# Properties

 Label 273.2.g.a Level $273$ Weight $2$ Character orbit 273.g Analytic conductor $2.180$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} - 16q^{16} - 16q^{25} + 16q^{30} - 32q^{36} - 48q^{42} + 48q^{43} - 32q^{49} - 16q^{51} - 80q^{64} + 32q^{78} + 80q^{79} - 48q^{81} - 96q^{88} + 32q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
272.1 −2.35085 −0.813690 1.52902i 3.52651 0.514868i 1.91287 + 3.59451i −1.49028 + 2.18611i −3.58861 −1.67582 + 2.48830i 1.21038i
272.2 −2.35085 −0.813690 + 1.52902i 3.52651 0.514868i 1.91287 3.59451i −1.49028 2.18611i −3.58861 −1.67582 2.48830i 1.21038i
272.3 −2.35085 0.813690 1.52902i 3.52651 0.514868i −1.91287 + 3.59451i 1.49028 2.18611i −3.58861 −1.67582 2.48830i 1.21038i
272.4 −2.35085 0.813690 + 1.52902i 3.52651 0.514868i −1.91287 3.59451i 1.49028 + 2.18611i −3.58861 −1.67582 + 2.48830i 1.21038i
272.5 −1.77583 −1.65032 0.525794i 1.15356 3.85624i 2.93067 + 0.933719i −1.56982 2.12971i 1.50313 2.44708 + 1.73545i 6.84801i
272.6 −1.77583 −1.65032 + 0.525794i 1.15356 3.85624i 2.93067 0.933719i −1.56982 + 2.12971i 1.50313 2.44708 1.73545i 6.84801i
272.7 −1.77583 1.65032 0.525794i 1.15356 3.85624i −2.93067 + 0.933719i 1.56982 + 2.12971i 1.50313 2.44708 1.73545i 6.84801i
272.8 −1.77583 1.65032 + 0.525794i 1.15356 3.85624i −2.93067 0.933719i 1.56982 2.12971i 1.50313 2.44708 + 1.73545i 6.84801i
272.9 −1.10741 −0.669305 1.59751i −0.773652 1.62666i 0.741192 + 1.76909i 2.53337 + 0.762910i 3.07156 −2.10406 + 2.13844i 1.80138i
272.10 −1.10741 −0.669305 + 1.59751i −0.773652 1.62666i 0.741192 1.76909i 2.53337 0.762910i 3.07156 −2.10406 2.13844i 1.80138i
272.11 −1.10741 0.669305 1.59751i −0.773652 1.62666i −0.741192 + 1.76909i −2.53337 0.762910i 3.07156 −2.10406 2.13844i 1.80138i
272.12 −1.10741 0.669305 + 1.59751i −0.773652 1.62666i −0.741192 1.76909i −2.53337 + 0.762910i 3.07156 −2.10406 + 2.13844i 1.80138i
272.13 −0.305902 −1.47187 0.913018i −1.90642 2.05385i 0.450247 + 0.279294i −0.946976 2.47047i 1.19498 1.33280 + 2.68769i 0.628276i
272.14 −0.305902 −1.47187 + 0.913018i −1.90642 2.05385i 0.450247 0.279294i −0.946976 + 2.47047i 1.19498 1.33280 2.68769i 0.628276i
272.15 −0.305902 1.47187 0.913018i −1.90642 2.05385i −0.450247 + 0.279294i 0.946976 + 2.47047i 1.19498 1.33280 2.68769i 0.628276i
272.16 −0.305902 1.47187 + 0.913018i −1.90642 2.05385i −0.450247 0.279294i 0.946976 2.47047i 1.19498 1.33280 + 2.68769i 0.628276i
272.17 0.305902 −1.47187 0.913018i −1.90642 2.05385i −0.450247 0.279294i 0.946976 + 2.47047i −1.19498 1.33280 + 2.68769i 0.628276i
272.18 0.305902 −1.47187 + 0.913018i −1.90642 2.05385i −0.450247 + 0.279294i 0.946976 2.47047i −1.19498 1.33280 2.68769i 0.628276i
272.19 0.305902 1.47187 0.913018i −1.90642 2.05385i 0.450247 0.279294i −0.946976 2.47047i −1.19498 1.33280 2.68769i 0.628276i
272.20 0.305902 1.47187 + 0.913018i −1.90642 2.05385i 0.450247 + 0.279294i −0.946976 + 2.47047i −1.19498 1.33280 + 2.68769i 0.628276i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 272.32 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
13.b even 2 1 inner
21.c even 2 1 inner
39.d odd 2 1 inner
91.b odd 2 1 inner
273.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.g.a 32
3.b odd 2 1 inner 273.2.g.a 32
7.b odd 2 1 inner 273.2.g.a 32
13.b even 2 1 inner 273.2.g.a 32
21.c even 2 1 inner 273.2.g.a 32
39.d odd 2 1 inner 273.2.g.a 32
91.b odd 2 1 inner 273.2.g.a 32
273.g even 2 1 inner 273.2.g.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.g.a 32 1.a even 1 1 trivial
273.2.g.a 32 3.b odd 2 1 inner
273.2.g.a 32 7.b odd 2 1 inner
273.2.g.a 32 13.b even 2 1 inner
273.2.g.a 32 21.c even 2 1 inner
273.2.g.a 32 39.d odd 2 1 inner
273.2.g.a 32 91.b odd 2 1 inner
273.2.g.a 32 273.g even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(273, [\chi])$$.