Embedded newform 441.2.y.a.184.47

• Label: 441.2.y.a.184.47
• Level: $441$
• Weight: $2$
• Character: 441.184
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.y (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			184.47
Character	\(\chi\)	\(=\)	441.184
Dual form			441.2.y.a.151.47

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.97274 + 0.950020i) q^{2} +(-0.880333 + 1.49165i) q^{3} +(1.74217 + 2.18461i) q^{4} +(-1.72336 + 1.59904i) q^{5} +(-3.15376 + 2.10629i) q^{6} +(0.702929 + 2.55066i) q^{7} +(0.386968 + 1.69542i) q^{8} +(-1.45003 - 2.62629i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.97274	+	0.950020i	1.39494	+	0.671765i	0.972127	−	0.234454i	\(-0.0753302\pi\)
							0.422808	+	0.906219i	\(0.361044\pi\)
\(3\)	−0.880333	+	1.49165i	−0.508261	+	0.861203i
							\(5\)	−1.72336	+	1.59904i	−0.770710	+	0.715114i	−0.963963	−	0.266035i	\(-0.914286\pi\)
0.193254	+	0.981149i	\(0.438096\pi\)
\(7\)	0.702929	+	2.55066i	0.265682	+	0.964061i
							\(11\)	1.75138	−	1.19407i	0.528061	−	0.360026i	−0.269753	−	0.962930i	\(-0.586942\pi\)
0.797814	+	0.602904i	\(0.205990\pi\)
\(13\)	0.407923	−	0.278117i	0.113138	−	0.0771358i	−0.505429	−	0.862868i	\(-0.668666\pi\)
							0.618567	+	0.785732i	\(0.287714\pi\)
\(17\)	−1.06254	−	0.160152i	−0.257704	−	0.0388426i	0.0189196	−	0.999821i	\(-0.493977\pi\)
							−0.276623	+	0.960978i	\(0.589215\pi\)
\(19\)	−1.55406	+	2.69170i	−0.356525	+	0.617519i	−0.987378	−	0.158383i	\(-0.949372\pi\)
							0.630853	+	0.775903i	\(0.282705\pi\)
\(23\)	−1.66191	+	4.23447i	−0.346532	+	0.882949i	0.646262	+	0.763116i	\(0.276331\pi\)
							−0.992794	+	0.119833i	\(0.961764\pi\)
\(29\)	0.161216	+	0.0242995i	0.0299371	+	0.00451230i	0.163994	−	0.986461i	\(-0.447562\pi\)
							−0.134057	+	0.990974i	\(0.542800\pi\)
\(31\)	6.34885			1.14029			0.570144	−	0.821545i	\(-0.306887\pi\)
							0.570144	+	0.821545i	\(0.306887\pi\)
\(37\)	2.77018	+	7.05831i	0.455415	+	1.16038i	0.955654	+	0.294492i	\(0.0951504\pi\)
							−0.500238	+	0.865888i	\(0.666754\pi\)
\(41\)	1.49270	+	0.460437i	0.233121	+	0.0719082i	0.409114	−	0.912483i	\(-0.365838\pi\)
							−0.175994	+	0.984391i	\(0.556314\pi\)
\(43\)	9.57531	−	2.95359i	1.46022	−	0.450419i	0.539949	−	0.841698i	\(-0.318444\pi\)
							0.920272	+	0.391279i	\(0.127967\pi\)
\(47\)	4.11892	+	1.98356i	0.600806	+	0.289333i	0.709459	−	0.704747i	\(-0.248940\pi\)
							−0.108653	+	0.994080i	\(0.534654\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.184.47	yes	648
9.7	even	3		441.2.z.a.331.8	yes	648
49.4	even	21		441.2.z.a.4.8	yes	648
441.151	even	21	inner	441.2.y.a.151.47	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.47	✓	648	441.151	even	21	inner
441.2.y.a.184.47	yes	648	1.1	even	1	trivial
441.2.z.a.4.8	yes	648	49.4	even	21
441.2.z.a.331.8	yes	648	9.7	even	3