Properties

Label 2015.1.h.a
Level $2015$
Weight $1$
Character orbit 2015.h
Analytic conductor $1.006$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -155
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2015,1,Mod(2014,2015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("2015.2014");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2015 = 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2015.h (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: no
Analytic conductor: \(1.00561600046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.130975.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{3} - q^{4} + \zeta_{8}^{2} q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{3} - q^{4} + \zeta_{8}^{2} q^{5} + q^{9} + (\zeta_{8}^{3} - \zeta_{8}) q^{12} - \zeta_{8}^{3} q^{13} + (\zeta_{8}^{3} + \zeta_{8}) q^{15} + q^{16} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{17} + \zeta_{8}^{2} q^{19} - \zeta_{8}^{2} q^{20} + (\zeta_{8}^{3} - \zeta_{8}) q^{23} - q^{25} - \zeta_{8}^{2} q^{31} - q^{36} + (\zeta_{8}^{3} + \zeta_{8}) q^{37} + ( - \zeta_{8}^{2} + 1) q^{39} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{43} + \zeta_{8}^{2} q^{45} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{48} - q^{49} + ( - \zeta_{8}^{2} + 2) q^{51} + \zeta_{8}^{3} q^{52} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{53} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{57} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{60} - q^{64} + \zeta_{8} q^{65} + (\zeta_{8}^{3} - \zeta_{8}) q^{68} + (\zeta_{8}^{2} - 2) q^{69} - \zeta_{8}^{2} q^{71} + (\zeta_{8}^{3} + \zeta_{8}) q^{73} + (\zeta_{8}^{3} - \zeta_{8}) q^{75} - 2 \zeta_{8}^{2} q^{76} + \zeta_{8}^{2} q^{80} - q^{81} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{83} + (\zeta_{8}^{3} + \zeta_{8}) q^{85} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{92} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{93} - 2 q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{9} + 4 q^{16} - 4 q^{25} - 4 q^{36} + 4 q^{39} - 4 q^{49} + 8 q^{51} - 4 q^{64} - 8 q^{69} - 4 q^{81} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2015\mathbb{Z}\right)^\times\).

\(n\) \(716\) \(807\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2014.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 −1.41421 −1.00000 1.00000i 0 0 0 1.00000 0
2014.2 0 −1.41421 −1.00000 1.00000i 0 0 0 1.00000 0
2014.3 0 1.41421 −1.00000 1.00000i 0 0 0 1.00000 0
2014.4 0 1.41421 −1.00000 1.00000i 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
155.c odd 2 1 CM by \(\Q(\sqrt{-155}) \)
5.b even 2 1 inner
13.b even 2 1 inner
31.b odd 2 1 inner
65.d even 2 1 inner
403.b odd 2 1 inner
2015.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2015.1.h.a 4
5.b even 2 1 inner 2015.1.h.a 4
13.b even 2 1 inner 2015.1.h.a 4
31.b odd 2 1 inner 2015.1.h.a 4
65.d even 2 1 inner 2015.1.h.a 4
155.c odd 2 1 CM 2015.1.h.a 4
403.b odd 2 1 inner 2015.1.h.a 4
2015.h odd 2 1 inner 2015.1.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.h.a 4 1.a even 1 1 trivial
2015.1.h.a 4 5.b even 2 1 inner
2015.1.h.a 4 13.b even 2 1 inner
2015.1.h.a 4 31.b odd 2 1 inner
2015.1.h.a 4 65.d even 2 1 inner
2015.1.h.a 4 155.c odd 2 1 CM
2015.1.h.a 4 403.b odd 2 1 inner
2015.1.h.a 4 2015.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2015, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{2} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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