Properties

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

Related objects

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: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ 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.

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

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 −2.44949 0 −3.86370 0 0 0
1.2 0 0 0 −2.44949 0 −1.03528 0 0 0
1.3 0 0 0 2.44949 0 1.03528 0 0 0
1.4 0 0 0 2.44949 0 3.86370 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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.bk 4
3.b odd 2 1 1024.2.a.g 4
4.b odd 2 1 9216.2.a.bb 4
8.b even 2 1 9216.2.a.bb 4
8.d odd 2 1 inner 9216.2.a.bk 4
12.b even 2 1 1024.2.a.j 4
24.f even 2 1 1024.2.a.g 4
24.h odd 2 1 1024.2.a.j 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 3.b odd 2 1
1024.2.a.g 4 24.f even 2 1
1024.2.a.j 4 12.b even 2 1
1024.2.a.j 4 24.h odd 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 4.b odd 2 1
9216.2.a.bb 4 8.b even 2 1
9216.2.a.bk 4 1.a even 1 1 trivial
9216.2.a.bk 4 8.d 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9216))$$:

 $$T_{5}^{2} - 6$$ $$T_{7}^{4} - 16 T_{7}^{2} + 16$$ $$T_{11}^{2} - 6 T_{11} + 6$$ $$T_{13}^{4} - 28 T_{13}^{2} + 4$$ $$T_{17}^{2} - 12$$ $$T_{19}^{2} + 6 T_{19} + 6$$ $$T_{67}^{2} + 2 T_{67} - 26$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -6 + T^{2} )^{2}$$
$7$ $$16 - 16 T^{2} + T^{4}$$
$11$ $$( 6 - 6 T + T^{2} )^{2}$$
$13$ $$4 - 28 T^{2} + T^{4}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$( 6 + 6 T + T^{2} )^{2}$$
$23$ $$144 - 48 T^{2} + T^{4}$$
$29$ $$( -6 + T^{2} )^{2}$$
$31$ $$( -32 + T^{2} )^{2}$$
$37$ $$4 - 28 T^{2} + T^{4}$$
$41$ $$( -48 + T^{2} )^{2}$$
$43$ $$( -18 + 6 T + T^{2} )^{2}$$
$47$ $$( -96 + T^{2} )^{2}$$
$53$ $$4356 - 156 T^{2} + T^{4}$$
$59$ $$( 78 - 18 T + T^{2} )^{2}$$
$61$ $$4 - 28 T^{2} + T^{4}$$
$67$ $$( -26 + 2 T + T^{2} )^{2}$$
$71$ $$144 - 48 T^{2} + T^{4}$$
$73$ $$( 24 + 12 T + T^{2} )^{2}$$
$79$ $$4096 - 256 T^{2} + T^{4}$$
$83$ $$( -18 - 6 T + T^{2} )^{2}$$
$89$ $$( 24 - 12 T + T^{2} )^{2}$$
$97$ $$( -12 + T^{2} )^{2}$$