# Properties

 Label 2394.2.a.y 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: $$\Q(\sqrt{29})$$ Defining polynomial: $$x^{2} - x - 7$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{29})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta q^{5} - q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + \beta q^{5} - q^{7} + q^{8} + \beta q^{10} + ( -1 - \beta ) q^{11} + ( -2 + 2 \beta ) q^{13} - q^{14} + q^{16} + 4 q^{17} + q^{19} + \beta q^{20} + ( -1 - \beta ) q^{22} + ( 4 - 2 \beta ) q^{23} + ( 2 + \beta ) q^{25} + ( -2 + 2 \beta ) q^{26} - q^{28} + ( -2 - \beta ) q^{29} + 10 q^{31} + q^{32} + 4 q^{34} -\beta q^{35} + ( 2 - \beta ) q^{37} + q^{38} + \beta q^{40} + 3 \beta q^{41} + ( 8 - \beta ) q^{43} + ( -1 - \beta ) q^{44} + ( 4 - 2 \beta ) q^{46} + ( 7 + \beta ) q^{47} + q^{49} + ( 2 + \beta ) q^{50} + ( -2 + 2 \beta ) q^{52} + ( 3 - \beta ) q^{53} + ( -7 - 2 \beta ) q^{55} - q^{56} + ( -2 - \beta ) q^{58} + ( -2 - 3 \beta ) q^{59} + ( -3 - \beta ) q^{61} + 10 q^{62} + q^{64} + 14 q^{65} -2 q^{67} + 4 q^{68} -\beta q^{70} + ( -1 - \beta ) q^{71} + 2 \beta q^{73} + ( 2 - \beta ) q^{74} + q^{76} + ( 1 + \beta ) q^{77} + ( 4 + \beta ) q^{79} + \beta q^{80} + 3 \beta q^{82} -8 q^{83} + 4 \beta q^{85} + ( 8 - \beta ) q^{86} + ( -1 - \beta ) q^{88} + ( -9 - 3 \beta ) q^{89} + ( 2 - 2 \beta ) q^{91} + ( 4 - 2 \beta ) q^{92} + ( 7 + \beta ) q^{94} + \beta q^{95} + ( 11 + \beta ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + q^{10} - 3 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 8 q^{17} + 2 q^{19} + q^{20} - 3 q^{22} + 6 q^{23} + 5 q^{25} - 2 q^{26} - 2 q^{28} - 5 q^{29} + 20 q^{31} + 2 q^{32} + 8 q^{34} - q^{35} + 3 q^{37} + 2 q^{38} + q^{40} + 3 q^{41} + 15 q^{43} - 3 q^{44} + 6 q^{46} + 15 q^{47} + 2 q^{49} + 5 q^{50} - 2 q^{52} + 5 q^{53} - 16 q^{55} - 2 q^{56} - 5 q^{58} - 7 q^{59} - 7 q^{61} + 20 q^{62} + 2 q^{64} + 28 q^{65} - 4 q^{67} + 8 q^{68} - q^{70} - 3 q^{71} + 2 q^{73} + 3 q^{74} + 2 q^{76} + 3 q^{77} + 9 q^{79} + q^{80} + 3 q^{82} - 16 q^{83} + 4 q^{85} + 15 q^{86} - 3 q^{88} - 21 q^{89} + 2 q^{91} + 6 q^{92} + 15 q^{94} + q^{95} + 23 q^{97} + 2 q^{98} + O(q^{100})$$

## 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
 −2.19258 3.19258
1.00000 0 1.00000 −2.19258 0 −1.00000 1.00000 0 −2.19258
1.2 1.00000 0 1.00000 3.19258 0 −1.00000 1.00000 0 3.19258
 $$n$$: 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.y 2
3.b odd 2 1 266.2.a.a 2
12.b even 2 1 2128.2.a.g 2
15.d odd 2 1 6650.2.a.bu 2
21.c even 2 1 1862.2.a.h 2
24.f even 2 1 8512.2.a.w 2
24.h odd 2 1 8512.2.a.p 2
57.d even 2 1 5054.2.a.n 2

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

 $$T_{5}^{2} - T_{5} - 7$$ $$T_{11}^{2} + 3 T_{11} - 5$$ $$T_{13}^{2} + 2 T_{13} - 28$$ $$T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-7 - T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-5 + 3 T + T^{2}$$
$13$ $$-28 + 2 T + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-20 - 6 T + T^{2}$$
$29$ $$-1 + 5 T + T^{2}$$
$31$ $$( -10 + T )^{2}$$
$37$ $$-5 - 3 T + T^{2}$$
$41$ $$-63 - 3 T + T^{2}$$
$43$ $$49 - 15 T + T^{2}$$
$47$ $$49 - 15 T + T^{2}$$
$53$ $$-1 - 5 T + T^{2}$$
$59$ $$-53 + 7 T + T^{2}$$
$61$ $$5 + 7 T + T^{2}$$
$67$ $$( 2 + T )^{2}$$
$71$ $$-5 + 3 T + T^{2}$$
$73$ $$-28 - 2 T + T^{2}$$
$79$ $$13 - 9 T + T^{2}$$
$83$ $$( 8 + T )^{2}$$
$89$ $$45 + 21 T + T^{2}$$
$97$ $$125 - 23 T + T^{2}$$