Properties

Label 1215.1.h.a
Level $1215$
Weight $1$
Character orbit 1215.h
Analytic conductor $0.606$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -15
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1215,1,Mod(404,1215)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1215, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1215.404");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1215 = 3^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1215.h (of order \(6\), degree \(2\), not 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: \(0.606363990349\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective 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)
 

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

\(f(q)\) \(=\) \( q + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{2} + (\zeta_{18}^{8} - \zeta_{18}^{7} - \zeta_{18}^{3}) q^{4} - \zeta_{18}^{3} q^{5} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{3} - \zeta_{18}^{2}) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{2} + (\zeta_{18}^{8} - \zeta_{18}^{7} - \zeta_{18}^{3}) q^{4} - \zeta_{18}^{3} q^{5} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{3} - \zeta_{18}^{2}) q^{8} + ( - \zeta_{18}^{7} + \zeta_{18}^{2}) q^{10} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{5} - \zeta_{18}^{2} + \zeta_{18}) q^{16} + (\zeta_{18}^{8} - \zeta_{18}) q^{17} + (\zeta_{18}^{8} - \zeta_{18}) q^{19} + (\zeta_{18}^{6} + \zeta_{18}^{2} - \zeta_{18}) q^{20} + (\zeta_{18}^{8} - \zeta_{18}^{7}) q^{23} + \zeta_{18}^{6} q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{31} + (\zeta_{18}^{6} - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{32} + ( - \zeta_{18}^{7} - \zeta_{18}^{5} - \zeta_{18}^{3} + 1) q^{34} + ( - \zeta_{18}^{7} - \zeta_{18}^{5} - \zeta_{18}^{3} + 1) q^{38} + (\zeta_{18}^{6} - \zeta_{18}^{5} - \zeta_{18} + 1) q^{40} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{46} - \zeta_{18}^{6} q^{47} - \zeta_{18}^{3} q^{49} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{50} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{53} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{61} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2) q^{62} + (\zeta_{18}^{8} + \zeta_{18}^{6} - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18} - 1) q^{64} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \zeta_{18}^{6} + \zeta_{18}^{4} + \zeta_{18}^{2} + 1) q^{68} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \zeta_{18}^{6} + \zeta_{18}^{4} + \zeta_{18}^{2} + 1) q^{76} + (\zeta_{18}^{2} - \zeta_{18}) q^{79} + (\zeta_{18}^{8} - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18} + 1) q^{80} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{83} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{85} + ( - \zeta_{18}^{7} + 2 \zeta_{18}^{6} - \zeta_{18}^{5} + \zeta_{18}^{2} - \zeta_{18}) q^{92} + (\zeta_{18}^{5} + \zeta_{18}) q^{94} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{95} + ( - \zeta_{18}^{7} + \zeta_{18}^{2}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{4} - 3 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{4} - 3 q^{5} - 6 q^{8} - 3 q^{16} - 3 q^{20} - 3 q^{25} + 3 q^{32} + 3 q^{34} + 3 q^{38} + 3 q^{40} - 6 q^{46} + 3 q^{47} - 3 q^{49} + 12 q^{62} + 3 q^{68} + 3 q^{76} + 6 q^{80} - 6 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(\zeta_{18}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
404.1
0.939693 0.342020i
−0.766044 0.642788i
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 + 0.642788i
−0.173648 0.984808i
−0.766044 1.32683i 0 −0.673648 + 1.16679i −0.500000 + 0.866025i 0 0 0.532089 0 1.53209
404.2 −0.173648 0.300767i 0 0.439693 0.761570i −0.500000 + 0.866025i 0 0 −0.652704 0 0.347296
404.3 0.939693 + 1.62760i 0 −1.26604 + 2.19285i −0.500000 + 0.866025i 0 0 −2.87939 0 −1.87939
809.1 −0.766044 + 1.32683i 0 −0.673648 1.16679i −0.500000 0.866025i 0 0 0.532089 0 1.53209
809.2 −0.173648 + 0.300767i 0 0.439693 + 0.761570i −0.500000 0.866025i 0 0 −0.652704 0 0.347296
809.3 0.939693 1.62760i 0 −1.26604 2.19285i −0.500000 0.866025i 0 0 −2.87939 0 −1.87939
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 404.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
9.c even 3 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1215.1.h.a 6
3.b odd 2 1 1215.1.h.b 6
5.b even 2 1 1215.1.h.b 6
9.c even 3 1 1215.1.d.b yes 3
9.c even 3 1 inner 1215.1.h.a 6
9.d odd 6 1 1215.1.d.a 3
9.d odd 6 1 1215.1.h.b 6
15.d odd 2 1 CM 1215.1.h.a 6
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 1 1215.1.d.b yes 3
45.h odd 6 1 inner 1215.1.h.a 6
45.j even 6 1 1215.1.d.a 3
45.j even 6 1 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 9.d odd 6 1
1215.1.d.a 3 45.j even 6 1
1215.1.d.b yes 3 9.c even 3 1
1215.1.d.b yes 3 45.h odd 6 1
1215.1.h.a 6 1.a even 1 1 trivial
1215.1.h.a 6 9.c even 3 1 inner
1215.1.h.a 6 15.d odd 2 1 CM
1215.1.h.a 6 45.h odd 6 1 inner
1215.1.h.b 6 3.b odd 2 1
1215.1.h.b 6 5.b even 2 1
1215.1.h.b 6 9.d odd 6 1
1215.1.h.b 6 45.j even 6 1
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 \( T_{2}^{6} + 3T_{2}^{4} + 2T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1215, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$53$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
show more
show less