Properties

Label 1575.1.x.a
Level $1575$
Weight $1$
Character orbit 1575.x
Analytic conductor $0.786$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,1,Mod(451,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.451");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.x (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: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.283618125.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{4} - \zeta_{6} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{4} - \zeta_{6} q^{7} + (\zeta_{6}^{2} + \zeta_{6}) q^{13} + \zeta_{6}^{2} q^{16} + (\zeta_{6} + 1) q^{19} - \zeta_{6}^{2} q^{28} - \zeta_{6}^{2} q^{37} + q^{43} + \zeta_{6}^{2} q^{49} + (\zeta_{6}^{2} - 1) q^{52} + ( - \zeta_{6} - 1) q^{61} - q^{64} - \zeta_{6} q^{67} + (\zeta_{6}^{2} - 1) q^{73} + (\zeta_{6}^{2} + \zeta_{6}) q^{76} - \zeta_{6}^{2} q^{79} + ( - \zeta_{6}^{2} + 1) q^{91} + (\zeta_{6}^{2} + \zeta_{6}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{4} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{4} - q^{7} - q^{16} + 3 q^{19} + q^{28} + q^{37} + 4 q^{43} - q^{49} - 3 q^{52} - 3 q^{61} - 2 q^{64} - q^{67} - 3 q^{73} + q^{79} + 3 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(-\zeta_{6}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0.500000 0.866025i 0 0 −0.500000 + 0.866025i 0 0 0
901.1 0 0 0.500000 + 0.866025i 0 0 −0.500000 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.x.a 2
3.b odd 2 1 CM 1575.1.x.a 2
5.b even 2 1 1575.1.x.b yes 2
5.c odd 4 2 1575.1.bj.a 4
7.d odd 6 1 inner 1575.1.x.a 2
15.d odd 2 1 1575.1.x.b yes 2
15.e even 4 2 1575.1.bj.a 4
21.g even 6 1 inner 1575.1.x.a 2
35.i odd 6 1 1575.1.x.b yes 2
35.k even 12 2 1575.1.bj.a 4
105.p even 6 1 1575.1.x.b yes 2
105.w odd 12 2 1575.1.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.x.a 2 1.a even 1 1 trivial
1575.1.x.a 2 3.b odd 2 1 CM
1575.1.x.a 2 7.d odd 6 1 inner
1575.1.x.a 2 21.g even 6 1 inner
1575.1.x.b yes 2 5.b even 2 1
1575.1.x.b yes 2 15.d odd 2 1
1575.1.x.b yes 2 35.i odd 6 1
1575.1.x.b yes 2 105.p even 6 1
1575.1.bj.a 4 5.c odd 4 2
1575.1.bj.a 4 15.e even 4 2
1575.1.bj.a 4 35.k even 12 2
1575.1.bj.a 4 105.w odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{37}^{2} - T_{37} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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