# Properties

 Label 2541.2.a.bb 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.

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

## 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 −0.381966 −0.618034 1.00000 2.23607 1.00000 0.236068
1.2 1.61803 1.00000 0.618034 −2.61803 1.61803 1.00000 −2.23607 1.00000 −4.23607
 $$n$$: 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.bb 2
3.b odd 2 1 7623.2.a.ba 2
11.b odd 2 1 2541.2.a.s 2
11.c even 5 2 231.2.j.d 4
33.d even 2 1 7623.2.a.bp 2
33.h odd 10 2 693.2.m.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.d 4 11.c even 5 2
693.2.m.b 4 33.h odd 10 2
2541.2.a.s 2 11.b odd 2 1
2541.2.a.bb 2 1.a even 1 1 trivial
7623.2.a.ba 2 3.b odd 2 1
7623.2.a.bp 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$$
$3$ $$( 1 - T )^{2}$$
$5$ $$1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T )^{2}$$
$11$ 
$13$ $$( 1 + T + 13 T^{2} )^{2}$$
$17$ $$1 + 4 T + 33 T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 + 2 T + 42 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 + 13 T^{2} + 841 T^{4}$$
$31$ $$1 + 16 T + 121 T^{2} + 496 T^{3} + 961 T^{4}$$
$37$ $$1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4}$$
$41$ $$1 + T + 51 T^{2} + 41 T^{3} + 1681 T^{4}$$
$43$ $$( 1 + T + 43 T^{2} )^{2}$$
$47$ $$1 - T + 33 T^{2} - 47 T^{3} + 2209 T^{4}$$
$53$ $$1 - 13 T + 147 T^{2} - 689 T^{3} + 2809 T^{4}$$
$59$ $$1 + 15 T + 163 T^{2} + 885 T^{3} + 3481 T^{4}$$
$61$ $$1 + 16 T + 181 T^{2} + 976 T^{3} + 3721 T^{4}$$
$67$ $$1 + 14 T + 178 T^{2} + 938 T^{3} + 4489 T^{4}$$
$71$ $$1 + 6 T + 131 T^{2} + 426 T^{3} + 5041 T^{4}$$
$73$ $$1 - 3 T + 47 T^{2} - 219 T^{3} + 5329 T^{4}$$
$79$ $$1 - 15 T + 203 T^{2} - 1185 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 6 T + 83 T^{2} )^{2}$$
$89$ $$1 + 15 T + 233 T^{2} + 1335 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 17 T + 97 T^{2} )^{2}$$