Embedded newform 441.2.g.g.67.5

• Label: 441.2.g.g.67.5
• Level: $441$
• Weight: $2$
• Character: 441.67
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			67.5
Root			\(0.474636 + 0.274031i\) of defining polynomial
Character	\(\chi\)	\(=\)	441.67
Dual form			441.2.g.g.79.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.849814 - 1.47192i) q^{2} +(-1.58016 + 0.709292i) q^{3} +(-0.444368 - 0.769668i) q^{4} -0.949271 q^{5} +(-0.298820 + 2.92864i) q^{6} +1.88874 q^{8} +(1.99381 - 2.24159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} - 6 q^{4} + 24 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.849814	−	1.47192i	0.600909	−	1.04081i	−0.391774	−	0.920061i	\(-0.628139\pi\)
							0.992684	−	0.120744i	\(-0.0385280\pi\)
\(3\)	−1.58016	+	0.709292i	−0.912306	+	0.409510i
							\(5\)	−0.949271			−0.424527			−0.212263	−	0.977212i	\(-0.568084\pi\)
−0.212263	+	0.977212i	\(0.568084\pi\)
\(7\)		0			0
							\(11\)	−0.588364			−0.177398			−0.0886992	−	0.996058i	\(-0.528271\pi\)
−0.0886992	+	0.996058i	\(0.528271\pi\)
\(13\)	2.50987	−	4.34722i	0.696112	−	1.20570i	−0.273692	−	0.961817i	\(-0.588245\pi\)
							0.969804	−	0.243885i	\(-0.0784218\pi\)
\(17\)	3.79121	−	6.56657i	0.919503	−	1.59263i	0.119332	−	0.992854i	\(-0.461925\pi\)
							0.800171	−	0.599772i	\(-0.204742\pi\)
\(19\)	2.23061	+	3.86353i	0.511737	+	0.886355i	0.999907	+	0.0136063i	\(0.00433116\pi\)
							−0.488170	+	0.872748i	\(0.662336\pi\)
\(23\)	2.47710			0.516511			0.258256	−	0.966077i	\(-0.416852\pi\)
							0.258256	+	0.966077i	\(0.416852\pi\)
\(29\)	−2.73855	−	4.74331i	−0.508536	−	0.880810i	−0.999951	−	0.00988468i	\(-0.996854\pi\)
							0.491415	−	0.870925i	\(-0.336480\pi\)
\(31\)	3.03731	+	5.26078i	0.545518	+	0.944865i	0.998574	+	0.0533826i	\(0.0170003\pi\)
							−0.453056	+	0.891482i	\(0.649666\pi\)
\(37\)	3.49381	+	6.05146i	0.574379	+	0.994853i	0.996109	+	0.0881319i	\(0.0280897\pi\)
							−0.421730	+	0.906721i	\(0.638577\pi\)
\(41\)	0.527445	−	0.913562i	0.0823731	−	0.142674i	−0.821896	−	0.569638i	\(-0.807083\pi\)
							0.904269	+	0.426964i	\(0.140417\pi\)
\(43\)	−3.49381	−	6.05146i	−0.532801	−	0.922838i	−0.999266	−	0.0382990i	\(-0.987806\pi\)
							0.466465	−	0.884540i	\(-0.345527\pi\)
\(47\)	3.73840	−	6.47510i	0.545301	−	0.944490i	−0.453286	−	0.891365i	\(-0.649749\pi\)
							0.998588	−	0.0531249i	\(-0.0169181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.g.g.67.5		12
3.2	odd	2		1323.2.g.g.361.2		12
7.2	even	3		441.2.h.g.373.2		12
7.3	odd	6		441.2.f.g.148.5	✓	12
7.4	even	3		441.2.f.g.148.6	yes	12
7.5	odd	6		441.2.h.g.373.1		12
7.6	odd	2	inner	441.2.g.g.67.6		12
9.2	odd	6		1323.2.h.g.802.5		12
9.7	even	3		441.2.h.g.214.2		12
21.2	odd	6		1323.2.h.g.226.5		12
21.5	even	6		1323.2.h.g.226.6		12
21.11	odd	6		1323.2.f.g.442.1		12
21.17	even	6		1323.2.f.g.442.2		12
21.20	even	2		1323.2.g.g.361.1		12
63.2	odd	6		1323.2.g.g.667.2		12
63.4	even	3		3969.2.a.be.1.1		6
63.11	odd	6		1323.2.f.g.883.1		12
63.16	even	3	inner	441.2.g.g.79.5		12
63.20	even	6		1323.2.h.g.802.6		12
63.25	even	3		441.2.f.g.295.6	yes	12
63.31	odd	6		3969.2.a.be.1.2		6
63.32	odd	6		3969.2.a.bd.1.6		6
63.34	odd	6		441.2.h.g.214.1		12
63.38	even	6		1323.2.f.g.883.2		12
63.47	even	6		1323.2.g.g.667.1		12
63.52	odd	6		441.2.f.g.295.5	yes	12
63.59	even	6		3969.2.a.bd.1.5		6
63.61	odd	6	inner	441.2.g.g.79.6		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.f.g.148.5	✓	12	7.3	odd	6
441.2.f.g.148.6	yes	12	7.4	even	3
441.2.f.g.295.5	yes	12	63.52	odd	6
441.2.f.g.295.6	yes	12	63.25	even	3
441.2.g.g.67.5		12	1.1	even	1	trivial
441.2.g.g.67.6		12	7.6	odd	2	inner
441.2.g.g.79.5		12	63.16	even	3	inner
441.2.g.g.79.6		12	63.61	odd	6	inner
441.2.h.g.214.1		12	63.34	odd	6
441.2.h.g.214.2		12	9.7	even	3
441.2.h.g.373.1		12	7.5	odd	6
441.2.h.g.373.2		12	7.2	even	3
1323.2.f.g.442.1		12	21.11	odd	6
1323.2.f.g.442.2		12	21.17	even	6
1323.2.f.g.883.1		12	63.11	odd	6
1323.2.f.g.883.2		12	63.38	even	6
1323.2.g.g.361.1		12	21.20	even	2
1323.2.g.g.361.2		12	3.2	odd	2
1323.2.g.g.667.1		12	63.47	even	6
1323.2.g.g.667.2		12	63.2	odd	6
1323.2.h.g.226.5		12	21.2	odd	6
1323.2.h.g.226.6		12	21.5	even	6
1323.2.h.g.802.5		12	9.2	odd	6
1323.2.h.g.802.6		12	63.20	even	6
3969.2.a.bd.1.5		6	63.59	even	6
3969.2.a.bd.1.6		6	63.32	odd	6
3969.2.a.be.1.1		6	63.4	even	3
3969.2.a.be.1.2		6	63.31	odd	6