# Properties

 Label 231.2.p.a Level 231 Weight 2 Character orbit 231.p Analytic conductor 1.845 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 12q^{4} - 12q^{5} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 12q^{4} - 12q^{5} + 16q^{9} + 2q^{11} - 32q^{14} + 8q^{15} - 20q^{16} + 8q^{22} + 24q^{23} - 24q^{26} - 12q^{31} - 18q^{33} + 24q^{36} - 32q^{37} + 24q^{38} - 24q^{42} - 28q^{44} - 12q^{45} + 24q^{47} - 36q^{49} + 36q^{53} - 56q^{56} + 12q^{58} - 48q^{59} + 8q^{64} - 36q^{66} + 20q^{67} + 24q^{70} + 72q^{71} + 24q^{75} - 48q^{78} + 72q^{80} - 16q^{81} - 48q^{82} + 64q^{86} + 24q^{88} + 60q^{89} - 28q^{91} - 16q^{92} + 16q^{93} + 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1 −2.23994 + 1.29323i −0.866025 0.500000i 2.34487 4.06144i −1.30115 + 0.751219i 2.58645 1.55637 + 2.13956i 6.95692i 0.500000 + 0.866025i 1.94300 3.36537i
10.2 −2.16769 + 1.25152i 0.866025 + 0.500000i 2.13259 3.69375i −0.849508 + 0.490464i −2.50303 1.62716 2.08623i 5.66981i 0.500000 + 0.866025i 1.22765 2.12635i
10.3 −1.71613 + 0.990810i 0.866025 + 0.500000i 0.963407 1.66867i 0.315938 0.182407i −1.98162 0.143176 + 2.64187i 0.145025i 0.500000 + 0.866025i −0.361461 + 0.626069i
10.4 −1.49489 + 0.863074i −0.866025 0.500000i 0.489794 0.848348i 1.43342 0.827587i 1.72615 −2.00786 + 1.72293i 1.76138i 0.500000 + 0.866025i −1.42854 + 2.47430i
10.5 −0.931635 + 0.537879i −0.866025 0.500000i −0.421371 + 0.729836i −2.91250 + 1.68153i 1.07576 0.879349 2.49534i 3.05811i 0.500000 + 0.866025i 1.80893 3.13315i
10.6 −0.567337 + 0.327552i 0.866025 + 0.500000i −0.785419 + 1.36039i −2.94602 + 1.70089i −0.655105 2.64073 + 0.162880i 2.33927i 0.500000 + 0.866025i 1.11426 1.92995i
10.7 −0.533044 + 0.307753i 0.866025 + 0.500000i −0.810576 + 1.40396i 2.84562 1.64292i −0.615506 0.613452 2.57365i 2.22884i 0.500000 + 0.866025i −1.01123 + 1.75150i
10.8 −0.360631 + 0.208210i −0.866025 0.500000i −0.913297 + 1.58188i 0.414204 0.239141i 0.416420 −2.50100 0.863126i 1.59347i 0.500000 + 0.866025i −0.0995832 + 0.172483i
10.9 0.360631 0.208210i −0.866025 0.500000i −0.913297 + 1.58188i 0.414204 0.239141i −0.416420 2.50100 + 0.863126i 1.59347i 0.500000 + 0.866025i 0.0995832 0.172483i
10.10 0.533044 0.307753i 0.866025 + 0.500000i −0.810576 + 1.40396i 2.84562 1.64292i 0.615506 −0.613452 + 2.57365i 2.22884i 0.500000 + 0.866025i 1.01123 1.75150i
10.11 0.567337 0.327552i 0.866025 + 0.500000i −0.785419 + 1.36039i −2.94602 + 1.70089i 0.655105 −2.64073 0.162880i 2.33927i 0.500000 + 0.866025i −1.11426 + 1.92995i
10.12 0.931635 0.537879i −0.866025 0.500000i −0.421371 + 0.729836i −2.91250 + 1.68153i −1.07576 −0.879349 + 2.49534i 3.05811i 0.500000 + 0.866025i −1.80893 + 3.13315i
10.13 1.49489 0.863074i −0.866025 0.500000i 0.489794 0.848348i 1.43342 0.827587i −1.72615 2.00786 1.72293i 1.76138i 0.500000 + 0.866025i 1.42854 2.47430i
10.14 1.71613 0.990810i 0.866025 + 0.500000i 0.963407 1.66867i 0.315938 0.182407i 1.98162 −0.143176 2.64187i 0.145025i 0.500000 + 0.866025i 0.361461 0.626069i
10.15 2.16769 1.25152i 0.866025 + 0.500000i 2.13259 3.69375i −0.849508 + 0.490464i 2.50303 −1.62716 + 2.08623i 5.66981i 0.500000 + 0.866025i −1.22765 + 2.12635i
10.16 2.23994 1.29323i −0.866025 0.500000i 2.34487 4.06144i −1.30115 + 0.751219i −2.58645 −1.55637 2.13956i 6.95692i 0.500000 + 0.866025i −1.94300 + 3.36537i
208.1 −2.23994 1.29323i −0.866025 + 0.500000i 2.34487 + 4.06144i −1.30115 0.751219i 2.58645 1.55637 2.13956i 6.95692i 0.500000 0.866025i 1.94300 + 3.36537i
208.2 −2.16769 1.25152i 0.866025 0.500000i 2.13259 + 3.69375i −0.849508 0.490464i −2.50303 1.62716 + 2.08623i 5.66981i 0.500000 0.866025i 1.22765 + 2.12635i
208.3 −1.71613 0.990810i 0.866025 0.500000i 0.963407 + 1.66867i 0.315938 + 0.182407i −1.98162 0.143176 2.64187i 0.145025i 0.500000 0.866025i −0.361461 0.626069i
208.4 −1.49489 0.863074i −0.866025 + 0.500000i 0.489794 + 0.848348i 1.43342 + 0.827587i 1.72615 −2.00786 1.72293i 1.76138i 0.500000 0.866025i −1.42854 2.47430i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 208.16 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.d odd 6 1 inner
11.b odd 2 1 inner
77.i 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.p.a 32
3.b odd 2 1 693.2.bg.b 32
7.c even 3 1 1617.2.c.a 32
7.d odd 6 1 inner 231.2.p.a 32
7.d odd 6 1 1617.2.c.a 32
11.b odd 2 1 inner 231.2.p.a 32
21.g even 6 1 693.2.bg.b 32
33.d even 2 1 693.2.bg.b 32
77.h odd 6 1 1617.2.c.a 32
77.i even 6 1 inner 231.2.p.a 32
77.i even 6 1 1617.2.c.a 32
231.k odd 6 1 693.2.bg.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.p.a 32 1.a even 1 1 trivial
231.2.p.a 32 7.d odd 6 1 inner
231.2.p.a 32 11.b odd 2 1 inner
231.2.p.a 32 77.i even 6 1 inner
693.2.bg.b 32 3.b odd 2 1
693.2.bg.b 32 21.g even 6 1
693.2.bg.b 32 33.d even 2 1
693.2.bg.b 32 231.k odd 6 1
1617.2.c.a 32 7.c even 3 1
1617.2.c.a 32 7.d odd 6 1
1617.2.c.a 32 77.h odd 6 1
1617.2.c.a 32 77.i even 6 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