Embedded newform 441.2.z.a.4.6

• Label: 441.2.z.a.4.6
• Level: $441$
• Weight: $2$
• Character: 441.4
• 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.z (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			4.6
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.173112 + 2.31002i) q^{2} +(-0.732646 + 1.56947i) q^{3} +(-3.32857 - 0.501701i) q^{4} +(-0.445172 + 1.95043i) q^{5} +(-3.49867 - 1.96412i) q^{6} +(0.693731 - 2.55318i) q^{7} +(0.704215 - 3.08537i) q^{8} +(-1.92646 - 2.29973i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.173112	+	2.31002i	−0.122409	+	1.63343i	0.510976	+	0.859595i	\(0.329284\pi\)
							−0.633385	+	0.773837i	\(0.718335\pi\)
\(3\)	−0.732646	+	1.56947i	−0.422994	+	0.906133i
							\(5\)	−0.445172	+	1.95043i	−0.199087	+	0.872258i	0.772395	+	0.635143i	\(0.219059\pi\)
−0.971482	+	0.237115i	\(0.923798\pi\)
\(7\)	0.693731	−	2.55318i	0.262206	−	0.965012i
							\(11\)	−2.98953	+	1.43968i	−0.901377	+	0.434080i	−0.826386	−	0.563104i	\(-0.809607\pi\)
−0.0749906	+	0.997184i	\(0.523893\pi\)
\(13\)	0.277238	−	3.69948i	0.0768919	−	1.02605i	−0.816668	−	0.577107i	\(-0.804181\pi\)
							0.893560	−	0.448944i	\(-0.148200\pi\)
\(17\)	−5.17133	+	0.779453i	−1.25423	+	0.189045i	−0.742351	−	0.670011i	\(-0.766289\pi\)
							−0.511881	+	0.859056i	\(0.671051\pi\)
\(19\)	0.930856	+	1.61229i	0.213553	+	0.369885i	0.952824	−	0.303523i	\(-0.0981630\pi\)
							−0.739271	+	0.673408i	\(0.764830\pi\)
\(23\)	−1.48381	+	1.86064i	−0.309395	+	0.387969i	−0.912082	−	0.410009i	\(-0.865526\pi\)
							0.602686	+	0.797978i	\(0.294097\pi\)
\(29\)	−1.83795	+	4.68303i	−0.341299	+	0.869617i	0.652434	+	0.757845i	\(0.273748\pi\)
							−0.993734	+	0.111772i	\(0.964347\pi\)
\(31\)	0.858768	+	1.48743i	0.154239	+	0.267150i	0.932782	−	0.360442i	\(-0.117374\pi\)
							−0.778542	+	0.627592i	\(0.784041\pi\)
\(37\)	−3.33948	+	8.50886i	−0.549008	+	1.39885i	0.339770	+	0.940509i	\(0.389651\pi\)
							−0.888777	+	0.458340i	\(0.848444\pi\)
\(41\)	−0.0655173	−	0.0607912i	−0.0102321	−	0.00949399i	0.675040	−	0.737781i	\(-0.264126\pi\)
							−0.685272	+	0.728287i	\(0.740317\pi\)
\(43\)	−3.16138	+	2.93333i	−0.482105	+	0.447328i	−0.883399	−	0.468621i	\(-0.844751\pi\)
							0.401294	+	0.915949i	\(0.368561\pi\)
\(47\)	0.723770	−	9.65804i	0.105573	−	1.40877i	−0.653087	−	0.757283i	\(-0.726527\pi\)
							0.758660	−	0.651487i	\(-0.225854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.4.6	yes	648
9.7	even	3		441.2.y.a.151.49	✓	648
49.37	even	21		441.2.y.a.184.49	yes	648
441.331	even	21	inner	441.2.z.a.331.6	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.49	✓	648	9.7	even	3
441.2.y.a.184.49	yes	648	49.37	even	21
441.2.z.a.4.6	yes	648	1.1	even	1	trivial
441.2.z.a.331.6	yes	648	441.331	even	21	inner