Properties

Label 9216.2.a.bb
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

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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 256)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_1 - 3) q^{11} + (\beta_{3} - 2 \beta_{2}) q^{13} + 2 \beta_1 q^{17} + (\beta_1 + 3) q^{19} + ( - \beta_{3} + 3 \beta_{2}) q^{23} + q^{25} - \beta_{3} q^{29} + 4 \beta_{2} q^{31} + ( - 2 \beta_1 - 6) q^{35} + (\beta_{3} + 2 \beta_{2}) q^{37} - 4 \beta_1 q^{41} + ( - 3 \beta_1 + 3) q^{43} - 4 \beta_{3} q^{47} + (4 \beta_1 + 1) q^{49} + (\beta_{3} + 6 \beta_{2}) q^{53} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{55} + (\beta_1 - 9) q^{59} + ( - \beta_{3} - 2 \beta_{2}) q^{61} + ( - 4 \beta_1 + 6) q^{65} + (3 \beta_1 + 1) q^{67} + (\beta_{3} - 3 \beta_{2}) q^{71} + (2 \beta_1 - 6) q^{73} + (4 \beta_{3} + 6 \beta_{2}) q^{77} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{79} + (3 \beta_1 - 3) q^{83} + 6 \beta_{2} q^{85} + ( - 2 \beta_1 + 6) q^{89} + (2 \beta_1 - 2) q^{91} + (3 \beta_{3} + 3 \beta_{2}) q^{95} + 2 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{11} + 12 q^{19} + 4 q^{25} - 24 q^{35} + 12 q^{43} + 4 q^{49} - 36 q^{59} + 24 q^{65} + 4 q^{67} - 24 q^{73} - 12 q^{83} + 24 q^{89} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} + 5\beta_{2} ) / 2 \) 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
−0.517638
−1.93185
1.93185
0.517638
0 0 0 −2.44949 0 1.03528 0 0 0
1.2 0 0 0 −2.44949 0 3.86370 0 0 0
1.3 0 0 0 2.44949 0 −3.86370 0 0 0
1.4 0 0 0 2.44949 0 −1.03528 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
8.d 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.bb 4
3.b odd 2 1 1024.2.a.j 4
4.b odd 2 1 9216.2.a.bk 4
8.b even 2 1 9216.2.a.bk 4
8.d odd 2 1 inner 9216.2.a.bb 4
12.b even 2 1 1024.2.a.g 4
24.f even 2 1 1024.2.a.j 4
24.h odd 2 1 1024.2.a.g 4
32.g even 8 2 2304.2.k.f 8
32.g even 8 2 2304.2.k.k 8
32.h odd 8 2 2304.2.k.f 8
32.h odd 8 2 2304.2.k.k 8
48.i odd 4 2 1024.2.b.h 8
48.k even 4 2 1024.2.b.h 8
96.o even 8 2 256.2.e.a 8
96.o even 8 2 256.2.e.b yes 8
96.p odd 8 2 256.2.e.a 8
96.p odd 8 2 256.2.e.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.2.e.a 8 96.o even 8 2
256.2.e.a 8 96.p odd 8 2
256.2.e.b yes 8 96.o even 8 2
256.2.e.b yes 8 96.p odd 8 2
1024.2.a.g 4 12.b even 2 1
1024.2.a.g 4 24.h odd 2 1
1024.2.a.j 4 3.b odd 2 1
1024.2.a.j 4 24.f even 2 1
1024.2.b.h 8 48.i odd 4 2
1024.2.b.h 8 48.k even 4 2
2304.2.k.f 8 32.g even 8 2
2304.2.k.f 8 32.h odd 8 2
2304.2.k.k 8 32.g even 8 2
2304.2.k.k 8 32.h odd 8 2
9216.2.a.bb 4 1.a even 1 1 trivial
9216.2.a.bb 4 8.d odd 2 1 inner
9216.2.a.bk 4 4.b odd 2 1
9216.2.a.bk 4 8.b even 2 1

Hecke kernels

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

\( T_{5}^{2} - 6 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 6 \) Copy content Toggle raw display
\( T_{13}^{4} - 28T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 12 \) Copy content Toggle raw display
\( T_{19}^{2} - 6T_{19} + 6 \) Copy content Toggle raw display
\( T_{67}^{2} - 2T_{67} - 26 \) 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
$5$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 28T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 48T^{2} + 144 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 28T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 156T^{2} + 4356 \) Copy content Toggle raw display
$59$ \( (T^{2} + 18 T + 78)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 28T^{2} + 4 \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 48T^{2} + 144 \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 256T^{2} + 4096 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
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