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Properties

Label 1584.2.d.d
Level $1584$
Weight $2$
Character orbit 1584.d
Analytic conductor $12.648$
Analytic rank $0$
Dimension $8$
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] = [1584,2,Mod(287,1584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1584, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1584.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1584.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: \(12.6483036802\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1768034304.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 9x^{6} - 2x^{5} + 34x^{4} - 18x^{3} + 51x^{2} + 18x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + \beta_{4} q^{7} + q^{11} + (\beta_{3} - 1) q^{13} - \beta_{6} q^{17} + ( - \beta_{7} + \beta_{6} - \beta_{5}) q^{19} + (\beta_1 - 2) q^{23} + (\beta_{3} + \beta_1 - 2) q^{25} - \beta_{5} q^{29} + ( - \beta_{6} + \beta_{5}) q^{31} + ( - \beta_{3} - \beta_1 + 7) q^{35} + (\beta_{2} + 1) q^{37} + (\beta_{6} - \beta_{5}) q^{41} + ( - \beta_{7} - \beta_{6} + \cdots - 2 \beta_{4}) q^{43}+ \cdots + (\beta_{3} + \beta_1 - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{11} - 8 q^{13} - 16 q^{23} - 16 q^{25} + 56 q^{35} + 8 q^{37} + 16 q^{47} + 16 q^{59} + 24 q^{61} - 16 q^{71} + 40 q^{83} + 16 q^{85} + 8 q^{95} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 9x^{6} - 2x^{5} + 34x^{4} - 18x^{3} + 51x^{2} + 18x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} + \nu^{6} - 18\nu^{5} - 30\nu^{4} - 110\nu^{3} - 72\nu^{2} - 27\nu - 141 ) / 30 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + \nu^{6} - 12\nu^{5} - 20\nu^{4} - 56\nu^{3} - 48\nu^{2} - 18\nu - 60 ) / 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} + 6\nu^{5} + 10\nu^{4} + 22\nu^{3} + 24\nu^{2} + 9\nu + 79 ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} - 10\nu^{6} + 42\nu^{5} - 26\nu^{4} + 143\nu^{3} - 102\nu^{2} + 222\nu + 27 ) / 45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8\nu^{7} - 18\nu^{6} + 72\nu^{5} - 28\nu^{4} + 252\nu^{3} - 252\nu^{2} + 360\nu + 36 ) / 45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{7} + 20\nu^{6} - 90\nu^{5} + 42\nu^{4} - 340\nu^{3} + 180\nu^{2} - 633\nu - 90 ) / 45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 17\nu^{6} - 66\nu^{5} + 38\nu^{4} - 220\nu^{3} + 216\nu^{2} - 291\nu - 27 ) / 30 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{6} - 6\beta_{4} - \beta_{2} + 2\beta _1 + 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{7} - 3\beta_{6} - 9\beta_{5} - 6\beta_{4} + 3\beta_{3} + 2\beta_{2} - \beta _1 - 21 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 3\beta_{2} - 3\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 54\beta_{7} + 21\beta_{6} + 54\beta_{5} + 78\beta_{4} + 21\beta_{3} + 22\beta_{2} - 11\beta _1 - 129 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 132\beta_{7} + 75\beta_{6} + 99\beta_{5} + 312\beta_{4} - 39\beta_{3} - 74\beta_{2} + 55\beta _1 + 273 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -26\beta_{3} - 33\beta_{2} + 18\beta _1 + 158 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1188\beta_{7} - 561\beta_{6} - 918\beta_{5} - 2454\beta_{4} - 372\beta_{3} - 607\beta_{2} + 398\beta _1 + 2397 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(353\) \(991\) \(1189\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−0.943680 − 1.63450i
1.44368 − 2.50053i
−0.197356 + 0.341831i
0.697356 + 1.20786i
0.697356 − 1.20786i
−0.197356 − 0.341831i
1.44368 + 2.50053i
−0.943680 + 1.63450i
0 0 0 − 4.08334i 0 4.08334i 0 0 0
287.2 0 0 0 − 2.66913i 0 2.66913i 0 0 0
287.3 0 0 0 − 1.97242i 0 1.97242i 0 0 0
287.4 0 0 0 − 0.558208i 0 0.558208i 0 0 0
287.5 0 0 0 0.558208i 0 − 0.558208i 0 0 0
287.6 0 0 0 1.97242i 0 − 1.97242i 0 0 0
287.7 0 0 0 2.66913i 0 − 2.66913i 0 0 0
287.8 0 0 0 4.08334i 0 − 4.08334i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.2.d.d yes 8
3.b odd 2 1 1584.2.d.c ✓ 8
4.b odd 2 1 1584.2.d.c ✓ 8
8.b even 2 1 6336.2.d.c 8
8.d odd 2 1 6336.2.d.e 8
12.b even 2 1 inner 1584.2.d.d yes 8
24.f even 2 1 6336.2.d.c 8
24.h odd 2 1 6336.2.d.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1584.2.d.c ✓ 8 3.b odd 2 1
1584.2.d.c ✓ 8 4.b odd 2 1
1584.2.d.d yes 8 1.a even 1 1 trivial
1584.2.d.d yes 8 12.b even 2 1 inner
6336.2.d.c 8 8.b even 2 1
6336.2.d.c 8 24.f even 2 1
6336.2.d.e 8 8.d odd 2 1
6336.2.d.e 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1584, [\chi])\):

\( T_{5}^{8} + 28T_{5}^{6} + 220T_{5}^{4} + 528T_{5}^{2} + 144 \) Copy content Toggle raw display
\( T_{23}^{4} + 8T_{23}^{3} - 26T_{23}^{2} - 216T_{23} + 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 28 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( (T - 1)^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} + \cdots - 180)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 88 T^{6} + \cdots + 57600 \) Copy content Toggle raw display
$19$ \( T^{8} + 112 T^{6} + \cdots + 318096 \) Copy content Toggle raw display
$23$ \( (T^{4} + 8 T^{3} + \cdots + 108)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 88 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + \cdots + 1200)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 88 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$43$ \( T^{8} + 240 T^{6} + \cdots + 2624400 \) Copy content Toggle raw display
$47$ \( (T^{4} - 8 T^{3} + \cdots + 5868)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 220 T^{6} + \cdots + 3027600 \) Copy content Toggle raw display
$59$ \( (T^{4} - 8 T^{3} + \cdots - 2880)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 12 T^{3} + \cdots - 20)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 168 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8 T^{3} + \cdots + 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 152 T^{2} + \cdots - 752)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 348 T^{6} + \cdots + 944784 \) Copy content Toggle raw display
$83$ \( (T^{4} - 20 T^{3} + \cdots - 2160)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 328 T^{6} + \cdots + 266256 \) Copy content Toggle raw display
$97$ \( (T^{4} + 12 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
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