LMFDB - Embedded newform 84.12.i.a.25.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(25,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("84.25");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.i (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:	$64.5408271670$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} + 198245134 x^{12} + 414863096508 x^{11} + \cdots + 37\!\cdots\!56 $ x^14 - 6x^13 + 198245134x^12 + 414863096508x^11 + 26949056203541719x^10 + 65398284799267941978x^9 + 2028387875108119977261582x^8 + 6445062169106156227737324792x^7 + 111668469386916870576107083796769x^6 + 325583881393416391390922284227512022x^5 + 3320063138939962847561145196113433362264x^4 + 9575836750537552288866777850271834960214456x^3 + 68507643214686757134474644335292207644539438816x^2 + 142358467144947535218510754697116911147938126144352*x + 378103215519089647001740821506658140294236360807649856
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{12}\cdot 7^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.1
Root			$5091.70 - 8819.09i$ of defining polynomial
Character	$\chi$	$=$	84.25
Dual form			84.12.i.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(121.500 - 210.444i) q^{3} +(-4575.70 - 7925.35i) q^{5} +(-7241.01 + 43873.6i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})$	\(q+(121.500 - 210.444i) q^{3} +(-4575.70 - 7925.35i) q^{5} +(-7241.01 + 43873.6i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-8822.08 + 15280.3i) q^{11} +177148. q^{13} -2.22379e6 q^{15} +(-2.65023e6 + 4.59033e6i) q^{17} +(2.46464e6 + 4.26889e6i) q^{19} +(8.35316e6 + 6.85447e6i) q^{21} +(1.65062e7 + 2.85896e7i) q^{23} +(-1.74601e7 + 3.02418e7i) q^{25} -1.43489e7 q^{27} +7.19461e7 q^{29} +(2.51367e7 - 4.35381e7i) q^{31} +(2.14377e6 + 3.71311e6i) q^{33} +(3.80847e8 - 1.43365e8i) q^{35} +(-2.53175e8 - 4.38512e8i) q^{37} +(2.15235e7 - 3.72798e7i) q^{39} +1.15088e8 q^{41} +9.01216e7 q^{43} +(-2.70191e8 + 4.67984e8i) q^{45} +(1.45935e9 + 2.52767e9i) q^{47} +(-1.87246e9 - 6.35379e8i) q^{49} +(6.44005e8 + 1.11545e9i) q^{51} +(4.76180e8 - 8.24767e8i) q^{53} +1.61469e8 q^{55} +1.19782e9 q^{57} +(1.54563e9 - 2.67711e9i) q^{59} +(-3.87030e9 - 6.70356e9i) q^{61} +(2.45739e9 - 9.25056e8i) q^{63} +(-8.10577e8 - 1.40396e9i) q^{65} +(6.27921e9 - 1.08759e10i) q^{67} +8.02201e9 q^{69} +1.32497e10 q^{71} +(9.55888e9 - 1.65565e10i) q^{73} +(4.24280e9 + 7.34875e9i) q^{75} +(-6.06521e8 - 4.97701e8i) q^{77} +(2.68746e10 + 4.65483e10i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} +1.79059e10 q^{83} +4.85066e10 q^{85} +(8.74146e9 - 1.51406e10i) q^{87} +(2.06112e10 + 3.56997e10i) q^{89} +(-1.28273e9 + 7.77213e9i) q^{91} +(-6.10823e9 - 1.05798e10i) q^{93} +(2.25550e10 - 3.90664e10i) q^{95} +9.70107e10 q^{97} +1.04187e9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 1701 q^{3} + 7218 q^{5} + 35001 q^{7} - 413343 q^{9}+O(q^{10}) $ 14 * q + 1701 * q^3 + 7218 * q^5 + 35001 * q^7 - 413343 * q^9	$ 14 q + 1701 q^{3} + 7218 q^{5} + 35001 q^{7} - 413343 q^{9} + 54450 q^{11} + 1534982 q^{13} + 3507948 q^{15} + 1478880 q^{17} - 22875935 q^{19} + 3394224 q^{21} + 62540568 q^{23} - 62136141 q^{25} - 200884698 q^{27} + 102097728 q^{29} + 188600405 q^{31} - 13231350 q^{33} - 253840734 q^{35} + 199685599 q^{37} + 186500313 q^{39} - 693868716 q^{41} - 620701754 q^{43} + 426215682 q^{45} + 2771987346 q^{47} - 5209147075 q^{49} - 359367840 q^{51} + 6487034184 q^{53} + 10046238656 q^{55} - 11117704410 q^{57} - 8183838888 q^{59} + 4069556330 q^{61} - 1241977617 q^{63} - 1520229906 q^{65} + 15766443531 q^{67} + 30394716048 q^{69} - 33183285444 q^{71} - 31685143839 q^{73} + 15099082263 q^{75} + 3261253500 q^{77} + 21999509987 q^{79} - 24407490807 q^{81} - 63053885988 q^{83} + 35204204624 q^{85} + 12404873952 q^{87} + 67041904680 q^{89} - 190876959523 q^{91} - 45829898415 q^{93} + 133488871470 q^{95} + 284083418100 q^{97} - 6430436100 q^{99}+O(q^{100}) $ 14 * q + 1701 * q^3 + 7218 * q^5 + 35001 * q^7 - 413343 * q^9 + 54450 * q^11 + 1534982 * q^13 + 3507948 * q^15 + 1478880 * q^17 - 22875935 * q^19 + 3394224 * q^21 + 62540568 * q^23 - 62136141 * q^25 - 200884698 * q^27 + 102097728 * q^29 + 188600405 * q^31 - 13231350 * q^33 - 253840734 * q^35 + 199685599 * q^37 + 186500313 * q^39 - 693868716 * q^41 - 620701754 * q^43 + 426215682 * q^45 + 2771987346 * q^47 - 5209147075 * q^49 - 359367840 * q^51 + 6487034184 * q^53 + 10046238656 * q^55 - 11117704410 * q^57 - 8183838888 * q^59 + 4069556330 * q^61 - 1241977617 * q^63 - 1520229906 * q^65 + 15766443531 * q^67 + 30394716048 * q^69 - 33183285444 * q^71 - 31685143839 * q^73 + 15099082263 * q^75 + 3261253500 * q^77 + 21999509987 * q^79 - 24407490807 * q^81 - 63053885988 * q^83 + 35204204624 * q^85 + 12404873952 * q^87 + 67041904680 * q^89 - 190876959523 * q^91 - 45829898415 * q^93 + 133488871470 * q^95 + 284083418100 * q^97 - 6430436100 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	121.500	−	210.444i	0.288675	−	0.500000i
$3$	121.500	−	210.444i	0.288675	−	0.500000i
$4$		0			0
$5$	−4575.70	−	7925.35i	−0.654822	−	1.13418i	−0.981939	−	0.189200i	$-0.939410\pi$
$5$	−4575.70	−	7925.35i	−0.654822	−	1.13418i	0.327117	−	0.944984i	$-0.393923\pi$
$6$		0			0
$7$	−7241.01	+	43873.6i	−0.162840	+	0.986653i
$7$	−7241.01	+	43873.6i	−0.162840	+	0.986653i
$8$		0			0
$9$	−29524.5	−	51137.9i	−0.166667	−	0.288675i
$10$		0			0
$11$	−8822.08	+	15280.3i	−0.0165162	+	0.0286070i	−0.874165	−	0.485628i	$-0.838591\pi$
$11$	−8822.08	+	15280.3i	−0.0165162	+	0.0286070i	0.857649	+	0.514235i	$0.171924\pi$
$12$		0			0
$13$	177148.			0.132327			0.0661634	−	0.997809i	$-0.478924\pi$
$13$	177148.			0.132327			0.0661634	+	0.997809i	$0.478924\pi$
$14$		0			0
$15$	−2.22379e6			−0.756123
$16$		0			0
$17$	−2.65023e6	+	4.59033e6i	−0.452704	+	0.784106i	−0.998553	−	0.0537776i	$-0.982874\pi$
$17$	−2.65023e6	+	4.59033e6i	−0.452704	+	0.784106i	0.545849	+	0.837883i	$0.316207\pi$
$18$		0			0
$19$	2.46464e6	+	4.26889e6i	0.228354	+	0.395521i	0.957321	−	0.289028i	$-0.0933322\pi$
$19$	2.46464e6	+	4.26889e6i	0.228354	+	0.395521i	−0.728966	+	0.684550i	$0.759999\pi$
$20$		0			0
$21$	8.35316e6	+	6.85447e6i	0.446319	+	0.366242i
$22$		0			0
$23$	1.65062e7	+	2.85896e7i	0.534741	+	0.926199i	0.999176	+	0.0405915i	$0.0129242\pi$
$23$	1.65062e7	+	2.85896e7i	0.534741	+	0.926199i	−0.464435	+	0.885607i	$0.653742\pi$
$24$		0			0
$25$	−1.74601e7	+	3.02418e7i	−0.357583	+	0.619351i
$26$		0			0
$27$	−1.43489e7			−0.192450
$28$		0			0
$29$	7.19461e7			0.651356			0.325678	−	0.945481i	$-0.394407\pi$
$29$	7.19461e7			0.651356			0.325678	+	0.945481i	$0.394407\pi$
$30$		0			0
$31$	2.51367e7	−	4.35381e7i	0.157696	−	0.273137i	−0.776342	−	0.630312i	$-0.782927\pi$
$31$	2.51367e7	−	4.35381e7i	0.157696	−	0.273137i	0.934037	+	0.357176i	$0.116260\pi$
$32$		0			0
$33$	2.14377e6	+	3.71311e6i	0.00953565	+	0.0165162i
$34$		0			0
$35$	3.80847e8	−	1.43365e8i	1.22568	−	0.461391i
$36$		0			0
$37$	−2.53175e8	−	4.38512e8i	−0.600222	−	1.03961i	−0.992787	−	0.119891i	$-0.961746\pi$
$37$	−2.53175e8	−	4.38512e8i	−0.600222	−	1.03961i	0.392565	−	0.919724i	$-0.371588\pi$
$38$		0			0
$39$	2.15235e7	−	3.72798e7i	0.0381995	−	0.0661634i
$40$		0			0
$41$	1.15088e8			0.155139			0.0775694	−	0.996987i	$-0.475284\pi$
$41$	1.15088e8			0.155139			0.0775694	+	0.996987i	$0.475284\pi$
$42$		0			0
$43$	9.01216e7			0.0934873			0.0467436	−	0.998907i	$-0.485116\pi$
$43$	9.01216e7			0.0934873			0.0467436	+	0.998907i	$0.485116\pi$
$44$		0			0
$45$	−2.70191e8	+	4.67984e8i	−0.218274	+	0.378061i
$46$		0			0
$47$	1.45935e9	+	2.52767e9i	0.928157	+	1.60761i	0.786404	+	0.617713i	$0.211941\pi$
$47$	1.45935e9	+	2.52767e9i	0.928157	+	1.60761i	0.141753	+	0.989902i	$0.454726\pi$
$48$		0			0
$49$	−1.87246e9	−	6.35379e8i	−0.946966	−	0.321332i
$50$		0			0
$51$	6.44005e8	+	1.11545e9i	0.261369	+	0.452704i
$52$		0			0
$53$	4.76180e8	−	8.24767e8i	0.156406	−	0.270903i	−0.777164	−	0.629298i	$-0.783342\pi$
$53$	4.76180e8	−	8.24767e8i	0.156406	−	0.270903i	0.933570	+	0.358395i	$0.116676\pi$
$54$		0			0
$55$	1.61469e8			0.0432608
$56$		0			0
$57$	1.19782e9			0.263681
$58$		0			0
$59$	1.54563e9	−	2.67711e9i	0.281462	−	0.487506i	−0.690283	−	0.723539i	$-0.742514\pi$
$59$	1.54563e9	−	2.67711e9i	0.281462	−	0.487506i	0.971745	+	0.236033i	$0.0758474\pi$
$60$		0			0
$61$	−3.87030e9	−	6.70356e9i	−0.586720	−	1.01623i	−0.994659	−	0.103219i	$-0.967086\pi$
$61$	−3.87030e9	−	6.70356e9i	−0.586720	−	1.01623i	0.407939	−	0.913009i	$-0.366248\pi$
$62$		0			0
$63$	2.45739e9	−	9.25056e8i	0.311962	−	0.117434i
$64$		0			0
$65$	−8.10577e8	−	1.40396e9i	−0.0866505	−	0.150083i
$66$		0			0
$67$	6.27921e9	−	1.08759e10i	0.568190	−	0.984135i	−0.428555	−	0.903516i	$-0.640977\pi$
$67$	6.27921e9	−	1.08759e10i	0.568190	−	0.984135i	0.996745	−	0.0806188i	$-0.0256896\pi$
$68$		0			0
$69$	8.02201e9			0.617466
$70$		0			0
$71$	1.32497e10			0.871537			0.435768	−	0.900059i	$-0.356477\pi$
$71$	1.32497e10			0.871537			0.435768	+	0.900059i	$0.356477\pi$
$72$		0			0
$73$	9.55888e9	−	1.65565e10i	0.539674	−	0.934743i	−0.459247	−	0.888308i	$-0.651881\pi$
$73$	9.55888e9	−	1.65565e10i	0.539674	−	0.934743i	0.998921	−	0.0464343i	$-0.0147858\pi$
$74$		0			0
$75$	4.24280e9	+	7.34875e9i	0.206450	+	0.357583i
$76$		0			0
$77$	−6.06521e8	−	4.97701e8i	−0.0255356	−	0.0209541i
$78$		0			0
$79$	2.68746e10	+	4.65483e10i	0.982639	+	1.70198i	0.651993	+	0.758225i	$0.273933\pi$
$79$	2.68746e10	+	4.65483e10i	0.982639	+	1.70198i	0.330645	+	0.943755i	$0.392734\pi$
$80$		0			0
$81$	−1.74339e9	+	3.01964e9i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	1.79059e10			0.498962			0.249481	−	0.968380i	$-0.419740\pi$
$83$	1.79059e10			0.498962			0.249481	+	0.968380i	$0.419740\pi$
$84$		0			0
$85$	4.85066e10			1.18576
$86$		0			0
$87$	8.74146e9	−	1.51406e10i	0.188030	−	0.325678i
$88$		0			0
$89$	2.06112e10	+	3.56997e10i	0.391254	+	0.677673i	0.992615	−	0.121305i	$-0.0387079\pi$
$89$	2.06112e10	+	3.56997e10i	0.391254	+	0.677673i	−0.601361	+	0.798978i	$0.705375\pi$
$90$		0			0
$91$	−1.28273e9	+	7.77213e9i	−0.0215481	+	0.130561i
$92$		0			0
$93$	−6.10823e9	−	1.05798e10i	−0.0910456	−	0.157696i
$94$		0			0
$95$	2.25550e10	−	3.90664e10i	0.299063	−	0.517992i
$96$		0			0
$97$	9.70107e10			1.14703			0.573515	−	0.819195i	$-0.305579\pi$
$97$	9.70107e10			1.14703			0.573515	+	0.819195i	$0.305579\pi$
$98$		0			0
$99$	1.04187e9			0.0110108
$100$		0			0
$101$	7.37798e10	−	1.27790e11i	0.698506	−	1.20985i	−0.270478	−	0.962726i	$-0.587182\pi$
$101$	7.37798e10	−	1.27790e11i	0.698506	−	1.20985i	0.968984	−	0.247122i	$-0.0794848\pi$
$102$		0			0
$103$	1.66921e10	+	2.89115e10i	0.141875	+	0.245734i	0.928203	−	0.372075i	$-0.121354\pi$
$103$	1.66921e10	+	2.89115e10i	0.141875	+	0.245734i	−0.786328	+	0.617809i	$0.788020\pi$
$104$		0			0
$105$	1.61025e10	−	9.75658e10i	0.123127	−	0.746031i
$106$		0			0
$107$	3.76372e10	+	6.51895e10i	0.259422	+	0.449332i	0.966087	−	0.258216i	$-0.0831347\pi$
$107$	3.76372e10	+	6.51895e10i	0.259422	+	0.449332i	−0.706665	+	0.707548i	$0.749801\pi$
$108$		0			0
$109$	1.05738e10	−	1.83144e10i	0.0658242	−	0.114011i	−0.831235	−	0.555921i	$-0.812366\pi$
$109$	1.05738e10	−	1.83144e10i	0.0658242	−	0.114011i	0.897059	+	0.441910i	$0.145699\pi$
$110$		0			0
$111$	−1.23043e11			−0.693077
$112$		0			0
$113$	2.99196e11			1.52765			0.763825	−	0.645424i	$-0.223319\pi$
$113$	2.99196e11			1.52765			0.763825	+	0.645424i	$0.223319\pi$
$114$		0			0
$115$	1.51055e11	−	2.61635e11i	0.700320	−	1.21299i
$116$		0			0
$117$	−5.23021e9	−	9.05898e9i	−0.0220545	−	0.0381995i
$118$		0			0
$119$	−1.82204e11	−	1.49514e11i	−0.699922	−	0.574345i
$120$		0			0
$121$	1.42500e11	+	2.46818e11i	0.499454	+	0.865080i
$122$		0			0
$123$	1.39832e10	−	2.42197e10i	0.0447847	−	0.0775694i
$124$		0			0
$125$	−1.27277e11			−0.373032
$126$		0			0
$127$	1.70035e11			0.456685			0.228342	−	0.973581i	$-0.426669\pi$
$127$	1.70035e11			0.456685			0.228342	+	0.973581i	$0.426669\pi$
$128$		0			0
$129$	1.09498e10	−	1.89656e10i	0.0269875	−	0.0467436i
$130$		0			0
$131$	2.03471e10	+	3.52423e10i	0.0460798	+	0.0798126i	0.888145	−	0.459563i	$-0.151994\pi$
$131$	2.03471e10	+	3.52423e10i	0.0460798	+	0.0798126i	−0.842066	+	0.539375i	$0.818660\pi$
$132$		0			0
$133$	−2.05138e11	+	7.72218e10i	−0.427427	+	0.160900i
$134$		0			0
$135$	6.56564e10	+	1.13720e11i	0.126020	+	0.218274i
$136$		0			0
$137$	−3.67925e11	+	6.37265e11i	−0.651323	+	1.12812i	0.331479	+	0.943462i	$0.392452\pi$
$137$	−3.67925e11	+	6.37265e11i	−0.651323	+	1.12812i	−0.982802	+	0.184662i	$0.940881\pi$
$138$		0			0
$139$	−7.92380e11			−1.29525			−0.647623	−	0.761961i	$-0.724237\pi$
$139$	−7.92380e11			−1.29525			−0.647623	+	0.761961i	$0.724237\pi$
$140$		0			0
$141$	7.09244e11			1.07174
$142$		0			0
$143$	−1.56281e9	+	2.70687e9i	−0.00218554	+	0.00378547i
$144$		0			0
$145$	−3.29204e11	−	5.70199e11i	−0.426522	−	0.738758i
$146$		0			0
$147$	−3.61216e11	+	3.16850e11i	−0.434032	+	0.380723i
$148$		0			0
$149$	4.80256e11	+	8.31827e11i	0.535733	+	0.927916i	0.999127	+	0.0417641i	$0.0132978\pi$
$149$	4.80256e11	+	8.31827e11i	0.535733	+	0.927916i	−0.463395	+	0.886152i	$0.653369\pi$
$150$		0			0
$151$	2.89691e10	−	5.01760e10i	0.0300305	−	0.0520143i	−0.850620	−	0.525782i	$-0.823773\pi$
$151$	2.89691e10	−	5.01760e10i	0.0300305	−	0.0520143i	0.880650	+	0.473767i	$0.157106\pi$
$152$		0			0
$153$	3.12987e11			0.301802
$154$		0			0
$155$	−4.60073e11			−0.413050
$156$		0			0
$157$	3.40172e11	−	5.89196e11i	0.284610	−	0.492960i	−0.687904	−	0.725802i	$-0.741469\pi$
$157$	3.40172e11	−	5.89196e11i	0.284610	−	0.492960i	0.972515	+	0.232842i	$0.0748024\pi$
$158$		0			0
$159$	−1.15712e11	−	2.00418e11i	−0.0903011	−	0.156406i
$160$		0			0
$161$	−1.37385e12	+	5.17169e11i	−1.00091	+	0.376782i
$162$		0			0
$163$	1.63049e9	+	2.82409e9i	0.00110990	+	0.00192241i	0.866580	−	0.499038i	$-0.166313\pi$
$163$	1.63049e9	+	2.82409e9i	0.00110990	+	0.00192241i	−0.865470	+	0.500961i	$0.832980\pi$
$164$		0			0
$165$	1.96185e10	−	3.39802e10i	0.0124883	−	0.0216304i
$166$		0			0
$167$	−1.14336e12			−0.681151			−0.340575	−	0.940217i	$-0.610622\pi$
$167$	−1.14336e12			−0.681151			−0.340575	+	0.940217i	$0.610622\pi$
$168$		0			0
$169$	−1.76078e12			−0.982490
$170$		0			0
$171$	1.45535e11	−	2.52074e11i	0.0761181	−	0.131840i
$172$		0			0
$173$	−2.75299e11	−	4.76831e11i	−0.135067	−	0.233944i	0.790556	−	0.612390i	$-0.209792\pi$
$173$	−2.75299e11	−	4.76831e11i	−0.135067	−	0.233944i	−0.925623	+	0.378446i	$0.876458\pi$
$174$		0			0
$175$	−1.20039e12	−	9.85018e11i	−0.552856	−	0.453665i
$176$		0			0
$177$	−3.75588e11	−	6.50538e11i	−0.162502	−	0.281462i
$178$		0			0
$179$	−1.58214e11	+	2.74035e11i	−0.0643508	+	0.111459i	−0.896406	−	0.443234i	$-0.853831\pi$
$179$	−1.58214e11	+	2.74035e11i	−0.0643508	+	0.111459i	0.832055	+	0.554693i	$0.187164\pi$
$180$		0			0
$181$	−1.83375e12			−0.701630			−0.350815	−	0.936445i	$-0.614095\pi$
$181$	−1.83375e12			−0.701630			−0.350815	+	0.936445i	$0.614095\pi$
$182$		0			0
$183$	−1.88097e12			−0.677486
$184$		0			0
$185$	−2.31691e12	+	4.01301e12i	−0.786077	+	1.36152i
$186$		0			0
$187$	−4.67610e10	−	8.09925e10i	−0.0149539	−	0.0259010i
$188$		0			0
$189$	1.03901e11	−	6.29538e11i	0.0313385	−	0.189881i
$190$		0			0
$191$	3.87779e11	+	6.71654e11i	0.110383	+	0.191189i	0.915925	−	0.401350i	$-0.131459\pi$
$191$	3.87779e11	+	6.71654e11i	0.110383	+	0.191189i	−0.805542	+	0.592539i	$0.798126\pi$
$192$		0			0
$193$	−6.18368e11	+	1.07105e12i	−0.166220	+	0.287901i	−0.937088	−	0.349094i	$-0.886489\pi$
$193$	−6.18368e11	+	1.07105e12i	−0.166220	+	0.287901i	0.770868	+	0.636995i	$0.219823\pi$
$194$		0			0
$195$	−3.93940e11			−0.100055
$196$		0			0
$197$	−4.30178e11			−0.103296			−0.0516481	−	0.998665i	$-0.516447\pi$
$197$	−4.30178e11			−0.103296			−0.0516481	+	0.998665i	$0.516447\pi$
$198$		0			0
$199$	−1.67173e12	+	2.89552e12i	−0.379730	+	0.657711i	−0.991023	−	0.133693i	$-0.957316\pi$
$199$	−1.67173e12	+	2.89552e12i	−0.379730	+	0.657711i	0.611293	+	0.791404i	$0.290650\pi$
$200$		0			0
$201$	−1.52585e12	−	2.64285e12i	−0.328045	−	0.568190i
$202$		0			0
$203$	−5.20963e11	+	3.15654e12i	−0.106067	+	0.642662i
$204$		0			0
$205$	−5.26611e11	−	9.12116e11i	−0.101588	−	0.175956i
$206$		0			0
$207$	9.74674e11	−	1.68818e12i	0.178247	−	0.308733i
$208$		0			0
$209$	−8.69731e10			−0.0150862
$210$		0			0
$211$	1.08640e13			1.78828			0.894140	−	0.447788i	$-0.147788\pi$
$211$	1.08640e13			1.78828			0.894140	+	0.447788i	$0.147788\pi$
$212$		0			0
$213$	1.60984e12	−	2.78833e12i	0.251591	−	0.435768i
$214$		0			0
$215$	−4.12370e11	−	7.14246e11i	−0.0612175	−	0.106032i
$216$		0			0
$217$	1.72816e12	+	1.41810e12i	0.243812	+	0.200068i
$218$		0			0
$219$	−2.32281e12	−	4.02322e12i	−0.311581	−	0.539674i
$220$		0			0
$221$	−4.69483e11	+	8.13168e11i	−0.0599048	+	0.103758i
$222$		0			0
$223$	1.19662e13			1.45305			0.726524	−	0.687141i	$-0.241134\pi$
$223$	1.19662e13			1.45305			0.726524	+	0.687141i	$0.241134\pi$
$224$		0			0
$225$	2.06200e12			0.238388
$226$		0			0
$227$	−3.69154e12	+	6.39393e12i	−0.406504	+	0.704086i	−0.994495	−	0.104782i	$-0.966586\pi$
$227$	−3.69154e12	+	6.39393e12i	−0.406504	+	0.704086i	0.587991	+	0.808867i	$0.299919\pi$
$228$		0			0
$229$	−5.14260e12	−	8.90724e12i	−0.539619	−	0.934648i	−0.998924	−	0.0463692i	$-0.985235\pi$
$229$	−5.14260e12	−	8.90724e12i	−0.539619	−	0.934648i	0.459305	−	0.888278i	$-0.348098\pi$
$230$		0			0
$231$	−1.78431e11	+	6.71680e10i	−0.0178486	+	0.00671888i
$232$		0			0
$233$	1.35422e12	+	2.34557e12i	0.129191	+	0.223765i	0.923363	−	0.383928i	$-0.125429\pi$
$233$	1.35422e12	+	2.34557e12i	0.129191	+	0.223765i	−0.794173	+	0.607692i	$0.792095\pi$
$234$		0			0
$235$	1.33551e13	−	2.31317e13i	1.21555	−	2.10540i
$236$		0			0
$237$	1.30611e13			1.13465
$238$		0			0
$239$	−4.90422e12			−0.406800			−0.203400	−	0.979096i	$-0.565199\pi$
$239$	−4.90422e12			−0.406800			−0.203400	+	0.979096i	$0.565199\pi$
$240$		0			0
$241$	−8.66407e12	+	1.50066e13i	−0.686481	+	1.18902i	0.286488	+	0.958084i	$0.407512\pi$
$241$	−8.66407e12	+	1.50066e13i	−0.686481	+	1.18902i	−0.972969	+	0.230936i	$0.925821\pi$
$242$		0			0
$243$	4.23644e11	+	7.33773e11i	0.0320750	+	0.0555556i
$244$		0			0
$245$	3.53223e12	+	1.77472e13i	0.255644	+	1.28445i
$246$		0			0
$247$	4.36607e11	+	7.56225e11i	0.0302174	+	0.0523381i
$248$		0			0
$249$	2.17557e12	−	3.76820e12i	0.144038	−	0.249481i
$250$		0			0
$251$	1.87669e13			1.18902			0.594508	−	0.804090i	$-0.297347\pi$
$251$	1.87669e13			1.18902			0.594508	+	0.804090i	$0.297347\pi$
$252$		0			0
$253$	−5.82475e11			−0.0353276
$254$		0			0
$255$	5.89356e12	−	1.02079e13i	0.342300	−	0.592880i
$256$		0			0
$257$	9.20234e12	+	1.59389e13i	0.511996	+	0.886803i	0.999903	+	0.0139073i	$0.00442696\pi$
$257$	9.20234e12	+	1.59389e13i	0.511996	+	0.886803i	−0.487908	+	0.872895i	$0.662240\pi$
$258$		0			0
$259$	2.10724e13	−	7.93244e12i	1.12348	−	0.422920i
$260$		0			0
$261$	−2.12417e12	−	3.67918e12i	−0.108559	−	0.188030i
$262$		0			0
$263$	−1.02304e13	+	1.77195e13i	−0.501343	+	0.868352i	0.498655	+	0.866800i	$0.333827\pi$
$263$	−1.02304e13	+	1.77195e13i	−0.501343	+	0.868352i	−0.999999	+	0.00155202i	$0.999506\pi$
$264$		0			0
$265$	−8.71543e12			−0.409672
$266$		0			0
$267$	1.00171e13			0.451782
$268$		0			0
$269$	−7.49665e12	+	1.29846e13i	−0.324511	+	0.562070i	−0.981413	−	0.191906i	$-0.938533\pi$
$269$	−7.49665e12	+	1.29846e13i	−0.324511	+	0.562070i	0.656902	+	0.753976i	$0.271866\pi$
$270$		0			0
$271$	−2.18982e13	−	3.79288e13i	−0.910075	−	1.57630i	−0.813956	−	0.580927i	$-0.802690\pi$
$271$	−2.18982e13	−	3.79288e13i	−0.910075	−	1.57630i	−0.0961200	−	0.995370i	$-0.530643\pi$
$272$		0			0
$273$	1.47975e12	+	1.21426e12i	0.0590599	+	0.0484636i
$274$		0			0
$275$	−3.08069e11	−	5.33590e11i	−0.0118118	−	0.0204587i
$276$		0			0
$277$	1.25675e13	−	2.17675e13i	0.463030	−	0.801992i	−0.536080	−	0.844167i	$-0.680095\pi$
$277$	1.25675e13	−	2.17675e13i	0.463030	−	0.801992i	0.999110	+	0.0421753i	$0.0134288\pi$
$278$		0			0
$279$	−2.96860e12			−0.105130
$280$		0			0
$281$	1.97470e13			0.672381			0.336190	−	0.941794i	$-0.390861\pi$
$281$	1.97470e13			0.672381			0.336190	+	0.941794i	$0.390861\pi$
$282$		0			0
$283$	2.54810e13	−	4.41344e13i	0.834432	−	1.44528i	−0.0600605	−	0.998195i	$-0.519129\pi$
$283$	2.54810e13	−	4.41344e13i	0.834432	−	1.44528i	0.894492	−	0.447083i	$-0.147537\pi$
$284$		0			0
$285$	−5.48086e12	−	9.49312e12i	−0.172664	−	0.299063i
$286$		0			0
$287$	−8.33357e11	+	5.04935e12i	−0.0252628	+	0.153068i
$288$		0			0
$289$	3.08854e12	+	5.34951e12i	0.0901187	+	0.156090i
$290$		0			0
$291$	1.17868e13	−	2.04153e13i	0.331119	−	0.573515i
$292$		0			0
$293$	1.84450e13			0.499007			0.249503	−	0.968374i	$-0.419733\pi$
$293$	1.84450e13			0.499007			0.249503	+	0.968374i	$0.419733\pi$
$294$		0			0
$295$	−2.82894e13			−0.737229
$296$		0			0
$297$	1.26587e11	−	2.19255e11i	0.00317855	−	0.00550541i
$298$		0			0
$299$	2.92404e12	+	5.06458e12i	0.0707606	+	0.122561i
$300$		0			0
$301$	−6.52572e11	+	3.95396e12i	−0.0152234	+	0.0922395i
$302$		0			0
$303$	−1.79285e13	−	3.10531e13i	−0.403283	−	0.698506i
$304$		0			0
$305$	−3.54187e13	+	6.13470e13i	−0.768394	+	1.33090i
$306$		0			0
$307$	−2.23604e13			−0.467970			−0.233985	−	0.972240i	$-0.575177\pi$
$307$	−2.23604e13			−0.467970			−0.233985	+	0.972240i	$0.575177\pi$
$308$		0			0
$309$	8.11234e12			0.163823
$310$		0			0
$311$	−2.58482e13	+	4.47704e13i	−0.503789	+	0.872588i	0.496202	+	0.868207i	$0.334728\pi$
$311$	−2.58482e13	+	4.47704e13i	−0.503789	+	0.872588i	−0.999990	+	0.00438050i	$0.998606\pi$
$312$		0			0
$313$	3.10552e13	+	5.37891e13i	0.584306	+	1.01205i	0.994962	+	0.100257i	$0.0319664\pi$
$313$	3.10552e13	+	5.37891e13i	0.584306	+	1.01205i	−0.410656	+	0.911790i	$0.634700\pi$
$314$		0			0
$315$	−1.85757e13	−	1.52429e13i	−0.337472	−	0.276924i
$316$		0			0
$317$	2.87446e13	+	4.97871e13i	0.504348	+	0.873556i	0.999987	+	0.00502776i	$0.00160039\pi$
$317$	2.87446e13	+	4.97871e13i	0.504348	+	0.873556i	−0.495640	+	0.868528i	$0.665066\pi$
$318$		0			0
$319$	−6.34715e11	+	1.09936e12i	−0.0107580	+	0.0186333i
$320$		0			0
$321$	1.82917e13			0.299554
$322$		0			0
$323$	−2.61275e13			−0.413508
$324$		0			0
$325$	−3.09302e12	+	5.35727e12i	−0.0473178	+	0.0819568i
$326$		0			0
$327$	−2.56944e12	−	4.45039e12i	−0.0380036	−	0.0658242i
$328$		0			0
$329$	−1.21465e14	+	4.57241e13i	−1.73730	+	0.653985i
$330$		0			0
$331$	6.50205e13	+	1.12619e14i	0.899489	+	1.55796i	0.828148	+	0.560509i	$0.189395\pi$
$331$	6.50205e13	+	1.12619e14i	0.899489	+	1.55796i	0.0713413	+	0.997452i	$0.477272\pi$
$332$		0			0
$333$	−1.49497e13	+	2.58937e13i	−0.200074	+	0.346538i
$334$		0			0
$335$	−1.14927e14			−1.48825
$336$		0			0
$337$	−4.15742e13			−0.521026			−0.260513	−	0.965470i	$-0.583892\pi$
$337$	−4.15742e13			−0.521026			−0.260513	+	0.965470i	$0.583892\pi$
$338$		0			0
$339$	3.63523e13	−	6.29640e13i	0.440994	−	0.763825i
$340$		0			0
$341$	4.43516e11	+	7.68193e11i	0.00520908	+	0.00902238i
$342$		0			0
$343$	4.14349e13	−	7.75509e13i	0.471247	−	0.882001i
$344$		0			0
$345$	−3.67063e13	−	6.35772e13i	−0.404330	−	0.700320i
$346$		0			0
$347$	−1.52788e13	+	2.64636e13i	−0.163033	+	0.282382i	−0.935955	−	0.352119i	$-0.885461\pi$
$347$	−1.52788e13	+	2.64636e13i	−0.163033	+	0.282382i	0.772922	+	0.634501i	$0.218794\pi$
$348$		0			0
$349$	−1.27725e14			−1.32049			−0.660247	−	0.751049i	$-0.729548\pi$
$349$	−1.27725e14			−1.32049			−0.660247	+	0.751049i	$0.729548\pi$
$350$		0			0
$351$	−2.54188e12			−0.0254663
$352$		0			0
$353$	1.58615e13	−	2.74729e13i	0.154022	−	0.266774i	−0.778680	−	0.627421i	$-0.784111\pi$
$353$	1.58615e13	−	2.74729e13i	0.154022	−	0.266774i	0.932702	+	0.360647i	$0.117444\pi$
$354$		0			0
$355$	−6.06268e13	−	1.05009e14i	−0.570701	−	0.988483i
$356$		0			0
$357$	−5.36021e13	+	2.01779e13i	−0.489222	+	0.184162i
$358$		0			0
$359$	−6.01401e13	−	1.04166e14i	−0.532285	−	0.921945i	−0.999289	−	0.0376900i	$-0.988000\pi$
$359$	−6.01401e13	−	1.04166e14i	−0.532285	−	0.921945i	0.467004	−	0.884255i	$-0.345333\pi$
$360$		0			0
$361$	4.60962e13	−	7.98409e13i	0.395709	−	0.685387i
$362$		0			0
$363$	6.92551e13			0.576720
$364$		0			0
$365$	−1.74955e14			−1.41356
$366$		0			0
$367$	5.41797e13	−	9.38419e13i	0.424789	−	0.735756i	−0.571612	−	0.820524i	$-0.693682\pi$
$367$	5.41797e13	−	9.38419e13i	0.424789	−	0.735756i	0.996401	+	0.0847685i	$0.0270151\pi$
$368$		0			0
$369$	−3.39793e12	−	5.88538e12i	−0.0258565	−	0.0447847i
$370$		0			0
$371$	3.27375e13	+	2.68639e13i	0.241818	+	0.198432i
$372$		0			0
$373$	1.00268e14	+	1.73670e14i	0.719060	+	1.24545i	0.961372	+	0.275251i	$0.0887609\pi$
$373$	1.00268e14	+	1.73670e14i	0.719060	+	1.24545i	−0.242312	+	0.970198i	$0.577906\pi$
$374$		0			0
$375$	−1.54642e13	+	2.67848e13i	−0.107685	+	0.186516i
$376$		0			0
$377$	1.27451e13			0.0861919
$378$		0			0
$379$	−1.58861e14			−1.04352			−0.521762	−	0.853091i	$-0.674725\pi$
$379$	−1.58861e14			−1.04352			−0.521762	+	0.853091i	$0.674725\pi$
$380$		0			0
$381$	2.06592e13	−	3.57828e13i	0.131834	−	0.228342i
$382$		0			0
$383$	1.48731e14	+	2.57609e14i	0.922163	+	1.59723i	0.796061	+	0.605217i	$0.206914\pi$
$383$	1.48731e14	+	2.57609e14i	0.922163	+	1.59723i	0.126103	+	0.992017i	$0.459753\pi$
$384$		0			0
$385$	−1.16920e12	+	7.08423e12i	−0.00704457	+	0.0426833i
$386$		0			0
$387$	−2.66080e12	−	4.60863e12i	−0.0155812	−	0.0269875i
$388$		0			0
$389$	−1.26529e14	+	2.19155e14i	−0.720224	+	1.24746i	0.240686	+	0.970603i	$0.422628\pi$
$389$	−1.26529e14	+	2.19155e14i	−0.720224	+	1.24746i	−0.960910	+	0.276861i	$0.910706\pi$
$390$		0			0
$391$	−1.74981e14			−0.968317
$392$		0			0
$393$	9.88870e12			0.0532084
$394$		0			0
$395$	2.45941e14	−	4.25982e14i	1.28691	−	2.22899i
$396$		0			0
$397$	−5.84457e13	−	1.01231e14i	−0.297443	−	0.515187i	0.678107	−	0.734963i	$-0.262801\pi$
$397$	−5.84457e13	−	1.01231e14i	−0.297443	−	0.515187i	−0.975550	+	0.219776i	$0.929467\pi$
$398$		0			0
$399$	−8.67341e12	+	5.25526e13i	−0.0429377	+	0.260162i
$400$		0			0
$401$	1.48856e14	+	2.57826e14i	0.716923	+	1.24175i	0.962213	+	0.272297i	$0.0877833\pi$
$401$	1.48856e14	+	2.57826e14i	0.716923	+	1.24175i	−0.245291	+	0.969450i	$0.578883\pi$
$402$		0			0
$403$	4.45292e12	−	7.71269e12i	0.0208674	−	0.0361433i
$404$		0			0
$405$	3.19090e13			0.145516
$406$		0			0
$407$	8.93413e12			0.0396536
$408$		0			0
$409$	−6.96695e13	+	1.20671e14i	−0.300999	+	0.521345i	−0.976362	−	0.216140i	$-0.930653\pi$
$409$	−6.96695e13	+	1.20671e14i	−0.300999	+	0.521345i	0.675364	+	0.737485i	$0.263987\pi$
$410$		0			0
$411$	8.94058e13	+	1.54855e14i	0.376041	+	0.651323i
$412$		0			0
$413$	1.06263e14	+	8.71974e13i	0.435166	+	0.357090i
$414$		0			0
$415$	−8.19323e13	−	1.41911e14i	−0.326731	−	0.565915i
$416$		0			0
$417$	−9.62741e13	+	1.66752e14i	−0.373905	+	0.647623i
$418$		0			0
$419$	3.76705e14			1.42503			0.712516	−	0.701656i	$-0.247556\pi$
$419$	3.76705e14			1.42503			0.712516	+	0.701656i	$0.247556\pi$
$420$		0			0
$421$	1.19372e14			0.439896			0.219948	−	0.975512i	$-0.429411\pi$
$421$	1.19372e14			0.439896			0.219948	+	0.975512i	$0.429411\pi$
$422$		0			0
$423$	8.61732e13	−	1.49256e14i	0.309386	−	0.535872i
$424$		0			0
$425$	−9.25464e13	−	1.60295e14i	−0.323758	−	0.560765i
$426$		0			0
$427$	3.22134e14	−	1.21264e14i	1.09821	−	0.413406i
$428$		0			0
$429$	3.79764e11	+	6.57770e11i	0.00126182	+	0.00218554i
$430$		0			0
$431$	2.80251e14	−	4.85409e14i	0.907659	−	1.57211i	0.0903505	−	0.995910i	$-0.471201\pi$
$431$	2.80251e14	−	4.85409e14i	0.907659	−	1.57211i	0.817308	−	0.576201i	$-0.195465\pi$
$432$		0			0
$433$	2.03277e14			0.641807			0.320903	−	0.947112i	$-0.396013\pi$
$433$	2.03277e14			0.641807			0.320903	+	0.947112i	$0.396013\pi$
$434$		0			0
$435$	−1.59993e14			−0.492505
$436$		0			0
$437$	−8.13637e13	+	1.40926e14i	−0.244221	+	0.423003i
$438$		0			0
$439$	7.77803e13	+	1.34719e14i	0.227675	+	0.394344i	0.957119	−	0.289697i	$-0.0935544\pi$
$439$	7.77803e13	+	1.34719e14i	0.227675	+	0.394344i	−0.729444	+	0.684041i	$0.760221\pi$
$440$		0			0
$441$	2.27915e13	+	1.14513e14i	0.0650671	+	0.326921i
$442$		0			0
$443$	−2.02312e14	−	3.50414e14i	−0.563378	−	0.975800i	−0.997199	−	0.0748005i	$-0.976168\pi$
$443$	−2.02312e14	−	3.50414e14i	−0.563378	−	0.975800i	0.433820	−	0.901000i	$-0.357165\pi$
$444$		0			0
$445$	1.88622e14	−	3.26703e14i	0.512404	−	0.887509i
$446$		0			0
$447$	2.33404e14			0.618611
$448$		0			0
$449$	8.13852e13			0.210470			0.105235	−	0.994447i	$-0.466440\pi$
$449$	8.13852e13			0.210470			0.105235	+	0.994447i	$0.466440\pi$
$450$		0			0
$451$	−1.01532e12	+	1.75858e12i	−0.00256231	+	0.00443805i
$452$		0			0
$453$	−7.03950e12	−	1.21928e13i	−0.0173381	−	0.0300305i
$454$		0			0
$455$	6.74662e13	−	2.53969e13i	0.162190	−	0.0610544i
$456$		0			0
$457$	4.02498e14	+	6.97147e14i	0.944549	+	1.63601i	0.756651	+	0.653819i	$0.226834\pi$
$457$	4.02498e14	+	6.97147e14i	0.944549	+	1.63601i	0.187898	+	0.982189i	$0.439832\pi$
$458$		0			0
$459$	3.80279e13	−	6.58662e13i	0.0871229	−	0.150901i
$460$		0			0
$461$	4.72909e14			1.05785			0.528923	−	0.848670i	$-0.322596\pi$
$461$	4.72909e14			1.05785			0.528923	+	0.848670i	$0.322596\pi$
$462$		0			0
$463$	−2.47830e14			−0.541325			−0.270663	−	0.962674i	$-0.587243\pi$
$463$	−2.47830e14			−0.541325			−0.270663	+	0.962674i	$0.587243\pi$
$464$		0			0
$465$	−5.58989e13	+	9.68197e13i	−0.119237	+	0.206525i
$466$		0			0
$467$	−2.56507e14	−	4.44284e14i	−0.534388	−	0.925587i	−0.999193	−	0.0401740i	$-0.987209\pi$
$467$	−2.56507e14	−	4.44284e14i	−0.534388	−	0.925587i	0.464805	−	0.885413i	$-0.346125\pi$
$468$		0			0
$469$	4.31698e14	+	3.54245e14i	0.878475	+	0.720863i
$470$		0			0
$471$	−8.26619e13	−	1.43175e14i	−0.164320	−	0.284610i
$472$		0			0
$473$	−7.95060e11	+	1.37708e12i	−0.00154406	+	0.00267439i
$474$		0			0
$475$	−1.72132e14			−0.326622
$476$		0			0
$477$	−5.62359e13			−0.104271
$478$		0			0
$479$	3.97046e14	−	6.87703e14i	0.719441	−	1.24611i	−0.241781	−	0.970331i	$-0.577731\pi$
$479$	3.97046e14	−	6.87703e14i	0.719441	−	1.24611i	0.961222	−	0.275777i	$-0.0889352\pi$
$480$		0			0
$481$	−4.48495e13	−	7.76816e13i	−0.0794255	−	0.137569i
$482$		0			0
$483$	−5.80875e13	+	3.51954e14i	−0.100548	+	0.609224i
$484$		0			0
$485$	−4.43892e14	−	7.68844e14i	−0.751100	−	1.30094i
$486$		0			0
$487$	3.42846e13	−	5.93827e13i	0.0567140	−	0.0982315i	−0.836274	−	0.548311i	$-0.815271\pi$
$487$	3.42846e13	−	5.93827e13i	0.0567140	−	0.0982315i	0.892989	+	0.450079i	$0.148604\pi$
$488$		0			0
$489$	7.92417e11			0.00128161
$490$		0			0
$491$	−9.27121e14			−1.46618			−0.733092	−	0.680130i	$-0.761923\pi$
$491$	−9.27121e14			−1.46618			−0.733092	+	0.680130i	$0.761923\pi$
$492$		0			0
$493$	−1.90674e14	+	3.30256e14i	−0.294871	+	0.510732i
$494$		0			0
$495$	−4.76729e12	−	8.25719e12i	−0.00721013	−	0.0124883i
$496$		0			0
$497$	−9.59414e13	+	5.81313e14i	−0.141921	+	0.859904i
$498$		0			0
$499$	3.86878e14	+	6.70092e14i	0.559785	+	0.969575i	0.997514	+	0.0704686i	$0.0224495\pi$
$499$	3.86878e14	+	6.70092e14i	0.559785	+	0.969575i	−0.437729	+	0.899107i	$0.644217\pi$
$500$		0			0
$501$	−1.38919e14	+	2.40614e14i	−0.196631	+	0.340575i
$502$		0			0
$503$	8.04685e14			1.11430			0.557150	−	0.830412i	$-0.311895\pi$
$503$	8.04685e14			1.11430			0.557150	+	0.830412i	$0.311895\pi$
$504$		0			0
$505$	−1.35038e15			−1.82959
$506$		0			0
$507$	−2.13935e14	+	3.70546e14i	−0.283620	+	0.491245i
$508$		0			0
$509$	−4.10768e14	−	7.11470e14i	−0.532903	−	0.923015i	−0.999262	−	0.0384197i	$-0.987768\pi$
$509$	−4.10768e14	−	7.11470e14i	−0.532903	−	0.923015i	0.466358	−	0.884596i	$-0.345566\pi$
$510$		0			0
$511$	6.57176e14	+	5.39269e14i	0.834386	+	0.684684i
$512$		0			0
$513$	−3.53650e13	−	6.12539e13i	−0.0439468	−	0.0761181i
$514$		0			0
$515$	1.52756e14	−	2.64581e14i	0.185805	−	0.321824i
$516$		0			0
$517$	−5.14980e13			−0.0613186
$518$		0			0
$519$	−1.33795e14			−0.155962
$520$		0			0
$521$	3.66754e14	−	6.35236e14i	0.418569	−	0.724983i	−0.577227	−	0.816584i	$-0.695865\pi$
$521$	3.66754e14	−	6.35236e14i	0.418569	−	0.724983i	0.995796	+	0.0916010i	$0.0291984\pi$
$522$		0			0
$523$	1.32086e14	+	2.28780e14i	0.147604	+	0.255657i	0.930341	−	0.366695i	$-0.119511\pi$
$523$	1.32086e14	+	2.28780e14i	0.147604	+	0.255657i	−0.782738	+	0.622352i	$0.786177\pi$
$524$		0			0
$525$	−3.53138e14	+	1.32935e14i	−0.386428	+	0.145466i
$526$		0			0
$527$	1.33236e14	+	2.30772e14i	0.142779	+	0.247300i
$528$		0			0
$529$	−6.85034e13	+	1.18651e14i	−0.0718962	+	0.124528i
$530$		0			0
$531$	−1.82536e14			−0.187641
$532$		0			0
$533$	2.03877e13			0.0205290
$534$		0			0
$535$	3.44433e14	−	5.96576e14i	0.339750	−	0.588464i
$536$		0			0
$537$	3.84461e13	+	6.65905e13i	0.0371530	+	0.0643508i
$538$		0			0
$539$	2.62278e13	−	2.30064e13i	0.0248327	−	0.0217826i
$540$		0			0
$541$	1.30594e14	+	2.26195e14i	0.121154	+	0.209844i	0.920223	−	0.391395i	$-0.128007\pi$
$541$	1.30594e14	+	2.26195e14i	0.121154	+	0.209844i	−0.799069	+	0.601239i	$0.794674\pi$
$542$		0			0
$543$	−2.22801e14	+	3.85902e14i	−0.202543	+	0.350815i
$544$		0			0
$545$	−1.93530e14			−0.172412
$546$		0			0
$547$	−1.99988e15			−1.74612			−0.873060	−	0.487612i	$-0.837868\pi$
$547$	−1.99988e15			−1.74612			−0.873060	+	0.487612i	$0.837868\pi$
$548$		0			0
$549$	−2.28537e14	+	3.95838e14i	−0.195573	+	0.338743i
$550$		0			0
$551$	1.77322e14	+	3.07130e14i	0.148740	+	0.257625i
$552$		0			0
$553$	−2.23684e15	+	8.42031e14i	−1.83928	+	0.692373i
$554$		0			0
$555$	5.63009e14	+	9.75161e14i	0.453841	+	0.786077i
$556$		0			0
$557$	1.07117e15	−	1.85533e15i	0.846557	−	1.46628i	−0.0377045	−	0.999289i	$-0.512005\pi$
$557$	1.07117e15	−	1.85533e15i	0.846557	−	1.46628i	0.884262	−	0.466991i	$-0.154662\pi$
$558$		0			0
$559$	1.59649e13			0.0123709
$560$		0			0
$561$	−2.27259e13			−0.0172673
$562$		0			0
$563$	1.48815e14	−	2.57756e14i	0.110880	−	0.192049i	−0.805246	−	0.592941i	$-0.797967\pi$
$563$	1.48815e14	−	2.57756e14i	0.110880	−	0.192049i	0.916125	+	0.400892i	$0.131300\pi$
$564$		0			0
$565$	−1.36903e15	−	2.37123e15i	−1.00034	−	1.73264i
$566$		0			0
$567$	−1.19859e14	−	9.83542e13i	−0.0858940	−	0.0704833i
$568$		0			0
$569$	9.10757e14	+	1.57748e15i	0.640155	+	1.10878i	0.985398	+	0.170268i	$0.0544633\pi$
$569$	9.10757e14	+	1.57748e15i	0.640155	+	1.10878i	−0.345243	+	0.938513i	$0.612203\pi$
$570$		0			0
$571$	−3.75274e14	+	6.49994e14i	−0.258732	+	0.448137i	−0.965903	−	0.258906i	$-0.916638\pi$
$571$	−3.75274e14	+	6.49994e14i	−0.258732	+	0.448137i	0.707170	+	0.707043i	$0.249971\pi$
$572$		0			0
$573$	1.88461e14			0.127459
$574$		0			0
$575$	−1.15280e15			−0.764857
$576$		0			0
$577$	−2.05467e14	+	3.55879e14i	−0.133744	+	0.231651i	−0.925117	−	0.379682i	$-0.876033\pi$
$577$	−2.05467e14	+	3.55879e14i	−0.133744	+	0.231651i	0.791373	+	0.611334i	$0.209367\pi$
$578$		0			0
$579$	1.50263e14	+	2.60264e14i	0.0959669	+	0.166220i
$580$		0			0
$581$	−1.29657e14	+	7.85599e14i	−0.0812508	+	0.492302i
$582$		0			0
$583$	8.40179e12	+	1.45523e13i	0.00516648	+	0.00894861i
$584$		0			0
$585$	−4.78638e13	+	8.29025e13i	−0.0288835	+	0.0500277i
$586$		0			0
$587$	−2.46218e15			−1.45818			−0.729090	−	0.684418i	$-0.760056\pi$
$587$	−2.46218e15			−1.45818			−0.729090	+	0.684418i	$0.760056\pi$
$588$		0			0
$589$	2.47812e14			0.144042
$590$		0			0
$591$	−5.22667e13	+	9.05285e13i	−0.0298190	+	0.0516481i
$592$		0			0
$593$	−1.44386e15	−	2.50084e15i	−0.808581	−	1.40050i	−0.913846	−	0.406060i	$-0.866902\pi$
$593$	−1.44386e15	−	2.50084e15i	−0.808581	−	1.40050i	0.105265	−	0.994444i	$-0.466431\pi$
$594$		0			0
$595$	−3.51237e14	+	2.12816e15i	−0.193089	+	1.16993i
$596$		0			0
$597$	4.06231e14	+	7.03612e14i	0.219237	+	0.379730i
$598$		0			0
$599$	−1.15324e15	+	1.99748e15i	−0.611046	+	1.05836i	0.380019	+	0.924979i	$0.375917\pi$
$599$	−1.15324e15	+	1.99748e15i	−0.611046	+	1.05836i	−0.991065	+	0.133383i	$0.957416\pi$
$600$		0			0
$601$	9.00075e14			0.468241			0.234120	−	0.972208i	$-0.424779\pi$
$601$	9.00075e14			0.468241			0.234120	+	0.972208i	$0.424779\pi$
$602$		0			0
$603$	−7.41563e14			−0.378794
$604$		0			0
$605$	1.30408e15	−	2.25873e15i	0.654107	−	1.13295i
$606$		0			0
$607$	−1.01914e14	−	1.76519e14i	−0.0501989	−	0.0869470i	0.839834	−	0.542843i	$-0.182652\pi$
$607$	−1.01914e14	−	1.76519e14i	−0.0501989	−	0.0869470i	−0.890033	+	0.455896i	$0.849319\pi$
$608$		0			0
$609$	6.00978e14	+	4.93153e14i	0.290712	+	0.238554i
$610$		0			0
$611$	2.58521e14	+	4.47772e14i	0.122820	+	0.212731i
$612$		0			0
$613$	1.37650e14	−	2.38417e14i	0.0642309	−	0.111251i	−0.832122	−	0.554593i	$-0.812874\pi$
$613$	1.37650e14	−	2.38417e14i	0.0642309	−	0.111251i	0.896353	+	0.443342i	$0.146207\pi$
$614$		0			0
$615$	−2.55933e14			−0.117304
$616$		0			0
$617$	4.59941e14			0.207078			0.103539	−	0.994625i	$-0.466983\pi$
$617$	4.59941e14			0.207078			0.103539	+	0.994625i	$0.466983\pi$
$618$		0			0
$619$	1.69884e15	−	2.94248e15i	0.751371	−	1.30141i	−0.195788	−	0.980646i	$-0.562726\pi$
$619$	1.69884e15	−	2.94248e15i	0.751371	−	1.30141i	0.947158	−	0.320766i	$-0.103940\pi$
$620$		0			0
$621$	−2.36846e14	−	4.10229e14i	−0.102911	−	0.178247i
$622$		0			0
$623$	−1.71552e15	+	6.45788e14i	−0.732339	+	0.275680i
$624$		0			0
$625$	1.43493e15	+	2.48537e15i	0.601852	+	1.04244i
$626$		0			0
$627$	−1.05672e13	+	1.83030e13i	−0.00435502	+	0.00754311i
$628$		0			0
$629$	2.68389e15			1.08689
$630$		0			0
$631$	1.21421e15			0.483207			0.241604	−	0.970375i	$-0.422327\pi$
$631$	1.21421e15			0.483207			0.241604	+	0.970375i	$0.422327\pi$
$632$		0			0
$633$	1.31997e15	−	2.28626e15i	0.516232	−	0.894140i
$634$		0			0
$635$	−7.78028e14	−	1.34758e15i	−0.299047	−	0.517965i
$636$		0			0
$637$	−3.31703e14	−	1.12556e14i	−0.125309	−	0.0425209i
$638$		0			0
$639$	−3.91191e14	−	6.77563e14i	−0.145256	−	0.251591i
$640$		0			0
$641$	1.42844e15	−	2.47412e15i	0.521365	−	0.903030i	−0.478327	−	0.878182i	$-0.658756\pi$
$641$	1.42844e15	−	2.47412e15i	0.521365	−	0.903030i	0.999691	−	0.0248480i	$-0.00791017\pi$
$642$		0			0
$643$	1.68754e15			0.605473			0.302737	−	0.953074i	$-0.402100\pi$
$643$	1.68754e15			0.605473			0.302737	+	0.953074i	$0.402100\pi$
$644$		0			0
$645$	−2.00412e14			−0.0706879
$646$		0			0
$647$	1.49548e15	−	2.59025e15i	0.518571	−	0.898191i	−0.481196	−	0.876613i	$-0.659798\pi$
$647$	1.49548e15	−	2.59025e15i	0.518571	−	0.898191i	0.999767	−	0.0215781i	$-0.00686905\pi$
$648$		0			0
$649$	2.72713e13	+	4.72354e13i	0.00929738	+	0.0161035i
$650$		0			0
$651$	5.08402e14	−	1.91382e14i	0.170417	−	0.0641513i
$652$		0			0
$653$	2.76374e15	+	4.78694e15i	0.910908	+	1.57774i	0.812784	+	0.582565i	$0.197951\pi$
$653$	2.76374e15	+	4.78694e15i	0.910908	+	1.57774i	0.0981236	+	0.995174i	$0.468716\pi$
$654$		0			0
$655$	1.86205e14	−	3.22516e14i	0.0603481	−	0.104526i
$656$		0			0
$657$	−1.12889e15			−0.359783
$658$		0			0
$659$	−3.22725e15			−1.01149			−0.505747	−	0.862682i	$-0.668783\pi$
$659$	−3.22725e15			−1.01149			−0.505747	+	0.862682i	$0.668783\pi$
$660$		0			0
$661$	2.64269e15	−	4.57728e15i	0.814589	−	1.41091i	−0.0950338	−	0.995474i	$-0.530296\pi$
$661$	2.64269e15	−	4.57728e15i	0.814589	−	1.41091i	0.909623	−	0.415435i	$-0.136371\pi$
$662$		0			0
$663$	1.14084e14	+	1.97600e14i	0.0345861	+	0.0599048i
$664$		0			0
$665$	1.55066e15	+	1.27245e15i	0.462379	+	0.379421i
$666$		0			0
$667$	1.18756e15	+	2.05691e15i	0.348307	+	0.603285i
$668$		0			0
$669$	1.45390e15	−	2.51822e15i	0.419459	−	0.726524i
$670$		0			0
$671$	1.36576e14			0.0387616
$672$		0			0
$673$	−9.98701e14			−0.278838			−0.139419	−	0.990233i	$-0.544524\pi$
$673$	−9.98701e14			−0.278838			−0.139419	+	0.990233i	$0.544524\pi$
$674$		0			0
$675$	2.50533e14	−	4.33936e14i	0.0688168	−	0.119194i
$676$		0			0
$677$	6.74651e14	+	1.16853e15i	0.182323	+	0.315793i	0.942671	−	0.333723i	$-0.108305\pi$
$677$	6.74651e14	+	1.16853e15i	0.182323	+	0.315793i	−0.760348	+	0.649516i	$0.774972\pi$
$678$		0			0
$679$	−7.02456e14	+	4.25621e15i	−0.186782	+	1.13172i
$680$		0			0
$681$	8.97043e14	+	1.55372e15i	0.234695	+	0.406504i
$682$		0			0
$683$	3.77576e14	−	6.53981e14i	0.0972055	−	0.168365i	−0.813321	−	0.581815i	$-0.802343\pi$
$683$	3.77576e14	−	6.53981e14i	0.0972055	−	0.168365i	0.910527	+	0.413450i	$0.135676\pi$
$684$		0			0
$685$	6.73407e15			1.70600
$686$		0			0
$687$	−2.49930e15			−0.623098
$688$		0			0
$689$	8.43543e13	−	1.46106e14i	0.0206967	−	0.0358478i
$690$		0			0
$691$	2.63507e15	+	4.56408e15i	0.636302	+	1.10211i	0.986238	+	0.165333i	$0.0528700\pi$
$691$	2.63507e15	+	4.56408e15i	0.636302	+	1.10211i	−0.349936	+	0.936774i	$0.613797\pi$
$692$		0			0
$693$	−7.54419e12	+	4.57106e13i	−0.00179300	+	0.0108639i
$694$		0			0
$695$	3.62570e15	+	6.27989e15i	0.848155	+	1.46905i
$696$		0			0
$697$	−3.05010e14	+	5.28294e14i	−0.0702319	+	0.121645i
$698$		0			0
$699$	6.58149e14			0.149176
$700$		0			0
$701$	−3.91411e15			−0.873341			−0.436670	−	0.899622i	$-0.643842\pi$
$701$	−3.91411e15			−0.873341			−0.436670	+	0.899622i	$0.643842\pi$
$702$		0			0
$703$	1.24797e15	−	2.16155e15i	0.274127	−	0.474801i
$704$		0			0
$705$	−3.24529e15	−	5.62101e15i	−0.701801	−	1.21555i
$706$		0			0
$707$	5.07239e15	+	4.16232e15i	1.07995	+	0.886194i
$708$		0			0
$709$	−4.37175e15	−	7.57210e15i	−0.916434	−	1.58731i	−0.804788	−	0.593562i	$-0.797721\pi$
$709$	−4.37175e15	−	7.57210e15i	−0.916434	−	1.58731i	−0.111646	−	0.993748i	$-0.535612\pi$
$710$		0			0
$711$	1.58692e15	−	2.74863e15i	0.327546	−	0.567327i
$712$		0			0
$713$	1.65965e15			0.337305
$714$		0			0
$715$	2.86039e13			0.00572456
$716$		0			0
$717$	−5.95862e14	+	1.03206e15i	−0.117433	+	0.203400i
$718$		0			0
$719$	1.00799e15	+	1.74588e15i	0.195634	+	0.338849i	0.947108	−	0.320914i	$-0.103990\pi$
$719$	1.00799e15	+	1.74588e15i	0.195634	+	0.338849i	−0.751474	+	0.659763i	$0.770657\pi$
$720$		0			0
$721$	−1.38932e15	+	5.22992e14i	−0.265557	+	0.0999658i
$722$		0			0
$723$	2.10537e15	+	3.64661e15i	0.396340	+	0.686481i
$724$		0			0
$725$	−1.25619e15	+	2.17578e15i	−0.232914	+	0.403418i
$726$		0			0
$727$	9.53644e14			0.174159			0.0870797	−	0.996201i	$-0.472247\pi$
$727$	9.53644e14			0.174159			0.0870797	+	0.996201i	$0.472247\pi$
$728$		0			0
$729$	2.05891e14			0.0370370
$730$		0			0
$731$	−2.38843e14	+	4.13688e14i	−0.0423220	+	0.0733039i
$732$		0			0
$733$	−4.77782e15	−	8.27543e15i	−0.833985	−	1.44450i	−0.894854	−	0.446359i	$-0.852721\pi$
$733$	−4.77782e15	−	8.27543e15i	−0.833985	−	1.44450i	0.0608693	−	0.998146i	$-0.480613\pi$
$734$		0			0
$735$	4.16397e15	+	1.41295e15i	0.716023	+	0.242967i
$736$		0			0
$737$	1.10791e14	+	1.91896e14i	0.0187687	+	0.0325084i
$738$		0			0
$739$	3.85057e14	−	6.66938e14i	0.0642658	−	0.111312i	−0.832102	−	0.554622i	$-0.812863\pi$
$739$	3.85057e14	−	6.66938e14i	0.0642658	−	0.111312i	0.896368	+	0.443311i	$0.146196\pi$
$740$		0			0
$741$	2.12191e14			0.0348921
$742$		0			0
$743$	−8.89205e15			−1.44067			−0.720333	−	0.693628i	$-0.756011\pi$
$743$	−8.89205e15			−1.44067			−0.720333	+	0.693628i	$0.756011\pi$
$744$		0			0
$745$	4.39502e15	−	7.61239e15i	0.701618	−	1.21524i
$746$		0			0
$747$	−5.28664e14	−	9.15673e14i	−0.0831604	−	0.144038i
$748$		0			0
$749$	−3.13263e15	+	1.17924e15i	−0.485578	+	0.182790i
$750$		0			0
$751$	−4.94769e15	−	8.56965e15i	−0.755759	−	1.30901i	−0.944996	−	0.327081i	$-0.893935\pi$
$751$	−4.94769e15	−	8.56965e15i	−0.755759	−	1.30901i	0.189238	−	0.981931i	$-0.439398\pi$
$752$		0			0
$753$	2.28018e15	−	3.94939e15i	0.343239	−	0.594508i
$754$		0			0
$755$	−5.30217e14			−0.0786584
$756$		0			0
$757$	−1.22777e16			−1.79510			−0.897551	−	0.440910i	$-0.854656\pi$
$757$	−1.22777e16			−1.79510			−0.897551	+	0.440910i	$0.854656\pi$
$758$		0			0
$759$	−7.07708e13	+	1.22579e14i	−0.0101982	+	0.0176638i
$760$		0			0
$761$	−9.77089e14	−	1.69237e15i	−0.138777	−	0.240369i	0.788257	−	0.615346i	$-0.210984\pi$
$761$	−9.77089e14	−	1.69237e15i	−0.138777	−	0.240369i	−0.927034	+	0.374977i	$0.877651\pi$
$762$		0			0
$763$	7.26953e14	+	5.96526e14i	0.101770	+	0.0835111i
$764$		0			0
$765$	−1.43213e15	−	2.48053e15i	−0.197627	−	0.342300i
$766$		0			0
$767$	2.73805e14	−	4.74245e14i	0.0372450	−	0.0645102i
$768$		0			0
$769$	6.27304e15			0.841168			0.420584	−	0.907254i	$-0.361825\pi$
$769$	6.27304e15			0.841168			0.420584	+	0.907254i	$0.361825\pi$
$770$		0			0
$771$	4.47234e15			0.591202
$772$		0			0
$773$	−6.28229e15	+	1.08812e16i	−0.818711	+	1.41805i	0.0879216	+	0.996127i	$0.471978\pi$
$773$	−6.28229e15	+	1.08812e16i	−0.818711	+	1.41805i	−0.906632	+	0.421921i	$0.861356\pi$
$774$		0			0
$775$	8.77779e14	+	1.52036e15i	0.112778	+	0.195338i
$776$		0			0
$777$	8.90958e14	−	5.39835e15i	0.112860	−	0.683826i
$778$		0			0
$779$	2.83652e14	+	4.91300e14i	0.0354266	+	0.0613607i
$780$		0			0
$781$	−1.16890e14	+	2.02459e14i	−0.0143945	+	0.0249320i
$782$		0			0
$783$	−1.03235e15			−0.125354
$784$		0			0
$785$	−6.22611e15			−0.745476
$786$		0			0
$787$	−4.75934e15	+	8.24341e15i	−0.561934	+	0.973298i	0.435394	+	0.900240i	$0.356609\pi$
$787$	−4.75934e15	+	8.24341e15i	−0.561934	+	0.973298i	−0.997328	+	0.0730582i	$0.976724\pi$
$788$		0			0
$789$	2.48598e15	+	4.30585e15i	0.289451	+	0.501343i
$790$		0			0
$791$	−2.16648e15	+	1.31268e16i	−0.248762	+	1.50726i
$792$		0			0
$793$	−6.85616e14	−	1.18752e15i	−0.0776388	−	0.134474i
$794$		0			0
$795$	−1.05892e15	+	1.83411e15i	−0.118262	+	0.204836i
$796$		0			0
$797$	−4.27134e15			−0.470482			−0.235241	−	0.971937i	$-0.575588\pi$
$797$	−4.27134e15			−0.470482			−0.235241	+	0.971937i	$0.575588\pi$
$798$		0			0
$799$	−1.54704e16			−1.68072
$800$		0			0
$801$	1.21707e15	−	2.10803e15i	0.130418	−	0.225891i
$802$		0			0
$803$	1.68658e14	+	2.92125e14i	0.0178268	+	0.0308769i
$804$		0			0
$805$	1.03851e16	+	8.52182e15i	1.08276	+	0.888495i
$806$		0			0
$807$	1.82169e15	+	3.15525e15i	0.187357	+	0.324511i
$808$		0			0
$809$	−8.23533e15	+	1.42640e16i	−0.835534	+	1.44719i	0.0580609	+	0.998313i	$0.481508\pi$
$809$	−8.23533e15	+	1.42640e16i	−0.835534	+	1.44719i	−0.893595	+	0.448874i	$0.851825\pi$
$810$		0			0
$811$	2.15420e15			0.215611			0.107805	−	0.994172i	$-0.465618\pi$
$811$	2.15420e15			0.215611			0.107805	+	0.994172i	$0.465618\pi$
$812$		0			0
$813$	−1.06425e16			−1.05086
$814$		0			0
$815$	1.49213e13	−	2.58444e13i	0.00145358	−	0.00251767i
$816$		0			0
$817$	2.22118e14	+	3.84719e14i	0.0213482	+	0.0369762i
$818$		0			0
$819$	4.35322e14	−	1.63872e14i	0.0412809	−	0.0155397i
$820$		0			0
$821$	5.67350e15	+	9.82680e15i	0.530840	+	0.919442i	0.999352	+	0.0359853i	$0.0114569\pi$
$821$	5.67350e15	+	9.82680e15i	0.530840	+	0.919442i	−0.468512	+	0.883457i	$0.655210\pi$
$822$		0			0
$823$	−7.79193e13	+	1.34960e14i	−0.00719359	+	0.0124597i	−0.869600	−	0.493757i	$-0.835623\pi$
$823$	−7.79193e13	+	1.34960e14i	−0.00719359	+	0.0124597i	0.862406	+	0.506217i	$0.168956\pi$
$824$		0			0
$825$	−1.49721e14			−0.0136391
$826$		0			0
$827$	−1.05148e16			−0.945195			−0.472598	−	0.881278i	$-0.656684\pi$
$827$	−1.05148e16			−0.945195			−0.472598	+	0.881278i	$0.656684\pi$
$828$		0			0
$829$	8.29261e15	−	1.43632e16i	0.735599	−	1.27409i	−0.218861	−	0.975756i	$-0.570234\pi$
$829$	8.29261e15	−	1.43632e16i	0.735599	−	1.27409i	0.954460	−	0.298339i	$-0.0964325\pi$
$830$		0			0
$831$	−3.05390e15	−	5.28950e15i	−0.267331	−	0.463030i
$832$		0			0
$833$	7.87905e15	−	6.91132e15i	0.680654	−	0.597054i
$834$		0			0
$835$	5.23169e15	+	9.06155e15i	0.446032	+	0.772551i
$836$		0			0
$837$	−3.60685e14	+	6.24724e14i	−0.0303485	+	0.0525652i
$838$		0			0
$839$	−1.72858e16			−1.43549			−0.717743	−	0.696308i	$-0.754825\pi$
$839$	−1.72858e16			−1.43549			−0.717743	+	0.696308i	$0.754825\pi$
$840$		0			0
$841$	−7.02426e15			−0.575735
$842$		0			0
$843$	2.39925e15	−	4.15563e15i	0.194100	−	0.336190i
$844$		0			0
$845$	8.05680e15	+	1.39548e16i	0.643355	+	1.11432i
$846$		0			0
$847$	−1.18606e16	+	4.46479e15i	−0.934865	+	0.351919i
$848$		0			0
$849$	−6.19188e15	−	1.07246e16i	−0.481759	−	0.834432i
$850$		0			0
$851$	8.35792e15	−	1.44763e16i	0.641927	−	1.11185i
$852$		0			0
$853$	5.29740e15			0.401645			0.200823	−	0.979628i	$-0.435638\pi$
$853$	5.29740e15			0.401645			0.200823	+	0.979628i	$0.435638\pi$
$854$		0			0
$855$	−2.66370e15			−0.199375
$856$		0			0
$857$	8.34828e15	−	1.44596e16i	0.616883	−	1.06847i	−0.373168	−	0.927764i	$-0.621729\pi$
$857$	8.34828e15	−	1.44596e16i	0.616883	−	1.06847i	0.990051	−	0.140708i	$-0.0449380\pi$
$858$		0			0
$859$	−3.39187e15	−	5.87488e15i	−0.247444	−	0.428585i	0.715372	−	0.698744i	$-0.246257\pi$
$859$	−3.39187e15	−	5.87488e15i	−0.247444	−	0.428585i	−0.962816	+	0.270159i	$0.912924\pi$
$860$		0			0
$861$	9.61352e14	+	7.88871e14i	0.0692413	+	0.0568183i
$862$		0			0
$863$	−4.58726e15	−	7.94537e15i	−0.326207	−	0.565008i	0.655549	−	0.755153i	$-0.272437\pi$
$863$	−4.58726e15	−	7.94537e15i	−0.326207	−	0.565008i	−0.981756	+	0.190145i	$0.939104\pi$
$864$		0			0
$865$	−2.51937e15	+	4.36368e15i	−0.176890	+	0.306383i
$866$		0			0
$867$	1.50103e15			0.104060
$868$		0			0
$869$	−9.48361e14			−0.0649180
$870$		0			0
$871$	1.11235e15	−	1.92665e15i	0.0751868	−	0.130227i
$872$		0			0
$873$	−2.86419e15	−	4.96093e15i	−0.191172	−	0.331119i
$874$		0			0
$875$	9.21617e14	−	5.58412e15i	0.0607444	−	0.368053i
$876$		0			0
$877$	−8.03268e14	−	1.39130e15i	−0.0522833	−	0.0905573i	0.838699	−	0.544595i	$-0.183317\pi$
$877$	−8.03268e14	−	1.39130e15i	−0.0522833	−	0.0905573i	−0.890983	+	0.454037i	$0.849983\pi$
$878$		0			0
$879$	2.24107e15	−	3.88164e15i	0.144051	−	0.249503i
$880$		0			0
$881$	1.42333e16			0.903523			0.451762	−	0.892139i	$-0.350796\pi$
$881$	1.42333e16			0.903523			0.451762	+	0.892139i	$0.350796\pi$
$882$		0			0
$883$	1.78837e16			1.12118			0.560588	−	0.828095i	$-0.310575\pi$
$883$	1.78837e16			1.12118			0.560588	+	0.828095i	$0.310575\pi$
$884$		0			0
$885$	−3.43716e15	+	5.95334e15i	−0.212820	+	0.368615i
$886$		0			0
$887$	−1.20429e16	−	2.08590e16i	−0.736465	−	1.27559i	−0.954078	−	0.299560i	$-0.903160\pi$
$887$	−1.20429e16	−	2.08590e16i	−0.736465	−	1.27559i	0.217613	−	0.976035i	$-0.430173\pi$
$888$		0			0
$889$	−1.23122e15	+	7.46003e15i	−0.0743664	+	0.450589i
$890$		0			0
$891$	−3.07607e13	−	5.32791e13i	−0.00183514	−	0.00317855i
$892$		0			0
$893$	−7.19356e15	+	1.24596e16i	−0.423897	+	0.734212i
$894$		0			0
$895$	2.89577e15			0.168553
$896$		0			0
$897$	1.42108e15			0.0817073
$898$		0			0
$899$	1.80849e15	−	3.13240e15i	0.102716	−	0.177909i
$900$		0			0
$901$	2.52397e15	+	4.37164e15i	0.141611	+	0.245278i
$902$		0			0
$903$	7.52801e14	+	6.17736e14i	0.0417251	+	0.0342390i
$904$		0			0
$905$	8.39070e15	+	1.45331e16i	0.459442	+	0.795777i
$906$		0			0
$907$	3.00137e15	−	5.19853e15i	0.162360	−	0.281216i	−0.773354	−	0.633974i	$-0.781423\pi$
$907$	3.00137e15	−	5.19853e15i	0.162360	−	0.281216i	0.935715	+	0.352758i	$0.114756\pi$
$908$		0			0
$909$	−8.71325e15			−0.465671
$910$		0			0
$911$	1.74578e16			0.921801			0.460901	−	0.887452i	$-0.347526\pi$
$911$	1.74578e16			0.921801			0.460901	+	0.887452i	$0.347526\pi$
$912$		0			0
$913$	−1.57968e14	+	2.73608e14i	−0.00824098	+	0.0142738i
$914$		0			0
$915$	8.60675e15	+	1.49073e16i	0.443632	+	0.768394i
$916$		0			0
$917$	−1.69354e15	+	6.37512e14i	−0.0862509	+	0.0324681i
$918$		0			0
$919$	3.85158e15	+	6.67114e15i	0.193822	+	0.335710i	0.946514	−	0.322663i	$-0.104578\pi$
$919$	3.85158e15	+	6.67114e15i	0.193822	+	0.335710i	−0.752691	+	0.658373i	$0.771245\pi$
$920$		0			0
$921$	−2.71678e15	+	4.70561e15i	−0.135091	+	0.233985i
$922$		0			0
$923$	2.34716e15			0.115328
$924$		0			0
$925$	1.76819e16			0.858516
$926$		0			0
$927$	9.85649e14	−	1.70719e15i	0.0472916	−	0.0819114i
$928$		0			0
$929$	4.56263e15	+	7.90270e15i	0.216336	+	0.374705i	0.953685	−	0.300807i	$-0.0972561\pi$
$929$	4.56263e15	+	7.90270e15i	0.216336	+	0.374705i	−0.737349	+	0.675512i	$0.763923\pi$
$930$		0			0
$931$	−1.90259e15	−	9.55932e15i	−0.0891501	−	0.447923i
$932$		0			0
$933$	6.28112e15	+	1.08792e16i	0.290863	+	0.503789i
$934$		0			0
$935$	−4.27929e14	+	7.41195e14i	−0.0195843	+	0.0339210i
$936$		0			0
$937$	−2.35949e16			−1.06721			−0.533607	−	0.845733i	$-0.679164\pi$
$937$	−2.35949e16			−1.06721			−0.533607	+	0.845733i	$0.679164\pi$
$938$		0			0
$939$	1.50928e16			0.674698
$940$		0			0
$941$	1.09105e16	−	1.88975e16i	0.482059	−	0.834950i	−0.517729	−	0.855545i	$-0.673223\pi$
$941$	1.09105e16	−	1.88975e16i	0.482059	−	0.834950i	0.999788	+	0.0205943i	$0.00655584\pi$
$942$		0			0
$943$	1.89967e15	+	3.29033e15i	0.0829591	+	0.143689i
$944$		0			0
$945$	−5.46473e15	+	2.05713e15i	−0.235882	+	0.0887948i
$946$		0			0
$947$	−2.92238e14	−	5.06170e14i	−0.0124684	−	0.0215959i	0.859724	−	0.510759i	$-0.170636\pi$
$947$	−2.92238e14	−	5.06170e14i	−0.0124684	−	0.0215959i	−0.872192	+	0.489163i	$0.837302\pi$
$948$		0			0
$949$	1.69334e15	−	2.93295e15i	0.0714133	−	0.123692i
$950$		0			0
$951$	1.39699e16			0.582371
$952$		0			0
$953$	1.71969e16			0.708662			0.354331	−	0.935120i	$-0.384709\pi$
$953$	1.71969e16			0.708662			0.354331	+	0.935120i	$0.384709\pi$
$954$		0			0
$955$	3.54873e15	−	6.14658e15i	0.144562	−	0.250389i
$956$		0			0
$957$	1.54236e14	+	2.67144e14i	0.00621111	+	0.0107580i
$958$		0			0
$959$	−2.52950e16	−	2.07566e16i	−1.00701	−	0.826333i
$960$		0			0
$961$	1.14405e16	+	1.98156e16i	0.450264	+	0.779880i
$962$		0			0
$963$	2.22244e15	−	3.84938e15i	0.0864739	−	0.149777i
$964$		0			0
$965$	1.13179e16			0.435376
$966$		0			0
$967$	1.93106e16			0.734430			0.367215	−	0.930136i	$-0.380311\pi$
$967$	1.93106e16			0.734430			0.367215	+	0.930136i	$0.380311\pi$
$968$		0			0
$969$	−3.17449e15	+	5.49837e15i	−0.119369	+	0.206754i
$970$		0			0
$971$	−1.85117e15	−	3.20631e15i	−0.0688240	−	0.119207i	0.829560	−	0.558418i	$-0.188591\pi$
$971$	−1.85117e15	−	3.20631e15i	−0.0688240	−	0.119207i	−0.898384	+	0.439211i	$0.855258\pi$
$972$		0			0
$973$	5.73763e15	−	3.47646e16i	0.210917	−	1.27796i
$974$		0			0
$975$	7.51604e14	+	1.30182e15i	0.0273189	+	0.0473178i
$976$		0			0
$977$	1.64079e16	−	2.84193e16i	0.589702	−	1.02139i	−0.404570	−	0.914507i	$-0.632579\pi$
$977$	1.64079e16	−	2.84193e16i	0.589702	−	1.02139i	0.994271	−	0.106886i	$-0.0340880\pi$
$978$		0			0
$979$	−7.27336e14			−0.0258482
$980$		0			0
$981$	−1.24875e15			−0.0438828
$982$		0			0
$983$	2.18466e16	−	3.78394e16i	0.759170	−	1.31492i	−0.184105	−	0.982907i	$-0.558939\pi$
$983$	2.18466e16	−	3.78394e16i	0.759170	−	1.31492i	0.943274	−	0.332014i	$-0.107728\pi$
$984$		0			0
$985$	1.96837e15	+	3.40931e15i	0.0676406	+	0.117157i
$986$		0			0
$987$	−5.13565e15	+	3.11171e16i	−0.174522	+	1.05744i
$988$		0			0
$989$	1.48756e15	+	2.57654e15i	0.0499915	+	0.0865878i
$990$		0			0
$991$	1.67634e16	−	2.90351e16i	0.557131	−	0.964978i	−0.440604	−	0.897702i	$-0.645236\pi$
$991$	1.67634e16	−	2.90351e16i	0.557131	−	0.964978i	0.997734	−	0.0672767i	$-0.0214310\pi$
$992$		0			0
$993$	3.15999e16			1.03864
$994$		0			0
$995$	3.05974e16			0.994621
$996$		0			0
$997$	−2.24203e16	+	3.88331e16i	−0.720805	+	1.24847i	0.239873	+	0.970804i	$0.422894\pi$
$997$	−2.24203e16	+	3.88331e16i	−0.720805	+	1.24847i	−0.960678	+	0.277666i	$0.910439\pi$
$998$		0			0
$999$	3.63279e15	+	6.29217e15i	0.115513	+	0.200074i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.i.a.25.1	✓	14
3.2	odd	2		252.12.k.b.109.7		14
7.2	even	3	inner	84.12.i.a.37.1	yes	14
21.2	odd	6		252.12.k.b.37.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.i.a.25.1	✓	14	1.1	even	1	trivial
84.12.i.a.37.1	yes	14	7.2	even	3	inner
252.12.k.b.37.7		14	21.2	odd	6
252.12.k.b.109.7		14	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(121.500 - 210.444i) q^{3} +(-4575.70 - 7925.35i) q^{5} +(-7241.01 + 43873.6i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})\)	\(q+(121.500 - 210.444i) q^{3} +(-4575.70 - 7925.35i) q^{5} +(-7241.01 + 43873.6i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-8822.08 + 15280.3i) q^{11} +177148. q^{13} -2.22379e6 q^{15} +(-2.65023e6 + 4.59033e6i) q^{17} +(2.46464e6 + 4.26889e6i) q^{19} +(8.35316e6 + 6.85447e6i) q^{21} +(1.65062e7 + 2.85896e7i) q^{23} +(-1.74601e7 + 3.02418e7i) q^{25} -1.43489e7 q^{27} +7.19461e7 q^{29} +(2.51367e7 - 4.35381e7i) q^{31} +(2.14377e6 + 3.71311e6i) q^{33} +(3.80847e8 - 1.43365e8i) q^{35} +(-2.53175e8 - 4.38512e8i) q^{37} +(2.15235e7 - 3.72798e7i) q^{39} +1.15088e8 q^{41} +9.01216e7 q^{43} +(-2.70191e8 + 4.67984e8i) q^{45} +(1.45935e9 + 2.52767e9i) q^{47} +(-1.87246e9 - 6.35379e8i) q^{49} +(6.44005e8 + 1.11545e9i) q^{51} +(4.76180e8 - 8.24767e8i) q^{53} +1.61469e8 q^{55} +1.19782e9 q^{57} +(1.54563e9 - 2.67711e9i) q^{59} +(-3.87030e9 - 6.70356e9i) q^{61} +(2.45739e9 - 9.25056e8i) q^{63} +(-8.10577e8 - 1.40396e9i) q^{65} +(6.27921e9 - 1.08759e10i) q^{67} +8.02201e9 q^{69} +1.32497e10 q^{71} +(9.55888e9 - 1.65565e10i) q^{73} +(4.24280e9 + 7.34875e9i) q^{75} +(-6.06521e8 - 4.97701e8i) q^{77} +(2.68746e10 + 4.65483e10i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} +1.79059e10 q^{83} +4.85066e10 q^{85} +(8.74146e9 - 1.51406e10i) q^{87} +(2.06112e10 + 3.56997e10i) q^{89} +(-1.28273e9 + 7.77213e9i) q^{91} +(-6.10823e9 - 1.05798e10i) q^{93} +(2.25550e10 - 3.90664e10i) q^{95} +9.70107e10 q^{97} +1.04187e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 1701 q^{3} + 7218 q^{5} + 35001 q^{7} - 413343 q^{9}+O(q^{10}) \) 14 * q + 1701 * q^3 + 7218 * q^5 + 35001 * q^7 - 413343 * q^9	\( 14 q + 1701 q^{3} + 7218 q^{5} + 35001 q^{7} - 413343 q^{9} + 54450 q^{11} + 1534982 q^{13} + 3507948 q^{15} + 1478880 q^{17} - 22875935 q^{19} + 3394224 q^{21} + 62540568 q^{23} - 62136141 q^{25} - 200884698 q^{27} + 102097728 q^{29} + 188600405 q^{31} - 13231350 q^{33} - 253840734 q^{35} + 199685599 q^{37} + 186500313 q^{39} - 693868716 q^{41} - 620701754 q^{43} + 426215682 q^{45} + 2771987346 q^{47} - 5209147075 q^{49} - 359367840 q^{51} + 6487034184 q^{53} + 10046238656 q^{55} - 11117704410 q^{57} - 8183838888 q^{59} + 4069556330 q^{61} - 1241977617 q^{63} - 1520229906 q^{65} + 15766443531 q^{67} + 30394716048 q^{69} - 33183285444 q^{71} - 31685143839 q^{73} + 15099082263 q^{75} + 3261253500 q^{77} + 21999509987 q^{79} - 24407490807 q^{81} - 63053885988 q^{83} + 35204204624 q^{85} + 12404873952 q^{87} + 67041904680 q^{89} - 190876959523 q^{91} - 45829898415 q^{93} + 133488871470 q^{95} + 284083418100 q^{97} - 6430436100 q^{99}+O(q^{100}) \) 14 * q + 1701 * q^3 + 7218 * q^5 + 35001 * q^7 - 413343 * q^9 + 54450 * q^11 + 1534982 * q^13 + 3507948 * q^15 + 1478880 * q^17 - 22875935 * q^19 + 3394224 * q^21 + 62540568 * q^23 - 62136141 * q^25 - 200884698 * q^27 + 102097728 * q^29 + 188600405 * q^31 - 13231350 * q^33 - 253840734 * q^35 + 199685599 * q^37 + 186500313 * q^39 - 693868716 * q^41 - 620701754 * q^43 + 426215682 * q^45 + 2771987346 * q^47 - 5209147075 * q^49 - 359367840 * q^51 + 6487034184 * q^53 + 10046238656 * q^55 - 11117704410 * q^57 - 8183838888 * q^59 + 4069556330 * q^61 - 1241977617 * q^63 - 1520229906 * q^65 + 15766443531 * q^67 + 30394716048 * q^69 - 33183285444 * q^71 - 31685143839 * q^73 + 15099082263 * q^75 + 3261253500 * q^77 + 21999509987 * q^79 - 24407490807 * q^81 - 63053885988 * q^83 + 35204204624 * q^85 + 12404873952 * q^87 + 67041904680 * q^89 - 190876959523 * q^91 - 45829898415 * q^93 + 133488871470 * q^95 + 284083418100 * q^97 - 6430436100 * q^99

Introduction

L-functions

Modular forms

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