Properties

Label 2304.4.a.q
Level $2304$
Weight $4$
Character orbit 2304.a
Self dual yes
Analytic conductor $135.940$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,4,Mod(1,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.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: \(135.940400653\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1152)
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 \(\beta = 8\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 6 q^{5} - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{5} - \beta q^{7} - 2 \beta q^{11} - 20 q^{13} - 8 q^{17} - 4 \beta q^{19} - 8 \beta q^{23} - 89 q^{25} + 46 q^{29} - \beta q^{31} + 6 \beta q^{35} + 164 q^{37} - 312 q^{41} - 20 \beta q^{43} - 8 \beta q^{47} + 105 q^{49} - 266 q^{53} + 12 \beta q^{55} + 12 \beta q^{59} + 132 q^{61} + 120 q^{65} - 24 \beta q^{67} + 32 \beta q^{71} + 246 q^{73} + 896 q^{77} + 11 \beta q^{79} - 46 \beta q^{83} + 48 q^{85} - 1392 q^{89} + 20 \beta q^{91} + 24 \beta q^{95} - 302 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{5} - 40 q^{13} - 16 q^{17} - 178 q^{25} + 92 q^{29} + 328 q^{37} - 624 q^{41} + 210 q^{49} - 532 q^{53} + 264 q^{61} + 240 q^{65} + 492 q^{73} + 1792 q^{77} + 96 q^{85} - 2784 q^{89} - 604 q^{97}+O(q^{100}) \) 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
2.64575
−2.64575
0 0 0 −6.00000 0 −21.1660 0 0 0
1.2 0 0 0 −6.00000 0 21.1660 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.4.a.q 2
3.b odd 2 1 2304.4.a.br 2
4.b odd 2 1 inner 2304.4.a.q 2
8.b even 2 1 2304.4.a.bs 2
8.d odd 2 1 2304.4.a.bs 2
12.b even 2 1 2304.4.a.br 2
16.e even 4 2 1152.4.d.k 4
16.f odd 4 2 1152.4.d.k 4
24.f even 2 1 2304.4.a.r 2
24.h odd 2 1 2304.4.a.r 2
48.i odd 4 2 1152.4.d.m yes 4
48.k even 4 2 1152.4.d.m yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.4.d.k 4 16.e even 4 2
1152.4.d.k 4 16.f odd 4 2
1152.4.d.m yes 4 48.i odd 4 2
1152.4.d.m yes 4 48.k even 4 2
2304.4.a.q 2 1.a even 1 1 trivial
2304.4.a.q 2 4.b odd 2 1 inner
2304.4.a.r 2 24.f even 2 1
2304.4.a.r 2 24.h odd 2 1
2304.4.a.br 2 3.b odd 2 1
2304.4.a.br 2 12.b even 2 1
2304.4.a.bs 2 8.b even 2 1
2304.4.a.bs 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2304))\):

\( T_{5} + 6 \) Copy content Toggle raw display
\( T_{7}^{2} - 448 \) Copy content Toggle raw display
\( T_{11}^{2} - 1792 \) Copy content Toggle raw display
\( T_{13} + 20 \) Copy content Toggle raw display
\( T_{17} + 8 \) Copy content Toggle raw display
\( T_{19}^{2} - 7168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 6)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 448 \) Copy content Toggle raw display
$11$ \( T^{2} - 1792 \) Copy content Toggle raw display
$13$ \( (T + 20)^{2} \) Copy content Toggle raw display
$17$ \( (T + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 7168 \) Copy content Toggle raw display
$23$ \( T^{2} - 28672 \) Copy content Toggle raw display
$29$ \( (T - 46)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 448 \) Copy content Toggle raw display
$37$ \( (T - 164)^{2} \) Copy content Toggle raw display
$41$ \( (T + 312)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 179200 \) Copy content Toggle raw display
$47$ \( T^{2} - 28672 \) Copy content Toggle raw display
$53$ \( (T + 266)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 64512 \) Copy content Toggle raw display
$61$ \( (T - 132)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 258048 \) Copy content Toggle raw display
$71$ \( T^{2} - 458752 \) Copy content Toggle raw display
$73$ \( (T - 246)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 54208 \) Copy content Toggle raw display
$83$ \( T^{2} - 947968 \) Copy content Toggle raw display
$89$ \( (T + 1392)^{2} \) Copy content Toggle raw display
$97$ \( (T + 302)^{2} \) Copy content Toggle raw display
show more
show less