LMFDB - Newform orbit 207.2.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,2,Mod(68,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("207.68");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	207.g (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:	$1.65290332184$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.57352136505929721.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3x^{9} + x^{6} - 24x^{3} + 64 $ x^12 - 3x^9 + x^6 - 24x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{10} q^{2} + \beta_{7} q^{3} + (\beta_{7} + 2 \beta_{5} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + ( - \beta_{10} + \beta_{9} - \beta_{6}) q^{9}+O(q^{10}) $ q + b10 * q^2 + b7 * q^3 + (b7 + 2b5 - b3 + b2 - b1 + 2) q^4 + (-b9 - b8 + b6 - b2 - b1) * q^6 + (-2b11 + 2b10 - 2b9 + b8 - 2b5 + b4 - 1) * q^8 + (-b10 + b9 - b6) * q^9	$ q + \beta_{10} q^{2} + \beta_{7} q^{3} + (\beta_{7} + 2 \beta_{5} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + (7 \beta_{11} - 7 \beta_{10} + 7 \beta_{9}) q^{98}+O(q^{100}) $ q + b10 * q^2 + b7 * q^3 + (b7 + 2b5 - b3 + b2 - b1 + 2) q^4 + (-b9 - b8 + b6 - b2 - b1) * q^6 + (-2b11 + 2b10 - 2b9 + b8 - 2b5 + b4 - 1) * q^8 + (-b10 + b9 - b6) * q^9 + (-b10 + b9 + b8 - 2b6 + b5 + b4 + 2b3 + 2b1 - 2) q^12 + (-2b11 + b10 - b6 + b3 + 3b2 + b1) * q^13 + (3b11 - 2b10 + 2b9 + 2b7 + 2b6 + 4b5 - 2b3 - b2) q^16 + (b10 + b9 - b7 - 4b5 - b4 + b3 - b2 - 4) q^18 + (-b5 - 2b4 - 1) q^23 + (3b11 - 4b10 + b9 - 2b8 + 2b7 + 2b6 - 2b5 - 2b4 - 3b3 - 3b2 - 2) q^24 - 5b5 q^25 + (-b11 - 2b9 - 2b8 + 3b6 - 2b5 - 2b4 - 3b3 - b2 - 1) * q^26 + (2b8 + 3b5) * q^27 + (2b11 - 2b10 + b9 + 3b3 - 3b2 + 3b1) q^29 + (-4b11 + 5b10 - 3b9 - b7 + b6 + 2b2) * q^31 + (-3b11 + b10 - 3b9 - 3b7 - 3b6 - 2b5 + 2b4 + 2b2 + 4) q^32 + (3b11 - 2b10 + 3b9 - 2b7 + 5b5 - b4 + b2 - 2b1 + 5) * q^36 + (-2b10 + 3b9 + 5b5 + 2b4 + b2 + b1 + 5) * q^39 + (2b10 - 3b9 - 3b7 + 3b3 + b2 - 3b1) q^41 + (-b11 + 3b10 - b9 - 4b7 + 2b2 - 4b1) * q^46 + (4b11 - 5b10 + 2b9 + 3b7 - 3b6 - 3b2 + 3b1) q^47 + (b10 - b9 + b8 - 4b7 - 4b6 - 3b5 + 3b4 + 4b3 + b1 + 3) q^48 + (-7b5 - 7) q^49 + (5b11 + 5b9) * q^50 + (b10 + 2b9 + 3b8 - 4b7 - 4b6 + 5b5 + 6b4 + 4b3 + 2b2 + 3) * q^52 + (-3b11 + 2b10 - 5b9 + 4b6 - 4b3 - 4b1) * q^54 + (5b11 - 7b10 + 3b9 - 6b8 - b7 + b6 - 2b5 - 3b4 - 4b2 + 1) q^58 + (7b5 + 2b4 - 5) * q^59 + (b10 - 2b9 + b8 + 3b7 + 3b6 + 16b5 + b4 - 3b3 + 4b2 - 3b1 + 8) q^62 + (4b11 + 4b9 + 3b8 + 4b7 - 3b4 - 2b2 + 4b1 - 6) q^64 + (-6b11 + 2b10 - 4b9 - b3 + 4b2) * q^69 + (b10 - 2b9 + 3b7 + 3b6 - 3b3 - 5b2 - 3b1) * q^71 + (-6b11 + 5b10 - 3b9 + 2b8 - 2b7 - 5b6 - 6b5 + 2b2 + 3b1) q^72 + (3b10 - 2b9 - 5b7 + 5b6 + 5b3 - b2 - 5b1) * q^73 + (5b7 - 5b3) * q^75 + (-6b11 + 3b10 - 4b9 - b8 - b7 - 9b5 - b2 + 4b1) q^78 + (-2b10 - 2b9 - b7 + b3 + 2b2) q^81 + (-b11 - 3b10 + 2b9 + 3b8 + 5b7 - 3b4 - b2 + 5b1 - 7) * q^82 + (2b10 + b9 - 2b8 - 2b5 - 2b4 + 3b2 - 5b1 + 7) * q^87 + (-5b11 + 4b10 + 2b9 + 4b8 + 3b7 - 3b6 + 2b5 + 6b2 + 3b1) q^92 + (-3b10 - b9 - 4b8 + 5b6 - 3b5 - 4b2) q^93 + (7b11 - 5b10 + 3b9 - 6b8 - b6 - 11b5 - 3b4 + b3 - 9b2 + b1 - 8) q^94 + (6b11 - 2b10 + 5b9 - b8 + 6b7 + 2b6 + 5b5 - 4b4 - 2b3 - 3b2 + 4b1 - 4) * q^96 + (7b11 - 7b10 + 7b9) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{4} - 3 q^{6}+O(q^{10}) $ 12 * q + 12 * q^4 - 3 * q^6	$ 12 q + 12 q^{4} - 3 q^{6} - 30 q^{12} - 24 q^{16} - 21 q^{18} - 12 q^{24} + 30 q^{25} - 12 q^{27} + 54 q^{32} + 33 q^{36} + 24 q^{39} + 48 q^{48} - 42 q^{49} - 3 q^{52} + 15 q^{58} - 108 q^{59} - 54 q^{64} + 42 q^{72} + 51 q^{78} - 66 q^{82} + 96 q^{87} + 6 q^{93} - 39 q^{94} - 69 q^{96}+O(q^{100}) $ 12 * q + 12 * q^4 - 3 * q^6 - 30 * q^12 - 24 * q^16 - 21 * q^18 - 12 * q^24 + 30 * q^25 - 12 * q^27 + 54 * q^32 + 33 * q^36 + 24 * q^39 + 48 * q^48 - 42 * q^49 - 3 * q^52 + 15 * q^58 - 108 * q^59 - 54 * q^64 + 42 * q^72 + 51 * q^78 - 66 * q^82 + 96 * q^87 + 6 * q^93 - 39 * q^94 - 69 * q^96

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 3x^{9} + x^{6} - 24x^{3} + 64 $ x^12 - 3*x^9 + x^6 - 24*x^3 + 64 :

$\beta_{1}$	$=$	$ ( -\nu^{11} + 3\nu^{8} - \nu^{5} - 8\nu^{2} ) / 32 $ (-v^11 + 3v^8 - v^5 - 8v^2) / 32
$\beta_{2}$	$=$	$ ( -\nu^{11} + 3\nu^{8} - \nu^{5} + 24\nu^{2} - 32\nu ) / 32 $ (-v^11 + 3v^8 - v^5 + 24v^2 - 32*v) / 32
$\beta_{3}$	$=$	$ ( 3\nu^{10} - \nu^{7} - 5\nu^{4} - 64\nu ) / 16 $ (3v^10 - v^7 - 5v^4 - 64*v) / 16
$\beta_{4}$	$=$	$ ( -\nu^{9} - \nu^{6} - \nu^{3} + 24 ) / 4 $ (-v^9 - v^6 - v^3 + 24) / 4
$\beta_{5}$	$=$	$ ( -3\nu^{9} + \nu^{6} - 3\nu^{3} + 64 ) / 8 $ (-3v^9 + v^6 - 3v^3 + 64) / 8
$\beta_{6}$	$=$	$ ( -\nu^{11} + \nu^{8} - 3\nu^{5} + 22\nu^{2} ) / 8 $ (-v^11 + v^8 - 3v^5 + 22v^2) / 8
$\beta_{7}$	$=$	$ ( -5\nu^{10} - \nu^{7} - 5\nu^{4} + 120\nu ) / 16 $ (-5v^10 - v^7 - 5v^4 + 120*v) / 16
$\beta_{8}$	$=$	$ ( -3\nu^{9} + \nu^{6} + 5\nu^{3} + 64 ) / 8 $ (-3v^9 + v^6 + 5v^3 + 64) / 8
$\beta_{9}$	$=$	$ ( \nu^{11} - 21\nu^{2} + 4\nu ) / 4 $ (v^11 - 21v^2 + 4v) / 4
$\beta_{10}$	$=$	$ ( -9\nu^{11} + 3\nu^{8} - \nu^{5} + 192\nu^{2} + 32\nu ) / 32 $ (-9v^11 + 3v^8 - v^5 + 192v^2 + 32v) / 32
$\beta_{11}$	$=$	$ ( -9\nu^{11} + 12\nu^{10} + 3\nu^{8} - 4\nu^{7} - \nu^{5} + 12\nu^{4} + 192\nu^{2} - 288\nu ) / 32 $ (-9v^11 + 12v^10 + 3v^8 - 4v^7 - v^5 + 12v^4 + 192v^2 - 288*v) / 32

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

$\nu$	$=$	$ ( \beta_{10} + \beta_{9} - \beta_{2} ) / 3 $ (b10 + b9 - b2) / 3
$\nu^{2}$	$=$	$ ( \beta_{10} + \beta_{9} + 2\beta_{2} - 3\beta_1 ) / 3 $ (b10 + b9 + 2b2 - 3b1) / 3
$\nu^{3}$	$=$	$ \beta_{8} - \beta_{5} $ b8 - b5
$\nu^{4}$	$=$	$ ( 3\beta_{11} - \beta_{10} + 2\beta_{9} - 6\beta_{3} - 2\beta_{2} ) / 3 $ (3b11 - b10 + 2b9 - 6b3 - 2b2) / 3
$\nu^{5}$	$=$	$ ( 5\beta_{10} + 2\beta_{9} - 9\beta_{6} + 7\beta_{2} ) / 3 $ (5b10 + 2b9 - 9b6 + 7b2) / 3
$\nu^{6}$	$=$	$ 2\beta_{5} - 3\beta_{4} + 2 $ 2b5 - 3b4 + 2
$\nu^{7}$	$=$	$ ( -15\beta_{11} + 10\beta_{10} - 5\beta_{9} - 18\beta_{7} + 5\beta_{2} ) / 3 $ (-15b11 + 10b10 - 5b9 - 18b7 + 5*b2) / 3
$\nu^{8}$	$=$	$ ( 10\beta_{10} + 13\beta_{9} - 3\beta_{6} + 23\beta_{2} + 3\beta_1 ) / 3 $ (10b10 + 13b9 - 3b6 + 23b2 + 3*b1) / 3
$\nu^{9}$	$=$	$ -\beta_{8} - \beta_{5} - \beta_{4} + 22 $ -b8 - b5 - b4 + 22
$\nu^{10}$	$=$	$ ( 23\beta_{10} + 23\beta_{9} - 6\beta_{7} + 6\beta_{3} - 23\beta_{2} ) / 3 $ (23b10 + 23b9 - 6b7 + 6b3 - 23*b2) / 3
$\nu^{11}$	$=$	$ ( 17\beta_{10} + 29\beta_{9} + 46\beta_{2} - 63\beta_1 ) / 3 $ (17b10 + 29b9 + 46b2 - 63b1) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/207\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$1 + \beta_{5}$

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} $

68.1

−1.32313	+	0.499333i
−0.721506	+	1.21632i
−0.692610	−	1.23300i
1.09400	+	0.896196i
0.229129	−	1.39553i
1.41412	+	0.0166831i
−1.32313	−	0.499333i
−0.721506	−	1.21632i
−0.692610	+	1.23300i
1.09400	−	0.896196i
0.229129	+	1.39553i
1.41412	−	0.0166831i

−2.41713

1.39553i

−1.73046

0.0741663i

2.89500

−

5.01429i

4.07924

−

2.59418i

10.5781i

2.98900

−

0.256684i

68.2

−2.13562

1.23300i

1.65147

−

0.522159i

2.04058

−

3.53440i

−2.88309

3.15140i

5.13217i

2.45470

−

1.72466i

68.3

0.0288960

−

0.0166831i

−0.373531

1.69129i

−0.999443

1.73109i

0.0174225

0.0551034i

0.133428i

−2.72095

−

1.26350i

68.4

0.864870

−

0.499333i

0.929461

1.46154i

−0.501333

0.868335i

1.53366

0.799932i

2.99866i

−1.27220

2.71689i

68.5

1.55226

−

0.896196i

0.801001

−

1.53571i

0.606333

−

1.05020i

−0.132935

−

3.10167i

1.41121i

−1.71679

−

2.46021i

68.6

2.10672

−

1.21632i

−1.27794

−

1.16913i

1.95886

−

3.39284i

−4.11430

−

0.908665i

−

4.66511i

0.266250

2.98816i

137.1