0.684982i
$850$		0			0
$851$	2.40345e12	−	4.16289e12i	0.157091	−	0.272090i
$852$		0			0
$853$	−7.38403e12			−0.477554			−0.238777	−	0.971074i	$-0.576747\pi$
$853$	−7.38403e12			−0.477554			−0.238777	+	0.971074i	$0.576747\pi$
$854$		0			0
$855$	2.20046e12			0.140821
$856$		0			0
$857$	−2.03170e12	+	3.51901e12i	−0.128661	+	0.222847i	−0.923158	−	0.384421i	$-0.874401\pi$
$857$	−2.03170e12	+	3.51901e12i	−0.128661	+	0.222847i	0.794497	+	0.607268i	$0.207735\pi$
$858$		0			0
$859$	−1.49386e13	−	2.58743e13i	−0.936137	−	1.62144i	−0.772594	−	0.634900i	$-0.781041\pi$
$859$	−1.49386e13	−	2.58743e13i	−0.936137	−	1.62144i	−0.163543	−	0.986536i	$-0.552292\pi$
$860$		0			0
$861$	8.07162e12	+	2.60232e12i	0.500549	+	0.161379i
$862$		0			0
$863$	1.49683e13	+	2.59259e13i	0.918595	+	1.59105i	0.801552	+	0.597926i	$0.204008\pi$
$863$	1.49683e13	+	2.59259e13i	0.918595	+	1.59105i	0.117043	+	0.993127i	$0.462658\pi$
$864$		0			0
$865$	−7.31567e11	+	1.26711e12i	−0.0444306	+	0.0769560i
$866$		0			0
$867$	−8.81394e12			−0.529766
$868$		0			0
$869$	−2.23861e11			−0.0133165
$870$		0			0
$871$	2.20411e12	−	3.81763e12i	0.129763	−	0.224756i
$872$		0			0
$873$	1.41366e12	+	2.44853e12i	0.0823721	+	0.142673i
$874$		0			0
$875$	−3.92167e12	−	1.82456e13i	−0.226170	−	1.05226i
$876$		0			0
$877$	3.77893e12	+	6.54529e12i	0.215710	+	0.373621i	0.953492	−	0.301418i	$-0.0974601\pi$
$877$	3.77893e12	+	6.54529e12i	0.215710	+	0.373621i	−0.737782	+	0.675039i	$0.764127\pi$
$878$		0			0
$879$	7.11410e12	−	1.23220e13i	0.401948	−	0.696194i
$880$		0			0
$881$	−8.96770e12			−0.501521			−0.250761	−	0.968049i	$-0.580681\pi$
$881$	−8.96770e12			−0.501521			−0.250761	+	0.968049i	$0.580681\pi$
$882$		0			0
$883$	−1.50740e13			−0.834460			−0.417230	−	0.908801i	$-0.636999\pi$
$883$	−1.50740e13			−0.834460			−0.417230	+	0.908801i	$0.636999\pi$
$884$		0			0
$885$	−1.93656e12	+	3.35422e12i	−0.106117	+	0.183800i
$886$		0			0
$887$	1.41624e12	+	2.45300e12i	0.0768210	+	0.133058i	0.901877	−	0.431994i	$-0.142190\pi$
$887$	1.41624e12	+	2.45300e12i	0.0768210	+	0.133058i	−0.825056	+	0.565051i	$0.808856\pi$
$888$		0			0
$889$	−8.37731e11	−	3.89755e12i	−0.0449828	−	0.209283i
$890$		0			0
$891$	−1.04404e10	−	1.80833e10i	−0.000554968	−	0.000961233i
$892$		0			0
$893$	5.76879e12	−	9.99184e12i	0.303566	−	0.525792i
$894$		0			0
$895$	−1.58958e13			−0.828093
$896$		0			0
$897$	−1.96937e12			−0.101569
$898$		0			0
$899$	3.87485e12	−	6.71144e12i	0.197850	−	0.342687i
$900$		0			0
$901$	−8.62312e11	−	1.49357e12i	−0.0435916	−	0.0755029i
$902$		0			0
$903$	−2.69506e12	−	8.68897e11i	−0.134888	−	0.0434884i
$904$		0			0
$905$	3.21578e12	+	5.56989e12i	0.159356	+	0.276012i
$906$		0			0
$907$	8.97176e12	−	1.55395e13i	0.440195	−	0.762440i	−0.557509	−	0.830171i	$-0.688243\pi$
$907$	8.97176e12	−	1.55395e13i	0.440195	−	0.762440i	0.997704	+	0.0677313i	$0.0215761\pi$
$908$		0			0
$909$	6.29235e12			0.305686
$910$		0			0
$911$	6.73803e12			0.324116			0.162058	−	0.986781i	$-0.448187\pi$
$911$	6.73803e12			0.324116			0.162058	+	0.986781i	$0.448187\pi$
$912$		0			0
$913$	−1.23167e11	+	2.13332e11i	−0.00586647	+	0.0101610i
$914$		0			0
$915$	−4.38313e12	−	7.59180e12i	−0.206723	−	0.358055i
$916$		0			0
$917$	7.68793e12	−	6.95401e12i	0.359044	−	0.324768i
$918$		0			0
$919$	1.69521e13	+	2.93620e13i	0.783979	+	1.35789i	0.929607	+	0.368552i	$0.120146\pi$
$919$	1.69521e13	+	2.93620e13i	0.783979	+	1.35789i	−0.145628	+	0.989339i	$0.546520\pi$
$920$		0			0
$921$	2.42950e12	−	4.20802e12i	0.111263	−	0.192712i
$922$		0			0
$923$	−1.22331e13			−0.554790
$924$		0			0
$925$	4.15651e12			0.186677
$926$		0			0
$927$	1.35496e12	−	2.34686e12i	0.0602652	−	0.104382i
$928$		0			0
$929$	−1.28324e13	−	2.22263e13i	−0.565243	−	0.979030i	−0.997027	−	0.0770530i	$-0.975449\pi$
$929$	−1.28324e13	−	2.22263e13i	−0.565243	−	0.979030i	0.431784	−	0.901977i	$-0.357884\pi$
$930$		0			0
$931$	−8.86612e12	−	6.38012e12i	−0.386777	−	0.278327i
$932$		0			0
$933$	4.40133e12	+	7.62333e12i	0.190159	+	0.329365i
$934$		0			0
$935$	2.97090e10	−	5.14575e10i	0.00127126	−	0.00220190i
$936$		0			0
$937$	−3.56885e13			−1.51252			−0.756259	−	0.654272i	$-0.772975\pi$
$937$	−3.56885e13			−1.51252			−0.756259	+	0.654272i	$0.772975\pi$
$938$		0			0
$939$	−1.54326e13			−0.647803
$940$		0			0
$941$	1.91463e13	−	3.31624e13i	0.796034	−	1.37877i	−0.126146	−	0.992012i	$-0.540261\pi$
$941$	1.91463e13	−	3.31624e13i	0.796034	−	1.37877i	0.922180	−	0.386760i	$-0.126406\pi$
$942$		0			0
$943$	−3.98318e12	−	6.89906e12i	−0.164031	−	0.284111i
$944$		0			0
$945$	3.10211e12	−	2.80597e12i	0.126536	−	0.114456i
$946$		0			0
$947$	−1.22498e13	−	2.12173e13i	−0.494942	−	0.857265i	0.505041	−	0.863095i	$-0.331477\pi$
$947$	−1.22498e13	−	2.12173e13i	−0.494942	−	0.857265i	−0.999983	+	0.00583048i	$0.998144\pi$
$948$		0			0
$949$	2.89358e12	−	5.01183e12i	0.115808	−	0.200585i
$950$		0			0
$951$	−4.56542e12			−0.180996
$952$		0			0
$953$	−1.50562e13			−0.591285			−0.295642	−	0.955299i	$-0.595534\pi$
$953$	−1.50562e13			−0.591285			−0.295642	+	0.955299i	$0.595534\pi$
$954$		0			0
$955$	8.14317e12	−	1.41044e13i	0.316795	−	0.548705i
$956$		0			0
$957$	−6.62115e10	−	1.14682e11i	−0.00255170	−	0.00441968i
$958$		0			0
$959$	3.15032e13	+	1.01567e13i	1.20274	+	0.387767i
$960$		0			0
$961$	1.05762e13	+	1.83185e13i	0.400013	+	0.692843i
$962$		0			0
$963$	4.23640e12	−	7.33767e12i	0.158737	−	0.274941i
$964$		0			0
$965$	−2.41807e13			−0.897626
$966$		0			0
$967$	−2.17413e13			−0.799588			−0.399794	−	0.916605i	$-0.630918\pi$
$967$	−2.17413e13			−0.799588			−0.399794	+	0.916605i	$0.630918\pi$
$968$		0			0
$969$	1.08380e12	−	1.87720e12i	0.0394905	−	0.0683995i
$970$		0			0
$971$	−1.51397e13	−	2.62227e13i	−0.546551	−	0.946654i	−0.998508	−	0.0546141i	$-0.982607\pi$
$971$	−1.51397e13	−	2.62227e13i	−0.546551	−	0.946654i	0.451957	−	0.892040i	$-0.350726\pi$
$972$		0			0
$973$	−1.01284e13	−	4.71224e13i	−0.362271	−	1.68547i
$974$		0			0
$975$	−8.51455e11	−	1.47476e12i	−0.0301745	−	0.0522638i
$976$		0			0
$977$	2.38603e13	−	4.13273e13i	0.837819	−	1.45115i	−0.0538947	−	0.998547i	$-0.517164\pi$
$977$	2.38603e13	−	4.13273e13i	0.837819	−	1.45115i	0.891714	−	0.452599i	$-0.149503\pi$
$978$		0			0
$979$	−5.32840e10			−0.00185385
$980$		0			0
$981$	7.71950e11			0.0266121
$982$		0			0
$983$	−6.82036e12	+	1.18132e13i	−0.232979	+	0.403531i	−0.958683	−	0.284475i	$-0.908181\pi$
$983$	−6.82036e12	+	1.18132e13i	−0.232979	+	0.403531i	0.725704	+	0.688007i	$0.241514\pi$
$984$		0			0
$985$	5.34141e12	+	9.25160e12i	0.180798	+	0.313151i
$986$		0			0
$987$	−4.60876e12	−	2.14423e13i	−0.154581	−	0.719190i
$988$		0			0
$989$	1.32995e12	+	2.30355e12i	0.0442032	+	0.0765622i
$990$		0			0
$991$	−1.79362e13	+	3.10664e13i	−0.590743	+	1.02320i	0.403389	+	0.915029i	$0.367832\pi$
$991$	−1.79362e13	+	3.10664e13i	−0.590743	+	1.02320i	−0.994133	+	0.108169i	$0.965501\pi$
$992$		0			0
$993$	3.19257e12			0.104200
$994$		0			0
$995$	2.81446e13			0.910315
$996$		0			0
$997$	−9.74162e12	+	1.68730e13i	−0.312250	+	0.540834i	−0.978849	−	0.204583i	$-0.934416\pi$
$997$	−9.74162e12	+	1.68730e13i	−0.312250	+	0.540834i	0.666599	+	0.745417i	$0.267749\pi$
$998$		0			0
$999$	2.64264e12	+	4.57719e12i	0.0839447	+	0.145396i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.10.q.a.25.6	✓	16
7.2	even	3	inner	168.10.q.a.121.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.10.q.a.25.6	✓	16	1.1	even	1	trivial
168.10.q.a.121.6	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)

\(f(q)\)	\(=\)	\(q+(-40.5000 + 70.1481i) q^{3} +(619.513 + 1073.03i) q^{5} +(-1334.90 - 6210.61i) q^{7} +(-3280.50 - 5681.99i) q^{9} +O(q^{10})\)	\(q+(-40.5000 + 70.1481i) q^{3} +(619.513 + 1073.03i) q^{5} +(-1334.90 - 6210.61i) q^{7} +(-3280.50 - 5681.99i) q^{9} +(-242.537 + 420.086i) q^{11} -50302.6 q^{13} -100361. q^{15} +(-49431.1 + 85617.3i) q^{17} +(135342. + 234420. i) q^{19} +(489725. + 157889. i) q^{21} +(-241669. - 418583. i) q^{23} +(208971. - 361948. i) q^{25} +531441. q^{27} +3.37032e6 q^{29} +(1.14970e6 - 1.99134e6i) q^{31} +(-19645.5 - 34027.0i) q^{33} +(5.83717e6 - 5.27993e6i) q^{35} +(4.97259e6 + 8.61278e6i) q^{37} +(2.03726e6 - 3.52863e6i) q^{39} +1.64819e7 q^{41} -5.50320e6 q^{43} +(4.06462e6 - 7.04013e6i) q^{45} +(-2.13119e7 - 3.69132e7i) q^{47} +(-3.67897e7 + 1.65810e7i) q^{49} +(-4.00392e6 - 6.93500e6i) q^{51} +(-8.72236e6 + 1.51076e7i) q^{53} -601018. q^{55} -2.19254e7 q^{57} +(1.92959e7 - 3.34215e7i) q^{59} +(4.36736e7 + 7.56449e7i) q^{61} +(-3.09095e7 + 2.79588e7i) q^{63} +(-3.11631e7 - 5.39761e7i) q^{65} +(-4.38170e7 + 7.58932e7i) q^{67} +3.91504e7 q^{69} +2.43190e8 q^{71} +(-5.75235e7 + 9.96336e7i) q^{73} +(1.69266e7 + 2.93178e7i) q^{75} +(2.93275e6 + 945530. i) q^{77} +(2.30749e8 + 3.99670e8i) q^{79} +(-2.15234e7 + 3.72796e7i) q^{81} +5.07829e8 q^{83} -1.22493e8 q^{85} +(-1.36498e8 + 2.36421e8i) q^{87} +(5.49237e7 + 9.51306e7i) q^{89} +(6.71488e7 + 3.12410e8i) q^{91} +(9.31256e7 + 1.61298e8i) q^{93} +(-1.67692e8 + 2.90452e8i) q^{95} -4.30927e8 q^{97} +3.18257e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) \) 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

Properties

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