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Properties

Label 1575.1.n.d
Level $1575$
Weight $1$
Character orbit 1575.n
Analytic conductor $0.786$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,1,Mod(818,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.818");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.n (of order \(4\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{10} - \zeta_{24}^{8}) q^{2} + ( - \zeta_{24}^{8} + \cdots - \zeta_{24}^{4}) q^{4}+ \cdots + ( - \zeta_{24}^{6} + \zeta_{24}^{4} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{24}^{10} - \zeta_{24}^{8}) q^{2} + ( - \zeta_{24}^{8} + \cdots - \zeta_{24}^{4}) q^{4}+ \cdots + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 12 q^{8} - 16 q^{16} - 4 q^{23} - 8 q^{32} + 8 q^{46} + 8 q^{53} - 4 q^{77} + 8 q^{92} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
818.1
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.366025 − 0.366025i 0 − 0.732051i 0 0 −0.707107 + 0.707107i −0.633975 + 0.633975i 0 0
818.2 −0.366025 − 0.366025i 0 − 0.732051i 0 0 0.707107 − 0.707107i −0.633975 + 0.633975i 0 0
818.3 1.36603 + 1.36603i 0 2.73205i 0 0 −0.707107 + 0.707107i −2.36603 + 2.36603i 0 0
818.4 1.36603 + 1.36603i 0 2.73205i 0 0 0.707107 − 0.707107i −2.36603 + 2.36603i 0 0
1007.1 −0.366025 + 0.366025i 0 0.732051i 0 0 −0.707107 − 0.707107i −0.633975 − 0.633975i 0 0
1007.2 −0.366025 + 0.366025i 0 0.732051i 0 0 0.707107 + 0.707107i −0.633975 − 0.633975i 0 0
1007.3 1.36603 − 1.36603i 0 − 2.73205i 0 0 −0.707107 − 0.707107i −2.36603 − 2.36603i 0 0
1007.4 1.36603 − 1.36603i 0 − 2.73205i 0 0 0.707107 + 0.707107i −2.36603 − 2.36603i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 818.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.c odd 4 1 inner
15.d odd 2 1 inner
15.e even 4 1 inner
35.f even 4 1 inner
105.g even 2 1 inner
105.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.n.d yes 8
3.b odd 2 1 1575.1.n.c ✓ 8
5.b even 2 1 1575.1.n.c ✓ 8
5.c odd 4 1 1575.1.n.c ✓ 8
5.c odd 4 1 inner 1575.1.n.d yes 8
7.b odd 2 1 CM 1575.1.n.d yes 8
15.d odd 2 1 inner 1575.1.n.d yes 8
15.e even 4 1 1575.1.n.c ✓ 8
15.e even 4 1 inner 1575.1.n.d yes 8
21.c even 2 1 1575.1.n.c ✓ 8
35.c odd 2 1 1575.1.n.c ✓ 8
35.f even 4 1 1575.1.n.c ✓ 8
35.f even 4 1 inner 1575.1.n.d yes 8
105.g even 2 1 inner 1575.1.n.d yes 8
105.k odd 4 1 1575.1.n.c ✓ 8
105.k odd 4 1 inner 1575.1.n.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.n.c ✓ 8 3.b odd 2 1
1575.1.n.c ✓ 8 5.b even 2 1
1575.1.n.c ✓ 8 5.c odd 4 1
1575.1.n.c ✓ 8 15.e even 4 1
1575.1.n.c ✓ 8 21.c even 2 1
1575.1.n.c ✓ 8 35.c odd 2 1
1575.1.n.c ✓ 8 35.f even 4 1
1575.1.n.c ✓ 8 105.k odd 4 1
1575.1.n.d yes 8 1.a even 1 1 trivial
1575.1.n.d yes 8 5.c odd 4 1 inner
1575.1.n.d yes 8 7.b odd 2 1 CM
1575.1.n.d yes 8 15.d odd 2 1 inner
1575.1.n.d yes 8 15.e even 4 1 inner
1575.1.n.d yes 8 35.f even 4 1 inner
1575.1.n.d yes 8 105.g even 2 1 inner
1575.1.n.d yes 8 105.k odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 2T_{2}^{2} + 2T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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