# Properties

 Label 5776.2.a.n Level $5776$ Weight $2$ Character orbit 5776.a Self dual yes Analytic conductor $46.122$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 4 q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + 4 * q^7 - 2 * q^9 $$q + q^{3} + 4 q^{7} - 2 q^{9} - 3 q^{11} - 2 q^{13} - 6 q^{17} + 4 q^{21} + 6 q^{23} - 5 q^{25} - 5 q^{27} + 2 q^{31} - 3 q^{33} + 10 q^{37} - 2 q^{39} - 9 q^{41} + 4 q^{43} + 9 q^{49} - 6 q^{51} - 6 q^{53} - 9 q^{59} - 4 q^{61} - 8 q^{63} - 7 q^{67} + 6 q^{69} - 6 q^{71} - q^{73} - 5 q^{75} - 12 q^{77} - 4 q^{79} + q^{81} - 3 q^{83} - 6 q^{89} - 8 q^{91} + 2 q^{93} - 17 q^{97} + 6 q^{99}+O(q^{100})$$ q + q^3 + 4 * q^7 - 2 * q^9 - 3 * q^11 - 2 * q^13 - 6 * q^17 + 4 * q^21 + 6 * q^23 - 5 * q^25 - 5 * q^27 + 2 * q^31 - 3 * q^33 + 10 * q^37 - 2 * q^39 - 9 * q^41 + 4 * q^43 + 9 * q^49 - 6 * q^51 - 6 * q^53 - 9 * q^59 - 4 * q^61 - 8 * q^63 - 7 * q^67 + 6 * q^69 - 6 * q^71 - q^73 - 5 * q^75 - 12 * q^77 - 4 * q^79 + q^81 - 3 * q^83 - 6 * q^89 - 8 * q^91 + 2 * q^93 - 17 * q^97 + 6 * q^99

## 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 0 0 4.00000 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.n 1
4.b odd 2 1 722.2.a.d 1
12.b even 2 1 6498.2.a.e 1
19.b odd 2 1 5776.2.a.g 1
19.d odd 6 2 304.2.i.c 2
57.f even 6 2 2736.2.s.m 2
76.d even 2 1 722.2.a.c 1
76.f even 6 2 38.2.c.a 2
76.g odd 6 2 722.2.c.b 2
76.k even 18 6 722.2.e.j 6
76.l odd 18 6 722.2.e.i 6
152.l odd 6 2 1216.2.i.d 2
152.o even 6 2 1216.2.i.h 2
228.b odd 2 1 6498.2.a.s 1
228.n odd 6 2 342.2.g.b 2
380.s even 6 2 950.2.e.d 2
380.w odd 12 4 950.2.j.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a 2 76.f even 6 2
304.2.i.c 2 19.d odd 6 2
342.2.g.b 2 228.n odd 6 2
722.2.a.c 1 76.d even 2 1
722.2.a.d 1 4.b odd 2 1
722.2.c.b 2 76.g odd 6 2
722.2.e.i 6 76.l odd 18 6
722.2.e.j 6 76.k even 18 6
950.2.e.d 2 380.s even 6 2
950.2.j.e 4 380.w odd 12 4
1216.2.i.d 2 152.l odd 6 2
1216.2.i.h 2 152.o even 6 2
2736.2.s.m 2 57.f even 6 2
5776.2.a.g 1 19.b odd 2 1
5776.2.a.n 1 1.a even 1 1 trivial
6498.2.a.e 1 12.b even 2 1
6498.2.a.s 1 228.b odd 2 1

## Hecke kernels

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

 $$T_{3} - 1$$ T3 - 1 $$T_{5}$$ T5 $$T_{7} - 4$$ T7 - 4 $$T_{11} + 3$$ T11 + 3 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T + 3$$
$13$ $$T + 2$$
$17$ $$T + 6$$
$19$ $$T$$
$23$ $$T - 6$$
$29$ $$T$$
$31$ $$T - 2$$
$37$ $$T - 10$$
$41$ $$T + 9$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T + 9$$
$61$ $$T + 4$$
$67$ $$T + 7$$
$71$ $$T + 6$$
$73$ $$T + 1$$
$79$ $$T + 4$$
$83$ $$T + 3$$
$89$ $$T + 6$$
$97$ $$T + 17$$