# Properties

 Label 288.2.c.a Level $288$ Weight $2$ Character orbit 288.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} +O(q^{10})$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} + 4 q^{13} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{17} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{23} + 3 q^{25} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{29} + 4 \zeta_{8}^{2} q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{35} -6 q^{37} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{41} -8 \zeta_{8}^{2} q^{43} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{47} -9 q^{49} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{53} -8 \zeta_{8}^{2} q^{55} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} -2 q^{61} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{65} -8 \zeta_{8}^{2} q^{67} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + ( 16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{77} -4 \zeta_{8}^{2} q^{79} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{83} -6 q^{85} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} + 16 \zeta_{8}^{2} q^{91} -8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 16q^{13} + 12q^{25} - 24q^{37} - 36q^{49} - 8q^{61} - 24q^{85} - 32q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
287.1
 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
0 0 0 1.41421i 0 4.00000i 0 0 0
287.2 0 0 0 1.41421i 0 4.00000i 0 0 0
287.3 0 0 0 1.41421i 0 4.00000i 0 0 0
287.4 0 0 0 1.41421i 0 4.00000i 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
4.b odd 2 1 inner
12.b 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.c.a 4
3.b odd 2 1 inner 288.2.c.a 4
4.b odd 2 1 inner 288.2.c.a 4
5.b even 2 1 7200.2.h.d 4
5.c odd 4 1 7200.2.o.a 4
5.c odd 4 1 7200.2.o.n 4
8.b even 2 1 576.2.c.c 4
8.d odd 2 1 576.2.c.c 4
9.c even 3 2 2592.2.s.d 8
9.d odd 6 2 2592.2.s.d 8
12.b even 2 1 inner 288.2.c.a 4
15.d odd 2 1 7200.2.h.d 4
15.e even 4 1 7200.2.o.a 4
15.e even 4 1 7200.2.o.n 4
16.e even 4 1 2304.2.f.c 4
16.e even 4 1 2304.2.f.e 4
16.f odd 4 1 2304.2.f.c 4
16.f odd 4 1 2304.2.f.e 4
20.d odd 2 1 7200.2.h.d 4
20.e even 4 1 7200.2.o.a 4
20.e even 4 1 7200.2.o.n 4
24.f even 2 1 576.2.c.c 4
24.h odd 2 1 576.2.c.c 4
36.f odd 6 2 2592.2.s.d 8
36.h even 6 2 2592.2.s.d 8
48.i odd 4 1 2304.2.f.c 4
48.i odd 4 1 2304.2.f.e 4
48.k even 4 1 2304.2.f.c 4
48.k even 4 1 2304.2.f.e 4
60.h even 2 1 7200.2.h.d 4
60.l odd 4 1 7200.2.o.a 4
60.l odd 4 1 7200.2.o.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.c.a 4 1.a even 1 1 trivial
288.2.c.a 4 3.b odd 2 1 inner
288.2.c.a 4 4.b odd 2 1 inner
288.2.c.a 4 12.b even 2 1 inner
576.2.c.c 4 8.b even 2 1
576.2.c.c 4 8.d odd 2 1
576.2.c.c 4 24.f even 2 1
576.2.c.c 4 24.h odd 2 1
2304.2.f.c 4 16.e even 4 1
2304.2.f.c 4 16.f odd 4 1
2304.2.f.c 4 48.i odd 4 1
2304.2.f.c 4 48.k even 4 1
2304.2.f.e 4 16.e even 4 1
2304.2.f.e 4 16.f odd 4 1
2304.2.f.e 4 48.i odd 4 1
2304.2.f.e 4 48.k even 4 1
2592.2.s.d 8 9.c even 3 2
2592.2.s.d 8 9.d odd 6 2
2592.2.s.d 8 36.f odd 6 2
2592.2.s.d 8 36.h even 6 2
7200.2.h.d 4 5.b even 2 1
7200.2.h.d 4 15.d odd 2 1
7200.2.h.d 4 20.d odd 2 1
7200.2.h.d 4 60.h even 2 1
7200.2.o.a 4 5.c odd 4 1
7200.2.o.a 4 15.e even 4 1
7200.2.o.a 4 20.e even 4 1
7200.2.o.a 4 60.l odd 4 1
7200.2.o.n 4 5.c odd 4 1
7200.2.o.n 4 15.e even 4 1
7200.2.o.n 4 20.e even 4 1
7200.2.o.n 4 60.l odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 2 + T^{2} )^{2}$$
$7$ $$( 16 + T^{2} )^{2}$$
$11$ $$( -32 + T^{2} )^{2}$$
$13$ $$( -4 + T )^{4}$$
$17$ $$( 18 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( -32 + T^{2} )^{2}$$
$29$ $$( 2 + T^{2} )^{2}$$
$31$ $$( 16 + T^{2} )^{2}$$
$37$ $$( 6 + T )^{4}$$
$41$ $$( 98 + T^{2} )^{2}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$( -32 + T^{2} )^{2}$$
$53$ $$( 18 + T^{2} )^{2}$$
$59$ $$( -128 + T^{2} )^{2}$$
$61$ $$( 2 + T )^{4}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( -32 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$( 16 + T^{2} )^{2}$$
$83$ $$( -32 + T^{2} )^{2}$$
$89$ $$( 18 + T^{2} )^{2}$$
$97$ $$( 8 + T )^{4}$$