Properties

Label 1215.1.d.b
Level $1215$
Weight $1$
Character orbit 1215.d
Self dual yes
Analytic conductor $0.606$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
CM discriminant -15
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1215,1,Mod(1214,1215)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1215, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1215.1214");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1215 = 3^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1215.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: \(0.606363990349\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.242137805625.3
Artin image: $D_9$
Artin field: Galois closure of 9.1.242137805625.3

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

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_1 - 1) q^{8} - \beta_1 q^{10} + (\beta_1 + 1) q^{16} + ( - \beta_{2} + \beta_1) q^{17} + ( - \beta_{2} + \beta_1) q^{19} + (\beta_{2} + 1) q^{20} + \beta_{2} q^{23} + q^{25} - \beta_1 q^{31} + ( - \beta_{2} - 1) q^{32} + ( - \beta_{2} + \beta_1 - 1) q^{34} + ( - \beta_{2} + \beta_1 - 1) q^{38} + ( - \beta_1 - 1) q^{40} + ( - \beta_1 - 1) q^{46} - q^{47} + q^{49} - \beta_1 q^{50} + \beta_{2} q^{53} - \beta_1 q^{61} + (\beta_{2} + 2) q^{62} + \beta_1 q^{64} + (\beta_1 - 1) q^{68} + (\beta_1 - 1) q^{76} + \beta_{2} q^{79} + (\beta_1 + 1) q^{80} + ( - \beta_{2} + \beta_1) q^{83} + ( - \beta_{2} + \beta_1) q^{85} + (\beta_1 + 2) q^{92} + \beta_1 q^{94} + ( - \beta_{2} + \beta_1) q^{95} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{4} + 3 q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{4} + 3 q^{5} - 3 q^{8} + 3 q^{16} + 3 q^{20} + 3 q^{25} - 3 q^{32} - 3 q^{34} - 3 q^{38} - 3 q^{40} - 3 q^{46} - 3 q^{47} + 3 q^{49} + 6 q^{62} - 3 q^{68} - 3 q^{76} + 3 q^{80} + 6 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1215\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1214.1
1.87939
−0.347296
−1.53209
−1.87939 0 2.53209 1.00000 0 0 −2.87939 0 −1.87939
1214.2 0.347296 0 −0.879385 1.00000 0 0 −0.652704 0 0.347296
1214.3 1.53209 0 1.34730 1.00000 0 0 0.532089 0 1.53209
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1215.1.d.b yes 3
3.b odd 2 1 1215.1.d.a 3
5.b even 2 1 1215.1.d.a 3
9.c even 3 2 1215.1.h.a 6
9.d odd 6 2 1215.1.h.b 6
15.d odd 2 1 CM 1215.1.d.b yes 3
27.e even 9 2 3645.1.n.b 6
27.e even 9 2 3645.1.n.c 6
27.e even 9 2 3645.1.n.h 6
27.f odd 18 2 3645.1.n.a 6
27.f odd 18 2 3645.1.n.f 6
27.f odd 18 2 3645.1.n.g 6
45.h odd 6 2 1215.1.h.a 6
45.j even 6 2 1215.1.h.b 6
135.n odd 18 2 3645.1.n.b 6
135.n odd 18 2 3645.1.n.c 6
135.n odd 18 2 3645.1.n.h 6
135.p even 18 2 3645.1.n.a 6
135.p even 18 2 3645.1.n.f 6
135.p even 18 2 3645.1.n.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1215.1.d.a 3 3.b odd 2 1
1215.1.d.a 3 5.b even 2 1
1215.1.d.b yes 3 1.a even 1 1 trivial
1215.1.d.b yes 3 15.d odd 2 1 CM
1215.1.h.a 6 9.c even 3 2
1215.1.h.a 6 45.h odd 6 2
1215.1.h.b 6 9.d odd 6 2
1215.1.h.b 6 45.j even 6 2
3645.1.n.a 6 27.f odd 18 2
3645.1.n.a 6 135.p even 18 2
3645.1.n.b 6 27.e even 9 2
3645.1.n.b 6 135.n odd 18 2
3645.1.n.c 6 27.e even 9 2
3645.1.n.c 6 135.n odd 18 2
3645.1.n.f 6 27.f odd 18 2
3645.1.n.f 6 135.p even 18 2
3645.1.n.g 6 27.f odd 18 2
3645.1.n.g 6 135.p even 18 2
3645.1.n.h 6 27.e even 9 2
3645.1.n.h 6 135.n odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1215, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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