# Properties

 Label 231.2.u.a Level 231 Weight 2 Character orbit 231.u Analytic conductor 1.845 Analytic rank 0 Dimension 112 CM no Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$28$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$112q + 12q^{4} - 2q^{7} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q + 12q^{4} - 2q^{7} - 12q^{9} - 60q^{16} + 2q^{18} - 80q^{22} - 36q^{25} + 8q^{28} + 24q^{30} - 34q^{36} + 4q^{37} - 66q^{39} - 30q^{42} + 36q^{46} + 26q^{49} + 38q^{51} - 86q^{57} - 116q^{58} + 88q^{60} + 44q^{63} - 32q^{64} + 110q^{70} + 86q^{72} - 52q^{78} - 156q^{79} + 68q^{81} - 30q^{84} + 8q^{85} - 112q^{88} + 42q^{91} + 56q^{93} + 70q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1 −2.38523 0.775008i −1.70548 0.302226i 3.47065 + 2.52157i −0.223123 0.686701i 3.83373 + 2.04264i −2.41111 + 1.08928i −3.37575 4.64631i 2.81732 + 1.03088i 1.81086i
20.2 −2.38523 0.775008i 1.70548 + 0.302226i 3.47065 + 2.52157i 0.223123 + 0.686701i −3.83373 2.04264i 0.290890 2.62971i −3.37575 4.64631i 2.81732 + 1.03088i 1.81086i
20.3 −2.06683 0.671554i −0.0708238 1.73060i 2.20277 + 1.60041i −1.17677 3.62171i −1.01581 + 3.62443i 1.06501 2.42193i −0.923256 1.27075i −2.98997 + 0.245136i 8.27573i
20.4 −2.06683 0.671554i 0.0708238 + 1.73060i 2.20277 + 1.60041i 1.17677 + 3.62171i 1.01581 3.62443i −1.97429 + 1.76130i −0.923256 1.27075i −2.98997 + 0.245136i 8.27573i
20.5 −1.64177 0.533443i −1.71497 0.242653i 0.792810 + 0.576010i 0.0901224 + 0.277368i 2.68614 + 1.31322i 2.64157 + 0.148745i 1.03500 + 1.42455i 2.88224 + 0.832286i 0.503450i
20.6 −1.64177 0.533443i 1.71497 + 0.242653i 0.792810 + 0.576010i −0.0901224 0.277368i −2.68614 1.31322i 0.957754 + 2.46631i 1.03500 + 1.42455i 2.88224 + 0.832286i 0.503450i
20.7 −1.48714 0.483201i −0.691398 1.58807i 0.360067 + 0.261604i 0.748085 + 2.30237i 0.260847 + 2.69577i −2.45500 0.986383i 1.42914 + 1.96705i −2.04394 + 2.19598i 3.78542i
20.8 −1.48714 0.483201i 0.691398 + 1.58807i 0.360067 + 0.261604i −0.748085 2.30237i −0.260847 2.69577i −1.69674 2.03004i 1.42914 + 1.96705i −2.04394 + 2.19598i 3.78542i
20.9 −1.17237 0.380926i −1.03573 + 1.38826i −0.388687 0.282397i −0.966595 2.97487i 1.74309 1.23301i 0.481870 + 2.60150i 1.79724 + 2.47369i −0.854517 2.87573i 3.85585i
20.10 −1.17237 0.380926i 1.03573 1.38826i −0.388687 0.282397i 0.966595 + 2.97487i −1.74309 + 1.23301i 2.62308 0.345622i 1.79724 + 2.47369i −0.854517 2.87573i 3.85585i
20.11 −0.606621 0.197103i −1.17667 + 1.27100i −1.28889 0.936436i 0.284064 + 0.874260i 0.964313 0.539091i −0.678241 2.55734i 1.34712 + 1.85415i −0.230890 2.99110i 0.586335i
20.12 −0.606621 0.197103i 1.17667 1.27100i −1.28889 0.936436i −0.284064 0.874260i −0.964313 + 0.539091i −2.64176 + 0.145216i 1.34712 + 1.85415i −0.230890 2.99110i 0.586335i
20.13 −0.126891 0.0412293i −0.347648 1.69680i −1.60363 1.16511i −0.887927 2.73276i −0.0258447 + 0.229642i 0.380318 + 2.61827i 0.312296 + 0.429838i −2.75828 + 1.17978i 0.383371i
20.14 −0.126891 0.0412293i 0.347648 + 1.69680i −1.60363 1.16511i 0.887927 + 2.73276i 0.0258447 0.229642i 2.60765 0.447387i 0.312296 + 0.429838i −2.75828 + 1.17978i 0.383371i
20.15 0.126891 + 0.0412293i −1.50633 0.854974i −1.60363 1.16511i 0.887927 + 2.73276i −0.155889 0.170593i 0.380318 + 2.61827i −0.312296 0.429838i 1.53804 + 2.57574i 0.383371i
20.16 0.126891 + 0.0412293i 1.50633 + 0.854974i −1.60363 1.16511i −0.887927 2.73276i 0.155889 + 0.170593i 2.60765 0.447387i −0.312296 0.429838i 1.53804 + 2.57574i 0.383371i
20.17 0.606621 + 0.197103i −1.57241 + 0.726320i −1.28889 0.936436i 0.284064 + 0.874260i −1.09702 + 0.130675i −2.64176 + 0.145216i −1.34712 1.85415i 1.94492 2.28414i 0.586335i
20.18 0.606621 + 0.197103i 1.57241 0.726320i −1.28889 0.936436i −0.284064 0.874260i 1.09702 0.130675i −0.678241 2.55734i −1.34712 1.85415i 1.94492 2.28414i 0.586335i
20.19 1.17237 + 0.380926i −1.64037 + 0.556045i −0.388687 0.282397i −0.966595 2.97487i −2.13493 + 0.0270304i 2.62308 0.345622i −1.79724 2.47369i 2.38163 1.82424i 3.85585i
20.20 1.17237 + 0.380926i 1.64037 0.556045i −0.388687 0.282397i 0.966595 + 2.97487i 2.13493 0.0270304i 0.481870 + 2.60150i −1.79724 2.47369i 2.38163 1.82424i 3.85585i
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 146.28 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.b odd 2 1 inner
11.c even 5 1 inner
21.c even 2 1 inner
33.h odd 10 1 inner
77.j odd 10 1 inner
231.u 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.u.a 112
3.b odd 2 1 inner 231.2.u.a 112
7.b odd 2 1 inner 231.2.u.a 112
11.c even 5 1 inner 231.2.u.a 112
21.c even 2 1 inner 231.2.u.a 112
33.h odd 10 1 inner 231.2.u.a 112
77.j odd 10 1 inner 231.2.u.a 112
231.u even 10 1 inner 231.2.u.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.u.a 112 1.a even 1 1 trivial
231.2.u.a 112 3.b odd 2 1 inner
231.2.u.a 112 7.b odd 2 1 inner
231.2.u.a 112 11.c even 5 1 inner
231.2.u.a 112 21.c even 2 1 inner
231.2.u.a 112 33.h odd 10 1 inner
231.2.u.a 112 77.j odd 10 1 inner
231.2.u.a 112 231.u 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