# Properties

 Label 2541.2.a.bj Level $2541$ Weight $2$ Character orbit 2541.a Self dual yes Analytic conductor $20.290$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{2} + 1) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + (-b2 + b1) * q^5 + b1 * q^6 + q^7 + (b2 + 1) * q^8 + q^9 $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{2} + 1) q^{8} + q^{9} + ( - \beta_1 + 2) q^{10} + (\beta_{2} + 1) q^{12} + (\beta_1 + 2) q^{13} + \beta_1 q^{14} + ( - \beta_{2} + \beta_1) q^{15} + ( - \beta_{2} + 2 \beta_1 - 1) q^{16} + ( - \beta_{2} + \beta_1) q^{17} + \beta_1 q^{18} + ( - \beta_{2} - \beta_1 + 3) q^{19} + (\beta_{2} - 3) q^{20} + q^{21} + (3 \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{2} + 1) q^{24} - 2 \beta_{2} q^{25} + (\beta_{2} + 2 \beta_1 + 3) q^{26} + q^{27} + (\beta_{2} + 1) q^{28} + (\beta_1 + 2) q^{29} + ( - \beta_1 + 2) q^{30} + (2 \beta_{2} - 2 \beta_1) q^{31} + ( - \beta_{2} - 2 \beta_1 + 3) q^{32} + ( - \beta_1 + 2) q^{34} + ( - \beta_{2} + \beta_1) q^{35} + (\beta_{2} + 1) q^{36} + ( - \beta_{2} - 1) q^{37} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{38} + (\beta_1 + 2) q^{39} + (\beta_{2} - 3) q^{40} + (\beta_{2} - 2 \beta_1 + 5) q^{41} + \beta_1 q^{42} + ( - \beta_{2} - 4 \beta_1 + 4) q^{43} + ( - \beta_{2} + \beta_1) q^{45} + (4 \beta_{2} + 2 \beta_1 + 6) q^{46} + ( - 2 \beta_{2} + 3 \beta_1 - 3) q^{47} + ( - \beta_{2} + 2 \beta_1 - 1) q^{48} + q^{49} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{50} + ( - \beta_{2} + \beta_1) q^{51} + (3 \beta_{2} + 2 \beta_1 + 3) q^{52} + (\beta_{2} - 7) q^{53} + \beta_1 q^{54} + (\beta_{2} + 1) q^{56} + ( - \beta_{2} - \beta_1 + 3) q^{57} + (\beta_{2} + 2 \beta_1 + 3) q^{58} + (3 \beta_1 + 1) q^{59} + (\beta_{2} - 3) q^{60} + (3 \beta_{2} - 3 \beta_1 + 5) q^{61} + (2 \beta_1 - 4) q^{62} + q^{63} + ( - \beta_{2} - 2 \beta_1 - 5) q^{64} + ( - 2 \beta_{2} + \beta_1 + 2) q^{65} + ( - 5 \beta_{2} + 2 \beta_1) q^{67} + (\beta_{2} - 3) q^{68} + (3 \beta_{2} + \beta_1 - 1) q^{69} + ( - \beta_1 + 2) q^{70} + (\beta_{2} - 3 \beta_1 + 1) q^{71} + (\beta_{2} + 1) q^{72} + (3 \beta_{2} + 5 \beta_1 - 3) q^{73} + ( - \beta_{2} - 2 \beta_1 - 1) q^{74} - 2 \beta_{2} q^{75} + (2 \beta_{2} - 4 \beta_1 - 2) q^{76} + (\beta_{2} + 2 \beta_1 + 3) q^{78} + ( - 2 \beta_{2} + 6 \beta_1 + 4) q^{79} + ( - \beta_{2} - 2 \beta_1 + 7) q^{80} + q^{81} + ( - \beta_{2} + 6 \beta_1 - 5) q^{82} + ( - 6 \beta_{2} + 3 \beta_1 - 5) q^{83} + (\beta_{2} + 1) q^{84} + ( - 2 \beta_{2} + 5) q^{85} + ( - 5 \beta_{2} + 3 \beta_1 - 13) q^{86} + (\beta_1 + 2) q^{87} + ( - 3 \beta_{2} - \beta_1) q^{89} + ( - \beta_1 + 2) q^{90} + (\beta_1 + 2) q^{91} + (8 \beta_1 + 12) q^{92} + (2 \beta_{2} - 2 \beta_1) q^{93} + (\beta_{2} - 5 \beta_1 + 7) q^{94} + ( - 5 \beta_{2} + 5 \beta_1 + 1) q^{95} + ( - \beta_{2} - 2 \beta_1 + 3) q^{96} + (2 \beta_{2} - 7 \beta_1 - 6) q^{97} + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + (-b2 + b1) * q^5 + b1 * q^6 + q^7 + (b2 + 1) * q^8 + q^9 + (-b1 + 2) * q^10 + (b2 + 1) * q^12 + (b1 + 2) * q^13 + b1 * q^14 + (-b2 + b1) * q^15 + (-b2 + 2*b1 - 1) * q^16 + (-b2 + b1) * q^17 + b1 * q^18 + (-b2 - b1 + 3) * q^19 + (b2 - 3) * q^20 + q^21 + (3*b2 + b1 - 1) * q^23 + (b2 + 1) * q^24 - 2*b2 * q^25 + (b2 + 2*b1 + 3) * q^26 + q^27 + (b2 + 1) * q^28 + (b1 + 2) * q^29 + (-b1 + 2) * q^30 + (2*b2 - 2*b1) * q^31 + (-b2 - 2*b1 + 3) * q^32 + (-b1 + 2) * q^34 + (-b2 + b1) * q^35 + (b2 + 1) * q^36 + (-b2 - 1) * q^37 + (-2*b2 + 2*b1 - 4) * q^38 + (b1 + 2) * q^39 + (b2 - 3) * q^40 + (b2 - 2*b1 + 5) * q^41 + b1 * q^42 + (-b2 - 4*b1 + 4) * q^43 + (-b2 + b1) * q^45 + (4*b2 + 2*b1 + 6) * q^46 + (-2*b2 + 3*b1 - 3) * q^47 + (-b2 + 2*b1 - 1) * q^48 + q^49 + (-2*b2 - 2*b1 - 2) * q^50 + (-b2 + b1) * q^51 + (3*b2 + 2*b1 + 3) * q^52 + (b2 - 7) * q^53 + b1 * q^54 + (b2 + 1) * q^56 + (-b2 - b1 + 3) * q^57 + (b2 + 2*b1 + 3) * q^58 + (3*b1 + 1) * q^59 + (b2 - 3) * q^60 + (3*b2 - 3*b1 + 5) * q^61 + (2*b1 - 4) * q^62 + q^63 + (-b2 - 2*b1 - 5) * q^64 + (-2*b2 + b1 + 2) * q^65 + (-5*b2 + 2*b1) * q^67 + (b2 - 3) * q^68 + (3*b2 + b1 - 1) * q^69 + (-b1 + 2) * q^70 + (b2 - 3*b1 + 1) * q^71 + (b2 + 1) * q^72 + (3*b2 + 5*b1 - 3) * q^73 + (-b2 - 2*b1 - 1) * q^74 - 2*b2 * q^75 + (2*b2 - 4*b1 - 2) * q^76 + (b2 + 2*b1 + 3) * q^78 + (-2*b2 + 6*b1 + 4) * q^79 + (-b2 - 2*b1 + 7) * q^80 + q^81 + (-b2 + 6*b1 - 5) * q^82 + (-6*b2 + 3*b1 - 5) * q^83 + (b2 + 1) * q^84 + (-2*b2 + 5) * q^85 + (-5*b2 + 3*b1 - 13) * q^86 + (b1 + 2) * q^87 + (-3*b2 - b1) * q^89 + (-b1 + 2) * q^90 + (b1 + 2) * q^91 + (8*b1 + 12) * q^92 + (2*b2 - 2*b1) * q^93 + (b2 - 5*b1 + 7) * q^94 + (-5*b2 + 5*b1 + 1) * q^95 + (-b2 - 2*b1 + 3) * q^96 + (2*b2 - 7*b1 - 6) * q^97 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 3 q^{3} + 3 q^{4} + q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + q^2 + 3 * q^3 + 3 * q^4 + q^5 + q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 $$3 q + q^{2} + 3 q^{3} + 3 q^{4} + q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} + 5 q^{10} + 3 q^{12} + 7 q^{13} + q^{14} + q^{15} - q^{16} + q^{17} + q^{18} + 8 q^{19} - 9 q^{20} + 3 q^{21} - 2 q^{23} + 3 q^{24} + 11 q^{26} + 3 q^{27} + 3 q^{28} + 7 q^{29} + 5 q^{30} - 2 q^{31} + 7 q^{32} + 5 q^{34} + q^{35} + 3 q^{36} - 3 q^{37} - 10 q^{38} + 7 q^{39} - 9 q^{40} + 13 q^{41} + q^{42} + 8 q^{43} + q^{45} + 20 q^{46} - 6 q^{47} - q^{48} + 3 q^{49} - 8 q^{50} + q^{51} + 11 q^{52} - 21 q^{53} + q^{54} + 3 q^{56} + 8 q^{57} + 11 q^{58} + 6 q^{59} - 9 q^{60} + 12 q^{61} - 10 q^{62} + 3 q^{63} - 17 q^{64} + 7 q^{65} + 2 q^{67} - 9 q^{68} - 2 q^{69} + 5 q^{70} + 3 q^{72} - 4 q^{73} - 5 q^{74} - 10 q^{76} + 11 q^{78} + 18 q^{79} + 19 q^{80} + 3 q^{81} - 9 q^{82} - 12 q^{83} + 3 q^{84} + 15 q^{85} - 36 q^{86} + 7 q^{87} - q^{89} + 5 q^{90} + 7 q^{91} + 44 q^{92} - 2 q^{93} + 16 q^{94} + 8 q^{95} + 7 q^{96} - 25 q^{97} + q^{98}+O(q^{100})$$ 3 * q + q^2 + 3 * q^3 + 3 * q^4 + q^5 + q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 + 5 * q^10 + 3 * q^12 + 7 * q^13 + q^14 + q^15 - q^16 + q^17 + q^18 + 8 * q^19 - 9 * q^20 + 3 * q^21 - 2 * q^23 + 3 * q^24 + 11 * q^26 + 3 * q^27 + 3 * q^28 + 7 * q^29 + 5 * q^30 - 2 * q^31 + 7 * q^32 + 5 * q^34 + q^35 + 3 * q^36 - 3 * q^37 - 10 * q^38 + 7 * q^39 - 9 * q^40 + 13 * q^41 + q^42 + 8 * q^43 + q^45 + 20 * q^46 - 6 * q^47 - q^48 + 3 * q^49 - 8 * q^50 + q^51 + 11 * q^52 - 21 * q^53 + q^54 + 3 * q^56 + 8 * q^57 + 11 * q^58 + 6 * q^59 - 9 * q^60 + 12 * q^61 - 10 * q^62 + 3 * q^63 - 17 * q^64 + 7 * q^65 + 2 * q^67 - 9 * q^68 - 2 * q^69 + 5 * q^70 + 3 * q^72 - 4 * q^73 - 5 * q^74 - 10 * q^76 + 11 * q^78 + 18 * q^79 + 19 * q^80 + 3 * q^81 - 9 * q^82 - 12 * q^83 + 3 * q^84 + 15 * q^85 - 36 * q^86 + 7 * q^87 - q^89 + 5 * q^90 + 7 * q^91 + 44 * q^92 - 2 * q^93 + 16 * q^94 + 8 * q^95 + 7 * q^96 - 25 * q^97 + q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 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
 −1.81361 0.470683 2.34292
−1.81361 1.00000 1.28917 −2.10278 −1.81361 1.00000 1.28917 1.00000 3.81361
1.2 0.470683 1.00000 −1.77846 3.24914 0.470683 1.00000 −1.77846 1.00000 1.52932
1.3 2.34292 1.00000 3.48929 −0.146365 2.34292 1.00000 3.48929 1.00000 −0.342923
 $$n$$: 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.bj yes 3
3.b odd 2 1 7623.2.a.ca 3
11.b odd 2 1 2541.2.a.bh 3
33.d even 2 1 7623.2.a.cc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bh 3 11.b odd 2 1
2541.2.a.bj yes 3 1.a even 1 1 trivial
7623.2.a.ca 3 3.b odd 2 1
7623.2.a.cc 3 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

 $$T_{2}^{3} - T_{2}^{2} - 4T_{2} + 2$$ T2^3 - T2^2 - 4*T2 + 2 $$T_{5}^{3} - T_{5}^{2} - 7T_{5} - 1$$ T5^3 - T5^2 - 7*T5 - 1 $$T_{13}^{3} - 7T_{13}^{2} + 12T_{13} - 2$$ T13^3 - 7*T13^2 + 12*T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 4T + 2$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} - T^{2} - 7T - 1$$
$7$ $$(T - 1)^{3}$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 7 T^{2} + 12 T - 2$$
$17$ $$T^{3} - T^{2} - 7T - 1$$
$19$ $$T^{3} - 8 T^{2} + 6 T + 44$$
$23$ $$T^{3} + 2 T^{2} - 78 T - 152$$
$29$ $$T^{3} - 7 T^{2} + 12 T - 2$$
$31$ $$T^{3} + 2 T^{2} - 28 T + 8$$
$37$ $$T^{3} + 3 T^{2} - 4 T - 8$$
$41$ $$T^{3} - 13 T^{2} + 40 T - 32$$
$43$ $$T^{3} - 8 T^{2} - 71 T + 422$$
$47$ $$T^{3} + 6 T^{2} - 31 T - 34$$
$53$ $$T^{3} + 21 T^{2} + 140 T + 296$$
$59$ $$T^{3} - 6 T^{2} - 27 T + 86$$
$61$ $$T^{3} - 12 T^{2} - 18 T + 292$$
$67$ $$T^{3} - 2 T^{2} - 151 T - 584$$
$71$ $$T^{3} - 34T - 76$$
$73$ $$T^{3} + 4 T^{2} - 226 T - 1628$$
$79$ $$T^{3} - 18 T^{2} - 28 T + 1208$$
$83$ $$T^{3} + 12 T^{2} - 171 T - 2056$$
$89$ $$T^{3} + T^{2} - 79 T + 73$$
$97$ $$T^{3} + 25 T^{2} + 24 T - 1882$$