Properties

Label 9216.2.a.bq
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: 8.8.10070523904.1
Defining polynomial: \(x^{8} - 4 x^{7} - 8 x^{6} + 24 x^{5} + 30 x^{4} - 16 x^{3} - 20 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 144)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( -1 - \beta_{5} ) q^{7} +O(q^{10})\) \( q + \beta_{2} q^{5} + ( -1 - \beta_{5} ) q^{7} + ( -\beta_{2} + \beta_{7} ) q^{11} + \beta_{1} q^{13} + \beta_{3} q^{17} + ( -\beta_{1} + \beta_{4} ) q^{19} + ( -\beta_{3} + \beta_{6} ) q^{23} + ( 1 + 2 \beta_{5} ) q^{25} + ( \beta_{2} - 2 \beta_{7} ) q^{29} + ( -3 + \beta_{5} ) q^{31} + ( -3 \beta_{2} - \beta_{7} ) q^{35} + ( -\beta_{1} - 2 \beta_{4} ) q^{37} + ( -\beta_{3} - 2 \beta_{6} ) q^{41} + ( -\beta_{1} - 3 \beta_{4} ) q^{43} + ( -\beta_{3} - 3 \beta_{6} ) q^{47} + ( 1 + 2 \beta_{5} ) q^{49} + ( -\beta_{2} - 2 \beta_{7} ) q^{53} -4 q^{55} + ( -2 \beta_{2} - 2 \beta_{7} ) q^{59} + ( \beta_{1} + 2 \beta_{4} ) q^{61} + ( -\beta_{3} + 2 \beta_{6} ) q^{65} + 4 \beta_{4} q^{67} + ( 2 \beta_{3} + 2 \beta_{6} ) q^{71} + ( 2 + 2 \beta_{5} ) q^{73} + 2 \beta_{7} q^{77} + ( -7 + \beta_{5} ) q^{79} + ( -\beta_{2} + \beta_{7} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{85} + 2 \beta_{6} q^{89} + ( -\beta_{1} - 7 \beta_{4} ) q^{91} + ( \beta_{3} - \beta_{6} ) q^{95} + 4 \beta_{5} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} + 8q^{25} - 24q^{31} + 8q^{49} - 32q^{55} + 16q^{73} - 56q^{79} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 8 x^{6} + 24 x^{5} + 30 x^{4} - 16 x^{3} - 20 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{7} - 9 \nu^{6} - 10 \nu^{5} + 47 \nu^{4} + 26 \nu^{3} - 12 \nu^{2} - 4 \nu - 10 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -18 \nu^{7} + 83 \nu^{6} + 94 \nu^{5} - 492 \nu^{4} - 246 \nu^{3} + 452 \nu^{2} + 106 \nu - 66 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 20 \nu^{7} - 90 \nu^{6} - 114 \nu^{5} + 532 \nu^{4} + 332 \nu^{3} - 462 \nu^{2} - 168 \nu + 64 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 20 \nu^{7} - 91 \nu^{6} - 110 \nu^{5} + 541 \nu^{4} + 302 \nu^{3} - 490 \nu^{2} - 128 \nu + 76 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -38 \nu^{7} + 174 \nu^{6} + 204 \nu^{5} - 1033 \nu^{4} - 548 \nu^{3} + 942 \nu^{2} + 240 \nu - 145 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( 44 \nu^{7} - 200 \nu^{6} - 242 \nu^{5} + 1184 \nu^{4} + 668 \nu^{3} - 1046 \nu^{2} - 284 \nu + 152 \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -44 \nu^{7} + 199 \nu^{6} + 248 \nu^{5} - 1186 \nu^{4} - 698 \nu^{3} + 1066 \nu^{2} + 314 \nu - 154 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} - \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} + \beta_{1} + 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{7} - 9 \beta_{6} + 20 \beta_{5} + 22 \beta_{4} + 3 \beta_{3} - 26 \beta_{2} + 6 \beta_{1} + 44\)\()/4\)
\(\nu^{4}\)\(=\)\(-6 \beta_{7} - 12 \beta_{6} + 17 \beta_{5} + 21 \beta_{4} + 2 \beta_{3} - 24 \beta_{2} + 9 \beta_{1} + 49\)
\(\nu^{5}\)\(=\)\((\)\(-39 \beta_{7} - 105 \beta_{6} + 144 \beta_{5} + 209 \beta_{4} + 25 \beta_{3} - 197 \beta_{2} + 75 \beta_{1} + 366\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-176 \beta_{7} - 473 \beta_{6} + 566 \beta_{5} + 890 \beta_{4} + 97 \beta_{3} - 786 \beta_{2} + 344 \beta_{1} + 1526\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-685 \beta_{7} - 2037 \beta_{6} + 2352 \beta_{5} + 3928 \beta_{4} + 448 \beta_{3} - 3245 \beta_{2} + 1470 \beta_{1} + 6168\)\()/2\)

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
2.79591
4.21012
0.305093
−1.10912
−0.536630
−1.95084
−0.564373
0.849841
0 0 0 −3.36028 0 −3.64575 0 0 0
1.2 0 0 0 −3.36028 0 −3.64575 0 0 0
1.3 0 0 0 −0.841723 0 1.64575 0 0 0
1.4 0 0 0 −0.841723 0 1.64575 0 0 0
1.5 0 0 0 0.841723 0 1.64575 0 0 0
1.6 0 0 0 0.841723 0 1.64575 0 0 0
1.7 0 0 0 3.36028 0 −3.64575 0 0 0
1.8 0 0 0 3.36028 0 −3.64575 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 inner
8.b even 2 1 inner
24.h 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.bq 8
3.b odd 2 1 inner 9216.2.a.bq 8
4.b odd 2 1 9216.2.a.bt 8
8.b even 2 1 inner 9216.2.a.bq 8
8.d odd 2 1 9216.2.a.bt 8
12.b even 2 1 9216.2.a.bt 8
24.f even 2 1 9216.2.a.bt 8
24.h odd 2 1 inner 9216.2.a.bq 8
32.g even 8 2 144.2.k.c 8
32.g even 8 2 1152.2.k.e 8
32.h odd 8 2 576.2.k.c 8
32.h odd 8 2 1152.2.k.d 8
96.o even 8 2 576.2.k.c 8
96.o even 8 2 1152.2.k.d 8
96.p odd 8 2 144.2.k.c 8
96.p odd 8 2 1152.2.k.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.k.c 8 32.g even 8 2
144.2.k.c 8 96.p odd 8 2
576.2.k.c 8 32.h odd 8 2
576.2.k.c 8 96.o even 8 2
1152.2.k.d 8 32.h odd 8 2
1152.2.k.d 8 96.o even 8 2
1152.2.k.e 8 32.g even 8 2
1152.2.k.e 8 96.p odd 8 2
9216.2.a.bq 8 1.a even 1 1 trivial
9216.2.a.bq 8 3.b odd 2 1 inner
9216.2.a.bq 8 8.b even 2 1 inner
9216.2.a.bq 8 24.h odd 2 1 inner
9216.2.a.bt 8 4.b odd 2 1
9216.2.a.bt 8 8.d odd 2 1
9216.2.a.bt 8 12.b even 2 1
9216.2.a.bt 8 24.f even 2 1

Hecke kernels

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

\( T_{5}^{4} - 12 T_{5}^{2} + 8 \)
\( T_{7}^{2} + 2 T_{7} - 6 \)
\( T_{11}^{4} - 24 T_{11}^{2} + 32 \)
\( T_{13}^{2} - 14 \)
\( T_{17}^{4} - 40 T_{17}^{2} + 288 \)
\( T_{19}^{4} - 32 T_{19}^{2} + 144 \)
\( T_{67}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 8 - 12 T^{2} + T^{4} )^{2} \)
$7$ \( ( -6 + 2 T + T^{2} )^{4} \)
$11$ \( ( 32 - 24 T^{2} + T^{4} )^{2} \)
$13$ \( ( -14 + T^{2} )^{4} \)
$17$ \( ( 288 - 40 T^{2} + T^{4} )^{2} \)
$19$ \( ( 144 - 32 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1152 - 80 T^{2} + T^{4} )^{2} \)
$29$ \( ( 72 - 76 T^{2} + T^{4} )^{2} \)
$31$ \( ( 2 + 6 T + T^{2} )^{4} \)
$37$ \( ( 36 - 44 T^{2} + T^{4} )^{2} \)
$41$ \( ( 2592 - 104 T^{2} + T^{4} )^{2} \)
$43$ \( ( 16 - 64 T^{2} + T^{4} )^{2} \)
$47$ \( ( 10368 - 208 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2888 - 108 T^{2} + T^{4} )^{2} \)
$59$ \( ( 4608 - 160 T^{2} + T^{4} )^{2} \)
$61$ \( ( 36 - 44 T^{2} + T^{4} )^{2} \)
$67$ \( ( -32 + T^{2} )^{4} \)
$71$ \( ( 2048 - 192 T^{2} + T^{4} )^{2} \)
$73$ \( ( -24 - 4 T + T^{2} )^{4} \)
$79$ \( ( 42 + 14 T + T^{2} )^{4} \)
$83$ \( ( 32 - 24 T^{2} + T^{4} )^{2} \)
$89$ \( ( 512 - 96 T^{2} + T^{4} )^{2} \)
$97$ \( ( -112 + T^{2} )^{4} \)
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