# Properties

 Label 2394.2.a.q 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: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 266) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} -\beta q^{5} + q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} -\beta q^{5} + q^{7} - q^{8} + \beta q^{10} + ( 3 - \beta ) q^{11} + ( 2 + 2 \beta ) q^{13} - q^{14} + q^{16} + q^{19} -\beta q^{20} + ( -3 + \beta ) q^{22} + 2 \beta q^{23} + ( -2 + \beta ) q^{25} + ( -2 - 2 \beta ) q^{26} + q^{28} + ( 6 - 3 \beta ) q^{29} + ( 2 - 4 \beta ) q^{31} - q^{32} -\beta q^{35} + ( 2 - 3 \beta ) q^{37} - q^{38} + \beta q^{40} -\beta q^{41} + ( -4 - \beta ) q^{43} + ( 3 - \beta ) q^{44} -2 \beta q^{46} + ( -3 + 3 \beta ) q^{47} + q^{49} + ( 2 - \beta ) q^{50} + ( 2 + 2 \beta ) q^{52} + ( -3 + 5 \beta ) q^{53} + ( 3 - 2 \beta ) q^{55} - q^{56} + ( -6 + 3 \beta ) q^{58} + ( 6 + \beta ) q^{59} + ( -1 - 3 \beta ) q^{61} + ( -2 + 4 \beta ) q^{62} + q^{64} + ( -6 - 4 \beta ) q^{65} + ( -10 + 4 \beta ) q^{67} + \beta q^{70} + ( 9 - 3 \beta ) q^{71} + ( 8 + 2 \beta ) q^{73} + ( -2 + 3 \beta ) q^{74} + q^{76} + ( 3 - \beta ) q^{77} + ( 8 + 3 \beta ) q^{79} -\beta q^{80} + \beta q^{82} + 4 \beta q^{83} + ( 4 + \beta ) q^{86} + ( -3 + \beta ) q^{88} + ( -9 - 3 \beta ) q^{89} + ( 2 + 2 \beta ) q^{91} + 2 \beta q^{92} + ( 3 - 3 \beta ) q^{94} -\beta q^{95} + ( 11 + \beta ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} - 2 q^{8} + q^{10} + 5 q^{11} + 6 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{19} - q^{20} - 5 q^{22} + 2 q^{23} - 3 q^{25} - 6 q^{26} + 2 q^{28} + 9 q^{29} - 2 q^{32} - q^{35} + q^{37} - 2 q^{38} + q^{40} - q^{41} - 9 q^{43} + 5 q^{44} - 2 q^{46} - 3 q^{47} + 2 q^{49} + 3 q^{50} + 6 q^{52} - q^{53} + 4 q^{55} - 2 q^{56} - 9 q^{58} + 13 q^{59} - 5 q^{61} + 2 q^{64} - 16 q^{65} - 16 q^{67} + q^{70} + 15 q^{71} + 18 q^{73} - q^{74} + 2 q^{76} + 5 q^{77} + 19 q^{79} - q^{80} + q^{82} + 4 q^{83} + 9 q^{86} - 5 q^{88} - 21 q^{89} + 6 q^{91} + 2 q^{92} + 3 q^{94} - q^{95} + 23 q^{97} - 2 q^{98} + O(q^{100})$$

## 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
 2.30278 −1.30278
−1.00000 0 1.00000 −2.30278 0 1.00000 −1.00000 0 2.30278
1.2 −1.00000 0 1.00000 1.30278 0 1.00000 −1.00000 0 −1.30278
 $$n$$: 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.q 2
3.b odd 2 1 266.2.a.c 2
12.b even 2 1 2128.2.a.h 2
15.d odd 2 1 6650.2.a.bl 2
21.c even 2 1 1862.2.a.l 2
24.f even 2 1 8512.2.a.u 2
24.h odd 2 1 8512.2.a.n 2
57.d even 2 1 5054.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.a.c 2 3.b odd 2 1
1862.2.a.l 2 21.c even 2 1
2128.2.a.h 2 12.b even 2 1
2394.2.a.q 2 1.a even 1 1 trivial
5054.2.a.e 2 57.d even 2 1
6650.2.a.bl 2 15.d odd 2 1
8512.2.a.n 2 24.h odd 2 1
8512.2.a.u 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2394))$$:

 $$T_{5}^{2} + T_{5} - 3$$ $$T_{11}^{2} - 5 T_{11} + 3$$ $$T_{13}^{2} - 6 T_{13} - 4$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-3 + T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$3 - 5 T + T^{2}$$
$13$ $$-4 - 6 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-12 - 2 T + T^{2}$$
$29$ $$-9 - 9 T + T^{2}$$
$31$ $$-52 + T^{2}$$
$37$ $$-29 - T + T^{2}$$
$41$ $$-3 + T + T^{2}$$
$43$ $$17 + 9 T + T^{2}$$
$47$ $$-27 + 3 T + T^{2}$$
$53$ $$-81 + T + T^{2}$$
$59$ $$39 - 13 T + T^{2}$$
$61$ $$-23 + 5 T + T^{2}$$
$67$ $$12 + 16 T + T^{2}$$
$71$ $$27 - 15 T + T^{2}$$
$73$ $$68 - 18 T + T^{2}$$
$79$ $$61 - 19 T + T^{2}$$
$83$ $$-48 - 4 T + T^{2}$$
$89$ $$81 + 21 T + T^{2}$$
$97$ $$129 - 23 T + T^{2}$$