Properties

Label 9216.2.a.bf
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $4$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2304)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 2 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{2} + \beta_1) q^{13} - \beta_{3} q^{19} - 5 q^{25} + ( - 3 \beta_{2} + 2 \beta_1) q^{31} + (2 \beta_{2} + 5 \beta_1) q^{37} + 3 \beta_{3} q^{43} + (4 \beta_{3} + 7) q^{49} + ( - 2 \beta_{2} + 7 \beta_1) q^{61} - 16 q^{67} - 4 \beta_{3} q^{73} + (5 \beta_{2} + 2 \beta_1) q^{79} + ( - 5 \beta_{3} - 16) q^{91} - 4 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{25} + 28 q^{49} - 64 q^{67} - 64 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{2} + 5\beta_1 ) / 2 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.93185
−0.517638
0.517638
−1.93185
0 0 0 0 0 −5.27792 0 0 0
1.2 0 0 0 0 0 −0.378937 0 0 0
1.3 0 0 0 0 0 0.378937 0 0 0
1.4 0 0 0 0 0 5.27792 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
8.d odd 2 1 inner
24.f even 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.bf 4
3.b odd 2 1 CM 9216.2.a.bf 4
4.b odd 2 1 9216.2.a.bg 4
8.b even 2 1 9216.2.a.bg 4
8.d odd 2 1 inner 9216.2.a.bf 4
12.b even 2 1 9216.2.a.bg 4
24.f even 2 1 inner 9216.2.a.bf 4
24.h odd 2 1 9216.2.a.bg 4
32.g even 8 2 2304.2.k.g 8
32.g even 8 2 2304.2.k.j yes 8
32.h odd 8 2 2304.2.k.g 8
32.h odd 8 2 2304.2.k.j yes 8
96.o even 8 2 2304.2.k.g 8
96.o even 8 2 2304.2.k.j yes 8
96.p odd 8 2 2304.2.k.g 8
96.p odd 8 2 2304.2.k.j yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.2.k.g 8 32.g even 8 2
2304.2.k.g 8 32.h odd 8 2
2304.2.k.g 8 96.o even 8 2
2304.2.k.g 8 96.p odd 8 2
2304.2.k.j yes 8 32.g even 8 2
2304.2.k.j yes 8 32.h odd 8 2
2304.2.k.j yes 8 96.o even 8 2
2304.2.k.j yes 8 96.p odd 8 2
9216.2.a.bf 4 1.a even 1 1 trivial
9216.2.a.bf 4 3.b odd 2 1 CM
9216.2.a.bf 4 8.d odd 2 1 inner
9216.2.a.bf 4 24.f even 2 1 inner
9216.2.a.bg 4 4.b odd 2 1
9216.2.a.bg 4 8.b even 2 1
9216.2.a.bg 4 12.b even 2 1
9216.2.a.bg 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{4} - 28T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{4} - 52T_{13}^{2} + 484 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19}^{2} - 12 \) Copy content Toggle raw display
\( T_{67} + 16 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 28T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 52T^{2} + 484 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 124T^{2} + 2116 \) Copy content Toggle raw display
$37$ \( T^{4} - 148T^{2} + 676 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 244T^{2} + 5476 \) Copy content Toggle raw display
$67$ \( (T + 16)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 316 T^{2} + 20164 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
show more
show less