Properties

Label 6080.2.a.bl
Level $6080$
Weight $2$
Character orbit 6080.a
Self dual yes
Analytic conductor $48.549$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
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 380)
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.

\(f(q)\) \(=\) \( q + (\beta + 2) q^{3} - q^{5} + (2 \beta - 2) q^{7} + (4 \beta + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 2) q^{3} - q^{5} + (2 \beta - 2) q^{7} + (4 \beta + 3) q^{9} + 2 q^{11} + (3 \beta + 2) q^{13} + ( - \beta - 2) q^{15} + (2 \beta - 2) q^{17} + q^{19} + 2 \beta q^{21} - 6 q^{23} + q^{25} + (8 \beta + 8) q^{27} + (6 \beta - 2) q^{29} + ( - 2 \beta - 4) q^{31} + (2 \beta + 4) q^{33} + ( - 2 \beta + 2) q^{35} + ( - 3 \beta + 6) q^{37} + (8 \beta + 10) q^{39} + (4 \beta - 2) q^{41} + ( - 2 \beta - 2) q^{43} + ( - 4 \beta - 3) q^{45} + ( - 2 \beta - 2) q^{47} + ( - 8 \beta + 5) q^{49} + 2 \beta q^{51} + ( - 5 \beta - 2) q^{53} - 2 q^{55} + (\beta + 2) q^{57} + ( - 4 \beta - 8) q^{59} + (4 \beta + 8) q^{61} + ( - 2 \beta + 10) q^{63} + ( - 3 \beta - 2) q^{65} + (\beta + 2) q^{67} + ( - 6 \beta - 12) q^{69} + ( - 2 \beta + 8) q^{71} + ( - 6 \beta + 6) q^{73} + (\beta + 2) q^{75} + (4 \beta - 4) q^{77} + (4 \beta - 4) q^{79} + (12 \beta + 23) q^{81} + (8 \beta + 2) q^{83} + ( - 2 \beta + 2) q^{85} + (10 \beta + 8) q^{87} + ( - 6 \beta + 2) q^{89} + ( - 2 \beta + 8) q^{91} + ( - 8 \beta - 12) q^{93} - q^{95} + ( - 3 \beta - 6) q^{97} + (8 \beta + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 6 q^{9} + 4 q^{11} + 4 q^{13} - 4 q^{15} - 4 q^{17} + 2 q^{19} - 12 q^{23} + 2 q^{25} + 16 q^{27} - 4 q^{29} - 8 q^{31} + 8 q^{33} + 4 q^{35} + 12 q^{37} + 20 q^{39} - 4 q^{41} - 4 q^{43} - 6 q^{45} - 4 q^{47} + 10 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{57} - 16 q^{59} + 16 q^{61} + 20 q^{63} - 4 q^{65} + 4 q^{67} - 24 q^{69} + 16 q^{71} + 12 q^{73} + 4 q^{75} - 8 q^{77} - 8 q^{79} + 46 q^{81} + 4 q^{83} + 4 q^{85} + 16 q^{87} + 4 q^{89} + 16 q^{91} - 24 q^{93} - 2 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 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 0.585786 0 −1.00000 0 −4.82843 0 −2.65685 0
1.2 0 3.41421 0 −1.00000 0 0.828427 0 8.65685 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.bl 2
4.b odd 2 1 6080.2.a.y 2
8.b even 2 1 380.2.a.c 2
8.d odd 2 1 1520.2.a.o 2
24.h odd 2 1 3420.2.a.g 2
40.e odd 2 1 7600.2.a.u 2
40.f even 2 1 1900.2.a.e 2
40.i odd 4 2 1900.2.c.d 4
152.g odd 2 1 7220.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.c 2 8.b even 2 1
1520.2.a.o 2 8.d odd 2 1
1900.2.a.e 2 40.f even 2 1
1900.2.c.d 4 40.i odd 4 2
3420.2.a.g 2 24.h odd 2 1
6080.2.a.y 2 4.b odd 2 1
6080.2.a.bl 2 1.a even 1 1 trivial
7220.2.a.m 2 152.g odd 2 1
7600.2.a.u 2 40.e odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 46 \) Copy content Toggle raw display
$59$ \( T^{2} + 16T + 32 \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T - 36 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$89$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
show more
show less