# Properties

 Label 1840.2.a.t Level $1840$ Weight $2$ Character orbit 1840.a Self dual yes Analytic conductor $14.692$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.6924739719$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6 x - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} - q^{5} + ( -1 - \beta_{1} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} - q^{5} + ( -1 - \beta_{1} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} + \beta_{1} q^{15} + ( -1 + \beta_{1} - \beta_{2} ) q^{17} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{19} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{21} - q^{23} + q^{25} -3 q^{27} + ( 1 + \beta_{2} ) q^{29} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{31} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{33} + ( 1 + \beta_{1} ) q^{35} + 4 q^{37} + ( -5 - 2 \beta_{2} ) q^{39} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -6 - 2 \beta_{2} ) q^{43} + ( -1 - \beta_{1} - \beta_{2} ) q^{45} + ( -5 - \beta_{2} ) q^{47} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{49} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{51} + ( 2 + 4 \beta_{2} ) q^{53} + ( 1 - \beta_{1} + \beta_{2} ) q^{55} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{57} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 9 - \beta_{1} + \beta_{2} ) q^{61} + ( -4 - 4 \beta_{1} - \beta_{2} ) q^{63} + ( -\beta_{1} + \beta_{2} ) q^{65} + ( -4 - 4 \beta_{1} + 4 \beta_{2} ) q^{67} + \beta_{1} q^{69} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -5 - 4 \beta_{1} + \beta_{2} ) q^{73} -\beta_{1} q^{75} + ( -4 - \beta_{2} ) q^{77} + ( -3 - 3 \beta_{2} ) q^{81} + ( 4 + 4 \beta_{1} + 2 \beta_{2} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} ) q^{85} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{87} + ( -10 - 2 \beta_{2} ) q^{89} + ( -5 - \beta_{1} - \beta_{2} ) q^{91} + ( -11 - 2 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{95} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{97} + ( -3 + 3 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{5} - 3q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{5} - 3q^{7} + 3q^{9} - 3q^{11} - 3q^{17} - 3q^{19} + 12q^{21} - 3q^{23} + 3q^{25} - 9q^{27} + 3q^{29} - 6q^{31} - 15q^{33} + 3q^{35} + 12q^{37} - 15q^{39} - 6q^{41} - 18q^{43} - 3q^{45} - 15q^{47} - 6q^{49} - 15q^{51} + 6q^{53} + 3q^{55} - 6q^{57} - 6q^{59} + 27q^{61} - 12q^{63} - 12q^{67} - 6q^{71} - 15q^{73} - 12q^{77} - 9q^{81} + 12q^{83} + 3q^{85} + 3q^{87} - 30q^{89} - 15q^{91} - 33q^{93} + 3q^{95} + 9q^{97} - 9q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 4$$

## 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
 2.66908 −0.523976 −2.14510
0 −2.66908 0 −1.00000 0 −3.66908 0 4.12398 0
1.2 0 0.523976 0 −1.00000 0 −0.476024 0 −2.72545 0
1.3 0 2.14510 0 −1.00000 0 1.14510 0 1.60147 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
$$2$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

## 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 1840.2.a.t 3
4.b odd 2 1 920.2.a.g 3
5.b even 2 1 9200.2.a.cd 3
8.b even 2 1 7360.2.a.ca 3
8.d odd 2 1 7360.2.a.cb 3
12.b even 2 1 8280.2.a.bo 3
20.d odd 2 1 4600.2.a.y 3
20.e even 4 2 4600.2.e.r 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.g 3 4.b odd 2 1
1840.2.a.t 3 1.a even 1 1 trivial
4600.2.a.y 3 20.d odd 2 1
4600.2.e.r 6 20.e even 4 2
7360.2.a.ca 3 8.b even 2 1
7360.2.a.cb 3 8.d odd 2 1
8280.2.a.bo 3 12.b even 2 1
9200.2.a.cd 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{3} - 6 T_{3} + 3$$ $$T_{7}^{3} + 3 T_{7}^{2} - 3 T_{7} - 2$$ $$T_{11}^{3} + 3 T_{11}^{2} - 15 T_{11} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$3 - 6 T + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$-2 - 3 T + 3 T^{2} + T^{3}$$
$11$ $$12 - 15 T + 3 T^{2} + T^{3}$$
$13$ $$29 - 18 T + T^{3}$$
$17$ $$12 - 15 T + 3 T^{2} + T^{3}$$
$19$ $$48 - 33 T + 3 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$12 - 6 T - 3 T^{2} + T^{3}$$
$31$ $$-249 - 42 T + 6 T^{2} + T^{3}$$
$37$ $$( -4 + T )^{3}$$
$41$ $$-69 - 60 T + 6 T^{2} + T^{3}$$
$43$ $$-32 + 72 T + 18 T^{2} + T^{3}$$
$47$ $$76 + 66 T + 15 T^{2} + T^{3}$$
$53$ $$536 - 132 T - 6 T^{2} + T^{3}$$
$59$ $$-32 - 36 T + 6 T^{2} + T^{3}$$
$61$ $$-596 + 225 T - 27 T^{2} + T^{3}$$
$67$ $$-2944 - 240 T + 12 T^{2} + T^{3}$$
$71$ $$-203 - 60 T + 6 T^{2} + T^{3}$$
$73$ $$-588 - 42 T + 15 T^{2} + T^{3}$$
$79$ $$T^{3}$$
$83$ $$64 - 60 T - 12 T^{2} + T^{3}$$
$89$ $$608 + 264 T + 30 T^{2} + T^{3}$$
$97$ $$138 - 69 T - 9 T^{2} + T^{3}$$