Newform orbit 231.2.n.a

• 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: $4$

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 \(\Q(\zeta_{6})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 24 q^{4} - 6 q^{7} - 4 q^{9}+O(q^{10}) \)

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

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 \(S_{2}^{\mathrm{new}}(231, [\chi])\).