# Properties

 Label 2394.2.e.d Level $2394$ Weight $2$ Character orbit 2394.e Analytic conductor $19.116$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$19.1161862439$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{4} - 8 q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{4} - 8 q^{7} + 24 q^{16} + 24 q^{25} + 8 q^{28} + 32 q^{43} - 24 q^{49} + 16 q^{58} - 24 q^{64} - 64 q^{85} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1063.1 1.00000i 0 −1.00000 0.575772i 0 −2.19869 + 1.47165i 1.00000i 0 −0.575772
1063.2 1.00000i 0 −1.00000 0.575772i 0 −2.19869 + 1.47165i 1.00000i 0 0.575772
1063.3 1.00000i 0 −1.00000 0.575772i 0 −2.19869 1.47165i 1.00000i 0 −0.575772
1063.4 1.00000i 0 −1.00000 0.575772i 0 −2.19869 1.47165i 1.00000i 0 0.575772
1063.5 1.00000i 0 −1.00000 1.63913i 0 1.91223 + 1.82849i 1.00000i 0 −1.63913
1063.6 1.00000i 0 −1.00000 1.63913i 0 1.91223 + 1.82849i 1.00000i 0 1.63913
1063.7 1.00000i 0 −1.00000 1.63913i 0 1.91223 1.82849i 1.00000i 0 −1.63913
1063.8 1.00000i 0 −1.00000 1.63913i 0 1.91223 1.82849i 1.00000i 0 1.63913
1063.9 1.00000i 0 −1.00000 2.99695i 0 −0.713538 2.54772i 1.00000i 0 2.99695
1063.10 1.00000i 0 −1.00000 2.99695i 0 −0.713538 2.54772i 1.00000i 0 −2.99695
1063.11 1.00000i 0 −1.00000 2.99695i 0 −0.713538 + 2.54772i 1.00000i 0 2.99695
1063.12 1.00000i 0 −1.00000 2.99695i 0 −0.713538 + 2.54772i 1.00000i 0 −2.99695
1063.13 1.00000i 0 −1.00000 1.63913i 0 1.91223 1.82849i 1.00000i 0 1.63913
1063.14 1.00000i 0 −1.00000 1.63913i 0 1.91223 1.82849i 1.00000i 0 −1.63913
1063.15 1.00000i 0 −1.00000 1.63913i 0 1.91223 + 1.82849i 1.00000i 0 1.63913
1063.16 1.00000i 0 −1.00000 1.63913i 0 1.91223 + 1.82849i 1.00000i 0 −1.63913
1063.17 1.00000i 0 −1.00000 2.99695i 0 −0.713538 + 2.54772i 1.00000i 0 −2.99695
1063.18 1.00000i 0 −1.00000 2.99695i 0 −0.713538 + 2.54772i 1.00000i 0 2.99695
1063.19 1.00000i 0 −1.00000 2.99695i 0 −0.713538 2.54772i 1.00000i 0 −2.99695
1063.20 1.00000i 0 −1.00000 2.99695i 0 −0.713538 2.54772i 1.00000i 0 2.99695
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1063.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
19.b odd 2 1 inner
21.c even 2 1 inner
57.d even 2 1 inner
133.c even 2 1 inner
399.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.e.d 24
3.b odd 2 1 inner 2394.2.e.d 24
7.b odd 2 1 inner 2394.2.e.d 24
19.b odd 2 1 inner 2394.2.e.d 24
21.c even 2 1 inner 2394.2.e.d 24
57.d even 2 1 inner 2394.2.e.d 24
133.c even 2 1 inner 2394.2.e.d 24
399.h odd 2 1 inner 2394.2.e.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.e.d 24 1.a even 1 1 trivial
2394.2.e.d 24 3.b odd 2 1 inner
2394.2.e.d 24 7.b odd 2 1 inner
2394.2.e.d 24 19.b odd 2 1 inner
2394.2.e.d 24 21.c even 2 1 inner
2394.2.e.d 24 57.d even 2 1 inner
2394.2.e.d 24 133.c even 2 1 inner
2394.2.e.d 24 399.h odd 2 1 inner

## Hecke kernels

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

 $$T_{5}^{6} + 12 T_{5}^{4} + 28 T_{5}^{2} + 8$$ $$T_{13}^{6} - 44 T_{13}^{4} + 140 T_{13}^{2} - 72$$