# Properties

 Label 232.2.a.d Level $232$ Weight $2$ Character orbit 232.a Self dual yes Analytic conductor $1.853$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$232 = 2^{3} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 232.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.85252932689$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} + ( - \beta_{2} + 1) q^{5} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 + (-b2 + 1) * q^5 + (b2 + 2) * q^9 $$q + ( - \beta_1 + 1) q^{3} + ( - \beta_{2} + 1) q^{5} + (\beta_{2} + 2) q^{9} + (2 \beta_{2} + \beta_1 + 1) q^{11} + (\beta_{2} + 2 \beta_1 + 1) q^{13} + ( - 2 \beta_{2} - \beta_1 - 1) q^{15} + 2 q^{17} + ( - 2 \beta_{2} - 2) q^{19} + (2 \beta_1 - 2) q^{23} + ( - 3 \beta_{2} - 2 \beta_1 + 2) q^{25} + (2 \beta_{2} + \beta_1 + 1) q^{27} + q^{29} + (\beta_1 - 5) q^{31} + (3 \beta_{2} - 2 \beta_1 + 1) q^{33} + 2 \beta_{2} q^{37} + ( - 3 \beta_1 - 5) q^{39} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{41} + ( - 2 \beta_{2} + \beta_1 - 3) q^{43} + (2 \beta_1 - 4) q^{45} + (2 \beta_{2} - 3 \beta_1 + 1) q^{47} - 7 q^{49} + ( - 2 \beta_1 + 2) q^{51} + ( - \beta_{2} + 2 \beta_1 - 1) q^{53} + (4 \beta_{2} + 5 \beta_1 - 9) q^{55} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{57} + (2 \beta_1 + 2) q^{59} + ( - 4 \beta_1 + 2) q^{61} + (3 \beta_{2} + 4 \beta_1 - 1) q^{65} + ( - 4 \beta_1 + 8) q^{67} + ( - 2 \beta_{2} - 10) q^{69} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{71} + (2 \beta_{2} + 4 \beta_1) q^{73} + ( - 4 \beta_{2} + 4) q^{75} + (2 \beta_{2} - \beta_1 - 9) q^{79} + ( - 2 \beta_1 - 5) q^{81} + (2 \beta_1 + 10) q^{83} + ( - 2 \beta_{2} + 2) q^{85} + ( - \beta_1 + 1) q^{87} + (6 \beta_{2} + 4 \beta_1 + 4) q^{89} + ( - \beta_{2} + 4 \beta_1 - 9) q^{93} + ( - 2 \beta_{2} - 4 \beta_1 + 10) q^{95} + (2 \beta_{2} - 4) q^{97} + (2 \beta_{2} - 2 \beta_1 + 12) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 + (-b2 + 1) * q^5 + (b2 + 2) * q^9 + (2*b2 + b1 + 1) * q^11 + (b2 + 2*b1 + 1) * q^13 + (-2*b2 - b1 - 1) * q^15 + 2 * q^17 + (-2*b2 - 2) * q^19 + (2*b1 - 2) * q^23 + (-3*b2 - 2*b1 + 2) * q^25 + (2*b2 + b1 + 1) * q^27 + q^29 + (b1 - 5) * q^31 + (3*b2 - 2*b1 + 1) * q^33 + 2*b2 * q^37 + (-3*b1 - 5) * q^39 + (-2*b2 - 4*b1 + 4) * q^41 + (-2*b2 + b1 - 3) * q^43 + (2*b1 - 4) * q^45 + (2*b2 - 3*b1 + 1) * q^47 - 7 * q^49 + (-2*b1 + 2) * q^51 + (-b2 + 2*b1 - 1) * q^53 + (4*b2 + 5*b1 - 9) * q^55 + (-4*b2 + 2*b1 - 6) * q^57 + (2*b1 + 2) * q^59 + (-4*b1 + 2) * q^61 + (3*b2 + 4*b1 - 1) * q^65 + (-4*b1 + 8) * q^67 + (-2*b2 - 10) * q^69 + (-4*b2 + 2*b1 + 2) * q^71 + (2*b2 + 4*b1) * q^73 + (-4*b2 + 4) * q^75 + (2*b2 - b1 - 9) * q^79 + (-2*b1 - 5) * q^81 + (2*b1 + 10) * q^83 + (-2*b2 + 2) * q^85 + (-b1 + 1) * q^87 + (6*b2 + 4*b1 + 4) * q^89 + (-b2 + 4*b1 - 9) * q^93 + (-2*b2 - 4*b1 + 10) * q^95 + (2*b2 - 4) * q^97 + (2*b2 - 2*b1 + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 4 q^{5} + 5 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 4 * q^5 + 5 * q^9 $$3 q + 2 q^{3} + 4 q^{5} + 5 q^{9} + 2 q^{11} + 4 q^{13} - 2 q^{15} + 6 q^{17} - 4 q^{19} - 4 q^{23} + 7 q^{25} + 2 q^{27} + 3 q^{29} - 14 q^{31} - 2 q^{33} - 2 q^{37} - 18 q^{39} + 10 q^{41} - 6 q^{43} - 10 q^{45} - 2 q^{47} - 21 q^{49} + 4 q^{51} - 26 q^{55} - 12 q^{57} + 8 q^{59} + 2 q^{61} - 2 q^{65} + 20 q^{67} - 28 q^{69} + 12 q^{71} + 2 q^{73} + 16 q^{75} - 30 q^{79} - 17 q^{81} + 32 q^{83} + 8 q^{85} + 2 q^{87} + 10 q^{89} - 22 q^{93} + 28 q^{95} - 14 q^{97} + 32 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 4 * q^5 + 5 * q^9 + 2 * q^11 + 4 * q^13 - 2 * q^15 + 6 * q^17 - 4 * q^19 - 4 * q^23 + 7 * q^25 + 2 * q^27 + 3 * q^29 - 14 * q^31 - 2 * q^33 - 2 * q^37 - 18 * q^39 + 10 * q^41 - 6 * q^43 - 10 * q^45 - 2 * q^47 - 21 * q^49 + 4 * q^51 - 26 * q^55 - 12 * q^57 + 8 * q^59 + 2 * q^61 - 2 * q^65 + 20 * q^67 - 28 * q^69 + 12 * q^71 + 2 * q^73 + 16 * q^75 - 30 * q^79 - 17 * q^81 + 32 * q^83 + 8 * q^85 + 2 * q^87 + 10 * q^89 - 22 * q^93 + 28 * q^95 - 14 * q^97 + 32 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 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
 3.12489 −0.363328 −1.76156
0 −2.12489 0 1.48486 0 0 0 1.51514 0
1.2 0 1.36333 0 4.14134 0 0 0 −1.14134 0
1.3 0 2.76156 0 −1.62620 0 0 0 4.62620 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$29$$ $$-1$$

## 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 232.2.a.d 3
3.b odd 2 1 2088.2.a.s 3
4.b odd 2 1 464.2.a.j 3
5.b even 2 1 5800.2.a.p 3
8.b even 2 1 1856.2.a.x 3
8.d odd 2 1 1856.2.a.y 3
12.b even 2 1 4176.2.a.bu 3
29.b even 2 1 6728.2.a.j 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.a.d 3 1.a even 1 1 trivial
464.2.a.j 3 4.b odd 2 1
1856.2.a.x 3 8.b even 2 1
1856.2.a.y 3 8.d odd 2 1
2088.2.a.s 3 3.b odd 2 1
4176.2.a.bu 3 12.b even 2 1
5800.2.a.p 3 5.b even 2 1
6728.2.a.j 3 29.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 2T_{3}^{2} - 5T_{3} + 8$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(232))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} - 5 T + 8$$
$5$ $$T^{3} - 4 T^{2} - 3 T + 10$$
$7$ $$T^{3}$$
$11$ $$T^{3} - 2 T^{2} - 29 T + 80$$
$13$ $$T^{3} - 4 T^{2} - 19 T + 2$$
$17$ $$(T - 2)^{3}$$
$19$ $$T^{3} + 4 T^{2} - 28 T - 32$$
$23$ $$T^{3} + 4 T^{2} - 20 T - 64$$
$29$ $$(T - 1)^{3}$$
$31$ $$T^{3} + 14 T^{2} + 59 T + 68$$
$37$ $$T^{3} + 2 T^{2} - 32 T - 32$$
$41$ $$T^{3} - 10 T^{2} - 64 T + 512$$
$43$ $$T^{3} + 6 T^{2} - 37 T + 32$$
$47$ $$T^{3} + 2 T^{2} - 117 T - 452$$
$53$ $$T^{3} - 43T + 58$$
$59$ $$T^{3} - 8 T^{2} - 4 T + 16$$
$61$ $$T^{3} - 2 T^{2} - 100 T + 328$$
$67$ $$T^{3} - 20 T^{2} + 32 T + 640$$
$71$ $$T^{3} - 12 T^{2} - 148 T + 1696$$
$73$ $$T^{3} - 2 T^{2} - 96 T - 160$$
$79$ $$T^{3} + 30 T^{2} + 251 T + 388$$
$83$ $$T^{3} - 32 T^{2} + 316 T - 976$$
$89$ $$T^{3} - 10 T^{2} - 256 T + 2816$$
$97$ $$T^{3} + 14 T^{2} + 32 T - 64$$