Embedded newform 441.2.bh.a.104.9

• Label: 441.2.bh.a.104.9
• 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			104.9
Character	\(\chi\)	\(=\)	441.104
Dual form			441.2.bh.a.335.9

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.41.9	✓	648	49.41	odd	14	inner
441.2.bh.a.104.9	yes	648	1.1	even	1	trivial
441.2.bh.a.335.9	yes	648	441.335	even	42	inner
441.2.bh.a.398.9	yes	648	9.2	odd	6	inner