Properties

Label 576.2.d.a
Level $576$
Weight $2$
Character orbit 576.d
Analytic conductor $4.599$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,2,Mod(289,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.d (of order \(2\), degree \(1\), 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: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 64)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta q^{11} + 6 q^{17} + \beta q^{19} + 5 q^{25} + 6 q^{41} + 5 \beta q^{43} - 7 q^{49} + 3 \beta q^{59} - 7 \beta q^{67} + 2 q^{73} - 9 \beta q^{83} - 18 q^{89} + 10 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{17} + 10 q^{25} + 12 q^{41} - 14 q^{49} + 4 q^{73} - 36 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
289.2 0 0 0 0 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.d.a 2
3.b odd 2 1 64.2.b.a 2
4.b odd 2 1 inner 576.2.d.a 2
8.b even 2 1 inner 576.2.d.a 2
8.d odd 2 1 CM 576.2.d.a 2
12.b even 2 1 64.2.b.a 2
15.d odd 2 1 1600.2.d.a 2
15.e even 4 1 1600.2.f.a 2
15.e even 4 1 1600.2.f.b 2
16.e even 4 1 2304.2.a.h 1
16.e even 4 1 2304.2.a.i 1
16.f odd 4 1 2304.2.a.h 1
16.f odd 4 1 2304.2.a.i 1
21.c even 2 1 3136.2.b.b 2
24.f even 2 1 64.2.b.a 2
24.h odd 2 1 64.2.b.a 2
48.i odd 4 1 256.2.a.a 1
48.i odd 4 1 256.2.a.d 1
48.k even 4 1 256.2.a.a 1
48.k even 4 1 256.2.a.d 1
60.h even 2 1 1600.2.d.a 2
60.l odd 4 1 1600.2.f.a 2
60.l odd 4 1 1600.2.f.b 2
84.h odd 2 1 3136.2.b.b 2
96.o even 8 4 1024.2.e.l 4
96.p odd 8 4 1024.2.e.l 4
120.i odd 2 1 1600.2.d.a 2
120.m even 2 1 1600.2.d.a 2
120.q odd 4 1 1600.2.f.a 2
120.q odd 4 1 1600.2.f.b 2
120.w even 4 1 1600.2.f.a 2
120.w even 4 1 1600.2.f.b 2
168.e odd 2 1 3136.2.b.b 2
168.i even 2 1 3136.2.b.b 2
240.t even 4 1 6400.2.a.a 1
240.t even 4 1 6400.2.a.x 1
240.bm odd 4 1 6400.2.a.a 1
240.bm odd 4 1 6400.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 3.b odd 2 1
64.2.b.a 2 12.b even 2 1
64.2.b.a 2 24.f even 2 1
64.2.b.a 2 24.h odd 2 1
256.2.a.a 1 48.i odd 4 1
256.2.a.a 1 48.k even 4 1
256.2.a.d 1 48.i odd 4 1
256.2.a.d 1 48.k even 4 1
576.2.d.a 2 1.a even 1 1 trivial
576.2.d.a 2 4.b odd 2 1 inner
576.2.d.a 2 8.b even 2 1 inner
576.2.d.a 2 8.d odd 2 1 CM
1024.2.e.l 4 96.o even 8 4
1024.2.e.l 4 96.p odd 8 4
1600.2.d.a 2 15.d odd 2 1
1600.2.d.a 2 60.h even 2 1
1600.2.d.a 2 120.i odd 2 1
1600.2.d.a 2 120.m even 2 1
1600.2.f.a 2 15.e even 4 1
1600.2.f.a 2 60.l odd 4 1
1600.2.f.a 2 120.q odd 4 1
1600.2.f.a 2 120.w even 4 1
1600.2.f.b 2 15.e even 4 1
1600.2.f.b 2 60.l odd 4 1
1600.2.f.b 2 120.q odd 4 1
1600.2.f.b 2 120.w even 4 1
2304.2.a.h 1 16.e even 4 1
2304.2.a.h 1 16.f odd 4 1
2304.2.a.i 1 16.e even 4 1
2304.2.a.i 1 16.f odd 4 1
3136.2.b.b 2 21.c even 2 1
3136.2.b.b 2 84.h odd 2 1
3136.2.b.b 2 168.e odd 2 1
3136.2.b.b 2 168.i even 2 1
6400.2.a.a 1 240.t even 4 1
6400.2.a.a 1 240.bm odd 4 1
6400.2.a.x 1 240.t even 4 1
6400.2.a.x 1 240.bm odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 324 \) Copy content Toggle raw display
$89$ \( (T + 18)^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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