Properties

Label 3584.2.b.f
Level $3584$
Weight $2$
Character orbit 3584.b
Analytic conductor $28.618$
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] = [3584,2,Mod(1793,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3584.1793");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.b (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: \(28.6183840844\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.46091337728.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 35x^{4} + 16x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + (\beta_{7} - \beta_{5}) q^{5} + q^{7} + ( - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + (\beta_{7} - \beta_{5}) q^{5} + q^{7} + ( - \beta_1 - 1) q^{9} + (\beta_{5} + \beta_{2}) q^{11} + ( - \beta_{7} - \beta_{4} + \beta_{2}) q^{13} + ( - \beta_1 - 2) q^{15} + ( - \beta_{6} - 1) q^{17} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_{2}) q^{19} + \beta_{7} q^{21} + ( - \beta_{6} + \beta_1 + 1) q^{23} + ( - \beta_{6} - \beta_{3} + \beta_1 - 2) q^{25} + ( - 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2}) q^{27} + ( - \beta_{5} + \beta_{4}) q^{29} + ( - \beta_{3} - 4) q^{31} + (\beta_{6} - 2 \beta_{3} - 1) q^{33} + (\beta_{7} - \beta_{5}) q^{35} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_{2}) q^{37} + (\beta_{6} - \beta_{3} + \beta_1 + 3) q^{39} + (2 \beta_1 + 2) q^{41} + (\beta_{5} + 2 \beta_{4} + \beta_{2}) q^{43} + ( - 3 \beta_{7} - 2 \beta_{5} - \beta_{4} + \beta_{2}) q^{45} + (\beta_{3} + 2 \beta_1) q^{47} + q^{49} + ( - 2 \beta_{7} - 2 \beta_{4} + 4 \beta_{2}) q^{51} + (2 \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{53} + (2 \beta_{6} - 2 \beta_1 + 6) q^{55} + (\beta_{6} - 3 \beta_{3} - \beta_1 + 1) q^{57} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2}) q^{59} + ( - 3 \beta_{7} + \beta_{5} - 2 \beta_{4}) q^{61} + ( - \beta_1 - 1) q^{63} + ( - 2 \beta_{3} + \beta_1 + 2) q^{65} + ( - 2 \beta_{7} - \beta_{5} - \beta_{2}) q^{67} + (4 \beta_{7} - \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{69} + ( - \beta_{6} + \beta_{3} - 2 \beta_1 + 3) q^{71} + (\beta_{6} + 1) q^{73} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{2}) q^{75} + (\beta_{5} + \beta_{2}) q^{77} + ( - \beta_{6} - 9) q^{79} + (\beta_{6} - \beta_{3} - \beta_1 + 2) q^{81} + (\beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{2}) q^{83} + ( - 2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{85} + ( - \beta_{3} + 4) q^{87} + (\beta_{6} - 4 \beta_{3} - 2 \beta_1 + 5) q^{89} + ( - \beta_{7} - \beta_{4} + \beta_{2}) q^{91} + ( - 4 \beta_{7} - 2 \beta_{2}) q^{93} + (\beta_{6} - \beta_{3} + \beta_1 - 5) q^{95} + ( - \beta_{6} - 2 \beta_{3} - 2 \beta_1 - 5) q^{97} + (3 \beta_{5} + 2 \beta_{4} - 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} - 8 q^{9} - 16 q^{15} - 8 q^{17} + 8 q^{23} - 16 q^{25} - 32 q^{31} - 8 q^{33} + 24 q^{39} + 16 q^{41} + 8 q^{49} + 48 q^{55} + 8 q^{57} - 8 q^{63} + 16 q^{65} + 24 q^{71} + 8 q^{73} - 72 q^{79} + 16 q^{81} + 32 q^{87} + 40 q^{89} - 40 q^{95} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 35x^{4} + 16x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{6} + 14\nu^{4} + 35\nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{7} + 14\nu^{5} + 35\nu^{3} + 16\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{6} - 41\nu^{4} - 92\nu^{2} - 22 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} + 41\nu^{5} + 92\nu^{3} + 26\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 41\nu^{5} - 92\nu^{3} - 22\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 5\nu^{6} + 69\nu^{4} + 160\nu^{2} + 39 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{7} - 151\nu^{5} - 344\nu^{3} - 84\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - \beta_{3} + 2\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 13\beta_{5} - 7\beta_{4} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{6} + 15\beta_{3} - 20\beta _1 + 63 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -26\beta_{7} + 147\beta_{5} + 65\beta_{4} - 20\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -147\beta_{6} - 175\beta_{3} + 212\beta _1 - 661 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 294\beta_{7} - 1619\beta_{5} - 681\beta_{4} + 212\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1793.1
0.481620i
0.558894i
1.58915i
3.30610i
3.30610i
1.58915i
0.558894i
0.481620i
0 2.93637i 0 2.25526i 0 1.00000 0 −5.62226 0
1793.2 0 2.53038i 0 1.73998i 0 1.00000 0 −3.40281 0
1793.3 0 0.889918i 0 1.35748i 0 1.00000 0 2.20805 0
1793.4 0 0.427759i 0 4.24777i 0 1.00000 0 2.81702 0
1793.5 0 0.427759i 0 4.24777i 0 1.00000 0 2.81702 0
1793.6 0 0.889918i 0 1.35748i 0 1.00000 0 2.20805 0
1793.7 0 2.53038i 0 1.73998i 0 1.00000 0 −3.40281 0
1793.8 0 2.93637i 0 2.25526i 0 1.00000 0 −5.62226 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1793.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.b.f 8
4.b odd 2 1 3584.2.b.e 8
8.b even 2 1 inner 3584.2.b.f 8
8.d odd 2 1 3584.2.b.e 8
16.e even 4 2 3584.2.a.m 8
16.f odd 4 2 3584.2.a.n yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.m 8 16.e even 4 2
3584.2.a.n yes 8 16.f odd 4 2
3584.2.b.e 8 4.b odd 2 1
3584.2.b.e 8 8.d odd 2 1
3584.2.b.f 8 1.a even 1 1 trivial
3584.2.b.f 8 8.b 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 \(S_{2}^{\mathrm{new}}(3584, [\chi])\):

\( T_{3}^{8} + 16T_{3}^{6} + 70T_{3}^{4} + 56T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{23}^{4} - 4T_{23}^{3} - 54T_{23}^{2} + 192T_{23} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 16 T^{6} + 70 T^{4} + 56 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{8} + 28 T^{6} + 210 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 60 T^{6} + 1156 T^{4} + \cdots + 6272 \) Copy content Toggle raw display
$13$ \( T^{8} + 44 T^{6} + 466 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} - 44 T^{2} - 96 T + 544)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 112 T^{6} + 3590 T^{4} + \cdots + 75272 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} - 54 T^{2} + 192 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 72 T^{6} + 816 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 8)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 264 T^{6} + 21808 T^{4} + \cdots + 5431808 \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} - 80 T^{2} + 320 T + 1904)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 252 T^{6} + 22148 T^{4} + \cdots + 9400448 \) Copy content Toggle raw display
$47$ \( (T^{4} - 104 T^{2} + 64 T + 2176)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 216 T^{6} + 7312 T^{4} + \cdots + 8192 \) Copy content Toggle raw display
$59$ \( T^{8} + 112 T^{6} + 3814 T^{4} + \cdots + 40328 \) Copy content Toggle raw display
$61$ \( T^{8} + 188 T^{6} + 9938 T^{4} + \cdots + 32768 \) Copy content Toggle raw display
$67$ \( T^{8} + 140 T^{6} + 5092 T^{4} + \cdots + 147968 \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} - 164 T^{2} + 2688 T - 8704)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 4 T^{3} - 44 T^{2} + 96 T + 544)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 36 T^{3} + 436 T^{2} + 2016 T + 3104)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 464 T^{6} + \cdots + 15523592 \) Copy content Toggle raw display
$89$ \( (T^{4} - 20 T^{3} - 164 T^{2} + \cdots - 19232)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 20 T^{3} - 68 T^{2} - 3168 T - 14624)^{2} \) Copy content Toggle raw display
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