Properties

Label 2304.2.f.b
Level $2304$
Weight $2$
Character orbit 2304.f
Analytic conductor $18.398$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,2,Mod(1151,2304)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.f (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: \(18.3975326257\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1152)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2) q^{5} + (2 \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2) q^{5} + (2 \beta_{2} - \beta_1) q^{7} - 2 \beta_{2} q^{11} + 2 \beta_{2} q^{13} + (\beta_{2} - 2 \beta_1) q^{17} - 4 \beta_{3} q^{19} + (2 \beta_{3} + 4) q^{23} + ( - 4 \beta_{3} + 1) q^{25} + (\beta_{3} + 2) q^{29} + ( - 2 \beta_{2} - 3 \beta_1) q^{31} + ( - 6 \beta_{2} + 4 \beta_1) q^{35} + ( - 4 \beta_{2} - \beta_1) q^{37} + (\beta_{2} + 2 \beta_1) q^{41} + (4 \beta_{3} - 4) q^{43} + (6 \beta_{3} - 4) q^{47} + (8 \beta_{3} - 5) q^{49} + ( - 5 \beta_{3} - 2) q^{53} + (4 \beta_{2} - 2 \beta_1) q^{55} + ( - 4 \beta_{2} - 4 \beta_1) q^{59} + (4 \beta_{2} - \beta_1) q^{61} + ( - 4 \beta_{2} + 2 \beta_1) q^{65} - 12 q^{67} + (6 \beta_{3} + 4) q^{71} - 4 q^{73} + ( - 4 \beta_{3} + 8) q^{77} + ( - 6 \beta_{2} - \beta_1) q^{79} + ( - 2 \beta_{2} - 4 \beta_1) q^{83} + ( - 6 \beta_{2} + 5 \beta_1) q^{85} + ( - 3 \beta_{2} + 4 \beta_1) q^{89} + (4 \beta_{3} - 8) q^{91} + (8 \beta_{3} - 8) q^{95} + ( - 4 \beta_{3} + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{5} + 16 q^{23} + 4 q^{25} + 8 q^{29} - 16 q^{43} - 16 q^{47} - 20 q^{49} - 8 q^{53} - 48 q^{67} + 16 q^{71} - 16 q^{73} + 32 q^{77} - 32 q^{91} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) 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
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) 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 \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 −3.41421 0 4.82843i 0 0 0
1151.2 0 0 0 −3.41421 0 4.82843i 0 0 0
1151.3 0 0 0 −0.585786 0 0.828427i 0 0 0
1151.4 0 0 0 −0.585786 0 0.828427i 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
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.f.b 4
3.b odd 2 1 2304.2.f.g 4
4.b odd 2 1 2304.2.f.a 4
8.b even 2 1 2304.2.f.h 4
8.d odd 2 1 2304.2.f.g 4
12.b even 2 1 2304.2.f.h 4
16.e even 4 1 1152.2.c.a 4
16.e even 4 1 1152.2.c.b yes 4
16.f odd 4 1 1152.2.c.c yes 4
16.f odd 4 1 1152.2.c.d yes 4
24.f even 2 1 inner 2304.2.f.b 4
24.h odd 2 1 2304.2.f.a 4
48.i odd 4 1 1152.2.c.c yes 4
48.i odd 4 1 1152.2.c.d yes 4
48.k even 4 1 1152.2.c.a 4
48.k even 4 1 1152.2.c.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.c.a 4 16.e even 4 1
1152.2.c.a 4 48.k even 4 1
1152.2.c.b yes 4 16.e even 4 1
1152.2.c.b yes 4 48.k even 4 1
1152.2.c.c yes 4 16.f odd 4 1
1152.2.c.c yes 4 48.i odd 4 1
1152.2.c.d yes 4 16.f odd 4 1
1152.2.c.d yes 4 48.i odd 4 1
2304.2.f.a 4 4.b odd 2 1
2304.2.f.a 4 24.h odd 2 1
2304.2.f.b 4 1.a even 1 1 trivial
2304.2.f.b 4 24.f even 2 1 inner
2304.2.f.g 4 3.b odd 2 1
2304.2.f.g 4 8.d odd 2 1
2304.2.f.h 4 8.b even 2 1
2304.2.f.h 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 4T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{4} + 24T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{23}^{2} - 8T_{23} + 8 \) Copy content Toggle raw display
\( T_{43}^{2} + 8T_{43} - 16 \) 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
$5$ \( (T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 36T^{2} + 196 \) Copy content Toggle raw display
$19$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$37$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$41$ \( T^{4} + 36T^{2} + 196 \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8 T - 56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4 T - 46)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 192T^{2} + 1024 \) Copy content Toggle raw display
$61$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$67$ \( (T + 12)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 152T^{2} + 4624 \) Copy content Toggle raw display
$83$ \( T^{4} + 144T^{2} + 3136 \) Copy content Toggle raw display
$89$ \( T^{4} + 164T^{2} + 2116 \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
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