Embedded newform 441.2.bh.a.104.42

• Label: 441.2.bh.a.104.42
• Level: $441$
• Weight: $2$
• Character: 441.104
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.bh (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			104.42
Character	\(\chi\)	\(=\)	441.104
Dual form			441.2.bh.a.335.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.244765 - 1.62391i) q^{2} +(-1.73002 - 0.0837976i) q^{3} +(-0.666033 - 0.205444i) q^{4} +(-0.532661 + 0.363162i) q^{5} +(-0.559529 + 2.78889i) q^{6} +(1.40053 + 2.24466i) q^{7} +(0.928450 - 1.92795i) q^{8} +(2.98596 + 0.289944i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.244765	−	1.62391i	0.173075	−	1.14828i	−0.718540	−	0.695485i	\(-0.755189\pi\)
							0.891615	−	0.452793i	\(-0.149572\pi\)
\(3\)	−1.73002	−	0.0837976i	−0.998829	−	0.0483806i
							\(5\)	−0.532661	+	0.363162i	−0.238213	+	0.162411i	−0.676543	−	0.736403i	\(-0.736523\pi\)
0.438330	+	0.898814i	\(0.355570\pi\)
\(7\)	1.40053	+	2.24466i	0.529351	+	0.848403i
							\(11\)	0.457227	−	3.03350i	0.137859	−	0.914635i	−0.807796	−	0.589463i	\(-0.799340\pi\)
0.945655	−	0.325172i	\(-0.105422\pi\)
\(13\)	2.65209	−	1.04087i	0.735559	−	0.288686i	0.0321616	−	0.999483i	\(-0.489761\pi\)
							0.703397	+	0.710797i	\(0.251666\pi\)
\(17\)	−0.255080	+	1.11758i	−0.0618660	+	0.271053i	−0.996395	−	0.0848340i	\(-0.972964\pi\)
							0.934529	+	0.355887i	\(0.115821\pi\)
\(19\)		−	2.78250i		−	0.638350i	−0.947696	−	0.319175i	\(-0.896594\pi\)
							0.947696	−	0.319175i	\(-0.103406\pi\)
\(23\)	2.56760	−	8.32397i	0.535382	−	1.73567i	−0.132289	−	0.991211i	\(-0.542233\pi\)
							0.667671	−	0.744456i	\(-0.267291\pi\)
\(29\)	0.370056	+	1.19969i	0.0687177	+	0.222777i	0.983417	−	0.181357i	\(-0.0580490\pi\)
							−0.914700	+	0.404134i	\(0.867573\pi\)
\(31\)	4.77494	+	2.75681i	0.857604	+	0.495138i	0.863209	−	0.504847i	\(-0.168451\pi\)
							−0.00560541	+	0.999984i	\(0.501784\pi\)
\(37\)	2.37473	−	10.4044i	0.390403	−	1.71047i	−0.272836	−	0.962061i	\(-0.587962\pi\)
							0.663239	−	0.748408i	\(-0.269181\pi\)
\(41\)	−3.15378	+	2.15021i	−0.492537	+	0.335806i	−0.783984	−	0.620781i	\(-0.786816\pi\)
							0.291447	+	0.956587i	\(0.405863\pi\)
\(43\)	1.21486	+	0.828275i	0.185264	+	0.126311i	0.652401	−	0.757874i	\(-0.273762\pi\)
							−0.467137	+	0.884185i	\(0.654714\pi\)
\(47\)	4.72501	+	0.712181i	0.689214	+	0.103882i	0.484306	−	0.874899i	\(-0.339072\pi\)
							0.204908	+	0.978781i	\(0.434310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.104.42	yes	648
9.2	odd	6	inner	441.2.bh.a.398.42	yes	648
49.41	odd	14	inner	441.2.bh.a.41.42	✓	648
441.335	even	42	inner	441.2.bh.a.335.42	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.41.42	✓	648	49.41	odd	14	inner
441.2.bh.a.104.42	yes	648	1.1	even	1	trivial
441.2.bh.a.335.42	yes	648	441.335	even	42	inner
441.2.bh.a.398.42	yes	648	9.2	odd	6	inner