↑

Properties

Label 3648.2.k.j
Level $3648$
Weight $2$
Character orbit 3648.k
Analytic conductor $29.129$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3648,2,Mod(2431,3648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3648, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3648.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.k (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: \(29.1294266574\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.28906271571968.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + x^{8} + 8x^{2} - 32x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 228)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - \beta_1 q^{5} - \beta_{5} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_1 q^{5} - \beta_{5} q^{7} + q^{9} + (\beta_{8} + \beta_{2}) q^{11} + \beta_{2} q^{13} - \beta_1 q^{15} - \beta_{4} q^{17} + (\beta_{6} + \beta_{3} + \beta_{2} - 1) q^{19} - \beta_{5} q^{21} + (\beta_{9} - \beta_{8} + \cdots - \beta_{3}) q^{23}+ \cdots + (\beta_{8} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} + 10 q^{9} - 6 q^{19} + 10 q^{25} + 10 q^{27} + 12 q^{31} - 6 q^{49} - 6 q^{57} + 28 q^{61} - 28 q^{67} + 16 q^{71} - 12 q^{73} + 10 q^{75} + 4 q^{77} + 10 q^{81} + 20 q^{85} - 16 q^{91} + 12 q^{93} - 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + x^{8} + 8x^{2} - 32x + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{9} - \nu^{7} + 2\nu^{6} + 4\nu^{5} + 4\nu^{4} - 8\nu^{2} - 8\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} - \nu^{7} - 2\nu^{6} - 4\nu^{5} + 8\nu^{4} - 8\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - \nu^{7} + 2\nu^{6} + 4\nu^{5} + 4\nu^{4} + 24\nu^{2} - 8\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} + \nu^{4} - 4\nu^{3} + 2\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + \nu^{7} + 2\nu^{4} - 4\nu^{2} + 8\nu - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{9} + 3\nu^{7} + 2\nu^{6} - 4\nu^{4} - 8\nu^{3} - 8\nu^{2} + 8\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} - \nu^{7} + 8\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{9} + 4\nu^{8} + \nu^{7} + 2\nu^{6} - 4\nu^{4} - 8\nu^{3} - 8\nu^{2} - 40\nu + 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{9} + 4\nu^{8} + 3\nu^{7} - 2\nu^{6} + 4\nu^{5} + 4\nu^{4} - 8\nu^{2} - 40\nu + 112 ) / 16 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{7} + \beta_{6} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + \beta_{3} + 2\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} - 3\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{9} + 2\beta_{8} + 2\beta_{6} + 2\beta_{5} - 2\beta_{4} + 3\beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{9} - \beta_{8} - 2\beta_{7} + 3\beta_{6} + 4\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} - 4\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4\beta_{8} + 4\beta_{7} - 4\beta_{6} + 6\beta_{5} + \beta_{3} - 2\beta_{2} - 3\beta _1 - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -3\beta_{9} + 5\beta_{8} - 2\beta_{7} - 7\beta_{6} + 8\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta _1 + 30 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(1921\) \(2053\) \(2623\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
0.391657 − 1.35890i
0.391657 + 1.35890i
1.39467 + 0.234296i
1.39467 − 0.234296i
−0.610711 + 1.27555i
−0.610711 − 1.27555i
−1.31523 + 0.519787i
−1.31523 − 0.519787i
1.13961 − 0.837430i
1.13961 + 0.837430i
0 1.00000 0 −3.42712 0 − 1.84067i 0 1.00000 0
2431.2 0 1.00000 0 −3.42712 0 1.84067i 0 1.00000 0
2431.3 0 1.00000 0 −1.25558 0 − 4.64552i 0 1.00000 0
2431.4 0 1.00000 0 −1.25558 0 4.64552i 0 1.00000 0
2431.5 0 1.00000 0 −0.399423 0 − 2.63719i 0 1.00000 0
2431.6 0 1.00000 0 −0.399423 0 2.63719i 0 1.00000 0
2431.7 0 1.00000 0 1.19853 0 − 0.204235i 0 1.00000 0
2431.8 0 1.00000 0 1.19853 0 0.204235i 0 1.00000 0
2431.9 0 1.00000 0 3.88360 0 − 2.45653i 0 1.00000 0
2431.10 0 1.00000 0 3.88360 0 2.45653i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2431.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3648.2.k.j 10
4.b odd 2 1 3648.2.k.i 10
8.b even 2 1 228.2.f.b yes 10
8.d odd 2 1 228.2.f.a ✓ 10
19.b odd 2 1 3648.2.k.i 10
24.f even 2 1 684.2.f.d 10
24.h odd 2 1 684.2.f.c 10
76.d even 2 1 inner 3648.2.k.j 10
152.b even 2 1 228.2.f.b yes 10
152.g odd 2 1 228.2.f.a ✓ 10
456.l odd 2 1 684.2.f.c 10
456.p even 2 1 684.2.f.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.f.a ✓ 10 8.d odd 2 1
228.2.f.a ✓ 10 152.g odd 2 1
228.2.f.b yes 10 8.b even 2 1
228.2.f.b yes 10 152.b even 2 1
684.2.f.c 10 24.h odd 2 1
684.2.f.c 10 456.l odd 2 1
684.2.f.d 10 24.f even 2 1
684.2.f.d 10 456.p even 2 1
3648.2.k.i 10 4.b odd 2 1
3648.2.k.i 10 19.b odd 2 1
3648.2.k.j 10 1.a even 1 1 trivial
3648.2.k.j 10 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3648, [\chi])\):

\( T_{5}^{5} - 15T_{5}^{3} - 6T_{5}^{2} + 20T_{5} + 8 \) Copy content Toggle raw display
\( T_{31}^{5} - 6T_{31}^{4} - 56T_{31}^{3} + 208T_{31}^{2} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} - 15 T^{3} - 6 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + 38 T^{8} + \cdots + 128 \) Copy content Toggle raw display
$11$ \( T^{10} + 64 T^{8} + \cdots + 59168 \) Copy content Toggle raw display
$13$ \( (T^{2} + 8)^{5} \) Copy content Toggle raw display
$17$ \( (T^{5} - 47 T^{3} - 98 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 6 T^{9} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( T^{10} + 114 T^{8} + \cdots + 946688 \) Copy content Toggle raw display
$29$ \( T^{10} + 146 T^{8} + \cdots + 1605632 \) Copy content Toggle raw display
$31$ \( (T^{5} - 6 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 176 T^{8} + \cdots + 8388608 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 134217728 \) Copy content Toggle raw display
$43$ \( T^{10} + 222 T^{8} + \cdots + 11214848 \) Copy content Toggle raw display
$47$ \( T^{10} + 192 T^{8} + \cdots + 892448 \) Copy content Toggle raw display
$53$ \( T^{10} + 274 T^{8} + \cdots + 131072 \) Copy content Toggle raw display
$59$ \( (T^{5} - 104 T^{3} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 14 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 14 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 8 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 6 T^{4} + \cdots - 2752)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 176 T^{3} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 306 T^{8} + \cdots + 13603328 \) Copy content Toggle raw display
$89$ \( T^{10} + 258 T^{8} + \cdots + 524288 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 717750272 \) Copy content Toggle raw display
show more
show less