# Properties

 Label 2394.2.a.ba Level $2394$ Weight $2$ Character orbit 2394.a Self dual yes Analytic conductor $19.116$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.469.1 Defining polynomial: $$x^{3} - x^{2} - 5 x + 4$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -2 + \beta_{1} ) q^{5} - q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -2 + \beta_{1} ) q^{5} - q^{7} - q^{8} + ( 2 - \beta_{1} ) q^{10} + ( -2 \beta_{1} + \beta_{2} ) q^{11} + ( -2 + 2 \beta_{1} ) q^{13} + q^{14} + q^{16} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{17} - q^{19} + ( -2 + \beta_{1} ) q^{20} + ( 2 \beta_{1} - \beta_{2} ) q^{22} + 2 \beta_{1} q^{23} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{25} + ( 2 - 2 \beta_{1} ) q^{26} - q^{28} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{31} - q^{32} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{34} + ( 2 - \beta_{1} ) q^{35} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{37} + q^{38} + ( 2 - \beta_{1} ) q^{40} + ( 2 + \beta_{1} ) q^{41} + ( \beta_{1} + 2 \beta_{2} ) q^{43} + ( -2 \beta_{1} + \beta_{2} ) q^{44} -2 \beta_{1} q^{46} + ( 4 + \beta_{2} ) q^{47} + q^{49} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{50} + ( -2 + 2 \beta_{1} ) q^{52} + ( -2 - 3 \beta_{2} ) q^{53} + ( -8 + 5 \beta_{1} - 3 \beta_{2} ) q^{55} + q^{56} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{58} + 3 \beta_{1} q^{59} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{61} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{62} + q^{64} + ( 12 - 6 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{68} + ( -2 + \beta_{1} ) q^{70} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{71} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{74} - q^{76} + ( 2 \beta_{1} - \beta_{2} ) q^{77} + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{79} + ( -2 + \beta_{1} ) q^{80} + ( -2 - \beta_{1} ) q^{82} + ( 4 + 4 \beta_{1} ) q^{83} + ( 12 - 4 \beta_{1} ) q^{85} + ( -\beta_{1} - 2 \beta_{2} ) q^{86} + ( 2 \beta_{1} - \beta_{2} ) q^{88} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{89} + ( 2 - 2 \beta_{1} ) q^{91} + 2 \beta_{1} q^{92} + ( -4 - \beta_{2} ) q^{94} + ( 2 - \beta_{1} ) q^{95} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8} + O(q^{10})$$ $$3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8} + 5 q^{10} - 3 q^{11} - 4 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{19} - 5 q^{20} + 3 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{26} - 3 q^{28} - 5 q^{29} + 4 q^{31} - 3 q^{32} + 6 q^{34} + 5 q^{35} + 7 q^{37} + 3 q^{38} + 5 q^{40} + 7 q^{41} - q^{43} - 3 q^{44} - 2 q^{46} + 11 q^{47} + 3 q^{49} - 4 q^{50} - 4 q^{52} - 3 q^{53} - 16 q^{55} + 3 q^{56} + 5 q^{58} + 3 q^{59} + 7 q^{61} - 4 q^{62} + 3 q^{64} + 28 q^{65} + 12 q^{67} - 6 q^{68} - 5 q^{70} - 9 q^{71} - 7 q^{74} - 3 q^{76} + 3 q^{77} + 15 q^{79} - 5 q^{80} - 7 q^{82} + 16 q^{83} + 32 q^{85} + q^{86} + 3 q^{88} + 3 q^{89} + 4 q^{91} + 2 q^{92} - 11 q^{94} + 5 q^{95} - 5 q^{97} - 3 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

## 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
 −2.16425 0.772866 2.39138
−1.00000 0 1.00000 −4.16425 0 −1.00000 −1.00000 0 4.16425
1.2 −1.00000 0 1.00000 −1.22713 0 −1.00000 −1.00000 0 1.22713
1.3 −1.00000 0 1.00000 0.391382 0 −1.00000 −1.00000 0 −0.391382
 $$n$$: 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.ba 3
3.b odd 2 1 266.2.a.d 3
12.b even 2 1 2128.2.a.s 3
15.d odd 2 1 6650.2.a.cd 3
21.c even 2 1 1862.2.a.r 3
24.f even 2 1 8512.2.a.bj 3
24.h odd 2 1 8512.2.a.bm 3
57.d even 2 1 5054.2.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.a.d 3 3.b odd 2 1
1862.2.a.r 3 21.c even 2 1
2128.2.a.s 3 12.b even 2 1
2394.2.a.ba 3 1.a even 1 1 trivial
5054.2.a.r 3 57.d even 2 1
6650.2.a.cd 3 15.d odd 2 1
8512.2.a.bj 3 24.f even 2 1
8512.2.a.bm 3 24.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{3} + 5 T_{5}^{2} + 3 T_{5} - 2$$ $$T_{11}^{3} + 3 T_{11}^{2} - 25 T_{11} - 76$$ $$T_{13}^{3} + 4 T_{13}^{2} - 16 T_{13} - 8$$ $$T_{17}^{3} + 6 T_{17}^{2} - 40 T_{17} - 224$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$T^{3}$$
$5$ $$-2 + 3 T + 5 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-76 - 25 T + 3 T^{2} + T^{3}$$
$13$ $$-8 - 16 T + 4 T^{2} + T^{3}$$
$17$ $$-224 - 40 T + 6 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$32 - 20 T - 2 T^{2} + T^{3}$$
$29$ $$-38 - 27 T + 5 T^{2} + T^{3}$$
$31$ $$64 - 44 T - 4 T^{2} + T^{3}$$
$37$ $$86 - 19 T - 7 T^{2} + T^{3}$$
$41$ $$2 + 11 T - 7 T^{2} + T^{3}$$
$43$ $$-28 - 35 T + T^{2} + T^{3}$$
$47$ $$-16 + 33 T - 11 T^{2} + T^{3}$$
$53$ $$-238 - 63 T + 3 T^{2} + T^{3}$$
$59$ $$108 - 45 T - 3 T^{2} + T^{3}$$
$61$ $$2 - 13 T - 7 T^{2} + T^{3}$$
$67$ $$16 - 4 T - 12 T^{2} + T^{3}$$
$71$ $$-8 - T + 9 T^{2} + T^{3}$$
$73$ $$-392 - 112 T + T^{3}$$
$79$ $$-16 + 41 T - 15 T^{2} + T^{3}$$
$83$ $$448 - 16 T^{2} + T^{3}$$
$89$ $$-22 - 25 T - 3 T^{2} + T^{3}$$
$97$ $$-98 - 21 T + 5 T^{2} + T^{3}$$