Properties

 Label 2394.2.a.bb Level $2394$ Weight $2$ Character orbit 2394.a Self dual yes Analytic conductor $19.116$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -1 - \beta_{1} ) q^{5} - q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -1 - \beta_{1} ) q^{5} - q^{7} - q^{8} + ( 1 + \beta_{1} ) q^{10} + ( -2 - \beta_{2} ) q^{11} + ( 3 + \beta_{1} ) q^{13} + q^{14} + q^{16} + \beta_{2} q^{17} - q^{19} + ( -1 - \beta_{1} ) q^{20} + ( 2 + \beta_{2} ) q^{22} + ( -1 + \beta_{1} ) q^{23} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{25} + ( -3 - \beta_{1} ) q^{26} - q^{28} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -1 - \beta_{1} - \beta_{2} ) q^{31} - q^{32} -\beta_{2} q^{34} + ( 1 + \beta_{1} ) q^{35} + ( 1 - \beta_{1} - \beta_{2} ) q^{37} + q^{38} + ( 1 + \beta_{1} ) q^{40} + 2 \beta_{1} q^{41} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -2 - \beta_{2} ) q^{44} + ( 1 - \beta_{1} ) q^{46} + ( -5 + \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( -5 - 2 \beta_{1} - 2 \beta_{2} ) q^{50} + ( 3 + \beta_{1} ) q^{52} -2 q^{53} + 4 \beta_{1} q^{55} + q^{56} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 1 + \beta_{1} + \beta_{2} ) q^{62} + q^{64} + ( -12 - 4 \beta_{1} - 2 \beta_{2} ) q^{65} + \beta_{2} q^{67} + \beta_{2} q^{68} + ( -1 - \beta_{1} ) q^{70} + ( -8 - 2 \beta_{2} ) q^{71} + ( -2 - 2 \beta_{2} ) q^{73} + ( -1 + \beta_{1} + \beta_{2} ) q^{74} - q^{76} + ( 2 + \beta_{2} ) q^{77} + ( -7 - 3 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -1 - \beta_{1} ) q^{80} -2 \beta_{1} q^{82} + ( -4 + 2 \beta_{1} ) q^{83} + ( 2 - 2 \beta_{1} ) q^{85} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{86} + ( 2 + \beta_{2} ) q^{88} + ( -6 + 2 \beta_{2} ) q^{89} + ( -3 - \beta_{1} ) q^{91} + ( -1 + \beta_{1} ) q^{92} + ( 5 - \beta_{1} + \beta_{2} ) q^{94} + ( 1 + \beta_{1} ) q^{95} + ( 2 \beta_{1} - \beta_{2} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} - 3 q^{7} - 3 q^{8} + O(q^{10})$$ $$3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} - 3 q^{7} - 3 q^{8} + 2 q^{10} - 6 q^{11} + 8 q^{13} + 3 q^{14} + 3 q^{16} - 3 q^{19} - 2 q^{20} + 6 q^{22} - 4 q^{23} + 13 q^{25} - 8 q^{26} - 3 q^{28} - 2 q^{29} - 2 q^{31} - 3 q^{32} + 2 q^{35} + 4 q^{37} + 3 q^{38} + 2 q^{40} - 2 q^{41} - 4 q^{43} - 6 q^{44} + 4 q^{46} - 16 q^{47} + 3 q^{49} - 13 q^{50} + 8 q^{52} - 6 q^{53} - 4 q^{55} + 3 q^{56} + 2 q^{58} + 10 q^{61} + 2 q^{62} + 3 q^{64} - 32 q^{65} - 2 q^{70} - 24 q^{71} - 6 q^{73} - 4 q^{74} - 3 q^{76} + 6 q^{77} - 18 q^{79} - 2 q^{80} + 2 q^{82} - 14 q^{83} + 8 q^{85} + 4 q^{86} + 6 q^{88} - 18 q^{89} - 8 q^{91} - 4 q^{92} + 16 q^{94} + 2 q^{95} - 2 q^{97} - 3 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 5$$$$)/2$$

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
 2.17009 0.311108 −1.48119
−1.00000 0 1.00000 −4.34017 0 −1.00000 −1.00000 0 4.34017
1.2 −1.00000 0 1.00000 −0.622216 0 −1.00000 −1.00000 0 0.622216
1.3 −1.00000 0 1.00000 2.96239 0 −1.00000 −1.00000 0 −2.96239
 $$n$$: 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.bb 3
3.b odd 2 1 2394.2.a.bc yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.a.bb 3 1.a even 1 1 trivial
2394.2.a.bc yes 3 3.b odd 2 1

Hecke kernels

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

 $$T_{5}^{3} + 2 T_{5}^{2} - 12 T_{5} - 8$$ $$T_{11}^{3} + 6 T_{11}^{2} - 4 T_{11} - 40$$ $$T_{13}^{3} - 8 T_{13}^{2} + 8 T_{13} + 16$$ $$T_{17}^{3} - 16 T_{17} + 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$T^{3}$$
$5$ $$-8 - 12 T + 2 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-40 - 4 T + 6 T^{2} + T^{3}$$
$13$ $$16 + 8 T - 8 T^{2} + T^{3}$$
$17$ $$16 - 16 T + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$-16 - 8 T + 4 T^{2} + T^{3}$$
$29$ $$-104 - 84 T + 2 T^{2} + T^{3}$$
$31$ $$-8 - 20 T + 2 T^{2} + T^{3}$$
$37$ $$32 - 16 T - 4 T^{2} + T^{3}$$
$41$ $$-40 - 52 T + 2 T^{2} + T^{3}$$
$43$ $$-64 - 80 T + 4 T^{2} + T^{3}$$
$47$ $$32 + 48 T + 16 T^{2} + T^{3}$$
$53$ $$( 2 + T )^{3}$$
$59$ $$T^{3}$$
$61$ $$200 - 52 T - 10 T^{2} + T^{3}$$
$67$ $$16 - 16 T + T^{3}$$
$71$ $$-128 + 128 T + 24 T^{2} + T^{3}$$
$73$ $$-248 - 52 T + 6 T^{2} + T^{3}$$
$79$ $$-2536 - 124 T + 18 T^{2} + T^{3}$$
$83$ $$-152 + 12 T + 14 T^{2} + T^{3}$$
$89$ $$-40 + 44 T + 18 T^{2} + T^{3}$$
$97$ $$232 - 84 T + 2 T^{2} + T^{3}$$