LMFDB - Embedded newform 84.12.i.a.37.4

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			37.4
Root			$-1534.38 - 2657.62i$ of defining polynomial
Character	$\chi$	$=$	84.37
Dual form			84.12.i.a.25.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(121.500 + 210.444i) q^{3} +(2050.38 - 3551.36i) q^{5} +(-44391.9 - 2586.27i) q^{7} +(-29524.5 + 51137.9i) q^{9} +O(q^{10})$	\(q+(121.500 + 210.444i) q^{3} +(2050.38 - 3551.36i) q^{5} +(-44391.9 - 2586.27i) q^{7} +(-29524.5 + 51137.9i) q^{9} +(-512347. - 887411. i) q^{11} +1.22222e6 q^{13} +996483. q^{15} +(292898. + 507314. i) q^{17} +(-9.14606e6 + 1.58414e7i) q^{19} +(-4.84935e6 - 9.65624e6i) q^{21} +(3.07687e6 - 5.32929e6i) q^{23} +(1.60060e7 + 2.77232e7i) q^{25} -1.43489e7 q^{27} +7.95883e7 q^{29} +(1.26268e8 + 2.18703e8i) q^{31} +(1.24500e8 - 2.15641e8i) q^{33} +(-1.00205e8 + 1.52349e8i) q^{35} +(3.20825e8 - 5.55685e8i) q^{37} +(1.48500e8 + 2.57209e8i) q^{39} -1.15191e9 q^{41} +5.12710e7 q^{43} +(1.21073e8 + 2.09704e8i) q^{45} +(-6.60692e8 + 1.14435e9i) q^{47} +(1.96395e9 + 2.29619e8i) q^{49} +(-7.11742e7 + 1.23277e8i) q^{51} +(2.83146e9 + 4.90423e9i) q^{53} -4.20202e9 q^{55} -4.44499e9 q^{57} +(-4.72220e9 - 8.17909e9i) q^{59} +(-1.56193e9 + 2.70534e9i) q^{61} +(1.44290e9 - 2.19375e9i) q^{63} +(2.50601e9 - 4.34054e9i) q^{65} +(8.85008e9 + 1.53288e10i) q^{67} +1.49536e9 q^{69} -1.47179e9 q^{71} +(9.06346e9 + 1.56984e10i) q^{73} +(-3.88945e9 + 6.73673e9i) q^{75} +(2.04490e10 + 4.07189e10i) q^{77} +(-1.42410e10 + 2.46661e10i) q^{79} +(-1.74339e9 - 3.01964e9i) q^{81} +4.11599e9 q^{83} +2.40220e9 q^{85} +(9.66998e9 + 1.67489e10i) q^{87} +(-2.87026e10 + 4.97144e10i) q^{89} +(-5.42566e10 - 3.16099e9i) q^{91} +(-3.06832e10 + 5.31449e10i) q^{93} +(3.75057e10 + 6.49619e10i) q^{95} +6.33525e10 q^{97} +6.05072e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	121.500	+	210.444i	0.288675	+	0.500000i
$3$	121.500	+	210.444i	0.288675	+	0.500000i
$4$		0			0
$5$	2050.38	−	3551.36i	0.293426	−	0.508229i	−0.681191	−	0.732105i	$-0.738538\pi$
$5$	2050.38	−	3551.36i	0.293426	−	0.508229i	0.974618	+	0.223876i	$0.0718712\pi$
$6$		0			0
$7$	−44391.9	−	2586.27i	−0.998307	−	0.0581614i
$7$	−44391.9	−	2586.27i	−0.998307	−	0.0581614i
$8$		0			0
$9$	−29524.5	+	51137.9i	−0.166667	+	0.288675i
$10$		0			0
$11$	−512347.	−	887411.i	−0.959190	−	1.66137i	−0.724475	−	0.689301i	$-0.757918\pi$
$11$	−512347.	−	887411.i	−0.959190	−	1.66137i	−0.234714	−	0.972064i	$-0.575415\pi$
$12$		0			0
$13$	1.22222e6			0.912979			0.456490	−	0.889729i	$-0.349107\pi$
$13$	1.22222e6			0.912979			0.456490	+	0.889729i	$0.349107\pi$
$14$		0			0
$15$	996483.			0.338819
$16$		0			0
$17$	292898.	+	507314.i	0.0500319	+	0.0866578i	0.889957	−	0.456045i	$-0.150734\pi$
$17$	292898.	+	507314.i	0.0500319	+	0.0866578i	−0.839925	+	0.542703i	$0.817401\pi$
$18$		0			0
$19$	−9.14606e6	+	1.58414e7i	−0.847402	+	1.46774i	0.0361177	+	0.999348i	$0.488501\pi$
$19$	−9.14606e6	+	1.58414e7i	−0.847402	+	1.46774i	−0.883519	+	0.468395i	$0.844832\pi$
$20$		0			0
$21$	−4.84935e6	−	9.65624e6i	−0.259106	−	0.515943i
$22$		0			0
$23$	3.07687e6	−	5.32929e6i	0.0996794	−	0.172650i	−0.811873	−	0.583835i	$-0.801552\pi$
$23$	3.07687e6	−	5.32929e6i	0.0996794	−	0.172650i	0.911552	+	0.411185i	$0.134885\pi$
$24$		0			0
$25$	1.60060e7	+	2.77232e7i	0.327802	+	0.567770i
$26$		0			0
$27$	−1.43489e7			−0.192450
$28$		0			0
$29$	7.95883e7			0.720544			0.360272	−	0.932847i	$-0.382684\pi$
$29$	7.95883e7			0.720544			0.360272	+	0.932847i	$0.382684\pi$
$30$		0			0
$31$	1.26268e8	+	2.18703e8i	0.792147	+	1.37204i	0.924635	+	0.380854i	$0.124370\pi$
$31$	1.26268e8	+	2.18703e8i	0.792147	+	1.37204i	−0.132489	+	0.991185i	$0.542297\pi$
$32$		0			0
$33$	1.24500e8	−	2.15641e8i	0.553788	−	0.959190i
$34$		0			0
$35$	−1.00205e8	+	1.52349e8i	−0.322489	+	0.490302i
$36$		0			0
$37$	3.20825e8	−	5.55685e8i	0.760604	−	1.31741i	−0.181935	−	0.983310i	$-0.558236\pi$
$37$	3.20825e8	−	5.55685e8i	0.760604	−	1.31741i	0.942540	−	0.334095i	$-0.108431\pi$
$38$		0			0
$39$	1.48500e8	+	2.57209e8i	0.263554	+	0.456490i
$40$		0			0
$41$	−1.15191e9			−1.55277			−0.776385	−	0.630259i	$-0.782949\pi$
$41$	−1.15191e9			−1.55277			−0.776385	+	0.630259i	$0.782949\pi$
$42$		0			0
$43$	5.12710e7			0.0531857			0.0265929	−	0.999646i	$-0.491534\pi$
$43$	5.12710e7			0.0531857			0.0265929	+	0.999646i	$0.491534\pi$
$44$		0			0
$45$	1.21073e8	+	2.09704e8i	0.0978087	+	0.169410i
$46$		0			0
$47$	−6.60692e8	+	1.14435e9i	−0.420205	+	0.727816i	−0.995959	−	0.0898066i	$-0.971375\pi$
$47$	−6.60692e8	+	1.14435e9i	−0.420205	+	0.727816i	0.575754	+	0.817623i	$0.304708\pi$
$48$		0			0
$49$	1.96395e9	+	2.29619e8i	0.993234	+	0.116126i
$50$		0			0
$51$	−7.11742e7	+	1.23277e8i	−0.0288859	+	0.0500319i
$52$		0			0
$53$	2.83146e9	+	4.90423e9i	0.930021	+	1.61084i	0.783281	+	0.621668i	$0.213545\pi$
$53$	2.83146e9	+	4.90423e9i	0.930021	+	1.61084i	0.146740	+	0.989175i	$0.453122\pi$
$54$		0			0
$55$	−4.20202e9			−1.12580
$56$		0			0
$57$	−4.44499e9			−0.978495
$58$		0			0
$59$	−4.72220e9	−	8.17909e9i	−0.859921	−	1.48943i	−0.872004	−	0.489499i	$-0.837180\pi$
$59$	−4.72220e9	−	8.17909e9i	−0.859921	−	1.48943i	0.0120832	−	0.999927i	$-0.496154\pi$
$60$		0			0
$61$	−1.56193e9	+	2.70534e9i	−0.236781	+	0.410117i	−0.959789	−	0.280723i	$-0.909426\pi$
$61$	−1.56193e9	+	2.70534e9i	−0.236781	+	0.410117i	0.723008	+	0.690840i	$0.242759\pi$
$62$		0			0
$63$	1.44290e9	−	2.19375e9i	0.183174	−	0.278493i
$64$		0			0
$65$	2.50601e9	−	4.34054e9i	0.267892	−	0.464002i
$66$		0			0
$67$	8.85008e9	+	1.53288e10i	0.800822	+	1.38706i	0.919076	+	0.394081i	$0.128937\pi$
$67$	8.85008e9	+	1.53288e10i	0.800822	+	1.38706i	−0.118254	+	0.992983i	$0.537730\pi$
$68$		0			0
$69$	1.49536e9			0.115100
$70$		0			0
$71$	−1.47179e9			−0.0968109			−0.0484054	−	0.998828i	$-0.515414\pi$
$71$	−1.47179e9			−0.0968109			−0.0484054	+	0.998828i	$0.515414\pi$
$72$		0			0
$73$	9.06346e9	+	1.56984e10i	0.511704	+	0.886297i	0.999908	+	0.0135672i	$0.00431871\pi$
$73$	9.06346e9	+	1.56984e10i	0.511704	+	0.886297i	−0.488204	+	0.872729i	$0.662348\pi$
$74$		0			0
$75$	−3.88945e9	+	6.73673e9i	−0.189257	+	0.327802i
$76$		0			0
$77$	2.04490e10	+	4.07189e10i	0.860938	+	1.71434i
$78$		0			0
$79$	−1.42410e10	+	2.46661e10i	−0.520703	+	0.901885i	0.479007	+	0.877811i	$0.340997\pi$
$79$	−1.42410e10	+	2.46661e10i	−0.520703	+	0.901885i	−0.999710	+	0.0240735i	$0.992336\pi$
$80$		0			0
$81$	−1.74339e9	−	3.01964e9i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	4.11599e9			0.114695			0.0573475	−	0.998354i	$-0.481736\pi$
$83$	4.11599e9			0.114695			0.0573475	+	0.998354i	$0.481736\pi$
$84$		0			0
$85$	2.40220e9			0.0587227
$86$		0			0
$87$	9.66998e9	+	1.67489e10i	0.208003	+	0.360272i
$88$		0			0
$89$	−2.87026e10	+	4.97144e10i	−0.544850	+	0.943707i	0.453767	+	0.891121i	$0.350080\pi$
$89$	−2.87026e10	+	4.97144e10i	−0.544850	+	0.943707i	−0.998616	+	0.0525867i	$0.983253\pi$
$90$		0			0
$91$	−5.42566e10	−	3.16099e9i	−0.911434	−	0.0531002i
$92$		0			0
$93$	−3.06832e10	+	5.31449e10i	−0.457346	+	0.792147i
$94$		0			0
$95$	3.75057e10	+	6.49619e10i	0.497299	+	0.861348i
$96$		0			0
$97$	6.33525e10			0.749065			0.374532	−	0.927214i	$-0.377803\pi$
$97$	6.33525e10			0.749065			0.374532	+	0.927214i	$0.377803\pi$
$98$		0			0
$99$	6.05072e10			0.639460
$100$		0			0
$101$	4.18506e10	+	7.24874e10i	0.396218	+	0.686270i	0.993256	−	0.115943i	$-0.0369890\pi$
$101$	4.18506e10	+	7.24874e10i	0.396218	+	0.686270i	−0.597038	+	0.802213i	$0.703656\pi$
$102$		0			0
$103$	−3.03329e10	+	5.25381e10i	−0.257816	+	0.446550i	−0.965656	−	0.259822i	$-0.916336\pi$
$103$	−3.03329e10	+	5.25381e10i	−0.257816	+	0.446550i	0.707841	+	0.706372i	$0.249669\pi$
$104$		0			0
$105$	−4.42358e10	−	2.57718e9i	−0.338246	−	0.0197062i
$106$		0			0
$107$	−1.47323e10	+	2.55171e10i	−0.101545	+	0.175881i	−0.912321	−	0.409475i	$-0.865712\pi$
$107$	−1.47323e10	+	2.55171e10i	−0.101545	+	0.175881i	0.810776	+	0.585356i	$0.199045\pi$
$108$		0			0
$109$	4.61930e9	+	8.00087e9i	0.0287562	+	0.0498071i	0.880045	−	0.474890i	$-0.157512\pi$
$109$	4.61930e9	+	8.00087e9i	0.0287562	+	0.0498071i	−0.851289	+	0.524697i	$0.824179\pi$
$110$		0			0
$111$	1.55921e11			0.878270
$112$		0			0
$113$	−2.59219e11			−1.32353			−0.661767	−	0.749709i	$-0.730193\pi$
$113$	−2.59219e11			−1.32353			−0.661767	+	0.749709i	$0.730193\pi$
$114$		0			0
$115$	−1.26175e10	−	2.18541e10i	−0.0584971	−	0.101320i
$116$		0			0
$117$	−3.60854e10	+	6.25018e10i	−0.152163	+	0.263554i
$118$		0			0
$119$	−1.16902e10	−	2.32781e10i	−0.0449071	−	0.0894210i
$120$		0			0
$121$	−3.82343e11	+	6.62238e11i	−1.34009	+	2.32110i
$122$		0			0
$123$	−1.39957e11	−	2.42413e11i	−0.448246	−	0.776385i
$124$		0			0
$125$	3.31505e11			0.971595
$126$		0			0
$127$	3.69162e11			0.991509			0.495754	−	0.868463i	$-0.334892\pi$
$127$	3.69162e11			0.991509			0.495754	+	0.868463i	$0.334892\pi$
$128$		0			0
$129$	6.22942e9	+	1.07897e10i	0.0153534	+	0.0265929i
$130$		0			0
$131$	−4.84044e10	+	8.38388e10i	−0.109621	+	0.189869i	−0.915617	−	0.402052i	$-0.868297\pi$
$131$	−4.84044e10	+	8.38388e10i	−0.109621	+	0.189869i	0.805996	+	0.591921i	$0.201630\pi$
$132$		0			0
$133$	4.46981e11	−	6.79577e11i	0.931333	−	1.41597i
$134$		0			0
$135$	−2.94207e10	+	5.09581e10i	−0.0564699	+	0.0978087i
$136$		0			0
$137$	3.24632e11	+	5.62279e11i	0.574682	+	0.995379i	0.996076	+	0.0885013i	$0.0282077\pi$
$137$	3.24632e11	+	5.62279e11i	0.574682	+	0.995379i	−0.421394	+	0.906878i	$0.638459\pi$
$138$		0			0
$139$	−8.83959e11			−1.44494			−0.722471	−	0.691401i	$-0.756994\pi$
$139$	−8.83959e11			−1.44494			−0.722471	+	0.691401i	$0.756994\pi$
$140$		0			0
$141$	−3.21096e11			−0.485211
$142$		0			0
$143$	−6.26201e11	−	1.08461e12i	−0.875720	−	1.51679i
$144$		0			0
$145$	1.63186e11	−	2.82646e11i	0.211426	−	0.366201i
$146$		0			0
$147$	1.90298e11	+	4.41200e11i	0.228659	+	0.530140i
$148$		0			0
$149$	7.34266e11	−	1.27179e12i	0.819085	−	1.41870i	−0.0872723	−	0.996184i	$-0.527815\pi$
$149$	7.34266e11	−	1.27179e12i	0.819085	−	1.41870i	0.906357	−	0.422512i	$-0.138852\pi$
$150$		0			0
$151$	4.26379e11	+	7.38511e11i	0.442001	+	0.765568i	0.997838	−	0.0657235i	$-0.0209355\pi$
$151$	4.26379e11	+	7.38511e11i	0.442001	+	0.765568i	−0.555837	+	0.831291i	$0.687602\pi$
$152$		0			0
$153$	−3.45907e10			−0.0333546
$154$		0			0
$155$	1.03559e12			0.929746
$156$		0			0
$157$	−1.62460e11	−	2.81390e11i	−0.135925	−	0.235429i	0.790025	−	0.613074i	$-0.210067\pi$
$157$	−1.62460e11	−	2.81390e11i	−0.135925	−	0.235429i	−0.925950	+	0.377645i	$0.876734\pi$
$158$		0			0
$159$	−6.88044e11	+	1.19173e12i	−0.536948	+	0.930021i
$160$		0			0
$161$	−1.50371e11	+	2.28619e11i	−0.109552	+	0.166560i
$162$		0			0
$163$	1.04403e12	−	1.80831e12i	0.710692	−	1.23095i	−0.253906	−	0.967229i	$-0.581715\pi$
$163$	1.04403e12	−	1.80831e12i	0.710692	−	1.23095i	0.964598	−	0.263725i	$-0.0849513\pi$
$164$		0			0
$165$	−5.10545e11	−	8.84290e11i	−0.324992	−	0.562902i
$166$		0			0
$167$	1.12451e12			0.669919			0.334959	−	0.942233i	$-0.391277\pi$
$167$	1.12451e12			0.669919			0.334959	+	0.942233i	$0.391277\pi$
$168$		0			0
$169$	−2.98339e11			−0.166469
$170$		0			0
$171$	−5.40066e11	−	9.35421e11i	−0.282467	−	0.489248i
$172$		0			0
$173$	8.74498e11	−	1.51467e12i	0.429047	−	0.743132i	−0.567742	−	0.823207i	$-0.692183\pi$
$173$	8.74498e11	−	1.51467e12i	0.429047	−	0.743132i	0.996789	+	0.0800752i	$0.0255160\pi$
$174$		0			0
$175$	−6.38835e11	−	1.27208e12i	−0.294225	−	0.585874i
$176$		0			0
$177$	1.14749e12	−	1.98752e12i	0.496475	−	0.859921i
$178$		0			0
$179$	−1.78769e12	−	3.09637e12i	−0.727110	−	1.25939i	−0.958100	−	0.286435i	$-0.907530\pi$
$179$	−1.78769e12	−	3.09637e12i	−0.727110	−	1.25939i	0.230990	−	0.972956i	$-0.425804\pi$
$180$		0			0
$181$	−3.34657e12			−1.28047			−0.640233	−	0.768181i	$-0.721162\pi$
$181$	−3.34657e12			−1.28047			−0.640233	+	0.768181i	$0.721162\pi$
$182$		0			0
$183$	−7.59097e11			−0.273411
$184$		0			0
$185$	−1.31562e12	−	2.27873e12i	−0.446362	−	0.773122i
$186$		0			0
$187$	3.00131e11	−	5.19842e11i	0.0959802	−	0.166243i
$188$		0			0
$189$	6.36975e11	+	3.71102e10i	0.192124	+	0.0111932i
$190$		0			0
$191$	−3.70482e11	+	6.41693e11i	−0.105459	+	0.182660i	−0.913926	−	0.405882i	$-0.866964\pi$
$191$	−3.70482e11	+	6.41693e11i	−0.105459	+	0.182660i	0.808467	+	0.588542i	$0.200298\pi$
$192$		0			0
$193$	1.38383e12	+	2.39686e12i	0.371977	+	0.644283i	0.989870	−	0.141979i	$-0.0453466\pi$
$193$	1.38383e12	+	2.39686e12i	0.371977	+	0.644283i	−0.617892	+	0.786263i	$0.712013\pi$
$194$		0			0
$195$	1.21792e12			0.309335
$196$		0			0
$197$	−6.53831e11			−0.157001			−0.0785003	−	0.996914i	$-0.525013\pi$
$197$	−6.53831e11			−0.157001			−0.0785003	+	0.996914i	$0.525013\pi$
$198$		0			0
$199$	−1.56622e12	−	2.71277e12i	−0.355762	−	0.616199i	0.631486	−	0.775388i	$-0.282445\pi$
$199$	−1.56622e12	−	2.71277e12i	−0.355762	−	0.616199i	−0.987248	+	0.159189i	$0.949112\pi$
$200$		0			0
$201$	−2.15057e12	+	3.72490e12i	−0.462355	+	0.800822i
$202$		0			0
$203$	−3.53307e12	−	2.05837e11i	−0.719324	−	0.0419078i
$204$		0			0
$205$	−2.36185e12	+	4.09084e12i	−0.455623	+	0.789162i
$206$		0			0
$207$	1.81686e11	+	3.14689e11i	0.0332265	+	0.0575499i
$208$		0			0
$209$	1.87438e13			3.25127
$210$		0			0
$211$	6.00401e12			0.988297			0.494149	−	0.869377i	$-0.335480\pi$
$211$	6.00401e12			0.988297			0.494149	+	0.869377i	$0.335480\pi$
$212$		0			0
$213$	−1.78822e11	−	3.09729e11i	−0.0279469	−	0.0484054i
$214$		0			0
$215$	1.05125e11	−	1.82082e11i	0.0156061	−	0.0270305i
$216$		0			0
$217$	−5.03967e12	−	1.00352e13i	−0.711006	−	1.41579i
$218$		0			0
$219$	−2.20242e12	+	3.81471e12i	−0.295432	+	0.511704i
$220$		0			0
$221$	3.57986e11	+	6.20049e11i	0.0456781	+	0.0791168i
$222$		0			0
$223$	−1.11476e13			−1.35365			−0.676825	−	0.736144i	$-0.736645\pi$
$223$	−1.11476e13			−1.35365			−0.676825	+	0.736144i	$0.736645\pi$
$224$		0			0
$225$	−1.89027e12			−0.218535
$226$		0			0
$227$	6.74610e12	+	1.16846e13i	0.742866	+	1.28668i	0.951185	+	0.308621i	$0.0998674\pi$
$227$	6.74610e12	+	1.16846e13i	0.742866	+	1.28668i	−0.208319	+	0.978061i	$0.566799\pi$
$228$		0			0
$229$	−2.32802e12	+	4.03225e12i	−0.244282	+	0.423109i	−0.961930	−	0.273297i	$-0.911886\pi$
$229$	−2.32802e12	+	4.03225e12i	−0.244282	+	0.423109i	0.717647	+	0.696407i	$0.245219\pi$
$230$		0			0
$231$	−6.08451e12	+	9.25071e12i	−0.608639	+	0.925357i
$232$		0			0
$233$	8.56539e12	−	1.48357e13i	0.817127	−	1.41531i	−0.0906630	−	0.995882i	$-0.528899\pi$
$233$	8.56539e12	−	1.48357e13i	0.817127	−	1.41531i	0.907790	−	0.419424i	$-0.137768\pi$
$234$		0			0
$235$	2.70934e12	+	4.69271e12i	0.246598	+	0.427120i
$236$		0			0
$237$	−6.92111e12			−0.601256
$238$		0			0
$239$	2.11708e13			1.75610			0.878049	−	0.478571i	$-0.158845\pi$
$239$	2.11708e13			1.75610			0.878049	+	0.478571i	$0.158845\pi$
$240$		0			0
$241$	1.08816e13	+	1.88474e13i	0.862180	+	1.49334i	0.869820	+	0.493369i	$0.164235\pi$
$241$	1.08816e13	+	1.88474e13i	0.862180	+	1.49334i	−0.00763983	+	0.999971i	$0.502432\pi$
$242$		0			0
$243$	4.23644e11	−	7.33773e11i	0.0320750	−	0.0555556i
$244$		0			0
$245$	4.84230e12	−	6.50388e12i	0.350459	−	0.470716i
$246$		0			0
$247$	−1.11785e13	+	1.93617e13i	−0.773660	+	1.34002i
$248$		0			0
$249$	5.00092e11	+	8.66185e11i	0.0331096	+	0.0573475i
$250$		0			0
$251$	−2.37154e13			−1.50253			−0.751267	−	0.659999i	$-0.770557\pi$
$251$	−2.37154e13			−1.50253			−0.751267	+	0.659999i	$0.770557\pi$
$252$		0			0
$253$	−6.30569e12			−0.382446
$254$		0			0
$255$	2.91868e11	+	5.05530e11i	0.0169518	+	0.0293613i
$256$		0			0
$257$	−6.81274e12	+	1.18000e13i	−0.379044	+	0.656524i	−0.990923	−	0.134427i	$-0.957081\pi$
$257$	−6.81274e12	+	1.18000e13i	−0.379044	+	0.656524i	0.611879	+	0.790951i	$0.290414\pi$
$258$		0			0
$259$	−1.56792e13	+	2.38382e13i	−0.835939	+	1.27094i
$260$		0			0
$261$	−2.34980e12	+	4.06998e12i	−0.120091	+	0.208003i
$262$		0			0
$263$	8.12682e12	+	1.40761e13i	0.398258	+	0.689803i	0.993511	−	0.113735i	$-0.0362816\pi$
$263$	8.12682e12	+	1.40761e13i	0.398258	+	0.689803i	−0.595253	+	0.803538i	$0.702948\pi$
$264$		0			0
$265$	2.32222e13			1.09157
$266$		0			0
$267$	−1.39495e13			−0.629138
$268$		0			0
$269$	−6.44748e12	−	1.11674e13i	−0.279095	−	0.483407i	0.692065	−	0.721835i	$-0.256701\pi$
$269$	−6.44748e12	−	1.11674e13i	−0.279095	−	0.483407i	−0.971160	+	0.238428i	$0.923368\pi$
$270$		0			0
$271$	−1.47643e13	+	2.55725e13i	−0.613593	+	1.06277i	0.377036	+	0.926199i	$0.376943\pi$
$271$	−1.47643e13	+	2.55725e13i	−0.613593	+	1.06277i	−0.990630	+	0.136576i	$0.956390\pi$
$272$		0			0
$273$	−5.92697e12	−	1.18021e13i	−0.236558	−	0.471046i
$274$		0			0
$275$	1.64012e13	−	2.84077e13i	0.628849	−	1.08920i
$276$		0			0
$277$	1.30044e12	+	2.25243e12i	0.0479129	+	0.0829876i	0.888987	−	0.457932i	$-0.151410\pi$
$277$	1.30044e12	+	2.25243e12i	0.0479129	+	0.0829876i	−0.841074	+	0.540920i	$0.818076\pi$
$278$		0			0
$279$	−1.49121e13			−0.528098
$280$		0			0
$281$	3.24767e13			1.10583			0.552913	−	0.833239i	$-0.313516\pi$
$281$	3.24767e13			1.10583			0.552913	+	0.833239i	$0.313516\pi$
$282$		0			0
$283$	−1.55411e13	−	2.69180e13i	−0.508929	−	0.881490i	−0.999947	−	0.0103407i	$-0.996708\pi$
$283$	−1.55411e13	−	2.69180e13i	−0.508929	−	0.881490i	0.491018	−	0.871149i	$-0.336625\pi$
$284$		0			0
$285$	−9.11390e12	+	1.57857e13i	−0.287116	+	0.497299i
$286$		0			0
$287$	5.11354e13	+	2.97915e12i	1.55014	+	0.0903113i
$288$		0			0
$289$	1.69644e13	−	2.93832e13i	0.494994	−	0.857354i
$290$		0			0
$291$	7.69733e12	+	1.33322e13i	0.216236	+	0.374532i
$292$		0			0
$293$	−4.90801e12			−0.132780			−0.0663901	−	0.997794i	$-0.521148\pi$
$293$	−4.90801e12			−0.132780			−0.0663901	+	0.997794i	$0.521148\pi$
$294$		0			0
$295$	−3.87292e13			−1.00929
$296$		0			0
$297$	7.35162e12	+	1.27334e13i	0.184596	+	0.319730i
$298$		0			0
$299$	3.76061e12	−	6.51356e12i	0.0910052	−	0.157626i
$300$		0			0
$301$	−2.27601e12	−	1.32601e11i	−0.0530957	−	0.00309336i
$302$		0			0
$303$	−1.01697e13	+	1.76144e13i	−0.228757	+	0.396218i
$304$		0			0
$305$	6.40508e12	+	1.10939e13i	0.138955	+	0.240678i
$306$		0			0
$307$	3.10113e13			0.649021			0.324510	−	0.945882i	$-0.394800\pi$
$307$	3.10113e13			0.649021			0.324510	+	0.945882i	$0.394800\pi$
$308$		0			0
$309$	−1.47418e13			−0.297700
$310$		0			0
$311$	−5.24032e12	−	9.07650e12i	−0.102135	−	0.176904i	0.810429	−	0.585837i	$-0.199234\pi$
$311$	−5.24032e12	−	9.07650e12i	−0.102135	−	0.176904i	−0.912564	+	0.408933i	$0.865901\pi$
$312$		0			0
$313$	−8.40968e12	+	1.45660e13i	−0.158229	+	0.274060i	−0.934230	−	0.356671i	$-0.883912\pi$
$313$	−8.40968e12	+	1.45660e13i	−0.158229	+	0.274060i	0.776001	+	0.630731i	$0.217245\pi$
$314$		0			0
$315$	−4.83229e12	−	9.62228e12i	−0.0877900	−	0.174812i
$316$		0			0
$317$	2.04646e13	−	3.54457e13i	0.359068	−	0.621924i	−0.628738	−	0.777618i	$-0.716428\pi$
$317$	2.04646e13	−	3.54457e13i	0.359068	−	0.621924i	0.987805	+	0.155694i	$0.0497614\pi$
$318$		0			0
$319$	−4.07768e13	−	7.06275e13i	−0.691138	−	1.19709i
$320$		0			0
$321$	−7.15989e12			−0.117254
$322$		0			0
$323$	−1.07154e13			−0.169588
$324$		0			0
$325$	1.95628e13	+	3.38838e13i	0.299277	+	0.518362i
$326$		0			0
$327$	−1.12249e12	+	1.94421e12i	−0.0166024	+	0.0287562i
$328$		0			0
$329$	3.22890e13	−	4.90912e13i	0.461824	−	0.702144i
$330$		0			0
$331$	−2.39631e13	+	4.15054e13i	−0.331505	+	0.574183i	−0.982807	−	0.184635i	$-0.940890\pi$
$331$	−2.39631e13	+	4.15054e13i	−0.331505	+	0.574183i	0.651303	+	0.758818i	$0.274223\pi$
$332$		0			0
$333$	1.89444e13	+	3.28127e13i	0.253535	+	0.439135i
$334$		0			0
$335$	7.25840e13			0.939928
$336$		0			0
$337$	−7.21409e13			−0.904101			−0.452051	−	0.891992i	$-0.649307\pi$
$337$	−7.21409e13			−0.904101			−0.452051	+	0.891992i	$0.649307\pi$
$338$		0			0
$339$	−3.14951e13	−	5.45511e13i	−0.382072	−	0.661767i
$340$		0			0
$341$	1.29387e14	−	2.24104e14i	1.51964	−	2.63209i
$342$		0			0
$343$	−8.65895e13	−	1.52725e13i	−0.984799	−	0.173697i
$344$		0			0
$345$	3.06604e12	−	5.31055e12i	0.0337733	−	0.0584971i
$346$		0			0
$347$	−1.50776e13	−	2.61151e13i	−0.160886	−	0.278663i	0.774301	−	0.632818i	$-0.218102\pi$
$347$	−1.50776e13	−	2.61151e13i	−0.160886	−	0.278663i	−0.935187	+	0.354155i	$0.884769\pi$
$348$		0			0
$349$	1.05571e14			1.09145			0.545727	−	0.837963i	$-0.316254\pi$
$349$	1.05571e14			1.09145			0.545727	+	0.837963i	$0.316254\pi$
$350$		0			0
$351$	−1.75375e13			−0.175703
$352$		0			0
$353$	1.02912e13	+	1.78249e13i	0.0999322	+	0.173088i	0.911656	−	0.410953i	$-0.134804\pi$
$353$	1.02912e13	+	1.78249e13i	0.0999322	+	0.173088i	−0.811724	+	0.584041i	$0.801471\pi$
$354$		0			0
$355$	−3.01772e12	+	5.22684e12i	−0.0284068	+	0.0492021i
$356$		0			0
$357$	3.47838e12	−	5.28843e12i	0.0317470	−	0.0482672i
$358$		0			0
$359$	2.54183e11	−	4.40258e11i	0.00224971	−	0.00389661i	−0.864898	−	0.501947i	$-0.832617\pi$
$359$	2.54183e11	−	4.40258e11i	0.00224971	−	0.00389661i	0.867148	+	0.498050i	$0.165951\pi$
$360$		0			0
$361$	−1.09056e14	−	1.88890e14i	−0.936179	−	1.62151i
$362$		0			0
$363$	−1.85819e14			−1.54740
$364$		0			0
$365$	7.43341e13			0.600589
$366$		0			0
$367$	−1.50900e13	−	2.61366e13i	−0.118311	−	0.204920i	0.800788	−	0.598949i	$-0.204415\pi$
$367$	−1.50900e13	−	2.61366e13i	−0.118311	−	0.204920i	−0.919098	+	0.394028i	$0.871081\pi$
$368$		0			0
$369$	3.40095e13	−	5.89063e13i	0.258795	−	0.448246i
$370$		0			0
$371$	−1.13010e14	−	2.25031e14i	−0.834757	−	1.66221i
$372$		0			0
$373$	−2.88832e13	+	5.00271e13i	−0.207131	+	0.358762i	−0.950810	−	0.309775i	$-0.899746\pi$
$373$	−2.88832e13	+	5.00271e13i	−0.207131	+	0.358762i	0.743678	+	0.668538i	$0.233079\pi$
$374$		0			0
$375$	4.02779e13	+	6.97633e13i	0.280475	+	0.485798i
$376$		0			0
$377$	9.72744e13			0.657841
$378$		0			0
$379$	−5.82671e13			−0.382743			−0.191372	−	0.981518i	$-0.561294\pi$
$379$	−5.82671e13			−0.382743			−0.191372	+	0.981518i	$0.561294\pi$
$380$		0			0
$381$	4.48532e13	+	7.76880e13i	0.286224	+	0.495754i
$382$		0			0
$383$	−1.05371e14	+	1.82508e14i	−0.653323	+	1.13159i	0.328988	+	0.944334i	$0.393292\pi$
$383$	−1.05371e14	+	1.82508e14i	−0.653323	+	1.13159i	−0.982311	+	0.187255i	$0.940041\pi$
$384$		0			0
$385$	1.86535e14	+	1.08676e13i	1.12390	+	0.0654784i
$386$		0			0
$387$	−1.51375e12	+	2.62189e12i	−0.00886429	+	0.0153534i
$388$		0			0
$389$	2.46027e13	+	4.26131e13i	0.140043	+	0.242561i	0.927512	−	0.373792i	$-0.121943\pi$
$389$	2.46027e13	+	4.26131e13i	0.140043	+	0.242561i	−0.787470	+	0.616353i	$0.788609\pi$
$390$		0			0
$391$	3.60483e12			0.0199486
$392$		0			0
$393$	−2.35245e13			−0.126579
$394$		0			0
$395$	5.83987e13	+	1.01150e14i	0.305576	+	0.529273i
$396$		0			0
$397$	7.42631e13	−	1.28628e14i	0.377942	−	0.654615i	−0.612820	−	0.790222i	$-0.709965\pi$
$397$	7.42631e13	−	1.28628e14i	0.377942	−	0.654615i	0.990763	+	0.135607i	$0.0432984\pi$
$398$		0			0
$399$	1.97321e14	+	1.14959e13i	0.976839	+	0.0569107i
$400$		0			0
$401$	−6.33705e12	+	1.09761e13i	−0.0305206	+	0.0528633i	−0.880882	−	0.473335i	$-0.843050\pi$
$401$	−6.33705e12	+	1.09761e13i	−0.0305206	+	0.0528633i	0.850362	+	0.526199i	$0.176383\pi$
$402$		0			0
$403$	1.54328e14	+	2.67304e14i	0.723213	+	1.25264i
$404$		0			0
$405$	−1.42984e13			−0.0652058
$406$		0			0
$407$	−6.57495e14			−2.91825
$408$		0			0
$409$	1.63665e13	+	2.83476e13i	0.0707095	+	0.122472i	0.899212	−	0.437512i	$-0.144140\pi$
$409$	1.63665e13	+	2.83476e13i	0.0707095	+	0.122472i	−0.828503	+	0.559985i	$0.810807\pi$
$410$		0			0
$411$	−7.88855e13	+	1.36634e14i	−0.331793	+	0.574682i
$412$		0			0
$413$	1.88474e14	+	3.75298e14i	0.771838	+	1.53692i
$414$		0			0
$415$	8.43933e12	−	1.46173e13i	0.0336545	−	0.0582913i
$416$		0			0
$417$	−1.07401e14	−	1.86024e14i	−0.417119	−	0.722471i
$418$		0			0
$419$	−2.35769e13			−0.0891886			−0.0445943	−	0.999005i	$-0.514200\pi$
$419$	−2.35769e13			−0.0891886			−0.0445943	+	0.999005i	$0.514200\pi$
$420$		0			0
$421$	−6.80343e13			−0.250713			−0.125356	−	0.992112i	$-0.540007\pi$
$421$	−6.80343e13			−0.250713			−0.125356	+	0.992112i	$0.540007\pi$
$422$		0			0
$423$	−3.90132e13	−	6.75729e13i	−0.140068	−	0.242605i
$424$		0			0
$425$	−9.37623e12	+	1.62401e13i	−0.0328011	+	0.0568133i
$426$		0			0
$427$	7.63336e13	−	1.16055e14i	0.260233	−	0.395651i
$428$		0			0
$429$	1.52167e14	−	2.63561e14i	0.505597	−	0.875720i
$430$		0			0
$431$	1.11259e14	+	1.92706e14i	0.360338	+	0.624124i	0.988016	−	0.154349i	$-0.0493279\pi$
$431$	1.11259e14	+	1.92706e14i	0.360338	+	0.624124i	−0.627678	+	0.778473i	$0.715995\pi$
$432$		0			0
$433$	8.86592e13			0.279924			0.139962	−	0.990157i	$-0.455302\pi$
$433$	8.86592e13			0.279924			0.139962	+	0.990157i	$0.455302\pi$
$434$		0			0
$435$	7.93084e13			0.244134
$436$		0			0
$437$	5.62824e13	+	9.74840e13i	0.168937	+	0.292607i
$438$		0			0
$439$	−2.34709e14	+	4.06528e14i	−0.687029	+	1.18997i	0.285765	+	0.958300i	$0.407752\pi$
$439$	−2.34709e14	+	4.06528e14i	−0.687029	+	1.18997i	−0.972794	+	0.231670i	$0.925581\pi$
$440$		0			0
$441$	−6.97269e13	+	9.36529e13i	−0.199062	+	0.267368i
$442$		0			0
$443$	7.72450e13	−	1.33792e14i	0.215105	−	0.372572i	−0.738200	−	0.674582i	$-0.764324\pi$
$443$	7.72450e13	−	1.33792e14i	0.215105	−	0.372572i	0.953305	+	0.302009i	$0.0976574\pi$
$444$		0			0
$445$	1.17702e14	+	2.03867e14i	0.319746	+	0.553817i
$446$		0			0
$447$	3.56853e14			0.945798
$448$		0			0
$449$	2.11044e14			0.545782			0.272891	−	0.962045i	$-0.412020\pi$
$449$	2.11044e14			0.545782			0.272891	+	0.962045i	$0.412020\pi$
$450$		0			0
$451$	5.90177e14	+	1.02222e15i	1.48940	+	2.57972i
$452$		0			0
$453$	−1.03610e14	+	1.79458e14i	−0.255189	+	0.442001i
$454$		0			0
$455$	−1.22472e14	+	1.86203e14i	−0.294425	+	0.447636i
$456$		0			0
$457$	2.39745e13	−	4.15250e13i	0.0562613	−	0.0974475i	−0.836523	−	0.547932i	$-0.815415\pi$
$457$	2.39745e13	−	4.15250e13i	0.0562613	−	0.0974475i	0.892784	+	0.450484i	$0.148749\pi$
$458$		0			0
$459$	−4.20276e12	−	7.27940e12i	−0.00962865	−	0.0166773i
$460$		0			0
$461$	−6.67393e14			−1.49289			−0.746443	−	0.665450i	$-0.768240\pi$
$461$	−6.67393e14			−1.49289			−0.746443	+	0.665450i	$0.768240\pi$
$462$		0			0
$463$	−1.71310e14			−0.374187			−0.187093	−	0.982342i	$-0.559907\pi$
$463$	−1.71310e14			−0.374187			−0.187093	+	0.982342i	$0.559907\pi$
$464$		0			0
$465$	1.25824e14	+	2.17934e14i	0.268395	+	0.464873i
$466$		0			0
$467$	−1.84137e14	+	3.18935e14i	−0.383618	+	0.664446i	−0.991576	−	0.129523i	$-0.958656\pi$
$467$	−1.84137e14	+	3.18935e14i	−0.383618	+	0.664446i	0.607958	+	0.793969i	$0.291989\pi$
$468$		0			0
$469$	−3.53227e14	−	7.03362e14i	−0.718793	−	1.43129i
$470$		0			0
$471$	3.94779e13	−	6.83777e13i	0.0784764	−	0.135925i
$472$		0			0
$473$	−2.62685e13	−	4.54984e13i	−0.0510152	−	0.0883609i
$474$		0			0
$475$	−5.85566e14			−1.11112
$476$		0			0
$477$	−3.34389e14			−0.620014
$478$		0			0
$479$	2.65227e14	+	4.59387e14i	0.480588	+	0.832403i	0.999752	−	0.0222715i	$-0.00708983\pi$
$479$	2.65227e14	+	4.59387e14i	0.480588	+	0.832403i	−0.519164	+	0.854675i	$0.673756\pi$
$480$		0			0
$481$	3.92119e14	−	6.79169e14i	0.694416	−	1.20276i
$482$		0			0
$483$	−6.63817e13	−	3.86740e12i	−0.114905	−	0.00669437i
$484$		0			0
$485$	1.29897e14	−	2.24987e14i	0.219795	−	0.380696i
$486$		0			0
$487$	−3.51698e14	−	6.09159e14i	−0.581783	−	1.00768i	−0.995268	−	0.0971665i	$-0.969022\pi$
$487$	−3.51698e14	−	6.09159e14i	−0.581783	−	1.00768i	0.413485	−	0.910511i	$-0.364311\pi$
$488$		0			0
$489$	5.07399e14			0.820636
$490$		0			0
$491$	−1.13616e15			−1.79677			−0.898385	−	0.439209i	$-0.855259\pi$
$491$	−1.13616e15			−1.79677			−0.898385	+	0.439209i	$0.855259\pi$
$492$		0			0
$493$	2.33112e13	+	4.03763e13i	0.0360502	+	0.0624407i
$494$		0			0
$495$	1.24062e14	−	2.14883e14i	0.187634	−	0.324992i
$496$		0			0
$497$	6.53354e13	+	3.80644e12i	0.0966470	+	0.00563066i
$498$		0			0
$499$	−6.70441e13	+	1.16124e14i	−0.0970081	+	0.168023i	−0.910445	−	0.413630i	$-0.864261\pi$
$499$	−6.70441e13	+	1.16124e14i	−0.0970081	+	0.168023i	0.813437	+	0.581653i	$0.197594\pi$
$500$		0			0
$501$	1.36628e14	+	2.36646e14i	0.193389	+	0.334959i
$502$		0			0
$503$	2.57702e14			0.356857			0.178428	−	0.983953i	$-0.442899\pi$
$503$	2.57702e14			0.356857			0.178428	+	0.983953i	$0.442899\pi$
$504$		0			0
$505$	3.43238e14			0.465043
$506$		0			0
$507$	−3.62482e13	−	6.27837e13i	−0.0480554	−	0.0832344i
$508$		0			0
$509$	−6.06889e14	+	1.05116e15i	−0.787338	+	1.36371i	0.140254	+	0.990116i	$0.455208\pi$
$509$	−6.06889e14	+	1.05116e15i	−0.787338	+	1.36371i	−0.927592	+	0.373594i	$0.878125\pi$
$510$		0			0
$511$	−3.61744e14	−	7.20321e14i	−0.459289	−	0.914558i
$512$		0			0
$513$	1.31236e14	−	2.27307e14i	0.163083	−	0.282467i
$514$		0			0
$515$	1.24388e14	+	2.15446e14i	0.151300	+	0.262059i
$516$		0			0
$517$	1.35401e15			1.61222
$518$		0			0
$519$	4.25006e14			0.495421
$520$		0			0
$521$	−2.67571e12	−	4.63446e12i	−0.00305373	−	0.00528922i	0.864494	−	0.502642i	$-0.167639\pi$
$521$	−2.67571e12	−	4.63446e12i	−0.00305373	−	0.00528922i	−0.867548	+	0.497353i	$0.834305\pi$
$522$		0			0
$523$	6.28521e14	−	1.08863e15i	0.702362	−	1.21653i	−0.265274	−	0.964173i	$-0.585462\pi$
$523$	6.28521e14	−	1.08863e15i	0.702362	−	1.21653i	0.967635	−	0.252353i	$-0.0812044\pi$
$524$		0			0
$525$	1.90083e14	−	2.88997e14i	0.208002	−	0.316240i
$526$		0			0
$527$	−7.39675e13	+	1.28116e14i	−0.0792652	+	0.137291i
$528$		0			0
$529$	4.57471e14	+	7.92362e14i	0.480128	+	0.831606i
$530$		0			0
$531$	5.57682e14			0.573280
$532$		0			0
$533$	−1.40789e15			−1.41765
$534$		0			0
$535$	6.04135e13	+	1.04639e14i	0.0595920	+	0.103216i
$536$		0			0
$537$	4.34408e14	−	7.52417e14i	0.419797	−	0.727110i
$538$		0			0
$539$	−8.02457e14	−	1.86047e15i	−0.759773	−	1.76151i
$540$		0			0
$541$	−7.25705e13	+	1.25696e14i	−0.0673248	+	0.116610i	−0.897723	−	0.440561i	$-0.854780\pi$
$541$	−7.25705e13	+	1.25696e14i	−0.0673248	+	0.116610i	0.830398	+	0.557171i	$0.188113\pi$
$542$		0			0
$543$	−4.06609e14	−	7.04267e14i	−0.369639	−	0.640233i
$544$		0			0
$545$	3.78853e13			0.0337512
$546$		0			0
$547$	3.40968e14			0.297703			0.148852	−	0.988860i	$-0.452442\pi$
$547$	3.40968e14			0.297703			0.148852	+	0.988860i	$0.452442\pi$
$548$		0			0
$549$	−9.22303e13	−	1.59747e14i	−0.0789270	−	0.136706i
$550$		0			0
$551$	−7.27919e14	+	1.26079e15i	−0.610590	+	1.05757i
$552$		0			0
$553$	6.95976e14	−	1.05814e15i	0.572277	−	0.870073i
$554$		0			0
$555$	3.19697e14	−	5.53731e14i	0.257707	−	0.446362i
$556$		0			0
$557$	−6.59236e14	−	1.14183e15i	−0.520999	−	0.902397i	−0.999702	−	0.0244200i	$-0.992226\pi$
$557$	−6.59236e14	−	1.14183e15i	−0.520999	−	0.902397i	0.478703	−	0.877977i	$-0.341107\pi$
$558$		0			0
$559$	6.26644e13			0.0485575
$560$		0			0
$561$	1.45864e14			0.110828
$562$		0			0
$563$	−5.99221e13	−	1.03788e14i	−0.0446469	−	0.0773307i	0.842838	−	0.538167i	$-0.180883\pi$
$563$	−5.99221e13	−	1.03788e14i	−0.0446469	−	0.0773307i	−0.887485	+	0.460836i	$0.847550\pi$
$564$		0			0
$565$	−5.31497e14	+	9.20579e14i	−0.388360	+	0.672659i
$566$		0			0
$567$	6.95828e13	+	1.38557e14i	0.0498649	+	0.0992933i
$568$		0			0
$569$	1.01872e15	−	1.76448e15i	0.716043	−	1.24022i	−0.246513	−	0.969140i	$-0.579285\pi$
$569$	1.01872e15	−	1.76448e15i	0.716043	−	1.24022i	0.962556	−	0.271084i	$-0.0873820\pi$
$570$		0			0
$571$	8.36908e14	+	1.44957e15i	0.577004	+	0.999401i	0.995821	+	0.0913300i	$0.0291118\pi$
$571$	8.36908e14	+	1.44957e15i	0.577004	+	0.999401i	−0.418816	+	0.908071i	$0.637555\pi$
$572$		0			0
$573$	−1.80054e14			−0.121773
$574$		0			0
$575$	1.96993e14			0.130701
$576$		0			0
$577$	−6.35440e13	−	1.10061e14i	−0.0413626	−	0.0716421i	0.844603	−	0.535393i	$-0.179837\pi$
$577$	−6.35440e13	−	1.10061e14i	−0.0413626	−	0.0716421i	−0.885966	+	0.463751i	$0.846503\pi$
$578$		0			0
$579$	−3.36270e14	+	5.82436e14i	−0.214761	+	0.371977i
$580$		0			0
$581$	−1.82716e14	−	1.06451e13i	−0.114501	−	0.00667082i
$582$		0			0
$583$	2.90138e15	−	5.02533e15i	1.78413	−	3.09021i
$584$		0			0
$585$	1.47977e14	+	2.56305e14i	0.0892973	+	0.154667i
$586$		0			0
$587$	2.11940e13			0.0125517			0.00627586	−	0.999980i	$-0.498002\pi$
$587$	2.11940e13			0.0125517			0.00627586	+	0.999980i	$0.498002\pi$
$588$		0			0
$589$	−4.61944e15			−2.68507
$590$		0			0
$591$	−7.94405e13	−	1.37595e14i	−0.0453222	−	0.0785003i
$592$		0			0
$593$	−6.47682e14	+	1.12182e15i	−0.362712	+	0.628235i	−0.988406	−	0.151833i	$-0.951482\pi$
$593$	−6.47682e14	+	1.12182e15i	−0.362712	+	0.628235i	0.625695	+	0.780068i	$0.284816\pi$
$594$		0			0
$595$	−1.06638e14	−	6.21275e12i	−0.0586233	−	0.00341539i
$596$		0			0
$597$	3.80591e14	−	6.59203e14i	0.205400	−	0.355762i
$598$		0			0
$599$	−4.97391e14	−	8.61507e14i	−0.263543	−	0.456469i	0.703638	−	0.710558i	$-0.251558\pi$
$599$	−4.97391e14	−	8.61507e14i	−0.263543	−	0.456469i	−0.967181	+	0.254089i	$0.918224\pi$
$600$		0			0
$601$	9.14209e14			0.475593			0.237797	−	0.971315i	$-0.423575\pi$
$601$	9.14209e14			0.475593			0.237797	+	0.971315i	$0.423575\pi$
$602$		0			0
$603$	−1.04518e15			−0.533881
$604$		0			0
$605$	1.56790e15	+	2.71567e15i	0.786434	+	1.36214i
$606$		0			0
$607$	−9.39478e14	+	1.62722e15i	−0.462753	+	0.801511i	−0.999097	−	0.0424880i	$-0.986472\pi$
$607$	−9.39478e14	+	1.62722e15i	−0.462753	+	0.801511i	0.536344	+	0.843999i	$0.319805\pi$
$608$		0			0
$609$	−3.85951e14	−	7.68524e14i	−0.186697	−	0.371760i
$610$		0			0
$611$	−8.07511e14	+	1.39865e15i	−0.383638	+	0.664481i
$612$		0			0
$613$	−7.49482e14	−	1.29814e15i	−0.349726	−	0.605744i	0.636474	−	0.771298i	$-0.280392\pi$
$613$	−7.49482e14	−	1.29814e15i	−0.349726	−	0.605744i	−0.986201	+	0.165554i	$0.947059\pi$
$614$		0			0
$615$	−1.14786e15			−0.526108
$616$		0			0
$617$	−3.24729e15			−1.46202			−0.731008	−	0.682369i	$-0.760950\pi$
$617$	−3.24729e15			−1.46202			−0.731008	+	0.682369i	$0.760950\pi$
$618$		0			0
$619$	−5.98350e14	−	1.03637e15i	−0.264641	−	0.458371i	0.702829	−	0.711359i	$-0.251920\pi$
$619$	−5.98350e14	−	1.03637e15i	−0.264641	−	0.458371i	−0.967469	+	0.252988i	$0.918587\pi$
$620$		0			0
$621$	−4.41497e13	+	7.64694e13i	−0.0191833	+	0.0332265i
$622$		0			0
$623$	1.40274e15	−	2.13268e15i	0.598815	−	0.910421i
$624$		0			0
$625$	−1.01831e14	+	1.76376e14i	−0.0427109	+	0.0739775i
$626$		0			0
$627$	2.27737e15	+	3.94453e15i	0.938562	+	1.62564i
$628$		0			0
$629$	3.75876e14			0.152218
$630$		0			0
$631$	1.53157e15			0.609504			0.304752	−	0.952432i	$-0.401426\pi$
$631$	1.53157e15			0.609504			0.304752	+	0.952432i	$0.401426\pi$
$632$		0			0
$633$	7.29487e14	+	1.26351e15i	0.285297	+	0.494149i
$634$		0			0
$635$	7.56922e14	−	1.31103e15i	0.290935	−	0.503914i
$636$		0			0
$637$	2.40038e15	+	2.80645e14i	0.906802	+	0.106021i
$638$		0			0
$639$	4.34538e13	−	7.52641e13i	0.0161351	−	0.0279469i
$640$		0			0
$641$	2.08689e15	+	3.61460e15i	0.761693	+	1.31929i	0.941977	+	0.335676i	$0.108965\pi$
$641$	2.08689e15	+	3.61460e15i	0.761693	+	1.31929i	−0.180285	+	0.983615i	$0.557702\pi$
$642$		0			0
$643$	2.51431e15			0.902106			0.451053	−	0.892497i	$-0.351049\pi$
$643$	2.51431e15			0.902106			0.451053	+	0.892497i	$0.351049\pi$
$644$		0			0
$645$	5.10907e13			0.0180203
$646$		0			0
$647$	−3.57128e14	−	6.18563e14i	−0.123837	−	0.214492i	0.797441	−	0.603397i	$-0.206187\pi$
$647$	−3.57128e14	−	6.18563e14i	−0.123837	−	0.214492i	−0.921278	+	0.388905i	$0.872853\pi$
$648$		0			0
$649$	−4.83881e15	+	8.38107e15i	−1.64965	+	2.85728i
$650$		0			0
$651$	1.49953e15	−	2.27985e15i	0.502644	−	0.764206i
$652$		0			0
$653$	−2.84291e14	+	4.92406e14i	−0.0937001	+	0.162293i	−0.909065	−	0.416654i	$-0.863203\pi$
$653$	−2.84291e14	+	4.92406e14i	−0.0937001	+	0.162293i	0.815365	+	0.578947i	$0.196536\pi$
$654$		0			0
$655$	1.98494e14	+	3.43802e14i	0.0643311	+	0.111425i
$656$		0			0
$657$	−1.07038e15			−0.341136
$658$		0			0
$659$	3.30453e15			1.03572			0.517858	−	0.855467i	$-0.326730\pi$
$659$	3.30453e15			1.03572			0.517858	+	0.855467i	$0.326730\pi$
$660$		0			0
$661$	1.00297e15	+	1.73720e15i	0.309159	+	0.535479i	0.978179	−	0.207766i	$-0.0666192\pi$
$661$	1.00297e15	+	1.73720e15i	0.309159	+	0.535479i	−0.669020	+	0.743245i	$0.733286\pi$
$662$		0			0
$663$	−8.69905e13	+	1.50672e14i	−0.0263723	+	0.0456781i
$664$		0			0
$665$	−1.49694e15	−	2.98078e15i	−0.446360	−	0.888813i
$666$		0			0
$667$	2.44882e14	−	4.24149e14i	0.0718233	−	0.124402i
$668$		0			0
$669$	−1.35444e15	−	2.34596e15i	−0.390765	−	0.676825i
$670$		0			0
$671$	3.20100e15			0.908471
$672$		0			0
$673$	−6.47988e15			−1.80919			−0.904594	−	0.426274i	$-0.859826\pi$
$673$	−6.47988e15			−1.80919			−0.904594	+	0.426274i	$0.859826\pi$
$674$		0			0
$675$	−2.29668e14	−	3.97797e14i	−0.0630856	−	0.109267i
$676$		0			0
$677$	5.20385e14	−	9.01334e14i	0.140633	−	0.243584i	−0.787102	−	0.616823i	$-0.788420\pi$
$677$	5.20385e14	−	9.01334e14i	0.140633	−	0.243584i	0.927735	+	0.373239i	$0.121753\pi$
$678$		0			0
$679$	−2.81234e15	−	1.63847e14i	−0.747797	−	0.0435667i
$680$		0			0
$681$	−1.63930e15	+	2.83935e15i	−0.428894	+	0.742866i
$682$		0			0
$683$	2.41341e15	+	4.18016e15i	0.621324	+	1.07616i	0.989240	+	0.146305i	$0.0467381\pi$
$683$	2.41341e15	+	4.18016e15i	0.621324	+	1.07616i	−0.367916	+	0.929859i	$0.619929\pi$
$684$		0			0
$685$	2.66247e15			0.674507
$686$		0			0
$687$	−1.13142e15			−0.282073
$688$		0			0
$689$	3.46066e15	+	5.99404e15i	0.849090	+	1.47067i
$690$		0			0
$691$	2.56288e14	−	4.43904e14i	0.0618869	−	0.107191i	−0.833422	−	0.552637i	$-0.813622\pi$
$691$	2.56288e14	−	4.43904e14i	0.0618869	−	0.107191i	0.895309	+	0.445446i	$0.146955\pi$
$692$		0			0
$693$	−2.68603e15	−	1.56488e14i	−0.638377	−	0.0371919i
$694$		0			0
$695$	−1.81245e15	+	3.13925e15i	−0.423984	+	0.734361i
$696$		0			0
$697$	−3.37392e14	−	5.84380e14i	−0.0776880	−	0.134560i
$698$		0			0
$699$	4.16278e15			0.943537
$700$		0			0
$701$	−4.81777e15			−1.07497			−0.537486	−	0.843273i	$-0.680626\pi$
$701$	−4.81777e15			−1.07497			−0.537486	+	0.843273i	$0.680626\pi$
$702$		0			0
$703$	5.86857e15	+	1.01647e16i	1.28907	+	2.23274i
$704$		0			0
$705$	−6.58369e14	+	1.14033e15i	−0.142373	+	0.246598i
$706$		0			0
$707$	−1.67036e15	−	3.32609e15i	−0.355633	−	0.708153i
$708$		0			0
$709$	−5.30564e14	+	9.18964e14i	−0.111220	+	0.192639i	−0.916262	−	0.400578i	$-0.868809\pi$
$709$	−5.30564e14	+	9.18964e14i	−0.111220	+	0.192639i	0.805042	+	0.593217i	$0.202143\pi$
$710$		0			0
$711$	−8.40915e14	−	1.45651e15i	−0.173568	−	0.300628i
$712$		0			0
$713$	1.55404e15			0.315843
$714$		0			0
$715$	−5.13579e15			−1.02784
$716$		0			0
$717$	2.57225e15	+	4.45527e15i	0.506942	+	0.878049i
$718$		0			0
$719$	4.52805e14	−	7.84282e14i	0.0878826	−	0.152217i	−0.818733	−	0.574174i	$-0.805323\pi$
$719$	4.52805e14	−	7.84282e14i	0.0878826	−	0.152217i	0.906616	+	0.421957i	$0.138657\pi$
$720$		0			0
$721$	1.48241e15	−	2.25382e15i	0.283351	−	0.430799i
$722$		0			0
$723$	−2.64422e15	+	4.57993e15i	−0.497780	+	0.862180i
$724$		0			0
$725$	1.27389e15	+	2.20644e15i	0.236196	+	0.409103i
$726$		0			0
$727$	−3.27963e15			−0.598944			−0.299472	−	0.954105i	$-0.596810\pi$
$727$	−3.27963e15			−0.598944			−0.299472	+	0.954105i	$0.596810\pi$
$728$		0			0
$729$	2.05891e14			0.0370370
$730$		0			0
$731$	1.50172e13	+	2.60105e13i	0.00266098	+	0.00460896i
$732$		0			0
$733$	3.03272e15	−	5.25283e15i	0.529372	−	0.916899i	−0.470041	−	0.882644i	$-0.655761\pi$
$733$	3.03272e15	−	5.25283e15i	0.529372	−	0.916899i	0.999413	−	0.0342543i	$-0.0109056\pi$
$734$		0			0
$735$	1.95704e15	+	2.28811e14i	0.336527	+	0.0393457i
$736$		0			0
$737$	9.06863e15	−	1.57073e16i	1.53628	−	2.66091i
$738$		0			0
$739$	5.22037e13	+	9.04195e13i	0.00871278	+	0.0150910i	0.870349	−	0.492436i	$-0.163893\pi$
$739$	5.22037e13	+	9.04195e13i	0.00871278	+	0.0150910i	−0.861636	+	0.507527i	$0.830560\pi$
$740$		0			0
$741$	−5.43275e15			−0.893346
$742$		0			0
$743$	−3.29447e15			−0.533762			−0.266881	−	0.963730i	$-0.585993\pi$
$743$	−3.29447e15			−0.533762			−0.266881	+	0.963730i	$0.585993\pi$
$744$		0			0
$745$	−3.01104e15	−	5.21528e15i	−0.480682	−	0.832565i
$746$		0			0
$747$	−1.21522e14	+	2.10483e14i	−0.0191158	+	0.0331096i
$748$		0			0
$749$	7.19988e14	−	1.09465e15i	0.111603	−	0.169678i
$750$		0			0
$751$	4.49675e15	−	7.78860e15i	0.686877	−	1.18971i	−0.285966	−	0.958240i	$-0.592314\pi$
$751$	4.49675e15	−	7.78860e15i	0.686877	−	1.18971i	0.972843	−	0.231466i	$-0.0743523\pi$
$752$		0			0
$753$	−2.88142e15	−	4.99076e15i	−0.433744	−	0.751267i
$754$		0			0
$755$	3.49695e15			0.518778
$756$		0			0
$757$	−5.75460e15			−0.841371			−0.420686	−	0.907206i	$-0.638210\pi$
$757$	−5.75460e15			−0.841371			−0.420686	+	0.907206i	$0.638210\pi$
$758$		0			0
$759$	−7.66141e14	−	1.32700e15i	−0.110403	−	0.191223i
$760$		0			0
$761$	−1.20159e15	+	2.08121e15i	−0.170663	+	0.295597i	−0.938652	−	0.344866i	$-0.887924\pi$
$761$	−1.20159e15	+	2.08121e15i	−0.170663	+	0.295597i	0.767989	+	0.640463i	$0.221258\pi$
$762$		0			0
$763$	−1.84367e14	−	3.67120e14i	−0.0258106	−	0.0513953i
$764$		0			0
$765$	−7.09239e13	+	1.22844e14i	−0.00978711	+	0.0169518i
$766$		0			0
$767$	−5.77157e15	−	9.99665e15i	−0.785090	−	1.35982i
$768$		0			0
$769$	−2.80094e15			−0.375586			−0.187793	−	0.982209i	$-0.560133\pi$
$769$	−2.80094e15			−0.375586			−0.187793	+	0.982209i	$0.560133\pi$
$770$		0			0
$771$	−3.31099e15			−0.437683
$772$		0			0
$773$	5.61359e15	+	9.72303e15i	0.731567	+	1.26711i	0.956213	+	0.292670i	$0.0945438\pi$
$773$	5.61359e15	+	9.72303e15i	0.731567	+	1.26711i	−0.224647	+	0.974440i	$0.572123\pi$
$774$		0			0
$775$	−4.04210e15	+	7.00112e15i	−0.519335	+	0.899514i
$776$		0			0
$777$	−6.92162e15	−	4.03254e14i	−0.876783	−	0.0510814i
$778$		0			0
$779$	1.05354e16	−	1.82479e16i	1.31582	−	2.27907i
$780$		0			0
$781$	7.54066e14	+	1.30608e15i	0.0928600	+	0.160838i
$782$		0			0
$783$	−1.14201e15			−0.138669
$784$		0			0
$785$	−1.33242e15			−0.159536
$786$		0			0
$787$	3.91239e15	+	6.77646e15i	0.461935	+	0.800095i	0.999057	−	0.0434092i	$-0.0138219\pi$
$787$	3.91239e15	+	6.77646e15i	0.461935	+	0.800095i	−0.537122	+	0.843504i	$0.680489\pi$
$788$		0			0
$789$	−1.97482e15	+	3.42049e15i	−0.229934	+	0.398258i
$790$		0			0
$791$	1.15072e16	+	6.70411e14i	1.32129	+	0.0769787i
$792$		0			0
$793$	−1.90902e15	+	3.30652e15i	−0.216176	+	0.374428i
$794$		0			0
$795$	2.82150e15	+	4.88698e15i	0.315109	+	0.545785i
$796$		0			0
$797$	9.88414e15			1.08872			0.544362	−	0.838850i	$-0.316772\pi$
$797$	9.88414e15			1.08872			0.544362	+	0.838850i	$0.316772\pi$
$798$		0			0
$799$	−7.74061e14			−0.0840946
$800$		0			0
$801$	−1.69486e15	−	2.93559e15i	−0.181617	−	0.314569i
$802$		0			0
$803$	9.28728e15	−	1.60860e16i	0.981641	−	1.70025i
$804$		0			0
$805$	5.03592e14	+	1.00278e15i	0.0525051	+	0.104551i
$806$		0			0
$807$	1.56674e15	−	2.71367e15i	0.161136	−	0.279095i
$808$		0			0
$809$	4.12534e12	+	7.14529e12i	0.000418545	+	0.000724942i	0.866235	−	0.499637i	$-0.166533\pi$
$809$	4.12534e12	+	7.14529e12i	0.000418545	+	0.000724942i	−0.865816	+	0.500362i	$0.833200\pi$
$810$		0			0
$811$	−1.09537e16			−1.09635			−0.548173	−	0.836365i	$-0.684676\pi$
$811$	−1.09537e16			−1.09635			−0.548173	+	0.836365i	$0.684676\pi$
$812$		0			0
$813$	−7.17543e15			−0.708517
$814$		0			0
$815$	−4.28131e15	−	7.41545e15i	−0.417071	−	0.722388i
$816$		0			0
$817$	−4.68927e14	+	8.12206e14i	−0.0450697	+	0.0780629i
$818$		0			0
$819$	1.76355e15	−	2.68124e15i	0.167234	−	0.254258i
$820$		0			0
$821$	−9.47266e13	+	1.64071e14i	−0.00886308	+	0.0153513i	−0.870423	−	0.492305i	$-0.836155\pi$
$821$	−9.47266e13	+	1.64071e14i	−0.00886308	+	0.0153513i	0.861560	+	0.507656i	$0.169488\pi$
$822$		0			0
$823$	5.25852e15	+	9.10802e15i	0.485472	+	0.840862i	0.999861	−	0.0166952i	$-0.00531449\pi$
$823$	5.25852e15	+	9.10802e15i	0.485472	+	0.840862i	−0.514389	+	0.857557i	$0.671981\pi$
$824$		0			0
$825$	7.97099e15			0.726132
$826$		0			0
$827$	−1.30003e16			−1.16862			−0.584308	−	0.811532i	$-0.698634\pi$
$827$	−1.30003e16			−1.16862			−0.584308	+	0.811532i	$0.698634\pi$
$828$		0			0
$829$	−6.35907e15	−	1.10142e16i	−0.564083	−	0.977021i	−0.997134	−	0.0756517i	$-0.975896\pi$
$829$	−6.35907e15	−	1.10142e16i	−0.564083	−	0.977021i	0.433051	−	0.901369i	$-0.357437\pi$
$830$		0			0
$831$	−3.16008e14	+	5.47341e14i	−0.0276625	+	0.0479129i
$832$		0			0
$833$	4.58748e14	+	1.06359e15i	0.0396302	+	0.0918815i
$834$		0			0
$835$	2.30567e15	−	3.99353e15i	0.196572	−	0.340472i
$836$		0			0
$837$	−1.81181e15	−	3.13815e15i	−0.152449	−	0.264049i
$838$		0			0
$839$	−6.10069e15			−0.506626			−0.253313	−	0.967384i	$-0.581520\pi$
$839$	−6.10069e15			−0.506626			−0.253313	+	0.967384i	$0.581520\pi$
$840$		0			0
$841$	−5.86621e15			−0.480817
$842$		0			0
$843$	3.94592e15	+	6.83453e15i	0.319225	+	0.552913i
$844$		0			0
$845$	−6.11707e14	+	1.05951e15i	−0.0488463	+	0.0846043i
$846$		0			0
$847$	1.86857e16	−	2.84091e16i	1.47282	−	2.23923i
$848$		0			0
$849$	3.77649e15	−	6.54107e15i	0.293830	−	0.508929i
$850$		0			0
$851$	−1.97427e15	−	3.41954e15i	−0.151633	−	0.262636i
$852$		0			0
$853$	−7.99289e15			−0.606016			−0.303008	−	0.952988i	$-0.597991\pi$
$853$	−7.99289e15			−0.606016			−0.303008	+	0.952988i	$0.597991\pi$
$854$		0			0
$855$	−4.42935e15			−0.331533
$856$		0			0
$857$	3.18706e15	+	5.52014e15i	0.235502	+	0.407902i	0.959419	−	0.281986i	$-0.0909931\pi$
$857$	3.18706e15	+	5.52014e15i	0.235502	+	0.407902i	−0.723916	+	0.689888i	$0.757660\pi$
$858$		0			0
$859$	1.07451e16	−	1.86110e16i	0.783876	−	1.35771i	−0.145792	−	0.989315i	$-0.546573\pi$
$859$	1.07451e16	−	1.86110e16i	0.783876	−	1.35771i	0.929668	−	0.368398i	$-0.120094\pi$
$860$		0			0
$861$	5.58601e15	+	1.11231e16i	0.402332	+	0.801141i
$862$		0			0
$863$	−1.04765e16	+	1.81458e16i	−0.745001	+	1.29038i	0.205193	+	0.978722i	$0.434218\pi$
$863$	−1.04765e16	+	1.81458e16i	−0.745001	+	1.29038i	−0.950194	+	0.311659i	$0.899115\pi$
$864$		0			0
$865$	−3.58610e15	−	6.21131e15i	−0.251787	−	0.436108i
$866$		0			0
$867$	8.24468e15			0.571569
$868$		0			0
$869$	2.91853e16			1.99781
$870$		0			0
$871$	1.08167e16	+	1.87352e16i	0.731134	+	1.26636i
$872$		0			0
$873$	−1.87045e15	+	3.23972e15i	−0.124844	+	0.216236i
$874$		0			0
$875$	−1.47161e16	−	8.57363e14i	−0.969950	−	0.0565094i
$876$		0			0
$877$	5.89581e14	−	1.02118e15i	0.0383748	−	0.0664671i	−0.846200	−	0.532865i	$-0.821115\pi$
$877$	5.89581e14	−	1.02118e15i	0.0383748	−	0.0664671i	0.884575	+	0.466398i	$0.154449\pi$
$878$		0			0
$879$	−5.96323e14	−	1.03286e15i	−0.0383304	−	0.0663901i
$880$		0			0
$881$	1.24949e16			0.793166			0.396583	−	0.917999i	$-0.370196\pi$
$881$	1.24949e16			0.793166			0.396583	+	0.917999i	$0.370196\pi$
$882$		0			0
$883$	−1.08012e16			−0.677155			−0.338577	−	0.940939i	$-0.609946\pi$
$883$	−1.08012e16			−0.677155			−0.338577	+	0.940939i	$0.609946\pi$
$884$		0			0
$885$	−4.70559e15	−	8.15033e15i	−0.291358	−	0.504646i
$886$		0			0
$887$	−8.75702e15	+	1.51676e16i	−0.535521	+	0.927549i	0.463617	+	0.886036i	$0.346551\pi$
$887$	−8.75702e15	+	1.51676e16i	−0.535521	+	0.927549i	−0.999138	+	0.0415136i	$0.986782\pi$
$888$		0			0
$889$	−1.63878e16	−	9.54754e14i	−0.989831	−	0.0576676i
$890$		0			0
$891$	−1.78644e15	+	3.09421e15i	−0.106577	+	0.184596i
$892$		0			0
$893$	−1.20855e16	−	2.09326e16i	−0.712164	−	1.23350i
$894$		0			0
$895$	−1.46617e16			−0.853412
$896$		0			0
$897$	1.82765e15			0.105084
$898$		0			0
$899$	1.00495e16	+	1.74062e16i	0.570776	+	0.988613i
$900$		0			0
$901$	−1.65866e15	+	2.87287e15i	−0.0930614	+	0.161187i
$902$		0			0
$903$	−2.48631e14	−	4.95085e14i	−0.0137807	−	0.0274408i
$904$		0			0
$905$	−6.86173e15	+	1.18849e16i	−0.375722	+	0.650770i
$906$		0			0
$907$	1.24394e16	+	2.15456e16i	0.672911	+	1.16552i	0.977075	+	0.212896i	$0.0682897\pi$
$907$	1.24394e16	+	2.15456e16i	0.672911	+	1.16552i	−0.304164	+	0.952620i	$0.598377\pi$
$908$		0			0
$909$	−4.94248e15			−0.264146
$910$		0			0
$911$	−1.22551e16			−0.647090			−0.323545	−	0.946213i	$-0.604875\pi$
$911$	−1.22551e16			−0.647090			−0.323545	+	0.946213i	$0.604875\pi$
$912$		0			0
$913$	−2.10881e15	−	3.65257e15i	−0.110014	−	0.190550i
$914$		0			0
$915$	−1.55643e15	+	2.69582e15i	−0.0802259	+	0.138955i
$916$		0			0
$917$	2.36559e15	−	3.59657e15i	0.120478	−	0.183171i
$918$		0			0
$919$	−4.83221e15	+	8.36964e15i	−0.243170	+	0.421184i	−0.961616	−	0.274400i	$-0.911521\pi$
$919$	−4.83221e15	+	8.36964e15i	−0.243170	+	0.421184i	0.718445	+	0.695584i	$0.244854\pi$
$920$		0			0
$921$	3.76787e15	+	6.52614e15i	0.187356	+	0.324510i
$922$		0			0
$923$	−1.79885e15			−0.0883863
$924$		0			0
$925$	2.05405e16			0.997311
$926$		0			0
$927$	−1.79113e15	−	3.10232e15i	−0.0859385	−	0.148850i
$928$		0			0
$929$	−8.47475e15	+	1.46787e16i	−0.401828	+	0.695987i	−0.993947	−	0.109864i	$-0.964959\pi$
$929$	−8.47475e15	+	1.46787e16i	−0.401828	+	0.695987i	0.592118	+	0.805851i	$0.298292\pi$
$930$		0			0
$931$	−2.15999e16	+	2.90117e16i	−1.01211	+	1.35941i
$932$		0			0
$933$	1.27340e15	−	2.20559e15i	0.0589678	−	0.102135i
$934$		0			0
$935$	−1.23076e15	−	2.13174e15i	−0.0563262	−	0.0975598i
$936$		0			0
$937$	2.83000e16			1.28003			0.640013	−	0.768364i	$-0.278929\pi$
$937$	2.83000e16			1.28003			0.640013	+	0.768364i	$0.278929\pi$
$938$		0			0
$939$	−4.08710e15			−0.182707
$940$		0			0
$941$	2.07956e15	+	3.60190e15i	0.0918814	+	0.159143i	0.908303	−	0.418313i	$-0.137379\pi$
$941$	2.07956e15	+	3.60190e15i	0.0918814	+	0.159143i	−0.816421	+	0.577457i	$0.804045\pi$
$942$		0			0
$943$	−3.54427e15	+	6.13885e15i	−0.154779	+	0.268085i
$944$		0			0
$945$	1.43783e15	−	2.18604e15i	0.0620630	−	0.0943588i
$946$		0			0
$947$	−3.90192e15	+	6.75833e15i	−0.166477	+	0.288346i	−0.937179	−	0.348850i	$-0.886572\pi$
$947$	−3.90192e15	+	6.75833e15i	−0.166477	+	0.288346i	0.770702	+	0.637196i	$0.219906\pi$
$948$		0			0
$949$	1.10775e16	+	1.91869e16i	0.467175	+	0.809170i
$950$		0			0
$951$	9.94577e15			0.414616
$952$		0			0
$953$	−5.63766e15			−0.232321			−0.116160	−	0.993230i	$-0.537059\pi$
$953$	−5.63766e15			−0.232321			−0.116160	+	0.993230i	$0.537059\pi$
$954$		0			0
$955$	1.51925e15	+	2.63143e15i	0.0618888	+	0.107195i
$956$		0			0
$957$	9.90877e15	−	1.71625e16i	0.399029	−	0.691138i
$958$		0			0
$959$	−1.29568e16	−	2.58002e16i	−0.515817	−	1.02712i
$960$		0			0
$961$	−1.91832e16	+	3.32263e16i	−0.754993	+	1.30769i
$962$		0			0
$963$	−8.69927e14	−	1.50676e15i	−0.0338484	−	0.0586271i
$964$		0			0
$965$	1.13495e16			0.436591
$966$		0			0
$967$	−7.87792e15			−0.299617			−0.149808	−	0.988715i	$-0.547866\pi$
$967$	−7.87792e15			−0.299617			−0.149808	+	0.988715i	$0.547866\pi$
$968$		0			0
$969$	−1.30193e15	−	2.25500e15i	−0.0489560	−	0.0847942i
$970$		0			0
$971$	−7.03435e15	+	1.21838e16i	−0.261528	+	0.452980i	−0.966648	−	0.256108i	$-0.917560\pi$
$971$	−7.03435e15	+	1.21838e16i	−0.261528	+	0.452980i	0.705120	+	0.709088i	$0.250893\pi$
$972$		0			0
$973$	3.92406e16	+	2.28616e15i	1.44250	+	0.0840399i
$974$		0			0
$975$	−4.75376e15	+	8.23376e15i	−0.172787	+	0.299277i
$976$		0			0
$977$	2.08584e16	+	3.61279e16i	0.749655	+	1.29844i	0.947988	+	0.318307i	$0.103114\pi$
$977$	2.08584e16	+	3.61279e16i	0.749655	+	1.29844i	−0.198332	+	0.980135i	$0.563553\pi$
$978$		0			0
$979$	5.88228e16			2.09046
$980$		0			0
$981$	−5.45531e14			−0.0191708
$982$		0			0
$983$	−3.82943e15	−	6.63276e15i	−0.133073	−	0.230489i	0.791787	−	0.610798i	$-0.209151\pi$
$983$	−3.82943e15	−	6.63276e15i	−0.133073	−	0.230489i	−0.924860	+	0.380309i	$0.875818\pi$
$984$		0			0
$985$	−1.34060e15	+	2.32199e15i	−0.0460681	+	0.0797922i
$986$		0			0
$987$	1.42541e16	+	8.30443e14i	0.484389	+	0.0282205i
$988$		0			0
$989$	1.57754e14	−	2.73238e14i	0.00530152	−	0.00918250i
$990$		0			0
$991$	−2.42943e16	−	4.20790e16i	−0.807420	−	1.39849i	−0.914645	−	0.404258i	$-0.867530\pi$
$991$	−2.42943e16	−	4.20790e16i	−0.807420	−	1.39849i	0.107225	−	0.994235i	$-0.465804\pi$
$992$		0			0
$993$	−1.16461e16			−0.382788
$994$		0			0
$995$	−1.28453e16			−0.417560
$996$		0			0
$997$	−1.89471e16	−	3.28173e16i	−0.609143	−	1.05507i	−0.991382	−	0.131003i	$-0.958180\pi$
$997$	−1.89471e16	−	3.28173e16i	−0.609143	−	1.05507i	0.382239	−	0.924063i	$-0.375153\pi$
$998$		0			0
$999$	−4.60349e15	+	7.97348e15i	−0.146378	+	0.253535i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.i.a.37.4	yes	14
3.2	odd	2		252.12.k.b.37.4		14
7.4	even	3	inner	84.12.i.a.25.4	✓	14
21.11	odd	6		252.12.k.b.109.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.i.a.25.4	✓	14	7.4	even	3	inner
84.12.i.a.37.4	yes	14	1.1	even	1	trivial
252.12.k.b.37.4		14	3.2	odd	2
252.12.k.b.109.4		14	21.11	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(121.500 + 210.444i) q^{3} +(2050.38 - 3551.36i) q^{5} +(-44391.9 - 2586.27i) q^{7} +(-29524.5 + 51137.9i) q^{9} +O(q^{10})\)	\(q+(121.500 + 210.444i) q^{3} +(2050.38 - 3551.36i) q^{5} +(-44391.9 - 2586.27i) q^{7} +(-29524.5 + 51137.9i) q^{9} +(-512347. - 887411. i) q^{11} +1.22222e6 q^{13} +996483. q^{15} +(292898. + 507314. i) q^{17} +(-9.14606e6 + 1.58414e7i) q^{19} +(-4.84935e6 - 9.65624e6i) q^{21} +(3.07687e6 - 5.32929e6i) q^{23} +(1.60060e7 + 2.77232e7i) q^{25} -1.43489e7 q^{27} +7.95883e7 q^{29} +(1.26268e8 + 2.18703e8i) q^{31} +(1.24500e8 - 2.15641e8i) q^{33} +(-1.00205e8 + 1.52349e8i) q^{35} +(3.20825e8 - 5.55685e8i) q^{37} +(1.48500e8 + 2.57209e8i) q^{39} -1.15191e9 q^{41} +5.12710e7 q^{43} +(1.21073e8 + 2.09704e8i) q^{45} +(-6.60692e8 + 1.14435e9i) q^{47} +(1.96395e9 + 2.29619e8i) q^{49} +(-7.11742e7 + 1.23277e8i) q^{51} +(2.83146e9 + 4.90423e9i) q^{53} -4.20202e9 q^{55} -4.44499e9 q^{57} +(-4.72220e9 - 8.17909e9i) q^{59} +(-1.56193e9 + 2.70534e9i) q^{61} +(1.44290e9 - 2.19375e9i) q^{63} +(2.50601e9 - 4.34054e9i) q^{65} +(8.85008e9 + 1.53288e10i) q^{67} +1.49536e9 q^{69} -1.47179e9 q^{71} +(9.06346e9 + 1.56984e10i) q^{73} +(-3.88945e9 + 6.73673e9i) q^{75} +(2.04490e10 + 4.07189e10i) q^{77} +(-1.42410e10 + 2.46661e10i) q^{79} +(-1.74339e9 - 3.01964e9i) q^{81} +4.11599e9 q^{83} +2.40220e9 q^{85} +(9.66998e9 + 1.67489e10i) q^{87} +(-2.87026e10 + 4.97144e10i) q^{89} +(-5.42566e10 - 3.16099e9i) q^{91} +(-3.06832e10 + 5.31449e10i) q^{93} +(3.75057e10 + 6.49619e10i) q^{95} +6.33525e10 q^{97} +6.05072e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 1701 q^{3} + 7218 q^{5} + 35001 q^{7} - 413343 q^{9}+O(q^{10}) \) 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

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