# Properties

 Label 231.2.l.b Level 231 Weight 2 Character orbit 231.l Analytic conductor 1.845 Analytic rank 0 Dimension 48 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.l (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 6q^{3} - 36q^{4} - 10q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 6q^{3} - 36q^{4} - 10q^{9} + 14q^{12} + 4q^{15} - 28q^{16} - 24q^{22} - 36q^{27} - 4q^{31} - 26q^{33} + 56q^{34} + 12q^{36} - 28q^{37} + 96q^{42} + 30q^{45} - 72q^{48} + 24q^{49} - 80q^{55} + 16q^{58} - 32q^{60} + 256q^{64} - 42q^{66} + 16q^{67} + 36q^{69} - 76q^{70} + 34q^{75} + 104q^{78} - 26q^{81} - 4q^{82} - 16q^{88} - 36q^{91} + 52q^{93} - 144q^{97} - 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1 −1.36539 2.36492i −0.516280 1.65332i −2.72858 + 4.72604i −1.96785 + 1.13614i −3.20504 + 3.47838i −1.90462 1.83641i 9.44073 −2.46691 + 1.70715i 5.37375 + 3.10254i
32.2 −1.36539 2.36492i 1.68995 0.379546i −2.72858 + 4.72604i 1.96785 1.13614i −3.20504 3.47838i 1.90462 + 1.83641i 9.44073 2.71189 1.28283i −5.37375 3.10254i
32.3 −1.19873 2.07626i −1.49003 + 0.883063i −1.87390 + 3.24569i −0.707611 + 0.408540i 3.61961 + 2.03514i 2.64562 + 0.0267152i 4.19028 1.44040 2.63159i 1.69647 + 0.979456i
32.4 −1.19873 2.07626i −0.0197381 + 1.73194i −1.87390 + 3.24569i 0.707611 0.408540i 3.61961 2.03514i −2.64562 0.0267152i 4.19028 −2.99922 0.0683703i −1.69647 0.979456i
32.5 −0.939478 1.62722i −1.69195 0.370556i −0.765237 + 1.32543i 2.63713 1.52255i 0.986571 + 3.10131i −0.738610 2.54056i −0.882219 2.72538 + 1.25392i −4.95505 2.86080i
32.6 −0.939478 1.62722i 1.16688 + 1.27999i −0.765237 + 1.32543i −2.63713 + 1.52255i 0.986571 3.10131i 0.738610 + 2.54056i −0.882219 −0.276760 + 2.98721i 4.95505 + 2.86080i
32.7 −0.794537 1.37618i 0.668961 1.59765i −0.262580 + 0.454801i 1.14794 0.662766i −2.73017 + 0.348784i 2.48641 0.904309i −2.34363 −2.10498 2.13753i −1.82417 1.05318i
32.8 −0.794537 1.37618i 1.04913 1.37816i −0.262580 + 0.454801i −1.14794 + 0.662766i −2.73017 0.348784i −2.48641 + 0.904309i −2.34363 −0.798666 2.89174i 1.82417 + 1.05318i
32.9 −0.468581 0.811606i −1.03189 + 1.39112i 0.560864 0.971445i 2.73832 1.58097i 1.61256 + 0.185636i −0.0402023 + 2.64545i −2.92556 −0.870410 2.87096i −2.56625 1.48162i
32.10 −0.468581 0.811606i −0.688798 + 1.58920i 0.560864 0.971445i −2.73832 + 1.58097i 1.61256 0.185636i 0.0402023 2.64545i −2.92556 −2.05112 2.18928i 2.56625 + 1.48162i
32.11 −0.463987 0.803650i −1.71325 0.254482i 0.569432 0.986284i −1.53501 + 0.886239i 0.590414 + 1.49493i −2.26798 + 1.36244i −2.91279 2.87048 + 0.871983i 1.42445 + 0.822407i
32.12 −0.463987 0.803650i 1.07701 + 1.35648i 0.569432 0.986284i 1.53501 0.886239i 0.590414 1.49493i 2.26798 1.36244i −2.91279 −0.680079 + 2.92190i −1.42445 0.822407i
32.13 0.463987 + 0.803650i −1.71325 0.254482i 0.569432 0.986284i −1.53501 + 0.886239i −0.590414 1.49493i 2.26798 1.36244i 2.91279 2.87048 + 0.871983i −1.42445 0.822407i
32.14 0.463987 + 0.803650i 1.07701 + 1.35648i 0.569432 0.986284i 1.53501 0.886239i −0.590414 + 1.49493i −2.26798 + 1.36244i 2.91279 −0.680079 + 2.92190i 1.42445 + 0.822407i
32.15 0.468581 + 0.811606i −1.03189 + 1.39112i 0.560864 0.971445i 2.73832 1.58097i −1.61256 0.185636i 0.0402023 2.64545i 2.92556 −0.870410 2.87096i 2.56625 + 1.48162i
32.16 0.468581 + 0.811606i −0.688798 + 1.58920i 0.560864 0.971445i −2.73832 + 1.58097i −1.61256 + 0.185636i −0.0402023 + 2.64545i 2.92556 −2.05112 2.18928i −2.56625 1.48162i
32.17 0.794537 + 1.37618i 0.668961 1.59765i −0.262580 + 0.454801i 1.14794 0.662766i 2.73017 0.348784i −2.48641 + 0.904309i 2.34363 −2.10498 2.13753i 1.82417 + 1.05318i
32.18 0.794537 + 1.37618i 1.04913 1.37816i −0.262580 + 0.454801i −1.14794 + 0.662766i 2.73017 + 0.348784i 2.48641 0.904309i 2.34363 −0.798666 2.89174i −1.82417 1.05318i
32.19 0.939478 + 1.62722i −1.69195 0.370556i −0.765237 + 1.32543i 2.63713 1.52255i −0.986571 3.10131i 0.738610 + 2.54056i 0.882219 2.72538 + 1.25392i 4.95505 + 2.86080i
32.20 0.939478 + 1.62722i 1.16688 + 1.27999i −0.765237 + 1.32543i −2.63713 + 1.52255i −0.986571 + 3.10131i −0.738610 2.54056i 0.882219 −0.276760 + 2.98721i −4.95505 2.86080i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 65.24 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.c even 3 1 inner
11.b odd 2 1 inner
21.h odd 6 1 inner
33.d even 2 1 inner
77.h odd 6 1 inner
231.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.l.b 48
3.b odd 2 1 inner 231.2.l.b 48
7.c even 3 1 inner 231.2.l.b 48
11.b odd 2 1 inner 231.2.l.b 48
21.h odd 6 1 inner 231.2.l.b 48
33.d even 2 1 inner 231.2.l.b 48
77.h odd 6 1 inner 231.2.l.b 48
231.l even 6 1 inner 231.2.l.b 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.l.b 48 1.a even 1 1 trivial
231.2.l.b 48 3.b odd 2 1 inner
231.2.l.b 48 7.c even 3 1 inner
231.2.l.b 48 11.b odd 2 1 inner
231.2.l.b 48 21.h odd 6 1 inner
231.2.l.b 48 33.d even 2 1 inner
231.2.l.b 48 77.h odd 6 1 inner
231.2.l.b 48 231.l even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database