# Properties

 Label 2541.2.a.v Level 2541 Weight 2 Character orbit 2541.a Self dual yes Analytic conductor 20.290 Analytic rank 0 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \beta ) q^{2} - q^{3} + 3 q^{4} + ( 1 - \beta ) q^{5} + ( -1 + 2 \beta ) q^{6} - q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \beta ) q^{2} - q^{3} + 3 q^{4} + ( 1 - \beta ) q^{5} + ( -1 + 2 \beta ) q^{6} - q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} + ( 3 - \beta ) q^{10} -3 q^{12} -2 \beta q^{13} + ( -1 + 2 \beta ) q^{14} + ( -1 + \beta ) q^{15} - q^{16} -3 \beta q^{17} + ( 1 - 2 \beta ) q^{18} + ( 2 - 3 \beta ) q^{19} + ( 3 - 3 \beta ) q^{20} + q^{21} + ( 6 - \beta ) q^{23} + ( -1 + 2 \beta ) q^{24} + ( -3 - \beta ) q^{25} + ( 4 + 2 \beta ) q^{26} - q^{27} -3 q^{28} + 6 q^{29} + ( -3 + \beta ) q^{30} + ( 5 - 5 \beta ) q^{31} + ( -3 + 6 \beta ) q^{32} + ( 6 + 3 \beta ) q^{34} + ( -1 + \beta ) q^{35} + 3 q^{36} + ( 3 + \beta ) q^{37} + ( 8 - \beta ) q^{38} + 2 \beta q^{39} + ( 3 - \beta ) q^{40} + ( -9 + \beta ) q^{41} + ( 1 - 2 \beta ) q^{42} -6 \beta q^{43} + ( 1 - \beta ) q^{45} + ( 8 - 11 \beta ) q^{46} + ( -2 + 4 \beta ) q^{47} + q^{48} + q^{49} + ( -1 + 7 \beta ) q^{50} + 3 \beta q^{51} -6 \beta q^{52} + ( 10 - 2 \beta ) q^{53} + ( -1 + 2 \beta ) q^{54} + ( -1 + 2 \beta ) q^{56} + ( -2 + 3 \beta ) q^{57} + ( 6 - 12 \beta ) q^{58} + 2 \beta q^{59} + ( -3 + 3 \beta ) q^{60} + ( 15 - 5 \beta ) q^{62} - q^{63} -13 q^{64} + 2 q^{65} -9 \beta q^{68} + ( -6 + \beta ) q^{69} + ( -3 + \beta ) q^{70} + ( 8 - 8 \beta ) q^{71} + ( 1 - 2 \beta ) q^{72} + ( -4 - 6 \beta ) q^{73} + ( 1 - 7 \beta ) q^{74} + ( 3 + \beta ) q^{75} + ( 6 - 9 \beta ) q^{76} + ( -4 - 2 \beta ) q^{78} + 10 q^{79} + ( -1 + \beta ) q^{80} + q^{81} + ( -11 + 17 \beta ) q^{82} + ( -10 - 4 \beta ) q^{83} + 3 q^{84} + 3 q^{85} + ( 12 + 6 \beta ) q^{86} -6 q^{87} + ( 4 + \beta ) q^{89} + ( 3 - \beta ) q^{90} + 2 \beta q^{91} + ( 18 - 3 \beta ) q^{92} + ( -5 + 5 \beta ) q^{93} -10 q^{94} + ( 5 - 2 \beta ) q^{95} + ( 3 - 6 \beta ) q^{96} -6 q^{97} + ( 1 - 2 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 6q^{4} + q^{5} - 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 6q^{4} + q^{5} - 2q^{7} + 2q^{9} + 5q^{10} - 6q^{12} - 2q^{13} - q^{15} - 2q^{16} - 3q^{17} + q^{19} + 3q^{20} + 2q^{21} + 11q^{23} - 7q^{25} + 10q^{26} - 2q^{27} - 6q^{28} + 12q^{29} - 5q^{30} + 5q^{31} + 15q^{34} - q^{35} + 6q^{36} + 7q^{37} + 15q^{38} + 2q^{39} + 5q^{40} - 17q^{41} - 6q^{43} + q^{45} + 5q^{46} + 2q^{48} + 2q^{49} + 5q^{50} + 3q^{51} - 6q^{52} + 18q^{53} - q^{57} + 2q^{59} - 3q^{60} + 25q^{62} - 2q^{63} - 26q^{64} + 4q^{65} - 9q^{68} - 11q^{69} - 5q^{70} + 8q^{71} - 14q^{73} - 5q^{74} + 7q^{75} + 3q^{76} - 10q^{78} + 20q^{79} - q^{80} + 2q^{81} - 5q^{82} - 24q^{83} + 6q^{84} + 6q^{85} + 30q^{86} - 12q^{87} + 9q^{89} + 5q^{90} + 2q^{91} + 33q^{92} - 5q^{93} - 20q^{94} + 8q^{95} - 12q^{97} + O(q^{100})$$

## 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
−2.23607 −1.00000 3.00000 −0.618034 2.23607 −1.00000 −2.23607 1.00000 1.38197
1.2 2.23607 −1.00000 3.00000 1.61803 −2.23607 −1.00000 2.23607 1.00000 3.61803
 $$n$$: 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.v 2
3.b odd 2 1 7623.2.a.bj 2
11.b odd 2 1 2541.2.a.w 2
11.d odd 10 2 231.2.j.e 4
33.d even 2 1 7623.2.a.bk 2
33.f even 10 2 693.2.m.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.e 4 11.d odd 10 2
693.2.m.a 4 33.f even 10 2
2541.2.a.v 2 1.a even 1 1 trivial
2541.2.a.w 2 11.b odd 2 1
7623.2.a.bj 2 3.b odd 2 1
7623.2.a.bk 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + 4 T^{4}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 - T + 9 T^{2} - 5 T^{3} + 25 T^{4}$$
$7$ $$( 1 + T )^{2}$$
$11$ 1
$13$ $$1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$1 + 3 T + 25 T^{2} + 51 T^{3} + 289 T^{4}$$
$19$ $$1 - T + 27 T^{2} - 19 T^{3} + 361 T^{4}$$
$23$ $$1 - 11 T + 75 T^{2} - 253 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$1 - 5 T + 37 T^{2} - 155 T^{3} + 961 T^{4}$$
$37$ $$1 - 7 T + 85 T^{2} - 259 T^{3} + 1369 T^{4}$$
$41$ $$1 + 17 T + 153 T^{2} + 697 T^{3} + 1681 T^{4}$$
$43$ $$1 + 6 T + 50 T^{2} + 258 T^{3} + 1849 T^{4}$$
$47$ $$1 + 74 T^{2} + 2209 T^{4}$$
$53$ $$1 - 18 T + 182 T^{2} - 954 T^{3} + 2809 T^{4}$$
$59$ $$1 - 2 T + 114 T^{2} - 118 T^{3} + 3481 T^{4}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$( 1 + 67 T^{2} )^{2}$$
$71$ $$1 - 8 T + 78 T^{2} - 568 T^{3} + 5041 T^{4}$$
$73$ $$1 + 14 T + 150 T^{2} + 1022 T^{3} + 5329 T^{4}$$
$79$ $$( 1 - 10 T + 79 T^{2} )^{2}$$
$83$ $$1 + 24 T + 290 T^{2} + 1992 T^{3} + 6889 T^{4}$$
$89$ $$1 - 9 T + 197 T^{2} - 801 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 6 T + 97 T^{2} )^{2}$$