# Properties

 Label 5775.2.a.bw Level $5775$ Weight $2$ Character orbit 5775.a Self dual yes Analytic conductor $46.114$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5775.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.1136071673$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.837.1 Defining polynomial: $$x^{3} - 6 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + q^{7} + ( -1 - 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + q^{7} + ( -1 - 2 \beta_{1} ) q^{8} + q^{9} + q^{11} + ( 2 + \beta_{2} ) q^{12} + ( -\beta_{1} + \beta_{2} ) q^{13} -\beta_{1} q^{14} + ( 4 + \beta_{1} ) q^{16} + 2 \beta_{1} q^{17} -\beta_{1} q^{18} + ( 4 - \beta_{1} - \beta_{2} ) q^{19} + q^{21} -\beta_{1} q^{22} + ( 2 + 2 \beta_{1} ) q^{23} + ( -1 - 2 \beta_{1} ) q^{24} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{26} + q^{27} + ( 2 + \beta_{2} ) q^{28} + ( 4 - \beta_{1} + \beta_{2} ) q^{29} + ( -2 + 2 \beta_{2} ) q^{31} + ( -2 - \beta_{2} ) q^{32} + q^{33} + ( -8 - 2 \beta_{2} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( -3 \beta_{1} - \beta_{2} ) q^{37} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{38} + ( -\beta_{1} + \beta_{2} ) q^{39} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{41} -\beta_{1} q^{42} + ( -2 + 2 \beta_{1} ) q^{43} + ( 2 + \beta_{2} ) q^{44} + ( -8 - 2 \beta_{1} - 2 \beta_{2} ) q^{46} + ( 8 - \beta_{1} - \beta_{2} ) q^{47} + ( 4 + \beta_{1} ) q^{48} + q^{49} + 2 \beta_{1} q^{51} + ( 7 - 3 \beta_{1} ) q^{52} + 2 \beta_{2} q^{53} -\beta_{1} q^{54} + ( -1 - 2 \beta_{1} ) q^{56} + ( 4 - \beta_{1} - \beta_{2} ) q^{57} + ( 3 - 6 \beta_{1} + \beta_{2} ) q^{58} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{59} + 6 q^{61} + ( -2 - 2 \beta_{1} ) q^{62} + q^{63} + ( -7 + 2 \beta_{1} ) q^{64} -\beta_{1} q^{66} + ( -4 - \beta_{1} - \beta_{2} ) q^{67} + ( 2 + 8 \beta_{1} ) q^{68} + ( 2 + 2 \beta_{1} ) q^{69} + ( 4 - 4 \beta_{1} ) q^{71} + ( -1 - 2 \beta_{1} ) q^{72} + ( -8 + \beta_{1} - \beta_{2} ) q^{73} + ( 13 + 2 \beta_{1} + 3 \beta_{2} ) q^{74} + ( -1 - 5 \beta_{1} + 4 \beta_{2} ) q^{76} + q^{77} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{78} + ( 4 + 4 \beta_{1} ) q^{79} + q^{81} + ( -10 - 6 \beta_{1} - 2 \beta_{2} ) q^{82} + ( 6 - 2 \beta_{2} ) q^{83} + ( 2 + \beta_{2} ) q^{84} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 4 - \beta_{1} + \beta_{2} ) q^{87} + ( -1 - 2 \beta_{1} ) q^{88} + ( 6 + 4 \beta_{2} ) q^{89} + ( -\beta_{1} + \beta_{2} ) q^{91} + ( 6 + 8 \beta_{1} + 2 \beta_{2} ) q^{92} + ( -2 + 2 \beta_{2} ) q^{93} + ( 5 - 6 \beta_{1} + \beta_{2} ) q^{94} + ( -2 - \beta_{2} ) q^{96} + ( -8 - 2 \beta_{2} ) q^{97} -\beta_{1} q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} + 6q^{4} + 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} + 6q^{4} + 3q^{7} - 3q^{8} + 3q^{9} + 3q^{11} + 6q^{12} + 12q^{16} + 12q^{19} + 3q^{21} + 6q^{23} - 3q^{24} + 9q^{26} + 3q^{27} + 6q^{28} + 12q^{29} - 6q^{31} - 6q^{32} + 3q^{33} - 24q^{34} + 6q^{36} + 15q^{38} + 6q^{41} - 6q^{43} + 6q^{44} - 24q^{46} + 24q^{47} + 12q^{48} + 3q^{49} + 21q^{52} - 3q^{56} + 12q^{57} + 9q^{58} - 24q^{59} + 18q^{61} - 6q^{62} + 3q^{63} - 21q^{64} - 12q^{67} + 6q^{68} + 6q^{69} + 12q^{71} - 3q^{72} - 24q^{73} + 39q^{74} - 3q^{76} + 3q^{77} + 9q^{78} + 12q^{79} + 3q^{81} - 30q^{82} + 18q^{83} + 6q^{84} - 24q^{86} + 12q^{87} - 3q^{88} + 18q^{89} + 18q^{92} - 6q^{93} + 15q^{94} - 6q^{96} - 24q^{97} + 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

## 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
 2.52892 −0.167449 −2.36147
−2.52892 1.00000 4.39543 0 −2.52892 1.00000 −6.05784 1.00000 0
1.2 0.167449 1.00000 −1.97196 0 0.167449 1.00000 −0.665102 1.00000 0
1.3 2.36147 1.00000 3.57653 0 2.36147 1.00000 3.72294 1.00000 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 5775.2.a.bw 3
5.b even 2 1 231.2.a.d 3
15.d odd 2 1 693.2.a.m 3
20.d odd 2 1 3696.2.a.bp 3
35.c odd 2 1 1617.2.a.s 3
55.d odd 2 1 2541.2.a.bi 3
105.g even 2 1 4851.2.a.bp 3
165.d even 2 1 7623.2.a.cb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.d 3 5.b even 2 1
693.2.a.m 3 15.d odd 2 1
1617.2.a.s 3 35.c odd 2 1
2541.2.a.bi 3 55.d odd 2 1
3696.2.a.bp 3 20.d odd 2 1
4851.2.a.bp 3 105.g even 2 1
5775.2.a.bw 3 1.a even 1 1 trivial
7623.2.a.cb 3 165.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5775))$$:

 $$T_{2}^{3} - 6 T_{2} + 1$$ $$T_{13}^{3} - 15 T_{13} - 2$$ $$T_{17}^{3} - 24 T_{17} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 6 T + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$-2 - 15 T + T^{3}$$
$17$ $$-8 - 24 T + T^{3}$$
$19$ $$36 + 27 T - 12 T^{2} + T^{3}$$
$23$ $$32 - 12 T - 6 T^{2} + T^{3}$$
$29$ $$-6 + 33 T - 12 T^{2} + T^{3}$$
$31$ $$32 - 36 T + 6 T^{2} + T^{3}$$
$37$ $$246 - 75 T + T^{3}$$
$41$ $$32 - 72 T - 6 T^{2} + T^{3}$$
$43$ $$-48 - 12 T + 6 T^{2} + T^{3}$$
$47$ $$-328 + 171 T - 24 T^{2} + T^{3}$$
$53$ $$120 - 48 T + T^{3}$$
$59$ $$-716 + 87 T + 24 T^{2} + T^{3}$$
$61$ $$( -6 + T )^{3}$$
$67$ $$-4 + 27 T + 12 T^{2} + T^{3}$$
$71$ $$384 - 48 T - 12 T^{2} + T^{3}$$
$73$ $$394 + 177 T + 24 T^{2} + T^{3}$$
$79$ $$256 - 48 T - 12 T^{2} + T^{3}$$
$83$ $$-48 + 60 T - 18 T^{2} + T^{3}$$
$89$ $$1896 - 84 T - 18 T^{2} + T^{3}$$
$97$ $$8 + 144 T + 24 T^{2} + T^{3}$$