Properties

Label 5904.2.a.bp
Level $5904$
Weight $2$
Character orbit 5904.a
Self dual yes
Analytic conductor $47.144$
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] = [5904,2,Mod(1,5904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5904, 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("5904.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5904 = 2^{4} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5904.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: \(47.1436773534\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 10x^{2} - 6x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2} - 2) q^{5} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2} - 2) q^{5} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{11} + 2 \beta_1 q^{13} - 2 \beta_{3} q^{17} + (\beta_{2} + \beta_1 - 2) q^{19} + ( - 2 \beta_{2} - 2) q^{23} + ( - 2 \beta_1 + 3) q^{25} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{31} + ( - \beta_{3} + 4 \beta_1 - 7) q^{35} + ( - \beta_{3} - \beta_{2} + 4) q^{37} + q^{41} + (2 \beta_{3} - 2 \beta_1) q^{43} + (\beta_{2} - 3 \beta_1 - 2) q^{47} + (3 \beta_{3} + \beta_{2} - 4 \beta_1 + 5) q^{49} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{53} + (\beta_{3} + 4 \beta_{2} - 4 \beta_1 - 1) q^{55} + (2 \beta_{2} + 2) q^{59} + (2 \beta_1 + 6) q^{61} + (4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{65} + ( - \beta_{3} - \beta_{2} + \beta_1 - 7) q^{67} + ( - 4 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{71} + (3 \beta_{3} - \beta_{2} + 4) q^{73} + ( - \beta_{3} - \beta_{2} + 4 \beta_1 - 2) q^{77} + ( - 3 \beta_{2} + \beta_1 + 6) q^{79} + (2 \beta_{3} - 4 \beta_1 - 2) q^{83} + ( - 2 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 4) q^{85} + ( - 2 \beta_{2} + 4 \beta_1) q^{89} + ( - 6 \beta_{3} + 2 \beta_1 - 12) q^{91} + (3 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 7) q^{95} + ( - 4 \beta_{3} + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 4 q^{11} + 4 q^{17} - 6 q^{19} - 12 q^{23} + 12 q^{25} + 4 q^{29} + 8 q^{31} - 26 q^{35} + 16 q^{37} + 4 q^{41} - 4 q^{43} - 6 q^{47} + 16 q^{49} + 16 q^{53} + 2 q^{55} + 12 q^{59} + 24 q^{61} - 4 q^{65} - 28 q^{67} - 2 q^{71} + 8 q^{73} - 8 q^{77} + 18 q^{79} - 12 q^{83} + 32 q^{85} - 4 q^{89} - 36 q^{91} + 14 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 10x^{2} - 6x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 7\nu - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 3\nu^{2} + 7\nu - 12 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + 7\beta _1 + 3 \) 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
−2.46810
0.707500
3.31526
−1.55466
0 0 0 −3.59669 0 5.06479 0 0 0
1.2 0 0 0 −2.56613 0 0.858626 0 0 0
1.3 0 0 0 −1.17025 0 −3.14501 0 0 0
1.4 0 0 0 3.33307 0 −2.77840 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(41\) \(-1\)

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 5904.2.a.bp 4
3.b odd 2 1 656.2.a.i 4
4.b odd 2 1 1476.2.a.g 4
12.b even 2 1 164.2.a.a 4
24.f even 2 1 2624.2.a.v 4
24.h odd 2 1 2624.2.a.y 4
60.h even 2 1 4100.2.a.c 4
60.l odd 4 2 4100.2.d.c 8
84.h odd 2 1 8036.2.a.i 4
492.d even 2 1 6724.2.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.a.a 4 12.b even 2 1
656.2.a.i 4 3.b odd 2 1
1476.2.a.g 4 4.b odd 2 1
2624.2.a.v 4 24.f even 2 1
2624.2.a.y 4 24.h odd 2 1
4100.2.a.c 4 60.h even 2 1
4100.2.d.c 8 60.l odd 4 2
5904.2.a.bp 4 1.a even 1 1 trivial
6724.2.a.c 4 492.d even 2 1
8036.2.a.i 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5904))\):

\( T_{5}^{4} + 4T_{5}^{3} - 8T_{5}^{2} - 44T_{5} - 36 \) Copy content Toggle raw display
\( T_{7}^{4} - 22T_{7}^{2} - 26T_{7} + 38 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} - 18T_{11}^{2} + 18T_{11} + 54 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} - 8 T^{2} - 44 T - 36 \) Copy content Toggle raw display
$7$ \( T^{4} - 22 T^{2} - 26 T + 38 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} - 18 T^{2} + 18 T + 54 \) Copy content Toggle raw display
$13$ \( T^{4} - 40 T^{2} - 48 T + 144 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} - 48 T^{2} + 80 T + 432 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} - 14 T^{2} - 134 T - 186 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + 16 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} - 40 T^{2} + 144 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} - 32 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$37$ \( T^{4} - 16 T^{3} + 64 T^{2} + \cdots - 324 \) Copy content Toggle raw display
$41$ \( (T - 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} - 48 T^{2} - 272 T - 288 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} - 62 T^{2} + \cdots + 1182 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + 720 T - 1296 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 16 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$61$ \( T^{4} - 24 T^{3} + 176 T^{2} + \cdots + 288 \) Copy content Toggle raw display
$67$ \( T^{4} + 28 T^{3} + 270 T^{2} + \cdots + 1094 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} - 186 T^{2} + \cdots - 426 \) Copy content Toggle raw display
$73$ \( T^{4} - 8 T^{3} - 80 T^{2} + 692 T - 404 \) Copy content Toggle raw display
$79$ \( T^{4} - 18 T^{3} + 50 T^{2} + 42 T - 18 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} - 80 T^{2} + \cdots - 3456 \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} - 128 T^{2} + \cdots + 4272 \) Copy content Toggle raw display
$97$ \( T^{4} - 16 T^{3} - 120 T^{2} + \cdots + 4944 \) Copy content Toggle raw display
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