# Properties

 Label 2394.2.a.f Level $2394$ Weight $2$ Character orbit 2394.a Self dual yes Analytic conductor $19.116$ Analytic rank $0$ 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: $$0$$ 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 $$q - q^{2} + q^{4} + 4 q^{5} - q^{7} - q^{8} - 4 q^{10} + 6 q^{11} - 4 q^{13} + q^{14} + q^{16} + 4 q^{17} - q^{19} + 4 q^{20} - 6 q^{22} - 2 q^{23} + 11 q^{25} + 4 q^{26} - q^{28} - 2 q^{29} + 4 q^{31} - q^{32} - 4 q^{34} - 4 q^{35} + 2 q^{37} + q^{38} - 4 q^{40} - 6 q^{41} + 6 q^{44} + 2 q^{46} + 8 q^{47} + q^{49} - 11 q^{50} - 4 q^{52} + 14 q^{53} + 24 q^{55} + q^{56} + 2 q^{58} + 4 q^{59} - 10 q^{61} - 4 q^{62} + q^{64} - 16 q^{65} + 10 q^{67} + 4 q^{68} + 4 q^{70} + 4 q^{71} - 14 q^{73} - 2 q^{74} - q^{76} - 6 q^{77} + 2 q^{79} + 4 q^{80} + 6 q^{82} + 16 q^{85} - 6 q^{88} - 6 q^{89} + 4 q^{91} - 2 q^{92} - 8 q^{94} - 4 q^{95} - q^{98}+O(q^{100})$$ q - q^2 + q^4 + 4 * q^5 - q^7 - q^8 - 4 * q^10 + 6 * q^11 - 4 * q^13 + q^14 + q^16 + 4 * q^17 - q^19 + 4 * q^20 - 6 * q^22 - 2 * q^23 + 11 * q^25 + 4 * q^26 - q^28 - 2 * q^29 + 4 * q^31 - q^32 - 4 * q^34 - 4 * q^35 + 2 * q^37 + q^38 - 4 * q^40 - 6 * q^41 + 6 * q^44 + 2 * q^46 + 8 * q^47 + q^49 - 11 * q^50 - 4 * q^52 + 14 * q^53 + 24 * q^55 + q^56 + 2 * q^58 + 4 * q^59 - 10 * q^61 - 4 * q^62 + q^64 - 16 * q^65 + 10 * q^67 + 4 * q^68 + 4 * q^70 + 4 * q^71 - 14 * q^73 - 2 * q^74 - q^76 - 6 * q^77 + 2 * q^79 + 4 * q^80 + 6 * q^82 + 16 * q^85 - 6 * q^88 - 6 * q^89 + 4 * q^91 - 2 * q^92 - 8 * q^94 - 4 * q^95 - q^98

## 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.f 1
3.b odd 2 1 798.2.a.h 1
12.b even 2 1 6384.2.a.c 1
21.c even 2 1 5586.2.a.x 1

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

 $$T_{5} - 4$$ T5 - 4 $$T_{11} - 6$$ T11 - 6 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 4$$ T17 - 4

## Hecke characteristic polynomials

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