LMFDB - Embedded newform 84.12.i.a.37.3

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			37.3
Root			$2987.06 + 5173.74i$ of defining polynomial
Character	$\chi$	$=$	84.37
Dual form			84.12.i.a.25.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(121.500 + 210.444i) q^{3} +(-2471.06 + 4280.00i) q^{5} +(-15790.8 + 41569.0i) q^{7} +(-29524.5 + 51137.9i) q^{9} +O(q^{10})$	\(q+(121.500 + 210.444i) q^{3} +(-2471.06 + 4280.00i) q^{5} +(-15790.8 + 41569.0i) q^{7} +(-29524.5 + 51137.9i) q^{9} +(288366. + 499465. i) q^{11} +1.93950e6 q^{13} -1.20093e6 q^{15} +(3.06112e6 + 5.30202e6i) q^{17} +(1.10763e6 - 1.91847e6i) q^{19} +(-1.06665e7 + 1.72755e6i) q^{21} +(-7.04861e6 + 1.22086e7i) q^{23} +(1.22018e7 + 2.11342e7i) q^{25} -1.43489e7 q^{27} +4.59423e7 q^{29} +(-3.58987e7 - 6.21784e7i) q^{31} +(-7.00730e7 + 1.21370e8i) q^{33} +(-1.38895e8 - 1.70304e8i) q^{35} +(-2.42110e8 + 4.19347e8i) q^{37} +(2.35649e8 + 4.08157e8i) q^{39} -3.28509e8 q^{41} +7.73052e8 q^{43} +(-1.45913e8 - 2.52730e8i) q^{45} +(5.37762e8 - 9.31431e8i) q^{47} +(-1.47863e9 - 1.31281e9i) q^{49} +(-7.43853e8 + 1.28839e9i) q^{51} +(9.58988e7 + 1.66102e8i) q^{53} -2.85028e9 q^{55} +5.38309e8 q^{57} +(1.91762e9 + 3.32142e9i) q^{59} +(2.37046e9 - 4.10575e9i) q^{61} +(-1.65954e9 - 2.03481e9i) q^{63} +(-4.79262e9 + 8.30106e9i) q^{65} +(-3.82226e9 - 6.62035e9i) q^{67} -3.42562e9 q^{69} -2.37639e10 q^{71} +(2.96220e8 + 5.13068e8i) q^{73} +(-2.96504e9 + 5.13560e9i) q^{75} +(-2.53158e10 + 4.10013e9i) q^{77} +(-7.44339e9 + 1.28923e10i) q^{79} +(-1.74339e9 - 3.01964e9i) q^{81} +3.27603e9 q^{83} -3.02568e10 q^{85} +(5.58199e9 + 9.66828e9i) q^{87} +(-3.21650e10 + 5.57114e10i) q^{89} +(-3.06263e10 + 8.06230e10i) q^{91} +(8.72339e9 - 1.51094e10i) q^{93} +(5.47404e9 + 9.48132e9i) q^{95} -1.14678e11 q^{97} -3.40555e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2471.06	+	4280.00i	−0.353629	+	0.612503i	−0.986882	−	0.161441i	$-0.948386\pi$
$5$	−2471.06	+	4280.00i	−0.353629	+	0.612503i	0.633253	+	0.773945i	$0.281719\pi$
$6$		0			0
$7$	−15790.8	+	41569.0i	−0.355111	+	0.934824i
$7$	−15790.8	+	41569.0i	−0.355111	+	0.934824i
$8$		0			0
$9$	−29524.5	+	51137.9i	−0.166667	+	0.288675i
$10$		0			0
$11$	288366.	+	499465.i	0.539864	+	0.935072i	0.998911	+	0.0466599i	$0.0148577\pi$
$11$	288366.	+	499465.i	0.539864	+	0.935072i	−0.459047	+	0.888412i	$0.651809\pi$
$12$		0			0
$13$	1.93950e6			1.44878			0.724389	−	0.689392i	$-0.242122\pi$
$13$	1.93950e6			1.44878			0.724389	+	0.689392i	$0.242122\pi$
$14$		0			0
$15$	−1.20093e6			−0.408336
$16$		0			0
$17$	3.06112e6	+	5.30202e6i	0.522891	+	0.905675i	0.999645	+	0.0266377i	$0.00848004\pi$
$17$	3.06112e6	+	5.30202e6i	0.522891	+	0.905675i	−0.476754	+	0.879037i	$0.658187\pi$
$18$		0			0
$19$	1.10763e6	−	1.91847e6i	0.102624	−	0.177751i	−0.810141	−	0.586235i	$-0.800609\pi$
$19$	1.10763e6	−	1.91847e6i	0.102624	−	0.177751i	0.912765	+	0.408485i	$0.133943\pi$
$20$		0			0
$21$	−1.06665e7	+	1.72755e6i	−0.569924	+	0.0923048i
$22$		0			0
$23$	−7.04861e6	+	1.22086e7i	−0.228350	+	0.395513i	−0.957319	−	0.289033i	$-0.906666\pi$
$23$	−7.04861e6	+	1.22086e7i	−0.228350	+	0.395513i	0.728969	+	0.684546i	$0.240000\pi$
$24$		0			0
$25$	1.22018e7	+	2.11342e7i	0.249893	+	0.432828i
$26$		0			0
$27$	−1.43489e7			−0.192450
$28$		0			0
$29$	4.59423e7			0.415933			0.207967	−	0.978136i	$-0.433315\pi$
$29$	4.59423e7			0.415933			0.207967	+	0.978136i	$0.433315\pi$
$30$		0			0
$31$	−3.58987e7	−	6.21784e7i	−0.225211	−	0.390077i	0.731172	−	0.682193i	$-0.238974\pi$
$31$	−3.58987e7	−	6.21784e7i	−0.225211	−	0.390077i	−0.956383	+	0.292117i	$0.905640\pi$
$32$		0			0
$33$	−7.00730e7	+	1.21370e8i	−0.311691	+	0.539864i
$34$		0			0
$35$	−1.38895e8	−	1.70304e8i	−0.447005	−	0.548088i
$36$		0			0
$37$	−2.42110e8	+	4.19347e8i	−0.573989	+	0.994177i	0.422162	+	0.906520i	$0.361271\pi$
$37$	−2.42110e8	+	4.19347e8i	−0.573989	+	0.994177i	−0.996151	+	0.0876570i	$0.972062\pi$
$38$		0			0
$39$	2.35649e8	+	4.08157e8i	0.418226	+	0.724389i
$40$		0			0
$41$	−3.28509e8			−0.442830			−0.221415	−	0.975180i	$-0.571067\pi$
$41$	−3.28509e8			−0.442830			−0.221415	+	0.975180i	$0.571067\pi$
$42$		0			0
$43$	7.73052e8			0.801922			0.400961	−	0.916095i	$-0.368676\pi$
$43$	7.73052e8			0.801922			0.400961	+	0.916095i	$0.368676\pi$
$44$		0			0
$45$	−1.45913e8	−	2.52730e8i	−0.117876	−	0.204168i
$46$		0			0
$47$	5.37762e8	−	9.31431e8i	0.342020	−	0.592396i	−0.642788	−	0.766045i	$-0.722222\pi$
$47$	5.37762e8	−	9.31431e8i	0.342020	−	0.592396i	0.984808	+	0.173648i	$0.0555555\pi$
$48$		0			0
$49$	−1.47863e9	−	1.31281e9i	−0.747792	−	0.663933i
$50$		0			0
$51$	−7.43853e8	+	1.28839e9i	−0.301892	+	0.522891i
$52$		0			0
$53$	9.58988e7	+	1.66102e8i	0.0314990	+	0.0545578i	0.881345	−	0.472473i	$-0.156639\pi$
$53$	9.58988e7	+	1.66102e8i	0.0314990	+	0.0545578i	−0.849846	+	0.527031i	$0.823305\pi$
$54$		0			0
$55$	−2.85028e9			−0.763646
$56$		0			0
$57$	5.38309e8			0.118500
$58$		0			0
$59$	1.91762e9	+	3.32142e9i	0.349203	+	0.604837i	0.986108	−	0.166105i	$-0.0531191\pi$
$59$	1.91762e9	+	3.32142e9i	0.349203	+	0.604837i	−0.636905	+	0.770942i	$0.719786\pi$
$60$		0			0
$61$	2.37046e9	−	4.10575e9i	0.359350	−	0.622413i	−0.628502	−	0.777808i	$-0.716332\pi$
$61$	2.37046e9	−	4.10575e9i	0.359350	−	0.622413i	0.987852	+	0.155395i	$0.0496649\pi$
$62$		0			0
$63$	−1.65954e9	−	2.03481e9i	−0.210675	−	0.258316i
$64$		0			0
$65$	−4.79262e9	+	8.30106e9i	−0.512330	+	0.887381i
$66$		0			0
$67$	−3.82226e9	−	6.62035e9i	−0.345867	−	0.599059i	0.639644	−	0.768671i	$-0.279082\pi$
$67$	−3.82226e9	−	6.62035e9i	−0.345867	−	0.599059i	−0.985511	+	0.169612i	$0.945748\pi$
$68$		0			0
$69$	−3.42562e9			−0.263675
$70$		0			0
$71$	−2.37639e10			−1.56313			−0.781566	−	0.623822i	$-0.785579\pi$
$71$	−2.37639e10			−1.56313			−0.781566	+	0.623822i	$0.785579\pi$
$72$		0			0
$73$	2.96220e8	+	5.13068e8i	0.0167240	+	0.0289667i	0.874266	−	0.485447i	$-0.161343\pi$
$73$	2.96220e8	+	5.13068e8i	0.0167240	+	0.0289667i	−0.857542	+	0.514413i	$0.828010\pi$
$74$		0			0
$75$	−2.96504e9	+	5.13560e9i	−0.144276	+	0.249893i
$76$		0			0
$77$	−2.53158e10	+	4.10013e9i	−1.06584	+	0.172623i
$78$		0			0
$79$	−7.44339e9	+	1.28923e10i	−0.272159	+	0.471392i	−0.969414	−	0.245430i	$-0.921071\pi$
$79$	−7.44339e9	+	1.28923e10i	−0.272159	+	0.471392i	0.697256	+	0.716822i	$0.254404\pi$
$80$		0			0
$81$	−1.74339e9	−	3.01964e9i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	3.27603e9			0.0912889			0.0456444	−	0.998958i	$-0.485466\pi$
$83$	3.27603e9			0.0912889			0.0456444	+	0.998958i	$0.485466\pi$
$84$		0			0
$85$	−3.02568e10			−0.739638
$86$		0			0
$87$	5.58199e9	+	9.66828e9i	0.120070	+	0.207967i
$88$		0			0
$89$	−3.21650e10	+	5.57114e10i	−0.610574	+	1.05755i	0.380569	+	0.924752i	$0.375728\pi$
$89$	−3.21650e10	+	5.57114e10i	−0.610574	+	1.05755i	−0.991144	+	0.132793i	$0.957605\pi$
$90$		0			0
$91$	−3.06263e10	+	8.06230e10i	−0.514477	+	1.35435i
$92$		0			0
$93$	8.72339e9	−	1.51094e10i	0.130026	−	0.225211i
$94$		0			0
$95$	5.47404e9	+	9.48132e9i	0.0725819	+	0.125715i
$96$		0			0
$97$	−1.14678e11			−1.35592			−0.677960	−	0.735099i	$-0.737136\pi$
$97$	−1.14678e11			−1.35592			−0.677960	+	0.735099i	$0.737136\pi$
$98$		0			0
$99$	−3.40555e10			−0.359909
$100$		0			0
$101$	−2.51977e10	−	4.36438e10i	−0.238558	−	0.413195i	0.721743	−	0.692161i	$-0.243341\pi$
$101$	−2.51977e10	−	4.36438e10i	−0.238558	−	0.413195i	−0.960301	+	0.278967i	$0.910008\pi$
$102$		0			0
$103$	−7.28558e10	+	1.26190e11i	−0.619241	+	1.07256i	0.370384	+	0.928879i	$0.379226\pi$
$103$	−7.28558e10	+	1.26190e11i	−0.619241	+	1.07256i	−0.989625	+	0.143678i	$0.954107\pi$
$104$		0			0
$105$	1.89637e10	−	4.99216e10i	0.145005	−	0.381722i
$106$		0			0
$107$	7.09380e10	−	1.22868e11i	0.488954	−	0.846894i	−0.510965	−	0.859602i	$-0.670712\pi$
$107$	7.09380e10	−	1.22868e11i	0.488954	−	0.846894i	0.999919	+	0.0127078i	$0.00404514\pi$
$108$		0			0
$109$	−6.23443e10	−	1.07984e11i	−0.388107	−	0.672221i	0.604088	−	0.796918i	$-0.293538\pi$
$109$	−6.23443e10	−	1.07984e11i	−0.388107	−	0.672221i	−0.992195	+	0.124697i	$0.960204\pi$
$110$		0			0
$111$	−1.17665e11			−0.662785
$112$		0			0
$113$	−6.65221e10			−0.339652			−0.169826	−	0.985474i	$-0.554321\pi$
$113$	−6.65221e10			−0.339652			−0.169826	+	0.985474i	$0.554321\pi$
$114$		0			0
$115$	−3.48350e10	−	6.03361e10i	−0.161502	−	0.279730i
$116$		0			0
$117$	−5.72628e10	+	9.91821e10i	−0.241463	+	0.418226i
$118$		0			0
$119$	−2.68737e11	+	4.35246e10i	−1.03233	+	0.167196i
$120$		0			0
$121$	−2.36542e10	+	4.09702e10i	−0.0829064	+	0.143598i
$122$		0			0
$123$	−3.99139e10	−	6.91329e10i	−0.127834	−	0.221415i
$124$		0			0
$125$	−3.61920e11			−1.06074
$126$		0			0
$127$	3.83525e11			1.03009			0.515043	−	0.857164i	$-0.327776\pi$
$127$	3.83525e11			1.03009			0.515043	+	0.857164i	$0.327776\pi$
$128$		0			0
$129$	9.39258e10	+	1.62684e11i	0.231495	+	0.400961i
$130$		0			0
$131$	3.08633e11	−	5.34568e11i	0.698956	−	1.21063i	−0.269873	−	0.962896i	$-0.586982\pi$
$131$	3.08633e11	−	5.34568e11i	0.698956	−	1.21063i	0.968829	−	0.247731i	$-0.0796850\pi$
$132$		0			0
$133$	6.22586e10	+	7.63373e10i	0.129722	+	0.159057i
$134$		0			0
$135$	3.54570e10	−	6.14133e10i	0.0680559	−	0.117876i
$136$		0			0
$137$	4.96935e11	+	8.60716e11i	0.879703	+	1.52369i	0.851667	+	0.524084i	$0.175592\pi$
$137$	4.96935e11	+	8.60716e11i	0.879703	+	1.52369i	0.0280364	+	0.999607i	$0.491075\pi$
$138$		0			0
$139$	−4.95409e10			−0.0809809			−0.0404904	−	0.999180i	$-0.512892\pi$
$139$	−4.95409e10			−0.0809809			−0.0404904	+	0.999180i	$0.512892\pi$
$140$		0			0
$141$	2.61352e11			0.394931
$142$		0			0
$143$	5.59286e11	+	9.68712e11i	0.782143	+	1.35471i
$144$		0			0
$145$	−1.13526e11	+	1.96633e11i	−0.147086	+	0.254760i
$146$		0			0
$147$	9.66205e10	−	4.70676e11i	0.116098	−	0.565557i
$148$		0			0
$149$	−1.63133e11	+	2.82555e11i	−0.181977	+	0.315194i	−0.942554	−	0.334054i	$-0.891583\pi$
$149$	−1.63133e11	+	2.82555e11i	−0.181977	+	0.315194i	0.760576	+	0.649248i	$0.224916\pi$
$150$		0			0
$151$	−6.67604e11	−	1.15632e12i	−0.692063	−	1.19869i	−0.971161	−	0.238426i	$-0.923369\pi$
$151$	−6.67604e11	−	1.15632e12i	−0.692063	−	1.19869i	0.279097	−	0.960263i	$-0.409965\pi$
$152$		0			0
$153$	−3.61512e11			−0.348594
$154$		0			0
$155$	3.54831e11			0.318565
$156$		0			0
$157$	−6.83144e10	−	1.18324e11i	−0.0571563	−	0.0989977i	0.836031	−	0.548681i	$-0.184870\pi$
$157$	−6.83144e10	−	1.18324e11i	−0.0571563	−	0.0989977i	−0.893188	+	0.449684i	$0.851537\pi$
$158$		0			0
$159$	−2.33034e10	+	4.03627e10i	−0.0181859	+	0.0314990i
$160$		0			0
$161$	−3.96194e11	−	4.85786e11i	−0.288646	−	0.353918i
$162$		0			0
$163$	1.17232e12	−	2.03051e12i	0.798019	−	1.38221i	−0.122885	−	0.992421i	$-0.539215\pi$
$163$	1.17232e12	−	2.03051e12i	0.798019	−	1.38221i	0.920904	−	0.389789i	$-0.127452\pi$
$164$		0			0
$165$	−3.46309e11	−	5.99824e11i	−0.220446	−	0.381823i
$166$		0			0
$167$	2.93412e12			1.74798			0.873991	−	0.485942i	$-0.161523\pi$
$167$	2.93412e12			1.74798			0.873991	+	0.485942i	$0.161523\pi$
$168$		0			0
$169$	1.96950e12			1.09896
$170$		0			0
$171$	6.54045e10	+	1.13284e11i	0.0342081	+	0.0592502i
$172$		0			0
$173$	4.60805e11	−	7.98137e11i	0.226081	−	0.391583i	−0.730562	−	0.682846i	$-0.760742\pi$
$173$	4.60805e11	−	7.98137e11i	0.226081	−	0.391583i	0.956643	+	0.291263i	$0.0940754\pi$
$174$		0			0
$175$	−1.07120e12	+	1.73492e11i	−0.493357	+	0.0799041i
$176$		0			0
$177$	−4.65983e11	+	8.07106e11i	−0.201612	+	0.349203i
$178$		0			0
$179$	1.48555e12	+	2.57306e12i	0.604223	+	1.04654i	0.992174	+	0.124864i	$0.0398496\pi$
$179$	1.48555e12	+	2.57306e12i	0.604223	+	1.04654i	−0.387951	+	0.921680i	$0.626817\pi$
$180$		0			0
$181$	4.98756e12			1.90834			0.954171	−	0.299263i	$-0.0967409\pi$
$181$	4.98756e12			1.90834			0.954171	+	0.299263i	$0.0967409\pi$
$182$		0			0
$183$	1.15204e12			0.414942
$184$		0			0
$185$	−1.19654e12	−	2.07246e12i	−0.405958	−	0.703140i
$186$		0			0
$187$	−1.76545e12	+	3.05784e12i	−0.564581	+	0.977882i
$188$		0			0
$189$	2.26581e11	−	5.96469e11i	0.0683412	−	0.179907i
$190$		0			0
$191$	2.25057e12	−	3.89810e12i	0.640632	−	1.10961i	−0.344660	−	0.938727i	$-0.612006\pi$
$191$	2.25057e12	−	3.89810e12i	0.640632	−	1.10961i	0.985292	−	0.170879i	$-0.0546608\pi$
$192$		0			0
$193$	−7.16261e11	−	1.24060e12i	−0.192533	−	0.333478i	0.753556	−	0.657384i	$-0.228337\pi$
$193$	−7.16261e11	−	1.24060e12i	−0.192533	−	0.333478i	−0.946089	+	0.323906i	$0.895004\pi$
$194$		0			0
$195$	−2.32921e12			−0.591587
$196$		0			0
$197$	6.39062e12			1.53454			0.767271	−	0.641323i	$-0.221614\pi$
$197$	6.39062e12			1.53454			0.767271	+	0.641323i	$0.221614\pi$
$198$		0			0
$199$	−5.19479e11	−	8.99764e11i	−0.117998	−	0.204379i	0.800976	−	0.598697i	$-0.204314\pi$
$199$	−5.19479e11	−	8.99764e11i	−0.117998	−	0.204379i	−0.918974	+	0.394317i	$0.870981\pi$
$200$		0			0
$201$	9.28809e11	−	1.60874e12i	0.199686	−	0.345867i
$202$		0			0
$203$	−7.25465e11	+	1.90977e12i	−0.147703	+	0.388824i
$204$		0			0
$205$	8.11766e11	−	1.40602e12i	0.156597	−	0.271235i
$206$		0			0
$207$	−4.16213e11	−	7.20903e11i	−0.0761166	−	0.131838i
$208$		0			0
$209$	1.27761e12			0.221613
$210$		0			0
$211$	−1.00726e13			−1.65801			−0.829004	−	0.559242i	$-0.811092\pi$
$211$	−1.00726e13			−1.65801			−0.829004	+	0.559242i	$0.811092\pi$
$212$		0			0
$213$	−2.88731e12	−	5.00096e12i	−0.451238	−	0.781566i
$214$		0			0
$215$	−1.91026e12	+	3.30866e12i	−0.283583	+	0.491180i
$216$		0			0
$217$	3.15156e12	−	5.10426e11i	0.444628	−	0.0720119i
$218$		0			0
$219$	−7.19815e10	+	1.24676e11i	−0.00965558	+	0.0167240i
$220$		0			0
$221$	5.93705e12	+	1.02833e13i	0.757553	+	1.31212i
$222$		0			0
$223$	−9.94192e12			−1.20724			−0.603620	−	0.797272i	$-0.706275\pi$
$223$	−9.94192e12			−1.20724			−0.603620	+	0.797272i	$0.706275\pi$
$224$		0			0
$225$	−1.44101e12			−0.166595
$226$		0			0
$227$	2.27724e12	+	3.94429e12i	0.250764	+	0.434337i	0.963736	−	0.266856i	$-0.0859847\pi$
$227$	2.27724e12	+	3.94429e12i	0.250764	+	0.434337i	−0.712972	+	0.701192i	$0.752651\pi$
$228$		0			0
$229$	−3.30508e12	+	5.72457e12i	−0.346806	+	0.600686i	−0.985680	−	0.168625i	$-0.946067\pi$
$229$	−3.30508e12	+	5.72457e12i	−0.346806	+	0.600686i	0.638874	+	0.769311i	$0.279401\pi$
$230$		0			0
$231$	−3.93871e12	−	4.82939e12i	−0.393993	−	0.483088i
$232$		0			0
$233$	−9.76232e12	+	1.69088e13i	−0.931312	+	1.61308i	−0.150231	+	0.988651i	$0.548002\pi$
$233$	−9.76232e12	+	1.69088e13i	−0.931312	+	1.61308i	−0.781081	+	0.624429i	$0.785332\pi$
$234$		0			0
$235$	2.65768e12	+	4.60324e12i	0.241897	+	0.418977i
$236$		0			0
$237$	−3.61749e12			−0.314262
$238$		0			0
$239$	2.23809e12			0.185647			0.0928237	−	0.995683i	$-0.470411\pi$
$239$	2.23809e12			0.185647			0.0928237	+	0.995683i	$0.470411\pi$
$240$		0			0
$241$	−8.18468e12	−	1.41763e13i	−0.648497	−	1.12323i	−0.983482	−	0.181006i	$-0.942065\pi$
$241$	−8.18468e12	−	1.41763e13i	−0.648497	−	1.12323i	0.334985	−	0.942223i	$-0.391269\pi$
$242$		0			0
$243$	4.23644e11	−	7.33773e11i	0.0320750	−	0.0555556i
$244$		0			0
$245$	9.27261e12	−	3.08449e12i	0.671102	−	0.223239i
$246$		0			0
$247$	2.14825e12	−	3.72088e12i	0.148680	−	0.257521i
$248$		0			0
$249$	3.98037e11	+	6.89421e11i	0.0263528	+	0.0456444i
$250$		0			0
$251$	−3.88009e12			−0.245831			−0.122916	−	0.992417i	$-0.539224\pi$
$251$	−3.88009e12			−0.245831			−0.122916	+	0.992417i	$0.539224\pi$
$252$		0			0
$253$	−8.13032e12			−0.493111
$254$		0			0
$255$	−3.67621e12	−	6.36737e12i	−0.213515	−	0.369819i
$256$		0			0
$257$	−1.44048e13	+	2.49498e13i	−0.801445	+	1.38814i	0.117220	+	0.993106i	$0.462602\pi$
$257$	−1.44048e13	+	2.49498e13i	−0.801445	+	1.38814i	−0.918665	+	0.395037i	$0.870732\pi$
$258$		0			0
$259$	−1.36087e13	−	1.66861e13i	−0.725551	−	0.889622i
$260$		0			0
$261$	−1.35642e12	+	2.34939e12i	−0.0693222	+	0.120070i
$262$		0			0
$263$	−6.31528e12	−	1.09384e13i	−0.309483	−	0.536040i	0.668767	−	0.743472i	$-0.266823\pi$
$263$	−6.31528e12	−	1.09384e13i	−0.309483	−	0.536040i	−0.978249	+	0.207433i	$0.933489\pi$
$264$		0			0
$265$	−9.47886e11			−0.0445558
$266$		0			0
$267$	−1.56322e13			−0.705030
$268$		0			0
$269$	1.27776e13	+	2.21314e13i	0.553109	+	0.958013i	0.998048	+	0.0624522i	$0.0198921\pi$
$269$	1.27776e13	+	2.21314e13i	0.553109	+	0.958013i	−0.444939	+	0.895561i	$0.646775\pi$
$270$		0			0
$271$	−1.36631e12	+	2.36652e12i	−0.0567831	+	0.0983512i	−0.893020	−	0.450018i	$-0.851418\pi$
$271$	−1.36631e12	+	2.36652e12i	−0.0567831	+	0.0983512i	0.836237	+	0.548369i	$0.184751\pi$
$272$		0			0
$273$	−2.06877e13	+	3.35058e12i	−0.825693	+	0.133729i
$274$		0			0
$275$	−7.03718e12	+	1.21887e13i	−0.269817	+	0.467336i
$276$		0			0
$277$	−3.09890e12	−	5.36745e12i	−0.114174	−	0.197756i	0.803275	−	0.595608i	$-0.203089\pi$
$277$	−3.09890e12	−	5.36745e12i	−0.114174	−	0.197756i	−0.917449	+	0.397853i	$0.869756\pi$
$278$		0			0
$279$	4.23957e12			0.150141
$280$		0			0
$281$	3.53102e13			1.20231			0.601154	−	0.799133i	$-0.294708\pi$
$281$	3.53102e13			1.20231			0.601154	+	0.799133i	$0.294708\pi$
$282$		0			0
$283$	−1.93259e13	−	3.34735e13i	−0.632871	−	1.09617i	−0.986962	−	0.160954i	$-0.948543\pi$
$283$	−1.93259e13	−	3.34735e13i	−0.632871	−	1.09617i	0.354091	−	0.935211i	$-0.384790\pi$
$284$		0			0
$285$	−1.33019e12	+	2.30396e12i	−0.0419052	+	0.0725819i
$286$		0			0
$287$	5.18742e12	−	1.36558e13i	0.157254	−	0.413968i
$288$		0			0
$289$	−1.60499e12	+	2.77992e12i	−0.0468310	+	0.0811137i
$290$		0			0
$291$	−1.39333e13	−	2.41332e13i	−0.391421	−	0.677960i
$292$		0			0
$293$	7.20643e13			1.94961			0.974806	−	0.223053i	$-0.0716023\pi$
$293$	7.20643e13			1.94961			0.974806	+	0.223053i	$0.0716023\pi$
$294$		0			0
$295$	−1.89542e13			−0.493953
$296$		0			0
$297$	−4.13774e12	−	7.16677e12i	−0.103897	−	0.179955i
$298$		0			0
$299$	−1.36708e13	+	2.36785e13i	−0.330828	+	0.573011i
$300$		0			0
$301$	−1.22071e13	+	3.21350e13i	−0.284772	+	0.749656i
$302$		0			0
$303$	6.12305e12	−	1.06054e13i	0.137732	−	0.238558i
$304$		0			0
$305$	1.17151e13	+	2.02911e13i	0.254153	+	0.440207i
$306$		0			0
$307$	−4.64065e13			−0.971221			−0.485610	−	0.874175i	$-0.661403\pi$
$307$	−4.64065e13			−0.971221			−0.485610	+	0.874175i	$0.661403\pi$
$308$		0			0
$309$	−3.54079e13			−0.715038
$310$		0			0
$311$	3.18923e13	+	5.52391e13i	0.621590	+	1.07663i	0.989190	+	0.146641i	$0.0468463\pi$
$311$	3.18923e13	+	5.52391e13i	0.621590	+	1.07663i	−0.367600	+	0.929984i	$0.619820\pi$
$312$		0			0
$313$	1.51243e13	−	2.61961e13i	0.284566	−	0.492882i	−0.687938	−	0.725769i	$-0.741484\pi$
$313$	1.51243e13	−	2.61961e13i	0.284566	−	0.492882i	0.972504	+	0.232887i	$0.0748173\pi$
$314$		0			0
$315$	1.28098e13	−	2.07467e12i	0.232720	−	0.0376913i
$316$		0			0
$317$	2.09893e13	−	3.63545e13i	0.368274	−	0.637870i	−0.621022	−	0.783793i	$-0.713282\pi$
$317$	2.09893e13	−	3.63545e13i	0.368274	−	0.637870i	0.989296	+	0.145924i	$0.0466155\pi$
$318$		0			0
$319$	1.32482e13	+	2.29465e13i	0.224547	+	0.388927i
$320$		0			0
$321$	3.44759e13			0.564596
$322$		0			0
$323$	1.35624e13			0.214646
$324$		0			0
$325$	2.36654e13	+	4.09897e13i	0.362039	+	0.627071i
$326$		0			0
$327$	1.51497e13	−	2.62400e13i	0.224074	−	0.388107i
$328$		0			0
$329$	3.02269e13	+	3.70622e13i	0.432331	+	0.530096i
$330$		0			0
$331$	3.47164e13	−	6.01306e13i	0.480265	−	0.831844i	−0.519478	−	0.854484i	$-0.673874\pi$
$331$	3.47164e13	−	6.01306e13i	0.480265	−	0.831844i	0.999744	+	0.0226397i	$0.00720707\pi$
$332$		0			0
$333$	−1.42964e13	−	2.47620e13i	−0.191330	−	0.331392i
$334$		0			0
$335$	3.77801e13			0.489234
$336$		0			0
$337$	9.32541e13			1.16870			0.584351	−	0.811501i	$-0.301349\pi$
$337$	9.32541e13			1.16870			0.584351	+	0.811501i	$0.301349\pi$
$338$		0			0
$339$	−8.08244e12	−	1.39992e13i	−0.0980492	−	0.169826i
$340$		0			0
$341$	2.07039e13	−	3.58603e13i	0.243167	−	0.421177i
$342$		0			0
$343$	7.79210e13	−	4.07347e13i	0.886210	−	0.463283i
$344$		0			0
$345$	8.46491e12	−	1.46617e13i	0.0932433	−	0.161502i
$346$		0			0
$347$	3.29833e12	+	5.71288e12i	0.0351951	+	0.0609597i	0.883086	−	0.469210i	$-0.155461\pi$
$347$	3.29833e12	+	5.71288e12i	0.0351951	+	0.0609597i	−0.847891	+	0.530170i	$0.822128\pi$
$348$		0			0
$349$	−1.17838e14			−1.21828			−0.609140	−	0.793063i	$-0.708485\pi$
$349$	−1.17838e14			−1.21828			−0.609140	+	0.793063i	$0.708485\pi$
$350$		0			0
$351$	−2.78297e13			−0.278817
$352$		0			0
$353$	7.87055e13	+	1.36322e14i	0.764266	+	1.32375i	0.940634	+	0.339423i	$0.110232\pi$
$353$	7.87055e13	+	1.36322e14i	0.764266	+	1.32375i	−0.176368	+	0.984324i	$0.556435\pi$
$354$		0			0
$355$	5.87218e13	−	1.01709e14i	0.552769	−	0.957424i
$356$		0			0
$357$	−4.18110e13	−	5.12659e13i	−0.381606	−	0.467900i
$358$		0			0
$359$	2.04125e13	−	3.53556e13i	0.180667	−	0.312924i	−0.761441	−	0.648234i	$-0.775508\pi$
$359$	2.04125e13	−	3.53556e13i	0.180667	−	0.312924i	0.942108	+	0.335310i	$0.108841\pi$
$360$		0			0
$361$	5.57914e13	+	9.66336e13i	0.478936	+	0.829542i
$362$		0			0
$363$	−1.14959e13			−0.0957321
$364$		0			0
$365$	−2.92791e12			−0.0236563
$366$		0			0
$367$	−2.38448e13	−	4.13004e13i	−0.186952	−	0.323811i	0.757280	−	0.653090i	$-0.226528\pi$
$367$	−2.38448e13	−	4.13004e13i	−0.186952	−	0.323811i	−0.944233	+	0.329279i	$0.893194\pi$
$368$		0			0
$369$	9.69908e12	−	1.67993e13i	0.0738049	−	0.127834i
$370$		0			0
$371$	−8.41899e12	+	1.36354e12i	−0.0621876	+	0.0100719i
$372$		0			0
$373$	8.85737e13	−	1.53414e14i	0.635194	−	1.10019i	−0.351281	−	0.936270i	$-0.614254\pi$
$373$	8.85737e13	−	1.53414e14i	0.635194	−	1.10019i	0.986474	−	0.163917i	$-0.0524130\pi$
$374$		0			0
$375$	−4.39732e13	−	7.61639e13i	−0.306208	−	0.530368i
$376$		0			0
$377$	8.91051e13			0.602594
$378$		0			0
$379$	−2.15968e14			−1.41864			−0.709322	−	0.704885i	$-0.750999\pi$
$379$	−2.15968e14			−1.41864			−0.709322	+	0.704885i	$0.750999\pi$
$380$		0			0
$381$	4.65983e13	+	8.07107e13i	0.297360	+	0.515043i
$382$		0			0
$383$	9.46300e13	−	1.63904e14i	0.586727	−	1.01624i	−0.407931	−	0.913013i	$-0.633750\pi$
$383$	9.46300e13	−	1.63904e14i	0.586727	−	1.01624i	0.994658	−	0.103228i	$-0.0329170\pi$
$384$		0			0
$385$	4.50081e13	−	1.18483e14i	0.271179	−	0.713875i
$386$		0			0
$387$	−2.28240e13	+	3.95323e13i	−0.133654	+	0.231495i
$388$		0			0
$389$	9.65224e13	+	1.67182e14i	0.549421	+	0.951625i	0.998314	+	0.0580396i	$0.0184850\pi$
$389$	9.65224e13	+	1.67182e14i	0.549421	+	0.951625i	−0.448893	+	0.893585i	$0.648182\pi$
$390$		0			0
$391$	−8.63066e13			−0.477608
$392$		0			0
$393$	1.49995e14			0.807085
$394$		0			0
$395$	−3.67861e13	−	6.37154e13i	−0.192486	−	0.333396i
$396$		0			0
$397$	−1.27136e14	+	2.20206e14i	−0.647023	+	1.12068i	0.336807	+	0.941574i	$0.390653\pi$
$397$	−1.27136e14	+	2.20206e14i	−0.647023	+	1.12068i	−0.983830	+	0.179104i	$0.942680\pi$
$398$		0			0
$399$	−8.50032e12	+	2.23769e13i	−0.0420808	+	0.110777i
$400$		0			0
$401$	7.31111e12	−	1.26632e13i	0.0352119	−	0.0609888i	−0.847882	−	0.530184i	$-0.822123\pi$
$401$	7.31111e12	−	1.26632e13i	0.0352119	−	0.0609888i	0.883094	+	0.469196i	$0.155456\pi$
$402$		0			0
$403$	−6.96256e13	−	1.20595e14i	−0.326281	−	0.565135i
$404$		0			0
$405$	1.72321e13			0.0785842
$406$		0			0
$407$	−2.79265e14			−1.23950
$408$		0			0
$409$	2.11985e14	+	3.67169e14i	0.915855	+	1.58631i	0.805645	+	0.592399i	$0.201819\pi$
$409$	2.11985e14	+	3.67169e14i	0.915855	+	1.58631i	0.110210	+	0.993908i	$0.464848\pi$
$410$		0			0
$411$	−1.20755e14	+	2.09154e14i	−0.507897	+	0.879703i
$412$		0			0
$413$	−1.68349e14	+	2.72658e13i	−0.689422	+	0.111659i
$414$		0			0
$415$	−8.09525e12	+	1.40214e13i	−0.0322824	+	0.0559147i
$416$		0			0
$417$	−6.01922e12	−	1.04256e13i	−0.0233772	−	0.0404904i
$418$		0			0
$419$	1.55741e13			0.0589151			0.0294576	−	0.999566i	$-0.490622\pi$
$419$	1.55741e13			0.0589151			0.0294576	+	0.999566i	$0.490622\pi$
$420$		0			0
$421$	−2.25694e14			−0.831702			−0.415851	−	0.909433i	$-0.636516\pi$
$421$	−2.25694e14			−0.831702			−0.415851	+	0.909433i	$0.636516\pi$
$422$		0			0
$423$	3.17543e13	+	5.50001e13i	0.114007	+	0.197465i
$424$		0			0
$425$	−7.47025e13	+	1.29388e14i	−0.261334	+	0.452644i
$426$		0			0
$427$	1.33240e14	+	1.63371e14i	0.454237	+	0.556955i
$428$		0			0
$429$	−1.35907e14	+	2.35397e14i	−0.451570	+	0.782143i
$430$		0			0
$431$	−2.58420e14	−	4.47597e14i	−0.836953	−	1.44965i	−0.892430	−	0.451185i	$-0.851001\pi$
$431$	−2.58420e14	−	4.47597e14i	−0.836953	−	1.44965i	0.0554772	−	0.998460i	$-0.482332\pi$
$432$		0			0
$433$	2.57437e14			0.812807			0.406404	−	0.913694i	$-0.366783\pi$
$433$	2.57437e14			0.812807			0.406404	+	0.913694i	$0.366783\pi$
$434$		0			0
$435$	−5.51736e13			−0.169840
$436$		0			0
$437$	1.56145e13	+	2.70451e13i	0.0468685	+	0.0811786i
$438$		0			0
$439$	−4.14292e13	+	7.17575e13i	−0.121269	+	0.210045i	−0.920269	−	0.391287i	$-0.872030\pi$
$439$	−4.14292e13	+	7.17575e13i	−0.121269	+	0.210045i	0.798999	+	0.601332i	$0.205363\pi$
$440$		0			0
$441$	1.10790e14	−	3.68539e13i	0.316293	−	0.105213i
$442$		0			0
$443$	−2.39990e14	+	4.15675e14i	−0.668303	+	1.15753i	0.310076	+	0.950712i	$0.399646\pi$
$443$	−2.39990e14	+	4.15675e14i	−0.668303	+	1.15753i	−0.978379	+	0.206822i	$0.933688\pi$
$444$		0			0
$445$	−1.58963e14	−	2.75332e14i	−0.431833	−	0.747958i
$446$		0			0
$447$	−7.92826e13			−0.210129
$448$		0			0
$449$	−6.34397e14			−1.64061			−0.820307	−	0.571923i	$-0.806197\pi$
$449$	−6.34397e14			−1.64061			−0.820307	+	0.571923i	$0.806197\pi$
$450$		0			0
$451$	−9.47310e13	−	1.64079e14i	−0.239068	−	0.414077i
$452$		0			0
$453$	1.62228e14	−	2.80987e14i	0.399563	−	0.692063i
$454$		0			0
$455$	−2.69387e14	−	3.30304e14i	−0.647611	−	0.794057i
$456$		0			0
$457$	3.03704e13	−	5.26031e13i	0.0712708	−	0.123445i	−0.828188	−	0.560451i	$-0.810628\pi$
$457$	3.03704e13	−	5.26031e13i	0.0712708	−	0.123445i	0.899459	+	0.437006i	$0.143961\pi$
$458$		0			0
$459$	−4.39238e13	−	7.60782e13i	−0.100631	−	0.174297i
$460$		0			0
$461$	−1.70103e14			−0.380501			−0.190251	−	0.981736i	$-0.560930\pi$
$461$	−1.70103e14			−0.380501			−0.190251	+	0.981736i	$0.560930\pi$
$462$		0			0
$463$	−7.91437e14			−1.72870			−0.864352	−	0.502887i	$-0.832271\pi$
$463$	−7.91437e14			−1.72870			−0.864352	+	0.502887i	$0.832271\pi$
$464$		0			0
$465$	4.31120e13	+	7.46722e13i	0.0919617	+	0.159282i
$466$		0			0
$467$	1.12472e14	−	1.94808e14i	0.234316	−	0.405848i	−0.724757	−	0.689004i	$-0.758048\pi$
$467$	1.12472e14	−	1.94808e14i	0.234316	−	0.405848i	0.959074	+	0.283156i	$0.0913815\pi$
$468$		0			0
$469$	3.35557e14	−	5.43468e13i	0.682836	−	0.110592i
$470$		0			0
$471$	1.66004e13	−	2.87527e13i	0.0329992	−	0.0571563i
$472$		0			0
$473$	2.22922e14	+	3.86112e14i	0.432929	+	0.749855i
$474$		0			0
$475$	5.40604e13			0.102580
$476$		0			0
$477$	−1.13255e13			−0.0209993
$478$		0			0
$479$	8.96814e12	+	1.55333e13i	0.0162501	+	0.0281461i	0.874036	−	0.485861i	$-0.161494\pi$
$479$	8.96814e12	+	1.55333e13i	0.0162501	+	0.0281461i	−0.857786	+	0.514007i	$0.828161\pi$
$480$		0			0
$481$	−4.69573e14	+	8.13324e14i	−0.831581	+	1.44034i
$482$		0			0
$483$	5.40933e13	−	1.42400e14i	0.0936342	−	0.246490i
$484$		0			0
$485$	2.83375e14	−	4.90820e14i	0.479493	−	0.830506i
$486$		0			0
$487$	5.31614e14	+	9.20783e14i	0.879402	+	1.52317i	0.851998	+	0.523545i	$0.175391\pi$
$487$	5.31614e14	+	9.20783e14i	0.879402	+	1.52317i	0.0274040	+	0.999624i	$0.491276\pi$
$488$		0			0
$489$	5.69746e14			0.921474
$490$		0			0
$491$	−4.03069e14			−0.637428			−0.318714	−	0.947851i	$-0.603251\pi$
$491$	−4.03069e14			−0.637428			−0.318714	+	0.947851i	$0.603251\pi$
$492$		0			0
$493$	1.40635e14	+	2.43587e14i	0.217488	+	0.376700i
$494$		0			0
$495$	8.41530e13	−	1.45757e14i	0.127274	−	0.220446i
$496$		0			0
$497$	3.75250e14	−	9.87838e14i	0.555086	−	1.46125i
$498$		0			0
$499$	3.87976e14	−	6.71994e14i	0.561373	−	0.972327i	−0.436004	−	0.899945i	$-0.643607\pi$
$499$	3.87976e14	−	6.71994e14i	0.561373	−	0.972327i	0.997377	−	0.0723820i	$-0.0230601\pi$
$500$		0			0
$501$	3.56495e14	+	6.17468e14i	0.504599	+	0.873991i
$502$		0			0
$503$	−6.98960e14			−0.967895			−0.483948	−	0.875097i	$-0.660798\pi$
$503$	−6.98960e14			−0.967895			−0.483948	+	0.875097i	$0.660798\pi$
$504$		0			0
$505$	2.49060e14			0.337444
$506$		0			0
$507$	2.39295e14	+	4.14471e14i	0.317241	+	0.549478i
$508$		0			0
$509$	−2.14655e14	+	3.71793e14i	−0.278479	+	0.482340i	−0.971007	−	0.239051i	$-0.923164\pi$
$509$	−2.14655e14	+	3.71793e14i	−0.278479	+	0.482340i	0.692528	+	0.721391i	$0.256497\pi$
$510$		0			0
$511$	−2.60053e13	+	4.21181e12i	−0.0330177	+	0.00534753i
$512$		0			0
$513$	−1.58933e13	+	2.75280e13i	−0.0197501	+	0.0342081i
$514$		0			0
$515$	−3.60062e14	−	6.23645e14i	−0.437963	−	0.758574i
$516$		0			0
$517$	6.20289e14			0.738578
$518$		0			0
$519$	2.23951e14			0.261055
$520$		0			0
$521$	−4.03095e14	−	6.98180e14i	−0.460044	−	0.796820i	0.538918	−	0.842358i	$-0.318833\pi$
$521$	−4.03095e14	−	6.98180e14i	−0.460044	−	0.796820i	−0.998963	+	0.0455380i	$0.985500\pi$
$522$		0			0
$523$	5.78099e13	−	1.00130e14i	0.0646016	−	0.111893i	−0.831916	−	0.554902i	$-0.812756\pi$
$523$	5.78099e13	−	1.00130e14i	0.0646016	−	0.111893i	0.896517	+	0.443009i	$0.146089\pi$
$524$		0			0
$525$	−1.66661e14	−	2.04349e14i	−0.182372	−	0.223612i
$526$		0			0
$527$	2.19781e14	−	3.80671e14i	0.235522	−	0.407936i
$528$		0			0
$529$	3.77039e14	+	6.53051e14i	0.395713	+	0.685395i
$530$		0			0
$531$	−2.26468e14			−0.232802
$532$		0			0
$533$	−6.37144e14			−0.641561
$534$		0			0
$535$	3.50584e14	+	6.07229e14i	0.345817	+	0.598972i
$536$		0			0
$537$	−3.60990e14	+	6.25253e14i	−0.348848	+	0.604223i
$538$		0			0
$539$	2.29317e14	−	1.11709e15i	0.217119	−	1.05767i
$540$		0			0
$541$	5.35734e14	−	9.27919e14i	0.497009	−	0.860846i	−0.502985	−	0.864295i	$-0.667765\pi$
$541$	5.35734e14	−	9.27919e14i	0.497009	−	0.860846i	0.999994	+	0.00344983i	$0.00109812\pi$
$542$		0			0
$543$	6.05989e14	+	1.04960e15i	0.550891	+	0.954171i
$544$		0			0
$545$	6.16226e14			0.548983
$546$		0			0
$547$	3.47087e14			0.303045			0.151523	−	0.988454i	$-0.451582\pi$
$547$	3.47087e14			0.303045			0.151523	+	0.988454i	$0.451582\pi$
$548$		0			0
$549$	1.39973e14	+	2.42441e14i	0.119783	+	0.207471i
$550$		0			0
$551$	5.08871e13	−	8.81390e13i	0.0426849	−	0.0739323i
$552$		0			0
$553$	−4.18384e14	−	5.12994e14i	−0.344022	−	0.421817i
$554$		0			0
$555$	2.90758e14	−	5.03608e14i	0.234380	−	0.405958i
$556$		0			0
$557$	−3.64218e14	−	6.30844e14i	−0.287844	−	0.498561i	0.685451	−	0.728119i	$-0.259605\pi$
$557$	−3.64218e14	−	6.30844e14i	−0.287844	−	0.498561i	−0.973295	+	0.229558i	$0.926272\pi$
$558$		0			0
$559$	1.49934e15			1.16181
$560$		0			0
$561$	−8.58008e14			−0.651922
$562$		0			0
$563$	1.19763e15	+	2.07435e15i	0.892330	+	1.54556i	0.837075	+	0.547089i	$0.184264\pi$
$563$	1.19763e15	+	2.07435e15i	0.892330	+	1.54556i	0.0552554	+	0.998472i	$0.482403\pi$
$564$		0			0
$565$	1.64380e14	−	2.84714e14i	0.120111	−	0.208038i
$566$		0			0
$567$	1.53053e14	−	2.47884e13i	0.109682	−	0.0177641i
$568$		0			0
$569$	8.72232e14	−	1.51075e15i	0.613077	−	1.06188i	−0.377642	−	0.925952i	$-0.623265\pi$
$569$	8.72232e14	−	1.51075e15i	0.613077	−	1.06188i	0.990719	−	0.135928i	$-0.0434017\pi$
$570$		0			0
$571$	3.09306e14	+	5.35734e14i	0.213250	+	0.369361i	0.952730	−	0.303818i	$-0.0982616\pi$
$571$	3.09306e14	+	5.35734e14i	0.213250	+	0.369361i	−0.739479	+	0.673179i	$0.764928\pi$
$572$		0			0
$573$	1.09378e15			0.739738
$574$		0			0
$575$	−3.44023e14			−0.228252
$576$		0			0
$577$	−8.91322e13	−	1.54382e14i	−0.0580187	−	0.100491i	0.835557	−	0.549403i	$-0.185145\pi$
$577$	−8.91322e13	−	1.54382e14i	−0.0580187	−	0.100491i	−0.893576	+	0.448912i	$0.851812\pi$
$578$		0			0
$579$	1.74051e14	−	3.01466e14i	0.111159	−	0.192533i
$580$		0			0
$581$	−5.17310e13	+	1.36181e14i	−0.0324177	+	0.0853390i
$582$		0			0
$583$	−5.53079e13	+	9.57962e13i	−0.0340103	+	0.0589076i
$584$		0			0
$585$	−2.82999e14	−	4.90169e14i	−0.170777	−	0.295794i
$586$		0			0
$587$	1.68835e15			0.999890			0.499945	−	0.866057i	$-0.333354\pi$
$587$	1.68835e15			0.999890			0.499945	+	0.866057i	$0.333354\pi$
$588$		0			0
$589$	−1.59050e14			−0.0924485
$590$		0			0
$591$	7.76460e14	+	1.34487e15i	0.442984	+	0.767271i
$592$		0			0
$593$	−1.58957e15	+	2.75321e15i	−0.890180	+	1.54184i	−0.0505200	+	0.998723i	$0.516088\pi$
$593$	−1.58957e15	+	2.75321e15i	−0.890180	+	1.54184i	−0.839660	+	0.543113i	$0.817245\pi$
$594$		0			0
$595$	4.77779e14	−	1.25774e15i	0.262654	−	0.691432i
$596$		0			0
$597$	1.26233e14	−	2.18643e14i	0.0681264	−	0.117998i
$598$		0			0
$599$	−1.00464e15	−	1.74009e15i	−0.532308	−	0.921985i	−0.999288	−	0.0377169i	$-0.987991\pi$
$599$	−1.00464e15	−	1.74009e15i	−0.532308	−	0.921985i	0.466980	−	0.884268i	$-0.345342\pi$
$600$		0			0
$601$	1.54370e15			0.803068			0.401534	−	0.915844i	$-0.368477\pi$
$601$	1.54370e15			0.803068			0.401534	+	0.915844i	$0.368477\pi$
$602$		0			0
$603$	4.51401e14			0.230578
$604$		0			0
$605$	−1.16902e14	−	2.02480e14i	−0.0586362	−	0.101561i
$606$		0			0
$607$	1.17649e14	−	2.03774e14i	0.0579495	−	0.100372i	−0.835595	−	0.549346i	$-0.814877\pi$
$607$	1.17649e14	−	2.03774e14i	0.0579495	−	0.100372i	0.893545	+	0.448974i	$0.148210\pi$
$608$		0			0
$609$	−4.90044e14	+	7.93675e13i	−0.237050	+	0.0383926i
$610$		0			0
$611$	1.04299e15	−	1.80651e15i	0.495511	−	0.858251i
$612$		0			0
$613$	−8.31432e14	−	1.44008e15i	−0.387967	−	0.671978i	0.604209	−	0.796826i	$-0.293489\pi$
$613$	−8.31432e14	−	1.44008e15i	−0.387967	−	0.671978i	−0.992176	+	0.124848i	$0.960156\pi$
$614$		0			0
$615$	3.94518e14			0.180823
$616$		0			0
$617$	−1.07750e15			−0.485121			−0.242560	−	0.970136i	$-0.577987\pi$
$617$	−1.07750e15			−0.485121			−0.242560	+	0.970136i	$0.577987\pi$
$618$		0			0
$619$	1.74230e15	+	3.01776e15i	0.770593	+	1.33471i	0.937238	+	0.348689i	$0.113373\pi$
$619$	1.74230e15	+	3.01776e15i	0.770593	+	1.33471i	−0.166646	+	0.986017i	$0.553294\pi$
$620$		0			0
$621$	1.01140e14	−	1.75179e14i	0.0439459	−	0.0761166i
$622$		0			0
$623$	−1.80795e15	−	2.21679e15i	−0.771797	−	0.946326i
$624$		0			0
$625$	2.98533e14	−	5.17074e14i	0.125214	−	0.216877i
$626$		0			0
$627$	1.55230e14	+	2.68866e14i	0.0639741	+	0.110806i
$628$		0			0
$629$	−2.96451e15			−1.20053
$630$		0			0
$631$	−4.02635e15			−1.60232			−0.801162	−	0.598447i	$-0.795785\pi$
$631$	−4.02635e15			−1.60232			−0.801162	+	0.598447i	$0.795785\pi$
$632$		0			0
$633$	−1.22382e15	−	2.11971e15i	−0.478626	−	0.829004i
$634$		0			0
$635$	−9.47713e14	+	1.64149e15i	−0.364268	+	0.630931i
$636$		0			0
$637$	−2.86780e15	−	2.54620e15i	−1.08338	−	0.961891i
$638$		0			0
$639$	7.01616e14	−	1.21523e15i	0.260522	−	0.451238i
$640$		0			0
$641$	1.22236e15	+	2.11720e15i	0.446150	+	0.772755i	0.998132	−	0.0611014i	$-0.0194613\pi$
$641$	1.22236e15	+	2.11720e15i	0.446150	+	0.772755i	−0.551981	+	0.833857i	$0.686128\pi$
$642$		0			0
$643$	−2.67405e15			−0.959420			−0.479710	−	0.877427i	$-0.659258\pi$
$643$	−2.67405e15			−0.959420			−0.479710	+	0.877427i	$0.659258\pi$
$644$		0			0
$645$	−9.28384e14			−0.327453
$646$		0			0
$647$	1.53200e15	+	2.65350e15i	0.531233	+	0.920123i	0.999336	+	0.0364485i	$0.0116045\pi$
$647$	1.53200e15	+	2.65350e15i	0.531233	+	0.920123i	−0.468102	+	0.883674i	$0.655062\pi$
$648$		0			0
$649$	−1.10596e15	+	1.91557e15i	−0.377044	+	0.653059i
$650$		0			0
$651$	4.90331e14	+	6.01211e14i	0.164359	+	0.201526i
$652$		0			0
$653$	8.02991e14	−	1.39082e15i	0.264660	−	0.458404i	−0.702815	−	0.711373i	$-0.748074\pi$
$653$	8.02991e14	−	1.39082e15i	0.264660	−	0.458404i	0.967474	+	0.252969i	$0.0814070\pi$
$654$		0			0
$655$	1.52530e15	+	2.64189e15i	0.494342	+	0.856226i
$656$		0			0
$657$	−3.49830e13			−0.0111493
$658$		0			0
$659$	−2.21949e15			−0.695638			−0.347819	−	0.937562i	$-0.613078\pi$
$659$	−2.21949e15			−0.695638			−0.347819	+	0.937562i	$0.613078\pi$
$660$		0			0
$661$	2.34314e15	+	4.05843e15i	0.722253	+	1.25098i	0.960095	+	0.279675i	$0.0902269\pi$
$661$	2.34314e15	+	4.05843e15i	0.722253	+	1.25098i	−0.237841	+	0.971304i	$0.576440\pi$
$662$		0			0
$663$	−1.44270e15	+	2.49883e15i	−0.437374	+	0.757553i
$664$		0			0
$665$	−4.80568e14	+	7.78327e13i	−0.143296	+	0.0232083i
$666$		0			0
$667$	−3.23829e14	+	5.60889e14i	−0.0949782	+	0.164507i
$668$		0			0
$669$	−1.20794e15	−	2.09222e15i	−0.348500	−	0.603620i
$670$		0			0
$671$	2.73424e15			0.776002
$672$		0			0
$673$	−1.61460e15			−0.450797			−0.225399	−	0.974267i	$-0.572368\pi$
$673$	−1.61460e15			−0.450797			−0.225399	+	0.974267i	$0.572368\pi$
$674$		0			0
$675$	−1.75083e14	−	3.03252e14i	−0.0480920	−	0.0832977i
$676$		0			0
$677$	−4.77762e14	+	8.27509e14i	−0.129114	+	0.223632i	−0.923334	−	0.383999i	$-0.874547\pi$
$677$	−4.77762e14	+	8.27509e14i	−0.129114	+	0.223632i	0.794219	+	0.607631i	$0.207880\pi$
$678$		0			0
$679$	1.81085e15	−	4.76703e15i	0.481503	−	1.26755i
$680$		0			0
$681$	−5.53368e14	+	9.58462e14i	−0.144779	+	0.250764i
$682$		0			0
$683$	3.66194e15	+	6.34267e15i	0.942752	+	1.63290i	0.760190	+	0.649701i	$0.225106\pi$
$683$	3.66194e15	+	6.34267e15i	0.942752	+	1.63290i	0.182562	+	0.983194i	$0.441561\pi$
$684$		0			0
$685$	−4.91182e15			−1.24435
$686$		0			0
$687$	−1.60627e15			−0.400457
$688$		0			0
$689$	1.85996e14	+	3.22154e14i	0.0456350	+	0.0790421i
$690$		0			0
$691$	−3.34330e15	+	5.79076e15i	−0.807320	+	1.39832i	0.107394	+	0.994217i	$0.465749\pi$
$691$	−3.34330e15	+	5.79076e15i	−0.807320	+	1.39832i	−0.914714	+	0.404102i	$0.867584\pi$
$692$		0			0
$693$	5.37763e14	−	1.41565e15i	0.127808	−	0.336452i
$694$		0			0
$695$	1.22418e14	−	2.12035e14i	0.0286372	−	0.0496011i
$696$		0			0
$697$	−1.00561e15	−	1.74176e15i	−0.231552	−	0.401059i
$698$		0			0
$699$	−4.74449e15			−1.07539
$700$		0			0
$701$	−6.81863e15			−1.52142			−0.760708	−	0.649094i	$-0.775148\pi$
$701$	−6.81863e15			−1.52142			−0.760708	+	0.649094i	$0.775148\pi$
$702$		0			0
$703$	5.36337e14	+	9.28963e14i	0.117810	+	0.204054i
$704$		0			0
$705$	−6.45817e14	+	1.11859e15i	−0.139659	+	0.241897i
$706$		0			0
$707$	2.21212e15	−	3.58274e14i	0.470979	−	0.0762796i
$708$		0			0
$709$	1.16186e15	−	2.01240e15i	0.243557	−	0.421853i	−0.718168	−	0.695870i	$-0.755019\pi$
$709$	1.16186e15	−	2.01240e15i	0.243557	−	0.421853i	0.961725	+	0.274017i	$0.0883524\pi$
$710$		0			0
$711$	−4.39525e14	−	7.61279e14i	−0.0907195	−	0.157131i
$712$		0			0
$713$	1.01214e15			0.205707
$714$		0			0
$715$	−5.52811e15			−1.10635
$716$		0			0
$717$	2.71928e14	+	4.70993e14i	0.0535918	+	0.0928237i
$718$		0			0
$719$	2.75377e15	−	4.76967e15i	0.534464	−	0.925719i	−0.464725	−	0.885455i	$-0.653847\pi$
$719$	2.75377e15	−	4.76967e15i	0.534464	−	0.925719i	0.999189	−	0.0402641i	$-0.0128199\pi$
$720$		0			0
$721$	−4.09513e15	−	5.02118e15i	−0.782752	−	0.959758i
$722$		0			0
$723$	1.98888e15	−	3.44483e15i	0.374410	−	0.648497i
$724$		0			0
$725$	5.60579e14	+	9.70951e14i	0.103939	+	0.180027i
$726$		0			0
$727$	6.22785e15			1.13736			0.568682	−	0.822558i	$-0.307454\pi$
$727$	6.22785e15			1.13736			0.568682	+	0.822558i	$0.307454\pi$
$728$		0			0
$729$	2.05891e14			0.0370370
$730$		0			0
$731$	2.36641e15	+	4.09874e15i	0.419318	+	0.726281i
$732$		0			0
$733$	3.23997e15	−	5.61179e15i	0.565548	−	0.979557i	−0.431451	−	0.902136i	$-0.641998\pi$
$733$	3.23997e15	−	5.61179e15i	0.565548	−	0.979557i	0.996999	−	0.0774207i	$-0.0246684\pi$
$734$		0			0
$735$	1.77574e15	+	1.57660e15i	0.305350	+	0.271108i
$736$		0			0
$737$	2.20442e15	−	3.81817e15i	0.373442	−	0.646821i
$738$		0			0
$739$	−6.53738e14	−	1.13231e15i	−0.109109	−	0.188982i	0.806301	−	0.591506i	$-0.201466\pi$
$739$	−6.53738e14	−	1.13231e15i	−0.109109	−	0.188982i	−0.915409	+	0.402524i	$0.868133\pi$
$740$		0			0
$741$	1.04405e15			0.171681
$742$		0			0
$743$	−3.01777e15			−0.488931			−0.244466	−	0.969658i	$-0.578612\pi$
$743$	−3.01777e15			−0.488931			−0.244466	+	0.969658i	$0.578612\pi$
$744$		0			0
$745$	−8.06222e14	−	1.39642e15i	−0.128705	−	0.222923i
$746$		0			0
$747$	−9.67230e13	+	1.67529e14i	−0.0152148	+	0.0263528i
$748$		0			0
$749$	3.98734e15	+	4.88901e15i	0.618063	+	0.757828i
$750$		0			0
$751$	−1.44807e15	+	2.50814e15i	−0.221193	+	0.383117i	−0.955170	−	0.296056i	$-0.904328\pi$
$751$	−1.44807e15	+	2.50814e15i	−0.221193	+	0.383117i	0.733978	+	0.679174i	$0.237662\pi$
$752$		0			0
$753$	−4.71431e14	−	8.16543e14i	−0.0709653	−	0.122916i
$754$		0			0
$755$	6.59875e15			0.978934
$756$		0			0
$757$	−9.86986e15			−1.44306			−0.721529	−	0.692384i	$-0.756560\pi$
$757$	−9.86986e15			−1.44306			−0.721529	+	0.692384i	$0.756560\pi$
$758$		0			0
$759$	−9.87834e14	−	1.71098e15i	−0.142349	−	0.246556i
$760$		0			0
$761$	−2.16846e15	+	3.75588e15i	−0.307989	+	0.533453i	−0.977922	−	0.208968i	$-0.932989\pi$
$761$	−2.16846e15	+	3.75588e15i	−0.307989	+	0.533453i	0.669933	+	0.742422i	$0.266323\pi$
$762$		0			0
$763$	5.47323e15	−	8.86443e14i	0.766229	−	0.124098i
$764$		0			0
$765$	8.93318e14	−	1.54727e15i	0.123273	−	0.213515i
$766$		0			0
$767$	3.71924e15	+	6.44190e15i	0.505917	+	0.876274i
$768$		0			0
$769$	1.08249e16			1.45155			0.725773	−	0.687934i	$-0.241482\pi$
$769$	1.08249e16			1.45155			0.725773	+	0.687934i	$0.241482\pi$
$770$		0			0
$771$	−7.00071e15			−0.925429
$772$		0			0
$773$	5.35271e15	+	9.27117e15i	0.697568	+	1.20822i	0.969307	+	0.245853i	$0.0790680\pi$
$773$	5.35271e15	+	9.27117e15i	0.697568	+	1.20822i	−0.271739	+	0.962371i	$0.587599\pi$
$774$		0			0
$775$	8.76059e14	−	1.51738e15i	0.112557	−	0.194955i
$776$		0			0
$777$	1.85803e15	−	4.89123e15i	0.235362	−	0.619587i
$778$		0			0
$779$	−3.63867e14	+	6.30236e14i	−0.0454451	+	0.0787132i
$780$		0			0
$781$	−6.85269e15	−	1.18692e16i	−0.843879	−	1.46164i
$782$		0			0
$783$	−6.59221e14			−0.0800464
$784$		0			0
$785$	6.75235e14			0.0808485
$786$		0			0
$787$	−1.68316e15	−	2.91532e15i	−0.198731	−	0.344212i	0.749386	−	0.662133i	$-0.230349\pi$
$787$	−1.68316e15	−	2.91532e15i	−0.198731	−	0.344212i	−0.948117	+	0.317921i	$0.897015\pi$
$788$		0			0
$789$	1.53461e15	−	2.65803e15i	0.178680	−	0.309483i
$790$		0			0
$791$	1.05044e15	−	2.76525e15i	0.120614	−	0.317515i
$792$		0			0
$793$	4.59751e15	−	7.96311e15i	0.520619	−	0.901738i
$794$		0			0
$795$	−1.15168e14	−	1.99477e14i	−0.0128621	−	0.0222779i
$796$		0			0
$797$	1.44656e16			1.59337			0.796683	−	0.604397i	$-0.206586\pi$
$797$	1.44656e16			1.59337			0.796683	+	0.604397i	$0.206586\pi$
$798$		0			0
$799$	6.58462e15			0.715358
$800$		0			0
$801$	−1.89931e15	−	3.28970e15i	−0.203525	−	0.352515i
$802$		0			0
$803$	−1.70840e14	+	2.95903e14i	−0.0180573	+	0.0312762i
$804$		0			0
$805$	3.05818e15	−	4.95302e14i	0.318849	−	0.0516408i
$806$		0			0
$807$	−3.10495e15	+	5.37793e15i	−0.319338	+	0.553109i
$808$		0			0
$809$	2.40564e15	+	4.16669e15i	0.244070	+	0.422741i	0.961870	−	0.273508i	$-0.0881841\pi$
$809$	2.40564e15	+	4.16669e15i	0.244070	+	0.422741i	−0.717800	+	0.696249i	$0.754851\pi$
$810$		0			0
$811$	8.22732e15			0.823463			0.411731	−	0.911305i	$-0.364924\pi$
$811$	8.22732e15			0.823463			0.411731	+	0.911305i	$0.364924\pi$
$812$		0			0
$813$	−6.64028e14			−0.0655675
$814$		0			0
$815$	5.79373e15	+	1.00350e16i	0.564406	+	0.977579i
$816$		0			0
$817$	8.56256e14	−	1.48308e15i	0.0822967	−	0.142542i
$818$		0			0
$819$	−3.21867e15	−	3.94652e15i	−0.305221	−	0.374242i
$820$		0			0
$821$	−7.14550e14	+	1.23764e15i	−0.0668568	+	0.115799i	−0.897516	−	0.440982i	$-0.854630\pi$
$821$	−7.14550e14	+	1.23764e15i	−0.0668568	+	0.115799i	0.830659	+	0.556781i	$0.187964\pi$
$822$		0			0
$823$	1.54576e15	+	2.67734e15i	0.142707	+	0.247175i	0.928515	−	0.371295i	$-0.121086\pi$
$823$	1.54576e15	+	2.67734e15i	0.142707	+	0.247175i	−0.785808	+	0.618470i	$0.787753\pi$
$824$		0			0
$825$	−3.42007e15			−0.311557
$826$		0			0
$827$	9.66280e15			0.868606			0.434303	−	0.900767i	$-0.356995\pi$
$827$	9.66280e15			0.868606			0.434303	+	0.900767i	$0.356995\pi$
$828$		0			0
$829$	7.93066e15	+	1.37363e16i	0.703493	+	1.21849i	0.967233	+	0.253892i	$0.0817106\pi$
$829$	7.93066e15	+	1.37363e16i	0.703493	+	1.21849i	−0.263740	+	0.964594i	$0.584956\pi$
$830$		0			0
$831$	7.53032e14	−	1.30429e15i	0.0659186	−	0.114174i
$832$		0			0
$833$	2.43430e15	−	1.18584e16i	0.210294	−	1.02442i
$834$		0			0
$835$	−7.25037e15	+	1.25580e16i	−0.618137	+	1.07064i
$836$		0			0
$837$	5.15107e14	+	8.92192e14i	0.0433419	+	0.0750703i
$838$		0			0
$839$	−4.12134e15			−0.342253			−0.171127	−	0.985249i	$-0.554741\pi$
$839$	−4.12134e15			−0.342253			−0.171127	+	0.985249i	$0.554741\pi$
$840$		0			0
$841$	−1.00898e16			−0.827000
$842$		0			0
$843$	4.29019e15	+	7.43083e15i	0.347076	+	0.601154i
$844$		0			0
$845$	−4.86676e15	+	8.42947e15i	−0.388622	+	0.673114i
$846$		0			0
$847$	−1.32957e15	−	1.63023e15i	−0.104798	−	0.128496i
$848$		0			0
$849$	4.69620e15	−	8.13406e15i	0.365388	−	0.632871i
$850$		0			0
$851$	−3.41308e15	−	5.91162e15i	−0.262140	−	0.454040i
$852$		0			0
$853$	−3.08245e15			−0.233709			−0.116855	−	0.993149i	$-0.537281\pi$
$853$	−3.08245e15			−0.233709			−0.116855	+	0.993149i	$0.537281\pi$
$854$		0			0
$855$	−6.46473e14			−0.0483879
$856$		0			0
$857$	2.44767e15	+	4.23950e15i	0.180867	+	0.313271i	0.942176	−	0.335118i	$-0.108776\pi$
$857$	2.44767e15	+	4.23950e15i	0.180867	+	0.313271i	−0.761309	+	0.648389i	$0.775443\pi$
$858$		0			0
$859$	−1.29228e15	+	2.23830e15i	−0.0942746	+	0.163288i	−0.909306	−	0.416129i	$-0.863387\pi$
$859$	−1.29228e15	+	2.23830e15i	−0.0942746	+	0.163288i	0.815031	+	0.579417i	$0.196720\pi$
$860$		0			0
$861$	3.50405e15	−	5.67516e14i	0.252379	−	0.0408753i
$862$		0			0
$863$	1.35637e16	−	2.34930e16i	0.964536	−	1.67063i	0.253679	−	0.967288i	$-0.418359\pi$
$863$	1.35637e16	−	2.34930e16i	0.964536	−	1.67063i	0.710857	−	0.703337i	$-0.248307\pi$
$864$		0			0
$865$	2.27735e15	+	3.94449e15i	0.159897	+	0.276950i
$866$		0			0
$867$	−7.80024e14			−0.0540758
$868$		0			0
$869$	−8.58569e15			−0.587714
$870$		0			0
$871$	−7.41328e15	−	1.28402e16i	−0.501084	−	0.867903i
$872$		0			0
$873$	3.38580e15	−	5.86438e15i	0.225987	−	0.391421i
$874$		0			0
$875$	5.71500e15	−	1.50446e16i	0.376679	−	0.991601i
$876$		0			0
$877$	8.22916e14	−	1.42533e15i	0.0535621	−	0.0927724i	−0.838001	−	0.545668i	$-0.816276\pi$
$877$	8.22916e14	−	1.42533e15i	0.0535621	−	0.0927724i	0.891563	+	0.452896i	$0.149609\pi$
$878$		0			0
$879$	8.75581e15	+	1.51655e16i	0.562805	+	0.974806i
$880$		0			0
$881$	−1.53280e16			−0.973009			−0.486504	−	0.873678i	$-0.661728\pi$
$881$	−1.53280e16			−0.973009			−0.486504	+	0.873678i	$0.661728\pi$
$882$		0			0
$883$	−1.61478e15			−0.101235			−0.0506174	−	0.998718i	$-0.516119\pi$
$883$	−1.61478e15			−0.101235			−0.0506174	+	0.998718i	$0.516119\pi$
$884$		0			0
$885$	−2.30294e15	−	3.98881e15i	−0.142592	−	0.246976i
$886$		0			0
$887$	1.17403e16	−	2.03349e16i	0.717961	−	1.24354i	−0.243846	−	0.969814i	$-0.578409\pi$
$887$	1.17403e16	−	2.03349e16i	0.717961	−	1.24354i	0.961806	−	0.273731i	$-0.0882577\pi$
$888$		0			0
$889$	−6.05617e15	+	1.59427e16i	−0.365795	+	0.962949i
$890$		0			0
$891$	1.00547e15	−	1.74153e15i	0.0599849	−	0.103897i
$892$		0			0
$893$	−1.19128e15	−	2.06336e15i	−0.0701992	−	0.121589i
$894$		0			0
$895$	−1.46836e16			−0.854682
$896$		0			0
$897$	−6.64400e15			−0.382007
$898$		0			0
$899$	−1.64927e15	−	2.85662e15i	−0.0936727	−	0.162246i
$900$		0			0
$901$	−5.87116e14	+	1.01691e15i	−0.0329411	+	0.0570556i
$902$		0			0
$903$	−8.24578e15	+	1.33548e15i	−0.457035	+	0.0740212i
$904$		0			0
$905$	−1.23245e16	+	2.13467e16i	−0.674845	+	1.16887i
$906$		0			0
$907$	3.31151e14	+	5.73570e14i	0.0179137	+	0.0310275i	0.874843	−	0.484406i	$-0.160964\pi$
$907$	3.31151e14	+	5.73570e14i	0.0179137	+	0.0310275i	−0.856930	+	0.515433i	$0.827631\pi$
$908$		0			0
$909$	2.97580e15			0.159039
$910$		0			0
$911$	−1.96775e15			−0.103901			−0.0519505	−	0.998650i	$-0.516544\pi$
$911$	−1.96775e15			−0.103901			−0.0519505	+	0.998650i	$0.516544\pi$
$912$		0			0
$913$	9.44695e14	+	1.63626e15i	0.0492836	+	0.0853617i
$914$		0			0
$915$	−2.84676e15	+	4.93074e15i	−0.146736	+	0.254153i
$916$		0			0
$917$	1.73479e16	+	2.12708e16i	0.883516	+	1.08331i
$918$		0			0
$919$	8.95709e15	−	1.55141e16i	0.450746	−	0.780715i	−0.547687	−	0.836683i	$-0.684491\pi$
$919$	8.95709e15	−	1.55141e16i	0.450746	−	0.780715i	0.998433	+	0.0559689i	$0.0178248\pi$
$920$		0			0
$921$	−5.63839e15	−	9.76598e15i	−0.280367	−	0.485610i
$922$		0			0
$923$	−4.60900e16			−2.26463
$924$		0			0
$925$	−1.18167e16			−0.573743
$926$		0			0
$927$	−4.30206e15	−	7.45139e15i	−0.206414	−	0.357519i
$928$		0			0
$929$	−1.31551e16	+	2.27853e16i	−0.623745	+	1.08036i	0.365037	+	0.930993i	$0.381056\pi$
$929$	−1.31551e16	+	2.27853e16i	−0.623745	+	1.08036i	−0.988782	+	0.149365i	$0.952277\pi$
$930$		0			0
$931$	−4.15637e15	+	1.38260e15i	−0.194756	+	0.0647847i
$932$		0			0
$933$	−7.74983e15	+	1.34231e16i	−0.358875	+	0.621590i
$934$		0			0
$935$	−8.72504e15	−	1.51122e16i	−0.399304	−	0.691615i
$936$		0			0
$937$	2.28163e16			1.03199			0.515996	−	0.856591i	$-0.327422\pi$
$937$	2.28163e16			1.03199			0.515996	+	0.856591i	$0.327422\pi$
$938$		0			0
$939$	7.35043e15			0.328588
$940$		0			0
$941$	8.84573e15	+	1.53213e16i	0.390832	+	0.676942i	0.992560	−	0.121760i	$-0.0388538\pi$
$941$	8.84573e15	+	1.53213e16i	0.390832	+	0.676942i	−0.601727	+	0.798702i	$0.705521\pi$
$942$		0			0
$943$	2.31553e15	−	4.01062e15i	0.101120	−	0.175145i
$944$		0			0
$945$	1.99299e15	+	2.44367e15i	0.0860262	+	0.105480i
$946$		0			0
$947$	−1.64962e14	+	2.85723e14i	−0.00703817	+	0.0121905i	−0.869523	−	0.493892i	$-0.835574\pi$
$947$	−1.64962e14	+	2.85723e14i	−0.00703817	+	0.0121905i	0.862485	+	0.506083i	$0.168907\pi$
$948$		0			0
$949$	5.74519e14	+	9.95097e14i	0.0242293	+	0.0419663i
$950$		0			0
$951$	1.02008e16			0.425246
$952$		0			0
$953$	1.63649e15			0.0674376			0.0337188	−	0.999431i	$-0.489265\pi$
$953$	1.63649e15			0.0674376			0.0337188	+	0.999431i	$0.489265\pi$
$954$		0			0
$955$	1.11226e16	+	1.92648e16i	0.453092	+	0.784778i
$956$		0			0
$957$	−3.21931e15	+	5.57601e15i	−0.129642	+	0.224547i
$958$		0			0
$959$	−4.36261e16	+	7.06567e15i	−1.73678	+	0.281288i
$960$		0			0
$961$	1.01268e16	−	1.75401e16i	0.398560	−	0.690326i
$962$		0			0
$963$	4.18882e15	+	7.25525e15i	0.162985	+	0.282298i
$964$		0			0
$965$	7.07969e15			0.272342
$966$		0			0
$967$	9.69847e14			0.0368857			0.0184428	−	0.999830i	$-0.494129\pi$
$967$	9.69847e14			0.0368857			0.0184428	+	0.999830i	$0.494129\pi$
$968$		0			0
$969$	1.64783e15	+	2.85412e15i	0.0619628	+	0.107323i
$970$		0			0
$971$	−1.35248e16	+	2.34256e16i	−0.502834	+	0.870934i	0.497161	+	0.867659i	$0.334376\pi$
$971$	−1.35248e16	+	2.34256e16i	−0.502834	+	0.870934i	−0.999995	+	0.00327555i	$0.998957\pi$
$972$		0			0
$973$	7.82290e14	−	2.05936e15i	0.0287572	−	0.0757029i
$974$		0			0
$975$	−5.75070e15	+	9.96050e15i	−0.209024	+	0.362039i
$976$		0			0
$977$	−1.44868e16	−	2.50918e16i	−0.520656	−	0.901803i	−0.999712	−	0.0240186i	$-0.992354\pi$
$977$	−1.44868e16	−	2.50918e16i	−0.520656	−	0.901803i	0.479055	−	0.877785i	$-0.340979\pi$
$978$		0			0
$979$	−3.71012e16			−1.31851
$980$		0			0
$981$	7.36274e15			0.258738
$982$		0			0
$983$	−2.37153e16	−	4.10761e16i	−0.824109	−	1.42740i	−0.902598	−	0.430484i	$-0.858343\pi$
$983$	−2.37153e16	−	4.10761e16i	−0.824109	−	1.42740i	0.0784891	−	0.996915i	$-0.474990\pi$
$984$		0			0
$985$	−1.57916e16	+	2.73518e16i	−0.542659	+	0.939912i
$986$		0			0
$987$	−4.12696e15	+	1.08641e16i	−0.140244	+	0.369191i
$988$		0			0
$989$	−5.44894e15	+	9.43784e15i	−0.183119	+	0.317171i
$990$		0			0
$991$	1.24138e16	+	2.15014e16i	0.412573	+	0.714598i	0.995170	−	0.0981633i	$-0.0312968\pi$
$991$	1.24138e16	+	2.15014e16i	0.412573	+	0.714598i	−0.582597	+	0.812761i	$0.697963\pi$
$992$		0			0
$993$	1.68722e16			0.554563
$994$		0			0
$995$	5.13465e15			0.166911
$996$		0			0
$997$	−2.01795e16	−	3.49519e16i	−0.648765	−	1.12369i	−0.983418	−	0.181352i	$-0.941953\pi$
$997$	−2.01795e16	−	3.49519e16i	−0.648765	−	1.12369i	0.334653	−	0.942341i	$-0.391381\pi$
$998$		0			0
$999$	3.47401e15	−	6.01717e15i	0.110464	−	0.191330i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.i.a.37.3	yes	14
3.2	odd	2		252.12.k.b.37.5		14
7.4	even	3	inner	84.12.i.a.25.3	✓	14
21.11	odd	6		252.12.k.b.109.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.i.a.25.3	✓	14	7.4	even	3	inner
84.12.i.a.37.3	yes	14	1.1	even	1	trivial
252.12.k.b.37.5		14	3.2	odd	2
252.12.k.b.109.5		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)

\(f(q)\)	\(=\)	\(q+(121.500 + 210.444i) q^{3} +(-2471.06 + 4280.00i) q^{5} +(-15790.8 + 41569.0i) q^{7} +(-29524.5 + 51137.9i) q^{9} +O(q^{10})\)	\(q+(121.500 + 210.444i) q^{3} +(-2471.06 + 4280.00i) q^{5} +(-15790.8 + 41569.0i) q^{7} +(-29524.5 + 51137.9i) q^{9} +(288366. + 499465. i) q^{11} +1.93950e6 q^{13} -1.20093e6 q^{15} +(3.06112e6 + 5.30202e6i) q^{17} +(1.10763e6 - 1.91847e6i) q^{19} +(-1.06665e7 + 1.72755e6i) q^{21} +(-7.04861e6 + 1.22086e7i) q^{23} +(1.22018e7 + 2.11342e7i) q^{25} -1.43489e7 q^{27} +4.59423e7 q^{29} +(-3.58987e7 - 6.21784e7i) q^{31} +(-7.00730e7 + 1.21370e8i) q^{33} +(-1.38895e8 - 1.70304e8i) q^{35} +(-2.42110e8 + 4.19347e8i) q^{37} +(2.35649e8 + 4.08157e8i) q^{39} -3.28509e8 q^{41} +7.73052e8 q^{43} +(-1.45913e8 - 2.52730e8i) q^{45} +(5.37762e8 - 9.31431e8i) q^{47} +(-1.47863e9 - 1.31281e9i) q^{49} +(-7.43853e8 + 1.28839e9i) q^{51} +(9.58988e7 + 1.66102e8i) q^{53} -2.85028e9 q^{55} +5.38309e8 q^{57} +(1.91762e9 + 3.32142e9i) q^{59} +(2.37046e9 - 4.10575e9i) q^{61} +(-1.65954e9 - 2.03481e9i) q^{63} +(-4.79262e9 + 8.30106e9i) q^{65} +(-3.82226e9 - 6.62035e9i) q^{67} -3.42562e9 q^{69} -2.37639e10 q^{71} +(2.96220e8 + 5.13068e8i) q^{73} +(-2.96504e9 + 5.13560e9i) q^{75} +(-2.53158e10 + 4.10013e9i) q^{77} +(-7.44339e9 + 1.28923e10i) q^{79} +(-1.74339e9 - 3.01964e9i) q^{81} +3.27603e9 q^{83} -3.02568e10 q^{85} +(5.58199e9 + 9.66828e9i) q^{87} +(-3.21650e10 + 5.57114e10i) q^{89} +(-3.06263e10 + 8.06230e10i) q^{91} +(8.72339e9 - 1.51094e10i) q^{93} +(5.47404e9 + 9.48132e9i) q^{95} -1.14678e11 q^{97} -3.40555e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 1701 q^{3} + 7218 q^{5} + 35001 q^{7} - 413343 q^{9}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more