Properties

 Label 2394.2.a.v Level $2394$ Weight $2$ Character orbit 2394.a Self dual yes Analytic conductor $19.116$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
 1.61803 −0.618034
1.00000 0 1.00000 −3.23607 0 −1.00000 1.00000 0 −3.23607
1.2 1.00000 0 1.00000 1.23607 0 −1.00000 1.00000 0 1.23607
 $$n$$: 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.v yes 2
3.b odd 2 1 2394.2.a.s 2

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

Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} - 4$$ $$T_{11}^{2} + 6 T_{11} + 4$$ $$T_{13}^{2} + 2 T_{13} - 4$$ $$T_{17}^{2} - 2 T_{17} - 44$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-4 + 2 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$4 + 6 T + T^{2}$$
$13$ $$-4 + 2 T + T^{2}$$
$17$ $$-44 - 2 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$4 + 6 T + T^{2}$$
$29$ $$-4 + 8 T + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$16 + 12 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$-80 + T^{2}$$
$47$ $$-80 + T^{2}$$
$53$ $$44 + 16 T + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$-4 - 8 T + T^{2}$$
$67$ $$4 - 14 T + T^{2}$$
$71$ $$16 + 12 T + T^{2}$$
$73$ $$44 + 16 T + T^{2}$$
$79$ $$4 - 6 T + T^{2}$$
$83$ $$( -10 + T )^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$44 - 14 T + T^{2}$$