# Properties

 Label 5776.2.a.o Level $5776$ Weight $2$ Character orbit 5776.a Self dual yes Analytic conductor $46.122$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5776 = 2^{4} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5776.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.1215922075$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 3q^{5} - 2q^{9} + O(q^{10})$$ $$q + q^{3} + 3q^{5} - 2q^{9} + 4q^{11} + 5q^{13} + 3q^{15} - 5q^{17} + q^{23} + 4q^{25} - 5q^{27} - 3q^{29} + 4q^{31} + 4q^{33} - 2q^{37} + 5q^{39} + 5q^{41} + 11q^{43} - 6q^{45} + 5q^{47} - 7q^{49} - 5q^{51} + 9q^{53} + 12q^{55} + 13q^{59} - q^{61} + 15q^{65} - 5q^{67} + q^{69} + q^{71} - 9q^{73} + 4q^{75} + 17q^{79} + q^{81} - 16q^{83} - 15q^{85} - 3q^{87} - 3q^{89} + 4q^{93} + 13q^{97} - 8q^{99} + O(q^{100})$$

## 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
 0
0 1.00000 0 3.00000 0 0 0 −2.00000 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
$$2$$ $$1$$
$$19$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5776.2.a.o 1
4.b odd 2 1 2888.2.a.c 1
19.b odd 2 1 5776.2.a.h 1
19.d odd 6 2 304.2.i.b 2
57.f even 6 2 2736.2.s.q 2
76.d even 2 1 2888.2.a.e 1
76.f even 6 2 152.2.i.a 2
152.l odd 6 2 1216.2.i.e 2
152.o even 6 2 1216.2.i.i 2
228.n odd 6 2 1368.2.s.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.a 2 76.f even 6 2
304.2.i.b 2 19.d odd 6 2
1216.2.i.e 2 152.l odd 6 2
1216.2.i.i 2 152.o even 6 2
1368.2.s.g 2 228.n odd 6 2
2736.2.s.q 2 57.f even 6 2
2888.2.a.c 1 4.b odd 2 1
2888.2.a.e 1 76.d even 2 1
5776.2.a.h 1 19.b odd 2 1
5776.2.a.o 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5776))$$:

 $$T_{3} - 1$$ $$T_{5} - 3$$ $$T_{7}$$ $$T_{11} - 4$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$-3 + T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$-5 + T$$
$17$ $$5 + T$$
$19$ $$T$$
$23$ $$-1 + T$$
$29$ $$3 + T$$
$31$ $$-4 + T$$
$37$ $$2 + T$$
$41$ $$-5 + T$$
$43$ $$-11 + T$$
$47$ $$-5 + T$$
$53$ $$-9 + T$$
$59$ $$-13 + T$$
$61$ $$1 + T$$
$67$ $$5 + T$$
$71$ $$-1 + T$$
$73$ $$9 + T$$
$79$ $$-17 + T$$
$83$ $$16 + T$$
$89$ $$3 + T$$
$97$ $$-13 + T$$