# Properties

 Label 2394.2.a.x Level $2394$ Weight $2$ Character orbit 2394.a Self dual yes Analytic conductor $19.116$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2394.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.1161862439$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 798) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta q^{5} - q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + \beta q^{5} - q^{7} + q^{8} + \beta q^{10} + ( 2 + \beta ) q^{11} -\beta q^{13} - q^{14} + q^{16} + ( -4 - \beta ) q^{17} + q^{19} + \beta q^{20} + ( 2 + \beta ) q^{22} + ( 2 + \beta ) q^{23} + 7 q^{25} -\beta q^{26} - q^{28} + 6 q^{29} + 2 \beta q^{31} + q^{32} + ( -4 - \beta ) q^{34} -\beta q^{35} + 10 q^{37} + q^{38} + \beta q^{40} + 2 q^{41} -2 \beta q^{43} + ( 2 + \beta ) q^{44} + ( 2 + \beta ) q^{46} + ( -4 - 2 \beta ) q^{47} + q^{49} + 7 q^{50} -\beta q^{52} -2 q^{53} + ( 12 + 2 \beta ) q^{55} - q^{56} + 6 q^{58} -4 q^{59} + ( 2 - 2 \beta ) q^{61} + 2 \beta q^{62} + q^{64} -12 q^{65} + ( 6 + \beta ) q^{67} + ( -4 - \beta ) q^{68} -\beta q^{70} + ( 4 - 2 \beta ) q^{71} + 10 q^{73} + 10 q^{74} + q^{76} + ( -2 - \beta ) q^{77} + ( -2 - 3 \beta ) q^{79} + \beta q^{80} + 2 q^{82} -8 q^{83} + ( -12 - 4 \beta ) q^{85} -2 \beta q^{86} + ( 2 + \beta ) q^{88} + 2 q^{89} + \beta q^{91} + ( 2 + \beta ) q^{92} + ( -4 - 2 \beta ) q^{94} + \beta q^{95} + ( 8 - \beta ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} + 14 q^{25} - 2 q^{28} + 12 q^{29} + 2 q^{32} - 8 q^{34} + 20 q^{37} + 2 q^{38} + 4 q^{41} + 4 q^{44} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 14 q^{50} - 4 q^{53} + 24 q^{55} - 2 q^{56} + 12 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} - 24 q^{65} + 12 q^{67} - 8 q^{68} + 8 q^{71} + 20 q^{73} + 20 q^{74} + 2 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{82} - 16 q^{83} - 24 q^{85} + 4 q^{88} + 4 q^{89} + 4 q^{92} - 8 q^{94} + 16 q^{97} + 2 q^{98} + O(q^{100})$$

## 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
 −1.73205 1.73205
1.00000 0 1.00000 −3.46410 0 −1.00000 1.00000 0 −3.46410
1.2 1.00000 0 1.00000 3.46410 0 −1.00000 1.00000 0 3.46410
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$19$$ $$-1$$

## 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 2394.2.a.x 2
3.b odd 2 1 798.2.a.k 2
12.b even 2 1 6384.2.a.br 2
21.c even 2 1 5586.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.a.k 2 3.b odd 2 1
2394.2.a.x 2 1.a even 1 1 trivial
5586.2.a.bd 2 21.c even 2 1
6384.2.a.br 2 12.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2394))$$:

 $$T_{5}^{2} - 12$$ $$T_{11}^{2} - 4 T_{11} - 8$$ $$T_{13}^{2} - 12$$ $$T_{17}^{2} + 8 T_{17} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-12 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-8 - 4 T + T^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$4 + 8 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-8 - 4 T + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$-48 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$-48 + T^{2}$$
$47$ $$-32 + 8 T + T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$-44 - 4 T + T^{2}$$
$67$ $$24 - 12 T + T^{2}$$
$71$ $$-32 - 8 T + T^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$-104 + 4 T + T^{2}$$
$83$ $$( 8 + T )^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$52 - 16 T + T^{2}$$