Properties

Label 9216.2.a.ba
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1536)
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.

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

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
0.765367
−1.84776
−0.765367
0 0 0 −2.49661 0 −0.917608 0 0 0
1.2 0 0 0 −1.19891 0 0.613126 0 0 0
1.3 0 0 0 −0.331821 0 −3.08239 0 0 0
1.4 0 0 0 4.02734 0 −4.61313 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9216.2.a.ba 4
3.b odd 2 1 3072.2.a.j 4
4.b odd 2 1 9216.2.a.bm 4
8.b even 2 1 9216.2.a.z 4
8.d odd 2 1 9216.2.a.bl 4
12.b even 2 1 3072.2.a.s 4
24.f even 2 1 3072.2.a.m 4
24.h odd 2 1 3072.2.a.p 4
32.g even 8 2 4608.2.k.bc 8
32.g even 8 2 4608.2.k.bj 8
32.h odd 8 2 4608.2.k.be 8
32.h odd 8 2 4608.2.k.bh 8
48.i odd 4 2 3072.2.d.j 8
48.k even 4 2 3072.2.d.e 8
96.o even 8 2 1536.2.j.e 8
96.o even 8 2 1536.2.j.j yes 8
96.p odd 8 2 1536.2.j.f yes 8
96.p odd 8 2 1536.2.j.i yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.e 8 96.o even 8 2
1536.2.j.f yes 8 96.p odd 8 2
1536.2.j.i yes 8 96.p odd 8 2
1536.2.j.j yes 8 96.o even 8 2
3072.2.a.j 4 3.b odd 2 1
3072.2.a.m 4 24.f even 2 1
3072.2.a.p 4 24.h odd 2 1
3072.2.a.s 4 12.b even 2 1
3072.2.d.e 8 48.k even 4 2
3072.2.d.j 8 48.i odd 4 2
4608.2.k.bc 8 32.g even 8 2
4608.2.k.be 8 32.h odd 8 2
4608.2.k.bh 8 32.h odd 8 2
4608.2.k.bj 8 32.g even 8 2
9216.2.a.z 4 8.b even 2 1
9216.2.a.ba 4 1.a even 1 1 trivial
9216.2.a.bl 4 8.d odd 2 1
9216.2.a.bm 4 4.b odd 2 1

Hecke kernels

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

\( T_{5}^{4} - 12T_{5}^{2} - 16T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{4} + 8T_{7}^{3} + 16T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{4} - 16T_{11}^{2} + 32 \) Copy content Toggle raw display
\( T_{13}^{4} - 8T_{13}^{3} - 12T_{13}^{2} + 176T_{13} - 188 \) Copy content Toggle raw display
\( T_{17}^{4} - 32T_{17}^{2} + 64T_{17} - 32 \) Copy content Toggle raw display
\( T_{19}^{4} - 32T_{19}^{2} + 64T_{19} - 32 \) Copy content Toggle raw display
\( T_{67}^{4} + 16T_{67}^{3} - 96T_{67}^{2} - 2304T_{67} - 7936 \) 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
$5$ \( T^{4} - 12 T^{2} - 16 T - 4 \) Copy content Toggle raw display
$7$ \( T^{4} + 8 T^{3} + 16 T^{2} - 8 \) Copy content Toggle raw display
$11$ \( T^{4} - 16T^{2} + 32 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} - 12 T^{2} + 176 T - 188 \) Copy content Toggle raw display
$17$ \( T^{4} - 32 T^{2} + 64 T - 32 \) Copy content Toggle raw display
$19$ \( T^{4} - 32 T^{2} + 64 T - 32 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 44 T^{2} - 112 T - 68 \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} - 16 T^{2} - 128 T + 248 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} - 44 T^{2} + 112 T + 388 \) Copy content Toggle raw display
$41$ \( T^{4} - 96 T^{2} - 64 T + 992 \) Copy content Toggle raw display
$43$ \( T^{4} - 96 T^{2} - 64 T + 992 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 108 T^{2} - 272 T + 188 \) Copy content Toggle raw display
$59$ \( T^{4} + 16 T^{3} - 32 T^{2} + \cdots - 4352 \) Copy content Toggle raw display
$61$ \( T^{4} - 24 T^{3} + 148 T^{2} - 176 T + 4 \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} - 96 T^{2} + \cdots - 7936 \) Copy content Toggle raw display
$71$ \( T^{4} - 16 T^{3} + 768 T - 2176 \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} - 120 T^{2} + \cdots + 1552 \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + 80 T^{2} + \cdots - 7688 \) Copy content Toggle raw display
$83$ \( T^{4} - 80 T^{2} - 128 T + 544 \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} - 168 T^{2} + \cdots + 272 \) Copy content Toggle raw display
$97$ \( T^{4} + 16 T^{3} - 32 T^{2} + \cdots - 256 \) Copy content Toggle raw display
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