# Properties

 Label 2541.2.a.x 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: $$\Q(\sqrt{5})$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + ( -1 + \beta ) q^{5} -\beta q^{6} + q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + ( -1 + \beta ) q^{5} -\beta q^{6} + q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} + q^{10} + ( 1 - \beta ) q^{12} + ( -3 + 2 \beta ) q^{13} + \beta q^{14} + ( 1 - \beta ) q^{15} -3 \beta q^{16} + ( 1 - 4 \beta ) q^{17} + \beta q^{18} + ( -4 + 4 \beta ) q^{19} + ( 2 - \beta ) q^{20} - q^{21} + ( 2 - 6 \beta ) q^{23} + ( -1 + 2 \beta ) q^{24} + ( -3 - \beta ) q^{25} + ( 2 - \beta ) q^{26} - q^{27} + ( -1 + \beta ) q^{28} + ( -1 - 2 \beta ) q^{29} - q^{30} + 3 q^{31} + ( -5 + \beta ) q^{32} + ( -4 - 3 \beta ) q^{34} + ( -1 + \beta ) q^{35} + ( -1 + \beta ) q^{36} + 6 q^{37} + 4 q^{38} + ( 3 - 2 \beta ) q^{39} + ( -3 + \beta ) q^{40} + ( 8 - 3 \beta ) q^{41} -\beta q^{42} + ( 9 - 4 \beta ) q^{43} + ( -1 + \beta ) q^{45} + ( -6 - 4 \beta ) q^{46} + ( -7 - 3 \beta ) q^{47} + 3 \beta q^{48} + q^{49} + ( -1 - 4 \beta ) q^{50} + ( -1 + 4 \beta ) q^{51} + ( 5 - 3 \beta ) q^{52} + ( -7 + 9 \beta ) q^{53} -\beta q^{54} + ( 1 - 2 \beta ) q^{56} + ( 4 - 4 \beta ) q^{57} + ( -2 - 3 \beta ) q^{58} + ( -4 + \beta ) q^{59} + ( -2 + \beta ) q^{60} + ( 3 - 8 \beta ) q^{61} + 3 \beta q^{62} + q^{63} + ( 1 + 2 \beta ) q^{64} + ( 5 - 3 \beta ) q^{65} + ( 12 - 2 \beta ) q^{67} + ( -5 + \beta ) q^{68} + ( -2 + 6 \beta ) q^{69} + q^{70} + ( -7 - 4 \beta ) q^{71} + ( 1 - 2 \beta ) q^{72} + ( -8 - \beta ) q^{73} + 6 \beta q^{74} + ( 3 + \beta ) q^{75} + ( 8 - 4 \beta ) q^{76} + ( -2 + \beta ) q^{78} + ( 3 - 5 \beta ) q^{79} -3 q^{80} + q^{81} + ( -3 + 5 \beta ) q^{82} + ( -6 + 4 \beta ) q^{83} + ( 1 - \beta ) q^{84} + ( -5 + \beta ) q^{85} + ( -4 + 5 \beta ) q^{86} + ( 1 + 2 \beta ) q^{87} + ( -3 - 7 \beta ) q^{89} + q^{90} + ( -3 + 2 \beta ) q^{91} + ( -8 + 2 \beta ) q^{92} -3 q^{93} + ( -3 - 10 \beta ) q^{94} + ( 8 - 4 \beta ) q^{95} + ( 5 - \beta ) q^{96} + ( -9 - 6 \beta ) q^{97} + \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 2q^{3} - q^{4} - q^{5} - q^{6} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} - 2q^{3} - q^{4} - q^{5} - q^{6} + 2q^{7} + 2q^{9} + 2q^{10} + q^{12} - 4q^{13} + q^{14} + q^{15} - 3q^{16} - 2q^{17} + q^{18} - 4q^{19} + 3q^{20} - 2q^{21} - 2q^{23} - 7q^{25} + 3q^{26} - 2q^{27} - q^{28} - 4q^{29} - 2q^{30} + 6q^{31} - 9q^{32} - 11q^{34} - q^{35} - q^{36} + 12q^{37} + 8q^{38} + 4q^{39} - 5q^{40} + 13q^{41} - q^{42} + 14q^{43} - q^{45} - 16q^{46} - 17q^{47} + 3q^{48} + 2q^{49} - 6q^{50} + 2q^{51} + 7q^{52} - 5q^{53} - q^{54} + 4q^{57} - 7q^{58} - 7q^{59} - 3q^{60} - 2q^{61} + 3q^{62} + 2q^{63} + 4q^{64} + 7q^{65} + 22q^{67} - 9q^{68} + 2q^{69} + 2q^{70} - 18q^{71} - 17q^{73} + 6q^{74} + 7q^{75} + 12q^{76} - 3q^{78} + q^{79} - 6q^{80} + 2q^{81} - q^{82} - 8q^{83} + q^{84} - 9q^{85} - 3q^{86} + 4q^{87} - 13q^{89} + 2q^{90} - 4q^{91} - 14q^{92} - 6q^{93} - 16q^{94} + 12q^{95} + 9q^{96} - 24q^{97} + q^{98} + O(q^{100})$$

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

## 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.x 2
3.b odd 2 1 7623.2.a.z 2
11.b odd 2 1 2541.2.a.p 2
11.d odd 10 2 231.2.j.b 4
33.d even 2 1 7623.2.a.bo 2
33.f even 10 2 693.2.m.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.b 4 11.d odd 10 2
693.2.m.d 4 33.f even 10 2
2541.2.a.p 2 11.b odd 2 1
2541.2.a.x 2 1.a even 1 1 trivial
7623.2.a.z 2 3.b odd 2 1
7623.2.a.bo 2 33.d even 2 1

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

 $$T_{2}^{2} - T_{2} - 1$$ $$T_{5}^{2} + T_{5} - 1$$ $$T_{13}^{2} + 4 T_{13} - 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + T + 9 T^{2} + 5 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T )^{2}$$
$11$ 
$13$ $$1 + 4 T + 25 T^{2} + 52 T^{3} + 169 T^{4}$$
$17$ $$1 + 2 T + 15 T^{2} + 34 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 + 4 T + 57 T^{2} + 116 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 3 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 6 T + 37 T^{2} )^{2}$$
$41$ $$1 - 13 T + 113 T^{2} - 533 T^{3} + 1681 T^{4}$$
$43$ $$1 - 14 T + 115 T^{2} - 602 T^{3} + 1849 T^{4}$$
$47$ $$1 + 17 T + 155 T^{2} + 799 T^{3} + 2209 T^{4}$$
$53$ $$1 + 5 T + 11 T^{2} + 265 T^{3} + 2809 T^{4}$$
$59$ $$1 + 7 T + 129 T^{2} + 413 T^{3} + 3481 T^{4}$$
$61$ $$1 + 2 T + 43 T^{2} + 122 T^{3} + 3721 T^{4}$$
$67$ $$1 - 22 T + 250 T^{2} - 1474 T^{3} + 4489 T^{4}$$
$71$ $$1 + 18 T + 203 T^{2} + 1278 T^{3} + 5041 T^{4}$$
$73$ $$1 + 17 T + 217 T^{2} + 1241 T^{3} + 5329 T^{4}$$
$79$ $$1 - T + 127 T^{2} - 79 T^{3} + 6241 T^{4}$$
$83$ $$1 + 8 T + 162 T^{2} + 664 T^{3} + 6889 T^{4}$$
$89$ $$1 + 13 T + 159 T^{2} + 1157 T^{3} + 7921 T^{4}$$
$97$ $$1 + 24 T + 293 T^{2} + 2328 T^{3} + 9409 T^{4}$$