# Properties

 Label 1932.2.a.j Level $1932$ Weight $2$ Character orbit 1932.a Self dual yes Analytic conductor $15.427$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1932.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.4270976705$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9x + 12$$ x^3 - x^2 - 9*x + 12 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta_1 q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 + b1 * q^5 - q^7 + q^9 $$q + q^{3} + \beta_1 q^{5} - q^{7} + q^{9} + q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{13} + \beta_1 q^{15} + ( - \beta_{2} + 1) q^{17} + 3 q^{19} - q^{21} + q^{23} + (\beta_{2} - \beta_1 + 2) q^{25} + q^{27} + (3 \beta_{2} - 2 \beta_1 + 3) q^{29} + (\beta_{2} + 2 \beta_1 + 1) q^{31} + q^{33} - \beta_1 q^{35} + (3 \beta_{2} + 3) q^{37} + ( - \beta_{2} + \beta_1 - 1) q^{39} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{41} + ( - \beta_{2} - \beta_1 + 5) q^{43} + \beta_1 q^{45} + ( - \beta_{2} + 4) q^{47} + q^{49} + ( - \beta_{2} + 1) q^{51} + (\beta_{2} - \beta_1) q^{53} + \beta_1 q^{55} + 3 q^{57} + (\beta_{2} + 3 \beta_1 + 4) q^{59} + ( - 2 \beta_{2} - \beta_1 - 1) q^{61} - q^{63} + ( - \beta_{2} - 2 \beta_1 + 5) q^{65} + ( - 2 \beta_{2} + \beta_1 + 2) q^{67} + q^{69} + ( - 3 \beta_{2} - \beta_1 + 3) q^{71} + ( - \beta_{2} - 3) q^{73} + (\beta_{2} - \beta_1 + 2) q^{75} - q^{77} + ( - \beta_{2} + 2 \beta_1 - 5) q^{79} + q^{81} + (3 \beta_{2} - 2 \beta_1 + 7) q^{83} + ( - 2 \beta_{2} + \beta_1 - 2) q^{85} + (3 \beta_{2} - 2 \beta_1 + 3) q^{87} + (\beta_{2} - 3 \beta_1 + 5) q^{89} + (\beta_{2} - \beta_1 + 1) q^{91} + (\beta_{2} + 2 \beta_1 + 1) q^{93} + 3 \beta_1 q^{95} + (\beta_{2} - 3) q^{97} + q^{99}+O(q^{100})$$ q + q^3 + b1 * q^5 - q^7 + q^9 + q^11 + (-b2 + b1 - 1) * q^13 + b1 * q^15 + (-b2 + 1) * q^17 + 3 * q^19 - q^21 + q^23 + (b2 - b1 + 2) * q^25 + q^27 + (3*b2 - 2*b1 + 3) * q^29 + (b2 + 2*b1 + 1) * q^31 + q^33 - b1 * q^35 + (3*b2 + 3) * q^37 + (-b2 + b1 - 1) * q^39 + (-2*b2 - 2*b1 + 1) * q^41 + (-b2 - b1 + 5) * q^43 + b1 * q^45 + (-b2 + 4) * q^47 + q^49 + (-b2 + 1) * q^51 + (b2 - b1) * q^53 + b1 * q^55 + 3 * q^57 + (b2 + 3*b1 + 4) * q^59 + (-2*b2 - b1 - 1) * q^61 - q^63 + (-b2 - 2*b1 + 5) * q^65 + (-2*b2 + b1 + 2) * q^67 + q^69 + (-3*b2 - b1 + 3) * q^71 + (-b2 - 3) * q^73 + (b2 - b1 + 2) * q^75 - q^77 + (-b2 + 2*b1 - 5) * q^79 + q^81 + (3*b2 - 2*b1 + 7) * q^83 + (-2*b2 + b1 - 2) * q^85 + (3*b2 - 2*b1 + 3) * q^87 + (b2 - 3*b1 + 5) * q^89 + (b2 - b1 + 1) * q^91 + (b2 + 2*b1 + 1) * q^93 + 3*b1 * q^95 + (b2 - 3) * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - q^{13} + q^{15} + 4 q^{17} + 9 q^{19} - 3 q^{21} + 3 q^{23} + 4 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 3 q^{33} - q^{35} + 6 q^{37} - q^{39} + 3 q^{41} + 15 q^{43} + q^{45} + 13 q^{47} + 3 q^{49} + 4 q^{51} - 2 q^{53} + q^{55} + 9 q^{57} + 14 q^{59} - 2 q^{61} - 3 q^{63} + 14 q^{65} + 9 q^{67} + 3 q^{69} + 11 q^{71} - 8 q^{73} + 4 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 16 q^{83} - 3 q^{85} + 4 q^{87} + 11 q^{89} + q^{91} + 4 q^{93} + 3 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - q^13 + q^15 + 4 * q^17 + 9 * q^19 - 3 * q^21 + 3 * q^23 + 4 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 3 * q^33 - q^35 + 6 * q^37 - q^39 + 3 * q^41 + 15 * q^43 + q^45 + 13 * q^47 + 3 * q^49 + 4 * q^51 - 2 * q^53 + q^55 + 9 * q^57 + 14 * q^59 - 2 * q^61 - 3 * q^63 + 14 * q^65 + 9 * q^67 + 3 * q^69 + 11 * q^71 - 8 * q^73 + 4 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 16 * q^83 - 3 * q^85 + 4 * q^87 + 11 * q^89 + q^91 + 4 * q^93 + 3 * q^95 - 10 * q^97 + 3 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$ v^2 + v - 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 7$$ b2 - b1 + 7

## 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
 −3.11903 1.43163 2.68740
0 1.00000 0 −3.11903 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 1.43163 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.68740 0 −1.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 1932.2.a.j 3
3.b odd 2 1 5796.2.a.o 3
4.b odd 2 1 7728.2.a.br 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.j 3 1.a even 1 1 trivial
5796.2.a.o 3 3.b odd 2 1
7728.2.a.br 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1932))$$:

 $$T_{5}^{3} - T_{5}^{2} - 9T_{5} + 12$$ T5^3 - T5^2 - 9*T5 + 12 $$T_{11} - 1$$ T11 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} - T^{2} - 9T + 12$$
$7$ $$(T + 1)^{3}$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} + T^{2} - 15 T - 18$$
$17$ $$T^{3} - 4 T^{2} - 5 T + 12$$
$19$ $$(T - 3)^{3}$$
$23$ $$(T - 1)^{3}$$
$29$ $$T^{3} - 4 T^{2} - 99 T + 534$$
$31$ $$T^{3} - 4 T^{2} - 51 T + 18$$
$37$ $$T^{3} - 6 T^{2} - 81 T + 162$$
$41$ $$T^{3} - 3 T^{2} - 93 T + 423$$
$43$ $$T^{3} - 15 T^{2} + 51 T + 36$$
$47$ $$T^{3} - 13 T^{2} + 46 T - 36$$
$53$ $$T^{3} + 2 T^{2} - 14 T + 3$$
$59$ $$T^{3} - 14 T^{2} - 42 T + 411$$
$61$ $$T^{3} + 2 T^{2} - 58 T + 127$$
$67$ $$T^{3} - 9 T^{2} - 15 T - 4$$
$71$ $$T^{3} - 11 T^{2} - 75 T + 744$$
$73$ $$T^{3} + 8 T^{2} + 11 T - 8$$
$79$ $$T^{3} + 12 T^{2} + 9 T - 38$$
$83$ $$T^{3} - 16 T^{2} - 19 T + 802$$
$89$ $$T^{3} - 11 T^{2} - 41 T - 6$$
$97$ $$T^{3} + 10 T^{2} + 23 T + 2$$