# Properties

 Label 5776.2.a.bi Level $5776$ Weight $2$ Character orbit 5776.a Self dual yes Analytic conductor $46.122$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{3} + (\beta_1 - 1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{2} + 2 \beta_1) q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^3 + (b1 - 1) * q^5 + (b2 - b1) * q^7 + (b2 + 2*b1) * q^9 $$q + ( - \beta_1 - 1) q^{3} + (\beta_1 - 1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{2} + 2 \beta_1) q^{9} + ( - \beta_{2} + 2 \beta_1) q^{11} + (2 \beta_{2} - 3 \beta_1) q^{13} + ( - \beta_{2} - 1) q^{15} + ( - \beta_{2} + 2) q^{17} + q^{21} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{23} + (\beta_{2} - 2 \beta_1 - 2) q^{25} + ( - 3 \beta_{2} - 2) q^{27} + (\beta_1 + 5) q^{29} + (\beta_{2} - 3 \beta_1 - 3) q^{31} + ( - \beta_{2} - \beta_1 - 3) q^{33} + ( - 2 \beta_{2} + 2 \beta_1 - 1) q^{35} + ( - 2 \beta_{2} - \beta_1) q^{37} + (\beta_{2} + \beta_1 + 4) q^{39} + ( - 3 \beta_{2} - \beta_1 + 4) q^{41} + ( - 5 \beta_{2} + 2 \beta_1) q^{43} + (\beta_{2} - \beta_1 + 5) q^{45} + (\beta_{2} + 2 \beta_1 + 2) q^{47} + ( - \beta_1 - 5) q^{49} + (\beta_{2} - \beta_1 - 1) q^{51} + (3 \beta_{2} - 2 \beta_1 + 2) q^{53} + (3 \beta_{2} - 3 \beta_1 + 3) q^{55} + ( - 2 \beta_{2} + 2 \beta_1 - 7) q^{59} + (4 \beta_{2} + 3) q^{61} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{63} + ( - 5 \beta_{2} + 5 \beta_1 - 4) q^{65} + (6 \beta_{2} - 4 \beta_1 + 6) q^{67} + ( - 2 \beta_1 - 4) q^{69} + ( - 2 \beta_{2} + 2 \beta_1 - 10) q^{71} + ( - 4 \beta_{2} + 4 \beta_1) q^{73} + (\beta_{2} + 3 \beta_1 + 5) q^{75} + ( - \beta_{2} + 2 \beta_1 - 3) q^{77} + ( - 7 \beta_{2} + \beta_1 - 3) q^{79} + ( - \beta_1 + 5) q^{81} + (6 \beta_{2} + 3 \beta_1) q^{83} + (\beta_{2} + \beta_1 - 3) q^{85} + ( - \beta_{2} - 6 \beta_1 - 7) q^{87} + (\beta_{2} + 2 \beta_1 + 5) q^{89} + (\beta_{2} - 3 \beta_1 + 5) q^{91} + (2 \beta_{2} + 5 \beta_1 + 8) q^{93} + (2 \beta_{2} - 4 \beta_1 - 5) q^{97} + (5 \beta_{2} - \beta_1 + 6) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^3 + (b1 - 1) * q^5 + (b2 - b1) * q^7 + (b2 + 2*b1) * q^9 + (-b2 + 2*b1) * q^11 + (2*b2 - 3*b1) * q^13 + (-b2 - 1) * q^15 + (-b2 + 2) * q^17 + q^21 + (-2*b2 + 2*b1 + 2) * q^23 + (b2 - 2*b1 - 2) * q^25 + (-3*b2 - 2) * q^27 + (b1 + 5) * q^29 + (b2 - 3*b1 - 3) * q^31 + (-b2 - b1 - 3) * q^33 + (-2*b2 + 2*b1 - 1) * q^35 + (-2*b2 - b1) * q^37 + (b2 + b1 + 4) * q^39 + (-3*b2 - b1 + 4) * q^41 + (-5*b2 + 2*b1) * q^43 + (b2 - b1 + 5) * q^45 + (b2 + 2*b1 + 2) * q^47 + (-b1 - 5) * q^49 + (b2 - b1 - 1) * q^51 + (3*b2 - 2*b1 + 2) * q^53 + (3*b2 - 3*b1 + 3) * q^55 + (-2*b2 + 2*b1 - 7) * q^59 + (4*b2 + 3) * q^61 + (-3*b2 + 2*b1 - 1) * q^63 + (-5*b2 + 5*b1 - 4) * q^65 + (6*b2 - 4*b1 + 6) * q^67 + (-2*b1 - 4) * q^69 + (-2*b2 + 2*b1 - 10) * q^71 + (-4*b2 + 4*b1) * q^73 + (b2 + 3*b1 + 5) * q^75 + (-b2 + 2*b1 - 3) * q^77 + (-7*b2 + b1 - 3) * q^79 + (-b1 + 5) * q^81 + (6*b2 + 3*b1) * q^83 + (b2 + b1 - 3) * q^85 + (-b2 - 6*b1 - 7) * q^87 + (b2 + 2*b1 + 5) * q^89 + (b2 - 3*b1 + 5) * q^91 + (2*b2 + 5*b1 + 8) * q^93 + (2*b2 - 4*b1 - 5) * q^97 + (5*b2 - b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{5}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^5 $$3 q - 3 q^{3} - 3 q^{5} - 3 q^{15} + 6 q^{17} + 3 q^{21} + 6 q^{23} - 6 q^{25} - 6 q^{27} + 15 q^{29} - 9 q^{31} - 9 q^{33} - 3 q^{35} + 12 q^{39} + 12 q^{41} + 15 q^{45} + 6 q^{47} - 15 q^{49} - 3 q^{51} + 6 q^{53} + 9 q^{55} - 21 q^{59} + 9 q^{61} - 3 q^{63} - 12 q^{65} + 18 q^{67} - 12 q^{69} - 30 q^{71} + 15 q^{75} - 9 q^{77} - 9 q^{79} + 15 q^{81} - 9 q^{85} - 21 q^{87} + 15 q^{89} + 15 q^{91} + 24 q^{93} - 15 q^{97} + 18 q^{99}+O(q^{100})$$ 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

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

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

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

## Hecke kernels

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

 $$T_{3}^{3} + 3T_{3}^{2} - 1$$ T3^3 + 3*T3^2 - 1 $$T_{5}^{3} + 3T_{5}^{2} - 3$$ T5^3 + 3*T5^2 - 3 $$T_{7}^{3} - 3T_{7} - 1$$ T7^3 - 3*T7 - 1 $$T_{11}^{3} - 9T_{11} + 9$$ T11^3 - 9*T11 + 9 $$T_{13}^{3} - 21T_{13} - 37$$ T13^3 - 21*T13 - 37

## Hecke characteristic polynomials

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