# Properties

 Label 2394.2.a.w 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 −3.85410 0 1.00000 1.00000 0 −3.85410
1.2 1.00000 0 1.00000 2.85410 0 1.00000 1.00000 0 2.85410
 $$n$$: 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.w 2
3.b odd 2 1 266.2.a.b 2
12.b even 2 1 2128.2.a.b 2
15.d odd 2 1 6650.2.a.bq 2
21.c even 2 1 1862.2.a.g 2
24.f even 2 1 8512.2.a.bc 2
24.h odd 2 1 8512.2.a.h 2
57.d even 2 1 5054.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.a.b 2 3.b odd 2 1
1862.2.a.g 2 21.c even 2 1
2128.2.a.b 2 12.b even 2 1
2394.2.a.w 2 1.a even 1 1 trivial
5054.2.a.k 2 57.d even 2 1
6650.2.a.bq 2 15.d odd 2 1
8512.2.a.h 2 24.h odd 2 1
8512.2.a.bc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2394))$$:

 $$T_{5}^{2} + T_{5} - 11$$ $$T_{11}^{2} + 7 T_{11} + 11$$ $$T_{13}^{2} - 6 T_{13} + 4$$ $$T_{17}^{2} - 4 T_{17} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-11 + T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$11 + 7 T + T^{2}$$
$13$ $$4 - 6 T + T^{2}$$
$17$ $$-16 - 4 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-44 + 2 T + T^{2}$$
$29$ $$19 - 11 T + T^{2}$$
$31$ $$-4 - 8 T + T^{2}$$
$37$ $$-9 - 3 T + T^{2}$$
$41$ $$-55 - 5 T + T^{2}$$
$43$ $$41 + 13 T + T^{2}$$
$47$ $$-11 - T + T^{2}$$
$53$ $$155 - 25 T + T^{2}$$
$59$ $$19 - 11 T + T^{2}$$
$61$ $$-31 - 11 T + T^{2}$$
$67$ $$-76 - 4 T + T^{2}$$
$71$ $$-209 + 3 T + T^{2}$$
$73$ $$20 + 10 T + T^{2}$$
$79$ $$-31 + 11 T + T^{2}$$
$83$ $$-80 + T^{2}$$
$89$ $$5 - 5 T + T^{2}$$
$97$ $$-11 - T + T^{2}$$