Properties

Label 6080.2.a.bg
Level $6080$
Weight $2$
Character orbit 6080.a
Self dual yes
Analytic conductor $48.549$
Analytic rank $1$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 760)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} - 2 \beta q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{5} - 2 \beta q^{7} - q^{9} + (2 \beta + 2) q^{11} + (\beta - 2) q^{13} + \beta q^{15} + ( - 2 \beta + 2) q^{17} - q^{19} - 4 q^{21} - 4 q^{23} + q^{25} - 4 \beta q^{27} + ( - 2 \beta - 2) q^{29} + (2 \beta + 4) q^{33} - 2 \beta q^{35} + (3 \beta - 6) q^{37} + ( - 2 \beta + 2) q^{39} + (2 \beta + 2) q^{41} - 2 \beta q^{43} - q^{45} - 6 \beta q^{47} + q^{49} + (2 \beta - 4) q^{51} + (5 \beta - 6) q^{53} + (2 \beta + 2) q^{55} - \beta q^{57} - 2 \beta q^{59} + ( - 4 \beta - 4) q^{61} + 2 \beta q^{63} + (\beta - 2) q^{65} + (\beta - 8) q^{67} - 4 \beta q^{69} + (8 \beta - 4) q^{71} + ( - 10 \beta - 2) q^{73} + \beta q^{75} + ( - 4 \beta - 8) q^{77} + (2 \beta + 12) q^{79} - 5 q^{81} + 8 q^{83} + ( - 2 \beta + 2) q^{85} + ( - 2 \beta - 4) q^{87} + ( - 2 \beta - 6) q^{89} + (4 \beta - 4) q^{91} - q^{95} + (3 \beta + 2) q^{97} + ( - 2 \beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} - 8 q^{21} - 8 q^{23} + 2 q^{25} - 4 q^{29} + 8 q^{33} - 12 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{45} + 2 q^{49} - 8 q^{51} - 12 q^{53} + 4 q^{55} - 8 q^{61} - 4 q^{65} - 16 q^{67} - 8 q^{71} - 4 q^{73} - 16 q^{77} + 24 q^{79} - 10 q^{81} + 16 q^{83} + 4 q^{85} - 8 q^{87} - 12 q^{89} - 8 q^{91} - 2 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 −1.41421 0 1.00000 0 2.82843 0 −1.00000 0
1.2 0 1.41421 0 1.00000 0 −2.82843 0 −1.00000 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.bg 2
4.b odd 2 1 6080.2.a.bf 2
8.b even 2 1 1520.2.a.m 2
8.d odd 2 1 760.2.a.f 2
24.f even 2 1 6840.2.a.z 2
40.e odd 2 1 3800.2.a.n 2
40.f even 2 1 7600.2.a.ba 2
40.k even 4 2 3800.2.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.f 2 8.d odd 2 1
1520.2.a.m 2 8.b even 2 1
3800.2.a.n 2 40.e odd 2 1
3800.2.d.i 4 40.k even 4 2
6080.2.a.bf 2 4.b odd 2 1
6080.2.a.bg 2 1.a even 1 1 trivial
6840.2.a.z 2 24.f even 2 1
7600.2.a.ba 2 40.f even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 72 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T - 14 \) Copy content Toggle raw display
$59$ \( T^{2} - 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 16T + 62 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 112 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 196 \) Copy content Toggle raw display
$79$ \( T^{2} - 24T + 136 \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
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