Properties

 Label 231.2.n.a Level 231 Weight 2 Character orbit 231.n Analytic conductor 1.845 Analytic rank 0 Dimension 52 CM no Inner twists 4

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$52$$ Relative dimension: $$26$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$52q + 24q^{4} - 6q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$52q + 24q^{4} - 6q^{7} - 4q^{9} - 30q^{12} - 4q^{15} - 20q^{16} - 18q^{18} + 6q^{19} - 2q^{21} + 36q^{24} - 22q^{25} - 40q^{28} - 20q^{30} - 54q^{31} + 36q^{36} + 2q^{37} + 20q^{39} + 60q^{40} - 36q^{42} + 36q^{43} - 54q^{45} + 4q^{46} + 22q^{49} - 16q^{51} + 12q^{52} + 66q^{54} - 44q^{57} - 16q^{58} + 20q^{60} - 36q^{61} - 24q^{63} - 48q^{64} - 30q^{66} - 6q^{67} - 116q^{70} + 36q^{72} - 6q^{73} + 72q^{75} + 112q^{78} + 14q^{79} + 52q^{81} - 84q^{82} + 8q^{84} + 8q^{85} + 18q^{87} - 12q^{88} + 46q^{91} + 6q^{93} + 48q^{94} - 18q^{96} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1 −2.30668 1.33176i −1.72498 + 0.156349i 2.54717 + 4.41184i −1.59815 + 2.76808i 4.18719 + 1.93662i −2.47756 + 0.928284i 8.24186i 2.95111 0.539396i 7.37285 4.25672i
89.2 −2.25970 1.30464i 1.02945 + 1.39292i 2.40418 + 4.16416i −0.626810 + 1.08567i −0.508986 4.49066i 0.806620 2.51979i 7.32779i −0.880469 + 2.86789i 2.83281 1.63552i
89.3 −1.98793 1.14773i 1.53938 0.793914i 1.63457 + 2.83116i 1.11343 1.92852i −3.97138 0.188554i −2.63461 0.242572i 2.91327i 1.73940 2.44428i −4.42685 + 2.55584i
89.4 −1.93305 1.11604i −0.373056 + 1.69140i 1.49111 + 2.58268i 1.42193 2.46286i 2.60881 2.85320i 0.445307 + 2.60801i 2.19241i −2.72166 1.26197i −5.49732 + 3.17388i
89.5 −1.81398 1.04730i −1.71944 0.208619i 1.19369 + 2.06753i 1.09285 1.89288i 2.90055 + 2.17921i 2.41465 1.08141i 0.811414i 2.91296 + 0.717415i −3.96484 + 2.28910i
89.6 −1.36585 0.788573i 1.68141 + 0.415762i 0.243696 + 0.422094i −1.12576 + 1.94987i −1.96869 1.89378i −1.68560 + 2.03930i 2.38560i 2.65428 + 1.39813i 3.07523 1.77549i
89.7 −1.33690 0.771860i −0.650126 1.60541i 0.191534 + 0.331747i −0.895146 + 1.55044i −0.369997 + 2.64808i −0.181616 + 2.63951i 2.49609i −2.15467 + 2.08744i 2.39344 1.38185i
89.8 −1.14233 0.659525i −0.365976 + 1.69294i −0.130055 0.225261i −0.422862 + 0.732419i 1.53460 1.69253i −0.990003 2.45355i 2.98120i −2.73212 1.23915i 0.966096 0.557776i
89.9 −0.942161 0.543957i 1.10516 1.33365i −0.408222 0.707062i 1.52055 2.63367i −1.76669 + 0.655346i 2.61395 + 0.408977i 3.06405i −0.557221 2.94780i −2.86521 + 1.65423i
89.10 −0.872412 0.503687i 1.39293 + 1.02944i −0.492598 0.853205i 0.168910 0.292561i −0.696692 1.59970i 2.63835 + 0.197774i 3.00721i 0.880503 + 2.86788i −0.294718 + 0.170156i
89.11 −0.477360 0.275604i −1.66351 0.482443i −0.848085 1.46893i −1.62959 + 2.82254i 0.661127 + 0.688767i 1.54713 2.14625i 2.03736i 2.53450 + 1.60509i 1.55580 0.898244i
89.12 −0.408229 0.235691i −1.32879 1.11100i −0.888900 1.53962i 1.95312 3.38291i 0.280598 + 0.766725i −2.47945 + 0.923218i 1.78079i 0.531368 + 2.95257i −1.59464 + 0.920666i
89.13 −0.304723 0.175932i 0.634130 1.61179i −0.938096 1.62483i −0.734005 + 1.27133i −0.476800 + 0.379587i −1.51718 2.16753i 1.36389i −2.19576 2.04417i 0.447337 0.258270i
89.14 0.304723 + 0.175932i 1.71292 + 0.256725i −0.938096 1.62483i 0.734005 1.27133i 0.476800 + 0.379587i −1.51718 2.16753i 1.36389i 2.86818 + 0.879497i 0.447337 0.258270i
89.15 0.408229 + 0.235691i 0.297757 + 1.70627i −0.888900 1.53962i −1.95312 + 3.38291i −0.280598 + 0.766725i −2.47945 + 0.923218i 1.78079i −2.82268 + 1.01610i −1.59464 + 0.920666i
89.16 0.477360 + 0.275604i −0.413945 + 1.68186i −0.848085 1.46893i 1.62959 2.82254i −0.661127 + 0.688767i 1.54713 2.14625i 2.03736i −2.65730 1.39239i 1.55580 0.898244i
89.17 0.872412 + 0.503687i −0.195057 1.72103i −0.492598 0.853205i −0.168910 + 0.292561i 0.696692 1.59970i 2.63835 + 0.197774i 3.00721i −2.92391 + 0.671400i −0.294718 + 0.170156i
89.18 0.942161 + 0.543957i 1.70755 0.290278i −0.408222 0.707062i −1.52055 + 2.63367i 1.76669 + 0.655346i 2.61395 + 0.408977i 3.06405i 2.83148 0.991331i −2.86521 + 1.65423i
89.19 1.14233 + 0.659525i −1.64912 0.529528i −0.130055 0.225261i 0.422862 0.732419i −1.53460 1.69253i −0.990003 2.45355i 2.98120i 2.43920 + 1.74651i 0.966096 0.557776i
89.20 1.33690 + 0.771860i 1.06526 + 1.36573i 0.191534 + 0.331747i 0.895146 1.55044i 0.369997 + 2.64808i −0.181616 + 2.63951i 2.49609i −0.730435 + 2.90972i 2.39344 1.38185i
See all 52 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 122.26 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.d odd 6 1 inner
21.g 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.n.a 52
3.b odd 2 1 inner 231.2.n.a 52
7.d odd 6 1 inner 231.2.n.a 52
21.g even 6 1 inner 231.2.n.a 52

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.n.a 52 1.a even 1 1 trivial
231.2.n.a 52 3.b odd 2 1 inner
231.2.n.a 52 7.d odd 6 1 inner
231.2.n.a 52 21.g even 6 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