# Properties

 Label 2394.2.f.b Level $2394$ Weight $2$ Character orbit 2394.f Analytic conductor $19.116$ Analytic rank $0$ Dimension $24$ 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.f (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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{4} + 4 q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{4} + 4 q^{7} + 4 q^{14} + 24 q^{16} - 32 q^{17} - 8 q^{22} + 16 q^{25} - 4 q^{28} + 24 q^{35} - 24 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 16 q^{49} - 4 q^{56} + 16 q^{58} - 16 q^{59} + 16 q^{62} - 24 q^{64} - 24 q^{67} + 32 q^{68} - 16 q^{70} - 8 q^{77} - 40 q^{79} + 64 q^{83} + 40 q^{85} + 8 q^{88} + 64 q^{89} + 8 q^{91} + 16 q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2015.1 1.00000i 0 −1.00000 1.90763 0 −0.638569 2.56753i 1.00000i 0 1.90763i
2015.2 1.00000i 0 −1.00000 1.90763 0 −0.638569 + 2.56753i 1.00000i 0 1.90763i
2015.3 1.00000i 0 −1.00000 4.23211 0 2.64176 0.145265i 1.00000i 0 4.23211i
2015.4 1.00000i 0 −1.00000 4.23211 0 2.64176 + 0.145265i 1.00000i 0 4.23211i
2015.5 1.00000i 0 −1.00000 −3.41122 0 0.143354 + 2.64186i 1.00000i 0 3.41122i
2015.6 1.00000i 0 −1.00000 −3.41122 0 0.143354 2.64186i 1.00000i 0 3.41122i
2015.7 1.00000i 0 −1.00000 −1.26222 0 1.09287 2.40949i 1.00000i 0 1.26222i
2015.8 1.00000i 0 −1.00000 −1.26222 0 1.09287 + 2.40949i 1.00000i 0 1.26222i
2015.9 1.00000i 0 −1.00000 −1.42576 0 2.62657 + 0.317999i 1.00000i 0 1.42576i
2015.10 1.00000i 0 −1.00000 −1.42576 0 2.62657 0.317999i 1.00000i 0 1.42576i
2015.11 1.00000i 0 −1.00000 1.62775 0 −2.31472 + 1.28144i 1.00000i 0 1.62775i
2015.12 1.00000i 0 −1.00000 1.62775 0 −2.31472 1.28144i 1.00000i 0 1.62775i
2015.13 1.00000i 0 −1.00000 0.335231 0 −2.35234 + 1.21100i 1.00000i 0 0.335231i
2015.14 1.00000i 0 −1.00000 0.335231 0 −2.35234 1.21100i 1.00000i 0 0.335231i
2015.15 1.00000i 0 −1.00000 −3.11350 0 −0.672508 + 2.55885i 1.00000i 0 3.11350i
2015.16 1.00000i 0 −1.00000 −3.11350 0 −0.672508 2.55885i 1.00000i 0 3.11350i
2015.17 1.00000i 0 −1.00000 −2.79254 0 −1.85020 1.89123i 1.00000i 0 2.79254i
2015.18 1.00000i 0 −1.00000 −2.79254 0 −1.85020 + 1.89123i 1.00000i 0 2.79254i
2015.19 1.00000i 0 −1.00000 3.20606 0 1.74307 + 1.99040i 1.00000i 0 3.20606i
2015.20 1.00000i 0 −1.00000 3.20606 0 1.74307 1.99040i 1.00000i 0 3.20606i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2015.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c 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.f.b yes 24
3.b odd 2 1 2394.2.f.a 24
7.b odd 2 1 2394.2.f.a 24
21.c even 2 1 inner 2394.2.f.b yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.f.a 24 3.b odd 2 1
2394.2.f.a 24 7.b odd 2 1
2394.2.f.b yes 24 1.a even 1 1 trivial
2394.2.f.b yes 24 21.c even 2 1 inner