# Properties

 Label 3432.2.a.n Level $3432$ Weight $2$ Character orbit 3432.a Self dual yes Analytic conductor $27.405$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3432.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.4046579737$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5 x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -1 - \beta_{2} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -1 - \beta_{2} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} + q^{11} + q^{13} + ( -1 - \beta_{2} ) q^{15} + ( -1 + 3 \beta_{1} ) q^{17} + ( -2 + \beta_{1} - \beta_{2} ) q^{19} + ( -2 - \beta_{1} + \beta_{2} ) q^{21} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} ) q^{25} + q^{27} + ( -1 + \beta_{1} ) q^{29} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{31} + q^{33} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{37} + q^{39} + ( -3 \beta_{1} - \beta_{2} ) q^{41} + ( 3 - 3 \beta_{1} ) q^{43} + ( -1 - \beta_{2} ) q^{45} + ( -4 + 2 \beta_{2} ) q^{47} + ( 3 + 2 \beta_{1} - 4 \beta_{2} ) q^{49} + ( -1 + 3 \beta_{1} ) q^{51} + ( -4 - 2 \beta_{1} ) q^{53} + ( -1 - \beta_{2} ) q^{55} + ( -2 + \beta_{1} - \beta_{2} ) q^{57} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 2 + 2 \beta_{1} - 4 \beta_{2} ) q^{61} + ( -2 - \beta_{1} + \beta_{2} ) q^{63} + ( -1 - \beta_{2} ) q^{65} + ( -9 + 4 \beta_{1} + \beta_{2} ) q^{67} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{69} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 2 - \beta_{1} - \beta_{2} ) q^{73} + ( 1 + \beta_{1} + \beta_{2} ) q^{75} + ( -2 - \beta_{1} + \beta_{2} ) q^{77} + ( -7 + \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} + ( 6 - 4 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -2 - 9 \beta_{1} + \beta_{2} ) q^{85} + ( -1 + \beta_{1} ) q^{87} + ( -1 + 4 \beta_{1} - 3 \beta_{2} ) q^{89} + ( -2 - \beta_{1} + \beta_{2} ) q^{91} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{93} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{95} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} - 4q^{5} - 6q^{7} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} - 4q^{5} - 6q^{7} + 3q^{9} + 3q^{11} + 3q^{13} - 4q^{15} - 6q^{19} - 6q^{21} + 5q^{25} + 3q^{27} - 2q^{29} - 14q^{31} + 3q^{33} - 2q^{35} - 8q^{37} + 3q^{39} - 4q^{41} + 6q^{43} - 4q^{45} - 10q^{47} + 7q^{49} - 14q^{53} - 4q^{55} - 6q^{57} + 4q^{59} + 4q^{61} - 6q^{63} - 4q^{65} - 22q^{67} - 6q^{71} + 4q^{73} + 5q^{75} - 6q^{77} - 22q^{79} + 3q^{81} + 16q^{83} - 14q^{85} - 2q^{87} - 2q^{89} - 6q^{91} - 14q^{93} + 18q^{95} - 24q^{97} + 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x - 1$$:

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

## 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
 2.86620 −1.65544 −0.210756
0 1.00000 0 −3.34889 0 −2.51730 0 1.00000 0
1.2 0 1.00000 0 −2.39593 0 1.05137 0 1.00000 0
1.3 0 1.00000 0 1.74483 0 −4.53407 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$-1$$
$$13$$ $$-1$$

## 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 3432.2.a.n 3
4.b odd 2 1 6864.2.a.bn 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.n 3 1.a even 1 1 trivial
6864.2.a.bn 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3432))$$:

 $$T_{5}^{3} + 4 T_{5}^{2} - 2 T_{5} - 14$$ $$T_{7}^{3} + 6 T_{7}^{2} + 4 T_{7} - 12$$ $$T_{17}^{3} - 48 T_{17} - 74$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-14 - 2 T + 4 T^{2} + T^{3}$$
$7$ $$-12 + 4 T + 6 T^{2} + T^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$( -1 + T )^{3}$$
$17$ $$-74 - 48 T + T^{3}$$
$19$ $$-4 + 4 T + 6 T^{2} + T^{3}$$
$23$ $$-108 - 72 T + T^{3}$$
$29$ $$-6 - 4 T + 2 T^{2} + T^{3}$$
$31$ $$42 + 46 T + 14 T^{2} + T^{3}$$
$37$ $$-288 - 48 T + 8 T^{2} + T^{3}$$
$41$ $$132 - 64 T + 4 T^{2} + T^{3}$$
$43$ $$162 - 36 T - 6 T^{2} + T^{3}$$
$47$ $$-8 + 4 T + 10 T^{2} + T^{3}$$
$53$ $$24 + 44 T + 14 T^{2} + T^{3}$$
$59$ $$-144 - 72 T - 4 T^{2} + T^{3}$$
$61$ $$-144 - 96 T - 4 T^{2} + T^{3}$$
$67$ $$-862 + 50 T + 22 T^{2} + T^{3}$$
$71$ $$-24 - 20 T + 6 T^{2} + T^{3}$$
$73$ $$36 - 12 T - 4 T^{2} + T^{3}$$
$79$ $$174 + 136 T + 22 T^{2} + T^{3}$$
$83$ $$16 + 8 T - 16 T^{2} + T^{3}$$
$89$ $$258 - 94 T + 2 T^{2} + T^{3}$$
$97$ $$288 + 160 T + 24 T^{2} + T^{3}$$
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