Properties

Label 576.3.e.c
Level $576$
Weight $3$
Character orbit 576.e
Analytic conductor $15.695$
Analytic rank $0$
Dimension $2$
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] = [576,3,Mod(449,576)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.e (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: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 18)
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 \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} - 4 q^{7} - 4 \beta q^{11} - 8 q^{13} - 3 \beta q^{17} + 16 q^{19} - 4 \beta q^{23} + 7 q^{25} - \beta q^{29} + 44 q^{31} - 4 \beta q^{35} + 34 q^{37} + 11 \beta q^{41} + 40 q^{43} - 20 \beta q^{47} - 33 q^{49} - 9 \beta q^{53} + 72 q^{55} - 8 \beta q^{59} - 50 q^{61} - 8 \beta q^{65} - 8 q^{67} - 12 \beta q^{71} - 16 q^{73} + 16 \beta q^{77} - 76 q^{79} - 28 \beta q^{83} + 54 q^{85} + 3 \beta q^{89} + 32 q^{91} + 16 \beta q^{95} + 176 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} - 16 q^{13} + 32 q^{19} + 14 q^{25} + 88 q^{31} + 68 q^{37} + 80 q^{43} - 66 q^{49} + 144 q^{55} - 100 q^{61} - 16 q^{67} - 32 q^{73} - 152 q^{79} + 108 q^{85} + 64 q^{91} + 352 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.41421i
1.41421i
0 0 0 4.24264i 0 −4.00000 0 0 0
449.2 0 0 0 4.24264i 0 −4.00000 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.e.c 2
3.b odd 2 1 inner 576.3.e.c 2
4.b odd 2 1 576.3.e.f 2
8.b even 2 1 18.3.b.a 2
8.d odd 2 1 144.3.e.b 2
12.b even 2 1 576.3.e.f 2
16.e even 4 2 2304.3.h.f 4
16.f odd 4 2 2304.3.h.c 4
24.f even 2 1 144.3.e.b 2
24.h odd 2 1 18.3.b.a 2
40.e odd 2 1 3600.3.l.d 2
40.f even 2 1 450.3.d.f 2
40.i odd 4 2 450.3.b.b 4
40.k even 4 2 3600.3.c.b 4
48.i odd 4 2 2304.3.h.f 4
48.k even 4 2 2304.3.h.c 4
56.h odd 2 1 882.3.b.a 2
56.j odd 6 2 882.3.s.d 4
56.p even 6 2 882.3.s.b 4
72.j odd 6 2 162.3.d.b 4
72.l even 6 2 1296.3.q.f 4
72.n even 6 2 162.3.d.b 4
72.p odd 6 2 1296.3.q.f 4
88.b odd 2 1 2178.3.c.d 2
104.e even 2 1 3042.3.c.e 2
104.j odd 4 2 3042.3.d.a 4
120.i odd 2 1 450.3.d.f 2
120.m even 2 1 3600.3.l.d 2
120.q odd 4 2 3600.3.c.b 4
120.w even 4 2 450.3.b.b 4
168.i even 2 1 882.3.b.a 2
168.s odd 6 2 882.3.s.b 4
168.ba even 6 2 882.3.s.d 4
264.m even 2 1 2178.3.c.d 2
312.b odd 2 1 3042.3.c.e 2
312.y even 4 2 3042.3.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 8.b even 2 1
18.3.b.a 2 24.h odd 2 1
144.3.e.b 2 8.d odd 2 1
144.3.e.b 2 24.f even 2 1
162.3.d.b 4 72.j odd 6 2
162.3.d.b 4 72.n even 6 2
450.3.b.b 4 40.i odd 4 2
450.3.b.b 4 120.w even 4 2
450.3.d.f 2 40.f even 2 1
450.3.d.f 2 120.i odd 2 1
576.3.e.c 2 1.a even 1 1 trivial
576.3.e.c 2 3.b odd 2 1 inner
576.3.e.f 2 4.b odd 2 1
576.3.e.f 2 12.b even 2 1
882.3.b.a 2 56.h odd 2 1
882.3.b.a 2 168.i even 2 1
882.3.s.b 4 56.p even 6 2
882.3.s.b 4 168.s odd 6 2
882.3.s.d 4 56.j odd 6 2
882.3.s.d 4 168.ba even 6 2
1296.3.q.f 4 72.l even 6 2
1296.3.q.f 4 72.p odd 6 2
2178.3.c.d 2 88.b odd 2 1
2178.3.c.d 2 264.m even 2 1
2304.3.h.c 4 16.f odd 4 2
2304.3.h.c 4 48.k even 4 2
2304.3.h.f 4 16.e even 4 2
2304.3.h.f 4 48.i odd 4 2
3042.3.c.e 2 104.e even 2 1
3042.3.c.e 2 312.b odd 2 1
3042.3.d.a 4 104.j odd 4 2
3042.3.d.a 4 312.y even 4 2
3600.3.c.b 4 40.k even 4 2
3600.3.c.b 4 120.q odd 4 2
3600.3.l.d 2 40.e odd 2 1
3600.3.l.d 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} + 18 \) Copy content Toggle raw display
\( T_{7} + 4 \) 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
$5$ \( T^{2} + 18 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 288 \) Copy content Toggle raw display
$13$ \( (T + 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 162 \) Copy content Toggle raw display
$19$ \( (T - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 288 \) Copy content Toggle raw display
$29$ \( T^{2} + 18 \) Copy content Toggle raw display
$31$ \( (T - 44)^{2} \) Copy content Toggle raw display
$37$ \( (T - 34)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2178 \) Copy content Toggle raw display
$43$ \( (T - 40)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 7200 \) Copy content Toggle raw display
$53$ \( T^{2} + 1458 \) Copy content Toggle raw display
$59$ \( T^{2} + 1152 \) Copy content Toggle raw display
$61$ \( (T + 50)^{2} \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 2592 \) Copy content Toggle raw display
$73$ \( (T + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T + 76)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14112 \) Copy content Toggle raw display
$89$ \( T^{2} + 162 \) Copy content Toggle raw display
$97$ \( (T - 176)^{2} \) Copy content Toggle raw display
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