Embedded newform 441.2.bh.a.20.13

• Label: 441.2.bh.a.20.13
• Level: $441$
• Weight: $2$
• Character: 441.20
• 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			20.13
Character	\(\chi\)	\(=\)	441.20
Dual form			441.2.bh.a.419.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.979822 - 1.43714i) q^{2} +(-1.66301 + 0.484159i) q^{3} +(-0.374624 + 0.954527i) q^{4} +(0.820873 - 0.761659i) q^{5} +(2.32525 + 1.91558i) q^{6} +(1.31110 + 2.29805i) q^{7} +(-1.65267 + 0.377211i) q^{8} +(2.53118 - 1.61032i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.979822	−	1.43714i	−0.692839	−	1.01621i	−0.997957	−	0.0638936i	\(-0.979648\pi\)
							0.305118	−	0.952315i	\(-0.401304\pi\)
\(3\)	−1.66301	+	0.484159i	−0.960137	+	0.279529i
							\(5\)	0.820873	−	0.761659i	0.367106	−	0.340624i	−0.474952	−	0.880012i	\(-0.657535\pi\)
0.842058	+	0.539387i	\(0.181344\pi\)
\(7\)	1.31110	+	2.29805i	0.495549	+	0.868580i
							\(11\)	−1.65353	−	2.42528i	−0.498558	−	0.731250i	0.491659	−	0.870788i	\(-0.336391\pi\)
−0.990217	+	0.139538i	\(0.955438\pi\)
\(13\)	−7.01455	+	0.525668i	−1.94549	+	0.145794i	−0.989193	−	0.146618i	\(-0.953161\pi\)
							−0.956293	+	0.292412i	\(0.905542\pi\)
\(17\)	−2.36842	+	2.96990i	−0.574426	+	0.720307i	−0.981151	−	0.193244i	\(-0.938099\pi\)
							0.406725	+	0.913551i	\(0.366671\pi\)
\(19\)			1.93904i			0.444847i	0.974950	+	0.222423i	\(0.0713967\pi\)
							−0.974950	+	0.222423i	\(0.928603\pi\)
\(23\)	−0.910453	−	0.357327i	−0.189843	−	0.0745078i	0.268515	−	0.963275i	\(-0.413467\pi\)
							−0.458358	+	0.888768i	\(0.651562\pi\)
\(29\)	3.86783	−	1.51801i	0.718238	−	0.281888i	0.0220601	−	0.999757i	\(-0.492977\pi\)
							0.696178	+	0.717869i	\(0.254882\pi\)
\(31\)	4.33207	−	2.50112i	0.778063	−	0.449215i	−0.0576802	−	0.998335i	\(-0.518370\pi\)
							0.835743	+	0.549120i	\(0.185037\pi\)
\(37\)	−6.35144	+	7.96446i	−1.04417	+	1.30935i	−0.0946954	+	0.995506i	\(0.530188\pi\)
							−0.949475	+	0.313842i	\(0.898384\pi\)
\(41\)	−8.74930	+	8.11816i	−1.36641	+	1.26784i	−0.436673	+	0.899620i	\(0.643843\pi\)
							−0.929737	+	0.368223i	\(0.879966\pi\)
\(43\)	5.43684	+	5.04465i	0.829110	+	0.769302i	0.975517	−	0.219923i	\(-0.0705805\pi\)
							−0.146407	+	0.989224i	\(0.546771\pi\)
\(47\)	4.92467	−	3.35758i	0.718337	−	0.489754i	−0.148143	−	0.988966i	\(-0.547330\pi\)
							0.866480	+	0.499212i	\(0.166377\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.20.13	✓	648
9.5	odd	6	inner	441.2.bh.a.167.13	yes	648
49.27	odd	14	inner	441.2.bh.a.272.13	yes	648
441.419	even	42	inner	441.2.bh.a.419.13	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.20.13	✓	648	1.1	even	1	trivial
441.2.bh.a.167.13	yes	648	9.5	odd	6	inner
441.2.bh.a.272.13	yes	648	49.27	odd	14	inner
441.2.bh.a.419.13	yes	648	441.419	even	42	inner