Embedded newform 441.2.bd.a.248.32

• Label: 441.2.bd.a.248.32
• Level: $441$
• Weight: $2$
• Character: 441.248
• 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			248.32
Character	\(\chi\)	\(=\)	441.248
Dual form			441.2.bd.a.425.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0799818 + 0.530645i) q^{2} +(-1.27082 + 1.17687i) q^{3} +(1.63596 - 0.504626i) q^{4} +(1.46542 - 0.705708i) q^{5} +(-0.726141 - 0.580226i) q^{6} +(0.713839 - 2.54763i) q^{7} +(0.864302 + 1.79474i) q^{8} +(0.229966 - 2.99117i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0799818	+	0.530645i	0.0565557	+	0.375223i	0.999148	+	0.0412762i	\(0.0131424\pi\)
							−0.942592	+	0.333946i	\(0.891620\pi\)
\(3\)	−1.27082	+	1.17687i	−0.733708	+	0.679465i
							\(5\)	1.46542	−	0.705708i	0.655355	−	0.315602i	−0.0764933	−	0.997070i	\(-0.524372\pi\)
0.731848	+	0.681468i	\(0.238658\pi\)
\(7\)	0.713839	−	2.54763i	0.269806	−	0.962915i
							\(11\)	1.29535	−	1.03300i	0.390561	−	0.311462i	−0.408447	−	0.912782i	\(-0.633930\pi\)
0.799009	+	0.601320i	\(0.205358\pi\)
\(13\)	−0.828284	−	5.49530i	−0.229725	−	1.52412i	−0.744980	−	0.667087i	\(-0.767541\pi\)
							0.515255	−	0.857037i	\(-0.327697\pi\)
\(17\)	−4.27109	−	1.31746i	−1.03589	−	0.319530i	−0.270245	−	0.962791i	\(-0.587105\pi\)
							−0.765647	+	0.643261i	\(0.777581\pi\)
\(19\)	−0.895437	+	0.516981i	−0.205427	+	0.118604i	−0.599184	−	0.800611i	\(-0.704508\pi\)
							0.393757	+	0.919214i	\(0.371175\pi\)
\(23\)	3.86676	+	0.882564i	0.806276	+	0.184027i	0.605744	−	0.795660i	\(-0.292876\pi\)
							0.200532	+	0.979687i	\(0.435733\pi\)
\(29\)	2.66342	+	2.87049i	0.494585	+	0.533036i	0.930178	−	0.367108i	\(-0.119652\pi\)
							−0.435593	+	0.900144i	\(0.643461\pi\)
\(31\)	1.88162	−	1.08635i	0.337949	−	0.195115i	−0.321416	−	0.946938i	\(-0.604159\pi\)
							0.659364	+	0.751823i	\(0.270825\pi\)
\(37\)	−4.26592	+	3.95820i	−0.701313	+	0.650724i	−0.947963	−	0.318381i	\(-0.896861\pi\)
							0.246650	+	0.969105i	\(0.420670\pi\)
\(41\)	−0.814616	+	10.8703i	−0.127222	+	1.69766i	0.459751	+	0.888048i	\(0.347939\pi\)
							−0.586972	+	0.809607i	\(0.699680\pi\)
\(43\)	0.271390	+	3.62145i	0.0413866	+	0.552265i	0.978903	+	0.204325i	\(0.0654999\pi\)
							−0.937517	+	0.347941i	\(0.886881\pi\)
\(47\)	4.53251	−	0.683166i	0.661135	−	0.0996501i	0.190100	−	0.981765i	\(-0.439119\pi\)
							0.471035	+	0.882115i	\(0.343881\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.248.32	✓	648
9.2	odd	6		441.2.bn.a.101.23	yes	648
49.33	odd	42		441.2.bn.a.131.23	yes	648
441.425	even	42	inner	441.2.bd.a.425.32	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.248.32	✓	648	1.1	even	1	trivial
441.2.bd.a.425.32	yes	648	441.425	even	42	inner
441.2.bn.a.101.23	yes	648	9.2	odd	6
441.2.bn.a.131.23	yes	648	49.33	odd	42