# Properties

 Label 2541.2.a.t Level $2541$ Weight $2$ Character orbit 2541.a Self dual yes Analytic conductor $20.290$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2541 = 3 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2541.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.2899871536$$ Analytic rank: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} + q^{5} -\beta q^{6} - q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} + q^{5} -\beta q^{6} - q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} -\beta q^{10} + ( -1 + \beta ) q^{12} + ( -1 + 4 \beta ) q^{13} + \beta q^{14} + q^{15} -3 \beta q^{16} + ( -4 + 2 \beta ) q^{17} -\beta q^{18} + ( 3 - 6 \beta ) q^{19} + ( -1 + \beta ) q^{20} - q^{21} + ( 2 - 6 \beta ) q^{23} + ( -1 + 2 \beta ) q^{24} -4 q^{25} + ( -4 - 3 \beta ) q^{26} + q^{27} + ( 1 - \beta ) q^{28} -5 q^{29} -\beta q^{30} + ( -4 + 2 \beta ) q^{31} + ( 5 - \beta ) q^{32} + ( -2 + 2 \beta ) q^{34} - q^{35} + ( -1 + \beta ) q^{36} -7 q^{37} + ( 6 + 3 \beta ) q^{38} + ( -1 + 4 \beta ) q^{39} + ( -1 + 2 \beta ) q^{40} -4 \beta q^{41} + \beta q^{42} + ( -2 + 6 \beta ) q^{43} + q^{45} + ( 6 + 4 \beta ) q^{46} + ( -1 - 2 \beta ) q^{47} -3 \beta q^{48} + q^{49} + 4 \beta q^{50} + ( -4 + 2 \beta ) q^{51} + ( 5 - \beta ) q^{52} + ( -6 + 10 \beta ) q^{53} -\beta q^{54} + ( 1 - 2 \beta ) q^{56} + ( 3 - 6 \beta ) q^{57} + 5 \beta q^{58} + ( -5 + 10 \beta ) q^{59} + ( -1 + \beta ) q^{60} -2 q^{61} + ( -2 + 2 \beta ) q^{62} - q^{63} + ( 1 + 2 \beta ) q^{64} + ( -1 + 4 \beta ) q^{65} + ( -11 - 2 \beta ) q^{67} + ( 6 - 4 \beta ) q^{68} + ( 2 - 6 \beta ) q^{69} + \beta q^{70} + 4 \beta q^{71} + ( -1 + 2 \beta ) q^{72} + ( -7 - 4 \beta ) q^{73} + 7 \beta q^{74} -4 q^{75} + ( -9 + 3 \beta ) q^{76} + ( -4 - 3 \beta ) q^{78} + ( 12 - 4 \beta ) q^{79} -3 \beta q^{80} + q^{81} + ( 4 + 4 \beta ) q^{82} + ( -8 - 2 \beta ) q^{83} + ( 1 - \beta ) q^{84} + ( -4 + 2 \beta ) q^{85} + ( -6 - 4 \beta ) q^{86} -5 q^{87} + ( -2 + 4 \beta ) q^{89} -\beta q^{90} + ( 1 - 4 \beta ) q^{91} + ( -8 + 2 \beta ) q^{92} + ( -4 + 2 \beta ) q^{93} + ( 2 + 3 \beta ) q^{94} + ( 3 - 6 \beta ) q^{95} + ( 5 - \beta ) q^{96} + ( 6 - 6 \beta ) q^{97} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{9} - q^{10} - q^{12} + 2 q^{13} + q^{14} + 2 q^{15} - 3 q^{16} - 6 q^{17} - q^{18} - q^{20} - 2 q^{21} - 2 q^{23} - 8 q^{25} - 11 q^{26} + 2 q^{27} + q^{28} - 10 q^{29} - q^{30} - 6 q^{31} + 9 q^{32} - 2 q^{34} - 2 q^{35} - q^{36} - 14 q^{37} + 15 q^{38} + 2 q^{39} - 4 q^{41} + q^{42} + 2 q^{43} + 2 q^{45} + 16 q^{46} - 4 q^{47} - 3 q^{48} + 2 q^{49} + 4 q^{50} - 6 q^{51} + 9 q^{52} - 2 q^{53} - q^{54} + 5 q^{58} - q^{60} - 4 q^{61} - 2 q^{62} - 2 q^{63} + 4 q^{64} + 2 q^{65} - 24 q^{67} + 8 q^{68} - 2 q^{69} + q^{70} + 4 q^{71} - 18 q^{73} + 7 q^{74} - 8 q^{75} - 15 q^{76} - 11 q^{78} + 20 q^{79} - 3 q^{80} + 2 q^{81} + 12 q^{82} - 18 q^{83} + q^{84} - 6 q^{85} - 16 q^{86} - 10 q^{87} - q^{90} - 2 q^{91} - 14 q^{92} - 6 q^{93} + 7 q^{94} + 9 q^{96} + 6 q^{97} - q^{98} + O(q^{100})$$

## 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 1.00000 −1.61803 −1.00000 2.23607 1.00000 −1.61803
1.2 0.618034 1.00000 −1.61803 1.00000 0.618034 −1.00000 −2.23607 1.00000 0.618034
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## 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 2541.2.a.t 2
3.b odd 2 1 7623.2.a.bm 2
11.b odd 2 1 231.2.a.c 2
33.d even 2 1 693.2.a.f 2
44.c even 2 1 3696.2.a.be 2
55.d odd 2 1 5775.2.a.be 2
77.b even 2 1 1617.2.a.p 2
231.h odd 2 1 4851.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.c 2 11.b odd 2 1
693.2.a.f 2 33.d even 2 1
1617.2.a.p 2 77.b even 2 1
2541.2.a.t 2 1.a even 1 1 trivial
3696.2.a.be 2 44.c even 2 1
4851.2.a.w 2 231.h odd 2 1
5775.2.a.be 2 55.d odd 2 1
7623.2.a.bm 2 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

 $$T_{2}^{2} + T_{2} - 1$$ $$T_{5} - 1$$ $$T_{13}^{2} - 2 T_{13} - 19$$

## Hecke characteristic polynomials

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