# Properties

 Label 2541.2.a.r 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

## 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
−1.61803 1.00000 0.618034 −4.23607 −1.61803 1.00000 2.23607 1.00000 6.85410
1.2 0.618034 1.00000 −1.61803 0.236068 0.618034 1.00000 −2.23607 1.00000 0.145898
 $$n$$: 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.r 2
3.b odd 2 1 7623.2.a.bq 2
11.b odd 2 1 2541.2.a.ba yes 2
33.d even 2 1 7623.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.r 2 1.a even 1 1 trivial
2541.2.a.ba yes 2 11.b odd 2 1
7623.2.a.bb 2 33.d even 2 1
7623.2.a.bq 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

 $$T_{2}^{2} + T_{2} - 1$$ T2^2 + T2 - 1 $$T_{5}^{2} + 4T_{5} - 1$$ T5^2 + 4*T5 - 1 $$T_{13}^{2} + 8T_{13} + 11$$ T13^2 + 8*T13 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 4T - 1$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 8T + 11$$
$17$ $$T^{2} - 20$$
$19$ $$(T + 3)^{2}$$
$23$ $$T^{2} - 8T - 4$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 2T - 19$$
$41$ $$T^{2} + 12T + 16$$
$43$ $$T^{2} + 4T - 76$$
$47$ $$(T + 3)^{2}$$
$53$ $$T^{2} - 80$$
$59$ $$T^{2} - 6T - 11$$
$61$ $$T^{2} + 16T + 44$$
$67$ $$T^{2} + 4T - 41$$
$71$ $$T^{2} - 12T + 16$$
$73$ $$T^{2} + 20T + 95$$
$79$ $$T^{2} - 180$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 20$$
$97$ $$(T - 2)^{2}$$