Embedded newform 441.2.bd.a.173.26

• Label: 441.2.bd.a.173.26
• Level: $441$
• Weight: $2$
• Character: 441.173
• 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.bd (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			173.26
Character	\(\chi\)	\(=\)	441.173
Dual form			441.2.bd.a.311.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.117375 + 0.172158i) q^{2} +(0.764766 - 1.55407i) q^{3} +(0.714821 + 1.82133i) q^{4} +(-0.967496 + 4.23888i) q^{5} +(0.177781 + 0.314069i) q^{6} +(1.34574 - 2.27794i) q^{7} +(-0.803736 - 0.183448i) q^{8} +(-1.83027 - 2.37700i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.117375	+	0.172158i	−0.0829967	+	0.121734i	−0.865492	−	0.500923i	\(-0.832994\pi\)
							0.782495	+	0.622656i	\(0.213947\pi\)
\(3\)	0.764766	−	1.55407i	0.441538	−	0.897243i
							\(5\)	−0.967496	+	4.23888i	−0.432677	+	1.89568i	0.0118722	+	0.999930i	\(0.496221\pi\)
−0.444550	+	0.895754i	\(0.646636\pi\)
\(7\)	1.34574	−	2.27794i	0.508641	−	0.860979i
							\(11\)	0.959649	+	1.99273i	0.289345	+	0.600831i	0.994081	−	0.108638i	\(-0.0346490\pi\)
−0.704736	+	0.709469i	\(0.748935\pi\)
\(13\)	−0.165869	+	0.243285i	−0.0460037	+	0.0674751i	−0.848552	−	0.529113i	\(-0.822525\pi\)
							0.802548	+	0.596588i	\(0.203477\pi\)
\(17\)	−1.14480	+	2.91691i	−0.277656	+	0.707456i	0.722219	+	0.691664i	\(0.243122\pi\)
							−0.999875	+	0.0157916i	\(0.994973\pi\)
\(19\)	6.44462	+	3.72081i	1.47850	+	0.853611i	0.999704	−	0.0243161i	\(-0.00774081\pi\)
							0.478794	+	0.877927i	\(0.341074\pi\)
\(23\)	2.62072	+	2.08995i	0.546457	+	0.435785i	0.857405	−	0.514642i	\(-0.172075\pi\)
							−0.310948	+	0.950427i	\(0.600647\pi\)
\(29\)	0.182606	+	1.21151i	0.0339091	+	0.224973i	0.999367	−	0.0355885i	\(-0.0113306\pi\)
							−0.965457	+	0.260561i	\(0.916092\pi\)
\(31\)	−3.75144	−	2.16590i	−0.673779	−	0.389007i	0.123728	−	0.992316i	\(-0.460515\pi\)
							−0.797507	+	0.603310i	\(0.793848\pi\)
\(37\)	1.40522	−	0.211804i	0.231018	−	0.0348203i	−0.0325131	−	0.999471i	\(-0.510351\pi\)
							0.263531	+	0.964651i	\(0.415113\pi\)
\(41\)	8.52410	−	2.62934i	1.33124	−	0.410633i	0.454136	−	0.890932i	\(-0.349948\pi\)
							0.877105	+	0.480299i	\(0.159472\pi\)
\(43\)	3.77612	+	1.16478i	0.575852	+	0.177627i	0.568983	−	0.822349i	\(-0.307337\pi\)
							0.00686964	+	0.999976i	\(0.497813\pi\)
\(47\)	1.43860	+	0.980820i	0.209841	+	0.143067i	0.663680	−	0.748016i	\(-0.268994\pi\)
							−0.453839	+	0.891084i	\(0.649946\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.173.26	✓	648	1.1	even	1	trivial
441.2.bd.a.311.26	yes	648	441.311	even	42	inner
441.2.bn.a.164.29	yes	648	49.17	odd	42
441.2.bn.a.320.29	yes	648	9.5	odd	6