# Properties

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

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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: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6x^{2} + 4$$ x^4 - 6*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + ( - \beta_{2} - \beta_1) q^{7}+O(q^{10})$$ q + b2 * q^5 + (-b2 - b1) * q^7 $$q + \beta_{2} q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 2) q^{11} - \beta_1 q^{13} + (\beta_{3} + 2) q^{17} + (\beta_{3} - 2) q^{19} - 4 \beta_1 q^{23} + 5 q^{25} + (\beta_{2} - 2 \beta_1) q^{29} + ( - \beta_{2} - 5 \beta_1) q^{31} + ( - \beta_{3} - 10) q^{35} + (2 \beta_{2} - 3 \beta_1) q^{37} + ( - \beta_{3} - 2) q^{41} + (\beta_{3} + 6) q^{43} + (2 \beta_{3} + 5) q^{49} + ( - \beta_{2} + 6 \beta_1) q^{53} + ( - 2 \beta_{2} + 10 \beta_1) q^{55} + 2 \beta_{3} q^{59} + (2 \beta_{2} + 3 \beta_1) q^{61} - \beta_{3} q^{65} + 12 q^{67} + ( - 2 \beta_{2} + 2 \beta_1) q^{71} + ( - 2 \beta_{3} - 6) q^{73} - 8 \beta_1 q^{77} + (\beta_{2} - 3 \beta_1) q^{79} + ( - \beta_{3} + 2) q^{83} + (2 \beta_{2} + 10 \beta_1) q^{85} + 10 q^{89} + (\beta_{3} + 2) q^{91} + ( - 2 \beta_{2} + 10 \beta_1) q^{95} + (2 \beta_{3} - 4) q^{97}+O(q^{100})$$ q + b2 * q^5 + (-b2 - b1) * q^7 + (b3 - 2) * q^11 - b1 * q^13 + (b3 + 2) * q^17 + (b3 - 2) * q^19 - 4*b1 * q^23 + 5 * q^25 + (b2 - 2*b1) * q^29 + (-b2 - 5*b1) * q^31 + (-b3 - 10) * q^35 + (2*b2 - 3*b1) * q^37 + (-b3 - 2) * q^41 + (b3 + 6) * q^43 + (2*b3 + 5) * q^49 + (-b2 + 6*b1) * q^53 + (-2*b2 + 10*b1) * q^55 + 2*b3 * q^59 + (2*b2 + 3*b1) * q^61 - b3 * q^65 + 12 * q^67 + (-2*b2 + 2*b1) * q^71 + (-2*b3 - 6) * q^73 - 8*b1 * q^77 + (b2 - 3*b1) * q^79 + (-b3 + 2) * q^83 + (2*b2 + 10*b1) * q^85 + 10 * q^89 + (b3 + 2) * q^91 + (-2*b2 + 10*b1) * q^95 + (2*b3 - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 8 q^{11} + 8 q^{17} - 8 q^{19} + 20 q^{25} - 40 q^{35} - 8 q^{41} + 24 q^{43} + 20 q^{49} + 48 q^{67} - 24 q^{73} + 8 q^{83} + 40 q^{89} + 8 q^{91} - 16 q^{97}+O(q^{100})$$ 4 * q - 8 * q^11 + 8 * q^17 - 8 * q^19 + 20 * q^25 - 40 * q^35 - 8 * q^41 + 24 * q^43 + 20 * q^49 + 48 * q^67 - 24 * q^73 + 8 * q^83 + 40 * q^89 + 8 * q^91 - 16 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 4\nu ) / 2$$ (v^3 - 4*v) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 8\nu ) / 2$$ (-v^3 + 8*v) / 2 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 6$$ 2*v^2 - 6
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 6 ) / 2$$ (b3 + 6) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_{2} + 4\beta_1$$ 2*b2 + 4*b1

## 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
 −0.874032 −2.28825 2.28825 0.874032
0 0 0 −3.16228 0 1.74806 0 0 0
1.2 0 0 0 −3.16228 0 4.57649 0 0 0
1.3 0 0 0 3.16228 0 −4.57649 0 0 0
1.4 0 0 0 3.16228 0 −1.74806 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.bd 4
3.b odd 2 1 3072.2.a.q 4
4.b odd 2 1 9216.2.a.bj 4
8.b even 2 1 9216.2.a.bj 4
8.d odd 2 1 inner 9216.2.a.bd 4
12.b even 2 1 3072.2.a.k 4
24.f even 2 1 3072.2.a.q 4
24.h odd 2 1 3072.2.a.k 4
32.g even 8 2 4608.2.k.bf 8
32.g even 8 2 4608.2.k.bg 8
32.h odd 8 2 4608.2.k.bf 8
32.h odd 8 2 4608.2.k.bg 8
48.i odd 4 2 3072.2.d.g 8
48.k even 4 2 3072.2.d.g 8
96.o even 8 2 1536.2.j.g 8
96.o even 8 2 1536.2.j.h yes 8
96.p odd 8 2 1536.2.j.g 8
96.p odd 8 2 1536.2.j.h yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.g 8 96.o even 8 2
1536.2.j.g 8 96.p odd 8 2
1536.2.j.h yes 8 96.o even 8 2
1536.2.j.h yes 8 96.p odd 8 2
3072.2.a.k 4 12.b even 2 1
3072.2.a.k 4 24.h odd 2 1
3072.2.a.q 4 3.b odd 2 1
3072.2.a.q 4 24.f even 2 1
3072.2.d.g 8 48.i odd 4 2
3072.2.d.g 8 48.k even 4 2
4608.2.k.bf 8 32.g even 8 2
4608.2.k.bf 8 32.h odd 8 2
4608.2.k.bg 8 32.g even 8 2
4608.2.k.bg 8 32.h odd 8 2
9216.2.a.bd 4 1.a even 1 1 trivial
9216.2.a.bd 4 8.d odd 2 1 inner
9216.2.a.bj 4 4.b odd 2 1
9216.2.a.bj 4 8.b even 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 10$$ T5^2 - 10 $$T_{7}^{4} - 24T_{7}^{2} + 64$$ T7^4 - 24*T7^2 + 64 $$T_{11}^{2} + 4T_{11} - 16$$ T11^2 + 4*T11 - 16 $$T_{13}^{2} - 2$$ T13^2 - 2 $$T_{17}^{2} - 4T_{17} - 16$$ T17^2 - 4*T17 - 16 $$T_{19}^{2} + 4T_{19} - 16$$ T19^2 + 4*T19 - 16 $$T_{67} - 12$$ T67 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 10)^{2}$$
$7$ $$T^{4} - 24T^{2} + 64$$
$11$ $$(T^{2} + 4 T - 16)^{2}$$
$13$ $$(T^{2} - 2)^{2}$$
$17$ $$(T^{2} - 4 T - 16)^{2}$$
$19$ $$(T^{2} + 4 T - 16)^{2}$$
$23$ $$(T^{2} - 32)^{2}$$
$29$ $$T^{4} - 36T^{2} + 4$$
$31$ $$T^{4} - 120T^{2} + 1600$$
$37$ $$T^{4} - 116T^{2} + 484$$
$41$ $$(T^{2} + 4 T - 16)^{2}$$
$43$ $$(T^{2} - 12 T + 16)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4} - 164T^{2} + 3844$$
$59$ $$(T^{2} - 80)^{2}$$
$61$ $$T^{4} - 116T^{2} + 484$$
$67$ $$(T - 12)^{4}$$
$71$ $$T^{4} - 96T^{2} + 1024$$
$73$ $$(T^{2} + 12 T - 44)^{2}$$
$79$ $$T^{4} - 56T^{2} + 64$$
$83$ $$(T^{2} - 4 T - 16)^{2}$$
$89$ $$(T - 10)^{4}$$
$97$ $$(T^{2} + 8 T - 64)^{2}$$
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