# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{5} + 2 q^{7} +O(q^{10})$$ $$q + 2 i q^{5} + 2 q^{7} + 4 i q^{13} + 2 q^{17} + 4 i q^{19} + 4 q^{23} + q^{25} -6 i q^{29} -2 q^{31} + 4 i q^{35} -8 i q^{37} -2 q^{41} -4 i q^{43} -12 q^{47} -3 q^{49} + 6 i q^{53} -4 i q^{59} -8 q^{65} -12 i q^{67} + 12 q^{71} -6 q^{73} -10 q^{79} -16 i q^{83} + 4 i q^{85} + 10 q^{89} + 8 i q^{91} -8 q^{95} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} + O(q^{10})$$ $$2q + 4q^{7} + 4q^{17} + 8q^{23} + 2q^{25} - 4q^{31} - 4q^{41} - 24q^{47} - 6q^{49} - 16q^{65} + 24q^{71} - 12q^{73} - 20q^{79} + 20q^{89} - 16q^{95} - 4q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 − 1.00000i 1.00000i
0 0 0 2.00000i 0 2.00000 0 0 0
145.2 0 0 0 2.00000i 0 2.00000 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
8.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.d.b 2
3.b odd 2 1 96.2.d.a 2
4.b odd 2 1 72.2.d.b 2
5.b even 2 1 7200.2.k.d 2
5.c odd 4 1 7200.2.d.d 2
5.c odd 4 1 7200.2.d.g 2
8.b even 2 1 inner 288.2.d.b 2
8.d odd 2 1 72.2.d.b 2
9.c even 3 2 2592.2.r.g 4
9.d odd 6 2 2592.2.r.f 4
12.b even 2 1 24.2.d.a 2
15.d odd 2 1 2400.2.k.a 2
15.e even 4 1 2400.2.d.b 2
15.e even 4 1 2400.2.d.c 2
16.e even 4 1 2304.2.a.b 1
16.e even 4 1 2304.2.a.l 1
16.f odd 4 1 2304.2.a.e 1
16.f odd 4 1 2304.2.a.o 1
20.d odd 2 1 1800.2.k.a 2
20.e even 4 1 1800.2.d.b 2
20.e even 4 1 1800.2.d.i 2
21.c even 2 1 4704.2.c.a 2
24.f even 2 1 24.2.d.a 2
24.h odd 2 1 96.2.d.a 2
36.f odd 6 2 648.2.n.c 4
36.h even 6 2 648.2.n.k 4
40.e odd 2 1 1800.2.k.a 2
40.f even 2 1 7200.2.k.d 2
40.i odd 4 1 7200.2.d.d 2
40.i odd 4 1 7200.2.d.g 2
40.k even 4 1 1800.2.d.b 2
40.k even 4 1 1800.2.d.i 2
48.i odd 4 1 768.2.a.d 1
48.i odd 4 1 768.2.a.e 1
48.k even 4 1 768.2.a.a 1
48.k even 4 1 768.2.a.h 1
60.h even 2 1 600.2.k.b 2
60.l odd 4 1 600.2.d.b 2
60.l odd 4 1 600.2.d.c 2
72.j odd 6 2 2592.2.r.f 4
72.l even 6 2 648.2.n.k 4
72.n even 6 2 2592.2.r.g 4
72.p odd 6 2 648.2.n.c 4
84.h odd 2 1 1176.2.c.a 2
120.i odd 2 1 2400.2.k.a 2
120.m even 2 1 600.2.k.b 2
120.q odd 4 1 600.2.d.b 2
120.q odd 4 1 600.2.d.c 2
120.w even 4 1 2400.2.d.b 2
120.w even 4 1 2400.2.d.c 2
168.e odd 2 1 1176.2.c.a 2
168.i even 2 1 4704.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 12.b even 2 1
24.2.d.a 2 24.f even 2 1
72.2.d.b 2 4.b odd 2 1
72.2.d.b 2 8.d odd 2 1
96.2.d.a 2 3.b odd 2 1
96.2.d.a 2 24.h odd 2 1
288.2.d.b 2 1.a even 1 1 trivial
288.2.d.b 2 8.b even 2 1 inner
600.2.d.b 2 60.l odd 4 1
600.2.d.b 2 120.q odd 4 1
600.2.d.c 2 60.l odd 4 1
600.2.d.c 2 120.q odd 4 1
600.2.k.b 2 60.h even 2 1
600.2.k.b 2 120.m even 2 1
648.2.n.c 4 36.f odd 6 2
648.2.n.c 4 72.p odd 6 2
648.2.n.k 4 36.h even 6 2
648.2.n.k 4 72.l even 6 2
768.2.a.a 1 48.k even 4 1
768.2.a.d 1 48.i odd 4 1
768.2.a.e 1 48.i odd 4 1
768.2.a.h 1 48.k even 4 1
1176.2.c.a 2 84.h odd 2 1
1176.2.c.a 2 168.e odd 2 1
1800.2.d.b 2 20.e even 4 1
1800.2.d.b 2 40.k even 4 1
1800.2.d.i 2 20.e even 4 1
1800.2.d.i 2 40.k even 4 1
1800.2.k.a 2 20.d odd 2 1
1800.2.k.a 2 40.e odd 2 1
2304.2.a.b 1 16.e even 4 1
2304.2.a.e 1 16.f odd 4 1
2304.2.a.l 1 16.e even 4 1
2304.2.a.o 1 16.f odd 4 1
2400.2.d.b 2 15.e even 4 1
2400.2.d.b 2 120.w even 4 1
2400.2.d.c 2 15.e even 4 1
2400.2.d.c 2 120.w even 4 1
2400.2.k.a 2 15.d odd 2 1
2400.2.k.a 2 120.i odd 2 1
2592.2.r.f 4 9.d odd 6 2
2592.2.r.f 4 72.j odd 6 2
2592.2.r.g 4 9.c even 3 2
2592.2.r.g 4 72.n even 6 2
4704.2.c.a 2 21.c even 2 1
4704.2.c.a 2 168.i even 2 1
7200.2.d.d 2 5.c odd 4 1
7200.2.d.d 2 40.i odd 4 1
7200.2.d.g 2 5.c odd 4 1
7200.2.d.g 2 40.i odd 4 1
7200.2.k.d 2 5.b even 2 1
7200.2.k.d 2 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$( 12 + T )^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$( 10 + T )^{2}$$
$83$ $$256 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$( 2 + T )^{2}$$