# Properties

 Label 2541.2.a.bk Level $2541$ Weight $2$ Character orbit 2541.a Self dual yes Analytic conductor $20.290$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.7488.1 Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 2x + 1$$ x^4 - 2*x^3 - 4*x^2 + 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{3} + 1) q^{5} - \beta_{3} q^{6} - q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + q^{9}+O(q^{10})$$ q + b3 * q^2 - q^3 + (-b2 + 1) * q^4 + (-b3 + 1) * q^5 - b3 * q^6 - q^7 + (b3 + b2 + b1) * q^8 + q^9 $$q + \beta_{3} q^{2} - q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{3} + 1) q^{5} - \beta_{3} q^{6} - q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + q^{9} + (\beta_{3} + \beta_{2} - 3) q^{10} + (\beta_{2} - 1) q^{12} + ( - 2 \beta_{2} - \beta_1 + 1) q^{13} - \beta_{3} q^{14} + (\beta_{3} - 1) q^{15} + ( - 2 \beta_{3} - 2 \beta_1) q^{16} + (\beta_{2} + \beta_1 - 1) q^{17} + \beta_{3} q^{18} + (\beta_{3} + \beta_{2} - 1) q^{19} + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{20} + q^{21} + (\beta_{2} + \beta_1 + 2) q^{23} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{24} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{25} + (5 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{26} - q^{27} + (\beta_{2} - 1) q^{28} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 4) q^{29} + ( - \beta_{3} - \beta_{2} + 3) q^{30} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{31} + ( - 2 \beta_{3} - 4) q^{32} + ( - 3 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{34} + (\beta_{3} - 1) q^{35} + ( - \beta_{2} + 1) q^{36} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{37} + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{38} + (2 \beta_{2} + \beta_1 - 1) q^{39} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 2) q^{40} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{41} + \beta_{3} q^{42} + (\beta_{3} - \beta_1 - 4) q^{43} + ( - \beta_{3} + 1) q^{45} + ( - \beta_{2} - 2 \beta_1 - 1) q^{46} + (2 \beta_{2} - 3 \beta_1 + 2) q^{47} + (2 \beta_{3} + 2 \beta_1) q^{48} + q^{49} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 6) q^{50} + ( - \beta_{2} - \beta_1 + 1) q^{51} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 10) q^{52} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{53} - \beta_{3} q^{54} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{56} + ( - \beta_{3} - \beta_{2} + 1) q^{57} + ( - 6 \beta_{3} + \beta_{2} - 5) q^{58} + (7 \beta_{3} + 6 \beta_1 + 1) q^{59} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{60} + (3 \beta_{3} + \beta_{2} + 1) q^{61} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{62} - q^{63} + (2 \beta_{2} + 4 \beta_1 - 6) q^{64} + ( - 5 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{65} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 5) q^{67} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 5) q^{68} + ( - \beta_{2} - \beta_1 - 2) q^{69} + ( - \beta_{3} - \beta_{2} + 3) q^{70} + ( - \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{71} + (\beta_{3} + \beta_{2} + \beta_1) q^{72} + ( - 2 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 1) q^{73} + ( - 5 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 4) q^{74} + (2 \beta_{3} + \beta_{2} + 1) q^{75} + (5 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 6) q^{76} + ( - 5 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 1) q^{78} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{79} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{80} + q^{81} + ( - 7 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{82} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{83} + ( - \beta_{2} + 1) q^{84} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{85} + ( - 4 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{86} + (2 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{87} + ( - 3 \beta_{3} + 4 \beta_{2} - 5) q^{89} + (\beta_{3} + \beta_{2} - 3) q^{90} + (2 \beta_{2} + \beta_1 - 1) q^{91} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{92} + (\beta_{3} + \beta_{2} + \beta_1) q^{93} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{94} + (4 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{95} + (2 \beta_{3} + 4) q^{96} + (2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 9) q^{97} + \beta_{3} q^{98}+O(q^{100})$$ q + b3 * q^2 - q^3 + (-b2 + 1) * q^4 + (-b3 + 1) * q^5 - b3 * q^6 - q^7 + (b3 + b2 + b1) * q^8 + q^9 + (b3 + b2 - 3) * q^10 + (b2 - 1) * q^12 + (-2*b2 - b1 + 1) * q^13 - b3 * q^14 + (b3 - 1) * q^15 + (-2*b3 - 2*b1) * q^16 + (b2 + b1 - 1) * q^17 + b3 * q^18 + (b3 + b2 - 1) * q^19 + (-3*b3 - 2*b2 - b1 + 1) * q^20 + q^21 + (b2 + b1 + 2) * q^23 + (-b3 - b2 - b1) * q^24 + (-2*b3 - b2 - 1) * q^25 + (5*b3 + 2*b2 + 3*b1 + 1) * q^26 - q^27 + (b2 - 1) * q^28 + (-2*b3 + b2 - b1 - 4) * q^29 + (-b3 - b2 + 3) * q^30 + (-b3 - b2 - b1) * q^31 + (-2*b3 - 4) * q^32 + (-3*b3 - b2 - 2*b1 - 1) * q^34 + (b3 - 1) * q^35 + (-b2 + 1) * q^36 + (2*b3 + b2 + 2*b1 - 3) * q^37 + (-3*b3 - 2*b2 - b1 + 3) * q^38 + (2*b2 + b1 - 1) * q^39 + (3*b3 + 3*b2 + 3*b1 - 2) * q^40 + (b3 + 2*b2 - b1 - 3) * q^41 + b3 * q^42 + (b3 - b1 - 4) * q^43 + (-b3 + 1) * q^45 + (-b2 - 2*b1 - 1) * q^46 + (2*b2 - 3*b1 + 2) * q^47 + (2*b3 + 2*b1) * q^48 + q^49 + (b3 + 3*b2 + b1 - 6) * q^50 + (-b2 - b1 + 1) * q^51 + (-3*b3 - 3*b2 - 3*b1 + 10) * q^52 + (-2*b3 + 2*b2 - 2*b1 - 4) * q^53 - b3 * q^54 + (-b3 - b2 - b1) * q^56 + (-b3 - b2 + 1) * q^57 + (-6*b3 + b2 - 5) * q^58 + (7*b3 + 6*b1 + 1) * q^59 + (3*b3 + 2*b2 + b1 - 1) * q^60 + (3*b3 + b2 + 1) * q^61 + (2*b3 + 2*b2 + 2*b1 - 2) * q^62 - q^63 + (2*b2 + 4*b1 - 6) * q^64 + (-5*b3 - 4*b2 - 4*b1) * q^65 + (-4*b3 + 2*b2 - 4*b1 - 5) * q^67 + (b3 + 2*b2 + b1 - 5) * q^68 + (-b2 - b1 - 2) * q^69 + (-b3 - b2 + 3) * q^70 + (-b3 + 2*b2 + 6*b1 - 6) * q^71 + (b3 + b2 + b1) * q^72 + (-2*b3 - 2*b2 + 5*b1 - 1) * q^73 + (-5*b3 - 3*b2 - 3*b1 + 4) * q^74 + (2*b3 + b2 + 1) * q^75 + (5*b3 + 3*b2 + 3*b1 - 6) * q^76 + (-5*b3 - 2*b2 - 3*b1 - 1) * q^78 + (-3*b2 - 2*b1 - 3) * q^79 + (-2*b3 - 2*b2 - 4*b1 + 4) * q^80 + q^81 + (-7*b3 - 3*b2 - b1 + 4) * q^82 + (-b3 - b2 + 2*b1 - 6) * q^83 + (-b2 + 1) * q^84 + (3*b3 + 2*b2 + 3*b1) * q^85 + (-4*b3 - b2 + b1 + 4) * q^86 + (2*b3 - b2 + b1 + 4) * q^87 + (-3*b3 + 4*b2 - 5) * q^89 + (b3 + b2 - 3) * q^90 + (2*b2 + b1 - 1) * q^91 + (b3 - b2 + b1 - 2) * q^92 + (b3 + b2 + b1) * q^93 + (-2*b3 - 2*b2 + b1 + 3) * q^94 + (4*b3 + 3*b2 + b1 - 4) * q^95 + (2*b3 + 4) * q^96 + (2*b3 - 2*b2 + 3*b1 + 9) * q^97 + b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9 $$4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9} - 14 q^{10} - 4 q^{12} + 2 q^{13} + 2 q^{14} - 6 q^{15} - 2 q^{17} - 2 q^{18} - 6 q^{19} + 8 q^{20} + 4 q^{21} + 10 q^{23} - 4 q^{27} - 4 q^{28} - 14 q^{29} + 14 q^{30} - 12 q^{32} - 2 q^{34} - 6 q^{35} + 4 q^{36} - 12 q^{37} + 16 q^{38} - 2 q^{39} - 8 q^{40} - 16 q^{41} - 2 q^{42} - 20 q^{43} + 6 q^{45} - 8 q^{46} + 2 q^{47} + 4 q^{49} - 24 q^{50} + 2 q^{51} + 40 q^{52} - 16 q^{53} + 2 q^{54} + 6 q^{57} - 8 q^{58} + 2 q^{59} - 8 q^{60} - 2 q^{61} - 8 q^{62} - 4 q^{63} - 16 q^{64} + 2 q^{65} - 20 q^{67} - 20 q^{68} - 10 q^{69} + 14 q^{70} - 10 q^{71} + 10 q^{73} + 20 q^{74} - 28 q^{76} - 16 q^{79} + 12 q^{80} + 4 q^{81} + 28 q^{82} - 18 q^{83} + 4 q^{84} + 26 q^{86} + 14 q^{87} - 14 q^{89} - 14 q^{90} - 2 q^{91} - 8 q^{92} + 18 q^{94} - 22 q^{95} + 12 q^{96} + 38 q^{97} - 2 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9 - 14 * q^10 - 4 * q^12 + 2 * q^13 + 2 * q^14 - 6 * q^15 - 2 * q^17 - 2 * q^18 - 6 * q^19 + 8 * q^20 + 4 * q^21 + 10 * q^23 - 4 * q^27 - 4 * q^28 - 14 * q^29 + 14 * q^30 - 12 * q^32 - 2 * q^34 - 6 * q^35 + 4 * q^36 - 12 * q^37 + 16 * q^38 - 2 * q^39 - 8 * q^40 - 16 * q^41 - 2 * q^42 - 20 * q^43 + 6 * q^45 - 8 * q^46 + 2 * q^47 + 4 * q^49 - 24 * q^50 + 2 * q^51 + 40 * q^52 - 16 * q^53 + 2 * q^54 + 6 * q^57 - 8 * q^58 + 2 * q^59 - 8 * q^60 - 2 * q^61 - 8 * q^62 - 4 * q^63 - 16 * q^64 + 2 * q^65 - 20 * q^67 - 20 * q^68 - 10 * q^69 + 14 * q^70 - 10 * q^71 + 10 * q^73 + 20 * q^74 - 28 * q^76 - 16 * q^79 + 12 * q^80 + 4 * q^81 + 28 * q^82 - 18 * q^83 + 4 * q^84 + 26 * q^86 + 14 * q^87 - 14 * q^89 - 14 * q^90 - 2 * q^91 - 8 * q^92 + 18 * q^94 - 22 * q^95 + 12 * q^96 + 38 * q^97 - 2 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 2$$ v^2 - 2*v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 4\nu + 1$$ v^3 - 2*v^2 - 4*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 2$$ b2 + 2*b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 8\beta _1 + 3$$ b3 + 2*b2 + 8*b1 + 3

## 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
 0.698857 3.05896 −1.43091 −0.326909
−2.43091 −1.00000 3.90931 3.43091 2.43091 −1.00000 −4.64136 1.00000 −8.34022
1.2 −1.32691 −1.00000 −0.239314 2.32691 1.32691 −1.00000 2.97136 1.00000 −3.08759
1.3 −0.301143 −1.00000 −1.90931 1.30114 0.301143 −1.00000 1.17726 1.00000 −0.391830
1.4 2.05896 −1.00000 2.23931 −1.05896 −2.05896 −1.00000 0.492737 1.00000 −2.18035
 $$n$$: 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.bk 4
3.b odd 2 1 7623.2.a.cm 4
11.b odd 2 1 2541.2.a.bo yes 4
33.d even 2 1 7623.2.a.cf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bk 4 1.a even 1 1 trivial
2541.2.a.bo yes 4 11.b odd 2 1
7623.2.a.cf 4 33.d even 2 1
7623.2.a.cm 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

 $$T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 8T_{2} - 2$$ T2^4 + 2*T2^3 - 4*T2^2 - 8*T2 - 2 $$T_{5}^{4} - 6T_{5}^{3} + 8T_{5}^{2} + 6T_{5} - 11$$ T5^4 - 6*T5^3 + 8*T5^2 + 6*T5 - 11 $$T_{13}^{4} - 2T_{13}^{3} - 40T_{13}^{2} + 32T_{13} + 358$$ T13^4 - 2*T13^3 - 40*T13^2 + 32*T13 + 358

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} - 4 T^{2} - 8 T - 2$$
$3$ $$(T + 1)^{4}$$
$5$ $$T^{4} - 6 T^{3} + 8 T^{2} + 6 T - 11$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 2 T^{3} - 40 T^{2} + 32 T + 358$$
$17$ $$T^{4} + 2 T^{3} - 12 T^{2} - 22 T + 13$$
$19$ $$T^{4} + 6 T^{3} - 4 T^{2} - 12 T - 2$$
$23$ $$T^{4} - 10 T^{3} + 24 T^{2} - 4 T - 2$$
$29$ $$T^{4} + 14 T^{3} + 48 T^{2} + 20 T - 74$$
$31$ $$T^{4} - 16 T^{2} - 24 T - 8$$
$37$ $$T^{4} + 12 T^{3} + 20 T^{2} - 48 T - 44$$
$41$ $$T^{4} + 16 T^{3} + 32 T^{2} + \cdots - 716$$
$43$ $$T^{4} + 20 T^{3} + 134 T^{2} + \cdots + 277$$
$47$ $$T^{4} - 2 T^{3} - 100 T^{2} + \cdots + 169$$
$53$ $$T^{4} + 16 T^{3} + 32 T^{2} + \cdots - 1664$$
$59$ $$T^{4} - 2 T^{3} - 256 T^{2} + \cdots + 15517$$
$61$ $$T^{4} + 2 T^{3} - 64 T^{2} + 52 T + 286$$
$67$ $$T^{4} + 20 T^{3} + 14 T^{2} + \cdots - 4103$$
$71$ $$T^{4} + 10 T^{3} - 208 T^{2} + \cdots - 8882$$
$73$ $$T^{4} - 10 T^{3} - 240 T^{2} + \cdots + 10894$$
$79$ $$T^{4} + 16 T^{3} - 4 T^{2} - 520 T + 676$$
$83$ $$T^{4} + 18 T^{3} + 68 T^{2} - 78 T + 13$$
$89$ $$T^{4} + 14 T^{3} - 112 T^{2} + \cdots + 4477$$
$97$ $$T^{4} - 38 T^{3} + 456 T^{2} + \cdots - 3938$$