# Properties

 Label 2394.2.a.g Level $2394$ Weight $2$ Character orbit 2394.a Self dual yes Analytic conductor $19.116$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 798) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
1.00000 0 1.00000 −4.00000 0 1.00000 1.00000 0 −4.00000
 $$n$$: 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.g 1
3.b odd 2 1 798.2.a.f 1
12.b even 2 1 6384.2.a.p 1
21.c even 2 1 5586.2.a.a 1

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

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

## Hecke characteristic polynomials

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