Properties

Label 9216.2.a.bt
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $0$
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: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.10070523904.1
Defining polynomial: \( x^{8} - 4x^{7} - 8x^{6} + 24x^{5} + 30x^{4} - 16x^{3} - 20x^{2} + 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} + ( - \beta_{6} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} + ( - \beta_{6} + 1) q^{7} + ( - \beta_{5} - \beta_{2}) q^{11} + \beta_1 q^{13} - \beta_{7} q^{17} + ( - \beta_{3} + \beta_1) q^{19} + ( - \beta_{7} + \beta_{4}) q^{23} + ( - 2 \beta_{6} + 1) q^{25} + ( - \beta_{5} - 2 \beta_{2}) q^{29} + (\beta_{6} + 3) q^{31} + ( - 3 \beta_{5} + \beta_{2}) q^{35} + ( - 2 \beta_{3} - \beta_1) q^{37} + (\beta_{7} + 2 \beta_{4}) q^{41} + (3 \beta_{3} + \beta_1) q^{43} + ( - \beta_{7} - 3 \beta_{4}) q^{47} + ( - 2 \beta_{6} + 1) q^{49} + (\beta_{5} - 2 \beta_{2}) q^{53} + 4 q^{55} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{59} + (2 \beta_{3} + \beta_1) q^{61} + (\beta_{7} - 2 \beta_{4}) q^{65} - 4 \beta_{3} q^{67} + (2 \beta_{7} + 2 \beta_{4}) q^{71} + ( - 2 \beta_{6} + 2) q^{73} + 2 \beta_{2} q^{77} + (\beta_{6} + 7) q^{79} + ( - \beta_{5} - \beta_{2}) q^{83} + ( - 2 \beta_{3} - 2 \beta_1) q^{85} - 2 \beta_{4} q^{89} + (7 \beta_{3} + \beta_1) q^{91} + (\beta_{7} - \beta_{4}) q^{95} - 4 \beta_{6} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} + 8 q^{25} + 24 q^{31} + 8 q^{49} + 32 q^{55} + 16 q^{73} + 56 q^{79}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -2\nu^{7} + 9\nu^{6} + 10\nu^{5} - 47\nu^{4} - 26\nu^{3} + 12\nu^{2} + 4\nu + 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -12\nu^{7} + 55\nu^{6} + 64\nu^{5} - 326\nu^{4} - 168\nu^{3} + 292\nu^{2} + 58\nu - 44 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20\nu^{7} - 91\nu^{6} - 110\nu^{5} + 541\nu^{4} + 302\nu^{3} - 490\nu^{2} - 128\nu + 76 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -26\nu^{7} + 116\nu^{6} + 154\nu^{5} - 694\nu^{4} - 452\nu^{3} + 614\nu^{2} + 208\nu - 88 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -32\nu^{7} + 145\nu^{6} + 178\nu^{5} - 858\nu^{4} - 500\nu^{3} + 754\nu^{2} + 226\nu - 108 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -38\nu^{7} + 174\nu^{6} + 204\nu^{5} - 1033\nu^{4} - 548\nu^{3} + 942\nu^{2} + 240\nu - 145 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 62\nu^{7} - 282\nu^{6} - 342\nu^{5} + 1678\nu^{4} + 944\nu^{3} - 1518\nu^{2} - 420\nu + 220 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} + \beta_{4} + 2\beta_{3} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{7} + 20\beta_{6} + 6\beta_{5} + 11\beta_{4} + 22\beta_{3} + 12\beta_{2} - 6\beta _1 + 44 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 15\beta_{7} + 17\beta_{6} + 10\beta_{5} + 9\beta_{4} + 21\beta_{3} + 14\beta_{2} - 9\beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 118\beta_{7} + 144\beta_{6} + 80\beta_{5} + 79\beta_{4} + 209\beta_{3} + 130\beta_{2} - 75\beta _1 + 366 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 481\beta_{7} + 566\beta_{6} + 376\beta_{5} + 305\beta_{4} + 890\beta_{3} + 570\beta_{2} - 344\beta _1 + 1526 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1965 \beta_{7} + 2352 \beta_{6} + 1589 \beta_{5} + 1280 \beta_{4} + 3928 \beta_{3} + 2485 \beta_{2} - 1470 \beta _1 + 6168 ) / 2 \) 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
−1.95084
0.305093
4.21012
−0.564373
0.849841
2.79591
−1.10912
−0.536630
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.bt 8
3.b odd 2 1 inner 9216.2.a.bt 8
4.b odd 2 1 9216.2.a.bq 8
8.b even 2 1 inner 9216.2.a.bt 8
8.d odd 2 1 9216.2.a.bq 8
12.b even 2 1 9216.2.a.bq 8
24.f even 2 1 9216.2.a.bq 8
24.h odd 2 1 inner 9216.2.a.bt 8
32.g even 8 2 576.2.k.c 8
32.g even 8 2 1152.2.k.d 8
32.h odd 8 2 144.2.k.c 8
32.h odd 8 2 1152.2.k.e 8
96.o even 8 2 144.2.k.c 8
96.o even 8 2 1152.2.k.e 8
96.p odd 8 2 576.2.k.c 8
96.p odd 8 2 1152.2.k.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.k.c 8 32.h odd 8 2
144.2.k.c 8 96.o even 8 2
576.2.k.c 8 32.g even 8 2
576.2.k.c 8 96.p odd 8 2
1152.2.k.d 8 32.g even 8 2
1152.2.k.d 8 96.p odd 8 2
1152.2.k.e 8 32.h odd 8 2
1152.2.k.e 8 96.o even 8 2
9216.2.a.bq 8 4.b odd 2 1
9216.2.a.bq 8 8.d odd 2 1
9216.2.a.bq 8 12.b even 2 1
9216.2.a.bq 8 24.f even 2 1
9216.2.a.bt 8 1.a even 1 1 trivial
9216.2.a.bt 8 3.b odd 2 1 inner
9216.2.a.bt 8 8.b even 2 1 inner
9216.2.a.bt 8 24.h odd 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

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