# Properties

 Label 2394.2.b.e Level $2394$ Weight $2$ Character orbit 2394.b Analytic conductor $19.116$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2394.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1161862439$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2 \beta q^{5} + q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + 2 \beta q^{5} + q^{7} + q^{8} + 2 \beta q^{10} -4 \beta q^{11} -3 \beta q^{13} + q^{14} + q^{16} -4 \beta q^{17} + ( 1 - 3 \beta ) q^{19} + 2 \beta q^{20} -4 \beta q^{22} -4 \beta q^{23} -3 q^{25} -3 \beta q^{26} + q^{28} -6 q^{29} + 6 \beta q^{31} + q^{32} -4 \beta q^{34} + 2 \beta q^{35} + ( 1 - 3 \beta ) q^{38} + 2 \beta q^{40} -4 q^{43} -4 \beta q^{44} -4 \beta q^{46} -7 \beta q^{47} + q^{49} -3 q^{50} -3 \beta q^{52} -6 q^{53} + 16 q^{55} + q^{56} -6 q^{58} + 12 q^{59} + 2 q^{61} + 6 \beta q^{62} + q^{64} + 12 q^{65} + 9 \beta q^{67} -4 \beta q^{68} + 2 \beta q^{70} + 6 q^{71} + 14 q^{73} + ( 1 - 3 \beta ) q^{76} -4 \beta q^{77} + 9 \beta q^{79} + 2 \beta q^{80} + 11 \beta q^{83} + 16 q^{85} -4 q^{86} -4 \beta q^{88} -6 q^{89} -3 \beta q^{91} -4 \beta q^{92} -7 \beta q^{94} + ( 12 + 2 \beta ) q^{95} -3 \beta q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 2 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{25} + 2 q^{28} - 12 q^{29} + 2 q^{32} + 2 q^{38} - 8 q^{43} + 2 q^{49} - 6 q^{50} - 12 q^{53} + 32 q^{55} + 2 q^{56} - 12 q^{58} + 24 q^{59} + 4 q^{61} + 2 q^{64} + 24 q^{65} + 12 q^{71} + 28 q^{73} + 2 q^{76} + 32 q^{85} - 8 q^{86} - 12 q^{89} + 24 q^{95} + 2 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$$.

 $$n$$ $$533$$ $$1009$$ $$1711$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1709.1
 − 1.41421i 1.41421i
1.00000 0 1.00000 2.82843i 0 1.00000 1.00000 0 2.82843i
1709.2 1.00000 0 1.00000 2.82843i 0 1.00000 1.00000 0 2.82843i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.b.e yes 2
3.b odd 2 1 2394.2.b.b 2
19.b odd 2 1 2394.2.b.b 2
57.d even 2 1 inner 2394.2.b.e yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.b 2 3.b odd 2 1
2394.2.b.b 2 19.b odd 2 1
2394.2.b.e yes 2 1.a even 1 1 trivial
2394.2.b.e yes 2 57.d even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} + 8$$ $$T_{29} + 6$$ $$T_{53} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$8 + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$32 + T^{2}$$
$13$ $$18 + T^{2}$$
$17$ $$32 + T^{2}$$
$19$ $$19 - 2 T + T^{2}$$
$23$ $$32 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$72 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$98 + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$162 + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$( -14 + T )^{2}$$
$79$ $$162 + T^{2}$$
$83$ $$242 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$18 + T^{2}$$