# Properties

 Label 2394.2.a.b 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

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

## 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 −2.00000 0 −1.00000 −1.00000 0 2.00000
 $$n$$: 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.b 1
3.b odd 2 1 798.2.a.i 1
12.b even 2 1 6384.2.a.m 1
21.c even 2 1 5586.2.a.t 1

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

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

## Hecke characteristic polynomials

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