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Properties

Label 1540.2.a.i
Level $1540$
Weight $2$
Character orbit 1540.a
Self dual yes
Analytic conductor $12.297$
Analytic rank $0$
Dimension $4$
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] = [1540,2,Mod(1,1540)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1540, 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("1540.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1540 = 2^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1540.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: \(12.2969619113\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.111028.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 10x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} - q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} - q^{7} + (\beta_{2} + 2) q^{9} + q^{11} + ( - \beta_{3} + 1) q^{13} - \beta_1 q^{15} + ( - \beta_1 + 2) q^{17} + (\beta_{3} - \beta_1 + 1) q^{19} + \beta_1 q^{21} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{23} + q^{25} + ( - \beta_{3} - 3 \beta_1 - 1) q^{27} + 2 q^{29} + ( - \beta_{2} + \beta_1 + 1) q^{31} - \beta_1 q^{33} - q^{35} + (\beta_{3} - \beta_{2} + \beta_1 + 2) q^{37} + (\beta_{2} + 1) q^{39} + ( - \beta_{2} - \beta_1 + 3) q^{41} + (\beta_{3} - \beta_{2} - \beta_1) q^{43} + (\beta_{2} + 2) q^{45} + ( - \beta_{3} - 2 \beta_1 - 1) q^{47} + q^{49} + (\beta_{2} - 2 \beta_1 + 5) q^{51} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{53} + q^{55} + ( - 2 \beta_1 + 4) q^{57} + ( - \beta_{2} + 3 \beta_1 + 1) q^{59} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 6) q^{61} + ( - \beta_{2} - 2) q^{63} + ( - \beta_{3} + 1) q^{65} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{67} + (\beta_{3} - 2 \beta_{2} + 5 \beta_1 - 5) q^{69} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{71} + (\beta_1 + 6) q^{73} - \beta_1 q^{75} - q^{77} + ( - 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 3) q^{79} + (\beta_{2} + 2 \beta_1 + 10) q^{81} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{83} + ( - \beta_1 + 2) q^{85} - 2 \beta_1 q^{87} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{89} + (\beta_{3} - 1) q^{91} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 4) q^{93} + (\beta_{3} - \beta_1 + 1) q^{95} + (\beta_{3} + \beta_1 - 1) q^{97} + (\beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 4 q^{7} + 8 q^{9} + 4 q^{11} + 2 q^{13} + 8 q^{17} + 6 q^{19} - 6 q^{23} + 4 q^{25} - 6 q^{27} + 8 q^{29} + 4 q^{31} - 4 q^{35} + 10 q^{37} + 4 q^{39} + 12 q^{41} + 2 q^{43} + 8 q^{45} - 6 q^{47} + 4 q^{49} + 20 q^{51} + 6 q^{53} + 4 q^{55} + 16 q^{57} + 4 q^{59} + 26 q^{61} - 8 q^{63} + 2 q^{65} - 10 q^{67} - 18 q^{69} + 4 q^{71} + 24 q^{73} - 4 q^{77} + 8 q^{79} + 40 q^{81} - 2 q^{83} + 8 q^{85} - 6 q^{89} - 2 q^{91} - 14 q^{93} + 6 q^{95} - 2 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 10x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 9\nu - 1 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 9\beta _1 + 1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
3.19911
0.547280
−0.766757
−2.97964
0 −3.19911 0 1.00000 0 −1.00000 0 7.23433 0
1.2 0 −0.547280 0 1.00000 0 −1.00000 0 −2.70049 0
1.3 0 0.766757 0 1.00000 0 −1.00000 0 −2.41208 0
1.4 0 2.97964 0 1.00000 0 −1.00000 0 5.87824 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 1540.2.a.i ✓ 4
4.b odd 2 1 6160.2.a.bs 4
5.b even 2 1 7700.2.a.bb 4
5.c odd 4 2 7700.2.e.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1540.2.a.i ✓ 4 1.a even 1 1 trivial
6160.2.a.bs 4 4.b odd 2 1
7700.2.a.bb 4 5.b even 2 1
7700.2.e.t 8 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 10T_{3}^{2} + 2T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1540))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 10 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 96 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + \cdots - 24 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + \cdots - 96 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + \cdots - 184 \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 176 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + \cdots - 264 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + \cdots - 668 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + \cdots - 144 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} + \cdots + 1044 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + \cdots + 1608 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + \cdots - 1448 \) Copy content Toggle raw display
$61$ \( T^{4} - 26 T^{3} + \cdots - 5636 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$71$ \( T^{4} - 4 T^{3} + \cdots + 8704 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + \cdots + 952 \) Copy content Toggle raw display
$79$ \( T^{4} - 8 T^{3} + \cdots + 10528 \) Copy content Toggle raw display
$83$ \( T^{4} + 2 T^{3} + \cdots + 3936 \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + \cdots + 2592 \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + \cdots + 544 \) Copy content Toggle raw display
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