Properties

Label 6080.2.a.bt
Level $6080$
Weight $2$
Character orbit 6080.a
Self dual yes
Analytic conductor $48.549$
Analytic rank $1$
Dimension $3$
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] = [6080,2,Mod(1,6080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.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: \(48.5490444289\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3040)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - \beta_1) q^{9} + (\beta_{2} + \beta_1 - 1) q^{11} + (\beta_1 + 1) q^{13} + \beta_{2} q^{15} + (2 \beta_{2} + 2 \beta_1 - 4) q^{17} - q^{19} + ( - 2 \beta_1 + 4) q^{21} + ( - \beta_{2} - 3 \beta_1 + 3) q^{23} + q^{25} + (2 \beta_{2} + 2) q^{27} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{29} + (2 \beta_{2} + 2 \beta_1 - 4) q^{31} + (2 \beta_{2} - 2) q^{33} + (\beta_{2} - \beta_1 + 1) q^{35} + (4 \beta_{2} - \beta_1 + 1) q^{37} + ( - \beta_{2} - \beta_1 + 1) q^{39} + (2 \beta_{2} - 2 \beta_1 - 6) q^{41} + ( - \beta_{2} - \beta_1 - 3) q^{43} + (\beta_{2} + \beta_1) q^{45} + ( - \beta_{2} - 3 \beta_1 + 3) q^{47} + (2 \beta_{2} - 4 \beta_1 + 1) q^{49} + (6 \beta_{2} - 4) q^{51} + (4 \beta_{2} + \beta_1 + 3) q^{53} + ( - \beta_{2} - \beta_1 + 1) q^{55} + \beta_{2} q^{57} + 6 \beta_1 q^{59} + (3 \beta_{2} + 3 \beta_1 + 5) q^{61} + ( - \beta_{2} - \beta_1 + 1) q^{63} + ( - \beta_1 - 1) q^{65} + (\beta_{2} - 4 \beta_1 + 2) q^{67} + ( - 4 \beta_{2} + 2 \beta_1) q^{69} + ( - 2 \beta_{2} + 6 \beta_1 - 4) q^{71} + ( - 6 \beta_{2} - 4 \beta_1 + 2) q^{73} - \beta_{2} q^{75} + 2 \beta_{2} q^{77} + (6 \beta_1 - 8) q^{79} + (3 \beta_{2} + 5 \beta_1 - 6) q^{81} + ( - 5 \beta_{2} - \beta_1 + 1) q^{83} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{85} + ( - 8 \beta_{2} + 2 \beta_1 + 2) q^{87} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{89} + 2 q^{91} + (6 \beta_{2} - 4) q^{93} + q^{95} + (6 \beta_{2} - 3 \beta_1 + 3) q^{97} + (\beta_{2} - \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} - 2 q^{7} - q^{9} - 2 q^{11} + 4 q^{13} - 10 q^{17} - 3 q^{19} + 10 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} + 14 q^{29} - 10 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} + 2 q^{39} - 20 q^{41} - 10 q^{43} + q^{45} + 6 q^{47} - q^{49} - 12 q^{51} + 10 q^{53} + 2 q^{55} + 6 q^{59} + 18 q^{61} + 2 q^{63} - 4 q^{65} + 2 q^{67} + 2 q^{69} - 6 q^{71} + 2 q^{73} - 18 q^{79} - 13 q^{81} + 2 q^{83} + 10 q^{85} + 8 q^{87} - 2 q^{89} + 6 q^{91} - 12 q^{93} + 3 q^{95} + 6 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
2.17009
0.311108
0 −1.67513 0 −1.00000 0 −4.15633 0 −0.193937 0
1.2 0 −0.539189 0 −1.00000 0 0.630898 0 −2.70928 0
1.3 0 2.21432 0 −1.00000 0 1.52543 0 1.90321 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(19\) \( +1 \)

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 6080.2.a.bt 3
4.b odd 2 1 6080.2.a.bu 3
8.b even 2 1 3040.2.a.l 3
8.d odd 2 1 3040.2.a.m yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.l 3 8.b even 2 1
3040.2.a.m yes 3 8.d odd 2 1
6080.2.a.bt 3 1.a even 1 1 trivial
6080.2.a.bu 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6080))\):

\( T_{3}^{3} - 4T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 8T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 4T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{3} + 10 T^{2} + \cdots - 40 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$29$ \( T^{3} - 14 T^{2} + \cdots + 296 \) Copy content Toggle raw display
$31$ \( T^{3} + 10 T^{2} + \cdots - 40 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} + \cdots + 74 \) Copy content Toggle raw display
$41$ \( T^{3} + 20 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$47$ \( T^{3} - 6 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$53$ \( T^{3} - 10 T^{2} + \cdots + 334 \) Copy content Toggle raw display
$59$ \( T^{3} - 6 T^{2} + \cdots + 216 \) Copy content Toggle raw display
$61$ \( T^{3} - 18 T^{2} + \cdots + 52 \) Copy content Toggle raw display
$67$ \( T^{3} - 2 T^{2} + \cdots - 86 \) Copy content Toggle raw display
$71$ \( T^{3} + 6 T^{2} + \cdots + 296 \) Copy content Toggle raw display
$73$ \( T^{3} - 2 T^{2} + \cdots - 296 \) Copy content Toggle raw display
$79$ \( T^{3} + 18 T^{2} + \cdots - 520 \) Copy content Toggle raw display
$83$ \( T^{3} - 2 T^{2} + \cdots - 268 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} + \cdots + 232 \) Copy content Toggle raw display
$97$ \( T^{3} - 6 T^{2} + \cdots - 54 \) Copy content Toggle raw display
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