# Properties

 Label 3432.2.a.w Level $3432$ Weight $2$ Character orbit 3432.a Self dual yes Analytic conductor $27.405$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## 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: $$5$$ Coefficient field: 5.5.2172244.1 Defining polynomial: $$x^{5} - 2 x^{4} - 8 x^{3} + 16 x^{2} + 5 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta_{1} q^{5} + ( -1 - \beta_{4} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta_{1} q^{5} + ( -1 - \beta_{4} ) q^{7} + q^{9} - q^{11} + q^{13} -\beta_{1} q^{15} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{17} + ( -1 - \beta_{1} + \beta_{2} ) q^{19} + ( 1 + \beta_{4} ) q^{21} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{23} + ( \beta_{2} - \beta_{3} + \beta_{4} ) q^{25} - q^{27} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{29} + ( -2 + \beta_{2} + \beta_{4} ) q^{31} + q^{33} + ( -3 \beta_{1} + \beta_{3} + \beta_{4} ) q^{35} + ( -\beta_{2} + 3 \beta_{3} ) q^{37} - q^{39} + ( 1 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{41} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{43} + \beta_{1} q^{45} + ( -4 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{47} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{49} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{53} -\beta_{1} q^{55} + ( 1 + \beta_{1} - \beta_{2} ) q^{57} + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{59} + ( 2 - 3 \beta_{1} - \beta_{3} - \beta_{4} ) q^{61} + ( -1 - \beta_{4} ) q^{63} + \beta_{1} q^{65} + ( 2 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{67} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{69} + ( -6 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{71} + ( -3 + 2 \beta_{1} - \beta_{4} ) q^{73} + ( -\beta_{2} + \beta_{3} - \beta_{4} ) q^{75} + ( 1 + \beta_{4} ) q^{77} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{79} + q^{81} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{85} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{87} + ( -2 + 2 \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{89} + ( -1 - \beta_{4} ) q^{91} + ( 2 - \beta_{2} - \beta_{4} ) q^{93} + ( -6 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{95} + ( -4 + \beta_{2} + \beta_{3} ) q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 5q^{3} + q^{5} - 5q^{7} + 5q^{9} + O(q^{10})$$ $$5q - 5q^{3} + q^{5} - 5q^{7} + 5q^{9} - 5q^{11} + 5q^{13} - q^{15} - 4q^{19} + 5q^{21} - 5q^{23} + 4q^{25} - 5q^{27} + 11q^{29} - 8q^{31} + 5q^{33} - 5q^{35} - 8q^{37} - 5q^{39} - q^{41} + q^{43} + q^{45} - 18q^{47} + 10q^{49} + 2q^{53} - q^{55} + 4q^{57} - 13q^{59} + 9q^{61} - 5q^{63} + q^{65} + 5q^{67} + 5q^{69} - 24q^{71} - 13q^{73} - 4q^{75} + 5q^{77} - 6q^{79} + 5q^{81} - 22q^{83} - 22q^{85} - 11q^{87} - 14q^{89} - 5q^{91} + 8q^{93} - 32q^{95} - 20q^{97} - 5q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 8 x^{3} + 16 x^{2} + 5 x - 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{4} + \nu^{3} - 7 \nu^{2} - 5 \nu + 4$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{4} + \nu^{3} + 7 \nu^{2} - 5 \nu - 2$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{4} + \nu^{3} + 7 \nu^{2} - 9 \nu - 2$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{4} + \nu^{3} + 9 \nu^{2} - 7 \nu - 10$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{4} - \beta_{3} - \beta_{2} + 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{3} + 7 \beta_{2} + 2 \beta_{1} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$14 \beta_{4} - 7 \beta_{3} - 9 \beta_{2} + 2 \beta_{1} + 50$$$$)/2$$

## 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
 2.19338 0.679950 −2.71661 −0.758834 2.60211
0 −1.00000 0 −3.47314 0 −3.67591 0 1.00000 0
1.2 0 −1.00000 0 −1.05398 0 4.24902 0 1.00000 0
1.3 0 −1.00000 0 0.169303 0 −1.46187 0 1.00000 0
1.4 0 −1.00000 0 1.82899 0 −0.862883 0 1.00000 0
1.5 0 −1.00000 0 3.52882 0 −3.24835 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.w 5
4.b odd 2 1 6864.2.a.cf 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.w 5 1.a even 1 1 trivial
6864.2.a.cf 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3432))$$:

 $$T_{5}^{5} - T_{5}^{4} - 14 T_{5}^{3} + 12 T_{5}^{2} + 22 T_{5} - 4$$ $$T_{7}^{5} + 5 T_{7}^{4} - 10 T_{7}^{3} - 88 T_{7}^{2} - 140 T_{7} - 64$$ $$T_{17}^{5} - 52 T_{17}^{3} - 22 T_{17}^{2} + 492 T_{17} - 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$( 1 + T )^{5}$$
$5$ $$-4 + 22 T + 12 T^{2} - 14 T^{3} - T^{4} + T^{5}$$
$7$ $$-64 - 140 T - 88 T^{2} - 10 T^{3} + 5 T^{4} + T^{5}$$
$11$ $$( 1 + T )^{5}$$
$13$ $$( -1 + T )^{5}$$
$17$ $$-256 + 492 T - 22 T^{2} - 52 T^{3} + T^{5}$$
$19$ $$-32 - 152 T - 164 T^{2} - 40 T^{3} + 4 T^{4} + T^{5}$$
$23$ $$544 + 188 T - 132 T^{2} - 40 T^{3} + 5 T^{4} + T^{5}$$
$29$ $$-4892 + 182 T + 462 T^{2} - 30 T^{3} - 11 T^{4} + T^{5}$$
$31$ $$-256 + 752 T - 234 T^{2} - 42 T^{3} + 8 T^{4} + T^{5}$$
$37$ $$17152 + 2496 T - 880 T^{2} - 124 T^{3} + 8 T^{4} + T^{5}$$
$41$ $$-2648 + 3796 T - 68 T^{2} - 138 T^{3} + T^{4} + T^{5}$$
$43$ $$-1208 + 7638 T + 118 T^{2} - 196 T^{3} - T^{4} + T^{5}$$
$47$ $$2048 - 1248 T - 920 T^{2} + 4 T^{3} + 18 T^{4} + T^{5}$$
$53$ $$89888 + 21200 T - 384 T^{2} - 280 T^{3} - 2 T^{4} + T^{5}$$
$59$ $$64 - 688 T - 200 T^{2} + 28 T^{3} + 13 T^{4} + T^{5}$$
$61$ $$-9056 + 6560 T + 840 T^{2} - 150 T^{3} - 9 T^{4} + T^{5}$$
$67$ $$-248 + 66 T + 916 T^{2} - 138 T^{3} - 5 T^{4} + T^{5}$$
$71$ $$3968 - 1584 T - 640 T^{2} + 104 T^{3} + 24 T^{4} + T^{5}$$
$73$ $$-8 + 68 T - 80 T^{2} - 2 T^{3} + 13 T^{4} + T^{5}$$
$79$ $$-16096 + 7020 T - 242 T^{2} - 152 T^{3} + 6 T^{4} + T^{5}$$
$83$ $$105088 - 2656 T - 3392 T^{2} - 80 T^{3} + 22 T^{4} + T^{5}$$
$89$ $$-6112 - 7108 T - 2394 T^{2} - 162 T^{3} + 14 T^{4} + T^{5}$$
$97$ $$-256 - 960 T - 16 T^{2} + 108 T^{3} + 20 T^{4} + T^{5}$$