# Properties

 Label 2541.2.a.bp Level 2541 Weight 2 Character orbit 2541.a Self dual yes Analytic conductor 20.290 Analytic rank 0 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.7488.1 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 4 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 3$$

## 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
 −0.326909 −1.43091 3.05896 0.698857
−2.05896 1.00000 2.23931 −3.05896 −2.05896 −1.00000 −0.492737 1.00000 6.29827
1.2 0.301143 1.00000 −1.90931 −0.698857 0.301143 −1.00000 −1.17726 1.00000 −0.210456
1.3 1.32691 1.00000 −0.239314 0.326909 1.32691 −1.00000 −2.97136 1.00000 0.433778
1.4 2.43091 1.00000 3.90931 1.43091 2.43091 −1.00000 4.64136 1.00000 3.47841
 $$n$$: 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.bp yes 4
3.b odd 2 1 7623.2.a.cg 4
11.b odd 2 1 2541.2.a.bl 4
33.d even 2 1 7623.2.a.cn 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bl 4 11.b odd 2 1
2541.2.a.bp yes 4 1.a even 1 1 trivial
7623.2.a.cg 4 3.b odd 2 1
7623.2.a.cn 4 33.d even 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2541))$$:

 $$T_{2}^{4} - 2 T_{2}^{3} - 4 T_{2}^{2} + 8 T_{2} - 2$$ $$T_{5}^{4} + 2 T_{5}^{3} - 4 T_{5}^{2} - 2 T_{5} + 1$$ $$T_{13}^{4} - 10 T_{13}^{3} + 32 T_{13}^{2} - 32 T_{13} - 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 4 T^{2} - 4 T^{3} + 6 T^{4} - 8 T^{5} + 16 T^{6} - 16 T^{7} + 16 T^{8}$$
$3$ $$( 1 - T )^{4}$$
$5$ $$1 + 2 T + 16 T^{2} + 28 T^{3} + 111 T^{4} + 140 T^{5} + 400 T^{6} + 250 T^{7} + 625 T^{8}$$
$7$ $$( 1 + T )^{4}$$
$11$ 
$13$ $$1 - 10 T + 84 T^{2} - 422 T^{3} + 1844 T^{4} - 5486 T^{5} + 14196 T^{6} - 21970 T^{7} + 28561 T^{8}$$
$17$ $$1 - 6 T + 28 T^{2} - 24 T^{3} + 111 T^{4} - 408 T^{5} + 8092 T^{6} - 29478 T^{7} + 83521 T^{8}$$
$19$ $$1 - 18 T + 184 T^{2} - 1278 T^{3} + 6468 T^{4} - 24282 T^{5} + 66424 T^{6} - 123462 T^{7} + 130321 T^{8}$$
$23$ $$1 + 2 T + 28 T^{2} + 190 T^{3} + 516 T^{4} + 4370 T^{5} + 14812 T^{6} + 24334 T^{7} + 279841 T^{8}$$
$29$ $$1 - 6 T + 20 T^{2} - 126 T^{3} + 1404 T^{4} - 3654 T^{5} + 16820 T^{6} - 146334 T^{7} + 707281 T^{8}$$
$31$ $$1 + 28 T^{2} - 216 T^{3} + 606 T^{4} - 6696 T^{5} + 26908 T^{6} + 923521 T^{8}$$
$37$ $$1 + 4 T + 72 T^{2} + 284 T^{3} + 4082 T^{4} + 10508 T^{5} + 98568 T^{6} + 202612 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 124 T^{2} + 48 T^{3} + 6810 T^{4} + 1968 T^{5} + 208444 T^{6} + 2825761 T^{8}$$
$43$ $$1 - 20 T + 282 T^{2} - 2632 T^{3} + 19943 T^{4} - 113176 T^{5} + 521418 T^{6} - 1590140 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 6 T + 44 T^{2} + 504 T^{3} + 4371 T^{4} + 23688 T^{5} + 97196 T^{6} + 622938 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 148 T^{2} - 192 T^{3} + 9942 T^{4} - 10176 T^{5} + 415732 T^{6} + 7890481 T^{8}$$
$59$ $$1 + 6 T + 208 T^{2} + 936 T^{3} + 17751 T^{4} + 55224 T^{5} + 724048 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 10 T + 108 T^{2} + 314 T^{3} + 3596 T^{4} + 19154 T^{5} + 401868 T^{6} + 2269810 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 4 T + 130 T^{2} + 56 T^{3} + 8299 T^{4} + 3752 T^{5} + 583570 T^{6} - 1203052 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 6 T + 220 T^{2} + 1014 T^{3} + 20916 T^{4} + 71994 T^{5} + 1109020 T^{6} + 2147466 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 34 T + 708 T^{2} - 9614 T^{3} + 96764 T^{4} - 701822 T^{5} + 3772932 T^{6} - 13226578 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 24 T + 280 T^{2} - 1584 T^{3} + 8754 T^{4} - 125136 T^{5} + 1747480 T^{6} - 11832936 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 6 T + 260 T^{2} - 1188 T^{3} + 30039 T^{4} - 98604 T^{5} + 1791140 T^{6} - 3430722 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 18 T + 352 T^{2} - 4068 T^{3} + 48255 T^{4} - 362052 T^{5} + 2788192 T^{6} - 12689442 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 10 T + 388 T^{2} + 2758 T^{3} + 56428 T^{4} + 267526 T^{5} + 3650692 T^{6} + 9126730 T^{7} + 88529281 T^{8}$$