# Properties

 Label 231.2.w.a Level 231 Weight 2 Character orbit 231.w Analytic conductor 1.845 Analytic rank 0 Dimension 64 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 12q^{4} + 10q^{7} - 20q^{8} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 12q^{4} + 10q^{7} - 20q^{8} + 16q^{9} - 16q^{11} + 12q^{14} - 12q^{15} - 16q^{16} + 10q^{18} - 40q^{22} - 24q^{23} + 44q^{25} - 30q^{28} - 40q^{29} - 40q^{35} - 12q^{36} + 32q^{37} - 2q^{42} + 22q^{44} - 70q^{46} - 50q^{49} - 40q^{51} - 64q^{53} + 80q^{56} + 2q^{58} - 36q^{60} + 10q^{63} + 72q^{64} - 8q^{67} - 26q^{70} + 68q^{71} - 10q^{72} + 80q^{74} + 90q^{77} - 72q^{78} + 40q^{79} - 16q^{81} + 60q^{84} - 40q^{85} - 62q^{86} + 140q^{88} + 54q^{91} + 18q^{92} - 20q^{93} + 20q^{95} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −2.36445 0.768257i −0.587785 0.809017i 3.38238 + 2.45745i 1.43953 0.467733i 0.768257 + 2.36445i −0.764664 + 2.53284i −3.18691 4.38641i −0.309017 + 0.951057i −3.76305
13.2 −2.36445 0.768257i 0.587785 + 0.809017i 3.38238 + 2.45745i −1.43953 + 0.467733i −0.768257 2.36445i −2.17258 + 1.50993i −3.18691 4.38641i −0.309017 + 0.951057i 3.76305
13.3 −1.82917 0.594332i −0.587785 0.809017i 1.37458 + 0.998694i −3.42067 + 1.11144i 0.594332 + 1.82917i 2.15042 + 1.54133i 0.340186 + 0.468226i −0.309017 + 0.951057i 6.91755
13.4 −1.82917 0.594332i 0.587785 + 0.809017i 1.37458 + 0.998694i 3.42067 1.11144i −0.594332 1.82917i −2.13041 1.56887i 0.340186 + 0.468226i −0.309017 + 0.951057i −6.91755
13.5 −1.50185 0.487979i −0.587785 0.809017i 0.399382 + 0.290168i 2.27762 0.740044i 0.487979 + 1.50185i 1.84555 1.89577i 1.39817 + 1.92441i −0.309017 + 0.951057i −3.78176
13.6 −1.50185 0.487979i 0.587785 + 0.809017i 0.399382 + 0.290168i −2.27762 + 0.740044i −0.487979 1.50185i 1.23267 2.34105i 1.39817 + 1.92441i −0.309017 + 0.951057i 3.78176
13.7 −0.725588 0.235758i −0.587785 0.809017i −1.14714 0.833444i −0.666245 + 0.216476i 0.235758 + 0.725588i −2.63718 + 0.212765i 1.53273 + 2.10963i −0.309017 + 0.951057i 0.534456
13.8 −0.725588 0.235758i 0.587785 + 0.809017i −1.14714 0.833444i 0.666245 0.216476i −0.235758 0.725588i 0.612583 + 2.57386i 1.53273 + 2.10963i −0.309017 + 0.951057i −0.534456
13.9 0.332589 + 0.108065i −0.587785 0.809017i −1.51910 1.10369i 3.96320 1.28772i −0.108065 0.332589i −2.37010 + 1.17584i −0.797068 1.09707i −0.309017 + 0.951057i 1.45728
13.10 0.332589 + 0.108065i 0.587785 + 0.809017i −1.51910 1.10369i −3.96320 + 1.28772i 0.108065 + 0.332589i −0.385893 + 2.61746i −0.797068 1.09707i −0.309017 + 0.951057i −1.45728
13.11 0.819861 + 0.266389i −0.587785 0.809017i −1.01682 0.738766i −2.84938 + 0.925820i −0.266389 0.819861i 0.656605 2.56298i −1.65026 2.27139i −0.309017 + 0.951057i −2.58273
13.12 0.819861 + 0.266389i 0.587785 + 0.809017i −1.01682 0.738766i 2.84938 0.925820i 0.266389 + 0.819861i 2.23464 1.41647i −1.65026 2.27139i −0.309017 + 0.951057i 2.58273
13.13 1.94435 + 0.631758i −0.587785 0.809017i 1.76335 + 1.28115i 2.31266 0.751430i −0.631758 1.94435i −1.00115 2.44902i 0.215840 + 0.297079i −0.309017 + 0.951057i 4.97135
13.14 1.94435 + 0.631758i 0.587785 + 0.809017i 1.76335 + 1.28115i −2.31266 + 0.751430i 0.631758 + 1.94435i 2.63853 + 0.195359i 0.215840 + 0.297079i −0.309017 + 0.951057i −4.97135
13.15 2.20622 + 0.716844i −0.587785 0.809017i 2.73550 + 1.98746i 0.0209622 0.00681102i −0.716844 2.20622i 1.69347 + 2.03277i 1.88338 + 2.59224i −0.309017 + 0.951057i 0.0511296
13.16 2.20622 + 0.716844i 0.587785 + 0.809017i 2.73550 + 1.98746i −0.0209622 + 0.00681102i 0.716844 + 2.20622i −2.45659 0.982422i 1.88338 + 2.59224i −0.309017 + 0.951057i −0.0511296
118.1 −1.44554 1.98962i −0.951057 + 0.309017i −1.25096 + 3.85006i 1.57908 2.17341i 1.98962 + 1.44554i 2.53314 0.763690i 4.79060 1.55656i 0.809017 0.587785i −6.60689
118.2 −1.44554 1.98962i 0.951057 0.309017i −1.25096 + 3.85006i −1.57908 + 2.17341i −1.98962 1.44554i 1.60046 + 2.10678i 4.79060 1.55656i 0.809017 0.587785i 6.60689
118.3 −0.917545 1.26289i −0.951057 + 0.309017i −0.134974 + 0.415407i 0.118521 0.163130i 1.26289 + 0.917545i −2.40543 + 1.10177i −2.32078 + 0.754067i 0.809017 0.587785i −0.314765
118.4 −0.917545 1.26289i 0.951057 0.309017i −0.134974 + 0.415407i −0.118521 + 0.163130i −1.26289 0.917545i −1.29844 2.30523i −2.32078 + 0.754067i 0.809017 0.587785i 0.314765
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 160.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l 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.w.a 64
3.b odd 2 1 693.2.bu.f 64
7.b odd 2 1 inner 231.2.w.a 64
11.d odd 10 1 inner 231.2.w.a 64
21.c even 2 1 693.2.bu.f 64
33.f even 10 1 693.2.bu.f 64
77.l even 10 1 inner 231.2.w.a 64
231.r odd 10 1 693.2.bu.f 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.w.a 64 1.a even 1 1 trivial
231.2.w.a 64 7.b odd 2 1 inner
231.2.w.a 64 11.d odd 10 1 inner
231.2.w.a 64 77.l even 10 1 inner
693.2.bu.f 64 3.b odd 2 1
693.2.bu.f 64 21.c even 2 1
693.2.bu.f 64 33.f even 10 1
693.2.bu.f 64 231.r odd 10 1

## 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