Properties

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

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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 2304) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} -\beta_{1} q^{7} +O(q^{10})$$ $$q + \beta_{2} q^{5} -\beta_{1} q^{7} + \beta_{3} q^{11} -\beta_{1} q^{13} -\beta_{3} q^{17} -6 q^{19} -2 \beta_{2} q^{23} + 7 q^{25} -\beta_{2} q^{29} + \beta_{1} q^{31} -\beta_{3} q^{35} + 7 \beta_{1} q^{37} -\beta_{3} q^{41} -6 q^{43} -2 \beta_{2} q^{47} -5 q^{49} + \beta_{2} q^{53} + 12 \beta_{1} q^{55} + 2 \beta_{3} q^{59} + 5 \beta_{1} q^{61} -\beta_{3} q^{65} + 8 q^{67} -4 \beta_{2} q^{71} -12 q^{73} -2 \beta_{2} q^{77} -11 \beta_{1} q^{79} -3 \beta_{3} q^{83} -12 \beta_{1} q^{85} -2 \beta_{3} q^{89} + 2 q^{91} -6 \beta_{2} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 24q^{19} + 28q^{25} - 24q^{43} - 20q^{49} + 32q^{67} - 48q^{73} + 8q^{91} + O(q^{100})$$

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
 −0.517638 0.517638 1.93185 −1.93185
0 0 0 −3.46410 0 −1.41421 0 0 0
1.2 0 0 0 −3.46410 0 1.41421 0 0 0
1.3 0 0 0 3.46410 0 −1.41421 0 0 0
1.4 0 0 0 3.46410 0 1.41421 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
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 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.be 4
3.b odd 2 1 inner 9216.2.a.be 4
4.b odd 2 1 9216.2.a.bh 4
8.b even 2 1 9216.2.a.bh 4
8.d odd 2 1 inner 9216.2.a.be 4
12.b even 2 1 9216.2.a.bh 4
24.f even 2 1 inner 9216.2.a.be 4
24.h odd 2 1 9216.2.a.bh 4
32.g even 8 2 2304.2.k.h 8
32.g even 8 2 2304.2.k.i yes 8
32.h odd 8 2 2304.2.k.h 8
32.h odd 8 2 2304.2.k.i yes 8
96.o even 8 2 2304.2.k.h 8
96.o even 8 2 2304.2.k.i yes 8
96.p odd 8 2 2304.2.k.h 8
96.p odd 8 2 2304.2.k.i yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.2.k.h 8 32.g even 8 2
2304.2.k.h 8 32.h odd 8 2
2304.2.k.h 8 96.o even 8 2
2304.2.k.h 8 96.p odd 8 2
2304.2.k.i yes 8 32.g even 8 2
2304.2.k.i yes 8 32.h odd 8 2
2304.2.k.i yes 8 96.o even 8 2
2304.2.k.i yes 8 96.p odd 8 2
9216.2.a.be 4 1.a even 1 1 trivial
9216.2.a.be 4 3.b odd 2 1 inner
9216.2.a.be 4 8.d odd 2 1 inner
9216.2.a.be 4 24.f even 2 1 inner
9216.2.a.bh 4 4.b odd 2 1
9216.2.a.bh 4 8.b even 2 1
9216.2.a.bh 4 12.b even 2 1
9216.2.a.bh 4 24.h odd 2 1

Hecke kernels

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

 $$T_{5}^{2} - 12$$ $$T_{7}^{2} - 2$$ $$T_{11}^{2} - 24$$ $$T_{13}^{2} - 2$$ $$T_{17}^{2} - 24$$ $$T_{19} + 6$$ $$T_{67} - 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -12 + T^{2} )^{2}$$
$7$ $$( -2 + T^{2} )^{2}$$
$11$ $$( -24 + T^{2} )^{2}$$
$13$ $$( -2 + T^{2} )^{2}$$
$17$ $$( -24 + T^{2} )^{2}$$
$19$ $$( 6 + T )^{4}$$
$23$ $$( -48 + T^{2} )^{2}$$
$29$ $$( -12 + T^{2} )^{2}$$
$31$ $$( -2 + T^{2} )^{2}$$
$37$ $$( -98 + T^{2} )^{2}$$
$41$ $$( -24 + T^{2} )^{2}$$
$43$ $$( 6 + T )^{4}$$
$47$ $$( -48 + T^{2} )^{2}$$
$53$ $$( -12 + T^{2} )^{2}$$
$59$ $$( -96 + T^{2} )^{2}$$
$61$ $$( -50 + T^{2} )^{2}$$
$67$ $$( -8 + T )^{4}$$
$71$ $$( -192 + T^{2} )^{2}$$
$73$ $$( 12 + T )^{4}$$
$79$ $$( -242 + T^{2} )^{2}$$
$83$ $$( -216 + T^{2} )^{2}$$
$89$ $$( -96 + T^{2} )^{2}$$
$97$ $$T^{4}$$