Properties

Label 3080.2.a.j
Level $3080$
Weight $2$
Character orbit 3080.a
Self dual yes
Analytic conductor $24.594$
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] = [3080,2,Mod(1,3080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3080.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: \(24.5939238226\)
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: 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + q^{5} + q^{7} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + q^{5} + q^{7} + (\beta_{2} + 3 \beta_1) q^{9} + q^{11} + ( - 2 \beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_1 - 1) q^{15} + (2 \beta_{2} + 3 \beta_1 - 3) q^{17} + ( - 2 \beta_{2} - 2) q^{19} + ( - \beta_1 - 1) q^{21} + (3 \beta_{2} - 3 \beta_1 - 1) q^{23} + q^{25} + ( - 4 \beta_{2} - 4 \beta_1 - 2) q^{27} + ( - 2 \beta_{2} - 2 \beta_1) q^{29} + (\beta_{2} + 2 \beta_1 - 6) q^{31} + ( - \beta_1 - 1) q^{33} + q^{35} + (\beta_{2} + 3 \beta_1 - 1) q^{37} + (3 \beta_{2} + 5 \beta_1 + 1) q^{39} + (\beta_{2} + 2 \beta_1) q^{41} + (3 \beta_{2} - \beta_1 + 1) q^{43} + (\beta_{2} + 3 \beta_1) q^{45} + ( - 4 \beta_{2} + \beta_1 - 5) q^{47} + q^{49} + ( - 5 \beta_{2} - 5 \beta_1 - 1) q^{51} + ( - 5 \beta_{2} + \beta_1 + 1) q^{53} + q^{55} + (2 \beta_{2} + 4 \beta_1) q^{57} + (\beta_{2} + 4 \beta_1 - 4) q^{59} + (\beta_{2} + 2 \beta_1 - 2) q^{61} + (\beta_{2} + 3 \beta_1) q^{63} + ( - 2 \beta_{2} - \beta_1 - 1) q^{65} + (3 \beta_{2} - \beta_1 - 3) q^{67} + (4 \beta_1 + 10) q^{69} + ( - 2 \beta_{2} + 6 \beta_1 - 2) q^{71} + (2 \beta_{2} - 3 \beta_1 - 1) q^{73} + ( - \beta_1 - 1) q^{75} + q^{77} + (7 \beta_{2} + 3 \beta_1 + 3) q^{79} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + ( - 4 \beta_{2} - 4 \beta_1 - 6) q^{83} + (2 \beta_{2} + 3 \beta_1 - 3) q^{85} + (4 \beta_{2} + 6 \beta_1 + 2) q^{87} + ( - 4 \beta_{2} - 4 \beta_1 + 8) q^{89} + ( - 2 \beta_{2} - \beta_1 - 1) q^{91} + ( - 3 \beta_{2} + \beta_1 + 3) q^{93} + ( - 2 \beta_{2} - 2) q^{95} + (2 \beta_{2} + 2 \beta_1 - 2) q^{97} + (\beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} - 6 q^{17} - 6 q^{19} - 4 q^{21} - 6 q^{23} + 3 q^{25} - 10 q^{27} - 2 q^{29} - 16 q^{31} - 4 q^{33} + 3 q^{35} + 8 q^{39} + 2 q^{41} + 2 q^{43} + 3 q^{45} - 14 q^{47} + 3 q^{49} - 8 q^{51} + 4 q^{53} + 3 q^{55} + 4 q^{57} - 8 q^{59} - 4 q^{61} + 3 q^{63} - 4 q^{65} - 10 q^{67} + 34 q^{69} - 6 q^{73} - 4 q^{75} + 3 q^{77} + 12 q^{79} + 23 q^{81} - 22 q^{83} - 6 q^{85} + 12 q^{87} + 20 q^{89} - 4 q^{91} + 10 q^{93} - 6 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) \) 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
2.17009
0.311108
−1.48119
0 −3.17009 0 1.00000 0 1.00000 0 7.04945 0
1.2 0 −1.31111 0 1.00000 0 1.00000 0 −1.28100 0
1.3 0 0.481194 0 1.00000 0 1.00000 0 −2.76845 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 3080.2.a.j 3
4.b odd 2 1 6160.2.a.bp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3080.2.a.j 3 1.a even 1 1 trivial
6160.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3080))\):

\( T_{3}^{3} + 4T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{13}^{3} + 4T_{13}^{2} - 10T_{13} - 38 \) Copy content Toggle raw display
\( T_{17}^{3} + 6T_{17}^{2} - 22T_{17} - 122 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} + 2 T - 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 4 T^{2} - 10 T - 38 \) Copy content Toggle raw display
$17$ \( T^{3} + 6 T^{2} - 22 T - 122 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} - 4 T - 40 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} - 72 T - 428 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 20 T - 8 \) Copy content Toggle raw display
$31$ \( T^{3} + 16 T^{2} + 72 T + 62 \) Copy content Toggle raw display
$37$ \( T^{3} - 28T - 52 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 12 T - 10 \) Copy content Toggle raw display
$43$ \( T^{3} - 2 T^{2} - 44 T + 20 \) Copy content Toggle raw display
$47$ \( T^{3} + 14 T^{2} - 10 T - 274 \) Copy content Toggle raw display
$53$ \( T^{3} - 4 T^{2} - 108 T + 52 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} - 28 T - 214 \) Copy content Toggle raw display
$61$ \( T^{3} + 4 T^{2} - 8 T - 34 \) Copy content Toggle raw display
$67$ \( T^{3} + 10 T^{2} - 12 T - 124 \) Copy content Toggle raw display
$71$ \( T^{3} - 160T + 608 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} - 46 T - 278 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} - 136 T + 1580 \) Copy content Toggle raw display
$83$ \( T^{3} + 22 T^{2} + 76 T - 184 \) Copy content Toggle raw display
$89$ \( T^{3} - 20 T^{2} + 48 T + 320 \) Copy content Toggle raw display
$97$ \( T^{3} + 4 T^{2} - 16 T - 32 \) Copy content Toggle raw display
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