Properties

Label 2304.3.g.n
Level $2304$
Weight $3$
Character orbit 2304.g
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(1279,2304)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1279");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.g (of order \(2\), degree \(1\), not 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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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 384)
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 \(\beta = 4\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{5} - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{5} - \beta q^{7} - \beta q^{11} + 18 q^{17} + 3 \beta q^{19} + 6 \beta q^{23} - 9 q^{25} - 4 q^{29} + 7 \beta q^{31} - 4 \beta q^{35} + 72 q^{37} + 18 q^{41} + 9 \beta q^{43} - 6 \beta q^{47} + q^{49} + 44 q^{53} - 4 \beta q^{55} - 9 \beta q^{59} - 72 q^{61} + 3 \beta q^{67} - 6 \beta q^{71} + 82 q^{73} - 48 q^{77} - 9 \beta q^{79} - 19 \beta q^{83} + 72 q^{85} + 126 q^{89} + 12 \beta q^{95} + 110 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{5} + 36 q^{17} - 18 q^{25} - 8 q^{29} + 144 q^{37} + 36 q^{41} + 2 q^{49} + 88 q^{53} - 144 q^{61} + 164 q^{73} - 96 q^{77} + 144 q^{85} + 252 q^{89} + 220 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 4.00000 0 6.92820i 0 0 0
1279.2 0 0 0 4.00000 0 6.92820i 0 0 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.g.n 2
3.b odd 2 1 768.3.g.a 2
4.b odd 2 1 inner 2304.3.g.n 2
8.b even 2 1 2304.3.g.g 2
8.d odd 2 1 2304.3.g.g 2
12.b even 2 1 768.3.g.a 2
16.e even 4 2 1152.3.b.h 4
16.f odd 4 2 1152.3.b.h 4
24.f even 2 1 768.3.g.b 2
24.h odd 2 1 768.3.g.b 2
48.i odd 4 2 384.3.b.a 4
48.k even 4 2 384.3.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.a 4 48.i odd 4 2
384.3.b.a 4 48.k even 4 2
768.3.g.a 2 3.b odd 2 1
768.3.g.a 2 12.b even 2 1
768.3.g.b 2 24.f even 2 1
768.3.g.b 2 24.h odd 2 1
1152.3.b.h 4 16.e even 4 2
1152.3.b.h 4 16.f odd 4 2
2304.3.g.g 2 8.b even 2 1
2304.3.g.g 2 8.d odd 2 1
2304.3.g.n 2 1.a even 1 1 trivial
2304.3.g.n 2 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 48 \) Copy content Toggle raw display
\( T_{11}^{2} + 48 \) Copy content Toggle raw display
\( T_{13} \) 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
$5$ \( (T - 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 48 \) Copy content Toggle raw display
$11$ \( T^{2} + 48 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 18)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 432 \) Copy content Toggle raw display
$23$ \( T^{2} + 1728 \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2352 \) Copy content Toggle raw display
$37$ \( (T - 72)^{2} \) Copy content Toggle raw display
$41$ \( (T - 18)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 3888 \) Copy content Toggle raw display
$47$ \( T^{2} + 1728 \) Copy content Toggle raw display
$53$ \( (T - 44)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 3888 \) Copy content Toggle raw display
$61$ \( (T + 72)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 432 \) Copy content Toggle raw display
$71$ \( T^{2} + 1728 \) Copy content Toggle raw display
$73$ \( (T - 82)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 3888 \) Copy content Toggle raw display
$83$ \( T^{2} + 17328 \) Copy content Toggle raw display
$89$ \( (T - 126)^{2} \) Copy content Toggle raw display
$97$ \( (T - 110)^{2} \) Copy content Toggle raw display
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