# Properties

 Label 231.2.s.a Level 231 Weight 2 Character orbit 231.s Analytic conductor 1.845 Analytic rank 0 Dimension 96 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.s (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$24$$ over $$\Q(\zeta_{10})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$96q + 6q^{3} - 24q^{4} - 10q^{6} - 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96q + 6q^{3} - 24q^{4} - 10q^{6} - 20q^{9} - 20q^{12} - 32q^{16} - 10q^{18} - 60q^{19} + 30q^{24} + 12q^{25} - 40q^{28} + 60q^{30} + 40q^{31} - 44q^{33} + 8q^{34} + 2q^{36} + 16q^{37} - 10q^{39} - 96q^{45} - 40q^{46} + 48q^{48} + 24q^{49} - 30q^{51} - 40q^{52} + 28q^{55} - 30q^{57} + 36q^{58} - 32q^{60} + 40q^{61} - 28q^{64} + 118q^{66} - 56q^{67} + 26q^{69} + 20q^{70} + 150q^{72} + 40q^{73} + 2q^{75} - 20q^{78} - 40q^{79} + 8q^{81} - 16q^{82} + 40q^{84} - 60q^{85} - 156q^{88} + 100q^{90} + 36q^{91} - 36q^{93} + 160q^{96} - 88q^{97} - 94q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −2.24267 + 1.62939i 1.07372 1.35909i 1.75661 5.40628i 0.476086 0.655276i −0.193509 + 4.79750i −0.951057 0.309017i 3.15622 + 9.71386i −0.694246 2.91857i 2.24530i
8.2 −2.11971 + 1.54006i 0.569669 + 1.63569i 1.50336 4.62686i 0.853755 1.17509i −3.72660 2.58986i 0.951057 + 0.309017i 2.31965 + 7.13914i −2.35095 + 1.86360i 3.80570i
8.3 −1.67788 + 1.21905i −1.27625 + 1.17097i 0.711169 2.18875i 0.764082 1.05167i 0.713926 3.52058i −0.951057 0.309017i 0.193159 + 0.594483i 0.257645 2.98892i 2.69603i
8.4 −1.65451 + 1.20207i −1.40677 1.01044i 0.674386 2.07555i 0.537508 0.739816i 3.54213 0.0192577i −0.951057 0.309017i 0.115245 + 0.354686i 0.958019 + 2.84292i 1.87015i
8.5 −1.65450 + 1.20206i −0.288346 1.70788i 0.674376 2.07552i −1.12543 + 1.54901i 2.53005 + 2.47908i 0.951057 + 0.309017i 0.115224 + 0.354623i −2.83371 + 0.984921i 3.91568i
8.6 −1.61087 + 1.17037i 1.32659 + 1.11363i 0.607116 1.86851i −2.20177 + 3.03048i −3.44031 0.241324i −0.951057 0.309017i −0.0217402 0.0669093i 0.519657 + 2.95465i 7.45860i
8.7 −1.32709 + 0.964186i 1.73091 + 0.0627395i 0.213475 0.657007i 1.68710 2.32209i −2.35757 + 1.58566i 0.951057 + 0.309017i −0.663628 2.04244i 2.99213 + 0.217193i 4.70830i
8.8 −0.723708 + 0.525805i −1.64598 0.539221i −0.370751 + 1.14105i −0.915619 + 1.26024i 1.47473 0.475224i 0.951057 + 0.309017i −0.884520 2.72227i 2.41848 + 1.77509i 1.39348i
8.9 −0.699320 + 0.508086i 0.0766728 + 1.73035i −0.387136 + 1.19148i −1.26605 + 1.74257i −0.932787 1.17111i 0.951057 + 0.309017i −0.868877 2.67413i −2.98824 + 0.265342i 1.86187i
8.10 −0.487641 + 0.354292i −1.64748 + 0.534620i −0.505763 + 1.55658i 2.33078 3.20804i 0.613966 0.844391i 0.951057 + 0.309017i −0.677377 2.08475i 2.42836 1.76155i 2.39015i
8.11 −0.372543 + 0.270668i −0.457102 1.67065i −0.552507 + 1.70044i 1.99416 2.74473i 0.622481 + 0.498664i −0.951057 0.309017i −0.539021 1.65893i −2.58212 + 1.52731i 1.56229i
8.12 −0.0636021 + 0.0462097i 1.52227 + 0.826261i −0.616124 + 1.89623i 0.403747 0.555710i −0.135001 + 0.0177914i −0.951057 0.309017i −0.0970253 0.298613i 1.63459 + 2.51558i 0.0540014i
8.13 0.0636021 0.0462097i −0.745875 + 1.56322i −0.616124 + 1.89623i −0.403747 + 0.555710i 0.0247968 + 0.133891i −0.951057 0.309017i 0.0970253 + 0.298613i −1.88734 2.33194i 0.0540014i
8.14 0.372543 0.270668i −0.612178 1.62026i −0.552507 + 1.70044i −1.99416 + 2.74473i −0.666615 0.437919i −0.951057 0.309017i 0.539021 + 1.65893i −2.25048 + 1.98377i 1.56229i
8.15 0.487641 0.354292i 1.64708 0.535846i −0.505763 + 1.55658i −2.33078 + 3.20804i 0.613338 0.844848i 0.951057 + 0.309017i 0.677377 + 2.08475i 2.42574 1.76516i 2.39015i
8.16 0.699320 0.508086i 0.955046 + 1.44495i −0.387136 + 1.19148i 1.26605 1.74257i 1.40204 + 0.525239i 0.951057 + 0.309017i 0.868877 + 2.67413i −1.17577 + 2.75999i 1.86187i
8.17 0.723708 0.525805i 1.01468 1.40372i −0.370751 + 1.14105i 0.915619 1.26024i −0.00375218 1.54941i 0.951057 + 0.309017i 0.884520 + 2.72227i −0.940859 2.84865i 1.39348i
8.18 1.32709 0.964186i −1.36346 + 1.06816i 0.213475 0.657007i −1.68710 + 2.32209i −0.779526 + 2.73218i 0.951057 + 0.309017i 0.663628 + 2.04244i 0.718055 2.91280i 4.70830i
8.19 1.61087 1.17037i −0.418655 + 1.68069i 0.607116 1.86851i 2.20177 3.03048i 1.29263 + 3.19736i −0.951057 0.309017i 0.0217402 + 0.0669093i −2.64946 1.40726i 7.45860i
8.20 1.65450 1.20206i −0.770590 1.55119i 0.674376 2.07552i 1.12543 1.54901i −3.13957 1.64014i 0.951057 + 0.309017i −0.115224 0.354623i −1.81238 + 2.39066i 3.91568i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 134.24 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.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.s.a 96
3.b odd 2 1 inner 231.2.s.a 96
11.d odd 10 1 inner 231.2.s.a 96
33.f even 10 1 inner 231.2.s.a 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.s.a 96 1.a even 1 1 trivial
231.2.s.a 96 3.b odd 2 1 inner
231.2.s.a 96 11.d odd 10 1 inner
231.2.s.a 96 33.f even 10 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