LMFDB - Newform orbit 252.2.bf.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,2,Mod(19,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 5]))

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

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

chi := DirichletCharacter("252.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	252.bf (of order $6$, 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:	$2.01223013094$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.562828176.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 $ x^8 - 2x^7 + x^6 + 2x^5 - 6x^4 + 4x^3 + 4x^2 - 16x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{5} - \beta_1) q^{2} + (\beta_{7} - \beta_{6} - \beta_{2}) q^{4} + (\beta_{6} - \beta_{5} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{8}+O(q^{10}) $ q + (-b5 - b1) * q^2 + (b7 - b6 - b2) * q^4 + (b6 - b5 - b4 - b2) * q^5 + (-b7 - b5 + b4 - b3 + b2 - b1 + 1) * q^7 + (-b4 - 2b3 + b2) q^8	$ q + ( - \beta_{5} - \beta_1) q^{2} + (\beta_{7} - \beta_{6} - \beta_{2}) q^{4} + (\beta_{6} - \beta_{5} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + 10) q^{98}+O(q^{100}) $ q + (-b5 - b1) * q^2 + (b7 - b6 - b2) * q^4 + (b6 - b5 - b4 - b2) * q^5 + (-b7 - b5 + b4 - b3 + b2 - b1 + 1) * q^7 + (-b4 - 2b3 + b2) q^8 + (b7 + b6 - 2b4 + 2b3 - b2 - 2) * q^10 + (-b7 - b6 - b1) * q^11 + (-b7 - b5 - b4 - 2b3 + b2 - b1 + 1) q^13 + (b7 - b6 - b5 + b4 - 2b2 + 2) q^14 + (-b7 - b6 + b4 + 2b1 - 2) q^16 + (2b5 - 2b4) * q^17 + (-2b7 + b6 + b5 - b4 + b3 + b2 - 1) q^19 + (2b5 - b4 + 2b3 + b2 - 4) * q^20 + (2b7 + b4 - 2b3 - b2) * q^22 + (2b4 + 2b2 - 2b1) q^23 + (b7 + 3b6 - 2b5 - 2b4 + b3 - 2b2 - 3b1) q^25 + (b7 - b6 - b5 + b4 - 2b2 + b1 - 2) q^26 + (b7 + b6 - 2b5 - 3b4 - 2b3 - 2b1 + 2) * q^28 + (b7 - 3b5 + 3b4 + b2 - 3b1 + 2) q^29 + (-2b6 + 4b5 + 4b4 + 3b3 + 2b1) q^31 + (-b7 + 3b6 + 2b5 - 2b4 - 2b3 - b2 + 2b1 + 2) q^32 + (2b4 + 2b2 - 4) * q^34 + (-b7 + 2b6 - b5 - b4 - 2b3 - b2 + 3b1 + 2) q^35 + (-2b7 + 3b6 + 3b5 - b4 - b3 + b2 + 2b1 + 1) * q^37 + (b7 + b6 + b5 + 2b4 + 2b3 - b2 - 2) * q^38 + (-b7 - b6 + 4b5 + 3b4 + 2b2 + 2b1 - 2) * q^40 + (-4b7 - 4b3 + 4b2 + 2) q^41 + (b7 - 2b6 - 3b5 - b4 + 2b3 - b2 - b1 - 1) q^43 + (-b7 - b6 - b4 + 2b2 + 2b1 + 2) * q^44 + (-4b6 + 4b4 + 4) * q^46 + (2b6 + 2b5 + 2b3 + 2b1 - 2) * q^47 + (b7 - 4b6 + b5 + 5b4 - b3 - b2 + b1 + 2) * q^49 + (4b7 - 2b6 - b4 + 6b3 - b2 - b1 - 4) q^50 + (2b6 + 2b5 - 4b4 - 2b3 + 2b1 + 2) q^52 + (b7 - 3b6 + 2b5 + 2b4 + 2b3 - 2b2 + 3b1) * q^53 + (b7 + b5 - b4 + b2 + b1 - 2) * q^55 + (b7 - 3b6 - 2b5 + 2b3 - b2 - 6) q^56 + (b7 - 5b6 - 2b5 + b4 - 2b2 - 2b1 + 6) * q^58 + (-b7 - 3b6 + 2b3 + 2b2 + 3b1) * q^59 + (-4b4 + 4b1) * q^61 + (-2b7 + 2b6 + 2b4 - 4b3 + 4b2 - 3b1 + 8) * q^62 + (4b6 - 2b5 - b4 + 6b3 + b2 - 4) q^64 + (2b6 + 2b5 + 2b3 + 2b1 - 2) * q^65 + (-b7 + 3b6 + 2b5 - 2b4 - 3b3 + 3b1 + 6) q^67 + (-2b7 - 2b6 + 4b5 + 2b4 + 4b1 + 4) q^68 + (-b7 + 5b6 - 2b5 - 5b4 + 4b3 - 2b2 - 2) q^70 + (4b7 + 2b6 - 2b4 + 4b3 - 4b2 - 2b1 - 2) * q^71 + (b7 - 3b6 - 4b5 + 4b4 + b3 - 3b1 - 2) * q^73 + (-b7 + 3b6 - b5 + 2b4 + 6b3 + b2 - 2) q^74 + (2b5 + 2b3 + 2b2 + 4) q^76 + (4b7 + b6 - b5 + b4 - 2b3 - 3b2 - 2b1) * q^77 + (6b6 - 6b5 - 2b4 - b3 - 4b1 - 1) * q^79 + (-3b7 + b6 + 2b5 + 4b4 - 2b3 + 3b2 + 2b1 + 6) * q^80 + (-2b5 + 4b4 - 4b2 + 2b1) * q^82 + (b7 - 4b6 + 3b5 + b4 + b2 - b1) * q^83 + (2b7 - 4b6 + 2b5 + 2b4 + 2b2 - 2b1 - 4) * q^85 + (b7 - b6 + b5 - 3b4 - 4b3 - 2b2 - b1 - 2) q^86 + (-3b7 + b6 - 2b5 - 2b3 - b2 - 2b1 - 2) * q^88 + (-2b6 + 2b5 - 2b4 - 2b2 + 4b1) q^89 + (2b7 - 3b6 - 3b5 + 3b4 - b3 - 3b2 - 3) q^91 + (-4b5 - 8b3 - 4b1 + 8) q^92 + (-2b7 + 2b6 + 2b5 + 4b3 + 2b2) q^94 + (2b7 - 2b6 - 2b5 + 2b4 - 2b3 - 2b1 + 4) * q^95 + (-3b7 + b5 + b4 + 4b3 + 3b2 + b1 - 2) q^97 + (-b7 + b6 - 2b5 - b4 - 8b3 + 2b2 - b1 + 10) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - q^{4} + 2 q^{7} - 4 q^{8}+O(q^{10}) $ 8 * q - q^2 - q^4 + 2 * q^7 - 4 * q^8	$ 8 q - q^{2} - q^{4} + 2 q^{7} - 4 q^{8} - 5 q^{10} - 6 q^{11} + 12 q^{14} - 17 q^{16} - 6 q^{19} - 22 q^{20} - 6 q^{22} + 2 q^{25} - 18 q^{26} + 13 q^{28} + 16 q^{29} + 6 q^{31} + 9 q^{32} - 28 q^{34} + 12 q^{35} + 6 q^{37} - 10 q^{38} - 17 q^{40} + 23 q^{44} + 24 q^{46} - 4 q^{47} + 4 q^{49} - 2 q^{50} + 16 q^{52} + 4 q^{53} - 8 q^{55} - 41 q^{56} + 37 q^{58} + 14 q^{59} + 12 q^{61} + 48 q^{62} + 2 q^{64} - 4 q^{65} + 42 q^{67} + 26 q^{68} + 3 q^{70} - 18 q^{73} + 10 q^{74} + 44 q^{76} - 8 q^{77} - 6 q^{79} + 39 q^{80} - 10 q^{82} - 4 q^{83} - 32 q^{85} - 36 q^{86} - 37 q^{88} - 34 q^{91} + 28 q^{92} + 16 q^{94} + 24 q^{95} + 53 q^{98}+O(q^{100}) $ 8 * q - q^2 - q^4 + 2 * q^7 - 4 * q^8 - 5 * q^10 - 6 * q^11 + 12 * q^14 - 17 * q^16 - 6 * q^19 - 22 * q^20 - 6 * q^22 + 2 * q^25 - 18 * q^26 + 13 * q^28 + 16 * q^29 + 6 * q^31 + 9 * q^32 - 28 * q^34 + 12 * q^35 + 6 * q^37 - 10 * q^38 - 17 * q^40 + 23 * q^44 + 24 * q^46 - 4 * q^47 + 4 * q^49 - 2 * q^50 + 16 * q^52 + 4 * q^53 - 8 * q^55 - 41 * q^56 + 37 * q^58 + 14 * q^59 + 12 * q^61 + 48 * q^62 + 2 * q^64 - 4 * q^65 + 42 * q^67 + 26 * q^68 + 3 * q^70 - 18 * q^73 + 10 * q^74 + 44 * q^76 - 8 * q^77 - 6 * q^79 + 39 * q^80 - 10 * q^82 - 4 * q^83 - 32 * q^85 - 36 * q^86 - 37 * q^88 - 34 * q^91 + 28 * q^92 + 16 * q^94 + 24 * q^95 + 53 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 $ x^8 - 2*x^7 + x^6 + 2*x^5 - 6*x^4 + 4*x^3 + 4*x^2 - 16*x + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{7} + \nu^{5} + 4\nu^{4} + 2\nu^{3} + 8\nu^{2} + 4\nu - 8 ) / 16 $ (v^7 + v^5 + 4v^4 + 2v^3 + 8v^2 + 4v - 8) / 16
$\beta_{3}$	$=$	$ ( \nu^{7} + \nu^{5} + 4\nu^{4} - 6\nu^{3} + 4\nu - 8 ) / 16 $ (v^7 + v^5 + 4v^4 - 6v^3 + 4*v - 8) / 16
$\beta_{4}$	$=$	$ ( -\nu^{7} - \nu^{5} - 4\nu^{4} - 2\nu^{3} + 8\nu^{2} - 4\nu + 8 ) / 16 $ (-v^7 - v^5 - 4v^4 - 2v^3 + 8v^2 - 4v + 8) / 16
$\beta_{5}$	$=$	$ ( -\nu^{7} - \nu^{5} + 2\nu^{3} - 4\nu + 8 ) / 8 $ (-v^7 - v^5 + 2v^3 - 4v + 8) / 8
$\beta_{6}$	$=$	$ ( -3\nu^{7} + 4\nu^{6} - 3\nu^{5} - 8\nu^{4} + 10\nu^{3} - 12\nu + 40 ) / 16 $ (-3v^7 + 4v^6 - 3v^5 - 8v^4 + 10v^3 - 12v + 40) / 16
$\beta_{7}$	$=$	$ ( -3\nu^{7} + 2\nu^{6} + \nu^{5} - 6\nu^{4} + 10\nu^{3} + 4\nu^{2} - 12\nu + 32 ) / 8 $ (-3v^7 + 2v^6 + v^5 - 6v^4 + 10v^3 + 4v^2 - 12v + 32) / 8

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

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

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$1 - \beta_{3}$	$-1$

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} $

19.1

0.0777157	+	1.41208i
1.40376	+	0.171630i
0.856419	−	1.12541i
−1.33790	−	0.458297i
0.0777157	−	1.41208i
1.40376	−	0.171630i
0.856419	+	1.12541i
−1.33790	+	0.458297i

−1.26175

−

0.638735i

1.18404

1.61185i

−0.380152

−

0.219481i

−2.02350

−

1.70453i

−0.464416

−

2.79004i

0.339468

0.519747i

19.2

−0.850516

1.12988i

−0.553244

−

1.92196i

−0.834598

−

0.481855i

−1.20103

2.35744i

2.64212

1.00956i

1.25428

−

0.533167i

19.3

0.546424

1.30439i

−1.40284

1.42549i

3.33878

1.92764i

1.59285

−

2.11254i

−2.62594

−

1.05092i

−0.690004

5.40836i

19.4

1.06584

−

0.929502i

0.272050

−

1.98141i

−2.12403

−

1.22631i

2.63169

−

0.272415i

−1.55176

−

2.36475i

−3.40374

0.667235i

199.1