Embedded newform 441.2.ba.a.337.45

• Label: 441.2.ba.a.337.45
• Level: $441$
• Weight: $2$
• Character: 441.337
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.ba (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			337.45
Character	\(\chi\)	\(=\)	441.337
Dual form			441.2.ba.a.106.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.89026 + 0.284911i) q^{2} +(1.05095 + 1.37677i) q^{3} +(1.58076 + 0.487598i) q^{4} +(-0.509493 + 0.347367i) q^{5} +(1.59432 + 2.90188i) q^{6} +(2.55194 + 0.698271i) q^{7} +(-0.595487 - 0.286771i) q^{8} +(-0.790990 + 2.89384i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.89026	+	0.284911i	1.33661	+	0.201462i	0.778156	−	0.628071i	\(-0.216155\pi\)
							0.558458	+	0.829533i	\(0.311393\pi\)
\(3\)	1.05095	+	1.37677i	0.606769	+	0.794878i
							\(5\)	−0.509493	+	0.347367i	−0.227852	+	0.155347i	−0.671866	−	0.740673i	\(-0.734507\pi\)
0.444013	+	0.896020i	\(0.353554\pi\)
\(7\)	2.55194	+	0.698271i	0.964544	+	0.263922i
							\(11\)	1.83914	+	0.277206i	0.554522	+	0.0835807i	0.420324	−	0.907374i	\(-0.361916\pi\)
0.134197	+	0.990955i	\(0.457154\pi\)
\(13\)	−1.35293	−	3.44722i	−0.375236	−	0.956086i	−0.986368	−	0.164557i	\(-0.947381\pi\)
							0.611132	−	0.791529i	\(-0.290715\pi\)
\(17\)	0.108842	−	0.476870i	0.0263982	−	0.115658i	−0.960012	−	0.279958i	\(-0.909679\pi\)
							0.986410	+	0.164300i	\(0.0525366\pi\)
\(19\)	−4.81600			−1.10487			−0.552433	−	0.833557i	\(-0.686300\pi\)
							−0.552433	+	0.833557i	\(0.686300\pi\)
\(23\)	2.49946	+	0.770981i	0.521173	+	0.160761i	0.544170	−	0.838975i	\(-0.316845\pi\)
							−0.0229968	+	0.999736i	\(0.507321\pi\)
\(29\)	9.29350	−	2.86667i	1.72576	−	0.532326i	0.736239	−	0.676721i	\(-0.236600\pi\)
							0.989520	+	0.144395i	\(0.0461235\pi\)
\(31\)	3.05299	−	5.28793i	0.548333	−	0.949741i	−0.450056	−	0.893000i	\(-0.648596\pi\)
							0.998389	−	0.0567403i	\(-0.0180707\pi\)
\(37\)	1.66442	−	7.29229i	0.273629	−	1.19885i	−0.632066	−	0.774915i	\(-0.717793\pi\)
							0.905695	−	0.423931i	\(-0.139350\pi\)
\(41\)	−7.15658	+	4.87927i	−1.11767	+	0.762015i	−0.973760	−	0.227576i	\(-0.926920\pi\)
							−0.143910	+	0.989591i	\(0.545968\pi\)
\(43\)	−5.57669	−	3.80212i	−0.850437	−	0.579818i	0.0577747	−	0.998330i	\(-0.481599\pi\)
							−0.908211	+	0.418512i	\(0.862552\pi\)
\(47\)	−0.579126	−	0.0872892i	−0.0844742	−	0.0127324i	0.106669	−	0.994295i	\(-0.465981\pi\)
							−0.191143	+	0.981562i	\(0.561220\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.337.45	yes	648
9.7	even	3	inner	441.2.ba.a.43.10	✓	648
49.8	even	7	inner	441.2.ba.a.400.10	yes	648
441.106	even	21	inner	441.2.ba.a.106.45	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.43.10	✓	648	9.7	even	3	inner
441.2.ba.a.106.45	yes	648	441.106	even	21	inner
441.2.ba.a.337.45	yes	648	1.1	even	1	trivial
441.2.ba.a.400.10	yes	648	49.8	even	7	inner