# Properties

 Label 2541.2.a.bn 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: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3x^{2} + x + 1$$ x^4 - x^3 - 3*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 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1 + 1) q^{2} - q^{3} + (\beta_{3} + \beta_1) q^{4} - 2 \beta_{2} q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} - q^{7} + ( - \beta_1 - 2) q^{8} + q^{9}+O(q^{10})$$ q + (-b2 - b1 + 1) * q^2 - q^3 + (b3 + b1) * q^4 - 2*b2 * q^5 + (b2 + b1 - 1) * q^6 - q^7 + (-b1 - 2) * q^8 + q^9 $$q + ( - \beta_{2} - \beta_1 + 1) q^{2} - q^{3} + (\beta_{3} + \beta_1) q^{4} - 2 \beta_{2} q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} - q^{7} + ( - \beta_1 - 2) q^{8} + q^{9} + (2 \beta_1 + 2) q^{10} + ( - \beta_{3} - \beta_1) q^{12} + (2 \beta_{2} + 2 \beta_1) q^{13} + (\beta_{2} + \beta_1 - 1) q^{14} + 2 \beta_{2} q^{15} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{16} + (2 \beta_{3} - 4 \beta_1 + 2) q^{17} + ( - \beta_{2} - \beta_1 + 1) q^{18} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{19} + ( - 2 \beta_{3} - 4 \beta_1 + 2) q^{20} + q^{21} + (3 \beta_{3} - 4 \beta_1 - 3) q^{23} + (\beta_1 + 2) q^{24} + ( - 4 \beta_{3} + 4 \beta_{2} + 3) q^{25} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{26} - q^{27} + ( - \beta_{3} - \beta_1) q^{28} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_1 - 2) q^{30} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{31} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{32} + (6 \beta_{3} - 2 \beta_1 + 2) q^{34} + 2 \beta_{2} q^{35} + (\beta_{3} + \beta_1) q^{36} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 3) q^{37} + ( - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{38} + ( - 2 \beta_{2} - 2 \beta_1) q^{39} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{40} + (6 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - \beta_{2} - \beta_1 + 1) q^{42} + ( - 5 \beta_{3} + 6 \beta_1) q^{43} - 2 \beta_{2} q^{45} + (7 \beta_{3} + 4 \beta_{2} + \beta_1 - 3) q^{46} + ( - 6 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{47} + (\beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{48} + q^{49} + ( - 4 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{50} + ( - 2 \beta_{3} + 4 \beta_1 - 2) q^{51} + (2 \beta_{3} + 4 \beta_{2} + 8 \beta_1) q^{52} + (5 \beta_{3} - 2 \beta_1 - 2) q^{53} + (\beta_{2} + \beta_1 - 1) q^{54} + (\beta_1 + 2) q^{56} + (2 \beta_{3} - 2 \beta_1 + 2) q^{57} + ( - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 6) q^{58} + (2 \beta_{3} + 2 \beta_{2} - 6) q^{59} + (2 \beta_{3} + 4 \beta_1 - 2) q^{60} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 2) q^{61} + (8 \beta_{2} + 8 \beta_1 - 6) q^{62} - q^{63} + ( - 2 \beta_{3} - 5 \beta_{2} - \beta_1 + 1) q^{64} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{65} + (3 \beta_{3} - 2 \beta_1 - 8) q^{67} + (4 \beta_{3} - 6 \beta_{2} - 4 \beta_1 - 2) q^{68} + ( - 3 \beta_{3} + 4 \beta_1 + 3) q^{69} + ( - 2 \beta_1 - 2) q^{70} + ( - 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 6) q^{71} + ( - \beta_1 - 2) q^{72} + (2 \beta_{2} - 2) q^{73} + ( - 5 \beta_{3} - 3 \beta_1 - 7) q^{74} + (4 \beta_{3} - 4 \beta_{2} - 3) q^{75} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{76} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{78} + ( - \beta_{3} - 2 \beta_1 - 4) q^{79} + (4 \beta_{3} - 2 \beta_{2} - 10) q^{80} + q^{81} + (8 \beta_{3} - 2 \beta_{2} - 12 \beta_1 - 6) q^{82} + ( - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 2) q^{83} + (\beta_{3} + \beta_1) q^{84} + (8 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 8) q^{85} + ( - 11 \beta_{3} - \beta_{2} + 4 \beta_1) q^{86} + (\beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{87} + (4 \beta_{3} - 6 \beta_{2} - 4 \beta_1 - 2) q^{89} + (2 \beta_1 + 2) q^{90} + ( - 2 \beta_{2} - 2 \beta_1) q^{91} + ( - 5 \beta_{2} - 8 \beta_1 - 1) q^{92} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{93} + ( - 10 \beta_{3} + 4 \beta_1) q^{94} + ( - 4 \beta_{3} + 4 \beta_{2} + 4) q^{95} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{96} + ( - 6 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 8) q^{97} + ( - \beta_{2} - \beta_1 + 1) q^{98}+O(q^{100})$$ q + (-b2 - b1 + 1) * q^2 - q^3 + (b3 + b1) * q^4 - 2*b2 * q^5 + (b2 + b1 - 1) * q^6 - q^7 + (-b1 - 2) * q^8 + q^9 + (2*b1 + 2) * q^10 + (-b3 - b1) * q^12 + (2*b2 + 2*b1) * q^13 + (b2 + b1 - 1) * q^14 + 2*b2 * q^15 + (-b3 + 3*b2 + b1 - 2) * q^16 + (2*b3 - 4*b1 + 2) * q^17 + (-b2 - b1 + 1) * q^18 + (-2*b3 + 2*b1 - 2) * q^19 + (-2*b3 - 4*b1 + 2) * q^20 + q^21 + (3*b3 - 4*b1 - 3) * q^23 + (b1 + 2) * q^24 + (-4*b3 + 4*b2 + 3) * q^25 + (-2*b3 - 2*b2 - 4*b1 - 2) * q^26 - q^27 + (-b3 - b1) * q^28 + (-b3 - 4*b2 + 2*b1 + 2) * q^29 + (-2*b1 - 2) * q^30 + (-2*b3 + 2*b2 - 2*b1 - 4) * q^31 + (-2*b3 + 2*b2 + 2*b1 - 1) * q^32 + (6*b3 - 2*b1 + 2) * q^34 + 2*b2 * q^35 + (b3 + b1) * q^36 + (-b3 + 4*b2 + 4*b1 - 3) * q^37 + (-4*b3 + 2*b2 + 4*b1 - 2) * q^38 + (-2*b2 - 2*b1) * q^39 + (2*b3 + 4*b2 + 2*b1 - 2) * q^40 + (6*b3 + 4*b2 - 2*b1 - 2) * q^41 + (-b2 - b1 + 1) * q^42 + (-5*b3 + 6*b1) * q^43 - 2*b2 * q^45 + (7*b3 + 4*b2 + b1 - 3) * q^46 + (-6*b3 + 2*b2 + 4*b1 + 2) * q^47 + (b3 - 3*b2 - b1 + 2) * q^48 + q^49 + (-4*b3 + b2 + b1 - 1) * q^50 + (-2*b3 + 4*b1 - 2) * q^51 + (2*b3 + 4*b2 + 8*b1) * q^52 + (5*b3 - 2*b1 - 2) * q^53 + (b2 + b1 - 1) * q^54 + (b1 + 2) * q^56 + (2*b3 - 2*b1 + 2) * q^57 + (-3*b3 - 3*b2 + 2*b1 + 6) * q^58 + (2*b3 + 2*b2 - 6) * q^59 + (2*b3 + 4*b1 - 2) * q^60 + (-2*b3 + 4*b2 + 2*b1 + 2) * q^61 + (8*b2 + 8*b1 - 6) * q^62 - q^63 + (-2*b3 - 5*b2 - b1 + 1) * q^64 + (-4*b2 - 4*b1 - 4) * q^65 + (3*b3 - 2*b1 - 8) * q^67 + (4*b3 - 6*b2 - 4*b1 - 2) * q^68 + (-3*b3 + 4*b1 + 3) * q^69 + (-2*b1 - 2) * q^70 + (-3*b3 + 4*b2 + 4*b1 - 6) * q^71 + (-b1 - 2) * q^72 + (2*b2 - 2) * q^73 + (-5*b3 - 3*b1 - 7) * q^74 + (4*b3 - 4*b2 - 3) * q^75 + (-4*b3 + 2*b2) * q^76 + (2*b3 + 2*b2 + 4*b1 + 2) * q^78 + (-b3 - 2*b1 - 4) * q^79 + (4*b3 - 2*b2 - 10) * q^80 + q^81 + (8*b3 - 2*b2 - 12*b1 - 6) * q^82 + (-2*b3 - 2*b2 + 6*b1 + 2) * q^83 + (b3 + b1) * q^84 + (8*b3 - 4*b2 + 4*b1 - 8) * q^85 + (-11*b3 - b2 + 4*b1) * q^86 + (b3 + 4*b2 - 2*b1 - 2) * q^87 + (4*b3 - 6*b2 - 4*b1 - 2) * q^89 + (2*b1 + 2) * q^90 + (-2*b2 - 2*b1) * q^91 + (-5*b2 - 8*b1 - 1) * q^92 + (2*b3 - 2*b2 + 2*b1 + 4) * q^93 + (-10*b3 + 4*b1) * q^94 + (-4*b3 + 4*b2 + 4) * q^95 + (2*b3 - 2*b2 - 2*b1 + 1) * q^96 + (-6*b3 + 4*b2 + 4*b1 + 8) * q^97 + (-b2 - b1 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + q^2 - 4 * q^3 + 3 * q^4 - 4 * q^5 - q^6 - 4 * q^7 - 9 * q^8 + 4 * q^9 $$4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9} + 10 q^{10} - 3 q^{12} + 6 q^{13} - q^{14} + 4 q^{15} - 3 q^{16} + 8 q^{17} + q^{18} - 10 q^{19} + 4 q^{21} - 10 q^{23} + 9 q^{24} + 12 q^{25} - 20 q^{26} - 4 q^{27} - 3 q^{28} - 10 q^{30} - 18 q^{31} - 2 q^{32} + 18 q^{34} + 4 q^{35} + 3 q^{36} - 2 q^{37} - 8 q^{38} - 6 q^{39} + 6 q^{40} + 10 q^{41} + q^{42} - 4 q^{43} - 4 q^{45} + 11 q^{46} + 4 q^{47} + 3 q^{48} + 4 q^{49} - 9 q^{50} - 8 q^{51} + 20 q^{52} - q^{54} + 9 q^{56} + 10 q^{57} + 14 q^{58} - 16 q^{59} + 14 q^{61} - 4 q^{63} - 11 q^{64} - 28 q^{65} - 28 q^{67} - 16 q^{68} + 10 q^{69} - 10 q^{70} - 18 q^{71} - 9 q^{72} - 4 q^{73} - 41 q^{74} - 12 q^{75} - 4 q^{76} + 20 q^{78} - 20 q^{79} - 36 q^{80} + 4 q^{81} - 24 q^{82} + 6 q^{83} + 3 q^{84} - 20 q^{85} - 20 q^{86} - 16 q^{89} + 10 q^{90} - 6 q^{91} - 22 q^{92} + 18 q^{93} - 16 q^{94} + 16 q^{95} + 2 q^{96} + 32 q^{97} + q^{98}+O(q^{100})$$ 4 * q + q^2 - 4 * q^3 + 3 * q^4 - 4 * q^5 - q^6 - 4 * q^7 - 9 * q^8 + 4 * q^9 + 10 * q^10 - 3 * q^12 + 6 * q^13 - q^14 + 4 * q^15 - 3 * q^16 + 8 * q^17 + q^18 - 10 * q^19 + 4 * q^21 - 10 * q^23 + 9 * q^24 + 12 * q^25 - 20 * q^26 - 4 * q^27 - 3 * q^28 - 10 * q^30 - 18 * q^31 - 2 * q^32 + 18 * q^34 + 4 * q^35 + 3 * q^36 - 2 * q^37 - 8 * q^38 - 6 * q^39 + 6 * q^40 + 10 * q^41 + q^42 - 4 * q^43 - 4 * q^45 + 11 * q^46 + 4 * q^47 + 3 * q^48 + 4 * q^49 - 9 * q^50 - 8 * q^51 + 20 * q^52 - q^54 + 9 * q^56 + 10 * q^57 + 14 * q^58 - 16 * q^59 + 14 * q^61 - 4 * q^63 - 11 * q^64 - 28 * q^65 - 28 * q^67 - 16 * q^68 + 10 * q^69 - 10 * q^70 - 18 * q^71 - 9 * q^72 - 4 * q^73 - 41 * q^74 - 12 * q^75 - 4 * q^76 + 20 * q^78 - 20 * q^79 - 36 * q^80 + 4 * q^81 - 24 * q^82 + 6 * q^83 + 3 * q^84 - 20 * q^85 - 20 * q^86 - 16 * q^89 + 10 * q^90 - 6 * q^91 - 22 * q^92 + 18 * q^93 - 16 * q^94 + 16 * q^95 + 2 * q^96 + 32 * q^97 + q^98

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

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

## 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
 2.09529 −1.35567 0.737640 −0.477260
−2.39026 −1.00000 3.71333 −2.58993 2.39026 −1.00000 −4.09529 1.00000 6.19059
1.2 0.162147 −1.00000 −1.97371 −4.38705 −0.162147 −1.00000 −0.644326 1.00000 −0.711349
1.3 1.45589 −1.00000 0.119606 2.38705 −1.45589 −1.00000 −2.73764 1.00000 3.47528
1.4 1.77222 −1.00000 1.14077 0.589926 −1.77222 −1.00000 −1.52274 1.00000 1.04548
 $$n$$: 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.bn 4
3.b odd 2 1 7623.2.a.ci 4
11.b odd 2 1 2541.2.a.bm 4
11.c even 5 2 231.2.j.f 8
33.d even 2 1 7623.2.a.cl 4
33.h odd 10 2 693.2.m.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.f 8 11.c even 5 2
693.2.m.f 8 33.h odd 10 2
2541.2.a.bm 4 11.b odd 2 1
2541.2.a.bn 4 1.a even 1 1 trivial
7623.2.a.ci 4 3.b odd 2 1
7623.2.a.cl 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

 $$T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + 7T_{2} - 1$$ T2^4 - T2^3 - 5*T2^2 + 7*T2 - 1 $$T_{5}^{4} + 4T_{5}^{3} - 8T_{5}^{2} - 24T_{5} + 16$$ T5^4 + 4*T5^3 - 8*T5^2 - 24*T5 + 16 $$T_{13}^{4} - 6T_{13}^{3} - 8T_{13}^{2} + 16T_{13} + 16$$ T13^4 - 6*T13^3 - 8*T13^2 + 16*T13 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} - 5 T^{2} + 7 T - 1$$
$3$ $$(T + 1)^{4}$$
$5$ $$T^{4} + 4 T^{3} - 8 T^{2} - 24 T + 16$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 6 T^{3} - 8 T^{2} + 16 T + 16$$
$17$ $$T^{4} - 8 T^{3} - 20 T^{2} + 144 T + 304$$
$19$ $$T^{4} + 10 T^{3} + 24 T^{2} - 16$$
$23$ $$T^{4} + 10 T^{3} - 9 T^{2} - 190 T + 109$$
$29$ $$T^{4} - 79T^{2} + 29$$
$31$ $$T^{4} + 18 T^{3} + 68 T^{2} + \cdots - 1744$$
$37$ $$T^{4} + 2 T^{3} - 77 T^{2} - 218 T + 281$$
$41$ $$T^{4} - 10 T^{3} - 104 T^{2} + \cdots - 4496$$
$43$ $$T^{4} + 4 T^{3} - 103 T^{2} + \cdots + 1861$$
$47$ $$T^{4} - 4 T^{3} - 80 T^{2} + 568 T - 976$$
$53$ $$T^{4} - 51 T^{2} - 20 T + 209$$
$59$ $$T^{4} + 16 T^{3} + 72 T^{2} + 104 T + 16$$
$61$ $$T^{4} - 14 T^{3} + 16 T^{2} + 256 T + 16$$
$67$ $$T^{4} + 28 T^{3} + 273 T^{2} + \cdots + 1301$$
$71$ $$T^{4} + 18 T^{3} + 43 T^{2} + \cdots - 179$$
$73$ $$T^{4} + 4 T^{3} - 8 T^{2} - 24 T + 16$$
$79$ $$T^{4} + 20 T^{3} + 129 T^{2} + \cdots + 149$$
$83$ $$T^{4} - 6 T^{3} - 120 T^{2} + \cdots + 2864$$
$89$ $$T^{4} + 16 T^{3} - 48 T^{2} + \cdots - 304$$
$97$ $$T^{4} - 32 T^{3} + 268 T^{2} + \cdots - 8464$$