LMFDB - Newform orbit 243.2.e.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [243,2,Mod(28,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([8]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("243.28");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$243 = 3^{5}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	243.e (of order $9$ , degree $6$ , not 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:	$1.94036476912$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{9})$
Coefficient field:	12.0.1952986685049.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} + \cdots + 3$ x^12 - 6x^11 + 27x^10 - 80x^9 + 186x^8 - 330x^7 + 463x^6 - 504x^5 + 420x^4 - 258x^3 + 108x^2 - 27*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$3$
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{11} - \beta_{8} - \beta_{2}) q^{2} + ( - \beta_{9} + \beta_{8} + \beta_{6} + \cdots - 1) q^{4} + ( - \beta_{8} + \beta_{6} - \beta_{5} + \cdots - 1) q^{5} + (2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \cdots - 1) q^{7}+ \cdots + (10 \beta_{11} + 3 \beta_{10} + \cdots + 1) q^{98}+O(q^{100})$ q + (-b11 - b8 - b2) * q^2 + (-b9 + b8 + b6 + b5 + b4 - b1 - 1) * q^4 + (-b8 + b6 - b5 - b3 - 1) * q^5 + (2b11 + b10 - 2b9 + b8 + b7 + b6 + b5 - b4 + b3 - b1 - 1) * q^7 + (2b11 + b10 - b8 + b5 - b4 + b3 - 1) q^8 + (b10 - 2b9 + b8 - b7 + 2b6 + b5 + b4 - b3 - b2 - 2b1 - 3) q^10 + (2b7 + b6 + b5 + b4 - b2 - b1 - 1) q^11 + (-b9 + b6 - b4 + b3 - b2 - b1) * q^13 + (-b11 - 2b10 + 2b9 - 3b7 - b6 - b5 + b2 + 3) q^14 + (-b11 + b9 + b8 - 2b7 + b6 + 2b4 - b3 - 2b1 - 1) q^16 + (-b11 + 2b10 + 2b8 + 2b7 - b6 - b4 + b2 + b1 + 1) q^17 + (2b11 - b10 - b9 + 2b8 + b7 + b5 - b4 + 2b3 + b2) q^19 + (-2b10 + 2b8 - 2b7 - b6 + b4 - 2) q^20 + (b11 + b10 - b7 - b6 + b5 - b3) * q^22 + (2b11 + 2b10 - 2b9 + 2b8 + 2b7 - b5 + b2 + b1 + 1) q^23 + (-2b11 - 2b10 - b9 - b8 + b4 - b2 + b1 + 2) * q^25 + (-3b10 - 3b9 + 2b8 - 2b7 + b5 + b4 + 2) * q^26 + (b10 + b9 - 2b8 + 2b7 - 1) * q^28 + (-b11 + 3b10 - 3b9 + 2b8 + 2b2 + 3) * q^29 + (b11 + b10 + b9 - b8 + 2b7 - b6 - 3b5 - 2b4 + 2b2 + 3b1 + 4) q^31 + (-b11 - 2b10 + b8 + b7 - 2b6 + 2b5 + 2b3 + b2 - b1 - 1) * q^32 + (-2b11 + 2b9 - b8 - b6 + b4 - 2b3 + 2b1 + 1) * q^34 + (-2b10 + 3b9 - b8 - 3b7 - b6 - 2b5 + 2b4 - b3 + b2 - 1) q^35 + (-2b11 - b10 + 2b9 - b8 + b7 - 3b6 - 2b5 - b4 + 2b3 + b2 + 3b1 + 5) * q^37 + (3b11 + 5b9 - 3b8 - 3b6 - 2b5 - 3b4 + 2b2 + 3b1 + 5) * q^38 + (3b9 - b6 - b5 + 2b4 - 2b3 + b2 - 2) q^40 + (-b11 + 3b10 + 2b7 + 3b6 + 3b5 + b4 - b3 - 3b2 - 3) q^41 + (b11 - b9 - b8 + 2b7 - 3b6 - b5 - 4b4 + b3 + b2 + 4b1 + 3) * q^43 + (-b11 + 2b9 + 2b7 + 3b6 + 2b4 - 2b2 - 3b1 - 3) * q^44 + (-3b11 - b10 + 5b9 - 4b8 - 5b7 - 2b6 - 2b5 + 2b4 - 3b3 - b2) * q^46 + (2b11 - b10 - 2b9 + 2b8 - b7 + 3b6 - 3b4 + b3 - b1 - 2) q^47 + (-b11 - 4b10 + b7 + 2b3 - b2 + b1 - 1) * q^49 + (3b11 + 3b10 - 2b9 + 2b8 + 2b5 - b4 - 2b2 - 2b1 - 5) q^50 + (2b11 + 2b10 + 4b9 - 2b8 - b6 - 2b4 + b3 - b1 - 4) q^52 + (-3b8 + 3b7 - 3b5 - 3b4 + 3) * q^53 + (-b10 - b9 + 2b8 - 2b7 - 3b5 - 2b4 - b3 + b2 + 2b1 + 1) q^55 + (-2b11 - b10 + 2b9 - 4b8 + b4 + b2 + b1 - 1) q^56 + (-3b11 - 3b10 + 3b9 - 3b8 - 6b7 - 2b6 + b5 + b4 - 3b2 - b1 - 4) q^58 + (-3b10 - 7b8 - 2b6 + 2b5 + 2b3 - 3b2 + 3b1 + 2) q^59 + (-3b11 - 2b10 + 3b9 - 2b7 - b6 - 5b5 + b4) q^61 + (-4b11 + 4b10 - 4b8 + 3b6 - 2b5 + 2b4 - 5b3 - 3b2 - 1) * q^62 + (2b11 - 2b10 + 4b9 - 2b8 + 2b7 - 3b6 + b5 - b4 - b3 + b2 + 3b1 + 2) q^64 + (b11 - b8 - 3b7 + b6 - b5 + 2b4 - b3 + b2 - 2b1 - 4) q^65 + (-b11 + 2b10 - b9 + b7 - 3b6 + b5 - b4 + b3 + 3b2 + 4b1 + 4) * q^67 + (-b11 + b10 - 2b9 - b5 - 2b4 + 2b3 - b1 + 1) q^68 + (3b11 - 3b8 + 3b7 - b5 - b4 + b3 + b2 + b1 - 1) q^70 + (-7b11 - 4b10 - 4b8 - 4b7 - b6 + 2b4 - 2b2 + b1 + 1) * q^71 + (-b11 + 6b10 - 6b9 + 6b7 + 3b6 - 3b3 - 3b2 - 2) * q^73 + (2b11 + 3b10 - 2b9 - b8 + 3b7 - b5 - 3b3 + 3b1 + 1) * q^74 + (-5b11 + 4b10 - 2b8 + 5b7 + 4b6 - 4b5 - 2b3 + 2) q^76 + (-3b11 - 3b10 + 3b9 - 3b8 - b7 - b6 + b5 + 2b4 - 2b2 - b1 + 1) * q^77 + (6b11 + 2b10 + 6b8 - b6 - b4 + b3 + 5b2 + 1) * q^79 + (-2b10 - 2b9 + b8 - b7 + b5 + b3 - b2 + b1 - 3) * q^80 + (3b8 - 3b7 + 4b5 + 3b4 + b3 - b2 - b1 - 3) * q^82 + (5b11 + 3b10 + 5b8 - 3b4 - b2 - 3b1 - 3) q^83 + (-2b11 - 2b10 + 2b7 - b6 + b5 - 2b2 - b1 + 3) * q^85 + (-4b11 + 6b10 + 4b8 + 4b7 + 4b6 - 4b5 - 3b3 + b2 - b1 + 2) q^86 + (-3b10 - 3b8 - 3b7 - 2b6 + 2b5 + 2b4 + 3) * q^88 + (5b11 - 3b10 - 2b9 + 5b8 + 2b7 - b6 + 4b5 - 4b4 + 4b3 - 1) * q^89 + (2b11 - 3b10 - 3b8 - 3b7 + b6 - b5 - b4 + b3 + b2 - b1) * q^91 + (-3b11 - 4b9 + 3b8 + 3b7 + 3b6 + b5 + 3b3 - b2 + 2) * q^92 + (2b11 - 4b10 - 2b7 + b5 - b4 + b3 + b1 + 4) q^94 + (3b11 - 3b10 + 5b9 - 5b6 - 3b5 - 2b4 + 2b3 + 5b2 + 2b1 + 4) q^95 + (-2b11 - b9 + 2b8 + 2b7 + 3b6 + 5b5 + 2b4 + b3 - 5b2 - 2b1) * q^97 + (10b11 + 3b10 - b9 + 3b8 + 2b7 - 2b6 + b5 - 3b4 - b3 + 3b2 + 2b1 + 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 6 q^{8} - 3 q^{10} + 6 q^{11} + 3 q^{13} + 21 q^{14} + 9 q^{16} - 9 q^{17} - 3 q^{19} - 24 q^{20} + 12 q^{22} + 12 q^{23} + 12 q^{25} + 30 q^{26} - 12 q^{28}+ \cdots + 45 q^{98}+O(q^{100})$ 12 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 6 * q^8 - 3 * q^10 + 6 * q^11 + 3 * q^13 + 21 * q^14 + 9 * q^16 - 9 * q^17 - 3 * q^19 - 24 * q^20 + 12 * q^22 + 12 * q^23 + 12 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 + 12 * q^31 - 27 * q^32 - 12 * q^35 - 3 * q^37 + 30 * q^38 - 15 * q^40 - 6 * q^41 - 15 * q^43 - 3 * q^44 - 3 * q^46 - 12 * q^47 - 33 * q^49 - 21 * q^50 - 45 * q^52 + 18 * q^53 - 12 * q^55 - 30 * q^56 - 51 * q^58 + 3 * q^59 - 33 * q^61 + 12 * q^62 + 12 * q^64 - 21 * q^65 - 6 * q^67 - 9 * q^68 - 15 * q^70 - 27 * q^71 + 6 * q^73 + 21 * q^74 + 6 * q^76 + 12 * q^77 + 21 * q^79 - 42 * q^80 - 12 * q^82 + 6 * q^83 + 36 * q^85 + 21 * q^86 + 42 * q^88 - 9 * q^89 + 6 * q^91 + 3 * q^92 + 48 * q^94 - 3 * q^95 + 39 * q^97 + 45 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} + \cdots + 3$ x^12 - 6*x^11 + 27*x^10 - 80*x^9 + 186*x^8 - 330*x^7 + 463*x^6 - 504*x^5 + 420*x^4 - 258*x^3 + 108*x^2 - 27*x + 3 :

$\beta_{1}$	$=$	$\nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + \cdots + 9$ v^10 - 5v^9 + 22v^8 - 58v^7 + 127v^6 - 199v^5 + 249v^4 - 224v^3 + 145v^2 - 58*v + 9
$\beta_{2}$	$=$	$3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} + \cdots - 25$ 3v^11 - 16v^10 + 71v^9 - 197v^8 + 445v^7 - 747v^6 + 1006v^5 - 1030v^4 + 803v^3 - 445v^2 + 155*v - 25
$\beta_{3}$	$=$	$- 6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + \cdots + 25$ -6v^11 + 32v^10 - 140v^9 + 384v^8 - 849v^7 + 1390v^6 - 1805v^5 + 1762v^4 - 1285v^3 + 649v^2 - 195*v + 25
$\beta_{4}$	$=$	$- 9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + \cdots + 49$ -9v^11 + 49v^10 - 216v^9 + 601v^8 - 1344v^7 + 2232v^6 - 2942v^5 + 2918v^4 - 2170v^3 + 1118v^2 - 348*v + 49
$\beta_{5}$	$=$	$9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} + \cdots - 62$ 9v^11 - 50v^10 + 221v^9 - 623v^8 + 1402v^7 - 2360v^6 + 3144v^5 - 3178v^4 + 2411v^3 - 1286v^2 + 421*v - 62
$\beta_{6}$	$=$	$11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} + \cdots - 61$ 11v^11 - 60v^10 + 265v^9 - 739v^8 + 1657v^7 - 2761v^6 + 3653v^5 - 3643v^4 + 2724v^3 - 1417v^2 + 442*v - 61
$\beta_{7}$	$=$	$- 16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + \cdots + 85$ -16v^11 + 87v^10 - 383v^9 + 1064v^8 - 2375v^7 + 3936v^6 - 5176v^5 + 5122v^4 - 3802v^3 + 1958v^2 - 610*v + 85
$\beta_{8}$	$=$	$- 16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + \cdots + 110$ -16v^11 + 89v^10 - 393v^9 + 1108v^8 - 2491v^7 + 4191v^6 - 5577v^5 + 5631v^4 - 4267v^3 + 2272v^2 - 742*v + 110
$\beta_{9}$	$=$	$36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} + \cdots - 209$ 36v^11 - 198v^10 + 873v^9 - 2443v^8 + 5472v^7 - 9134v^6 + 12076v^5 - 12058v^4 + 9024v^3 - 4708v^2 + 1486*v - 209
$\beta_{10}$	$=$	$- 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + \cdots + 217$ -36v^11 + 198v^10 - 873v^9 + 2444v^8 - 5476v^7 + 9150v^6 - 12110v^5 + 12120v^4 - 9096v^3 + 4772v^2 - 1519*v + 217
$\beta_{11}$	$=$	$- 42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + \cdots + 257$ -42v^11 + 231v^10 - 1019v^9 + 2853v^8 - 6396v^7 + 10689v^6 - 14157v^5 + 14172v^4 - 10648v^3 + 5589v^2 - 1785*v + 257

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \cdots + 3 ) / 3$ (b11 - b10 + b9 + b8 + b7 - 2b6 + b5 - 2b4 + b3 + b2 + b1 + 3) / 3
$\nu^{2}$	$=$	$( \beta_{11} - \beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \cdots - 6 ) / 3$ (b11 - b10 + b9 + 4b8 - 2b7 - 2b6 + 4b5 + b4 + b3 + b2 - 2*b1 - 6) / 3
$\nu^{3}$	$=$	$( - 5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + \cdots - 18 ) / 3$ (-5b11 + 5b10 - 5b9 + b8 - 8b7 + 7b6 + 4b5 + 10b4 - 5b3 - 5b2 - 8b1 - 18) / 3
$\nu^{4}$	$=$	$( - 11 \beta_{11} + 17 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} - 14 \beta_{5} + \cdots + 6 ) / 3$ (-11b11 + 17b10 - 5b9 - 20b8 + 4b7 + 16b6 - 14b5 + 4b4 - 14b3 - 8b2 + b1 + 6) / 3
$\nu^{5}$	$=$	$( 19 \beta_{11} - \beta_{10} + 31 \beta_{9} - 32 \beta_{8} + 43 \beta_{7} - 20 \beta_{6} - 41 \beta_{5} + \cdots + 87 ) / 3$ (19b11 - b10 + 31b9 - 32b8 + 43b7 - 20b6 - 41b5 - 44b4 + 10b3 + 25b2 + 40b1 + 87) / 3
$\nu^{6}$	$=$	$( 85 \beta_{11} - 97 \beta_{10} + 55 \beta_{9} + 64 \beta_{8} + 10 \beta_{7} - 101 \beta_{6} + 19 \beta_{5} + \cdots + 60 ) / 3$ (85b11 - 97b10 + 55b9 + 64b8 + 10b7 - 101b6 + 19b5 - 62b4 + 91b3 + 70b2 + 31*b1 + 60) / 3
$\nu^{7}$	$=$	$( - 20 \beta_{11} - 118 \beta_{10} - 134 \beta_{9} + 244 \beta_{8} - 218 \beta_{7} + \beta_{6} + \cdots - 357 ) / 3$ (-20b11 - 118b10 - 134b9 + 244b8 - 218b7 + b6 + 232b5 + 157b4 + 52b3 - 74b2 - 179b1 - 357) / 3
$\nu^{8}$	$=$	$( - 503 \beta_{11} + 386 \beta_{10} - 440 \beta_{9} - 47 \beta_{8} - 233 \beta_{7} + 514 \beta_{6} + \cdots - 639 ) / 3$ (-503b11 + 386b10 - 440b9 - 47b8 - 233b7 + 514b6 + 163b5 + 466b4 - 431b3 - 461b2 - 329*b1 - 639) / 3
$\nu^{9}$	$=$	$( - 425 \beta_{11} + 1076 \beta_{10} + 319 \beta_{9} - 1313 \beta_{8} + 955 \beta_{7} + 502 \beta_{6} + \cdots + 1164 ) / 3$ (-425b11 + 1076b10 + 319b9 - 1313b8 + 955b7 + 502b6 - 1013b5 - 332b4 - 743b3 - 59b2 + 631*b1 + 1164) / 3
$\nu^{10}$	$=$	$( 2299 \beta_{11} - 862 \beta_{10} + 2725 \beta_{9} - 1193 \beta_{8} + 2104 \beta_{7} - 2135 \beta_{6} + \cdots + 4356 ) / 3$ (2299b11 - 862b10 + 2725b9 - 1193b8 + 2104b7 - 2135b6 - 1907b5 - 2705b4 + 1495b3 + 2425b2 + 2344*b1 + 4356) / 3
$\nu^{11}$	$=$	$( 4708 \beta_{11} - 6628 \beta_{10} + 985 \beta_{9} + 5506 \beta_{8} - 3107 \beta_{7} - 4679 \beta_{6} + \cdots - 1698 ) / 3$ (4708b11 - 6628b10 + 985b9 + 5506b8 - 3107b7 - 4679b6 + 3238b5 - 992b4 + 5476b3 + 2770b2 - 1043*b1 - 1698) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$ .

$n$	$2$
$\chi(n)$	$\beta_{8} - \beta_{9}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

28.1

0.500000	+	0.258654i
0.500000	−	2.22827i
0.500000	+	1.00210i
0.500000	−	1.68614i
0.500000	+	0.0126039i
0.500000	+	1.27297i
0.500000	−	0.0126039i
0.500000	−	1.27297i
0.500000	−	1.00210i
0.500000	+	1.68614i
0.500000	−	0.258654i
0.500000	+	2.22827i

0.0721450

0.409154i

1.71718

−

0.625003i

−1.69693

−

1.42389i

1.24005

0.451340i

0.795075

1.37711i

0.460168

−

0.797034i

28.2

0.367548

2.08447i

−2.33052

0.848241i

2.05537

1.72466i

−0.913694

−

0.332557i

−0.508086

−

0.880031i

−2.83955

4.91825i

55.1

−2.25679

0.821403i

2.88629

−

2.42189i

−0.0161638

−

0.0916693i

−0.444200

−

0.372728i

−2.12277

3.67675i

0.111776

0.193601i

55.2

0.990741

−

0.360600i

−0.680553

0.571052i

0.303153

1.71926i

1.88389

1.58077i

−1.52266

2.63732i

0.920313

1.59403i

109.1

−1.28765

1.08047i

0.143341

−

0.812925i

−1.06142

0.386327i

−0.678777

−

3.84954i

−0.987144

−

1.70978i

0.949332

−

1.64429i

109.2

0.614005

−

0.515212i

−0.235737

1.33693i

−2.58401

0.940501i

0.412733

2.34072i

1.34559

2.33062i

−1.10204

1.90878i

136.1