Embedded newform 441.2.bn.a.164.29

• Label: 441.2.bn.a.164.29
• Level: $441$
• Weight: $2$
• Character: 441.164
• 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.bn (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			164.29
Character	\(\chi\)	\(=\)	441.164
Dual form			441.2.bn.a.320.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0904053 - 0.187729i) q^{2} +(-1.50732 + 0.853225i) q^{3} +(1.21991 + 1.52972i) q^{4} +(-4.15472 - 1.28156i) q^{5} +(0.0239052 + 0.360103i) q^{6} +(1.29988 - 2.30441i) q^{7} +(0.803736 - 0.183448i) q^{8} +(1.54401 - 2.57216i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0904053	−	0.187729i	0.0639262	−	0.132744i	−0.866558	−	0.499077i	\(-0.833673\pi\)
							0.930484	+	0.366333i	\(0.119387\pi\)
\(3\)	−1.50732	+	0.853225i	−0.870250	+	0.492610i
							\(5\)	−4.15472	−	1.28156i	−1.85805	−	0.573132i	−0.998021	−	0.0628858i	\(-0.979970\pi\)
−0.860028	−	0.510246i	\(-0.829554\pi\)
\(7\)	1.29988	−	2.30441i	0.491309	−	0.870985i
							\(11\)	2.20558	−	0.165285i	0.665007	−	0.0498354i	0.262050	−	0.965054i	\(-0.415601\pi\)
0.402957	+	0.915219i	\(0.367982\pi\)
\(13\)	0.293625	−	0.0220042i	0.0814370	−	0.00610286i	−0.0339493	−	0.999424i	\(-0.510808\pi\)
							0.115386	+	0.993321i	\(0.463189\pi\)
\(17\)	1.14480	+	2.91691i	0.277656	+	0.707456i	0.999875	+	0.0157916i	\(0.00502682\pi\)
							−0.722219	+	0.691664i	\(0.756878\pi\)
\(19\)	6.44462	−	3.72081i	1.47850	−	0.853611i	0.478794	−	0.877927i	\(-0.341074\pi\)
							0.999704	+	0.0243161i	\(0.00774081\pi\)
\(23\)	−0.499593	−	3.31458i	−0.104172	−	0.691138i	−0.978568	−	0.205925i	\(-0.933980\pi\)
							0.874396	−	0.485214i	\(-0.161258\pi\)
\(29\)	1.14051	−	0.447615i	0.211786	−	0.0831201i	−0.257076	−	0.966391i	\(-0.582759\pi\)
							0.468863	+	0.883271i	\(0.344664\pi\)
\(31\)		−	4.33179i		−	0.778013i	−0.921235	−	0.389007i	\(-0.872818\pi\)
							0.921235	−	0.389007i	\(-0.127182\pi\)
\(37\)	1.40522	+	0.211804i	0.231018	+	0.0348203i	0.263531	−	0.964651i	\(-0.415113\pi\)
							−0.0325131	+	0.999471i	\(0.510351\pi\)
\(41\)	6.53912	−	6.06742i	1.02124	−	0.947572i	0.0226012	−	0.999745i	\(-0.492805\pi\)
							0.998638	+	0.0521730i	\(0.0166147\pi\)
\(43\)	−2.89679	−	2.68782i	−0.441756	−	0.409889i	0.427684	−	0.903928i	\(-0.359330\pi\)
							−0.869440	+	0.494039i	\(0.835520\pi\)
\(47\)	1.56871	+	0.755453i	0.228820	+	0.110194i	0.544781	−	0.838578i	\(-0.316613\pi\)
							−0.315961	+	0.948772i	\(0.602327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.164.29	yes	648
9.5	odd	6		441.2.bd.a.311.26	yes	648
49.26	odd	42		441.2.bd.a.173.26	✓	648
441.320	even	42	inner	441.2.bn.a.320.29	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.173.26	✓	648	49.26	odd	42
441.2.bd.a.311.26	yes	648	9.5	odd	6
441.2.bn.a.164.29	yes	648	1.1	even	1	trivial
441.2.bn.a.320.29	yes	648	441.320	even	42	inner