# Properties

 Label 231.2.g.a Level 231 Weight 2 Character orbit 231.g Analytic conductor 1.845 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$24$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{3} + 24q^{4} + 10q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{3} + 24q^{4} + 10q^{9} - 20q^{12} - 10q^{15} - 8q^{16} - 12q^{25} - 20q^{31} + 14q^{33} - 8q^{34} - 12q^{36} + 4q^{37} + 6q^{45} - 48q^{48} - 24q^{49} - 28q^{55} + 44q^{58} + 32q^{60} - 52q^{64} + 12q^{66} - 4q^{67} + 54q^{69} - 20q^{70} + 68q^{75} - 20q^{78} + 2q^{81} + 16q^{82} - 44q^{88} + 24q^{91} + 26q^{93} - 12q^{97} - 6q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1 −2.45155 −1.72491 0.157123i 4.01010 3.63499i 4.22870 + 0.385196i 1.00000i −4.92788 2.95062 + 0.542048i 8.91135i
197.2 −2.45155 −1.72491 + 0.157123i 4.01010 3.63499i 4.22870 0.385196i 1.00000i −4.92788 2.95062 0.542048i 8.91135i
197.3 −2.30127 −0.220342 1.71798i 3.29586 1.52521i 0.507068 + 3.95354i 1.00000i −2.98213 −2.90290 + 0.757087i 3.50993i
197.4 −2.30127 −0.220342 + 1.71798i 3.29586 1.52521i 0.507068 3.95354i 1.00000i −2.98213 −2.90290 0.757087i 3.50993i
197.5 −1.92194 1.23500 1.21441i 1.69384 0.413489i −2.37358 + 2.33402i 1.00000i 0.588415 0.0504282 2.99958i 0.794699i
197.6 −1.92194 1.23500 + 1.21441i 1.69384 0.413489i −2.37358 2.33402i 1.00000i 0.588415 0.0504282 + 2.99958i 0.794699i
197.7 −1.36666 −0.859522 1.50374i −0.132249 3.05646i 1.17467 + 2.05509i 1.00000i 2.91405 −1.52244 + 2.58499i 4.17714i
197.8 −1.36666 −0.859522 + 1.50374i −0.132249 3.05646i 1.17467 2.05509i 1.00000i 2.91405 −1.52244 2.58499i 4.17714i
197.9 −0.858468 1.60964 0.639565i −1.26303 2.44714i −1.38183 + 0.549047i 1.00000i 2.80121 2.18191 2.05895i 2.10079i
197.10 −0.858468 1.60964 + 0.639565i −1.26303 2.44714i −1.38183 0.549047i 1.00000i 2.80121 2.18191 + 2.05895i 2.10079i
197.11 −0.628865 −1.53987 0.792976i −1.60453 1.39971i 0.968369 + 0.498675i 1.00000i 2.26676 1.74238 + 2.44216i 0.880226i
197.12 −0.628865 −1.53987 + 0.792976i −1.60453 1.39971i 0.968369 0.498675i 1.00000i 2.26676 1.74238 2.44216i 0.880226i
197.13 0.628865 −1.53987 0.792976i −1.60453 1.39971i −0.968369 0.498675i 1.00000i −2.26676 1.74238 + 2.44216i 0.880226i
197.14 0.628865 −1.53987 + 0.792976i −1.60453 1.39971i −0.968369 + 0.498675i 1.00000i −2.26676 1.74238 2.44216i 0.880226i
197.15 0.858468 1.60964 0.639565i −1.26303 2.44714i 1.38183 0.549047i 1.00000i −2.80121 2.18191 2.05895i 2.10079i
197.16 0.858468 1.60964 + 0.639565i −1.26303 2.44714i 1.38183 + 0.549047i 1.00000i −2.80121 2.18191 + 2.05895i 2.10079i
197.17 1.36666 −0.859522 1.50374i −0.132249 3.05646i −1.17467 2.05509i 1.00000i −2.91405 −1.52244 + 2.58499i 4.17714i
197.18 1.36666 −0.859522 + 1.50374i −0.132249 3.05646i −1.17467 + 2.05509i 1.00000i −2.91405 −1.52244 2.58499i 4.17714i
197.19 1.92194 1.23500 1.21441i 1.69384 0.413489i 2.37358 2.33402i 1.00000i −0.588415 0.0504282 2.99958i 0.794699i
197.20 1.92194 1.23500 + 1.21441i 1.69384 0.413489i 2.37358 + 2.33402i 1.00000i −0.588415 0.0504282 + 2.99958i 0.794699i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.24 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
11.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.g.a 24
3.b odd 2 1 inner 231.2.g.a 24
11.b odd 2 1 inner 231.2.g.a 24
33.d even 2 1 inner 231.2.g.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.g.a 24 1.a even 1 1 trivial
231.2.g.a 24 3.b odd 2 1 inner
231.2.g.a 24 11.b odd 2 1 inner
231.2.g.a 24 33.d even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database