# Properties

 Label 2541.2.a.bi 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: 3.3.837.1 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

## 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
 2.52892 −0.167449 −2.36147
−2.52892 −1.00000 4.39543 0.133492 2.52892 1.00000 −6.05784 1.00000 −0.337590
1.2 0.167449 −1.00000 −1.97196 3.80451 −0.167449 1.00000 −0.665102 1.00000 0.637062
1.3 2.36147 −1.00000 3.57653 −3.93800 −2.36147 1.00000 3.72294 1.00000 −9.29947
 $$n$$: 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.bi 3
3.b odd 2 1 7623.2.a.cb 3
11.b odd 2 1 231.2.a.d 3
33.d even 2 1 693.2.a.m 3
44.c even 2 1 3696.2.a.bp 3
55.d odd 2 1 5775.2.a.bw 3
77.b even 2 1 1617.2.a.s 3
231.h odd 2 1 4851.2.a.bp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.d 3 11.b odd 2 1
693.2.a.m 3 33.d even 2 1
1617.2.a.s 3 77.b even 2 1
2541.2.a.bi 3 1.a even 1 1 trivial
3696.2.a.bp 3 44.c even 2 1
4851.2.a.bp 3 231.h odd 2 1
5775.2.a.bw 3 55.d odd 2 1
7623.2.a.cb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

 $$T_{2}^{3} - 6 T_{2} + 1$$ $$T_{5}^{3} - 15 T_{5} + 2$$ $$T_{13}^{3} - 15 T_{13} - 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{3} + 8 T^{6}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$1 + 2 T^{3} + 125 T^{6}$$
$7$ $$( 1 - T )^{3}$$
$11$ 
$13$ $$1 + 24 T^{2} - 2 T^{3} + 312 T^{4} + 2197 T^{6}$$
$17$ $$1 + 27 T^{2} - 8 T^{3} + 459 T^{4} + 4913 T^{6}$$
$19$ $$1 + 12 T + 84 T^{2} + 420 T^{3} + 1596 T^{4} + 4332 T^{5} + 6859 T^{6}$$
$23$ $$1 + 6 T + 57 T^{2} + 244 T^{3} + 1311 T^{4} + 3174 T^{5} + 12167 T^{6}$$
$29$ $$1 + 12 T + 120 T^{2} + 702 T^{3} + 3480 T^{4} + 10092 T^{5} + 24389 T^{6}$$
$31$ $$1 + 6 T + 57 T^{2} + 404 T^{3} + 1767 T^{4} + 5766 T^{5} + 29791 T^{6}$$
$37$ $$1 + 36 T^{2} - 246 T^{3} + 1332 T^{4} + 50653 T^{6}$$
$41$ $$1 + 6 T + 51 T^{2} + 460 T^{3} + 2091 T^{4} + 10086 T^{5} + 68921 T^{6}$$
$43$ $$1 + 6 T + 117 T^{2} + 468 T^{3} + 5031 T^{4} + 11094 T^{5} + 79507 T^{6}$$
$47$ $$1 + 24 T + 312 T^{2} + 2584 T^{3} + 14664 T^{4} + 53016 T^{5} + 103823 T^{6}$$
$53$ $$1 + 111 T^{2} - 120 T^{3} + 5883 T^{4} + 148877 T^{6}$$
$59$ $$1 + 24 T + 264 T^{2} + 2116 T^{3} + 15576 T^{4} + 83544 T^{5} + 205379 T^{6}$$
$61$ $$( 1 + 6 T + 61 T^{2} )^{3}$$
$67$ $$1 - 12 T + 228 T^{2} - 1604 T^{3} + 15276 T^{4} - 53868 T^{5} + 300763 T^{6}$$
$71$ $$1 - 12 T + 165 T^{2} - 1320 T^{3} + 11715 T^{4} - 60492 T^{5} + 357911 T^{6}$$
$73$ $$1 + 24 T + 396 T^{2} + 3898 T^{3} + 28908 T^{4} + 127896 T^{5} + 389017 T^{6}$$
$79$ $$1 + 12 T + 189 T^{2} + 1640 T^{3} + 14931 T^{4} + 74892 T^{5} + 493039 T^{6}$$
$83$ $$1 - 18 T + 309 T^{2} - 3036 T^{3} + 25647 T^{4} - 124002 T^{5} + 571787 T^{6}$$
$89$ $$1 - 18 T + 183 T^{2} - 1308 T^{3} + 16287 T^{4} - 142578 T^{5} + 704969 T^{6}$$
$97$ $$1 - 24 T + 435 T^{2} - 4664 T^{3} + 42195 T^{4} - 225816 T^{5} + 912673 T^{6}$$