Properties

Label 115.5.c.b
Level $115$
Weight $5$
Character orbit 115.c
Self dual yes
Analytic conductor $11.888$
Analytic rank $0$
Dimension $1$
CM discriminant -115
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,5,Mod(114,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.114");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 115.c (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: \(11.8875457546\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 16 q^{4} + 25 q^{5} - 17 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{4} + 25 q^{5} - 17 q^{7} + 81 q^{9} + 256 q^{16} - 457 q^{17} + 400 q^{20} + 529 q^{23} + 625 q^{25} - 272 q^{28} + 1567 q^{29} + 887 q^{31} - 425 q^{35} + 1296 q^{36} - 137 q^{37} - 2273 q^{41} - 3662 q^{43} + 2025 q^{45} - 2112 q^{49} + 4583 q^{53} - 6953 q^{59} - 1377 q^{63} + 4096 q^{64} + 3343 q^{67} - 7312 q^{68} - 9353 q^{71} + 6400 q^{80} + 6561 q^{81} - 12097 q^{83} - 11425 q^{85} + 8464 q^{92} + 11458 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
114.1
0
0 0 16.0000 25.0000 0 −17.0000 0 81.0000 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
115.c odd 2 1 CM by \(\Q(\sqrt{-115}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.5.c.b yes 1
5.b even 2 1 115.5.c.a 1
23.b odd 2 1 115.5.c.a 1
115.c odd 2 1 CM 115.5.c.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.5.c.a 1 5.b even 2 1
115.5.c.a 1 23.b odd 2 1
115.5.c.b yes 1 1.a even 1 1 trivial
115.5.c.b yes 1 115.c 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 \(S_{5}^{\mathrm{new}}(115, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7} + 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T + 17 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 457 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 529 \) Copy content Toggle raw display
$29$ \( T - 1567 \) Copy content Toggle raw display
$31$ \( T - 887 \) Copy content Toggle raw display
$37$ \( T + 137 \) Copy content Toggle raw display
$41$ \( T + 2273 \) Copy content Toggle raw display
$43$ \( T + 3662 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 4583 \) Copy content Toggle raw display
$59$ \( T + 6953 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 3343 \) Copy content Toggle raw display
$71$ \( T + 9353 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 12097 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 11458 \) Copy content Toggle raw display
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