Properties

Label 1764.1.y.d
Level $1764$
Weight $1$
Character orbit 1764.y
Analytic conductor $0.880$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,1,Mod(667,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.667");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1764.y (of order \(6\), 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: \(0.880350682285\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.49392.1
Artin image: $C_3\times D_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} - \beta_1) q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} - \beta_1) q^{5} - q^{8} - \beta_1 q^{10} + \beta_{3} q^{13} + \beta_{2} q^{16} - \beta_1 q^{17} + \beta_{3} q^{20} + ( - \beta_{2} - 1) q^{25} + (\beta_{3} + \beta_1) q^{26} + 2 q^{29} + (\beta_{2} + 1) q^{32} + \beta_{3} q^{34} + (\beta_{3} + \beta_1) q^{40} - \beta_{3} q^{41} - q^{50} + \beta_1 q^{52} - 2 \beta_{2} q^{58} + (\beta_{3} + \beta_1) q^{61} + q^{64} + 2 \beta_{2} q^{65} + (\beta_{3} + \beta_1) q^{68} - \beta_1 q^{73} + \beta_1 q^{80} + ( - \beta_{3} - \beta_1) q^{82} - 2 q^{85} + ( - \beta_{3} - \beta_1) q^{89} - \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 2 q^{16} - 2 q^{25} + 8 q^{29} + 2 q^{32} - 4 q^{50} + 4 q^{58} + 4 q^{64} - 4 q^{65} - 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
667.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0.500000 0.866025i 0 −0.500000 0.866025i −0.707107 + 1.22474i 0 0 −1.00000 0 0.707107 + 1.22474i
667.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.707107 1.22474i 0 0 −1.00000 0 −0.707107 1.22474i
1243.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.707107 1.22474i 0 0 −1.00000 0 0.707107 1.22474i
1243.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.707107 + 1.22474i 0 0 −1.00000 0 −0.707107 + 1.22474i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.y.d 4
3.b odd 2 1 1764.1.y.b 4
4.b odd 2 1 CM 1764.1.y.d 4
7.b odd 2 1 inner 1764.1.y.d 4
7.c even 3 1 1764.1.g.b 2
7.c even 3 1 inner 1764.1.y.d 4
7.d odd 6 1 1764.1.g.b 2
7.d odd 6 1 inner 1764.1.y.d 4
12.b even 2 1 1764.1.y.b 4
21.c even 2 1 1764.1.y.b 4
21.g even 6 1 1764.1.g.d yes 2
21.g even 6 1 1764.1.y.b 4
21.h odd 6 1 1764.1.g.d yes 2
21.h odd 6 1 1764.1.y.b 4
28.d even 2 1 inner 1764.1.y.d 4
28.f even 6 1 1764.1.g.b 2
28.f even 6 1 inner 1764.1.y.d 4
28.g odd 6 1 1764.1.g.b 2
28.g odd 6 1 inner 1764.1.y.d 4
84.h odd 2 1 1764.1.y.b 4
84.j odd 6 1 1764.1.g.d yes 2
84.j odd 6 1 1764.1.y.b 4
84.n even 6 1 1764.1.g.d yes 2
84.n even 6 1 1764.1.y.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.g.b 2 7.c even 3 1
1764.1.g.b 2 7.d odd 6 1
1764.1.g.b 2 28.f even 6 1
1764.1.g.b 2 28.g odd 6 1
1764.1.g.d yes 2 21.g even 6 1
1764.1.g.d yes 2 21.h odd 6 1
1764.1.g.d yes 2 84.j odd 6 1
1764.1.g.d yes 2 84.n even 6 1
1764.1.y.b 4 3.b odd 2 1
1764.1.y.b 4 12.b even 2 1
1764.1.y.b 4 21.c even 2 1
1764.1.y.b 4 21.g even 6 1
1764.1.y.b 4 21.h odd 6 1
1764.1.y.b 4 84.h odd 2 1
1764.1.y.b 4 84.j odd 6 1
1764.1.y.b 4 84.n even 6 1
1764.1.y.d 4 1.a even 1 1 trivial
1764.1.y.d 4 4.b odd 2 1 CM
1764.1.y.d 4 7.b odd 2 1 inner
1764.1.y.d 4 7.c even 3 1 inner
1764.1.y.d 4 7.d odd 6 1 inner
1764.1.y.d 4 28.d even 2 1 inner
1764.1.y.d 4 28.f even 6 1 inner
1764.1.y.d 4 28.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} + 2T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{29} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
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