Properties

Label 2448.4.a.bo
Level $2448$
Weight $4$
Character orbit 2448.a
Self dual yes
Analytic conductor $144.437$
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] = [2448,4,Mod(1,2448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2448.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2448.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: \(144.436675694\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1506848.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 17x^{2} + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 153)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} + \beta_1 - 5) q^{5} + (\beta_{2} + 3 \beta_1 + 6) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} + \beta_1 - 5) q^{5} + (\beta_{2} + 3 \beta_1 + 6) q^{7} + (5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 15) q^{11} + (5 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 9) q^{13} + 17 q^{17} + (7 \beta_{3} - 3 \beta_{2} - 5) q^{19} + ( - \beta_{3} - 4 \beta_{2} + \cdots + 95) q^{23}+ \cdots + ( - 138 \beta_{3} - 19 \beta_{2} + \cdots - 88) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 22 q^{5} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 22 q^{5} + 24 q^{7} + 50 q^{11} + 26 q^{13} + 68 q^{17} - 34 q^{19} + 382 q^{23} + 138 q^{25} - 540 q^{29} + 356 q^{31} + 304 q^{35} - 404 q^{37} + 114 q^{41} - 570 q^{43} + 496 q^{47} - 224 q^{49} - 92 q^{53} + 482 q^{55} - 48 q^{59} - 1036 q^{61} + 342 q^{65} - 812 q^{67} + 1044 q^{71} - 1212 q^{73} + 564 q^{77} - 488 q^{79} + 1708 q^{83} - 374 q^{85} + 8 q^{89} - 716 q^{91} + 1010 q^{95} - 76 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 17x^{2} + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 34\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{2} + 34\beta_1 ) / 4 \) 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
−1.06515
−3.98315
1.06515
3.98315
0 0 0 −20.6288 0 5.24213 0 0 0
1.2 0 0 0 −7.60718 0 −16.3925 0 0 0
1.3 0 0 0 −5.10214 0 6.75787 0 0 0
1.4 0 0 0 11.3381 0 28.3925 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(17\) \( -1 \)

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 2448.4.a.bo 4
3.b odd 2 1 2448.4.a.bs 4
4.b odd 2 1 153.4.a.h 4
12.b even 2 1 153.4.a.i yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.4.a.h 4 4.b odd 2 1
153.4.a.i yes 4 12.b even 2 1
2448.4.a.bo 4 1.a even 1 1 trivial
2448.4.a.bs 4 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2448))\):

\( T_{5}^{4} + 22T_{5}^{3} - 77T_{5}^{2} - 2612T_{5} - 9078 \) Copy content Toggle raw display
\( T_{7}^{4} - 24T_{7}^{3} - 286T_{7}^{2} + 5160T_{7} - 16488 \) 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
$5$ \( T^{4} + 22 T^{3} + \cdots - 9078 \) Copy content Toggle raw display
$7$ \( T^{4} - 24 T^{3} + \cdots - 16488 \) Copy content Toggle raw display
$11$ \( T^{4} - 50 T^{3} + \cdots - 669528 \) Copy content Toggle raw display
$13$ \( T^{4} - 26 T^{3} + \cdots + 1030444 \) Copy content Toggle raw display
$17$ \( (T - 17)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 34 T^{3} + \cdots + 4368204 \) Copy content Toggle raw display
$23$ \( T^{4} - 382 T^{3} + \cdots + 49041294 \) Copy content Toggle raw display
$29$ \( T^{4} + 540 T^{3} + \cdots - 232426584 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 2282423072 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 6443365976 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 9365221188 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 6091569648 \) Copy content Toggle raw display
$47$ \( T^{4} - 496 T^{3} + \cdots - 153753984 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 2914187328 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 23071976496 \) Copy content Toggle raw display
$61$ \( T^{4} + 1036 T^{3} + \cdots + 531291272 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 1798725056 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 82941464832 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 188519703168 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 23511973896 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 13135194432 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 113358314448 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 896966722688 \) Copy content Toggle raw display
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