Embedded newform 441.2.bh.a.335.9

• Label: 441.2.bh.a.335.9
• Level: $441$
• Weight: $2$
• Character: 441.335
• 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			335.9
Character	\(\chi\)	\(=\)	441.335
Dual form			441.2.bh.a.104.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.314877 - 2.08907i) q^{2} +(-1.18752 + 1.26087i) q^{3} +(-2.35394 + 0.726094i) q^{4} +(0.866082 + 0.590485i) q^{5} +(3.00798 + 2.08380i) q^{6} +(-1.38205 + 2.25609i) q^{7} +(0.424760 + 0.882023i) q^{8} +(-0.179587 - 2.99462i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.314877	−	2.08907i	−0.222652	−	1.47720i	−0.768789	−	0.639503i	\(-0.779140\pi\)
							0.546137	−	0.837696i	\(-0.316098\pi\)
\(3\)	−1.18752	+	1.26087i	−0.685616	+	0.727964i
							\(5\)	0.866082	+	0.590485i	0.387324	+	0.264073i	0.741289	−	0.671186i	\(-0.234215\pi\)
−0.353966	+	0.935258i	\(0.615167\pi\)
\(7\)	−1.38205	+	2.25609i	−0.522366	+	0.852722i
							\(11\)	−0.706724	−	4.68881i	−0.213085	−	1.41373i	−0.798011	−	0.602642i	\(-0.794115\pi\)
0.584926	−	0.811087i	\(-0.301124\pi\)
\(13\)	1.79939	+	0.706207i	0.499060	+	0.195867i	0.601500	−	0.798873i	\(-0.294570\pi\)
							−0.102440	+	0.994739i	\(0.532665\pi\)
\(17\)	−0.344193	−	1.50801i	−0.0834790	−	0.365745i	0.915884	−	0.401444i	\(-0.131492\pi\)
							−0.999363	+	0.0356986i	\(0.988634\pi\)
\(19\)		−	7.78644i		−	1.78633i	−0.449728	−	0.893165i	\(-0.648479\pi\)
							0.449728	−	0.893165i	\(-0.351521\pi\)
\(23\)	−1.65634	−	5.36972i	−0.345370	−	1.11966i	−0.947093	−	0.320958i	\(-0.895995\pi\)
							0.601723	−	0.798705i	\(-0.294481\pi\)
\(29\)	0.367391	−	1.19105i	0.0682229	−	0.221173i	−0.915041	−	0.403361i	\(-0.867842\pi\)
							0.983264	+	0.182188i	\(0.0583180\pi\)
\(31\)	−3.80980	+	2.19959i	−0.684260	+	0.395058i	−0.801458	−	0.598051i	\(-0.795942\pi\)
							0.117198	+	0.993109i	\(0.462609\pi\)
\(37\)	0.460052	+	2.01562i	0.0756321	+	0.331366i	0.998562	−	0.0536011i	\(-0.0170700\pi\)
							−0.922930	+	0.384967i	\(0.874213\pi\)
\(41\)	−0.976167	−	0.665539i	−0.152452	−	0.103940i	0.484692	−	0.874685i	\(-0.338932\pi\)
							−0.637143	+	0.770745i	\(0.719884\pi\)
\(43\)	−6.36991	+	4.34293i	−0.971402	+	0.662291i	−0.941412	−	0.337259i	\(-0.890500\pi\)
							−0.0299906	+	0.999550i	\(0.509548\pi\)
\(47\)	7.44329	−	1.12190i	1.08572	−	0.163645i	0.418284	−	0.908316i	\(-0.362632\pi\)
							0.667432	+	0.744671i	\(0.267394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.335.9	yes	648
9.5	odd	6	inner	441.2.bh.a.41.9	✓	648
49.6	odd	14	inner	441.2.bh.a.398.9	yes	648
441.104	even	42	inner	441.2.bh.a.104.9	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.41.9	✓	648	9.5	odd	6	inner
441.2.bh.a.104.9	yes	648	441.104	even	42	inner
441.2.bh.a.335.9	yes	648	1.1	even	1	trivial
441.2.bh.a.398.9	yes	648	49.6	odd	14	inner