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Properties

Label 6048.2.c.d
Level $6048$
Weight $2$
Character orbit 6048.c
Analytic conductor $48.294$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6048,2,Mod(3025,6048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6048, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6048.3025");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6048.c (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: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{12} + \beta_{11}) q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} + \beta_{11}) q^{5} + q^{7} - \beta_{12} q^{11} + (\beta_{8} + \beta_{2}) q^{13} + (2 \beta_{6} + \beta_{3}) q^{17} - \beta_{8} q^{19} + ( - \beta_{15} - \beta_{6} - 2 \beta_{3}) q^{23} + (\beta_{9} + \beta_{7} + \beta_{4} - 2) q^{25} + ( - \beta_{14} - \beta_{12} + \cdots + 3 \beta_{10}) q^{29}+ \cdots + (5 \beta_{4} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{7} - 32 q^{25} - 40 q^{31} + 16 q^{49} + 72 q^{55} + 24 q^{73} - 24 q^{79} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{40}^{10} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{40}^{14} + \zeta_{40}^{12} + \zeta_{40}^{8} + 3\zeta_{40}^{6} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{40}^{15} + \zeta_{40}^{5} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{40}^{12} - 2\zeta_{40}^{8} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{40}^{8} + 2\zeta_{40}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{40}^{9} - \zeta_{40}^{7} - \zeta_{40}^{5} + \zeta_{40}^{3} + \zeta_{40} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\zeta_{40}^{14} - \zeta_{40}^{12} + 3\zeta_{40}^{10} + \zeta_{40}^{8} - 3\zeta_{40}^{6} + 6\zeta_{40}^{2} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -2\zeta_{40}^{12} - 2\zeta_{40}^{4} + 1 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -3\zeta_{40}^{14} + \zeta_{40}^{12} - \zeta_{40}^{8} + 3\zeta_{40}^{6} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( \zeta_{40}^{15} - 2\zeta_{40}^{11} - \zeta_{40}^{9} + \zeta_{40}^{7} - \zeta_{40}^{3} + \zeta_{40} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -2\zeta_{40}^{13} + \zeta_{40}^{9} - \zeta_{40}^{7} - \zeta_{40}^{5} - \zeta_{40}^{3} + \zeta_{40} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( 2\zeta_{40}^{15} + \zeta_{40}^{13} - \zeta_{40}^{11} - \zeta_{40}^{9} + \zeta_{40}^{7} + 2\zeta_{40}^{5} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( -2\zeta_{40}^{15} - 3\zeta_{40}^{13} + 3\zeta_{40}^{11} - \zeta_{40}^{5} + 3\zeta_{40}^{3} + 3\zeta_{40} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( 2\zeta_{40}^{15} + \zeta_{40}^{13} - \zeta_{40}^{11} + 2\zeta_{40}^{9} + 4\zeta_{40}^{7} - \zeta_{40}^{5} - 3\zeta_{40}^{3} + 3\zeta_{40} \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( -2\zeta_{40}^{15} + 3\zeta_{40}^{13} + 3\zeta_{40}^{11} - 3\zeta_{40}^{9} - 3\zeta_{40}^{7} + 2\zeta_{40}^{5} \) Copy content Toggle raw display

Display \(\zeta_{40}^j\) in terms of \(\beta_i\)

\(\zeta_{40}\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + 2\beta_{10} + 3\beta_{6} + 2\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\zeta_{40}^{2}\)\(=\) \( ( \beta_{8} + 2\beta_{7} + \beta_{5} + \beta_{4} + 2\beta_{2} - 2\beta _1 + 1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{40}^{3}\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - 3\beta_{11} + 3\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\zeta_{40}^{4}\)\(=\) \( ( -\beta_{8} + \beta_{5} - \beta_{4} + 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{40}^{5}\)\(=\) \( ( 2\beta_{12} + \beta_{11} - \beta_{10} + 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\zeta_{40}^{6}\)\(=\) \( ( 2\beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} + 2\beta_{2} - 1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{40}^{7}\)\(=\) \( ( -\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - 3\beta_{11} - 3\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\zeta_{40}^{8}\)\(=\) \( ( -\beta_{8} - \beta_{5} - \beta_{4} - 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{40}^{9}\)\(=\) \( ( -\beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - 4\beta_{10} + 3\beta_{6} + 4\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\zeta_{40}^{10}\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{40}^{11}\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - 4\beta_{10} - 3\beta_{6} - 4\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\zeta_{40}^{12}\)\(=\) \( ( -\beta_{8} - \beta_{5} + \beta_{4} + 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{40}^{13}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - 3\beta_{11} + 3\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\zeta_{40}^{14}\)\(=\) \( ( -2\beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} + 2\beta_{2} + 1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{40}^{15}\)\(=\) \( ( 2\beta_{12} + \beta_{11} - \beta_{10} - 3\beta_{3} ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{40}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3025.1
0.156434 − 0.987688i
−0.156434 − 0.987688i
−0.891007 − 0.453990i
0.891007 − 0.453990i
−0.453990 + 0.891007i
0.453990 + 0.891007i
−0.987688 + 0.156434i
0.987688 + 0.156434i
0.987688 − 0.156434i
−0.987688 − 0.156434i
0.453990 − 0.891007i
−0.453990 − 0.891007i
0.891007 + 0.453990i
−0.891007 + 0.453990i
−0.156434 + 0.987688i
0.156434 + 0.987688i
0 0 0 − 4.29757i 0 1.00000 0 0 0
3025.2 0 0 0 − 4.29757i 0 1.00000 0 0 0
3025.3 0 0 0 − 2.63506i 0 1.00000 0 0 0
3025.4 0 0 0 − 2.63506i 0 1.00000 0 0 0
3025.5 0 0 0 − 1.60758i 0 1.00000 0 0 0
3025.6 0 0 0 − 1.60758i 0 1.00000 0 0 0
3025.7 0 0 0 − 0.0549306i 0 1.00000 0 0 0
3025.8 0 0 0 − 0.0549306i 0 1.00000 0 0 0
3025.9 0 0 0 0.0549306i 0 1.00000 0 0 0
3025.10 0 0 0 0.0549306i 0 1.00000 0 0 0
3025.11 0 0 0 1.60758i 0 1.00000 0 0 0
3025.12 0 0 0 1.60758i 0 1.00000 0 0 0
3025.13 0 0 0 2.63506i 0 1.00000 0 0 0
3025.14 0 0 0 2.63506i 0 1.00000 0 0 0
3025.15 0 0 0 4.29757i 0 1.00000 0 0 0
3025.16 0 0 0 4.29757i 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3025.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6048.2.c.d 16
3.b odd 2 1 inner 6048.2.c.d 16
4.b odd 2 1 1512.2.c.d ✓ 16
8.b even 2 1 inner 6048.2.c.d 16
8.d odd 2 1 1512.2.c.d ✓ 16
12.b even 2 1 1512.2.c.d ✓ 16
24.f even 2 1 1512.2.c.d ✓ 16
24.h odd 2 1 inner 6048.2.c.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.d ✓ 16 4.b odd 2 1
1512.2.c.d ✓ 16 8.d odd 2 1
1512.2.c.d ✓ 16 12.b even 2 1
1512.2.c.d ✓ 16 24.f even 2 1
6048.2.c.d 16 1.a even 1 1 trivial
6048.2.c.d 16 3.b odd 2 1 inner
6048.2.c.d 16 8.b even 2 1 inner
6048.2.c.d 16 24.h 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 \(S_{2}^{\mathrm{new}}(6048, [\chi])\):

\( T_{5}^{8} + 28T_{5}^{6} + 194T_{5}^{4} + 332T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 28 T^{6} + 194 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 28 T^{6} + 194 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 64 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 10)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 10 T^{2} + 5)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 132 T^{6} + \cdots + 491401)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 188 T^{6} + \cdots + 633616)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{3} + \cdots + 205)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 104 T^{6} + \cdots + 361)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 108 T^{6} + \cdots + 78961)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 376 T^{6} + \cdots + 55413136)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 252 T^{6} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 208 T^{6} + \cdots + 952576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 92 T^{6} + \cdots + 26896)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 184 T^{6} + \cdots + 430336)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 304 T^{6} + \cdots + 524176)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 172 T^{6} + \cdots + 1256641)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + \cdots - 324)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} + \cdots + 6156)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 412 T^{6} + \cdots + 75968656)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 348 T^{6} + \cdots + 25816561)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 2 T - 124)^{8} \) Copy content Toggle raw display
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