# Properties

 Label 2541.2.a.bc 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + (\beta + 2) q^{4} + q^{5} + \beta q^{6} + q^{7} + (\beta + 4) q^{8} + q^{9}+O(q^{10})$$ q + b * q^2 + q^3 + (b + 2) * q^4 + q^5 + b * q^6 + q^7 + (b + 4) * q^8 + q^9 $$q + \beta q^{2} + q^{3} + (\beta + 2) q^{4} + q^{5} + \beta q^{6} + q^{7} + (\beta + 4) q^{8} + q^{9} + \beta q^{10} + (\beta + 2) q^{12} + ( - \beta + 4) q^{13} + \beta q^{14} + q^{15} + 3 \beta q^{16} + ( - 2 \beta + 1) q^{17} + \beta q^{18} + 6 q^{19} + (\beta + 2) q^{20} + q^{21} - 4 q^{23} + (\beta + 4) q^{24} - 4 q^{25} + (3 \beta - 4) q^{26} + q^{27} + (\beta + 2) q^{28} + ( - 3 \beta + 2) q^{29} + \beta q^{30} + ( - 4 \beta + 2) q^{31} + (\beta + 4) q^{32} + ( - \beta - 8) q^{34} + q^{35} + (\beta + 2) q^{36} + (3 \beta + 2) q^{37} + 6 \beta q^{38} + ( - \beta + 4) q^{39} + (\beta + 4) q^{40} + (\beta - 6) q^{41} + \beta q^{42} + (3 \beta - 1) q^{43} + q^{45} - 4 \beta q^{46} + (\beta + 5) q^{47} + 3 \beta q^{48} + q^{49} - 4 \beta q^{50} + ( - 2 \beta + 1) q^{51} + (\beta + 4) q^{52} - 5 \beta q^{53} + \beta q^{54} + (\beta + 4) q^{56} + 6 q^{57} + ( - \beta - 12) q^{58} + (\beta - 11) q^{59} + (\beta + 2) q^{60} + (6 \beta - 4) q^{61} + ( - 2 \beta - 16) q^{62} + q^{63} + ( - \beta + 4) q^{64} + ( - \beta + 4) q^{65} + ( - 5 \beta + 1) q^{67} + ( - 5 \beta - 6) q^{68} - 4 q^{69} + \beta q^{70} - 2 \beta q^{71} + (\beta + 4) q^{72} + ( - 2 \beta + 4) q^{73} + (5 \beta + 12) q^{74} - 4 q^{75} + (6 \beta + 12) q^{76} + (3 \beta - 4) q^{78} - 2 \beta q^{79} + 3 \beta q^{80} + q^{81} + ( - 5 \beta + 4) q^{82} + (5 \beta - 1) q^{83} + (\beta + 2) q^{84} + ( - 2 \beta + 1) q^{85} + (2 \beta + 12) q^{86} + ( - 3 \beta + 2) q^{87} + (4 \beta - 5) q^{89} + \beta q^{90} + ( - \beta + 4) q^{91} + ( - 4 \beta - 8) q^{92} + ( - 4 \beta + 2) q^{93} + (6 \beta + 4) q^{94} + 6 q^{95} + (\beta + 4) q^{96} + (3 \beta + 6) q^{97} + \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^3 + (b + 2) * q^4 + q^5 + b * q^6 + q^7 + (b + 4) * q^8 + q^9 + b * q^10 + (b + 2) * q^12 + (-b + 4) * q^13 + b * q^14 + q^15 + 3*b * q^16 + (-2*b + 1) * q^17 + b * q^18 + 6 * q^19 + (b + 2) * q^20 + q^21 - 4 * q^23 + (b + 4) * q^24 - 4 * q^25 + (3*b - 4) * q^26 + q^27 + (b + 2) * q^28 + (-3*b + 2) * q^29 + b * q^30 + (-4*b + 2) * q^31 + (b + 4) * q^32 + (-b - 8) * q^34 + q^35 + (b + 2) * q^36 + (3*b + 2) * q^37 + 6*b * q^38 + (-b + 4) * q^39 + (b + 4) * q^40 + (b - 6) * q^41 + b * q^42 + (3*b - 1) * q^43 + q^45 - 4*b * q^46 + (b + 5) * q^47 + 3*b * q^48 + q^49 - 4*b * q^50 + (-2*b + 1) * q^51 + (b + 4) * q^52 - 5*b * q^53 + b * q^54 + (b + 4) * q^56 + 6 * q^57 + (-b - 12) * q^58 + (b - 11) * q^59 + (b + 2) * q^60 + (6*b - 4) * q^61 + (-2*b - 16) * q^62 + q^63 + (-b + 4) * q^64 + (-b + 4) * q^65 + (-5*b + 1) * q^67 + (-5*b - 6) * q^68 - 4 * q^69 + b * q^70 - 2*b * q^71 + (b + 4) * q^72 + (-2*b + 4) * q^73 + (5*b + 12) * q^74 - 4 * q^75 + (6*b + 12) * q^76 + (3*b - 4) * q^78 - 2*b * q^79 + 3*b * q^80 + q^81 + (-5*b + 4) * q^82 + (5*b - 1) * q^83 + (b + 2) * q^84 + (-2*b + 1) * q^85 + (2*b + 12) * q^86 + (-3*b + 2) * q^87 + (4*b - 5) * q^89 + b * q^90 + (-b + 4) * q^91 + (-4*b - 8) * q^92 + (-4*b + 2) * q^93 + (6*b + 4) * q^94 + 6 * q^95 + (b + 4) * q^96 + (3*b + 6) * q^97 + b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} + 5 q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 9 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 + 2 * q^3 + 5 * q^4 + 2 * q^5 + q^6 + 2 * q^7 + 9 * q^8 + 2 * q^9 $$2 q + q^{2} + 2 q^{3} + 5 q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 9 q^{8} + 2 q^{9} + q^{10} + 5 q^{12} + 7 q^{13} + q^{14} + 2 q^{15} + 3 q^{16} + q^{18} + 12 q^{19} + 5 q^{20} + 2 q^{21} - 8 q^{23} + 9 q^{24} - 8 q^{25} - 5 q^{26} + 2 q^{27} + 5 q^{28} + q^{29} + q^{30} + 9 q^{32} - 17 q^{34} + 2 q^{35} + 5 q^{36} + 7 q^{37} + 6 q^{38} + 7 q^{39} + 9 q^{40} - 11 q^{41} + q^{42} + q^{43} + 2 q^{45} - 4 q^{46} + 11 q^{47} + 3 q^{48} + 2 q^{49} - 4 q^{50} + 9 q^{52} - 5 q^{53} + q^{54} + 9 q^{56} + 12 q^{57} - 25 q^{58} - 21 q^{59} + 5 q^{60} - 2 q^{61} - 34 q^{62} + 2 q^{63} + 7 q^{64} + 7 q^{65} - 3 q^{67} - 17 q^{68} - 8 q^{69} + q^{70} - 2 q^{71} + 9 q^{72} + 6 q^{73} + 29 q^{74} - 8 q^{75} + 30 q^{76} - 5 q^{78} - 2 q^{79} + 3 q^{80} + 2 q^{81} + 3 q^{82} + 3 q^{83} + 5 q^{84} + 26 q^{86} + q^{87} - 6 q^{89} + q^{90} + 7 q^{91} - 20 q^{92} + 14 q^{94} + 12 q^{95} + 9 q^{96} + 15 q^{97} + q^{98}+O(q^{100})$$ 2 * q + q^2 + 2 * q^3 + 5 * q^4 + 2 * q^5 + q^6 + 2 * q^7 + 9 * q^8 + 2 * q^9 + q^10 + 5 * q^12 + 7 * q^13 + q^14 + 2 * q^15 + 3 * q^16 + q^18 + 12 * q^19 + 5 * q^20 + 2 * q^21 - 8 * q^23 + 9 * q^24 - 8 * q^25 - 5 * q^26 + 2 * q^27 + 5 * q^28 + q^29 + q^30 + 9 * q^32 - 17 * q^34 + 2 * q^35 + 5 * q^36 + 7 * q^37 + 6 * q^38 + 7 * q^39 + 9 * q^40 - 11 * q^41 + q^42 + q^43 + 2 * q^45 - 4 * q^46 + 11 * q^47 + 3 * q^48 + 2 * q^49 - 4 * q^50 + 9 * q^52 - 5 * q^53 + q^54 + 9 * q^56 + 12 * q^57 - 25 * q^58 - 21 * q^59 + 5 * q^60 - 2 * q^61 - 34 * q^62 + 2 * q^63 + 7 * q^64 + 7 * q^65 - 3 * q^67 - 17 * q^68 - 8 * q^69 + q^70 - 2 * q^71 + 9 * q^72 + 6 * q^73 + 29 * q^74 - 8 * q^75 + 30 * q^76 - 5 * q^78 - 2 * q^79 + 3 * q^80 + 2 * q^81 + 3 * q^82 + 3 * q^83 + 5 * q^84 + 26 * q^86 + q^87 - 6 * q^89 + q^90 + 7 * q^91 - 20 * q^92 + 14 * q^94 + 12 * q^95 + 9 * q^96 + 15 * 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
 −1.56155 2.56155
−1.56155 1.00000 0.438447 1.00000 −1.56155 1.00000 2.43845 1.00000 −1.56155
1.2 2.56155 1.00000 4.56155 1.00000 2.56155 1.00000 6.56155 1.00000 2.56155
 $$n$$: 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.bc yes 2
3.b odd 2 1 7623.2.a.be 2
11.b odd 2 1 2541.2.a.u 2
33.d even 2 1 7623.2.a.bt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.u 2 11.b odd 2 1
2541.2.a.bc yes 2 1.a even 1 1 trivial
7623.2.a.be 2 3.b odd 2 1
7623.2.a.bt 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 7T + 8$$
$17$ $$T^{2} - 17$$
$19$ $$(T - 6)^{2}$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} - 68$$
$37$ $$T^{2} - 7T - 26$$
$41$ $$T^{2} + 11T + 26$$
$43$ $$T^{2} - T - 38$$
$47$ $$T^{2} - 11T + 26$$
$53$ $$T^{2} + 5T - 100$$
$59$ $$T^{2} + 21T + 106$$
$61$ $$T^{2} + 2T - 152$$
$67$ $$T^{2} + 3T - 104$$
$71$ $$T^{2} + 2T - 16$$
$73$ $$T^{2} - 6T - 8$$
$79$ $$T^{2} + 2T - 16$$
$83$ $$T^{2} - 3T - 104$$
$89$ $$T^{2} + 6T - 59$$
$97$ $$T^{2} - 15T + 18$$