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Properties

Label 1224.2.w.j
Level $1224$
Weight $2$
Character orbit 1224.w
Analytic conductor $9.774$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,2,Mod(217,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 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("1224.217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1224.w (of order \(4\), degree \(2\), 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: \(9.77368920740\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.269485056.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{3} + 81x^{2} - 72x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + (\beta_{4} - 1) q^{7} + ( - 2 \beta_{4} - \beta_{2} + 2) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{13} + (\beta_{3} + \beta_1 - 1) q^{17} + (\beta_{5} - 2 \beta_{4} + \cdots + \beta_1) q^{19}+ \cdots + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{7} + 12 q^{11} - 6 q^{17} - 6 q^{23} + 6 q^{31} + 12 q^{37} - 6 q^{41} - 12 q^{47} + 36 q^{55} + 12 q^{61} + 24 q^{65} - 24 q^{67} + 18 q^{71} - 18 q^{73} - 18 q^{79} + 12 q^{85} - 72 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 8x^{3} + 81x^{2} - 72x + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{5} - 4\nu^{4} - 81\nu^{3} + 36\nu^{2} - 713\nu + 648 ) / 713 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{5} + 85\nu^{4} + 117\nu^{3} - 52\nu^{2} + 3342 ) / 713 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 81\nu^{5} + 36\nu^{4} + 16\nu^{3} - 324\nu^{2} + 6417\nu - 2980 ) / 2852 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -225\nu^{5} - 100\nu^{4} + 114\nu^{3} + 2326\nu^{2} - 16399\nu + 7644 ) / 1426 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 6\beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{4} - 9\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{3} + 13\beta_{2} + 13\beta _1 - 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} + 60\beta_{4} - 4\beta_{3} - 89\beta _1 + 60 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
1.84885 + 1.84885i
0.467090 + 0.467090i
−2.31594 − 2.31594i
1.84885 − 1.84885i
0.467090 − 0.467090i
−2.31594 + 2.31594i
0 0 0 −1.84885 − 1.84885i 0 −1.00000 + 1.00000i 0 0 0
217.2 0 0 0 −0.467090 − 0.467090i 0 −1.00000 + 1.00000i 0 0 0
217.3 0 0 0 2.31594 + 2.31594i 0 −1.00000 + 1.00000i 0 0 0
361.1 0 0 0 −1.84885 + 1.84885i 0 −1.00000 − 1.00000i 0 0 0
361.2 0 0 0 −0.467090 + 0.467090i 0 −1.00000 − 1.00000i 0 0 0
361.3 0 0 0 2.31594 − 2.31594i 0 −1.00000 − 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.2.w.j yes 6
3.b odd 2 1 1224.2.w.i ✓ 6
4.b odd 2 1 2448.2.be.v 6
12.b even 2 1 2448.2.be.w 6
17.c even 4 1 inner 1224.2.w.j yes 6
51.f odd 4 1 1224.2.w.i ✓ 6
68.f odd 4 1 2448.2.be.v 6
204.l even 4 1 2448.2.be.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.2.w.i ✓ 6 3.b odd 2 1
1224.2.w.i ✓ 6 51.f odd 4 1
1224.2.w.j yes 6 1.a even 1 1 trivial
1224.2.w.j yes 6 17.c even 4 1 inner
2448.2.be.v 6 4.b odd 2 1
2448.2.be.v 6 68.f odd 4 1
2448.2.be.w 6 12.b even 2 1
2448.2.be.w 6 204.l even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{6} + 8T_{5}^{3} + 81T_{5}^{2} + 72T_{5} + 32 \) Copy content Toggle raw display
\( T_{11}^{6} - 12T_{11}^{5} + 72T_{11}^{4} - 176T_{11}^{3} + 225T_{11}^{2} - 60T_{11} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 8 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 2)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} - 12 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( (T^{3} - 27 T + 22)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + \cdots + 4913 \) Copy content Toggle raw display
$19$ \( T^{6} + 114 T^{4} + \cdots + 53824 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 1682 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 6 T^{5} + \cdots + 18432 \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + \cdots + 36992 \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$43$ \( T^{6} + 114 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$47$ \( (T^{3} + 6 T^{2} + \cdots - 144)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 180 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$59$ \( T^{6} + 84 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{6} - 12 T^{5} + \cdots + 430592 \) Copy content Toggle raw display
$67$ \( (T^{3} + 12 T^{2} + \cdots - 192)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + \cdots + 1002528 \) Copy content Toggle raw display
$73$ \( T^{6} + 18 T^{5} + \cdots + 557568 \) Copy content Toggle raw display
$79$ \( T^{6} + 18 T^{5} + \cdots + 29768 \) Copy content Toggle raw display
$83$ \( T^{6} + 192 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$89$ \( (T^{3} + 36 T^{2} + \cdots + 1496)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 6 T^{5} + \cdots + 5484672 \) Copy content Toggle raw display
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