# Properties

 Label 231.2.y.b Level 231 Weight 2 Character orbit 231.y 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.y (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$8$$ over $$\Q(\zeta_{15})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 4q^{2} + 8q^{3} + 10q^{4} - 4q^{5} - 8q^{6} - q^{7} + 8q^{8} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 4q^{2} + 8q^{3} + 10q^{4} - 4q^{5} - 8q^{6} - q^{7} + 8q^{8} + 8q^{9} - 14q^{10} + 11q^{11} - 30q^{12} - 8q^{13} + 6q^{14} - 12q^{15} - 4q^{17} - q^{18} - 2q^{19} + 24q^{20} - 2q^{21} - 14q^{22} + 16q^{24} - 10q^{25} + 4q^{26} - 16q^{27} + 29q^{28} - 58q^{29} + 11q^{30} - 19q^{31} - 64q^{32} + 6q^{33} - 88q^{34} + 17q^{35} - 20q^{36} - 20q^{37} + 29q^{38} + 4q^{39} + 51q^{40} - 68q^{41} - 11q^{42} + 92q^{43} - 21q^{44} - 4q^{45} - 5q^{46} - 26q^{47} - 10q^{48} + 37q^{49} - 10q^{50} + 6q^{51} - 14q^{52} - 3q^{53} - 6q^{54} - 32q^{55} + 24q^{56} - 36q^{57} + 52q^{58} + 7q^{59} - 12q^{60} - 21q^{61} + 92q^{62} - 7q^{63} - 72q^{64} - 66q^{65} - 23q^{66} - 4q^{67} - 17q^{68} + 40q^{69} - q^{70} + 58q^{71} + 16q^{72} - 3q^{73} - 28q^{74} + 20q^{75} + 168q^{76} - 34q^{77} + 132q^{78} + 9q^{79} - 5q^{80} + 8q^{81} - 42q^{82} + 60q^{83} - 39q^{84} + 110q^{85} + 13q^{86} - 46q^{87} + 92q^{88} - 10q^{89} + 8q^{90} + 10q^{91} + 110q^{92} - 19q^{93} - 46q^{94} + 43q^{95} - 4q^{96} + 64q^{97} - 88q^{98} + 28q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −0.266544 + 2.53600i 0.669131 + 0.743145i −4.40395 0.936088i −2.83093 1.26041i −2.06297 + 1.49883i −2.61901 0.375200i 1.97180 6.06857i −0.104528 + 0.994522i 3.95097 6.84328i
4.2 −0.257299 + 2.44803i 0.669131 + 0.743145i −3.97037 0.843927i 3.02928 + 1.34872i −1.99141 + 1.44684i 2.02429 + 1.70360i 1.56623 4.82036i −0.104528 + 0.994522i −4.08115 + 7.06876i
4.3 −0.116737 + 1.11067i 0.669131 + 0.743145i 0.736325 + 0.156511i −0.640590 0.285209i −0.903504 + 0.656434i −1.42180 + 2.23125i −0.950004 + 2.92381i −0.104528 + 0.994522i 0.391555 0.678193i
4.4 −0.100056 + 0.951974i 0.669131 + 0.743145i 1.06005 + 0.225321i −1.67824 0.747201i −0.774405 + 0.562638i 2.58488 0.564263i −0.912158 + 2.80733i −0.104528 + 0.994522i 0.879234 1.52288i
4.5 −0.0223018 + 0.212187i 0.669131 + 0.743145i 1.91177 + 0.406359i 2.20829 + 0.983193i −0.172609 + 0.125407i −0.215060 2.63700i −0.260721 + 0.802417i −0.104528 + 0.994522i −0.257870 + 0.446643i
4.6 0.121266 1.15377i 0.669131 + 0.743145i 0.639819 + 0.135998i 0.991091 + 0.441262i 0.938560 0.681904i −2.60174 0.480599i 0.951494 2.92840i −0.104528 + 0.994522i 0.629300 1.08998i
4.7 0.164728 1.56729i 0.669131 + 0.743145i −0.472952 0.100529i −3.79103 1.68787i 1.27494 0.926302i 2.12208 1.58012i 0.738505 2.27288i −0.104528 + 0.994522i −3.26987 + 5.66358i
4.8 0.203284 1.93412i 0.669131 + 0.743145i −1.74320 0.370529i 1.58292 + 0.704762i 1.57336 1.14311i 2.12258 + 1.57945i 0.130923 0.402938i −0.104528 + 0.994522i 1.68488 2.91829i
16.1 −2.41978 0.514340i −0.104528 + 0.994522i 3.76371 + 1.67571i 2.72281 + 3.02398i 0.764459 2.35276i 2.18271 1.49525i −4.24270 3.08250i −0.978148 0.207912i −5.03324 8.71783i
16.2 −2.17404 0.462106i −0.104528 + 0.994522i 2.68580 + 1.19580i −0.640870 0.711758i 0.686823 2.11382i −1.08656 + 2.41234i −1.69019 1.22800i −0.978148 0.207912i 1.06437 + 1.84354i
16.3 −1.21132 0.257474i −0.104528 + 0.994522i −0.426088 0.189706i 0.0859924 + 0.0955042i 0.382681 1.17777i −1.04835 2.42919i 2.47103 + 1.79531i −0.978148 0.207912i −0.0795744 0.137827i
16.4 −0.466730 0.0992066i −0.104528 + 0.994522i −1.61910 0.720868i −0.127761 0.141892i 0.147450 0.453804i 2.46763 + 0.954366i 1.45622 + 1.05801i −0.978148 0.207912i 0.0455530 + 0.0789002i
16.5 0.300702 + 0.0639162i −0.104528 + 0.994522i −1.74075 0.775034i −1.51090 1.67802i −0.0949980 + 0.292374i −2.44866 + 1.00203i −0.971327 0.705711i −0.978148 0.207912i −0.347077 0.601155i
16.6 1.15881 + 0.246312i −0.104528 + 0.994522i −0.544926 0.242617i 2.21930 + 2.46479i −0.366091 + 1.12671i −1.26723 + 2.32253i −2.48858 1.80806i −0.978148 0.207912i 1.96464 + 3.40285i
16.7 2.18402 + 0.464228i −0.104528 + 0.994522i 2.72735 + 1.21429i 0.687753 + 0.763827i −0.689977 + 2.12353i −0.0599597 2.64507i 1.78011 + 1.29333i −0.978148 0.207912i 1.14748 + 1.98749i
16.8 2.25472 + 0.479255i −0.104528 + 0.994522i 3.02697 + 1.34770i −1.27098 1.41156i −0.712312 + 2.19227i 1.99414 + 1.73879i 2.44937 + 1.77957i −0.978148 0.207912i −2.18920 3.79180i
25.1 −1.77664 0.791011i −0.978148 0.207912i 1.19249 + 1.32439i −0.181119 + 1.72323i 1.57336 + 1.14311i −0.788829 2.52542i 0.130923 + 0.402938i 0.913545 + 0.406737i 1.68488 2.91829i
25.2 −1.43967 0.640984i −0.978148 0.207912i 0.323537 + 0.359324i 0.433772 4.12706i 1.27494 + 0.926302i −2.64557 + 0.0310171i 0.738505 + 2.27288i 0.913545 + 0.406737i −3.26987 + 5.66358i
25.3 −1.05983 0.471865i −0.978148 0.207912i −0.437687 0.486101i −0.113401 + 1.07894i 0.938560 + 0.681904i 1.82236 + 1.91807i 0.951494 + 2.92840i 0.913545 + 0.406737i 0.629300 1.08998i
25.4 0.194910 + 0.0867797i −0.978148 0.207912i −1.30780 1.45246i −0.252674 + 2.40403i −0.172609 0.125407i −1.37600 + 2.25978i −0.260721 0.802417i 0.913545 + 0.406737i −0.257870 + 0.446643i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 214.8 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.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.y.b 64
3.b odd 2 1 693.2.by.c 64
7.c even 3 1 inner 231.2.y.b 64
11.c even 5 1 inner 231.2.y.b 64
21.h odd 6 1 693.2.by.c 64
33.h odd 10 1 693.2.by.c 64
77.m even 15 1 inner 231.2.y.b 64
231.z odd 30 1 693.2.by.c 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.y.b 64 1.a even 1 1 trivial
231.2.y.b 64 7.c even 3 1 inner
231.2.y.b 64 11.c even 5 1 inner
231.2.y.b 64 77.m even 15 1 inner
693.2.by.c 64 3.b odd 2 1
693.2.by.c 64 21.h odd 6 1
693.2.by.c 64 33.h odd 10 1
693.2.by.c 64 231.z odd 30 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{64} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database