Embedded newform 441.2.z.a.331.6

• Label: 441.2.z.a.331.6
• Level: $441$
• Weight: $2$
• Character: 441.331
• 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			331.6
Character	\(\chi\)	\(=\)	441.331
Dual form			441.2.z.a.4.6

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.49	✓	648	49.4	even	21
441.2.y.a.184.49	yes	648	9.4	even	3
441.2.z.a.4.6	yes	648	441.4	even	21	inner
441.2.z.a.331.6	yes	648	1.1	even	1	trivial