LMFDB - Newform orbit 819.2.j.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(235,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("819.235");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.j (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:	$6.53974792554$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.4868829729.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - x^{6} + 5x^{5} - 8x^{4} + 15x^{3} - 9x^{2} - 54x + 81 $ x^8 - 2x^7 - x^6 + 5x^5 - 8x^4 + 15x^3 - 9x^2 - 54x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{6} - \beta_{3} + \beta_1) q^{2} + ( - \beta_{6} + 2 \beta_{4} + \beta_{3}) q^{4} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots + 2) q^{8}+O(q^{10}) $ q + (-b6 - b3 + b1) * q^2 + (-b6 + 2b4 + b3) q^4 + (-b6 + b4 - b3 + b1 + 1) * q^5 + (-b7 + b5 + b3 + b2 + 1) * q^7 + (-b7 + b6 + b5 + b3 - b2 + 2) * q^8	$ q + ( - \beta_{6} - \beta_{3} + \beta_1) q^{2} + ( - \beta_{6} + 2 \beta_{4} + \beta_{3}) q^{4} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \cdots + 1) q^{5}+ \cdots + (2 \beta_{7} - 4 \beta_{5} - 8 \beta_{4} + \cdots - 10) q^{98}+O(q^{100}) $ q + (-b6 - b3 + b1) * q^2 + (-b6 + 2b4 + b3) q^4 + (-b6 + b4 - b3 + b1 + 1) * q^5 + (-b7 + b5 + b3 + b2 + 1) * q^7 + (-b7 + b6 + b5 + b3 - b2 + 2) * q^8 + (-b6 + b5 + 4b4 + b3 + b1) q^10 + (-b6 - b5 - b4 + b3 - b1) * q^11 - q^13 + (-2b6 - b4 + 3b1 - 3) * q^14 + (-b7 - b6 - b5 - 3b4 - 2b3 + 2b2 + 2b1 - 3) * q^16 + (2b7 + b6 - b3 - b2) q^17 + (2b5 - b4 + 2b3 - 2b1 - 1) q^19 + (-b7 + 2b6 + 2b5 + 3b3 - b2 + b1) q^20 + (-b7 + b6 + b5 - b2 - b1 + 6) * q^22 + (b7 + 2b6 - b5 - b4 + b3 - 2b2 - b1 - 1) * q^23 + (-b6 + 2b5 + b3 + 2b1) * q^25 + (b6 + b3 - b1) * q^26 + (b7 + 2b6 + 2b4 + 4b3 - b2 - 3b1 - 2) * q^28 + (b6 + b5 - b3 - 2b1 + 4) q^29 + (2b7 - b5 + b4 - b2 - b1) q^31 + (-3b6 - b5 + 5b4 + 3b3 - b1) q^32 + (-2b3 - 2b1 + 1) * q^34 + (-b7 - 2b6 + b3 + 3b1 - 2) * q^35 + (2b7 - 2b6 - b5 + 3b4 - 3b3 - 4b2 + 3b1 + 3) * q^37 + (-4b7 + b5 - 4b4 + 2b2 + b1) q^38 + (-3b5 - 5b4 - 3b3 + 3b1 - 5) * q^40 + (b7 + b6 + b5 + b3 + b2 - 3) * q^41 + (-b7 + b6 + b5 + 4b3 - b2 + 3b1 + 3) * q^43 + (-2b7 - 4b6 - 7b4 - 4b3 + 4b2 + 4b1 - 7) * q^44 + (4b6 - 3b5 - 9b4 - 4b3 - 3b1) q^46 + (b7 - 2b6 + 5b4 - 2b3 - 2b2 + 2b1 + 5) q^47 + (-2b7 + 3b6 + 2b5 - 2b4 + 2b3 + 2b2 - b1 - 4) * q^49 + (-b7 - b6 - b5 + b3 - b2 + 2b1 - 6) q^50 + (b6 - 2b4 - b3) q^52 + (3b6 - 3b5 - b4 - 3b3 - 3b1) * q^53 + (-b7 - b6 - b5 - b3 - b2 + 7) * q^55 + (-2b7 + 3b6 + 3b5 - 9b4 + b3 - b2 - b1 + 1) * q^56 + (-2b7 - 6b6 + b5 - 5b3 + 4b2 + 5b1) q^58 + (-2b7 - b5 - 6b4 + b2 - b1) * q^59 + (b7 - 2b6 - b5 - 4b4 - 3b3 - 2b2 + 3b1 - 4) q^61 + (-b7 + 3b6 + 3b5 - b2 - 3b1 + 7) q^62 + (-b7 + 5b6 + 5b5 + 6b3 - b2 + b1 + 4) q^64 + (b6 - b4 + b3 - b1 - 1) * q^65 + (-b6 + b5 + 3b4 + b3 + b1) q^67 + (-3b6 + 4b4 - 3b3 + 3b1 + 4) * q^68 + (b7 + 3b6 - 2b5 + b4 + 4b3 - 3b2 - b1 - 4) * q^70 + (b7 - b6 - b5 + b3 + b2 + 2b1 - 2) q^71 + (-2b7 - b5 - b4 + b2 - b1) q^73 + (-2b7 + 2b6 + 4b4 - 2b3 + b2) * q^74 + (2b7 - 3b6 - 3b5 - 2b3 + 2b2 + b1 - 12) q^76 + (b7 + b6 + 3b5 + 2b4 + 4b3 + 2b2 - 5b1) q^77 + (b7 - b6 - 2b5 - b4 - 3b3 - 2b2 + 3b1 - 1) * q^79 + (2b7 - 2b6 - 2b5 + 6b4 + 2b3 - b2 - 2b1) * q^80 + (b7 + 2b6 + b5 - b4 + 3b3 - 2b2 - 3b1 - 1) * q^82 + (2b6 + 2b5 - 3b3 - 5b1 - 2) * q^83 + (b7 + b6 + b5 - 2b3 + b2 - 3b1 + 1) * q^85 + (2b7 + b6 - 6b5 - 13b4 - 5b3 - 4b2 + 5b1 - 13) * q^86 + (-6b6 - 5b5 + 10b4 + 6b3 - 5b1) q^88 + (b7 - b6 + b5 + 14b4 - 2b2 + 14) * q^89 + (b7 - b5 - b3 - b2 - 1) * q^91 + (2b7 - 8b6 - 8b5 - 9b3 + 2b2 - b1 + 2) q^92 + (-2b7 + 4b5 + 5b4 + b2 + 4b1) * q^94 + (-4b7 - 2b6 + b5 - 5b4 + 2b3 + 2b2 + b1) q^95 + (-2b7 - 3b6 - 3b5 - 2b3 - 2b2 + b1 + 5) q^97 + (2b7 - 4b5 - 8b4 + b3 + b2 - 10) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 7 q^{4} + 3 q^{5} + 9 q^{7} + 12 q^{8}+O(q^{10}) $ 8 * q - q^2 - 7 * q^4 + 3 * q^5 + 9 * q^7 + 12 * q^8	$ 8 q - q^{2} - 7 q^{4} + 3 q^{5} + 9 q^{7} + 12 q^{8} - 14 q^{10} + 4 q^{11} - 8 q^{13} - 16 q^{14} - 9 q^{16} + 2 q^{17} - 6 q^{19} + 2 q^{20} + 40 q^{22} - 4 q^{23} + 3 q^{25} + q^{26} - 20 q^{28} + 26 q^{29} - 2 q^{31} - 18 q^{32} - 13 q^{35} + 5 q^{37} + 11 q^{38} - 17 q^{40} - 16 q^{41} + 32 q^{43} - 26 q^{44} + 29 q^{46} + 15 q^{47} - 21 q^{49} - 48 q^{50} + 7 q^{52} - 2 q^{53} + 48 q^{55} + 35 q^{56} - q^{58} + 20 q^{59} - 20 q^{61} + 44 q^{62} + 40 q^{64} - 3 q^{65} - 10 q^{67} + 13 q^{68} - 31 q^{70} - 4 q^{71} - 21 q^{74} - 86 q^{76} - 3 q^{77} - 6 q^{79} - 21 q^{80} - 6 q^{82} - 32 q^{83} + 4 q^{85} - 51 q^{86} - 39 q^{88} + 51 q^{89} - 9 q^{91} + 8 q^{92} - 19 q^{94} + 17 q^{95} + 26 q^{97} - 33 q^{98}+O(q^{100}) $ 8 * q - q^2 - 7 * q^4 + 3 * q^5 + 9 * q^7 + 12 * q^8 - 14 * q^10 + 4 * q^11 - 8 * q^13 - 16 * q^14 - 9 * q^16 + 2 * q^17 - 6 * q^19 + 2 * q^20 + 40 * q^22 - 4 * q^23 + 3 * q^25 + q^26 - 20 * q^28 + 26 * q^29 - 2 * q^31 - 18 * q^32 - 13 * q^35 + 5 * q^37 + 11 * q^38 - 17 * q^40 - 16 * q^41 + 32 * q^43 - 26 * q^44 + 29 * q^46 + 15 * q^47 - 21 * q^49 - 48 * q^50 + 7 * q^52 - 2 * q^53 + 48 * q^55 + 35 * q^56 - q^58 + 20 * q^59 - 20 * q^61 + 44 * q^62 + 40 * q^64 - 3 * q^65 - 10 * q^67 + 13 * q^68 - 31 * q^70 - 4 * q^71 - 21 * q^74 - 86 * q^76 - 3 * q^77 - 6 * q^79 - 21 * q^80 - 6 * q^82 - 32 * q^83 + 4 * q^85 - 51 * q^86 - 39 * q^88 + 51 * q^89 - 9 * q^91 + 8 * q^92 - 19 * q^94 + 17 * q^95 + 26 * q^97 - 33 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{7} - x^{6} + 5x^{5} - 8x^{4} + 15x^{3} - 9x^{2} - 54x + 81 $ x^8 - 2*x^7 - x^6 + 5*x^5 - 8*x^4 + 15*x^3 - 9*x^2 - 54*x + 81 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{7} + 4\nu^{5} + 3\nu^{4} - 7\nu^{3} + 26\nu^{2} + 3\nu - 18 ) / 45 $ (v^7 + 4v^5 + 3v^4 - 7v^3 + 26v^2 + 3*v - 18) / 45
$\beta_{3}$	$=$	$ ( -\nu^{7} + 2\nu^{6} + \nu^{5} - 5\nu^{4} + 8\nu^{3} - 15\nu^{2} + 9\nu + 54 ) / 27 $ (-v^7 + 2v^6 + v^5 - 5v^4 + 8v^3 - 15v^2 + 9*v + 54) / 27
$\beta_{4}$	$=$	$ ( 11\nu^{7} - 7\nu^{6} - 14\nu^{5} + 13\nu^{4} - 13\nu^{3} + 126\nu^{2} - 9\nu - 567 ) / 270 $ (11v^7 - 7v^6 - 14v^5 + 13v^4 - 13v^3 + 126v^2 - 9*v - 567) / 270
$\beta_{5}$	$=$	$ ( -11\nu^{7} - 11\nu^{6} + 32\nu^{5} - 13\nu^{4} + 49\nu^{3} - 126\nu^{2} - 279\nu + 621 ) / 270 $ (-11v^7 - 11v^6 + 32v^5 - 13v^4 + 49v^3 - 126v^2 - 279*v + 621) / 270
$\beta_{6}$	$=$	$ ( 5\nu^{7} - \nu^{6} - 14\nu^{5} + 25\nu^{4} - 13\nu^{3} + 30\nu^{2} + 99\nu - 297 ) / 90 $ (5v^7 - v^6 - 14v^5 + 25v^4 - 13v^3 + 30v^2 + 99v - 297) / 90
$\beta_{7}$	$=$	$ ( -16\nu^{7} + 8\nu^{6} + 28\nu^{5} - 38\nu^{4} + 71\nu^{3} - 66\nu^{2} - 90\nu + 729 ) / 135 $ (-16v^7 + 8v^6 + 28v^5 - 38v^4 + 71v^3 - 66v^2 - 90*v + 729) / 135

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 1 $ b7 - b5 + b4 - b3 + 1
$\nu^{3}$	$=$	$ \beta_{7} + 2\beta_{6} + 2\beta_{5} + 4\beta_{4} + 2\beta_{3} + 1 $ b7 + 2b6 + 2b5 + 4b4 + 2b3 + 1
$\nu^{4}$	$=$	$ \beta_{7} + 5\beta_{6} + 2\beta_{5} - 3\beta_{4} + 2\beta_{2} - 3\beta _1 + 1 $ b7 + 5b6 + 2b5 - 3b4 + 2b2 - 3*b1 + 1
$\nu^{5}$	$=$	$ -2\beta_{7} + 5\beta_{5} - \beta_{4} + 4\beta_{3} + 7\beta_{2} + 2\beta _1 - 8 $ -2b7 + 5b5 - b4 + 4b3 + 7b2 + 2*b1 - 8
$\nu^{6}$	$=$	$ 4\beta_{6} - 6\beta_{5} - 8\beta_{4} + 8\beta_{3} + 7\beta_{2} - 14\beta _1 - 3 $ 4b6 - 6b5 - 8b4 + 8b3 + 7b2 - 14b1 - 3
$\nu^{7}$	$=$	$ -14\beta_{7} - \beta_{6} + 14\beta_{5} + 15\beta_{4} + 24\beta_{3} + 11\beta_{2} - 2\beta _1 + 28 $ -14b7 - b6 + 14b5 + 15b4 + 24b3 + 11b2 - 2b1 + 28

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$.

$n$	$92$	$379$	$703$
$\chi(n)$	$1$	$1$	$-1 - \beta_{4}$

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} $

235.1

−0.272725	−	1.71044i
1.72192	+	0.187090i
1.25184	+	1.19703i
−1.70103	+	0.326320i
−0.272725	+	1.71044i
1.72192	−	0.187090i
1.25184	−	1.19703i
−1.70103	−	0.326320i

−1.34493

−

2.32948i

−2.61765

4.53390i

−0.844926

−

1.46345i

2.34493

−

1.22528i

8.70248

−2.27273

3.93648i

235.2

−0.698934

−

1.21059i

0.0229829

−

0.0398076i

−0.198934

−

0.344564i

1.69893

2.02821i

−2.85999

−0.278083

0.481654i

235.3

0.410741

0.711425i

0.662583

−

1.14763i

0.910741

1.57745i

0.589259

−

2.57930i

2.73157

−0.748158

1.29585i

235.4

1.13312

1.96262i

−1.56792

2.71571i

1.63312

2.82864i

−0.133118

2.64240i

−2.57406

−3.70103

6.41038i

352.1