Embedded newform 441.2.i.a.227.1

• Label: 441.2.i.a.227.1
• Level: $441$
• Weight: $2$
• Character: 441.227
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			227.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	441.227
Dual form			441.2.i.a.68.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} +1.73205i q^{3} -1.00000 q^{4} +(1.50000 - 2.59808i) q^{5} +3.00000 q^{6} -1.73205i q^{8} -3.00000 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 3 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.73205i		−	1.22474i	−0.790569	−	0.612372i	\(-0.790215\pi\)
							0.790569	−	0.612372i	\(-0.209785\pi\)
\(3\)			1.73205i			1.00000i
							\(5\)	1.50000	−	2.59808i	0.670820	−	1.16190i	−0.306851	−	0.951757i	\(-0.599275\pi\)
0.977672	−	0.210138i	\(-0.0673912\pi\)
\(7\)		0			0
							\(11\)	1.50000	−	0.866025i	0.452267	−	0.261116i	−0.256520	−	0.966539i	\(-0.582576\pi\)
0.708787	+	0.705422i	\(0.249243\pi\)
\(13\)	−1.50000	+	0.866025i	−0.416025	+	0.240192i	−0.693375	−	0.720577i	\(-0.743877\pi\)
							0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	\(-0.714811\pi\)
							0.988583	+	0.150675i	\(0.0481447\pi\)
\(19\)	4.50000	−	2.59808i	1.03237	−	0.596040i	0.114708	−	0.993399i	\(-0.463407\pi\)
							0.917663	+	0.397360i	\(0.130073\pi\)
\(23\)	4.50000	+	2.59808i	0.938315	+	0.541736i	0.889432	−	0.457068i	\(-0.151100\pi\)
							0.0488832	+	0.998805i	\(0.484434\pi\)
\(29\)	−4.50000	−	2.59808i	−0.835629	−	0.482451i	0.0201471	−	0.999797i	\(-0.493587\pi\)
							−0.855776	+	0.517346i	\(0.826920\pi\)
\(31\)			3.46410i			0.622171i	0.950382	+	0.311086i	\(0.100693\pi\)
							−0.950382	+	0.311086i	\(0.899307\pi\)
\(37\)	−3.50000	−	6.06218i	−0.575396	−	0.996616i	−0.995998	−	0.0893706i	\(-0.971514\pi\)
							0.420602	−	0.907245i	\(-0.361819\pi\)
\(41\)	1.50000	+	2.59808i	0.234261	+	0.405751i	0.959058	−	0.283211i	\(-0.0913998\pi\)
							−0.724797	+	0.688963i	\(0.758066\pi\)
\(43\)	−0.500000	+	0.866025i	−0.0762493	+	0.132068i	−0.901629	−	0.432511i	\(-0.857628\pi\)
							0.825380	+	0.564578i	\(0.190961\pi\)
\(47\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.i.a.227.1		2
3.2	odd	2		1323.2.i.a.521.1		2
7.2	even	3		63.2.s.a.47.1	yes	2
7.3	odd	6		441.2.o.a.146.1		2
7.4	even	3		441.2.o.b.146.1		2
7.5	odd	6		441.2.s.a.362.1		2
7.6	odd	2		63.2.i.a.38.1	yes	2
9.4	even	3		1323.2.s.a.962.1		2
9.5	odd	6		441.2.s.a.374.1		2
21.2	odd	6		189.2.s.a.89.1		2
21.5	even	6		1323.2.s.a.656.1		2
21.11	odd	6		1323.2.o.a.440.1		2
21.17	even	6		1323.2.o.b.440.1		2
21.20	even	2		189.2.i.a.143.1		2
28.23	odd	6		1008.2.df.a.929.1		2
28.27	even	2		1008.2.ca.a.353.1		2
63.2	odd	6		567.2.p.a.404.1		2
63.4	even	3		1323.2.o.b.881.1		2
63.5	even	6	inner	441.2.i.a.68.1		2
63.13	odd	6		189.2.s.a.17.1		2
63.16	even	3		567.2.p.b.404.1		2
63.20	even	6		567.2.p.b.80.1		2
63.23	odd	6		63.2.i.a.5.1	✓	2
63.31	odd	6		1323.2.o.a.881.1		2
63.32	odd	6		441.2.o.a.293.1		2
63.34	odd	6		567.2.p.a.80.1		2
63.40	odd	6		1323.2.i.a.1097.1		2
63.41	even	6		63.2.s.a.59.1	yes	2
63.58	even	3		189.2.i.a.152.1		2
63.59	even	6		441.2.o.b.293.1		2
84.23	even	6		3024.2.df.a.1601.1		2
84.83	odd	2		3024.2.ca.a.2033.1		2
252.23	even	6		1008.2.ca.a.257.1		2
252.139	even	6		3024.2.df.a.17.1		2
252.167	odd	6		1008.2.df.a.689.1		2
252.247	odd	6		3024.2.ca.a.2609.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.i.a.5.1	✓	2	63.23	odd	6
63.2.i.a.38.1	yes	2	7.6	odd	2
63.2.s.a.47.1	yes	2	7.2	even	3
63.2.s.a.59.1	yes	2	63.41	even	6
189.2.i.a.143.1		2	21.20	even	2
189.2.i.a.152.1		2	63.58	even	3
189.2.s.a.17.1		2	63.13	odd	6
189.2.s.a.89.1		2	21.2	odd	6
441.2.i.a.68.1		2	63.5	even	6	inner
441.2.i.a.227.1		2	1.1	even	1	trivial
441.2.o.a.146.1		2	7.3	odd	6
441.2.o.a.293.1		2	63.32	odd	6
441.2.o.b.146.1		2	7.4	even	3
441.2.o.b.293.1		2	63.59	even	6
441.2.s.a.362.1		2	7.5	odd	6
441.2.s.a.374.1		2	9.5	odd	6
567.2.p.a.80.1		2	63.34	odd	6
567.2.p.a.404.1		2	63.2	odd	6
567.2.p.b.80.1		2	63.20	even	6
567.2.p.b.404.1		2	63.16	even	3
1008.2.ca.a.257.1		2	252.23	even	6
1008.2.ca.a.353.1		2	28.27	even	2
1008.2.df.a.689.1		2	252.167	odd	6
1008.2.df.a.929.1		2	28.23	odd	6
1323.2.i.a.521.1		2	3.2	odd	2
1323.2.i.a.1097.1		2	63.40	odd	6
1323.2.o.a.440.1		2	21.11	odd	6
1323.2.o.a.881.1		2	63.31	odd	6
1323.2.o.b.440.1		2	21.17	even	6
1323.2.o.b.881.1		2	63.4	even	3
1323.2.s.a.656.1		2	21.5	even	6
1323.2.s.a.962.1		2	9.4	even	3
3024.2.ca.a.2033.1		2	84.83	odd	2
3024.2.ca.a.2609.1		2	252.247	odd	6
3024.2.df.a.17.1		2	252.139	even	6
3024.2.df.a.1601.1		2	84.23	even	6