Embedded newform 441.2.y.a.88.52

• Label: 441.2.y.a.88.52
• Level: $441$
• Weight: $2$
• Character: 441.88
• 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			88.52
Character	\(\chi\)	\(=\)	441.88
Dual form			441.2.y.a.436.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.65537 - 2.07577i) q^{2} +(-0.0384742 + 1.73162i) q^{3} +(-1.12353 - 4.92251i) q^{4} +(2.48042 - 1.69112i) q^{5} +(3.53077 + 2.94635i) q^{6} +(-2.34690 - 1.22150i) q^{7} +(-7.29370 - 3.51246i) q^{8} +(-2.99704 - 0.133246i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.65537	−	2.07577i	1.17053	−	1.46779i	0.315715	−	0.948854i	\(-0.397756\pi\)
							0.854811	−	0.518939i	\(-0.173673\pi\)
\(3\)	−0.0384742	+	1.73162i	−0.0222131	+	0.999753i
							\(5\)	2.48042	−	1.69112i	1.10928	−	0.756292i	0.137104	−	0.990557i	\(-0.456221\pi\)
0.972173	+	0.234265i	\(0.0752682\pi\)
\(7\)	−2.34690	−	1.22150i	−0.887044	−	0.461684i
							\(11\)	−0.224407	−	0.0338239i	−0.0676612	−	0.0101983i	0.115124	−	0.993351i	\(-0.463273\pi\)
−0.182786	+	0.983153i	\(0.558511\pi\)
\(13\)	5.02628	+	0.757589i	1.39404	+	0.210117i	0.802737	−	0.596333i	\(-0.203376\pi\)
							0.591301	+	0.806451i	\(0.298614\pi\)
\(17\)	−3.43055	−	3.18309i	−0.832032	−	0.772013i	0.144015	−	0.989576i	\(-0.453999\pi\)
							−0.976046	+	0.217563i	\(0.930189\pi\)
\(19\)	2.97385	+	5.15086i	0.682248	+	1.18169i	0.974293	+	0.225283i	\(0.0723308\pi\)
							−0.292046	+	0.956404i	\(0.594336\pi\)
\(23\)	5.34475	+	1.64864i	1.11446	+	0.343765i	0.796680	−	0.604401i	\(-0.206587\pi\)
							0.317777	+	0.948165i	\(0.397064\pi\)
\(29\)	0.593212	+	0.550420i	0.110157	+	0.102210i	0.733328	−	0.679875i	\(-0.237966\pi\)
							−0.623172	+	0.782085i	\(0.714156\pi\)
\(31\)	−0.722977			−0.129851			−0.0649253	−	0.997890i	\(-0.520681\pi\)
							−0.0649253	+	0.997890i	\(0.520681\pi\)
\(37\)	1.12677	−	0.347564i	0.185240	−	0.0571391i	−0.200747	−	0.979643i	\(-0.564337\pi\)
							0.385988	+	0.922504i	\(0.373861\pi\)
\(41\)	0.121521	+	1.62158i	0.0189783	+	0.253248i	0.998621	+	0.0524990i	\(0.0167186\pi\)
							−0.979643	+	0.200749i	\(0.935662\pi\)
\(43\)	−0.572889	+	7.64467i	−0.0873647	+	1.16580i	0.765417	+	0.643534i	\(0.222533\pi\)
							−0.852782	+	0.522267i	\(0.825086\pi\)
\(47\)	−5.05365	+	6.33707i	−0.737150	+	0.924357i	−0.999171	−	0.0407053i	\(-0.987040\pi\)
							0.262021	+	0.965062i	\(0.415611\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.88.52	✓	648
9.4	even	3		441.2.z.a.382.52	yes	648
49.44	even	21		441.2.z.a.142.52	yes	648
441.436	even	21	inner	441.2.y.a.436.52	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.52	✓	648	1.1	even	1	trivial
441.2.y.a.436.52	yes	648	441.436	even	21	inner
441.2.z.a.142.52	yes	648	49.44	even	21
441.2.z.a.382.52	yes	648	9.4	even	3