# Properties

 Label 2541.2.a.s 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: 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} + ( - \beta - 1) q^{5} - \beta q^{6} - q^{7} + (2 \beta - 1) q^{8} + q^{9} +O(q^{10})$$ q - b * q^2 + q^3 + (b - 1) * q^4 + (-b - 1) * q^5 - b * q^6 - q^7 + (2*b - 1) * q^8 + q^9 $$q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} + ( - \beta - 1) q^{5} - \beta q^{6} - q^{7} + (2 \beta - 1) q^{8} + q^{9} + (2 \beta + 1) q^{10} + (\beta - 1) q^{12} + q^{13} + \beta q^{14} + ( - \beta - 1) q^{15} - 3 \beta q^{16} + ( - 2 \beta + 3) q^{17} - \beta q^{18} - \beta q^{20} - q^{21} + (2 \beta - 2) q^{23} + (2 \beta - 1) q^{24} + (3 \beta - 3) q^{25} - \beta q^{26} + q^{27} + ( - \beta + 1) q^{28} + (6 \beta - 3) q^{29} + (2 \beta + 1) q^{30} + (2 \beta - 9) q^{31} + ( - \beta + 5) q^{32} + ( - \beta + 2) q^{34} + (\beta + 1) q^{35} + (\beta - 1) q^{36} + ( - 8 \beta + 2) q^{37} + q^{39} + ( - 3 \beta - 1) q^{40} + (5 \beta - 2) q^{41} + \beta q^{42} + q^{43} + ( - \beta - 1) q^{45} - 2 q^{46} + (7 \beta - 3) q^{47} - 3 \beta q^{48} + q^{49} - 3 q^{50} + ( - 2 \beta + 3) q^{51} + (\beta - 1) q^{52} + ( - \beta + 7) q^{53} - \beta q^{54} + ( - 2 \beta + 1) q^{56} + ( - 3 \beta - 6) q^{58} + ( - 3 \beta - 6) q^{59} - \beta q^{60} + (2 \beta + 7) q^{61} + (7 \beta - 2) q^{62} - q^{63} + (2 \beta + 1) q^{64} + ( - \beta - 1) q^{65} + (2 \beta - 8) q^{67} + (3 \beta - 5) q^{68} + (2 \beta - 2) q^{69} + ( - 2 \beta - 1) q^{70} + (4 \beta - 5) q^{71} + (2 \beta - 1) q^{72} + (9 \beta - 6) q^{73} + (6 \beta + 8) q^{74} + (3 \beta - 3) q^{75} - \beta q^{78} + (3 \beta - 9) q^{79} + (6 \beta + 3) q^{80} + q^{81} + ( - 3 \beta - 5) q^{82} + 6 q^{83} + ( - \beta + 1) q^{84} + (\beta - 1) q^{85} - \beta q^{86} + (6 \beta - 3) q^{87} + ( - \beta - 7) q^{89} + (2 \beta + 1) q^{90} - q^{91} + ( - 2 \beta + 4) q^{92} + (2 \beta - 9) q^{93} + ( - 4 \beta - 7) q^{94} + ( - \beta + 5) q^{96} - 17 q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + q^3 + (b - 1) * q^4 + (-b - 1) * q^5 - b * q^6 - q^7 + (2*b - 1) * q^8 + q^9 + (2*b + 1) * q^10 + (b - 1) * q^12 + q^13 + b * q^14 + (-b - 1) * q^15 - 3*b * q^16 + (-2*b + 3) * q^17 - b * q^18 - b * q^20 - q^21 + (2*b - 2) * q^23 + (2*b - 1) * q^24 + (3*b - 3) * q^25 - b * q^26 + q^27 + (-b + 1) * q^28 + (6*b - 3) * q^29 + (2*b + 1) * q^30 + (2*b - 9) * q^31 + (-b + 5) * q^32 + (-b + 2) * q^34 + (b + 1) * q^35 + (b - 1) * q^36 + (-8*b + 2) * q^37 + q^39 + (-3*b - 1) * q^40 + (5*b - 2) * q^41 + b * q^42 + q^43 + (-b - 1) * q^45 - 2 * q^46 + (7*b - 3) * q^47 - 3*b * q^48 + q^49 - 3 * q^50 + (-2*b + 3) * q^51 + (b - 1) * q^52 + (-b + 7) * q^53 - b * q^54 + (-2*b + 1) * q^56 + (-3*b - 6) * q^58 + (-3*b - 6) * q^59 - b * q^60 + (2*b + 7) * q^61 + (7*b - 2) * q^62 - q^63 + (2*b + 1) * q^64 + (-b - 1) * q^65 + (2*b - 8) * q^67 + (3*b - 5) * q^68 + (2*b - 2) * q^69 + (-2*b - 1) * q^70 + (4*b - 5) * q^71 + (2*b - 1) * q^72 + (9*b - 6) * q^73 + (6*b + 8) * q^74 + (3*b - 3) * q^75 - b * q^78 + (3*b - 9) * q^79 + (6*b + 3) * q^80 + q^81 + (-3*b - 5) * q^82 + 6 * q^83 + (-b + 1) * q^84 + (b - 1) * q^85 - b * q^86 + (6*b - 3) * q^87 + (-b - 7) * q^89 + (2*b + 1) * q^90 - q^91 + (-2*b + 4) * q^92 + (2*b - 9) * q^93 + (-4*b - 7) * q^94 + (-b + 5) * q^96 - 17 * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 + 2 * q^3 - q^4 - 3 * q^5 - q^6 - 2 * q^7 + 2 * q^9 $$2 q - q^{2} + 2 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 2 q^{9} + 4 q^{10} - q^{12} + 2 q^{13} + q^{14} - 3 q^{15} - 3 q^{16} + 4 q^{17} - q^{18} - q^{20} - 2 q^{21} - 2 q^{23} - 3 q^{25} - q^{26} + 2 q^{27} + q^{28} + 4 q^{30} - 16 q^{31} + 9 q^{32} + 3 q^{34} + 3 q^{35} - q^{36} - 4 q^{37} + 2 q^{39} - 5 q^{40} + q^{41} + q^{42} + 2 q^{43} - 3 q^{45} - 4 q^{46} + q^{47} - 3 q^{48} + 2 q^{49} - 6 q^{50} + 4 q^{51} - q^{52} + 13 q^{53} - q^{54} - 15 q^{58} - 15 q^{59} - q^{60} + 16 q^{61} + 3 q^{62} - 2 q^{63} + 4 q^{64} - 3 q^{65} - 14 q^{67} - 7 q^{68} - 2 q^{69} - 4 q^{70} - 6 q^{71} - 3 q^{73} + 22 q^{74} - 3 q^{75} - q^{78} - 15 q^{79} + 12 q^{80} + 2 q^{81} - 13 q^{82} + 12 q^{83} + q^{84} - q^{85} - q^{86} - 15 q^{89} + 4 q^{90} - 2 q^{91} + 6 q^{92} - 16 q^{93} - 18 q^{94} + 9 q^{96} - 34 q^{97} - q^{98}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 - q^4 - 3 * q^5 - q^6 - 2 * q^7 + 2 * q^9 + 4 * q^10 - q^12 + 2 * q^13 + q^14 - 3 * q^15 - 3 * q^16 + 4 * q^17 - q^18 - q^20 - 2 * q^21 - 2 * q^23 - 3 * q^25 - q^26 + 2 * q^27 + q^28 + 4 * q^30 - 16 * q^31 + 9 * q^32 + 3 * q^34 + 3 * q^35 - q^36 - 4 * q^37 + 2 * q^39 - 5 * q^40 + q^41 + q^42 + 2 * q^43 - 3 * q^45 - 4 * q^46 + q^47 - 3 * q^48 + 2 * q^49 - 6 * q^50 + 4 * q^51 - q^52 + 13 * q^53 - q^54 - 15 * q^58 - 15 * q^59 - q^60 + 16 * q^61 + 3 * q^62 - 2 * q^63 + 4 * q^64 - 3 * q^65 - 14 * q^67 - 7 * q^68 - 2 * q^69 - 4 * q^70 - 6 * q^71 - 3 * q^73 + 22 * q^74 - 3 * q^75 - q^78 - 15 * q^79 + 12 * q^80 + 2 * q^81 - 13 * q^82 + 12 * q^83 + q^84 - q^85 - q^86 - 15 * q^89 + 4 * q^90 - 2 * q^91 + 6 * q^92 - 16 * q^93 - 18 * q^94 + 9 * q^96 - 34 * 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 −2.61803 −1.61803 −1.00000 2.23607 1.00000 4.23607
1.2 0.618034 1.00000 −1.61803 −0.381966 0.618034 −1.00000 −2.23607 1.00000 −0.236068
 $$n$$: 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.s 2
3.b odd 2 1 7623.2.a.bp 2
11.b odd 2 1 2541.2.a.bb 2
11.d odd 10 2 231.2.j.d 4
33.d even 2 1 7623.2.a.ba 2
33.f even 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.d odd 10 2
693.2.m.b 4 33.f even 10 2
2541.2.a.s 2 1.a even 1 1 trivial
2541.2.a.bb 2 11.b odd 2 1
7623.2.a.ba 2 33.d even 2 1
7623.2.a.bp 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 $$T_{5}^{2} + 3T_{5} + 1$$ T5^2 + 3*T5 + 1 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 3T + 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 4T - 1$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 2T - 4$$
$29$ $$T^{2} - 45$$
$31$ $$T^{2} + 16T + 59$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} - T - 31$$
$43$ $$(T - 1)^{2}$$
$47$ $$T^{2} - T - 61$$
$53$ $$T^{2} - 13T + 41$$
$59$ $$T^{2} + 15T + 45$$
$61$ $$T^{2} - 16T + 59$$
$67$ $$T^{2} + 14T + 44$$
$71$ $$T^{2} + 6T - 11$$
$73$ $$T^{2} + 3T - 99$$
$79$ $$T^{2} + 15T + 45$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 15T + 55$$
$97$ $$(T + 17)^{2}$$