# Properties

 Label 5776.2.a.br Level $5776$ Weight $2$ Character orbit 5776.a Self dual yes Analytic conductor $46.122$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} ) q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} ) q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( 2 \beta_{1} + \beta_{2} ) q^{9} + ( 2 \beta_{1} - \beta_{2} ) q^{11} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 1 + \beta_{2} ) q^{15} + ( 2 - \beta_{2} ) q^{17} - q^{21} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{23} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{25} + ( 2 + 3 \beta_{2} ) q^{27} + ( -5 - \beta_{1} ) q^{29} + ( 3 + 3 \beta_{1} - \beta_{2} ) q^{31} + ( 3 + \beta_{1} + \beta_{2} ) q^{33} + ( -1 + 2 \beta_{1} - 2 \beta_{2} ) q^{35} + ( \beta_{1} + 2 \beta_{2} ) q^{37} + ( 4 + \beta_{1} + \beta_{2} ) q^{39} + ( -4 + \beta_{1} + 3 \beta_{2} ) q^{41} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{43} + ( 5 - \beta_{1} + \beta_{2} ) q^{45} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{47} + ( -5 - \beta_{1} ) q^{49} + ( 1 + \beta_{1} - \beta_{2} ) q^{51} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{53} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{55} + ( 7 - 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 3 + 4 \beta_{2} ) q^{61} + ( -1 + 2 \beta_{1} - 3 \beta_{2} ) q^{63} + ( 4 - 5 \beta_{1} + 5 \beta_{2} ) q^{65} + ( -6 + 4 \beta_{1} - 6 \beta_{2} ) q^{67} + ( 4 + 2 \beta_{1} ) q^{69} + ( 10 - 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -5 - 3 \beta_{1} - \beta_{2} ) q^{75} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{77} + ( 3 - \beta_{1} + 7 \beta_{2} ) q^{79} + ( 5 - \beta_{1} ) q^{81} + ( 3 \beta_{1} + 6 \beta_{2} ) q^{83} + ( -3 + \beta_{1} + \beta_{2} ) q^{85} + ( -7 - 6 \beta_{1} - \beta_{2} ) q^{87} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{89} + ( -5 + 3 \beta_{1} - \beta_{2} ) q^{91} + ( 8 + 5 \beta_{1} + 2 \beta_{2} ) q^{93} + ( 5 + 4 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 6 - \beta_{1} + 5 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 3 q^{5} + O(q^{10})$$ $$3 q + 3 q^{3} - 3 q^{5} + 3 q^{15} + 6 q^{17} - 3 q^{21} + 6 q^{23} - 6 q^{25} + 6 q^{27} - 15 q^{29} + 9 q^{31} + 9 q^{33} - 3 q^{35} + 12 q^{39} - 12 q^{41} + 15 q^{45} + 6 q^{47} - 15 q^{49} + 3 q^{51} - 6 q^{53} + 9 q^{55} + 21 q^{59} + 9 q^{61} - 3 q^{63} + 12 q^{65} - 18 q^{67} + 12 q^{69} + 30 q^{71} - 15 q^{75} - 9 q^{77} + 9 q^{79} + 15 q^{81} - 9 q^{85} - 21 q^{87} - 15 q^{89} - 15 q^{91} + 24 q^{93} + 15 q^{97} + 18 q^{99} + O(q^{100})$$

## 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
 −1.53209 −0.347296 1.87939
0 −0.532089 0 −2.53209 0 1.87939 0 −2.71688 0
1.2 0 0.652704 0 −1.34730 0 −1.53209 0 −2.57398 0
1.3 0 2.87939 0 0.879385 0 −0.347296 0 5.29086 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.br 3
4.b odd 2 1 361.2.a.g 3
12.b even 2 1 3249.2.a.z 3
19.b odd 2 1 5776.2.a.bi 3
19.e even 9 2 304.2.u.b 6
20.d odd 2 1 9025.2.a.bd 3
76.d even 2 1 361.2.a.h 3
76.f even 6 2 361.2.c.h 6
76.g odd 6 2 361.2.c.i 6
76.k even 18 2 361.2.e.a 6
76.k even 18 2 361.2.e.b 6
76.k even 18 2 361.2.e.h 6
76.l odd 18 2 19.2.e.a 6
76.l odd 18 2 361.2.e.f 6
76.l odd 18 2 361.2.e.g 6
228.b odd 2 1 3249.2.a.s 3
228.v even 18 2 171.2.u.c 6
380.d even 2 1 9025.2.a.x 3
380.ba odd 18 2 475.2.l.a 6
380.bj even 36 4 475.2.u.a 12
532.br even 18 2 931.2.x.b 6
532.bt odd 18 2 931.2.x.a 6
532.cc even 18 2 931.2.w.a 6
532.cd even 18 2 931.2.v.a 6
532.cf odd 18 2 931.2.v.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 76.l odd 18 2
171.2.u.c 6 228.v even 18 2
304.2.u.b 6 19.e even 9 2
361.2.a.g 3 4.b odd 2 1
361.2.a.h 3 76.d even 2 1
361.2.c.h 6 76.f even 6 2
361.2.c.i 6 76.g odd 6 2
361.2.e.a 6 76.k even 18 2
361.2.e.b 6 76.k even 18 2
361.2.e.f 6 76.l odd 18 2
361.2.e.g 6 76.l odd 18 2
361.2.e.h 6 76.k even 18 2
475.2.l.a 6 380.ba odd 18 2
475.2.u.a 12 380.bj even 36 4
931.2.v.a 6 532.cd even 18 2
931.2.v.b 6 532.cf odd 18 2
931.2.w.a 6 532.cc even 18 2
931.2.x.a 6 532.bt odd 18 2
931.2.x.b 6 532.br even 18 2
3249.2.a.s 3 228.b odd 2 1
3249.2.a.z 3 12.b even 2 1
5776.2.a.bi 3 19.b odd 2 1
5776.2.a.br 3 1.a even 1 1 trivial
9025.2.a.x 3 380.d even 2 1
9025.2.a.bd 3 20.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{3} - 3 T_{3}^{2} + 1$$ $$T_{5}^{3} + 3 T_{5}^{2} - 3$$ $$T_{7}^{3} - 3 T_{7} - 1$$ $$T_{11}^{3} - 9 T_{11} + 9$$ $$T_{13}^{3} - 21 T_{13} + 37$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$1 - 3 T^{2} + T^{3}$$
$5$ $$-3 + 3 T^{2} + T^{3}$$
$7$ $$-1 - 3 T + T^{3}$$
$11$ $$9 - 9 T + T^{3}$$
$13$ $$37 - 21 T + T^{3}$$
$17$ $$-3 + 9 T - 6 T^{2} + T^{3}$$
$19$ $$T^{3}$$
$23$ $$24 - 6 T^{2} + T^{3}$$
$29$ $$111 + 72 T + 15 T^{2} + T^{3}$$
$31$ $$53 + 6 T - 9 T^{2} + T^{3}$$
$37$ $$-17 - 21 T + T^{3}$$
$41$ $$-111 + 9 T + 12 T^{2} + T^{3}$$
$43$ $$-163 - 57 T + T^{3}$$
$47$ $$-3 - 9 T - 6 T^{2} + T^{3}$$
$53$ $$-51 - 9 T + 6 T^{2} + T^{3}$$
$59$ $$-267 + 135 T - 21 T^{2} + T^{3}$$
$61$ $$181 - 21 T - 9 T^{2} + T^{3}$$
$67$ $$-424 + 24 T + 18 T^{2} + T^{3}$$
$71$ $$-888 + 288 T - 30 T^{2} + T^{3}$$
$73$ $$64 - 48 T + T^{3}$$
$79$ $$809 - 102 T - 9 T^{2} + T^{3}$$
$83$ $$-459 - 189 T + T^{3}$$
$89$ $$57 + 54 T + 15 T^{2} + T^{3}$$
$97$ $$127 + 39 T - 15 T^{2} + T^{3}$$