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Properties

Label 1399.1.b.d
Level $1399$
Weight $1$
Character orbit 1399.b
Self dual yes
Analytic conductor $0.698$
Analytic rank $0$
Dimension $9$
Projective image $D_{27}$
CM discriminant -1399
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1399,1,Mod(1398,1399)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1399, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1399.1398");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1399 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1399.b (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: \(0.698191952674\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} + ( - \beta_{6} + \beta_{3}) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} + ( - \beta_{6} + \beta_{3}) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + ( - \beta_{7} - 1) q^{10} - \beta_{5} q^{11} + (\beta_{7} + \beta_{5} + \cdots - \beta_{2}) q^{14}+ \cdots - \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{4} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{4} + 9 q^{9} - 9 q^{10} + 9 q^{16} + 9 q^{25} + 9 q^{36} - 9 q^{37} - 9 q^{38} - 9 q^{40} + 9 q^{49} - 9 q^{56} - 9 q^{59} + 9 q^{64} + 9 q^{81} - 9 q^{90} - 9 q^{95}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/1399\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1398.1
1.98648
1.67098
1.37248
0.573606
0.116290
−0.792160
−1.19432
−1.78727
−1.94609
−1.98648 0 2.94609 1.19432 0 0.347296 −3.86586 1.00000 −2.37248
1398.2 −1.67098 0 1.79216 −0.116290 0 1.53209 −1.32368 1.00000 0.194317
1398.3 −1.37248 0 0.883710 1.94609 0 −1.87939 0.159606 1.00000 −2.67098
1398.4 −0.573606 0 −0.670976 −1.37248 0 −1.87939 0.958482 1.00000 0.787265
1398.5 −0.116290 0 −0.986477 1.78727 0 1.53209 0.231007 1.00000 −0.207840
1398.6 0.792160 0 −0.372483 −1.98648 0 0.347296 −1.08723 1.00000 −1.57361
1398.7 1.19432 0 0.426394 0.792160 0 0.347296 −0.685068 1.00000 0.946090
1398.8 1.78727 0 2.19432 −1.67098 0 1.53209 2.13456 1.00000 −2.98648
1398.9 1.94609 0 2.78727 −0.573606 0 −1.87939 3.47818 1.00000 −1.11629
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1398.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
1399.b odd 2 1 CM by \(\Q(\sqrt{-1399}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1399.1.b.d ✓ 9
1399.b odd 2 1 CM 1399.1.b.d ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1399.1.b.d ✓ 9 1.a even 1 1 trivial
1399.1.b.d ✓ 9 1399.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1399, [\chi])\):

\( T_{2}^{9} - 9T_{2}^{7} + 27T_{2}^{5} - 30T_{2}^{3} + 9T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

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