# Properties

 Label 9216.2.a.bs Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Coefficient field: 8.8.3288334336.1 Defining polynomial: $$x^{8} - 12x^{6} - 8x^{5} + 24x^{4} + 8x^{3} - 16x^{2} + 2$$ x^8 - 12*x^6 - 8*x^5 + 24*x^4 + 8*x^3 - 16*x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 4608) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{5} + \beta_{7} q^{7}+O(q^{10})$$ q - b4 * q^5 + b7 * q^7 $$q - \beta_{4} q^{5} + \beta_{7} q^{7} - \beta_{2} q^{11} + ( - \beta_{5} + 2) q^{13} + ( - \beta_{6} - \beta_{4}) q^{17} + (2 \beta_{7} + \beta_1) q^{19} + (\beta_{3} - \beta_{2}) q^{23} + ( - 2 \beta_{5} + 1) q^{25} + ( - 2 \beta_{6} + \beta_{4}) q^{29} + (\beta_{7} + 2 \beta_1) q^{31} + (2 \beta_{3} + \beta_{2}) q^{35} + (\beta_{5} + 2) q^{37} + (\beta_{6} - 3 \beta_{4}) q^{41} + (2 \beta_{7} - \beta_1) q^{43} + ( - \beta_{3} - 3 \beta_{2}) q^{47} + (2 \beta_{5} + 1) q^{49} + (2 \beta_{6} - \beta_{4}) q^{53} + ( - 2 \beta_{7} - 2 \beta_1) q^{55} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{59} + ( - \beta_{5} + 6) q^{61} + (\beta_{6} - 3 \beta_{4}) q^{65} - 2 \beta_1 q^{67} - 4 \beta_{3} q^{71} + ( - 6 \beta_{5} + 2) q^{73} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{77} - \beta_{7} q^{79} + ( - 4 \beta_{3} - \beta_{2}) q^{83} + (2 \beta_{5} + 8) q^{85} - 4 \beta_{4} q^{89} + (2 \beta_{7} + \beta_1) q^{91} + (3 \beta_{3} + 5 \beta_{2}) q^{95} + (8 \beta_{5} - 4) q^{97}+O(q^{100})$$ q - b4 * q^5 + b7 * q^7 - b2 * q^11 + (-b5 + 2) * q^13 + (-b6 - b4) * q^17 + (2*b7 + b1) * q^19 + (b3 - b2) * q^23 + (-2*b5 + 1) * q^25 + (-2*b6 + b4) * q^29 + (b7 + 2*b1) * q^31 + (2*b3 + b2) * q^35 + (b5 + 2) * q^37 + (b6 - 3*b4) * q^41 + (2*b7 - b1) * q^43 + (-b3 - 3*b2) * q^47 + (2*b5 + 1) * q^49 + (2*b6 - b4) * q^53 + (-2*b7 - 2*b1) * q^55 + (-2*b3 + 2*b2) * q^59 + (-b5 + 6) * q^61 + (b6 - 3*b4) * q^65 - 2*b1 * q^67 - 4*b3 * q^71 + (-6*b5 + 2) * q^73 + (-2*b6 + 2*b4) * q^77 - b7 * q^79 + (-4*b3 - b2) * q^83 + (2*b5 + 8) * q^85 - 4*b4 * q^89 + (2*b7 + b1) * q^91 + (3*b3 + 5*b2) * q^95 + (8*b5 - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 16 q^{13} + 8 q^{25} + 16 q^{37} + 8 q^{49} + 48 q^{61} + 16 q^{73} + 64 q^{85} - 32 q^{97}+O(q^{100})$$ 8 * q + 16 * q^13 + 8 * q^25 + 16 * q^37 + 8 * q^49 + 48 * q^61 + 16 * q^73 + 64 * q^85 - 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 12x^{6} - 8x^{5} + 24x^{4} + 8x^{3} - 16x^{2} + 2$$ :

 $$\beta_{1}$$ $$=$$ $$-2\nu^{6} + 2\nu^{5} + 18\nu^{4} + 4\nu^{3} - 18\nu^{2} - 8\nu + 4$$ -2*v^6 + 2*v^5 + 18*v^4 + 4*v^3 - 18*v^2 - 8*v + 4 $$\beta_{2}$$ $$=$$ $$-2\nu^{7} + 22\nu^{5} + 20\nu^{4} - 32\nu^{3} - 30\nu^{2} + 12\nu + 8$$ -2*v^7 + 22*v^5 + 20*v^4 - 32*v^3 - 30*v^2 + 12*v + 8 $$\beta_{3}$$ $$=$$ $$2\nu^{7} + 2\nu^{6} - 22\nu^{5} - 40\nu^{4} + 12\nu^{3} + 42\nu^{2} - 8$$ 2*v^7 + 2*v^6 - 22*v^5 - 40*v^4 + 12*v^3 + 42*v^2 - 8 $$\beta_{4}$$ $$=$$ $$-2\nu^{7} - \nu^{6} + 24\nu^{5} + 27\nu^{4} - 38\nu^{3} - 32\nu^{2} + 16\nu + 6$$ -2*v^7 - v^6 + 24*v^5 + 27*v^4 - 38*v^3 - 32*v^2 + 16*v + 6 $$\beta_{5}$$ $$=$$ $$2\nu^{7} + 2\nu^{6} - 24\nu^{5} - 37\nu^{4} + 28\nu^{3} + 38\nu^{2} - 12\nu - 6$$ 2*v^7 + 2*v^6 - 24*v^5 - 37*v^4 + 28*v^3 + 38*v^2 - 12*v - 6 $$\beta_{6}$$ $$=$$ $$-4\nu^{7} - \nu^{6} + 44\nu^{5} + 47\nu^{4} - 50\nu^{3} - 42\nu^{2} + 16\nu + 4$$ -4*v^7 - v^6 + 44*v^5 + 47*v^4 - 50*v^3 - 42*v^2 + 16*v + 4 $$\beta_{7}$$ $$=$$ $$4\nu^{7} + 3\nu^{6} - 46\nu^{5} - 66\nu^{4} + 48\nu^{3} + 68\nu^{2} - 14\nu - 14$$ 4*v^7 + 3*v^6 - 46*v^5 - 66*v^4 + 48*v^3 + 68*v^2 - 14*v - 14
 $$\nu$$ $$=$$ $$( -2\beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} ) / 4$$ (-2*b5 - 2*b4 + b3 + b2) / 4 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 6 ) / 2$$ (2*b7 + b6 - 2*b5 - b4 + b2 + b1 + 6) / 2 $$\nu^{3}$$ $$=$$ $$( 6\beta_{7} + 4\beta_{6} - 16\beta_{5} - 16\beta_{4} + 7\beta_{3} + 11\beta_{2} + 6\beta _1 + 12 ) / 4$$ (6*b7 + 4*b6 - 16*b5 - 16*b4 + 7*b3 + 11*b2 + 6*b1 + 12) / 4 $$\nu^{4}$$ $$=$$ $$10\beta_{7} + 5\beta_{6} - 13\beta_{5} - 9\beta_{4} + 2\beta_{3} + 8\beta_{2} + 6\beta _1 + 24$$ 10*b7 + 5*b6 - 13*b5 - 9*b4 + 2*b3 + 8*b2 + 6*b1 + 24 $$\nu^{5}$$ $$=$$ $$( 50\beta_{7} + 29\beta_{6} - 94\beta_{5} - 83\beta_{4} + 32\beta_{3} + 63\beta_{2} + 40\beta _1 + 110 ) / 2$$ (50*b7 + 29*b6 - 94*b5 - 83*b4 + 32*b3 + 63*b2 + 40*b1 + 110) / 2 $$\nu^{6}$$ $$=$$ $$( 218\beta_{7} + 114\beta_{6} - 322\beta_{5} - 248\beta_{4} + 73\beta_{3} + 207\beta_{2} + 144\beta _1 + 504 ) / 2$$ (218*b7 + 114*b6 - 322*b5 - 248*b4 + 73*b3 + 207*b2 + 144*b1 + 504) / 2 $$\nu^{7}$$ $$=$$ $$( 672\beta_{7} + 372\beta_{6} - 1142\beta_{5} - 956\beta_{4} + 339\beta_{3} + 752\beta_{2} + 497\beta _1 + 1512 ) / 2$$ (672*b7 + 372*b6 - 1142*b5 - 956*b4 + 339*b3 + 752*b2 + 497*b1 + 1512) / 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
 0.528036 −2.08509 −1.05636 −2.13875 −0.357857 0.724535 0.886177 3.49930
0 0 0 −2.97127 0 −2.27411 0 0 0
1.2 0 0 0 −2.97127 0 2.27411 0 0 0
1.3 0 0 0 −1.78089 0 −3.29066 0 0 0
1.4 0 0 0 −1.78089 0 3.29066 0 0 0
1.5 0 0 0 1.78089 0 −3.29066 0 0 0
1.6 0 0 0 1.78089 0 3.29066 0 0 0
1.7 0 0 0 2.97127 0 −2.27411 0 0 0
1.8 0 0 0 2.97127 0 2.27411 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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
4.b odd 2 1 inner
12.b 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.bs 8
3.b odd 2 1 inner 9216.2.a.bs 8
4.b odd 2 1 inner 9216.2.a.bs 8
8.b even 2 1 9216.2.a.br 8
8.d odd 2 1 9216.2.a.br 8
12.b even 2 1 inner 9216.2.a.bs 8
24.f even 2 1 9216.2.a.br 8
24.h odd 2 1 9216.2.a.br 8
32.g even 8 2 4608.2.k.bk 16
32.g even 8 2 4608.2.k.bl yes 16
32.h odd 8 2 4608.2.k.bk 16
32.h odd 8 2 4608.2.k.bl yes 16
96.o even 8 2 4608.2.k.bk 16
96.o even 8 2 4608.2.k.bl yes 16
96.p odd 8 2 4608.2.k.bk 16
96.p odd 8 2 4608.2.k.bl yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.k.bk 16 32.g even 8 2
4608.2.k.bk 16 32.h odd 8 2
4608.2.k.bk 16 96.o even 8 2
4608.2.k.bk 16 96.p odd 8 2
4608.2.k.bl yes 16 32.g even 8 2
4608.2.k.bl yes 16 32.h odd 8 2
4608.2.k.bl yes 16 96.o even 8 2
4608.2.k.bl yes 16 96.p odd 8 2
9216.2.a.br 8 8.b even 2 1
9216.2.a.br 8 8.d odd 2 1
9216.2.a.br 8 24.f even 2 1
9216.2.a.br 8 24.h odd 2 1
9216.2.a.bs 8 1.a even 1 1 trivial
9216.2.a.bs 8 3.b odd 2 1 inner
9216.2.a.bs 8 4.b odd 2 1 inner
9216.2.a.bs 8 12.b even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{4} - 12T_{5}^{2} + 28$$ T5^4 - 12*T5^2 + 28 $$T_{7}^{4} - 16T_{7}^{2} + 56$$ T7^4 - 16*T7^2 + 56 $$T_{11}^{4} - 16T_{11}^{2} + 32$$ T11^4 - 16*T11^2 + 32 $$T_{13}^{2} - 4T_{13} + 2$$ T13^2 - 4*T13 + 2 $$T_{17}^{4} - 40T_{17}^{2} + 112$$ T17^4 - 40*T17^2 + 112 $$T_{19}^{4} - 64T_{19}^{2} + 224$$ T19^4 - 64*T19^2 + 224 $$T_{67}^{4} - 128T_{67}^{2} + 3584$$ T67^4 - 128*T67^2 + 3584

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 12 T^{2} + 28)^{2}$$
$7$ $$(T^{4} - 16 T^{2} + 56)^{2}$$
$11$ $$(T^{4} - 16 T^{2} + 32)^{2}$$
$13$ $$(T^{2} - 4 T + 2)^{4}$$
$17$ $$(T^{4} - 40 T^{2} + 112)^{2}$$
$19$ $$(T^{4} - 64 T^{2} + 224)^{2}$$
$23$ $$(T^{4} - 32 T^{2} + 128)^{2}$$
$29$ $$(T^{4} - 76 T^{2} + 1372)^{2}$$
$31$ $$(T^{4} - 112 T^{2} + 2744)^{2}$$
$37$ $$(T^{2} - 4 T + 2)^{4}$$
$41$ $$(T^{4} - 104 T^{2} + 112)^{2}$$
$43$ $$(T^{4} - 128 T^{2} + 224)^{2}$$
$47$ $$(T^{4} - 160 T^{2} + 128)^{2}$$
$53$ $$(T^{4} - 76 T^{2} + 1372)^{2}$$
$59$ $$(T^{4} - 128 T^{2} + 2048)^{2}$$
$61$ $$(T^{2} - 12 T + 34)^{4}$$
$67$ $$(T^{4} - 128 T^{2} + 3584)^{2}$$
$71$ $$(T^{4} - 256 T^{2} + 8192)^{2}$$
$73$ $$(T^{2} - 4 T - 68)^{4}$$
$79$ $$(T^{4} - 16 T^{2} + 56)^{2}$$
$83$ $$(T^{4} - 272 T^{2} + 16928)^{2}$$
$89$ $$(T^{4} - 192 T^{2} + 7168)^{2}$$
$97$ $$(T^{2} + 8 T - 112)^{4}$$