# Properties

 Label 2541.2.a.bi Level $2541$ Weight $2$ Character orbit 2541.a Self dual yes Analytic conductor $20.290$ Analytic rank $1$ Dimension $3$ 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: $$3$$ 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.

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

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 0.133492 2.52892 1.00000 −6.05784 1.00000 −0.337590
1.2 0.167449 −1.00000 −1.97196 3.80451 −0.167449 1.00000 −0.665102 1.00000 0.637062
1.3 2.36147 −1.00000 3.57653 −3.93800 −2.36147 1.00000 3.72294 1.00000 −9.29947
 $$n$$: 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.bi 3
3.b odd 2 1 7623.2.a.cb 3
11.b odd 2 1 231.2.a.d 3
33.d even 2 1 693.2.a.m 3
44.c even 2 1 3696.2.a.bp 3
55.d odd 2 1 5775.2.a.bw 3
77.b even 2 1 1617.2.a.s 3
231.h odd 2 1 4851.2.a.bp 3

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

 $$T_{2}^{3} - 6 T_{2} + 1$$ $$T_{5}^{3} - 15 T_{5} + 2$$ $$T_{13}^{3} - 15 T_{13} - 2$$

## Hecke characteristic polynomials

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