LMFDB - Newform orbit 729.2.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(244,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

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

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

chi := DirichletCharacter("729.244");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	729.c (of order $3$, degree $2$, 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:	$5.82109430735$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
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^{3} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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_{10} + \beta_{8} + \beta_{6}) q^{2} + (\beta_{11} + \beta_{10} - \beta_{6} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + 1) q^{8}+O(q^{10}) $ q + (-b10 + b8 + b6) * q^2 + (b11 + b10 - b6 + b4 - b3 + b2 - 1) * q^4 + (-b9 + b7 + b5 + b1 - 1) * q^5 + (b11 - b10 + b8 - b7) * q^7 + (-b5 - b4 - 2b2 + 1) q^8	$ q + ( - \beta_{10} + \beta_{8} + \beta_{6}) q^{2} + (\beta_{11} + \beta_{10} - \beta_{6} + \cdots - 1) q^{4}+ \cdots + (4 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \cdots - 9) q^{98}+O(q^{100}) $ q + (-b10 + b8 + b6) * q^2 + (b11 + b10 - b6 + b4 - b3 + b2 - 1) * q^4 + (-b9 + b7 + b5 + b1 - 1) * q^5 + (b11 - b10 + b8 - b7) * q^7 + (-b5 - b4 - 2b2 + 1) q^8 + (b5 + b3 + b2 - b1 + 1) * q^10 + (2b9 + b7) q^11 + (b9 - b7 - 2b6 - b5 - 2b3 - b1 - 1) * q^13 + (b11 - 2b10 - b9 + b8 + b5 + b4 - b2 - 1) q^14 + (b11 - b10 - b9 + b8 + b7 + b6) * q^16 + (b4 - b3 - b2 + 1) * q^17 + (-b4 - b3 + b2 - b1) * q^19 + (-b11 - b10 + b9 + 2b8) q^20 + (2b10 - b8 - b7 - 2b6 - 2b5 - 2b3 + b2 - b1 - 2) * q^22 + (-2b10 - 2b9 + 2b8 - b7 - b6 + b5 - b3 - b1 - 3) q^23 + (b11 - b9 + 2b8 - b7) q^25 + (-b5 + b3 - 3b2 + b1 + 3) q^26 + (2b5 + b2 - 1) q^28 + (2b10 + 3b9 + b8 - 2b6) q^29 + (-b11 - 2b10 + 2b8 - 2b7 - b4 - 2b1) * q^31 + (-b11 - 4b10 + 2b8 - 2b7 - b4 - 2b2 - 2b1) q^32 + (-b11 + b10 - b9 - b8 + 2b7 + 2b6) * q^34 + (b5 + 2b4 + b2 - b1 + 2) q^35 + (-b4 - b3 - 2b2 + 2b1) * q^37 + (-2b10 + 5b8 - 3b7 - b6) q^38 + (b11 + b10 - 2b9 + 2b6 + b4 + 2b3 + b2) q^40 + (-3b11 - b10 - 3b9 + b8 + b6 + b5 - 3b4 + b3 - 2) q^41 + (-b11 + b10 + b9 - b8 - 3b7 - 2b6) * q^43 + (-2b5 - 2b4 + 3b3 - 4b2 + 2) * q^44 + (2b5 + 2b4 + 3b3 - 2b2 + b1 + 2) * q^46 + (3b11 + b10 + 4b9 + 2b8 - b7 - b6) q^47 + (b11 - 4b10 + b9 + 4b8 + 2b6 + 4b5 + b4 + 2b3 + 3) q^49 + (-3b10 - 2b9 + 2b7 + 3b6 + 2b5 + 3b3 - 3b2 + 2b1 + 1) * q^50 + (-b11 - 3b10 + b9 - 2b8 + 2b7 + 2b6) * q^52 + (-3b3 - 3b1) * q^53 + (-4b5 - b4 - 2b2 - 2b1 - 1) q^55 + (b11 - 3b10 + 2b8 - b7 - 2b6) q^56 + (-2b11 + b10 + b9 - 6b8 + 3b7 + 2b6 - 3b5 - 2b4 + 2b3 - 5b2 + 3b1 + 3) q^58 + (3b11 - 4b9 + 3b8 - 2b7 + b5 + 3b4 + 3b2 - 2b1 - 4) q^59 + (-b11 + 2b10 + b9 - 5b6) * q^61 + (-b5 + 2b4 - 2b2 + 3b1 - 2) q^62 + (-5b5 + 2b4 - 4b3 - b1 - 4) q^64 + (b11 + 2b10 - 2b9 + b7 + 2b6) q^65 + (-b11 + b8 + 4b7 + 3b6 + 5b5 - b4 + 3b3 + b2 + 4b1 + 3) q^67 + (b11 + b10 + 3b9 + b8 - b7 - 3b6 + b4 - 3b3 + 2b2 - b1) * q^68 + (-b11 + 2b10 + 2b9 + b6) * q^70 + (-2b4 - b3 + 2b2 + 4) * q^71 + (-3b5 + 3b1 - 1) * q^73 + (-2b10 - 3b9 - b8 + 3b7 + 2b6) * q^74 + (-b10 - 2b9 - 4b8 + 4b7 + 2b6 + 2b3 - 5b2 + 4b1) q^76 + (-b11 + 2b10 + 2b9 - b8 + 2b7 + b5 - b4 + b2 + 2b1 + 2) * q^77 + (6b10 + 2b9 - 6b8 + b7 - 4b6) * q^79 + (-b5 + b4 + b3 - b2 - 3) * q^80 + (b4 + 2b3 + b2 + 3b1) * q^82 + (-3b11 + 2b10 - 5b8 + 3b7 + 4b6) q^83 + (-b11 + b10 - b9 + 2b8 + 2b7 + 2b6 + 4b5 - b4 + 2b3 + 3b2 + 2b1 + 1) q^85 + (-b11 + b10 + 3b9 - 6b8 + 4b7 + b6 - 2b5 - b4 + b3 - 5b2 + 4b1 + 4) * q^86 + (-2b11 - 2b10 + b9 - 3b8 + 2b6) * q^88 + (3b5 - 4b4 + b3 + b2 + 2) * q^89 + (b5 + 2b3 + 3b2 + b1) * q^91 + (-3b11 - 2b10 - 4b8 + 3b7 + 2b6) q^92 + (-b11 - 3b10 - b9 - 2b8 + b7 - b5 - b4 - 5b2 + b1 - 1) q^94 + (3b11 + 3b10 - 2b9 - 3b8 + 2b7 - b5 + 3b4 + 2b1 - 2) q^95 + (-b11 - 2b10 + b9 - b8 + 3b7 - 2b6) q^97 + (4b4 - 3b3 + 2b2 - b1 - 9) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 12 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 - 3 * q^4 - 6 * q^5 + 12 * q^8	$ 12 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 12 q^{8} + 6 q^{10} - 12 q^{11} - 6 q^{14} + 3 q^{16} + 18 q^{17} + 6 q^{19} - 6 q^{20} - 6 q^{22} - 15 q^{23} + 6 q^{25} + 30 q^{26} - 12 q^{28} - 12 q^{29} + 24 q^{35} + 6 q^{37} + 3 q^{38} - 6 q^{40} - 15 q^{41} + 6 q^{44} + 6 q^{46} - 21 q^{47} + 12 q^{49} - 3 q^{50} - 12 q^{52} + 18 q^{53} - 12 q^{55} + 6 q^{56} + 12 q^{58} - 24 q^{59} + 9 q^{61} - 24 q^{62} - 24 q^{64} + 6 q^{65} + 9 q^{67} + 9 q^{68} - 15 q^{70} + 54 q^{71} - 12 q^{73} + 12 q^{74} - 6 q^{76} + 12 q^{77} - 42 q^{80} - 12 q^{82} - 12 q^{83} + 21 q^{86} - 12 q^{88} + 18 q^{89} - 12 q^{91} - 6 q^{92} - 6 q^{94} - 12 q^{95} - 90 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 - 3 * q^4 - 6 * q^5 + 12 * q^8 + 6 * q^10 - 12 * q^11 - 6 * q^14 + 3 * q^16 + 18 * q^17 + 6 * q^19 - 6 * q^20 - 6 * q^22 - 15 * q^23 + 6 * q^25 + 30 * q^26 - 12 * q^28 - 12 * q^29 + 24 * q^35 + 6 * q^37 + 3 * q^38 - 6 * q^40 - 15 * q^41 + 6 * q^44 + 6 * q^46 - 21 * q^47 + 12 * q^49 - 3 * q^50 - 12 * q^52 + 18 * q^53 - 12 * q^55 + 6 * q^56 + 12 * q^58 - 24 * q^59 + 9 * q^61 - 24 * q^62 - 24 * q^64 + 6 * q^65 + 9 * q^67 + 9 * q^68 - 15 * q^70 + 54 * q^71 - 12 * q^73 + 12 * q^74 - 6 * q^76 + 12 * q^77 - 42 * q^80 - 12 * q^82 - 12 * q^83 + 21 * q^86 - 12 * q^88 + 18 * q^89 - 12 * q^91 - 6 * q^92 - 6 * q^94 - 12 * q^95 - 90 * q^98

Display 100 coefficients 10 coefficients

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^{2} - \nu + 2 $ v^2 - v + 2
$\beta_{2}$	$=$	$ \nu^{8} - 4\nu^{7} + 16\nu^{6} - 34\nu^{5} + 62\nu^{4} - 72\nu^{3} + 64\nu^{2} - 33\nu + 8 $ v^8 - 4v^7 + 16v^6 - 34v^5 + 62v^4 - 72v^3 + 64v^2 - 33*v + 8
$\beta_{3}$	$=$	$ \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
$\beta_{4}$	$=$	$ - 2 \nu^{10} + 10 \nu^{9} - 43 \nu^{8} + 112 \nu^{7} - 239 \nu^{6} + 367 \nu^{5} - 448 \nu^{4} + \cdots - 20 $ -2v^10 + 10v^9 - 43v^8 + 112v^7 - 239v^6 + 367v^5 - 448v^4 + 395v^3 - 256v^2 + 104v - 20
$\beta_{5}$	$=$	$ - 2 \nu^{10} + 10 \nu^{9} - 44 \nu^{8} + 116 \nu^{7} - 255 \nu^{6} + 401 \nu^{5} - 509 \nu^{4} + \cdots - 25 $ -2v^10 + 10v^9 - 44v^8 + 116v^7 - 255v^6 + 401v^5 - 509v^4 + 465v^3 - 314v^2 + 132v - 25
$\beta_{6}$	$=$	$ 11 \nu^{11} - 61 \nu^{10} + 270 \nu^{9} - 761 \nu^{8} + 1715 \nu^{7} - 2888 \nu^{6} + 3852 \nu^{5} + \cdots - 71 $ 11v^11 - 61v^10 + 270v^9 - 761v^8 + 1715v^7 - 2888v^6 + 3852v^5 - 3892v^4 + 2948v^3 - 1562v^2 + 500*v - 71
$\beta_{7}$	$=$	$ - 14 \nu^{11} + 78 \nu^{10} - 344 \nu^{9} + 970 \nu^{8} - 2177 \nu^{7} + 3659 \nu^{6} - 4853 \nu^{5} + \cdots + 85 $ -14v^11 + 78v^10 - 344v^9 + 970v^8 - 2177v^7 + 3659v^6 - 4853v^5 + 4884v^4 - 3671v^3 + 1934v^2 - 613*v + 85
$\beta_{8}$	$=$	$ - 20 \nu^{11} + 111 \nu^{10} - 490 \nu^{9} + 1379 \nu^{8} - 3097 \nu^{7} + 5198 \nu^{6} - 6900 \nu^{5} + \cdots + 124 $ -20v^11 + 111v^10 - 490v^9 + 1379v^8 - 3097v^7 + 5198v^6 - 6900v^5 + 6936v^4 - 5222v^3 + 2750v^2 - 876*v + 124
$\beta_{9}$	$=$	$ 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
$\beta_{10}$	$=$	$ 68 \nu^{11} - 374 \nu^{10} + 1649 \nu^{9} - 4616 \nu^{8} + 10342 \nu^{7} - 17277 \nu^{6} + 22863 \nu^{5} + \cdots - 412 $ 68v^11 - 374v^10 + 1649v^9 - 4616v^8 + 10342v^7 - 17277v^6 + 22863v^5 - 22873v^4 + 17165v^3 - 9002v^2 + 2871*v - 412
$\beta_{11}$	$=$	$ - 98 \nu^{11} + 540 \nu^{10} - 2381 \nu^{9} + 6671 \nu^{8} - 14949 \nu^{7} + 24987 \nu^{6} - 33072 \nu^{5} + \cdots + 595 $ -98v^11 + 540v^10 - 2381v^9 + 6671v^8 - 14949v^7 + 24987v^6 - 33072v^5 + 33097v^4 - 24840v^3 + 13028v^2 - 4153*v + 595

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

$\nu$	$=$	$ ( 2\beta_{11} + 2\beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 1 ) / 3 $ (2b11 + 2b10 + b9 - 2b8 - 2b6 - b5 + b4 - b3 + 1) / 3
$\nu^{2}$	$=$	$ ( 2\beta_{11} + 2\beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 3\beta _1 - 5 ) / 3 $ (2b11 + 2b10 + b9 - 2b8 - 2b6 - b5 + b4 - b3 + 3*b1 - 5) / 3
$\nu^{3}$	$=$	$ ( - 4 \beta_{11} - 5 \beta_{10} + \beta_{9} + 9 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + \cdots - 5 ) / 3 $ (-4b11 - 5b10 + b9 + 9b8 - 3b7 + 4b6 + 3b5 - 2b4 + 2b3 + 2b2 + 3b1 - 5) / 3
$\nu^{4}$	$=$	$ ( - 10 \beta_{11} - 12 \beta_{10} + \beta_{9} + 20 \beta_{8} - 6 \beta_{7} + 10 \beta_{6} + 10 \beta_{5} + \cdots + 16 ) / 3 $ (-10b11 - 12b10 + b9 + 20b8 - 6b7 + 10b6 + 10b5 - 8b4 + 5b3 + 7b2 - 12b1 + 16) / 3
$\nu^{5}$	$=$	$ ( 7 \beta_{11} + 14 \beta_{10} - 12 \beta_{9} - 25 \beta_{8} + 14 \beta_{7} - 6 \beta_{6} + 2 \beta_{5} + \cdots + 30 ) / 3 $ (7b11 + 14b10 - 12b9 - 25b8 + 14b7 - 6b6 + 2b5 - 4b4 - 3b3 + 2b2 - 23*b1 + 30) / 3
$\nu^{6}$	$=$	$ ( 47 \beta_{11} + 73 \beta_{10} - 38 \beta_{9} - 126 \beta_{8} + 57 \beta_{7} - 44 \beta_{6} - 48 \beta_{5} + \cdots - 56 ) / 3 $ (47b11 + 73b10 - 38b9 - 126b8 + 57b7 - 44b6 - 48b5 + 34b4 - 28b3 - 37b2 + 42*b1 - 56) / 3
$\nu^{7}$	$=$	$ ( 8 \beta_{11} - 6 \beta_{10} + 28 \beta_{9} + 2 \beta_{8} - 15 \beta_{7} - 14 \beta_{6} - 80 \beta_{5} + \cdots - 191 ) / 3 $ (8b11 - 6b10 + 28b9 + 2b8 - 15b7 - 14b6 - 80b5 + 67b4 - 28b3 - 65b2 + 150*b1 - 191) / 3
$\nu^{8}$	$=$	$ ( - 212 \beta_{11} - 394 \beta_{10} + 291 \beta_{9} + 644 \beta_{8} - 340 \beta_{7} + 174 \beta_{6} + \cdots + 129 ) / 3 $ (-212b11 - 394b10 + 291b9 + 644b8 - 340b7 + 174b6 + 143b5 - 91b4 + 99b3 + 113b2 - 86*b1 + 129) / 3
$\nu^{9}$	$=$	$ ( - 226 \beta_{11} - 359 \beta_{10} + 199 \beta_{9} + 624 \beta_{8} - 288 \beta_{7} + 214 \beta_{6} + \cdots + 1108 ) / 3 $ (-226b11 - 359b10 + 199b9 + 624b8 - 288b7 + 214b6 + 600b5 - 455b4 + 287b3 + 488b2 - 846*b1 + 1108) / 3
$\nu^{10}$	$=$	$ ( 842 \beta_{11} + 1734 \beta_{10} - 1457 \beta_{9} - 2695 \beta_{8} + 1539 \beta_{7} - 596 \beta_{6} + \cdots + 358 ) / 3 $ (842b11 + 1734b10 - 1457b9 - 2695b8 + 1539b7 - 596b6 - 23b5 - 44b4 - 115b3 - 14b2 - 324*b1 + 358) / 3
$\nu^{11}$	$=$	$ ( 1918 \beta_{11} + 3656 \beta_{10} - 2790 \beta_{9} - 5875 \beta_{8} + 3164 \beta_{7} - 1515 \beta_{6} + \cdots - 5451 ) / 3 $ (1918b11 + 3656b10 - 2790b9 - 5875b8 + 3164b7 - 1515b6 - 3154b5 + 2312b4 - 1632b3 - 2578b2 + 4090*b1 - 5451) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1 - \beta_{9}$

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} $

244.1

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

−1.20081

−

2.07986i

−1.88389

3.26300i

0.0465417

−

0.0806126i

0.289931

0.502175i

4.24555

−0.223551

244.2

−1.05831

−

1.83305i

−1.24005

2.14782i

−1.34155

2.32363i

−0.486166

−

0.842065i

1.01617

5.67911

244.3

−0.400763

−

0.694143i

0.678777

−

1.17568i

−1.37492

2.38143i

−1.18842

−

2.05840i

−2.69117

2.20407

244.4

−0.207733

−

0.359804i

0.913694

−

1.58256i

1.10759

−

1.91841i

0.659815

1.14283i

−1.59015

−0.920335

244.5

0.527162

0.913072i

0.444200

−

0.769376i

−0.872894

1.51190i

−1.22962

−

2.12977i

3.04531

−1.84063

244.6

0.840456

1.45571i

−0.412733

0.714874i

−0.564772

0.978214i

1.95446

3.38523i

1.97429

−1.89866

487.1