# Properties

 Label 87.2.a.b Level $87$ Weight $2$ Character orbit 87.a Self dual yes Analytic conductor $0.695$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$87 = 3 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 87.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.694698497585$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{2} - q^{3} + ( 2 + \beta_{1} ) q^{4} -2 \beta_{1} q^{5} + ( -1 - \beta_{2} ) q^{6} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{2} - q^{3} + ( 2 + \beta_{1} ) q^{4} -2 \beta_{1} q^{5} + ( -1 - \beta_{2} ) q^{6} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} + ( -2 - 4 \beta_{1} ) q^{10} + ( -3 + \beta_{1} - \beta_{2} ) q^{11} + ( -2 - \beta_{1} ) q^{12} + ( 1 - \beta_{1} - \beta_{2} ) q^{13} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{14} + 2 \beta_{1} q^{15} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{16} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{17} + ( 1 + \beta_{2} ) q^{18} + 2 \beta_{2} q^{19} + ( -6 - 4 \beta_{1} - 2 \beta_{2} ) q^{20} + ( -1 - \beta_{1} + \beta_{2} ) q^{21} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{22} + ( 2 - 2 \beta_{1} ) q^{23} + ( -1 - 2 \beta_{1} ) q^{24} + ( 7 + 4 \beta_{2} ) q^{25} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{26} - q^{27} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{28} + q^{29} + ( 2 + 4 \beta_{1} ) q^{30} + ( 2 - 2 \beta_{1} ) q^{31} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{32} + ( 3 - \beta_{1} + \beta_{2} ) q^{33} + ( 1 + 5 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -4 - 2 \beta_{2} ) q^{35} + ( 2 + \beta_{1} ) q^{36} + ( 2 - 2 \beta_{2} ) q^{37} + ( 6 + 2 \beta_{1} - 2 \beta_{2} ) q^{38} + ( -1 + \beta_{1} + \beta_{2} ) q^{39} + ( -12 - 2 \beta_{1} - 4 \beta_{2} ) q^{40} + ( -2 + 4 \beta_{1} - 4 \beta_{2} ) q^{41} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{42} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{43} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{44} -2 \beta_{1} q^{45} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{46} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{48} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{49} + ( 19 + 4 \beta_{1} + 3 \beta_{2} ) q^{50} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{51} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{52} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{54} + ( -4 + 8 \beta_{1} - 2 \beta_{2} ) q^{55} + ( 5 + \beta_{1} + \beta_{2} ) q^{56} -2 \beta_{2} q^{57} + ( 1 + \beta_{2} ) q^{58} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 6 + 4 \beta_{1} + 2 \beta_{2} ) q^{60} + ( 2 + 2 \beta_{2} ) q^{61} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{62} + ( 1 + \beta_{1} - \beta_{2} ) q^{63} + ( -1 - 4 \beta_{1} + 2 \beta_{2} ) q^{64} + ( 8 + 2 \beta_{2} ) q^{65} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{66} + ( -1 + 3 \beta_{1} - 3 \beta_{2} ) q^{67} + ( 10 + 6 \beta_{1} + \beta_{2} ) q^{68} + ( -2 + 2 \beta_{1} ) q^{69} + ( -10 - 2 \beta_{1} - 2 \beta_{2} ) q^{70} + ( -6 - 2 \beta_{1} - 4 \beta_{2} ) q^{71} + ( 1 + 2 \beta_{1} ) q^{72} + ( -2 + 2 \beta_{2} ) q^{73} + ( -4 - 2 \beta_{1} + 4 \beta_{2} ) q^{74} + ( -7 - 4 \beta_{2} ) q^{75} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{76} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{77} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{78} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -14 - 4 \beta_{2} ) q^{80} + q^{81} + ( -10 + 4 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{84} + ( -16 - 6 \beta_{2} ) q^{85} + ( 8 - 4 \beta_{1} - 4 \beta_{2} ) q^{86} - q^{87} + ( 1 - 7 \beta_{1} + \beta_{2} ) q^{88} + ( -1 + \beta_{1} + 5 \beta_{2} ) q^{89} + ( -2 - 4 \beta_{1} ) q^{90} + ( 1 + \beta_{1} - 5 \beta_{2} ) q^{91} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{92} + ( -2 + 2 \beta_{1} ) q^{93} + ( 3 - 3 \beta_{1} - 6 \beta_{2} ) q^{94} + ( -4 - 4 \beta_{1} ) q^{95} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{96} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{97} + ( -10 - \beta_{1} + \beta_{2} ) q^{98} + ( -3 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 2q^{2} - 3q^{3} + 6q^{4} - 2q^{6} + 4q^{7} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 2q^{2} - 3q^{3} + 6q^{4} - 2q^{6} + 4q^{7} + 3q^{8} + 3q^{9} - 6q^{10} - 8q^{11} - 6q^{12} + 4q^{13} - 5q^{14} - 4q^{16} + 4q^{17} + 2q^{18} - 2q^{19} - 16q^{20} - 4q^{21} - 13q^{22} + 6q^{23} - 3q^{24} + 17q^{25} - 11q^{26} - 3q^{27} + 13q^{28} + 3q^{29} + 6q^{30} + 6q^{31} + 8q^{32} + 8q^{33} + q^{34} - 10q^{35} + 6q^{36} + 8q^{37} + 20q^{38} - 4q^{39} - 32q^{40} - 2q^{41} + 5q^{42} - 4q^{43} - 11q^{44} - 2q^{46} - 12q^{47} + 4q^{48} - 3q^{49} + 54q^{50} - 4q^{51} - 3q^{52} + 8q^{53} - 2q^{54} - 10q^{55} + 14q^{56} + 2q^{57} + 2q^{58} - 20q^{59} + 16q^{60} + 4q^{61} - 2q^{62} + 4q^{63} - 5q^{64} + 22q^{65} + 13q^{66} + 29q^{68} - 6q^{69} - 28q^{70} - 14q^{71} + 3q^{72} - 8q^{73} - 16q^{74} - 17q^{75} + 2q^{76} + 2q^{77} + 11q^{78} - 2q^{79} - 38q^{80} + 3q^{81} - 32q^{82} - 8q^{83} - 13q^{84} - 42q^{85} + 28q^{86} - 3q^{87} + 2q^{88} - 8q^{89} - 6q^{90} + 8q^{91} - 4q^{92} - 6q^{93} + 15q^{94} - 12q^{95} - 8q^{96} + 4q^{97} - 31q^{98} - 8q^{99} + O(q^{100})$$

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

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

## 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
 −0.254102 −1.86081 2.11491
−1.93543 −1.00000 1.74590 0.508203 1.93543 3.68133 0.491797 1.00000 −0.983593
1.2 1.46260 −1.00000 0.139194 3.72161 −1.46260 −1.32340 −2.72161 1.00000 5.44322
1.3 2.47283 −1.00000 4.11491 −4.22982 −2.47283 1.64207 5.22982 1.00000 −10.4596
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$29$$ $$-1$$

## 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 87.2.a.b 3
3.b odd 2 1 261.2.a.e 3
4.b odd 2 1 1392.2.a.u 3
5.b even 2 1 2175.2.a.t 3
5.c odd 4 2 2175.2.c.l 6
7.b odd 2 1 4263.2.a.m 3
8.b even 2 1 5568.2.a.cb 3
8.d odd 2 1 5568.2.a.bx 3
12.b even 2 1 4176.2.a.bx 3
15.d odd 2 1 6525.2.a.bg 3
29.b even 2 1 2523.2.a.h 3
87.d odd 2 1 7569.2.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.a.b 3 1.a even 1 1 trivial
261.2.a.e 3 3.b odd 2 1
1392.2.a.u 3 4.b odd 2 1
2175.2.a.t 3 5.b even 2 1
2175.2.c.l 6 5.c odd 4 2
2523.2.a.h 3 29.b even 2 1
4176.2.a.bx 3 12.b even 2 1
4263.2.a.m 3 7.b odd 2 1
5568.2.a.bx 3 8.d odd 2 1
5568.2.a.cb 3 8.b even 2 1
6525.2.a.bg 3 15.d odd 2 1
7569.2.a.t 3 87.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 2 T_{2}^{2} - 4 T_{2} + 7$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(87))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$7 - 4 T - 2 T^{2} + T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$8 - 16 T + T^{3}$$
$7$ $$8 - T - 4 T^{2} + T^{3}$$
$11$ $$4 + 15 T + 8 T^{2} + T^{3}$$
$13$ $$26 - 7 T - 4 T^{2} + T^{3}$$
$17$ $$94 - 27 T - 4 T^{2} + T^{3}$$
$19$ $$16 - 20 T + 2 T^{2} + T^{3}$$
$23$ $$32 - 4 T - 6 T^{2} + T^{3}$$
$29$ $$( -1 + T )^{3}$$
$31$ $$32 - 4 T - 6 T^{2} + T^{3}$$
$37$ $$8 - 8 T^{2} + T^{3}$$
$41$ $$56 - 100 T + 2 T^{2} + T^{3}$$
$43$ $$-256 - 96 T + 4 T^{2} + T^{3}$$
$47$ $$-216 - 9 T + 12 T^{2} + T^{3}$$
$53$ $$248 - 104 T - 8 T^{2} + T^{3}$$
$59$ $$112 + 108 T + 20 T^{2} + T^{3}$$
$61$ $$56 - 16 T - 4 T^{2} + T^{3}$$
$67$ $$52 - 57 T + T^{3}$$
$71$ $$-416 - 60 T + 14 T^{2} + T^{3}$$
$73$ $$-8 + 8 T^{2} + T^{3}$$
$79$ $$-224 - 60 T + 2 T^{2} + T^{3}$$
$83$ $$-208 - 28 T + 8 T^{2} + T^{3}$$
$89$ $$-74 - 131 T + 8 T^{2} + T^{3}$$
$97$ $$-104 - 72 T - 4 T^{2} + T^{3}$$