# Properties

 Label 2541.2.a.n 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + ( - 3 \beta + 2) q^{5} + q^{6} + q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 - q^4 + (-3*b + 2) * q^5 + q^6 + q^7 + 3 * q^8 + q^9 $$q - q^{2} - q^{3} - q^{4} + ( - 3 \beta + 2) q^{5} + q^{6} + q^{7} + 3 q^{8} + q^{9} + (3 \beta - 2) q^{10} + q^{12} + ( - 2 \beta + 2) q^{13} - q^{14} + (3 \beta - 2) q^{15} - q^{16} + (3 \beta + 3) q^{17} - q^{18} + (\beta + 1) q^{19} + (3 \beta - 2) q^{20} - q^{21} + ( - 5 \beta + 5) q^{23} - 3 q^{24} + ( - 3 \beta + 8) q^{25} + (2 \beta - 2) q^{26} - q^{27} - q^{28} - 2 q^{29} + ( - 3 \beta + 2) q^{30} + ( - 7 \beta + 4) q^{31} - 5 q^{32} + ( - 3 \beta - 3) q^{34} + ( - 3 \beta + 2) q^{35} - q^{36} + ( - 5 \beta - 4) q^{37} + ( - \beta - 1) q^{38} + (2 \beta - 2) q^{39} + ( - 9 \beta + 6) q^{40} + ( - 3 \beta - 6) q^{41} + q^{42} + ( - 2 \beta + 2) q^{43} + ( - 3 \beta + 2) q^{45} + (5 \beta - 5) q^{46} + 2 q^{47} + q^{48} + q^{49} + (3 \beta - 8) q^{50} + ( - 3 \beta - 3) q^{51} + (2 \beta - 2) q^{52} + (10 \beta - 4) q^{53} + q^{54} + 3 q^{56} + ( - \beta - 1) q^{57} + 2 q^{58} + ( - 2 \beta + 10) q^{59} + ( - 3 \beta + 2) q^{60} + (8 \beta - 4) q^{61} + (7 \beta - 4) q^{62} + q^{63} + 7 q^{64} + ( - 4 \beta + 10) q^{65} + 8 q^{67} + ( - 3 \beta - 3) q^{68} + (5 \beta - 5) q^{69} + (3 \beta - 2) q^{70} + (8 \beta - 4) q^{71} + 3 q^{72} + ( - 2 \beta - 2) q^{73} + (5 \beta + 4) q^{74} + (3 \beta - 8) q^{75} + ( - \beta - 1) q^{76} + ( - 2 \beta + 2) q^{78} - 14 q^{79} + (3 \beta - 2) q^{80} + q^{81} + (3 \beta + 6) q^{82} + (8 \beta - 10) q^{83} + q^{84} + ( - 12 \beta - 3) q^{85} + (2 \beta - 2) q^{86} + 2 q^{87} + (5 \beta - 7) q^{89} + (3 \beta - 2) q^{90} + ( - 2 \beta + 2) q^{91} + (5 \beta - 5) q^{92} + (7 \beta - 4) q^{93} - 2 q^{94} + ( - 4 \beta - 1) q^{95} + 5 q^{96} + (4 \beta - 10) q^{97} - q^{98}+O(q^{100})$$ q - q^2 - q^3 - q^4 + (-3*b + 2) * q^5 + q^6 + q^7 + 3 * q^8 + q^9 + (3*b - 2) * q^10 + q^12 + (-2*b + 2) * q^13 - q^14 + (3*b - 2) * q^15 - q^16 + (3*b + 3) * q^17 - q^18 + (b + 1) * q^19 + (3*b - 2) * q^20 - q^21 + (-5*b + 5) * q^23 - 3 * q^24 + (-3*b + 8) * q^25 + (2*b - 2) * q^26 - q^27 - q^28 - 2 * q^29 + (-3*b + 2) * q^30 + (-7*b + 4) * q^31 - 5 * q^32 + (-3*b - 3) * q^34 + (-3*b + 2) * q^35 - q^36 + (-5*b - 4) * q^37 + (-b - 1) * q^38 + (2*b - 2) * q^39 + (-9*b + 6) * q^40 + (-3*b - 6) * q^41 + q^42 + (-2*b + 2) * q^43 + (-3*b + 2) * q^45 + (5*b - 5) * q^46 + 2 * q^47 + q^48 + q^49 + (3*b - 8) * q^50 + (-3*b - 3) * q^51 + (2*b - 2) * q^52 + (10*b - 4) * q^53 + q^54 + 3 * q^56 + (-b - 1) * q^57 + 2 * q^58 + (-2*b + 10) * q^59 + (-3*b + 2) * q^60 + (8*b - 4) * q^61 + (7*b - 4) * q^62 + q^63 + 7 * q^64 + (-4*b + 10) * q^65 + 8 * q^67 + (-3*b - 3) * q^68 + (5*b - 5) * q^69 + (3*b - 2) * q^70 + (8*b - 4) * q^71 + 3 * q^72 + (-2*b - 2) * q^73 + (5*b + 4) * q^74 + (3*b - 8) * q^75 + (-b - 1) * q^76 + (-2*b + 2) * q^78 - 14 * q^79 + (3*b - 2) * q^80 + q^81 + (3*b + 6) * q^82 + (8*b - 10) * q^83 + q^84 + (-12*b - 3) * q^85 + (2*b - 2) * q^86 + 2 * q^87 + (5*b - 7) * q^89 + (3*b - 2) * q^90 + (-2*b + 2) * q^91 + (5*b - 5) * q^92 + (7*b - 4) * q^93 - 2 * q^94 + (-4*b - 1) * q^95 + 5 * q^96 + (4*b - 10) * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + q^5 + 2 * q^6 + 2 * q^7 + 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9} - q^{10} + 2 q^{12} + 2 q^{13} - 2 q^{14} - q^{15} - 2 q^{16} + 9 q^{17} - 2 q^{18} + 3 q^{19} - q^{20} - 2 q^{21} + 5 q^{23} - 6 q^{24} + 13 q^{25} - 2 q^{26} - 2 q^{27} - 2 q^{28} - 4 q^{29} + q^{30} + q^{31} - 10 q^{32} - 9 q^{34} + q^{35} - 2 q^{36} - 13 q^{37} - 3 q^{38} - 2 q^{39} + 3 q^{40} - 15 q^{41} + 2 q^{42} + 2 q^{43} + q^{45} - 5 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{49} - 13 q^{50} - 9 q^{51} - 2 q^{52} + 2 q^{53} + 2 q^{54} + 6 q^{56} - 3 q^{57} + 4 q^{58} + 18 q^{59} + q^{60} - q^{62} + 2 q^{63} + 14 q^{64} + 16 q^{65} + 16 q^{67} - 9 q^{68} - 5 q^{69} - q^{70} + 6 q^{72} - 6 q^{73} + 13 q^{74} - 13 q^{75} - 3 q^{76} + 2 q^{78} - 28 q^{79} - q^{80} + 2 q^{81} + 15 q^{82} - 12 q^{83} + 2 q^{84} - 18 q^{85} - 2 q^{86} + 4 q^{87} - 9 q^{89} - q^{90} + 2 q^{91} - 5 q^{92} - q^{93} - 4 q^{94} - 6 q^{95} + 10 q^{96} - 16 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + q^5 + 2 * q^6 + 2 * q^7 + 6 * q^8 + 2 * q^9 - q^10 + 2 * q^12 + 2 * q^13 - 2 * q^14 - q^15 - 2 * q^16 + 9 * q^17 - 2 * q^18 + 3 * q^19 - q^20 - 2 * q^21 + 5 * q^23 - 6 * q^24 + 13 * q^25 - 2 * q^26 - 2 * q^27 - 2 * q^28 - 4 * q^29 + q^30 + q^31 - 10 * q^32 - 9 * q^34 + q^35 - 2 * q^36 - 13 * q^37 - 3 * q^38 - 2 * q^39 + 3 * q^40 - 15 * q^41 + 2 * q^42 + 2 * q^43 + q^45 - 5 * q^46 + 4 * q^47 + 2 * q^48 + 2 * q^49 - 13 * q^50 - 9 * q^51 - 2 * q^52 + 2 * q^53 + 2 * q^54 + 6 * q^56 - 3 * q^57 + 4 * q^58 + 18 * q^59 + q^60 - q^62 + 2 * q^63 + 14 * q^64 + 16 * q^65 + 16 * q^67 - 9 * q^68 - 5 * q^69 - q^70 + 6 * q^72 - 6 * q^73 + 13 * q^74 - 13 * q^75 - 3 * q^76 + 2 * q^78 - 28 * q^79 - q^80 + 2 * q^81 + 15 * q^82 - 12 * q^83 + 2 * q^84 - 18 * q^85 - 2 * q^86 + 4 * q^87 - 9 * q^89 - q^90 + 2 * q^91 - 5 * q^92 - q^93 - 4 * q^94 - 6 * q^95 + 10 * q^96 - 16 * q^97 - 2 * 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
−1.00000 −1.00000 −1.00000 −2.85410 1.00000 1.00000 3.00000 1.00000 2.85410
1.2 −1.00000 −1.00000 −1.00000 3.85410 1.00000 1.00000 3.00000 1.00000 −3.85410
 $$n$$: 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.n 2
3.b odd 2 1 7623.2.a.bu 2
11.b odd 2 1 2541.2.a.bd 2
11.c even 5 2 231.2.j.c 4
33.d even 2 1 7623.2.a.w 2
33.h odd 10 2 693.2.m.c 4

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - T - 11$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 2T - 4$$
$17$ $$T^{2} - 9T + 9$$
$19$ $$T^{2} - 3T + 1$$
$23$ $$T^{2} - 5T - 25$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} - T - 61$$
$37$ $$T^{2} + 13T + 11$$
$41$ $$T^{2} + 15T + 45$$
$43$ $$T^{2} - 2T - 4$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} - 2T - 124$$
$59$ $$T^{2} - 18T + 76$$
$61$ $$T^{2} - 80$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} - 80$$
$73$ $$T^{2} + 6T + 4$$
$79$ $$(T + 14)^{2}$$
$83$ $$T^{2} + 12T - 44$$
$89$ $$T^{2} + 9T - 11$$
$97$ $$T^{2} + 16T + 44$$