Properties

Label 1584.2.o.g
Level $1584$
Weight $2$
Character orbit 1584.o
Analytic conductor $12.648$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1584,2,Mod(703,1584)]
 
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, 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("1584.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1584.o (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: 8.0.454201344.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 528)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{5} + \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{5} + \beta_{6} q^{7} + ( - \beta_{5} + \beta_{3}) q^{11} - \beta_{4} q^{13} - \beta_{7} q^{17} - 2 \beta_{3} q^{19} + ( - \beta_{5} + 2 \beta_{2}) q^{23} + (2 \beta_1 - 1) q^{25} + (\beta_{5} - \beta_{2}) q^{31} + ( - 2 \beta_{6} - 2 \beta_{3}) q^{35} + ( - 2 \beta_1 + 4) q^{37} - 2 \beta_{4} q^{41} - 2 \beta_{3} q^{43} + ( - \beta_{5} - 2 \beta_{2}) q^{47} + (6 \beta_1 + 9) q^{49} + (\beta_1 + 9) q^{53} + ( - \beta_{6} + 3 \beta_{5} - \beta_{2}) q^{55} + ( - 4 \beta_{5} + 2 \beta_{2}) q^{59} + ( - 2 \beta_{7} - \beta_{4}) q^{61} + ( - \beta_{7} + 2 \beta_{4}) q^{65} + ( - 3 \beta_{5} - 3 \beta_{2}) q^{67} + (5 \beta_{5} - 4 \beta_{2}) q^{71} - 2 \beta_{4} q^{73} + (\beta_{7} - 2 \beta_{4} + 5 \beta_1 + 1) q^{77} - \beta_{6} q^{79} + 2 \beta_{6} q^{83} - 2 \beta_{4} q^{85} + 6 \beta_1 q^{89} + ( - 11 \beta_{5} + 5 \beta_{2}) q^{91} + 2 \beta_{6} q^{95} + (2 \beta_1 + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 8 q^{25} + 32 q^{37} + 72 q^{49} + 72 q^{53} + 8 q^{77} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 20\nu^{7} + 402\nu^{6} + 43\nu^{5} - 760\nu^{4} + 4607\nu^{3} - 5750\nu^{2} + 6903\nu + 37211 ) / 21903 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1606 \nu^{7} + 13298 \nu^{6} - 54225 \nu^{5} + 26584 \nu^{4} + 186687 \nu^{3} - 439822 \nu^{2} + \cdots - 336102 ) / 284739 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4786 \nu^{7} - 17697 \nu^{6} - 40087 \nu^{5} + 168580 \nu^{4} - 95639 \nu^{3} - 434146 \nu^{2} + \cdots - 1041560 ) / 284739 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -720\nu^{7} + 130\nu^{6} + 5753\nu^{5} - 16446\nu^{4} + 2071\nu^{3} + 31776\nu^{2} - 153595\nu + 120604 ) / 21903 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10546 \nu^{7} + 3166 \nu^{6} - 86111 \nu^{5} + 124924 \nu^{4} + 238241 \nu^{3} - 337906 \nu^{2} + \cdots + 8684 ) / 284739 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5862 \nu^{7} - 15052 \nu^{6} - 33393 \nu^{5} + 171498 \nu^{4} - 114999 \nu^{3} - 305436 \nu^{2} + \cdots - 848796 ) / 94913 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1712\nu^{7} - 3554\nu^{6} - 2160\nu^{5} + 22556\nu^{4} - 37860\nu^{3} + 33472\nu^{2} + 59384\nu - 68064 ) / 21903 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} - \beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 4\beta_{2} - 6\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} - 2\beta_{6} + 7\beta_{5} - 2\beta_{4} + 6\beta_{3} - 13\beta_{2} + 6\beta _1 - 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{7} + 12\beta_{6} + 7\beta_{5} - 4\beta_{4} - 24\beta_{3} - 17\beta_{2} - 8\beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_{4} + 4\beta_{3} - 30\beta_{2} + 89\beta _1 - 149 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{6} - 9\beta_{5} - 45\beta_{3} + 18\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 106\beta_{7} - 31\beta_{6} - 50\beta_{5} + 81\beta_{4} + 50\beta_{3} + 161\beta_{2} + 433\beta _1 - 755 ) / 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
1.02715 + 1.10132i
1.02715 1.10132i
0.338876 1.46735i
0.338876 + 1.46735i
2.02641 1.27503i
2.02641 + 1.27503i
−2.39244 0.0909984i
−2.39244 + 0.0909984i
0 0 0 −2.73205 0 −5.13734 0 0 0
703.2 0 0 0 −2.73205 0 −5.13734 0 0 0
703.3 0 0 0 −2.73205 0 5.13734 0 0 0
703.4 0 0 0 −2.73205 0 5.13734 0 0 0
703.5 0 0 0 0.732051 0 −2.36806 0 0 0
703.6 0 0 0 0.732051 0 −2.36806 0 0 0
703.7 0 0 0 0.732051 0 2.36806 0 0 0
703.8 0 0 0 0.732051 0 2.36806 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c 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.o.g 8
3.b odd 2 1 528.2.o.b 8
4.b odd 2 1 inner 1584.2.o.g 8
11.b odd 2 1 inner 1584.2.o.g 8
12.b even 2 1 528.2.o.b 8
24.f even 2 1 2112.2.o.e 8
24.h odd 2 1 2112.2.o.e 8
33.d even 2 1 528.2.o.b 8
44.c even 2 1 inner 1584.2.o.g 8
132.d odd 2 1 528.2.o.b 8
264.m even 2 1 2112.2.o.e 8
264.p odd 2 1 2112.2.o.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.o.b 8 3.b odd 2 1
528.2.o.b 8 12.b even 2 1
528.2.o.b 8 33.d even 2 1
528.2.o.b 8 132.d odd 2 1
1584.2.o.g 8 1.a even 1 1 trivial
1584.2.o.g 8 4.b odd 2 1 inner
1584.2.o.g 8 11.b odd 2 1 inner
1584.2.o.g 8 44.c even 2 1 inner
2112.2.o.e 8 24.f even 2 1
2112.2.o.e 8 24.h odd 2 1
2112.2.o.e 8 264.m even 2 1
2112.2.o.e 8 264.p odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 2T_{5} - 2 \) Copy content Toggle raw display
\( T_{83}^{4} - 128T_{83}^{2} + 2368 \) 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
$5$ \( (T^{2} + 2 T - 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 32 T^{2} + 148)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{2} + 148)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 56 T^{2} + 592)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 56 T^{2} + 592)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 24 T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 128 T^{2} + 2368)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 56 T^{2} + 592)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 56 T^{2} + 676)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 18 T + 78)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 240 T^{2} + 1332)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 168 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 128 T^{2} + 2368)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 32 T^{2} + 148)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 128 T^{2} + 2368)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 8)^{4} \) Copy content Toggle raw display
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