Properties

Label 9216.2.a.w
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9216,2,Mod(1,9216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

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: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 512)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 2) q^{5} + (\beta_{3} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 2) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - 2 \beta_{3} - \beta_1) q^{11} + ( - \beta_{2} + 2) q^{13} + 2 \beta_{2} q^{17} + ( - 2 \beta_{3} + \beta_1) q^{19} + (3 \beta_{3} + \beta_1) q^{23} + (4 \beta_{2} + 1) q^{25} + (\beta_{2} - 6) q^{29} + (2 \beta_{3} + 2 \beta_1) q^{31} + 2 \beta_1 q^{35} + ( - 5 \beta_{2} + 2) q^{37} + 4 q^{41} + \beta_1 q^{43} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} + ( - 4 \beta_{2} + 1) q^{49} + ( - \beta_{2} - 6) q^{53} + (3 \beta_{3} + 5 \beta_1) q^{55} + \beta_1 q^{59} + (5 \beta_{2} + 6) q^{61} - 2 q^{65} + (2 \beta_{3} + 3 \beta_1) q^{67} + (\beta_{3} - 5 \beta_1) q^{71} + (6 \beta_{2} + 2) q^{73} + (8 \beta_{2} - 4) q^{77} + ( - 4 \beta_{3} + 4 \beta_1) q^{79} + ( - 4 \beta_{3} + 3 \beta_1) q^{83} + ( - 4 \beta_{2} - 4) q^{85} + (2 \beta_{2} - 2) q^{89} + (4 \beta_{3} - 2 \beta_1) q^{91} + (\beta_{3} - \beta_1) q^{95} + (2 \beta_{2} + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{5} + 8 q^{13} + 4 q^{25} - 24 q^{29} + 8 q^{37} + 16 q^{41} + 4 q^{49} - 24 q^{53} + 24 q^{61} - 8 q^{65} + 8 q^{73} - 16 q^{77} - 16 q^{85} - 8 q^{89} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu^{3} - 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.84776
−1.84776
−0.765367
0.765367
0 0 0 −3.41421 0 −1.53073 0 0 0
1.2 0 0 0 −3.41421 0 1.53073 0 0 0
1.3 0 0 0 −0.585786 0 −3.69552 0 0 0
1.4 0 0 0 −0.585786 0 3.69552 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9216.2.a.w 4
3.b odd 2 1 1024.2.a.i 4
4.b odd 2 1 inner 9216.2.a.w 4
8.b even 2 1 9216.2.a.bp 4
8.d odd 2 1 9216.2.a.bp 4
12.b even 2 1 1024.2.a.i 4
24.f even 2 1 1024.2.a.h 4
24.h odd 2 1 1024.2.a.h 4
32.g even 8 2 4608.2.k.bd 8
32.g even 8 2 4608.2.k.bi 8
32.h odd 8 2 4608.2.k.bd 8
32.h odd 8 2 4608.2.k.bi 8
48.i odd 4 2 1024.2.b.g 8
48.k even 4 2 1024.2.b.g 8
96.o even 8 2 512.2.e.i 8
96.o even 8 2 512.2.e.j yes 8
96.p odd 8 2 512.2.e.i 8
96.p odd 8 2 512.2.e.j yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.i 8 96.o even 8 2
512.2.e.i 8 96.p odd 8 2
512.2.e.j yes 8 96.o even 8 2
512.2.e.j yes 8 96.p odd 8 2
1024.2.a.h 4 24.f even 2 1
1024.2.a.h 4 24.h odd 2 1
1024.2.a.i 4 3.b odd 2 1
1024.2.a.i 4 12.b even 2 1
1024.2.b.g 8 48.i odd 4 2
1024.2.b.g 8 48.k even 4 2
4608.2.k.bd 8 32.g even 8 2
4608.2.k.bd 8 32.h odd 8 2
4608.2.k.bi 8 32.g even 8 2
4608.2.k.bi 8 32.h odd 8 2
9216.2.a.w 4 1.a even 1 1 trivial
9216.2.a.w 4 4.b odd 2 1 inner
9216.2.a.bp 4 8.b even 2 1
9216.2.a.bp 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5}^{2} + 4T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 32 \) Copy content Toggle raw display
\( T_{11}^{4} - 40T_{11}^{2} + 392 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 8 \) Copy content Toggle raw display
\( T_{19}^{4} - 40T_{19}^{2} + 8 \) Copy content Toggle raw display
\( T_{67}^{4} - 104T_{67}^{2} + 392 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 32 \) Copy content Toggle raw display
$11$ \( T^{4} - 40T^{2} + 392 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 40T^{2} + 8 \) Copy content Toggle raw display
$23$ \( T^{4} - 80T^{2} + 1568 \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 34)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 64T^{2} + 512 \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 46)^{2} \) Copy content Toggle raw display
$41$ \( (T - 4)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} - 8T^{2} + 8 \) Copy content Toggle raw display
$47$ \( T^{4} - 64T^{2} + 512 \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T + 34)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 8T^{2} + 8 \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T - 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 104T^{2} + 392 \) Copy content Toggle raw display
$71$ \( T^{4} - 208T^{2} + 9248 \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 68)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 256T^{2} + 8192 \) Copy content Toggle raw display
$83$ \( T^{4} - 200T^{2} + 2312 \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 56)^{2} \) Copy content Toggle raw display
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