# Properties

 Label 2394.2.a.j 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{7} + q^{8} - 6 q^{11} - 4 q^{13} + q^{14} + q^{16} + q^{19} - 6 q^{22} - 6 q^{23} - 5 q^{25} - 4 q^{26} + q^{28} - 6 q^{29} - 4 q^{31} + q^{32} + 2 q^{37} + q^{38} + 6 q^{41} + 8 q^{43} - 6 q^{44} - 6 q^{46} - 12 q^{47} + q^{49} - 5 q^{50} - 4 q^{52} - 6 q^{53} + q^{56} - 6 q^{58} + 12 q^{59} - 10 q^{61} - 4 q^{62} + q^{64} + 14 q^{67} + 12 q^{71} + 2 q^{73} + 2 q^{74} + q^{76} - 6 q^{77} - 10 q^{79} + 6 q^{82} - 12 q^{83} + 8 q^{86} - 6 q^{88} - 18 q^{89} - 4 q^{91} - 6 q^{92} - 12 q^{94} + 8 q^{97} + q^{98} + O(q^{100})$$

## 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 0 0 1.00000 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$19$$ $$-1$$

## 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.j 1
3.b odd 2 1 798.2.a.d 1
12.b even 2 1 6384.2.a.i 1
21.c even 2 1 5586.2.a.g 1

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

 $$T_{5}$$ $$T_{11} + 6$$ $$T_{13} + 4$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$6 + T$$
$13$ $$4 + T$$
$17$ $$T$$
$19$ $$-1 + T$$
$23$ $$6 + T$$
$29$ $$6 + T$$
$31$ $$4 + T$$
$37$ $$-2 + T$$
$41$ $$-6 + T$$
$43$ $$-8 + T$$
$47$ $$12 + T$$
$53$ $$6 + T$$
$59$ $$-12 + T$$
$61$ $$10 + T$$
$67$ $$-14 + T$$
$71$ $$-12 + T$$
$73$ $$-2 + T$$
$79$ $$10 + T$$
$83$ $$12 + T$$
$89$ $$18 + T$$
$97$ $$-8 + T$$