# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} -\beta_{3} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} -\beta_{3} q^{7} -2 \beta_{1} q^{11} -\beta_{3} q^{13} -\beta_{1} q^{17} + 4 q^{19} -2 \beta_{2} q^{23} + q^{25} + \beta_{2} q^{29} + \beta_{3} q^{31} + 6 \beta_{1} q^{35} -\beta_{1} q^{41} -8 q^{43} + 2 \beta_{2} q^{47} -5 q^{49} + 3 \beta_{2} q^{53} + 2 \beta_{3} q^{55} -8 \beta_{1} q^{59} + 4 \beta_{3} q^{61} + 6 \beta_{1} q^{65} + 4 q^{67} + 6 \beta_{2} q^{71} -4 q^{73} -4 \beta_{2} q^{77} -\beta_{3} q^{79} + 10 \beta_{1} q^{83} + \beta_{3} q^{85} + 5 \beta_{1} q^{89} -12 q^{91} -4 \beta_{2} q^{95} + 8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 16q^{19} + 4q^{25} - 32q^{43} - 20q^{49} + 16q^{67} - 16q^{73} - 48q^{91} + 32q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i −1.22474 + 0.707107i
0 0 0 −2.44949 0 3.46410i 0 0 0
143.2 0 0 0 −2.44949 0 3.46410i 0 0 0
143.3 0 0 0 2.44949 0 3.46410i 0 0 0
143.4 0 0 0 2.44949 0 3.46410i 0 0 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
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.f.a 4
3.b odd 2 1 inner 288.2.f.a 4
4.b odd 2 1 72.2.f.a 4
5.b even 2 1 7200.2.b.c 4
5.c odd 4 2 7200.2.m.c 8
8.b even 2 1 72.2.f.a 4
8.d odd 2 1 inner 288.2.f.a 4
9.c even 3 1 2592.2.p.a 4
9.c even 3 1 2592.2.p.c 4
9.d odd 6 1 2592.2.p.a 4
9.d odd 6 1 2592.2.p.c 4
12.b even 2 1 72.2.f.a 4
15.d odd 2 1 7200.2.b.c 4
15.e even 4 2 7200.2.m.c 8
16.e even 4 2 2304.2.c.i 8
16.f odd 4 2 2304.2.c.i 8
20.d odd 2 1 1800.2.b.c 4
20.e even 4 2 1800.2.m.c 8
24.f even 2 1 inner 288.2.f.a 4
24.h odd 2 1 72.2.f.a 4
36.f odd 6 1 648.2.l.a 4
36.f odd 6 1 648.2.l.c 4
36.h even 6 1 648.2.l.a 4
36.h even 6 1 648.2.l.c 4
40.e odd 2 1 7200.2.b.c 4
40.f even 2 1 1800.2.b.c 4
40.i odd 4 2 1800.2.m.c 8
40.k even 4 2 7200.2.m.c 8
48.i odd 4 2 2304.2.c.i 8
48.k even 4 2 2304.2.c.i 8
60.h even 2 1 1800.2.b.c 4
60.l odd 4 2 1800.2.m.c 8
72.j odd 6 1 648.2.l.a 4
72.j odd 6 1 648.2.l.c 4
72.l even 6 1 2592.2.p.a 4
72.l even 6 1 2592.2.p.c 4
72.n even 6 1 648.2.l.a 4
72.n even 6 1 648.2.l.c 4
72.p odd 6 1 2592.2.p.a 4
72.p odd 6 1 2592.2.p.c 4
120.i odd 2 1 1800.2.b.c 4
120.m even 2 1 7200.2.b.c 4
120.q odd 4 2 7200.2.m.c 8
120.w even 4 2 1800.2.m.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.f.a 4 4.b odd 2 1
72.2.f.a 4 8.b even 2 1
72.2.f.a 4 12.b even 2 1
72.2.f.a 4 24.h odd 2 1
288.2.f.a 4 1.a even 1 1 trivial
288.2.f.a 4 3.b odd 2 1 inner
288.2.f.a 4 8.d odd 2 1 inner
288.2.f.a 4 24.f even 2 1 inner
648.2.l.a 4 36.f odd 6 1
648.2.l.a 4 36.h even 6 1
648.2.l.a 4 72.j odd 6 1
648.2.l.a 4 72.n even 6 1
648.2.l.c 4 36.f odd 6 1
648.2.l.c 4 36.h even 6 1
648.2.l.c 4 72.j odd 6 1
648.2.l.c 4 72.n even 6 1
1800.2.b.c 4 20.d odd 2 1
1800.2.b.c 4 40.f even 2 1
1800.2.b.c 4 60.h even 2 1
1800.2.b.c 4 120.i odd 2 1
1800.2.m.c 8 20.e even 4 2
1800.2.m.c 8 40.i odd 4 2
1800.2.m.c 8 60.l odd 4 2
1800.2.m.c 8 120.w even 4 2
2304.2.c.i 8 16.e even 4 2
2304.2.c.i 8 16.f odd 4 2
2304.2.c.i 8 48.i odd 4 2
2304.2.c.i 8 48.k even 4 2
2592.2.p.a 4 9.c even 3 1
2592.2.p.a 4 9.d odd 6 1
2592.2.p.a 4 72.l even 6 1
2592.2.p.a 4 72.p odd 6 1
2592.2.p.c 4 9.c even 3 1
2592.2.p.c 4 9.d odd 6 1
2592.2.p.c 4 72.l even 6 1
2592.2.p.c 4 72.p odd 6 1
7200.2.b.c 4 5.b even 2 1
7200.2.b.c 4 15.d odd 2 1
7200.2.b.c 4 40.e odd 2 1
7200.2.b.c 4 120.m even 2 1
7200.2.m.c 8 5.c odd 4 2
7200.2.m.c 8 15.e even 4 2
7200.2.m.c 8 40.k even 4 2
7200.2.m.c 8 120.q odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -6 + T^{2} )^{2}$$
$7$ $$( 12 + T^{2} )^{2}$$
$11$ $$( 8 + T^{2} )^{2}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$( 2 + T^{2} )^{2}$$
$19$ $$( -4 + T )^{4}$$
$23$ $$( -24 + T^{2} )^{2}$$
$29$ $$( -6 + T^{2} )^{2}$$
$31$ $$( 12 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( 2 + T^{2} )^{2}$$
$43$ $$( 8 + T )^{4}$$
$47$ $$( -24 + T^{2} )^{2}$$
$53$ $$( -54 + T^{2} )^{2}$$
$59$ $$( 128 + T^{2} )^{2}$$
$61$ $$( 192 + T^{2} )^{2}$$
$67$ $$( -4 + T )^{4}$$
$71$ $$( -216 + T^{2} )^{2}$$
$73$ $$( 4 + T )^{4}$$
$79$ $$( 12 + T^{2} )^{2}$$
$83$ $$( 200 + T^{2} )^{2}$$
$89$ $$( 50 + T^{2} )^{2}$$
$97$ $$( -8 + T )^{4}$$