Properties

 Label 2880.2.a.bk Level $2880$ Weight $2$ Character orbit 2880.a Self dual yes Analytic conductor $22.997$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2880.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$22.9969157821$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + \beta q^{7} +O(q^{10})$$ $$q + q^{5} + \beta q^{7} + 2 \beta q^{11} + 2 q^{13} -2 q^{17} + \beta q^{23} + q^{25} + 6 q^{29} -2 \beta q^{31} + \beta q^{35} + 10 q^{37} -2 q^{41} -3 \beta q^{43} -\beta q^{47} + q^{49} + 6 q^{53} + 2 \beta q^{55} -4 \beta q^{59} + 2 q^{61} + 2 q^{65} -\beta q^{67} + 2 \beta q^{71} -6 q^{73} + 16 q^{77} -4 \beta q^{79} -\beta q^{83} -2 q^{85} -10 q^{89} + 2 \beta q^{91} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + O(q^{10})$$ $$2 q + 2 q^{5} + 4 q^{13} - 4 q^{17} + 2 q^{25} + 12 q^{29} + 20 q^{37} - 4 q^{41} + 2 q^{49} + 12 q^{53} + 4 q^{61} + 4 q^{65} - 12 q^{73} + 32 q^{77} - 4 q^{85} - 20 q^{89} + 4 q^{97} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 0 0 1.00000 0 −2.82843 0 0 0
1.2 0 0 0 1.00000 0 2.82843 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.a.bk 2
3.b odd 2 1 320.2.a.g 2
4.b odd 2 1 inner 2880.2.a.bk 2
8.b even 2 1 1440.2.a.o 2
8.d odd 2 1 1440.2.a.o 2
12.b even 2 1 320.2.a.g 2
15.d odd 2 1 1600.2.a.bc 2
15.e even 4 2 1600.2.c.n 4
24.f even 2 1 160.2.a.c 2
24.h odd 2 1 160.2.a.c 2
40.e odd 2 1 7200.2.a.cm 2
40.f even 2 1 7200.2.a.cm 2
40.i odd 4 2 7200.2.f.bh 4
40.k even 4 2 7200.2.f.bh 4
48.i odd 4 2 1280.2.d.l 4
48.k even 4 2 1280.2.d.l 4
60.h even 2 1 1600.2.a.bc 2
60.l odd 4 2 1600.2.c.n 4
120.i odd 2 1 800.2.a.m 2
120.m even 2 1 800.2.a.m 2
120.q odd 4 2 800.2.c.f 4
120.w even 4 2 800.2.c.f 4
168.e odd 2 1 7840.2.a.bf 2
168.i even 2 1 7840.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 24.f even 2 1
160.2.a.c 2 24.h odd 2 1
320.2.a.g 2 3.b odd 2 1
320.2.a.g 2 12.b even 2 1
800.2.a.m 2 120.i odd 2 1
800.2.a.m 2 120.m even 2 1
800.2.c.f 4 120.q odd 4 2
800.2.c.f 4 120.w even 4 2
1280.2.d.l 4 48.i odd 4 2
1280.2.d.l 4 48.k even 4 2
1440.2.a.o 2 8.b even 2 1
1440.2.a.o 2 8.d odd 2 1
1600.2.a.bc 2 15.d odd 2 1
1600.2.a.bc 2 60.h even 2 1
1600.2.c.n 4 15.e even 4 2
1600.2.c.n 4 60.l odd 4 2
2880.2.a.bk 2 1.a even 1 1 trivial
2880.2.a.bk 2 4.b odd 2 1 inner
7200.2.a.cm 2 40.e odd 2 1
7200.2.a.cm 2 40.f even 2 1
7200.2.f.bh 4 40.i odd 4 2
7200.2.f.bh 4 40.k even 4 2
7840.2.a.bf 2 168.e odd 2 1
7840.2.a.bf 2 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2880))$$:

 $$T_{7}^{2} - 8$$ $$T_{11}^{2} - 32$$ $$T_{13} - 2$$ $$T_{17} + 2$$ $$T_{19}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$-32 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$-32 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$-72 + T^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$-128 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$-8 + T^{2}$$
$71$ $$-32 + T^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$-128 + T^{2}$$
$83$ $$-8 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$