# Properties

 Label 288.2.i.f Level $288$ Weight $2$ Character orbit 288.i Analytic conductor $2.300$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.170772624.1 Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16$$ x^8 - 3*x^7 + 5*x^6 - 6*x^5 + 6*x^4 - 12*x^3 + 20*x^2 - 24*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_1) q^{3} + (\beta_{7} - \beta_{4} + 1) q^{5} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{7} + \beta_{6}) q^{9}+O(q^{10})$$ q + (b5 - b1) * q^3 + (b7 - b4 + 1) * q^5 + (-2*b5 - b3 - b2) * q^7 + (b7 + b6) * q^9 $$q + (\beta_{5} - \beta_1) q^{3} + (\beta_{7} - \beta_{4} + 1) q^{5} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{7} + \beta_{6}) q^{9} + ( - \beta_{5} + \beta_{3} + 2 \beta_1) q^{11} + (\beta_{7} + \beta_{4} - 1) q^{13} + (2 \beta_{5} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{15} + (\beta_{6} + 1) q^{17} + (\beta_{5} - \beta_{3} + \beta_1) q^{19} + (\beta_{7} - \beta_{6} + \beta_{4} + 3) q^{21} + (2 \beta_{5} + \beta_{3} - \beta_1) q^{23} + (\beta_{7} + \beta_{6} - 4 \beta_{4} + 1) q^{25} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{27} + ( - \beta_{7} - \beta_{6} + 3 \beta_{4} - 1) q^{29} + ( - 2 \beta_{5} - \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{31} + ( - 2 \beta_{7} - \beta_{6} - 5 \beta_{4} + 3) q^{33} + ( - 2 \beta_{5} + 2 \beta_{3} - \beta_{2} - 3 \beta_1) q^{35} + 4 q^{37} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{39} + (\beta_{4} - 1) q^{41} + (\beta_{5} - \beta_{3} - 2 \beta_1) q^{43} + ( - \beta_{7} + \beta_{4} - 9) q^{45} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{47} + ( - 3 \beta_{7} + 8 \beta_{4} - 8) q^{49} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2}) q^{51} - 4 q^{53} + (2 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} - \beta_1) q^{55} + ( - \beta_{7} + \beta_{6} + 8 \beta_{4} - 3) q^{57} + (4 \beta_{5} + 2 \beta_{3} - \beta_{2}) q^{59} + ( - 3 \beta_{7} - 3 \beta_{6} - 5 \beta_{4} - 3) q^{61} + (6 \beta_{5} + 3 \beta_{3} + 3 \beta_{2} - 6 \beta_1) q^{63} + ( - \beta_{7} - \beta_{6} - 7 \beta_{4} - 1) q^{65} + ( - 4 \beta_{5} - 2 \beta_{3} + \beta_{2}) q^{67} + (\beta_{7} + 2 \beta_{6} - 5 \beta_{4} + 3) q^{69} + (2 \beta_{2} + 2 \beta_1) q^{71} + ( - \beta_{6} + 7) q^{73} + ( - \beta_{3} + \beta_{2} + 3 \beta_1) q^{75} + ( - 3 \beta_{7} - 3 \beta_{4} + 3) q^{77} + ( - 4 \beta_{5} + \beta_{3} - \beta_{2} + 4 \beta_1) q^{79} + ( - \beta_{7} - 2 \beta_{6} + 8 \beta_{4} - 9) q^{81} + (3 \beta_{3} + \beta_{2} + 4 \beta_1) q^{83} + (8 \beta_{4} - 8) q^{85} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{87} + ( - 2 \beta_{6} + 6) q^{89} + ( - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{91} + ( - 3 \beta_{7} + 3 \beta_{4} + 9) q^{93} + ( - 8 \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{95} - 9 \beta_{4} q^{97} + (3 \beta_{5} - 3 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100})$$ q + (b5 - b1) * q^3 + (b7 - b4 + 1) * q^5 + (-2*b5 - b3 - b2) * q^7 + (b7 + b6) * q^9 + (-b5 + b3 + 2*b1) * q^11 + (b7 + b4 - 1) * q^13 + (2*b5 + b3 + 2*b2 - b1) * q^15 + (b6 + 1) * q^17 + (b5 - b3 + b1) * q^19 + (b7 - b6 + b4 + 3) * q^21 + (2*b5 + b3 - b1) * q^23 + (b7 + b6 - 4*b4 + 1) * q^25 + (-b5 - b3 + b2) * q^27 + (-b7 - b6 + 3*b4 - 1) * q^29 + (-2*b5 - b3 - 2*b2 + 5*b1) * q^31 + (-2*b7 - b6 - 5*b4 + 3) * q^33 + (-2*b5 + 2*b3 - b2 - 3*b1) * q^35 + 4 * q^37 + (b3 + 2*b2 - b1) * q^39 + (b4 - 1) * q^41 + (b5 - b3 - 2*b1) * q^43 + (-b7 + b4 - 9) * q^45 + (-2*b5 - b3 - b2) * q^47 + (-3*b7 + 8*b4 - 8) * q^49 + (-b5 - 2*b3 - b2) * q^51 - 4 * q^53 + (2*b5 - 2*b3 - 3*b2 - b1) * q^55 + (-b7 + b6 + 8*b4 - 3) * q^57 + (4*b5 + 2*b3 - b2) * q^59 + (-3*b7 - 3*b6 - 5*b4 - 3) * q^61 + (6*b5 + 3*b3 + 3*b2 - 6*b1) * q^63 + (-b7 - b6 - 7*b4 - 1) * q^65 + (-4*b5 - 2*b3 + b2) * q^67 + (b7 + 2*b6 - 5*b4 + 3) * q^69 + (2*b2 + 2*b1) * q^71 + (-b6 + 7) * q^73 + (-b3 + b2 + 3*b1) * q^75 + (-3*b7 - 3*b4 + 3) * q^77 + (-4*b5 + b3 - b2 + 4*b1) * q^79 + (-b7 - 2*b6 + 8*b4 - 9) * q^81 + (3*b3 + b2 + 4*b1) * q^83 + (8*b4 - 8) * q^85 + (b3 - b2 - 2*b1) * q^87 + (-2*b6 + 6) * q^89 + (-2*b5 + 2*b3 + b2 - b1) * q^91 + (-3*b7 + 3*b4 + 9) * q^93 + (-8*b5 - 4*b3 + 4*b1) * q^95 - 9*b4 * q^97 + (3*b5 - 3*b2 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{5} - 6 q^{9}+O(q^{10})$$ 8 * q + 2 * q^5 - 6 * q^9 $$8 q + 2 q^{5} - 6 q^{9} - 6 q^{13} + 4 q^{17} + 30 q^{21} - 14 q^{25} + 10 q^{29} + 12 q^{33} + 32 q^{37} - 4 q^{41} - 66 q^{45} - 26 q^{49} - 32 q^{53} + 6 q^{57} - 26 q^{61} - 30 q^{65} - 6 q^{69} + 60 q^{73} + 18 q^{77} - 30 q^{81} - 32 q^{85} + 56 q^{89} + 90 q^{93} - 36 q^{97}+O(q^{100})$$ 8 * q + 2 * q^5 - 6 * q^9 - 6 * q^13 + 4 * q^17 + 30 * q^21 - 14 * q^25 + 10 * q^29 + 12 * q^33 + 32 * q^37 - 4 * q^41 - 66 * q^45 - 26 * q^49 - 32 * q^53 + 6 * q^57 - 26 * q^61 - 30 * q^65 - 6 * q^69 + 60 * q^73 + 18 * q^77 - 30 * q^81 - 32 * q^85 + 56 * q^89 + 90 * q^93 - 36 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + \nu^{6} + \nu^{5} + 2\nu^{3} + 4\nu^{2} + 4\nu - 8 ) / 8$$ (-v^7 + v^6 + v^5 + 2*v^3 + 4*v^2 + 4*v - 8) / 8 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - \nu^{6} - \nu^{5} + 6\nu^{4} - 2\nu^{3} + 8\nu^{2} + 4\nu + 8 ) / 8$$ (-v^7 - v^6 - v^5 + 6*v^4 - 2*v^3 + 8*v^2 + 4*v + 8) / 8 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + \nu^{6} - 3\nu^{5} - 6\nu^{3} + 4\nu + 8 ) / 8$$ (-v^7 + v^6 - 3*v^5 - 6*v^3 + 4*v + 8) / 8 $$\beta_{4}$$ $$=$$ $$( -3\nu^{7} + 5\nu^{6} - 7\nu^{5} + 6\nu^{4} - 10\nu^{3} + 24\nu^{2} - 28\nu + 32 ) / 8$$ (-3*v^7 + 5*v^6 - 7*v^5 + 6*v^4 - 10*v^3 + 24*v^2 - 28*v + 32) / 8 $$\beta_{5}$$ $$=$$ $$( -2\nu^{7} + 3\nu^{6} - 5\nu^{5} + 3\nu^{4} - 4\nu^{3} + 16\nu^{2} - 16\nu + 20 ) / 4$$ (-2*v^7 + 3*v^6 - 5*v^5 + 3*v^4 - 4*v^3 + 16*v^2 - 16*v + 20) / 4 $$\beta_{6}$$ $$=$$ $$( \nu^{7} - 4\nu^{6} + 6\nu^{5} - 5\nu^{4} + 6\nu^{3} - 14\nu^{2} + 20\nu - 24 ) / 4$$ (v^7 - 4*v^6 + 6*v^5 - 5*v^4 + 6*v^3 - 14*v^2 + 20*v - 24) / 4 $$\beta_{7}$$ $$=$$ $$( 5\nu^{7} - 7\nu^{6} + 13\nu^{5} - 10\nu^{4} + 10\nu^{3} - 28\nu^{2} + 44\nu - 56 ) / 8$$ (5*v^7 - 7*v^6 + 13*v^5 - 10*v^4 + 10*v^3 - 28*v^2 + 44*v - 56) / 8
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 3 ) / 6$$ (b7 - b6 + b5 - 2*b4 + 2*b3 + b2 + 2*b1 + 3) / 6 $$\nu^{2}$$ $$=$$ $$( 3\beta_{7} + 4\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 ) / 6$$ (3*b7 + 4*b5 - b3 + b2 - b1) / 6 $$\nu^{3}$$ $$=$$ $$( -2\beta_{7} - \beta_{6} + 3\beta_{5} - 8\beta_{4} - 3\beta_{3} + 3\beta _1 + 3 ) / 6$$ (-2*b7 - b6 + 3*b5 - 8*b4 - 3*b3 + 3*b1 + 3) / 6 $$\nu^{4}$$ $$=$$ $$( -3\beta_{7} - 3\beta_{6} - 7\beta_{5} - 2\beta_{3} + 5\beta_{2} + 4\beta _1 - 3 ) / 6$$ (-3*b7 - 3*b6 - 7*b5 - 2*b3 + 5*b2 + 4*b1 - 3) / 6 $$\nu^{5}$$ $$=$$ $$( \beta_{7} + 2\beta_{6} - 10\beta_{5} + 16\beta_{4} - 5\beta_{3} - \beta_{2} + 7\beta _1 + 18 ) / 6$$ (b7 + 2*b6 - 10*b5 + 16*b4 - 5*b3 - b2 + 7*b1 + 18) / 6 $$\nu^{6}$$ $$=$$ $$( -3\beta_{6} - 3\beta_{5} + \beta_{3} - 2\beta_{2} + 7\beta _1 + 5 ) / 2$$ (-3*b6 - 3*b5 + b3 - 2*b2 + 7*b1 + 5) / 2 $$\nu^{7}$$ $$=$$ $$( 13\beta_{7} - 13\beta_{6} + 7\beta_{5} - 8\beta_{4} - 4\beta_{3} + \beta_{2} - 10\beta _1 + 3 ) / 6$$ (13*b7 - 13*b6 + 7*b5 - 8*b4 - 4*b3 + b2 - 10*b1 + 3) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.335728 − 1.37379i 1.41203 + 0.0786378i 0.774115 + 1.18353i −1.02187 + 0.977642i 0.335728 + 1.37379i 1.41203 − 0.0786378i 0.774115 − 1.18353i −1.02187 − 0.977642i
0 −1.35760 + 1.07561i 0 −1.18614 2.05446i 0 −1.10489 + 1.91373i 0 0.686141 2.92048i 0
97.2 0 −0.637910 1.61030i 0 1.68614 + 2.92048i 0 −2.35143 + 4.07279i 0 −2.18614 + 2.05446i 0
97.3 0 0.637910 + 1.61030i 0 1.68614 + 2.92048i 0 2.35143 4.07279i 0 −2.18614 + 2.05446i 0
97.4 0 1.35760 1.07561i 0 −1.18614 2.05446i 0 1.10489 1.91373i 0 0.686141 2.92048i 0
193.1 0 −1.35760 1.07561i 0 −1.18614 + 2.05446i 0 −1.10489 1.91373i 0 0.686141 + 2.92048i 0
193.2 0 −0.637910 + 1.61030i 0 1.68614 2.92048i 0 −2.35143 4.07279i 0 −2.18614 2.05446i 0
193.3 0 0.637910 1.61030i 0 1.68614 2.92048i 0 2.35143 + 4.07279i 0 −2.18614 2.05446i 0
193.4 0 1.35760 + 1.07561i 0 −1.18614 + 2.05446i 0 1.10489 + 1.91373i 0 0.686141 + 2.92048i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.i.f 8
3.b odd 2 1 864.2.i.f 8
4.b odd 2 1 inner 288.2.i.f 8
8.b even 2 1 576.2.i.n 8
8.d odd 2 1 576.2.i.n 8
9.c even 3 1 inner 288.2.i.f 8
9.c even 3 1 2592.2.a.u 4
9.d odd 6 1 864.2.i.f 8
9.d odd 6 1 2592.2.a.x 4
12.b even 2 1 864.2.i.f 8
24.f even 2 1 1728.2.i.n 8
24.h odd 2 1 1728.2.i.n 8
36.f odd 6 1 inner 288.2.i.f 8
36.f odd 6 1 2592.2.a.u 4
36.h even 6 1 864.2.i.f 8
36.h even 6 1 2592.2.a.x 4
72.j odd 6 1 1728.2.i.n 8
72.j odd 6 1 5184.2.a.cc 4
72.l even 6 1 1728.2.i.n 8
72.l even 6 1 5184.2.a.cc 4
72.n even 6 1 576.2.i.n 8
72.n even 6 1 5184.2.a.cf 4
72.p odd 6 1 576.2.i.n 8
72.p odd 6 1 5184.2.a.cf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.f 8 1.a even 1 1 trivial
288.2.i.f 8 4.b odd 2 1 inner
288.2.i.f 8 9.c even 3 1 inner
288.2.i.f 8 36.f odd 6 1 inner
576.2.i.n 8 8.b even 2 1
576.2.i.n 8 8.d odd 2 1
576.2.i.n 8 72.n even 6 1
576.2.i.n 8 72.p odd 6 1
864.2.i.f 8 3.b odd 2 1
864.2.i.f 8 9.d odd 6 1
864.2.i.f 8 12.b even 2 1
864.2.i.f 8 36.h even 6 1
1728.2.i.n 8 24.f even 2 1
1728.2.i.n 8 24.h odd 2 1
1728.2.i.n 8 72.j odd 6 1
1728.2.i.n 8 72.l even 6 1
2592.2.a.u 4 9.c even 3 1
2592.2.a.u 4 36.f odd 6 1
2592.2.a.x 4 9.d odd 6 1
2592.2.a.x 4 36.h even 6 1
5184.2.a.cc 4 72.j odd 6 1
5184.2.a.cc 4 72.l even 6 1
5184.2.a.cf 4 72.n even 6 1
5184.2.a.cf 4 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{3} + 9T_{5}^{2} + 8T_{5} + 64$$ T5^4 - T5^3 + 9*T5^2 + 8*T5 + 64 $$T_{7}^{8} + 27T_{7}^{6} + 621T_{7}^{4} + 2916T_{7}^{2} + 11664$$ T7^8 + 27*T7^6 + 621*T7^4 + 2916*T7^2 + 11664

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 3 T^{6} + 12 T^{4} + 27 T^{2} + \cdots + 81$$
$5$ $$(T^{4} - T^{3} + 9 T^{2} + 8 T + 64)^{2}$$
$7$ $$T^{8} + 27 T^{6} + 621 T^{4} + \cdots + 11664$$
$11$ $$T^{8} + 36 T^{6} + 1269 T^{4} + \cdots + 729$$
$13$ $$(T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36)^{2}$$
$17$ $$(T^{2} - T - 8)^{4}$$
$19$ $$(T^{4} - 45 T^{2} + 432)^{2}$$
$23$ $$T^{8} + 27 T^{6} + 621 T^{4} + \cdots + 11664$$
$29$ $$(T^{4} - 5 T^{3} + 27 T^{2} + 10 T + 4)^{2}$$
$31$ $$T^{8} + 135 T^{6} + \cdots + 15116544$$
$37$ $$(T - 4)^{8}$$
$41$ $$(T^{2} + T + 1)^{4}$$
$43$ $$T^{8} + 36 T^{6} + 1269 T^{4} + \cdots + 729$$
$47$ $$T^{8} + 27 T^{6} + 621 T^{4} + \cdots + 11664$$
$53$ $$(T + 4)^{8}$$
$59$ $$T^{8} + 180 T^{6} + \cdots + 60886809$$
$61$ $$(T^{4} + 13 T^{3} + 201 T^{2} - 416 T + 1024)^{2}$$
$67$ $$T^{8} + 180 T^{6} + \cdots + 60886809$$
$71$ $$(T^{4} - 108 T^{2} + 1728)^{2}$$
$73$ $$(T^{2} - 15 T + 48)^{4}$$
$79$ $$T^{8} + 135 T^{6} + \cdots + 15116544$$
$83$ $$T^{8} + 207 T^{6} + 41121 T^{4} + \cdots + 2985984$$
$89$ $$(T^{2} - 14 T + 16)^{4}$$
$97$ $$(T^{2} + 9 T + 81)^{4}$$