# 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 $4$

# 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: 8.8.10070523904.1 Defining polynomial: $$x^{8} - 4x^{7} - 8x^{6} + 24x^{5} + 30x^{4} - 16x^{3} - 20x^{2} + 2$$ x^8 - 4*x^7 - 8*x^6 + 24*x^5 + 30*x^4 - 16*x^3 - 20*x^2 + 2 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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{5} + ( - \beta_{6} + 1) q^{7}+O(q^{10})$$ q - b5 * q^5 + (-b6 + 1) * q^7 $$q - \beta_{5} q^{5} + ( - \beta_{6} + 1) q^{7} + ( - \beta_{5} - \beta_{2}) q^{11} + \beta_1 q^{13} - \beta_{7} q^{17} + ( - \beta_{3} + \beta_1) q^{19} + ( - \beta_{7} + \beta_{4}) q^{23} + ( - 2 \beta_{6} + 1) q^{25} + ( - \beta_{5} - 2 \beta_{2}) q^{29} + (\beta_{6} + 3) q^{31} + ( - 3 \beta_{5} + \beta_{2}) q^{35} + ( - 2 \beta_{3} - \beta_1) q^{37} + (\beta_{7} + 2 \beta_{4}) q^{41} + (3 \beta_{3} + \beta_1) q^{43} + ( - \beta_{7} - 3 \beta_{4}) q^{47} + ( - 2 \beta_{6} + 1) q^{49} + (\beta_{5} - 2 \beta_{2}) q^{53} + 4 q^{55} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{59} + (2 \beta_{3} + \beta_1) q^{61} + (\beta_{7} - 2 \beta_{4}) q^{65} - 4 \beta_{3} q^{67} + (2 \beta_{7} + 2 \beta_{4}) q^{71} + ( - 2 \beta_{6} + 2) q^{73} + 2 \beta_{2} q^{77} + (\beta_{6} + 7) q^{79} + ( - \beta_{5} - \beta_{2}) q^{83} + ( - 2 \beta_{3} - 2 \beta_1) q^{85} - 2 \beta_{4} q^{89} + (7 \beta_{3} + \beta_1) q^{91} + (\beta_{7} - \beta_{4}) q^{95} - 4 \beta_{6} q^{97}+O(q^{100})$$ q - b5 * q^5 + (-b6 + 1) * q^7 + (-b5 - b2) * q^11 + b1 * q^13 - b7 * q^17 + (-b3 + b1) * q^19 + (-b7 + b4) * q^23 + (-2*b6 + 1) * q^25 + (-b5 - 2*b2) * q^29 + (b6 + 3) * q^31 + (-3*b5 + b2) * q^35 + (-2*b3 - b1) * q^37 + (b7 + 2*b4) * q^41 + (3*b3 + b1) * q^43 + (-b7 - 3*b4) * q^47 + (-2*b6 + 1) * q^49 + (b5 - 2*b2) * q^53 + 4 * q^55 + (-2*b5 + 2*b2) * q^59 + (2*b3 + b1) * q^61 + (b7 - 2*b4) * q^65 - 4*b3 * q^67 + (2*b7 + 2*b4) * q^71 + (-2*b6 + 2) * q^73 + 2*b2 * q^77 + (b6 + 7) * q^79 + (-b5 - b2) * q^83 + (-2*b3 - 2*b1) * q^85 - 2*b4 * q^89 + (7*b3 + b1) * q^91 + (b7 - b4) * q^95 - 4*b6 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{7}+O(q^{10})$$ 8 * q + 8 * q^7 $$8 q + 8 q^{7} + 8 q^{25} + 24 q^{31} + 8 q^{49} + 32 q^{55} + 16 q^{73} + 56 q^{79}+O(q^{100})$$ 8 * q + 8 * q^7 + 8 * q^25 + 24 * q^31 + 8 * q^49 + 32 * q^55 + 16 * q^73 + 56 * q^79

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 8x^{6} + 24x^{5} + 30x^{4} - 16x^{3} - 20x^{2} + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{7} + 9\nu^{6} + 10\nu^{5} - 47\nu^{4} - 26\nu^{3} + 12\nu^{2} + 4\nu + 10 ) / 3$$ (-2*v^7 + 9*v^6 + 10*v^5 - 47*v^4 - 26*v^3 + 12*v^2 + 4*v + 10) / 3 $$\beta_{2}$$ $$=$$ $$( -12\nu^{7} + 55\nu^{6} + 64\nu^{5} - 326\nu^{4} - 168\nu^{3} + 292\nu^{2} + 58\nu - 44 ) / 3$$ (-12*v^7 + 55*v^6 + 64*v^5 - 326*v^4 - 168*v^3 + 292*v^2 + 58*v - 44) / 3 $$\beta_{3}$$ $$=$$ $$( 20\nu^{7} - 91\nu^{6} - 110\nu^{5} + 541\nu^{4} + 302\nu^{3} - 490\nu^{2} - 128\nu + 76 ) / 3$$ (20*v^7 - 91*v^6 - 110*v^5 + 541*v^4 + 302*v^3 - 490*v^2 - 128*v + 76) / 3 $$\beta_{4}$$ $$=$$ $$( -26\nu^{7} + 116\nu^{6} + 154\nu^{5} - 694\nu^{4} - 452\nu^{3} + 614\nu^{2} + 208\nu - 88 ) / 3$$ (-26*v^7 + 116*v^6 + 154*v^5 - 694*v^4 - 452*v^3 + 614*v^2 + 208*v - 88) / 3 $$\beta_{5}$$ $$=$$ $$( -32\nu^{7} + 145\nu^{6} + 178\nu^{5} - 858\nu^{4} - 500\nu^{3} + 754\nu^{2} + 226\nu - 108 ) / 3$$ (-32*v^7 + 145*v^6 + 178*v^5 - 858*v^4 - 500*v^3 + 754*v^2 + 226*v - 108) / 3 $$\beta_{6}$$ $$=$$ $$( -38\nu^{7} + 174\nu^{6} + 204\nu^{5} - 1033\nu^{4} - 548\nu^{3} + 942\nu^{2} + 240\nu - 145 ) / 3$$ (-38*v^7 + 174*v^6 + 204*v^5 - 1033*v^4 - 548*v^3 + 942*v^2 + 240*v - 145) / 3 $$\beta_{7}$$ $$=$$ $$( 62\nu^{7} - 282\nu^{6} - 342\nu^{5} + 1678\nu^{4} + 944\nu^{3} - 1518\nu^{2} - 420\nu + 220 ) / 3$$ (62*v^7 - 282*v^6 - 342*v^5 + 1678*v^4 + 944*v^3 - 1518*v^2 - 420*v + 220) / 3
 $$\nu$$ $$=$$ $$( \beta_{7} + 2\beta_{6} + \beta_{4} + 2\beta_{3} + 2 ) / 4$$ (b7 + 2*b6 + b4 + 2*b3 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 8 ) / 2$$ (2*b7 + 2*b6 + b5 + b4 + b3 + b2 - b1 + 8) / 2 $$\nu^{3}$$ $$=$$ $$( 15\beta_{7} + 20\beta_{6} + 6\beta_{5} + 11\beta_{4} + 22\beta_{3} + 12\beta_{2} - 6\beta _1 + 44 ) / 4$$ (15*b7 + 20*b6 + 6*b5 + 11*b4 + 22*b3 + 12*b2 - 6*b1 + 44) / 4 $$\nu^{4}$$ $$=$$ $$15\beta_{7} + 17\beta_{6} + 10\beta_{5} + 9\beta_{4} + 21\beta_{3} + 14\beta_{2} - 9\beta _1 + 49$$ 15*b7 + 17*b6 + 10*b5 + 9*b4 + 21*b3 + 14*b2 - 9*b1 + 49 $$\nu^{5}$$ $$=$$ $$( 118\beta_{7} + 144\beta_{6} + 80\beta_{5} + 79\beta_{4} + 209\beta_{3} + 130\beta_{2} - 75\beta _1 + 366 ) / 2$$ (118*b7 + 144*b6 + 80*b5 + 79*b4 + 209*b3 + 130*b2 - 75*b1 + 366) / 2 $$\nu^{6}$$ $$=$$ $$( 481\beta_{7} + 566\beta_{6} + 376\beta_{5} + 305\beta_{4} + 890\beta_{3} + 570\beta_{2} - 344\beta _1 + 1526 ) / 2$$ (481*b7 + 566*b6 + 376*b5 + 305*b4 + 890*b3 + 570*b2 - 344*b1 + 1526) / 2 $$\nu^{7}$$ $$=$$ $$( 1965 \beta_{7} + 2352 \beta_{6} + 1589 \beta_{5} + 1280 \beta_{4} + 3928 \beta_{3} + 2485 \beta_{2} - 1470 \beta _1 + 6168 ) / 2$$ (1965*b7 + 2352*b6 + 1589*b5 + 1280*b4 + 3928*b3 + 2485*b2 - 1470*b1 + 6168) / 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
 −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: 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.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9216))$$:

 $$T_{5}^{4} - 12T_{5}^{2} + 8$$ T5^4 - 12*T5^2 + 8 $$T_{7}^{2} - 2T_{7} - 6$$ T7^2 - 2*T7 - 6 $$T_{11}^{4} - 24T_{11}^{2} + 32$$ T11^4 - 24*T11^2 + 32 $$T_{13}^{2} - 14$$ T13^2 - 14 $$T_{17}^{4} - 40T_{17}^{2} + 288$$ T17^4 - 40*T17^2 + 288 $$T_{19}^{4} - 32T_{19}^{2} + 144$$ T19^4 - 32*T19^2 + 144 $$T_{67}^{2} - 32$$ T67^2 - 32

## Hecke characteristic polynomials

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