# Properties

 Label 5775.2.a.be Level $5775$ Weight $2$ Character orbit 5775.a Self dual yes Analytic conductor $46.114$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + \beta q^{6} - q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + \beta q^{6} - q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} + q^{11} + ( 1 - \beta ) q^{12} + ( -1 + 4 \beta ) q^{13} + \beta q^{14} -3 \beta q^{16} + ( -4 + 2 \beta ) q^{17} -\beta q^{18} + ( -3 + 6 \beta ) q^{19} + q^{21} -\beta q^{22} + ( -2 + 6 \beta ) q^{23} + ( 1 - 2 \beta ) q^{24} + ( -4 - 3 \beta ) q^{26} - q^{27} + ( 1 - \beta ) q^{28} + 5 q^{29} + ( -4 + 2 \beta ) q^{31} + ( 5 - \beta ) q^{32} - q^{33} + ( -2 + 2 \beta ) q^{34} + ( -1 + \beta ) q^{36} + 7 q^{37} + ( -6 - 3 \beta ) q^{38} + ( 1 - 4 \beta ) q^{39} + 4 \beta q^{41} -\beta q^{42} + ( -2 + 6 \beta ) q^{43} + ( -1 + \beta ) q^{44} + ( -6 - 4 \beta ) q^{46} + ( 1 + 2 \beta ) q^{47} + 3 \beta q^{48} + q^{49} + ( 4 - 2 \beta ) q^{51} + ( 5 - \beta ) q^{52} + ( 6 - 10 \beta ) q^{53} + \beta q^{54} + ( 1 - 2 \beta ) q^{56} + ( 3 - 6 \beta ) q^{57} -5 \beta q^{58} + ( -5 + 10 \beta ) q^{59} + 2 q^{61} + ( -2 + 2 \beta ) q^{62} - q^{63} + ( 1 + 2 \beta ) q^{64} + \beta q^{66} + ( 11 + 2 \beta ) q^{67} + ( 6 - 4 \beta ) q^{68} + ( 2 - 6 \beta ) q^{69} + 4 \beta q^{71} + ( -1 + 2 \beta ) q^{72} + ( -7 - 4 \beta ) q^{73} -7 \beta q^{74} + ( 9 - 3 \beta ) q^{76} - q^{77} + ( 4 + 3 \beta ) q^{78} + ( -12 + 4 \beta ) q^{79} + q^{81} + ( -4 - 4 \beta ) q^{82} + ( -8 - 2 \beta ) q^{83} + ( -1 + \beta ) q^{84} + ( -6 - 4 \beta ) q^{86} -5 q^{87} + ( -1 + 2 \beta ) q^{88} + ( -2 + 4 \beta ) q^{89} + ( 1 - 4 \beta ) q^{91} + ( 8 - 2 \beta ) q^{92} + ( 4 - 2 \beta ) q^{93} + ( -2 - 3 \beta ) q^{94} + ( -5 + \beta ) q^{96} + ( -6 + 6 \beta ) q^{97} -\beta q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 2q^{7} + 2q^{9} + 2q^{11} + q^{12} + 2q^{13} + q^{14} - 3q^{16} - 6q^{17} - q^{18} + 2q^{21} - q^{22} + 2q^{23} - 11q^{26} - 2q^{27} + q^{28} + 10q^{29} - 6q^{31} + 9q^{32} - 2q^{33} - 2q^{34} - q^{36} + 14q^{37} - 15q^{38} - 2q^{39} + 4q^{41} - q^{42} + 2q^{43} - q^{44} - 16q^{46} + 4q^{47} + 3q^{48} + 2q^{49} + 6q^{51} + 9q^{52} + 2q^{53} + q^{54} - 5q^{58} + 4q^{61} - 2q^{62} - 2q^{63} + 4q^{64} + q^{66} + 24q^{67} + 8q^{68} - 2q^{69} + 4q^{71} - 18q^{73} - 7q^{74} + 15q^{76} - 2q^{77} + 11q^{78} - 20q^{79} + 2q^{81} - 12q^{82} - 18q^{83} - q^{84} - 16q^{86} - 10q^{87} - 2q^{91} + 14q^{92} + 6q^{93} - 7q^{94} - 9q^{96} - 6q^{97} - q^{98} + 2q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
−1.61803 −1.00000 0.618034 0 1.61803 −1.00000 2.23607 1.00000 0
1.2 0.618034 −1.00000 −1.61803 0 −0.618034 −1.00000 −2.23607 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.be 2
5.b even 2 1 231.2.a.c 2
15.d odd 2 1 693.2.a.f 2
20.d odd 2 1 3696.2.a.be 2
35.c odd 2 1 1617.2.a.p 2
55.d odd 2 1 2541.2.a.t 2
105.g even 2 1 4851.2.a.w 2
165.d even 2 1 7623.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.c 2 5.b even 2 1
693.2.a.f 2 15.d odd 2 1
1617.2.a.p 2 35.c odd 2 1
2541.2.a.t 2 55.d odd 2 1
3696.2.a.be 2 20.d odd 2 1
4851.2.a.w 2 105.g even 2 1
5775.2.a.be 2 1.a even 1 1 trivial
7623.2.a.bm 2 165.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5775))$$:

 $$T_{2}^{2} + T_{2} - 1$$ $$T_{13}^{2} - 2 T_{13} - 19$$ $$T_{17}^{2} + 6 T_{17} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-19 - 2 T + T^{2}$$
$17$ $$4 + 6 T + T^{2}$$
$19$ $$-45 + T^{2}$$
$23$ $$-44 - 2 T + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$4 + 6 T + T^{2}$$
$37$ $$( -7 + T )^{2}$$
$41$ $$-16 - 4 T + T^{2}$$
$43$ $$-44 - 2 T + T^{2}$$
$47$ $$-1 - 4 T + T^{2}$$
$53$ $$-124 - 2 T + T^{2}$$
$59$ $$-125 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$139 - 24 T + T^{2}$$
$71$ $$-16 - 4 T + T^{2}$$
$73$ $$61 + 18 T + T^{2}$$
$79$ $$80 + 20 T + T^{2}$$
$83$ $$76 + 18 T + T^{2}$$
$89$ $$-20 + T^{2}$$
$97$ $$-36 + 6 T + T^{2}$$