# Properties

 Label 288.2.p.a Level $288$ Weight $2$ Character orbit 288.p Analytic conductor $2.300$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} + ( 6 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{11} + ( 3 - 6 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 1 - 6 \beta_{1} + 3 \beta_{3} ) q^{19} + 5 \beta_{2} q^{25} + ( -5 + \beta_{3} ) q^{27} + ( -1 + 2 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} ) q^{33} + ( 3 + 4 \beta_{1} + 3 \beta_{2} ) q^{41} + ( 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{43} + ( -7 + 7 \beta_{2} ) q^{49} + ( -6 - \beta_{1} + 7 \beta_{2} - 5 \beta_{3} ) q^{51} + ( -12 + 4 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{57} + ( -3 + 5 \beta_{1} - 3 \beta_{2} ) q^{59} + ( -7 + 3 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} ) q^{67} + ( -1 - 12 \beta_{1} + 6 \beta_{3} ) q^{73} + ( 5 + 5 \beta_{1} - 5 \beta_{2} ) q^{75} + ( -4 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{83} -4 \beta_{3} q^{89} + ( 6 \beta_{1} - 5 \beta_{2} + 6 \beta_{3} ) q^{97} + ( -1 - 12 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{3} + 2q^{9} + 18q^{11} + 4q^{19} + 10q^{25} - 20q^{27} - 14q^{33} + 18q^{41} + 10q^{43} - 14q^{49} - 10q^{51} - 38q^{57} - 18q^{59} - 14q^{67} - 4q^{73} + 10q^{75} + 14q^{81} - 10q^{97} - 16q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 −1.72474 + 0.158919i 0 0 0 0 0 2.94949 0.548188i 0
47.2 0 0.724745 + 1.57313i 0 0 0 0 0 −1.94949 + 2.28024i 0
239.1 0 −1.72474 0.158919i 0 0 0 0 0 2.94949 + 0.548188i 0
239.2 0 0.724745 1.57313i 0 0 0 0 0 −1.94949 2.28024i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
9.d odd 6 1 inner
72.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.p.a 4
3.b odd 2 1 864.2.p.a 4
4.b odd 2 1 72.2.l.a 4
8.b even 2 1 72.2.l.a 4
8.d odd 2 1 CM 288.2.p.a 4
9.c even 3 1 864.2.p.a 4
9.c even 3 1 2592.2.f.a 4
9.d odd 6 1 inner 288.2.p.a 4
9.d odd 6 1 2592.2.f.a 4
12.b even 2 1 216.2.l.a 4
24.f even 2 1 864.2.p.a 4
24.h odd 2 1 216.2.l.a 4
36.f odd 6 1 216.2.l.a 4
36.f odd 6 1 648.2.f.a 4
36.h even 6 1 72.2.l.a 4
36.h even 6 1 648.2.f.a 4
72.j odd 6 1 72.2.l.a 4
72.j odd 6 1 648.2.f.a 4
72.l even 6 1 inner 288.2.p.a 4
72.l even 6 1 2592.2.f.a 4
72.n even 6 1 216.2.l.a 4
72.n even 6 1 648.2.f.a 4
72.p odd 6 1 864.2.p.a 4
72.p odd 6 1 2592.2.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.a 4 4.b odd 2 1
72.2.l.a 4 8.b even 2 1
72.2.l.a 4 36.h even 6 1
72.2.l.a 4 72.j odd 6 1
216.2.l.a 4 12.b even 2 1
216.2.l.a 4 24.h odd 2 1
216.2.l.a 4 36.f odd 6 1
216.2.l.a 4 72.n even 6 1
288.2.p.a 4 1.a even 1 1 trivial
288.2.p.a 4 8.d odd 2 1 CM
288.2.p.a 4 9.d odd 6 1 inner
288.2.p.a 4 72.l even 6 1 inner
648.2.f.a 4 36.f odd 6 1
648.2.f.a 4 36.h even 6 1
648.2.f.a 4 72.j odd 6 1
648.2.f.a 4 72.n even 6 1
864.2.p.a 4 3.b odd 2 1
864.2.p.a 4 9.c even 3 1
864.2.p.a 4 24.f even 2 1
864.2.p.a 4 72.p odd 6 1
2592.2.f.a 4 9.c even 3 1
2592.2.f.a 4 9.d odd 6 1
2592.2.f.a 4 72.l even 6 1
2592.2.f.a 4 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 6 T + T^{2} + 2 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$625 - 450 T + 133 T^{2} - 18 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$361 + 70 T^{2} + T^{4}$$
$19$ $$( -53 - 2 T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$25 + 90 T + 103 T^{2} - 18 T^{3} + T^{4}$$
$43$ $$841 + 290 T + 129 T^{2} - 10 T^{3} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$529 - 414 T + 85 T^{2} + 18 T^{3} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$25 - 70 T + 201 T^{2} + 14 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -215 + 2 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$64 - 8 T^{2} + T^{4}$$
$89$ $$( 32 + T^{2} )^{2}$$
$97$ $$36481 - 1910 T + 291 T^{2} + 10 T^{3} + T^{4}$$