# Properties

 Label 2541.2.a.bh Level 2541 Weight 2 Character orbit 2541.a Self dual yes Analytic conductor 20.290 Analytic rank 1 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 2.34292 0.470683 −1.81361
−2.34292 1.00000 3.48929 −0.146365 −2.34292 −1.00000 −3.48929 1.00000 0.342923
1.2 −0.470683 1.00000 −1.77846 3.24914 −0.470683 −1.00000 1.77846 1.00000 −1.52932
1.3 1.81361 1.00000 1.28917 −2.10278 1.81361 −1.00000 −1.28917 1.00000 −3.81361
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.bh 3
3.b odd 2 1 7623.2.a.cc 3
11.b odd 2 1 2541.2.a.bj yes 3
33.d even 2 1 7623.2.a.ca 3

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

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## Hecke kernels

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

 $$T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 2$$ $$T_{5}^{3} - T_{5}^{2} - 7 T_{5} - 1$$ $$T_{13}^{3} + 7 T_{13}^{2} + 12 T_{13} + 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 2 T^{2} + 2 T^{3} + 4 T^{4} + 4 T^{5} + 8 T^{6}$$
$3$ $$( 1 - T )^{3}$$
$5$ $$1 - T + 8 T^{2} - 11 T^{3} + 40 T^{4} - 25 T^{5} + 125 T^{6}$$
$7$ $$( 1 + T )^{3}$$
$11$ 
$13$ $$1 + 7 T + 51 T^{2} + 184 T^{3} + 663 T^{4} + 1183 T^{5} + 2197 T^{6}$$
$17$ $$1 + T + 44 T^{2} + 35 T^{3} + 748 T^{4} + 289 T^{5} + 4913 T^{6}$$
$19$ $$1 + 8 T + 63 T^{2} + 260 T^{3} + 1197 T^{4} + 2888 T^{5} + 6859 T^{6}$$
$23$ $$1 + 2 T - 9 T^{2} - 60 T^{3} - 207 T^{4} + 1058 T^{5} + 12167 T^{6}$$
$29$ $$1 + 7 T + 99 T^{2} + 408 T^{3} + 2871 T^{4} + 5887 T^{5} + 24389 T^{6}$$
$31$ $$1 + 2 T + 65 T^{2} + 132 T^{3} + 2015 T^{4} + 1922 T^{5} + 29791 T^{6}$$
$37$ $$1 + 3 T + 107 T^{2} + 214 T^{3} + 3959 T^{4} + 4107 T^{5} + 50653 T^{6}$$
$41$ $$1 + 13 T + 163 T^{2} + 1098 T^{3} + 6683 T^{4} + 21853 T^{5} + 68921 T^{6}$$
$43$ $$1 + 8 T + 58 T^{2} + 266 T^{3} + 2494 T^{4} + 14792 T^{5} + 79507 T^{6}$$
$47$ $$1 + 6 T + 110 T^{2} + 530 T^{3} + 5170 T^{4} + 13254 T^{5} + 103823 T^{6}$$
$53$ $$1 + 21 T + 299 T^{2} + 2522 T^{3} + 15847 T^{4} + 58989 T^{5} + 148877 T^{6}$$
$59$ $$1 - 6 T + 150 T^{2} - 622 T^{3} + 8850 T^{4} - 20886 T^{5} + 205379 T^{6}$$
$61$ $$1 + 12 T + 165 T^{2} + 1172 T^{3} + 10065 T^{4} + 44652 T^{5} + 226981 T^{6}$$
$67$ $$1 - 2 T + 50 T^{2} - 852 T^{3} + 3350 T^{4} - 8978 T^{5} + 300763 T^{6}$$
$71$ $$1 + 179 T^{2} - 76 T^{3} + 12709 T^{4} + 357911 T^{6}$$
$73$ $$1 - 4 T - 7 T^{2} + 1044 T^{3} - 511 T^{4} - 21316 T^{5} + 389017 T^{6}$$
$79$ $$1 + 18 T + 209 T^{2} + 1636 T^{3} + 16511 T^{4} + 112338 T^{5} + 493039 T^{6}$$
$83$ $$1 - 12 T + 78 T^{2} + 64 T^{3} + 6474 T^{4} - 82668 T^{5} + 571787 T^{6}$$
$89$ $$1 + T + 188 T^{2} + 251 T^{3} + 16732 T^{4} + 7921 T^{5} + 704969 T^{6}$$
$97$ $$1 + 25 T + 315 T^{2} + 2968 T^{3} + 30555 T^{4} + 235225 T^{5} + 912673 T^{6}$$