# Properties

 Label 9216.2.a.bq Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + ( - \beta_{5} - 1) q^{7}+O(q^{10})$$ q + b2 * q^5 + (-b5 - 1) * q^7 $$q + \beta_{2} q^{5} + ( - \beta_{5} - 1) q^{7} + (\beta_{7} - \beta_{2}) q^{11} + \beta_1 q^{13} + \beta_{3} q^{17} + (\beta_{4} - \beta_1) q^{19} + (\beta_{6} - \beta_{3}) q^{23} + (2 \beta_{5} + 1) q^{25} + ( - 2 \beta_{7} + \beta_{2}) q^{29} + (\beta_{5} - 3) q^{31} + ( - \beta_{7} - 3 \beta_{2}) q^{35} + ( - 2 \beta_{4} - \beta_1) q^{37} + ( - 2 \beta_{6} - \beta_{3}) q^{41} + ( - 3 \beta_{4} - \beta_1) q^{43} + ( - 3 \beta_{6} - \beta_{3}) q^{47} + (2 \beta_{5} + 1) q^{49} + ( - 2 \beta_{7} - \beta_{2}) q^{53} - 4 q^{55} + ( - 2 \beta_{7} - 2 \beta_{2}) q^{59} + (2 \beta_{4} + \beta_1) q^{61} + (2 \beta_{6} - \beta_{3}) q^{65} + 4 \beta_{4} q^{67} + (2 \beta_{6} + 2 \beta_{3}) q^{71} + (2 \beta_{5} + 2) q^{73} + 2 \beta_{7} q^{77} + (\beta_{5} - 7) q^{79} + (\beta_{7} - \beta_{2}) q^{83} + ( - 2 \beta_{4} - 2 \beta_1) q^{85} + 2 \beta_{6} q^{89} + ( - 7 \beta_{4} - \beta_1) q^{91} + ( - \beta_{6} + \beta_{3}) q^{95} + 4 \beta_{5} q^{97}+O(q^{100})$$ q + b2 * q^5 + (-b5 - 1) * q^7 + (b7 - b2) * q^11 + b1 * q^13 + b3 * q^17 + (b4 - b1) * q^19 + (b6 - b3) * q^23 + (2*b5 + 1) * q^25 + (-2*b7 + b2) * q^29 + (b5 - 3) * q^31 + (-b7 - 3*b2) * q^35 + (-2*b4 - b1) * q^37 + (-2*b6 - b3) * q^41 + (-3*b4 - b1) * q^43 + (-3*b6 - b3) * q^47 + (2*b5 + 1) * q^49 + (-2*b7 - b2) * q^53 - 4 * q^55 + (-2*b7 - 2*b2) * q^59 + (2*b4 + b1) * q^61 + (2*b6 - b3) * q^65 + 4*b4 * q^67 + (2*b6 + 2*b3) * q^71 + (2*b5 + 2) * q^73 + 2*b7 * q^77 + (b5 - 7) * q^79 + (b7 - b2) * q^83 + (-2*b4 - 2*b1) * q^85 + 2*b6 * q^89 + (-7*b4 - b1) * q^91 + (-b6 + b3) * q^95 + 4*b5 * 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 $$\beta_{2}$$ $$=$$ $$( -18\nu^{7} + 83\nu^{6} + 94\nu^{5} - 492\nu^{4} - 246\nu^{3} + 452\nu^{2} + 106\nu - 66 ) / 3$$ (-18*v^7 + 83*v^6 + 94*v^5 - 492*v^4 - 246*v^3 + 452*v^2 + 106*v - 66) / 3 $$\beta_{3}$$ $$=$$ $$( 20\nu^{7} - 90\nu^{6} - 114\nu^{5} + 532\nu^{4} + 332\nu^{3} - 462\nu^{2} - 168\nu + 64 ) / 3$$ (20*v^7 - 90*v^6 - 114*v^5 + 532*v^4 + 332*v^3 - 462*v^2 - 168*v + 64) / 3 $$\beta_{4}$$ $$=$$ $$( 20\nu^{7} - 91\nu^{6} - 110\nu^{5} + 541\nu^{4} + 302\nu^{3} - 490\nu^{2} - 128\nu + 76 ) / 3$$ (20*v^7 - 91*v^6 - 110*v^5 + 541*v^4 + 302*v^3 - 490*v^2 - 128*v + 76) / 3 $$\beta_{5}$$ $$=$$ $$( -38\nu^{7} + 174\nu^{6} + 204\nu^{5} - 1033\nu^{4} - 548\nu^{3} + 942\nu^{2} + 240\nu - 145 ) / 3$$ (-38*v^7 + 174*v^6 + 204*v^5 - 1033*v^4 - 548*v^3 + 942*v^2 + 240*v - 145) / 3 $$\beta_{6}$$ $$=$$ $$( 44\nu^{7} - 200\nu^{6} - 242\nu^{5} + 1184\nu^{4} + 668\nu^{3} - 1046\nu^{2} - 284\nu + 152 ) / 3$$ (44*v^7 - 200*v^6 - 242*v^5 + 1184*v^4 + 668*v^3 - 1046*v^2 - 284*v + 152) / 3 $$\beta_{7}$$ $$=$$ $$( -44\nu^{7} + 199\nu^{6} + 248\nu^{5} - 1186\nu^{4} - 698\nu^{3} + 1066\nu^{2} + 314\nu - 154 ) / 3$$ (-44*v^7 + 199*v^6 + 248*v^5 - 1186*v^4 - 698*v^3 + 1066*v^2 + 314*v - 154) / 3
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} - \beta_{2} + 1 ) / 2$$ (b5 + b4 - b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} + \beta _1 + 8 ) / 2$$ (-b7 - b6 + 2*b5 + b4 - 3*b2 + b1 + 8) / 2 $$\nu^{3}$$ $$=$$ $$( -4\beta_{7} - 9\beta_{6} + 20\beta_{5} + 22\beta_{4} + 3\beta_{3} - 26\beta_{2} + 6\beta _1 + 44 ) / 4$$ (-4*b7 - 9*b6 + 20*b5 + 22*b4 + 3*b3 - 26*b2 + 6*b1 + 44) / 4 $$\nu^{4}$$ $$=$$ $$-6\beta_{7} - 12\beta_{6} + 17\beta_{5} + 21\beta_{4} + 2\beta_{3} - 24\beta_{2} + 9\beta _1 + 49$$ -6*b7 - 12*b6 + 17*b5 + 21*b4 + 2*b3 - 24*b2 + 9*b1 + 49 $$\nu^{5}$$ $$=$$ $$( -39\beta_{7} - 105\beta_{6} + 144\beta_{5} + 209\beta_{4} + 25\beta_{3} - 197\beta_{2} + 75\beta _1 + 366 ) / 2$$ (-39*b7 - 105*b6 + 144*b5 + 209*b4 + 25*b3 - 197*b2 + 75*b1 + 366) / 2 $$\nu^{6}$$ $$=$$ $$( -176\beta_{7} - 473\beta_{6} + 566\beta_{5} + 890\beta_{4} + 97\beta_{3} - 786\beta_{2} + 344\beta _1 + 1526 ) / 2$$ (-176*b7 - 473*b6 + 566*b5 + 890*b4 + 97*b3 - 786*b2 + 344*b1 + 1526) / 2 $$\nu^{7}$$ $$=$$ $$( - 685 \beta_{7} - 2037 \beta_{6} + 2352 \beta_{5} + 3928 \beta_{4} + 448 \beta_{3} - 3245 \beta_{2} + 1470 \beta _1 + 6168 ) / 2$$ (-685*b7 - 2037*b6 + 2352*b5 + 3928*b4 + 448*b3 - 3245*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
 2.79591 4.21012 0.305093 −1.10912 −0.536630 −1.95084 −0.564373 0.849841
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.bq 8
3.b odd 2 1 inner 9216.2.a.bq 8
4.b odd 2 1 9216.2.a.bt 8
8.b even 2 1 inner 9216.2.a.bq 8
8.d odd 2 1 9216.2.a.bt 8
12.b even 2 1 9216.2.a.bt 8
24.f even 2 1 9216.2.a.bt 8
24.h odd 2 1 inner 9216.2.a.bq 8
32.g even 8 2 144.2.k.c 8
32.g even 8 2 1152.2.k.e 8
32.h odd 8 2 576.2.k.c 8
32.h odd 8 2 1152.2.k.d 8
96.o even 8 2 576.2.k.c 8
96.o even 8 2 1152.2.k.d 8
96.p odd 8 2 144.2.k.c 8
96.p odd 8 2 1152.2.k.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.k.c 8 32.g even 8 2
144.2.k.c 8 96.p odd 8 2
576.2.k.c 8 32.h odd 8 2
576.2.k.c 8 96.o even 8 2
1152.2.k.d 8 32.h odd 8 2
1152.2.k.d 8 96.o even 8 2
1152.2.k.e 8 32.g even 8 2
1152.2.k.e 8 96.p odd 8 2
9216.2.a.bq 8 1.a even 1 1 trivial
9216.2.a.bq 8 3.b odd 2 1 inner
9216.2.a.bq 8 8.b even 2 1 inner
9216.2.a.bq 8 24.h odd 2 1 inner
9216.2.a.bt 8 4.b odd 2 1
9216.2.a.bt 8 8.d odd 2 1
9216.2.a.bt 8 12.b even 2 1
9216.2.a.bt 8 24.f even 2 1

## Hecke kernels

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

 $$T_{5}^{4} - 12T_{5}^{2} + 8$$ 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

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