# Properties

 Label 2541.2.a.m 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{2} - q^{3} + 3 \beta q^{4} + \beta q^{5} + ( 1 + \beta ) q^{6} - q^{7} + ( -1 - 4 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{2} - q^{3} + 3 \beta q^{4} + \beta q^{5} + ( 1 + \beta ) q^{6} - q^{7} + ( -1 - 4 \beta ) q^{8} + q^{9} + ( -1 - 2 \beta ) q^{10} -3 \beta q^{12} + ( 1 - 4 \beta ) q^{13} + ( 1 + \beta ) q^{14} -\beta q^{15} + ( 5 + 3 \beta ) q^{16} + 3 q^{17} + ( -1 - \beta ) q^{18} -4 q^{19} + ( 3 + 3 \beta ) q^{20} + q^{21} + ( -4 - 2 \beta ) q^{23} + ( 1 + 4 \beta ) q^{24} + ( -4 + \beta ) q^{25} + ( 3 + 7 \beta ) q^{26} - q^{27} -3 \beta q^{28} -3 q^{29} + ( 1 + 2 \beta ) q^{30} + ( -3 + 2 \beta ) q^{31} + ( -6 - 3 \beta ) q^{32} + ( -3 - 3 \beta ) q^{34} -\beta q^{35} + 3 \beta q^{36} + ( -2 - 4 \beta ) q^{37} + ( 4 + 4 \beta ) q^{38} + ( -1 + 4 \beta ) q^{39} + ( -4 - 5 \beta ) q^{40} + ( -5 + 5 \beta ) q^{41} + ( -1 - \beta ) q^{42} + 9 q^{43} + \beta q^{45} + ( 6 + 8 \beta ) q^{46} + ( 2 + 5 \beta ) q^{47} + ( -5 - 3 \beta ) q^{48} + q^{49} + ( 3 + 2 \beta ) q^{50} -3 q^{51} + ( -12 - 9 \beta ) q^{52} + ( -4 + 5 \beta ) q^{53} + ( 1 + \beta ) q^{54} + ( 1 + 4 \beta ) q^{56} + 4 q^{57} + ( 3 + 3 \beta ) q^{58} + ( 11 + \beta ) q^{59} + ( -3 - 3 \beta ) q^{60} + ( -3 + 6 \beta ) q^{61} + ( 1 - \beta ) q^{62} - q^{63} + ( -1 + 6 \beta ) q^{64} + ( -4 - 3 \beta ) q^{65} + ( 6 - 6 \beta ) q^{67} + 9 \beta q^{68} + ( 4 + 2 \beta ) q^{69} + ( 1 + 2 \beta ) q^{70} + ( 9 + 2 \beta ) q^{71} + ( -1 - 4 \beta ) q^{72} + ( -1 + 3 \beta ) q^{73} + ( 6 + 10 \beta ) q^{74} + ( 4 - \beta ) q^{75} -12 \beta q^{76} + ( -3 - 7 \beta ) q^{78} + ( -2 - 3 \beta ) q^{79} + ( 3 + 8 \beta ) q^{80} + q^{81} -5 \beta q^{82} + ( 10 - 8 \beta ) q^{83} + 3 \beta q^{84} + 3 \beta q^{85} + ( -9 - 9 \beta ) q^{86} + 3 q^{87} + ( 14 - 7 \beta ) q^{89} + ( -1 - 2 \beta ) q^{90} + ( -1 + 4 \beta ) q^{91} + ( -6 - 18 \beta ) q^{92} + ( 3 - 2 \beta ) q^{93} + ( -7 - 12 \beta ) q^{94} -4 \beta q^{95} + ( 6 + 3 \beta ) q^{96} + ( 3 - 12 \beta ) q^{97} + ( -1 - \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - 2q^{3} + 3q^{4} + q^{5} + 3q^{6} - 2q^{7} - 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 3q^{2} - 2q^{3} + 3q^{4} + q^{5} + 3q^{6} - 2q^{7} - 6q^{8} + 2q^{9} - 4q^{10} - 3q^{12} - 2q^{13} + 3q^{14} - q^{15} + 13q^{16} + 6q^{17} - 3q^{18} - 8q^{19} + 9q^{20} + 2q^{21} - 10q^{23} + 6q^{24} - 7q^{25} + 13q^{26} - 2q^{27} - 3q^{28} - 6q^{29} + 4q^{30} - 4q^{31} - 15q^{32} - 9q^{34} - q^{35} + 3q^{36} - 8q^{37} + 12q^{38} + 2q^{39} - 13q^{40} - 5q^{41} - 3q^{42} + 18q^{43} + q^{45} + 20q^{46} + 9q^{47} - 13q^{48} + 2q^{49} + 8q^{50} - 6q^{51} - 33q^{52} - 3q^{53} + 3q^{54} + 6q^{56} + 8q^{57} + 9q^{58} + 23q^{59} - 9q^{60} + q^{62} - 2q^{63} + 4q^{64} - 11q^{65} + 6q^{67} + 9q^{68} + 10q^{69} + 4q^{70} + 20q^{71} - 6q^{72} + q^{73} + 22q^{74} + 7q^{75} - 12q^{76} - 13q^{78} - 7q^{79} + 14q^{80} + 2q^{81} - 5q^{82} + 12q^{83} + 3q^{84} + 3q^{85} - 27q^{86} + 6q^{87} + 21q^{89} - 4q^{90} + 2q^{91} - 30q^{92} + 4q^{93} - 26q^{94} - 4q^{95} + 15q^{96} - 6q^{97} - 3q^{98} + O(q^{100})$$

## 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
−2.61803 −1.00000 4.85410 1.61803 2.61803 −1.00000 −7.47214 1.00000 −4.23607
1.2 −0.381966 −1.00000 −1.85410 −0.618034 0.381966 −1.00000 1.47214 1.00000 0.236068
 $$n$$: 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.m 2
3.b odd 2 1 7623.2.a.by 2
11.b odd 2 1 2541.2.a.bf 2
11.d odd 10 2 231.2.j.a 4
33.d even 2 1 7623.2.a.u 2
33.f even 10 2 693.2.m.e 4

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

 $$T_{2}^{2} + 3 T_{2} + 1$$ $$T_{5}^{2} - T_{5} - 1$$ $$T_{13}^{2} + 2 T_{13} - 19$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 - T + 9 T^{2} - 5 T^{3} + 25 T^{4}$$
$7$ $$( 1 + T )^{2}$$
$11$ 
$13$ $$1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 3 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 + 10 T + 66 T^{2} + 230 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 3 T + 29 T^{2} )^{2}$$
$31$ $$1 + 4 T + 61 T^{2} + 124 T^{3} + 961 T^{4}$$
$37$ $$1 + 8 T + 70 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$1 + 5 T + 57 T^{2} + 205 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 9 T + 43 T^{2} )^{2}$$
$47$ $$1 - 9 T + 83 T^{2} - 423 T^{3} + 2209 T^{4}$$
$53$ $$1 + 3 T + 77 T^{2} + 159 T^{3} + 2809 T^{4}$$
$59$ $$1 - 23 T + 249 T^{2} - 1357 T^{3} + 3481 T^{4}$$
$61$ $$1 + 77 T^{2} + 3721 T^{4}$$
$67$ $$1 - 6 T + 98 T^{2} - 402 T^{3} + 4489 T^{4}$$
$71$ $$1 - 20 T + 237 T^{2} - 1420 T^{3} + 5041 T^{4}$$
$73$ $$1 - T + 135 T^{2} - 73 T^{3} + 5329 T^{4}$$
$79$ $$1 + 7 T + 159 T^{2} + 553 T^{3} + 6241 T^{4}$$
$83$ $$1 - 12 T + 122 T^{2} - 996 T^{3} + 6889 T^{4}$$
$89$ $$1 - 21 T + 227 T^{2} - 1869 T^{3} + 7921 T^{4}$$
$97$ $$1 + 6 T + 23 T^{2} + 582 T^{3} + 9409 T^{4}$$