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Properties

Label 2252.1.d.a
Level $2252$
Weight $1$
Character orbit 2252.d
Self dual yes
Analytic conductor $1.124$
Analytic rank $0$
Dimension $9$
Projective image $D_{27}$
CM discriminant -563
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2252,1,Mod(1125,2252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2252.1125");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2252 = 2^{2} \cdot 563 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2252.d (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(1.12389440845\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\Q(\zeta_{54})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{27}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{27} - \cdots)\)

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{4} q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{4} q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{6} + \beta_{3}) q^{11} + \beta_{6} q^{13} - \beta_{7} q^{17} + (\beta_{7} - \beta_{2}) q^{19} + ( - \beta_{5} - \beta_{3}) q^{21} + (\beta_{5} - \beta_{4}) q^{23} + q^{25} + ( - \beta_{3} - \beta_1) q^{27} + (\beta_{7} + \beta_{5} + \cdots - \beta_{2}) q^{33}+ \cdots + ( - \beta_{8} - \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{9} + 9 q^{25} + 9 q^{49} + 9 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{54} + \zeta_{54}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2252\mathbb{Z}\right)^\times\).

\(n\) \(565\) \(1127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1125.1
1.98648
1.67098
1.37248
0.573606
0.116290
−0.792160
−1.19432
−1.78727
−1.94609
0 −1.98648 0 0 0 1.78727 0 2.94609 0
1125.2 0 −1.67098 0 0 0 −1.37248 0 1.79216 0
1125.3 0 −1.37248 0 0 0 −1.98648 0 0.883710 0
1125.4 0 −0.573606 0 0 0 0.792160 0 −0.670976 0
1125.5 0 −0.116290 0 0 0 1.94609 0 −0.986477 0
1125.6 0 0.792160 0 0 0 −0.116290 0 −0.372483 0
1125.7 0 1.19432 0 0 0 −1.67098 0 0.426394 0
1125.8 0 1.78727 0 0 0 −0.573606 0 2.19432 0
1125.9 0 1.94609 0 0 0 1.19432 0 2.78727 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1125.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
563.b odd 2 1 CM by \(\Q(\sqrt{-563}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2252.1.d.a ✓ 9
563.b odd 2 1 CM 2252.1.d.a ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2252.1.d.a ✓ 9 1.a even 1 1 trivial
2252.1.d.a ✓ 9 563.b odd 2 1 CM

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2252, [\chi])\).

Hecke characteristic polynomials

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