LMFDB - Newform orbit 729.2.c.d

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:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1 $ x^12 + 18x^10 + 105x^8 + 266x^6 + 306x^4 + 132*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
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_{9} - \beta_{6} + \cdots + \beta_{2}) q^{2}+ \cdots + (\beta_{10} - 3 \beta_{9} - \beta_{5} + \cdots - 1) q^{8}+O(q^{10}) $ q + (b9 - b6 + b5 - b3 + b2) * q^2 + (-b11 + b10 - b7 - b3 - 1) * q^4 + (b10 - b7 + b6 - b2 + b1 - 1) * q^5 + (-b9 + b8 - b7 + b6 - b5 + b3 - b2) * q^7 + (b10 - 3b9 - b5 - 3b1 - 1) * q^8	$ q + (\beta_{9} - \beta_{6} + \cdots + \beta_{2}) q^{2}+ \cdots + ( - 4 \beta_{10} - 4 \beta_{9} + \cdots - 1) q^{98}+O(q^{100}) $ q + (b9 - b6 + b5 - b3 + b2) * q^2 + (-b11 + b10 - b7 - b3 - 1) * q^4 + (b10 - b7 + b6 - b2 + b1 - 1) * q^5 + (-b9 + b8 - b7 + b6 - b5 + b3 - b2) * q^7 + (b10 - 3b9 - b5 - 3b1 - 1) * q^8 + (-b9 - b5 + b4 - 3b1 + 2) q^10 + (b11 - b9 + b7 + b6 - b4 + b2 - 1) * q^11 + (b11 - b8 - b3 + b2 - b1 - 1) * q^13 + (b11 + 3b8 + b3 + 2b2 - 2b1 + 3) q^14 + (b11 - b9 + b8 + 2b7 + b6 - b4 - 3b2 - 1) * q^16 + (b10 + b9 - b5 - b4 + 1) * q^17 + (-b10 + b9 - b4 + b1 + 2) * q^19 + (b11 + 3b9 + 3b8 + b7 - 3b6 + b5 - b4 - b3 - b2 - 1) q^20 + (b11 - 2b8 + 5b6 + 2b3 - 3b2 + 3b1 - 2) q^22 + (b11 - 3b8 + b3 + 2b2 - 2b1 - 3) q^23 + (b9 + 3b8 - b7 - b6 + 2b5 - 2b3) q^25 + (b10 + 3b9 - b5 - 4) q^26 + (5b9 + 3b5 + b1 - 1) * q^28 + (2b11 - 2b9 - 6b8 + b7 + 2b6 - 2b5 - 2b4 + 2b3 - b2 - 2) q^29 + (-b10 - 4b8 + b7 + 2b6 + 2b3 - 2b2 + 2b1 - 3) q^31 + (b11 - 3b10 - 3b8 + 3b7 + b6 + 2b3 - 5b2 + 5b1) * q^32 + (-b11 - 2b9 + b7 + 2b6 + b4 - 3b2 + 1) q^34 + (-b10 - b9 + 2b5 - b4 + b1 - 6) q^35 + (2b10 + 4b9 - b4 + b1 - 1) * q^37 + (-b11 - b9 - 2b7 + b6 + 2b5 + b4 - 2b3 + 3b2 + 1) * q^38 + (-b11 - 2b10 - 2b8 + 2b7 + 6b6 + 2b3 - 6b2 + 6b1) q^40 + (-2b11 + b10 - b7 - b6 - b3 + 3b2 - 3b1 - 1) q^41 + (-2b11 + 4b9 + 4b8 - 2b7 - 4b6 + 2b5 + 2b4 - 2b3 + 2b2 + 2) q^43 + (-b10 + b5 + 3b4 + 7) q^44 + (2b10 + b9 - b5 - 3b1 - 1) * q^46 + (3b9 + 3b8 - 2b7 - 3b6 - b5 + b3 + 3b2) q^47 + (-b11 - 2b10 - 2b8 + 2b7 - 2b6 - b3 - 2b2 + 2b1) * q^49 + (-b11 + b10 - 3b8 - b7 + 2b6 - b2 + b1 - 4) * q^50 + (-b11 - 4b9 - b8 + b7 + 4b6 - 3b5 + b4 + 3b3 - 7b2 + 1) q^52 + (b10 - 2b4 + 4b1) * q^53 + (-b10 - 2b9 - 4b5 + 5) * q^55 + (-3b11 - b9 - b7 + b6 + 3b4 - b2 + 3) * q^56 + (2b11 - 2b10 + b8 + 2b7 + 3b6 - b3 + 3) * q^58 + (b10 + 3b8 - b7 + b6 - 3b3 - b2 + b1 + 2) * q^59 + (b11 + 3b9 + 4b8 + b7 - 3b6 + 2b5 - b4 - 2b3 - 1) q^61 + (-3b10 - 3b9 - b5 + 2b4 - b1 + 8) q^62 + (-4b10 + 2b9 + 3b5 - b4 + 6b1 + 1) * q^64 + (-2b11 - 2b9 + 3b8 - b7 + 2b6 - 2b5 + 2b4 + 2b3 - 3b2 + 2) * q^65 + (2b10 - b8 - 2b7 - 5b6 - b3 + 3b2 - 3b1 - 3) q^67 + (-2b10 + 6b8 + 2b7 - b6 - 2b3 - 2b2 + 2b1 + 8) * q^68 + (b11 - 9b9 - 4b8 - 2b7 + 9b6 - 4b5 - b4 + 4b3 - 3b2 - 1) q^70 + (4b9 + 2b5 - 2b4 + 2b1 - 2) * q^71 + (3b9 + 3b5 + 3b4 + 3b1 + 2) * q^73 + (-4b11 - 4b9 - 3b8 + b7 + 4b6 - b5 + 4b4 + b3 - 9b2 + 4) * q^74 + (-b11 + 3b10 + 3b8 - 3b7 - 4b6 - 5b3 + 10b2 - 10b1) q^76 + (-3b11 + 6b8 - 7b6 - 4b3 + b2 - b1 + 6) * q^77 + (2b11 + 3b9 + b8 + 3b7 - 3b6 - b5 - 2b4 + b3 + 5b2 - 2) * q^79 + (-6b10 + 2b9 + 3b5 + 4b4 + 7) * q^80 + (4b10 - 9b9 - 4b5 - b4 - 9b1 - 4) * q^82 + (-2b11 + b9 + 3b8 - b7 - b6 + b5 + 2b4 - b3 - 3b2 + 2) * q^83 + (2b11 + 2b10 + 3b8 - 2b7 - b6 - 3b2 + 3b1 + 1) * q^85 + (-4b11 + 4b10 - 4b7 - 6b6 - 2b3 + 4b2 - 4b1 - 4) q^86 + (2b11 + 6b9 - 2b8 - b7 - 6b6 + 4b5 - 2b4 - 4b3 + 6b2 - 2) * q^88 + (4b10 + 2b9 + b5 - b1 - 1) * q^89 + (-6b9 + b5 - b4 - 2b1 + 2) * q^91 + (b11 - b9 - 6b8 + 5b7 + b6 - 4b5 - b4 + 4b3 - 5b2 - 1) q^92 + (-3b11 + 5b10 + 4b8 - 5b7 + b6 + 2b3 + 3b2 - 3b1 - 1) q^94 + (2b11 + b10 - 6b8 - b7 - 4b6 + 2b2 - 2b1 - 7) q^95 + (-2b11 + b9 - 2b8 - 2b7 - b6 - b5 + 2b4 + b3 - b2 + 2) * q^97 + (-4b10 - 4b9 - 2b5 - 2b4 + 6b1 - 1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} - 9 q^{4} - 3 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q + 3 * q^2 - 9 * q^4 - 3 * q^5 - 6 * q^7 - 12 * q^8	$ 12 q + 3 q^{2} - 9 q^{4} - 3 q^{5} - 6 q^{7} - 12 q^{8} + 12 q^{10} - 6 q^{11} - 6 q^{13} + 24 q^{14} - 15 q^{16} + 18 q^{17} + 24 q^{19} - 21 q^{20} - 3 q^{22} - 12 q^{23} - 9 q^{25} - 48 q^{26} + 6 q^{28} + 21 q^{29} - 15 q^{31} - 60 q^{35} + 6 q^{37} + 15 q^{38} - 3 q^{40} - 12 q^{41} - 6 q^{43} + 66 q^{44} - 6 q^{46} - 15 q^{47} - 12 q^{49} - 24 q^{50} - 3 q^{52} + 18 q^{53} + 30 q^{55} + 12 q^{56} + 15 q^{58} + 6 q^{59} - 24 q^{61} + 60 q^{62} + 12 q^{64} - 15 q^{65} - 15 q^{67} + 36 q^{68} + 15 q^{70} + 24 q^{73} + 24 q^{74} - 9 q^{76} + 15 q^{77} - 24 q^{79} + 42 q^{80} - 42 q^{82} - 6 q^{83} + 18 q^{85} - 30 q^{86} + 21 q^{88} + 18 q^{89} + 36 q^{91} + 6 q^{92} + 6 q^{94} - 33 q^{95} + 21 q^{97} - 36 q^{98}+O(q^{100}) $ 12 * q + 3 * q^2 - 9 * q^4 - 3 * q^5 - 6 * q^7 - 12 * q^8 + 12 * q^10 - 6 * q^11 - 6 * q^13 + 24 * q^14 - 15 * q^16 + 18 * q^17 + 24 * q^19 - 21 * q^20 - 3 * q^22 - 12 * q^23 - 9 * q^25 - 48 * q^26 + 6 * q^28 + 21 * q^29 - 15 * q^31 - 60 * q^35 + 6 * q^37 + 15 * q^38 - 3 * q^40 - 12 * q^41 - 6 * q^43 + 66 * q^44 - 6 * q^46 - 15 * q^47 - 12 * q^49 - 24 * q^50 - 3 * q^52 + 18 * q^53 + 30 * q^55 + 12 * q^56 + 15 * q^58 + 6 * q^59 - 24 * q^61 + 60 * q^62 + 12 * q^64 - 15 * q^65 - 15 * q^67 + 36 * q^68 + 15 * q^70 + 24 * q^73 + 24 * q^74 - 9 * q^76 + 15 * q^77 - 24 * q^79 + 42 * q^80 - 42 * q^82 - 6 * q^83 + 18 * q^85 - 30 * q^86 + 21 * q^88 + 18 * q^89 + 36 * q^91 + 6 * q^92 + 6 * q^94 - 33 * q^95 + 21 * q^97 - 36 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1 $ x^12 + 18*x^10 + 105*x^8 + 266*x^6 + 306*x^4 + 132*x^2 + 1 :

$\beta_{1}$	$=$	$ ( \nu^{8} + 15\nu^{6} + 58\nu^{4} + 63\nu^{2} + 8 ) / 4 $ (v^8 + 15v^6 + 58v^4 + 63*v^2 + 8) / 4
$\beta_{2}$	$=$	$ ( \nu^{11} + 16\nu^{9} + \nu^{8} + 70\nu^{7} + 15\nu^{6} + 76\nu^{5} + 58\nu^{4} - 95\nu^{3} + 63\nu^{2} - 149\nu + 8 ) / 8 $ (v^11 + 16v^9 + v^8 + 70v^7 + 15v^6 + 76v^5 + 58v^4 - 95v^3 + 63v^2 - 149v + 8) / 8
$\beta_{3}$	$=$	$ ( \nu^{11} - 2 \nu^{10} + 19 \nu^{9} - 34 \nu^{8} + 119 \nu^{7} - 175 \nu^{6} + 310 \nu^{5} - 343 \nu^{4} + \cdots + 3 ) / 8 $ (v^11 - 2v^10 + 19v^9 - 34v^8 + 119v^7 - 175v^6 + 310v^5 - 343v^4 + 322v^3 - 222v^2 + 95v + 3) / 8
$\beta_{4}$	$=$	$ ( 2\nu^{10} + 33\nu^{8} + 160\nu^{6} + 285\nu^{4} + 163\nu^{2} + 1 ) / 4 $ (2v^10 + 33v^8 + 160v^6 + 285v^4 + 163*v^2 + 1) / 4
$\beta_{5}$	$=$	$ ( -2\nu^{10} - 34\nu^{8} - 175\nu^{6} - 343\nu^{4} - 222\nu^{2} + 3 ) / 4 $ (-2v^10 - 34v^8 - 175v^6 - 343v^4 - 222*v^2 + 3) / 4
$\beta_{6}$	$=$	$ ( - \nu^{11} - 5 \nu^{10} - 16 \nu^{9} - 83 \nu^{8} - 73 \nu^{7} - 409 \nu^{6} - 121 \nu^{5} - 760 \nu^{4} + \cdots - 5 ) / 8 $ (-v^11 - 5v^10 - 16v^9 - 83v^8 - 73v^7 - 409v^6 - 121v^5 - 760v^4 - 79v^3 - 474v^2 - 28v - 5) / 8
$\beta_{7}$	$=$	$ ( - 6 \nu^{10} + 3 \nu^{9} - 99 \nu^{8} + 49 \nu^{7} - 482 \nu^{6} + 234 \nu^{5} - 881 \nu^{4} + \cdots - 17 ) / 8 $ (-6v^10 + 3v^9 - 99v^8 + 49v^7 - 482v^6 + 234v^5 - 881v^4 + 417v^3 - 549v^2 + 256v - 17) / 8
$\beta_{8}$	$=$	$ ( 3\nu^{11} + 50\nu^{9} + 249\nu^{7} + 477\nu^{5} + 335\nu^{3} + 42\nu - 2 ) / 4 $ (3v^11 + 50v^9 + 249v^7 + 477v^5 + 335v^3 + 42v - 2) / 4
$\beta_{9}$	$=$	$ ( -5\nu^{10} - 83\nu^{8} - 409\nu^{6} - 760\nu^{4} - 474\nu^{2} - 5 ) / 4 $ (-5v^10 - 83v^8 - 409v^6 - 760v^4 - 474*v^2 - 5) / 4
$\beta_{10}$	$=$	$ ( -6\nu^{10} - 99\nu^{8} - 482\nu^{6} - 881\nu^{4} - 549\nu^{2} - 13 ) / 4 $ (-6v^10 - 99v^8 - 482v^6 - 881v^4 - 549*v^2 - 13) / 4
$\beta_{11}$	$=$	$ ( 16 \nu^{11} + 2 \nu^{10} + 265 \nu^{9} + 33 \nu^{8} + 1301 \nu^{7} + 160 \nu^{6} + 2416 \nu^{5} + \cdots + 5 ) / 8 $ (16v^11 + 2v^10 + 265v^9 + 33v^8 + 1301v^7 + 160v^6 + 2416v^5 + 285v^4 + 1555v^3 + 163v^2 + 86*v + 5) / 8

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

$\nu$	$=$	$ ( -\beta_{10} + 2\beta_{7} + \beta_{5} - 2\beta_{3} + 2\beta_{2} - \beta _1 + 1 ) / 3 $ (-b10 + 2b7 + b5 - 2b3 + 2*b2 - b1 + 1) / 3
$\nu^{2}$	$=$	$ \beta_{5} + \beta_{4} + \beta _1 - 3 $ b5 + b4 + b1 - 3
$\nu^{3}$	$=$	$ ( 6 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} - 18 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} - 7 \beta_{5} + \cdots - 16 ) / 3 $ (6b11 + 4b10 + 4b9 - 18b8 - 8b7 - 8b6 - 7b5 - 3b4 + 14b3 - 10b2 + 5*b1 - 16) / 3
$\nu^{4}$	$=$	$ -3\beta_{10} + 4\beta_{9} - 10\beta_{5} - 9\beta_{4} - 8\beta _1 + 21 $ -3b10 + 4b9 - 10b5 - 9b4 - 8*b1 + 21
$\nu^{5}$	$=$	$ ( - 76 \beta_{11} - 27 \beta_{10} - 38 \beta_{9} + 222 \beta_{8} + 54 \beta_{7} + 76 \beta_{6} + \cdots + 176 ) / 3 $ (-76b11 - 27b10 - 38b9 + 222b8 + 54b7 + 76b6 + 61b5 + 38b4 - 122b3 + 82b2 - 41*b1 + 176) / 3
$\nu^{6}$	$=$	$ 37\beta_{10} - 52\beta_{9} + 100\beta_{5} + 81\beta_{4} + 74\beta _1 - 188 $ 37b10 - 52b9 + 100b5 + 81b4 + 74*b1 - 188
$\nu^{7}$	$=$	$ ( 792 \beta_{11} + 232 \beta_{10} + 342 \beta_{9} - 2286 \beta_{8} - 464 \beta_{7} - 684 \beta_{6} + \cdots - 1771 ) / 3 $ (792b11 + 232b10 + 342b9 - 2286b8 - 464b7 - 684b6 - 568b5 - 396b4 + 1136b3 - 776b2 + 388*b1 - 1771) / 3
$\nu^{8}$	$=$	$ -381\beta_{10} + 548\beta_{9} - 983\beta_{5} - 756\beta_{4} - 705\beta _1 + 1783 $ -381b10 + 548b9 - 983b5 - 756b4 - 705*b1 + 1783
$\nu^{9}$	$=$	$ ( - 7842 \beta_{11} - 2158 \beta_{10} - 3178 \beta_{9} + 22524 \beta_{8} + 4316 \beta_{7} + 6356 \beta_{6} + \cdots + 17341 ) / 3 $ (-7842b11 - 2158b10 - 3178b9 + 22524b8 + 4316b7 + 6356b6 + 5407b5 + 3921b4 - 10814b3 + 7498b2 - 3749*b1 + 17341) / 3
$\nu^{10}$	$=$	$ 3754\beta_{10} - 5452\beta_{9} + 9563\beta_{5} + 7197\beta_{4} + 6771\beta _1 - 17128 $ 3754b10 - 5452b9 + 9563b5 + 7197b4 + 6771*b1 - 17128
$\nu^{11}$	$=$	$ ( 76378 \beta_{11} + 20571 \beta_{10} + 30176 \beta_{9} - 218946 \beta_{8} - 41142 \beta_{7} + \cdots - 168233 ) / 3 $ (76378b11 + 20571b10 + 30176b9 - 218946b8 - 41142b7 - 60352b6 - 51904b5 - 38189b4 + 103808b3 - 72508b2 + 36254*b1 - 168233) / 3

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

Character values

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

$n$	$2$
$\chi(n)$	$-1 - \beta_{8}$

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

	−	1.22778i
	−	3.10658i
		1.37340i
		1.91182i
		1.13697i
	−	0.0878222i
		1.22778i
		3.10658i
	−	1.37340i
	−	1.91182i
	−	1.13697i
		0.0878222i

−1.22889

−

2.12851i

−2.02036

3.49937i

−1.54013

2.66759i

1.32933

2.30247i

5.01568

7.57064

244.2

−0.388866

−

0.673536i

0.697566

−

1.20822i

−1.18817

2.05798i

1.25069

2.16626i

−2.64050

1.84816

244.3

−0.0864880

−

0.149802i

0.985040

−

1.70614i

1.86828

−

3.23596i

−1.51575

−

2.62535i

−0.686728

−0.646335

244.4

0.789202

1.36694i

−0.245680

0.425530i

−0.839254

1.45363i

−1.38964

−

2.40693i

2.38124

−2.64936

244.5

1.06251

1.84033i

−1.25787

2.17870i

1.03547

−

1.79349i

−2.42434

−

4.19907i

−1.09598

4.40081

244.6

1.35253

2.34265i

−2.65869

4.60498i

−0.836192

1.44833i

−0.250296

−

0.433525i

−8.97372

−4.52391

487.1