Properties

Label 3040.2.j.b
Level $3040$
Weight $2$
Character orbit 3040.j
Analytic conductor $24.275$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3040,2,Mod(2431,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3040.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.j (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + q^{5} + 4 \beta_1 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + q^{5} + 4 \beta_1 q^{7} - q^{9} + 4 \beta_1 q^{11} - \beta_{2} q^{13} + \beta_{3} q^{15} + 2 q^{17} + ( - 3 \beta_{3} - \beta_1) q^{19} + 4 \beta_{2} q^{21} + 4 \beta_1 q^{23} + q^{25} - 4 \beta_{3} q^{27} - 4 \beta_{2} q^{29} + 4 \beta_{2} q^{33} + 4 \beta_1 q^{35} + 3 \beta_{2} q^{37} - 2 \beta_1 q^{39} + 2 \beta_{2} q^{41} + 12 \beta_1 q^{43} - q^{45} - 8 \beta_1 q^{47} - 9 q^{49} + 2 \beta_{3} q^{51} + \beta_{2} q^{53} + 4 \beta_1 q^{55} + ( - \beta_{2} - 6) q^{57} + 8 \beta_{3} q^{59} - 8 q^{61} - 4 \beta_1 q^{63} - \beta_{2} q^{65} - 7 \beta_{3} q^{67} + 4 \beta_{2} q^{69} - 4 \beta_{3} q^{71} + 2 q^{73} + \beta_{3} q^{75} - 16 q^{77} + 2 \beta_{3} q^{79} - 5 q^{81} + 8 \beta_1 q^{83} + 2 q^{85} - 8 \beta_1 q^{87} + 2 \beta_{2} q^{89} + 4 \beta_{3} q^{91} + ( - 3 \beta_{3} - \beta_1) q^{95} + 9 \beta_{2} q^{97} - 4 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 4 q^{9} + 8 q^{17} + 4 q^{25} - 4 q^{45} - 36 q^{49} - 24 q^{57} - 32 q^{61} + 8 q^{73} - 64 q^{77} - 20 q^{81} + 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 −1.41421 0 1.00000 0 4.00000i 0 −1.00000 0
2431.2 0 −1.41421 0 1.00000 0 4.00000i 0 −1.00000 0
2431.3 0 1.41421 0 1.00000 0 4.00000i 0 −1.00000 0
2431.4 0 1.41421 0 1.00000 0 4.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.2.j.b 4
4.b odd 2 1 inner 3040.2.j.b 4
19.b odd 2 1 inner 3040.2.j.b 4
76.d even 2 1 inner 3040.2.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.j.b 4 1.a even 1 1 trivial
3040.2.j.b 4 4.b odd 2 1 inner
3040.2.j.b 4 19.b odd 2 1 inner
3040.2.j.b 4 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(3040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T - 2)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 34T^{2} + 361 \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
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