Properties

Label 115.6.a.a
Level $115$
Weight $6$
Character orbit 115.a
Self dual yes
Analytic conductor $18.444$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,6,Mod(1,115)]
 
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([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)

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: \(18.4441392785\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1821}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 455 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{1821})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta + 2) q^{3} - 28 q^{4} - 25 q^{5} + ( - 2 \beta + 4) q^{6} + ( - 9 \beta + 31) q^{7} - 120 q^{8} + ( - 3 \beta + 216) q^{9} - 50 q^{10} + ( - 3 \beta - 280) q^{11} + (28 \beta - 56) q^{12}+ \cdots + (201 \beta - 56385) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 3 q^{3} - 56 q^{4} - 50 q^{5} + 6 q^{6} + 53 q^{7} - 240 q^{8} + 429 q^{9} - 100 q^{10} - 563 q^{11} - 84 q^{12} + 833 q^{13} + 106 q^{14} - 75 q^{15} + 1312 q^{16} + 1887 q^{17} + 858 q^{18}+ \cdots - 112569 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
21.8366
−20.8366
2.00000 −19.8366 −28.0000 −25.0000 −39.6732 −165.529 −120.000 150.490 −50.0000
1.2 2.00000 22.8366 −28.0000 −25.0000 45.6732 218.529 −120.000 278.510 −50.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(23\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.6.a.a 2
3.b odd 2 1 1035.6.a.a 2
5.b even 2 1 575.6.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.6.a.a 2 1.a even 1 1 trivial
575.6.a.a 2 5.b even 2 1
1035.6.a.a 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(115))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T - 453 \) Copy content Toggle raw display
$5$ \( (T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 53T - 36173 \) Copy content Toggle raw display
$11$ \( T^{2} + 563T + 75145 \) Copy content Toggle raw display
$13$ \( T^{2} - 833T - 27293 \) Copy content Toggle raw display
$17$ \( T^{2} - 1887 T + 394425 \) Copy content Toggle raw display
$19$ \( T^{2} + 1507T + 71995 \) Copy content Toggle raw display
$23$ \( (T - 529)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 5870 T - 1628900 \) Copy content Toggle raw display
$31$ \( T^{2} + 11683 T + 24285625 \) Copy content Toggle raw display
$37$ \( T^{2} - 8044 T + 1426384 \) Copy content Toggle raw display
$41$ \( T^{2} - 2855 T - 112230449 \) Copy content Toggle raw display
$43$ \( T^{2} + 30600 T + 221241024 \) Copy content Toggle raw display
$47$ \( T^{2} + 27570 T + 132976116 \) Copy content Toggle raw display
$53$ \( T^{2} + 500 T - 358922156 \) Copy content Toggle raw display
$59$ \( T^{2} - 15358 T - 616407260 \) Copy content Toggle raw display
$61$ \( T^{2} + 9169 T - 849734027 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 4096155824 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 1795332271 \) Copy content Toggle raw display
$73$ \( T^{2} + 47946 T + 562757148 \) Copy content Toggle raw display
$79$ \( T^{2} + 35484 T - 136836720 \) Copy content Toggle raw display
$83$ \( T^{2} + 56952 T + 799803612 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 4700059840 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 11218930093 \) Copy content Toggle raw display
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