Embedded newform 441.2.ba.a.337.37

• Label: 441.2.ba.a.337.37
• Level: $441$
• Weight: $2$
• Character: 441.337
• 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.ba (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			337.37
Character	\(\chi\)	\(=\)	441.337
Dual form			441.2.ba.a.106.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14224 + 0.172165i) q^{2} +(-1.17379 - 1.27366i) q^{3} +(-0.636071 - 0.196202i) q^{4} +(0.529157 - 0.360773i) q^{5} +(-1.12147 - 1.65691i) q^{6} +(-2.14355 + 1.55086i) q^{7} +(-2.77426 - 1.33602i) q^{8} +(-0.244420 + 2.99003i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14224	+	0.172165i	0.807687	+	0.121739i	0.539889	−	0.841736i	\(-0.318466\pi\)
							0.267797	+	0.963475i	\(0.413704\pi\)
\(3\)	−1.17379	−	1.27366i	−0.677690	−	0.735348i
							\(5\)	0.529157	−	0.360773i	0.236646	−	0.161343i	−0.439191	−	0.898394i	\(-0.644735\pi\)
0.675837	+	0.737051i	\(0.263782\pi\)
\(7\)	−2.14355	+	1.55086i	−0.810187	+	0.586171i
							\(11\)	−2.10343	−	0.317042i	−0.634209	−	0.0955917i	−0.175934	−	0.984402i	\(-0.556295\pi\)
−0.458275	+	0.888810i	\(0.651533\pi\)
\(13\)	−1.22608	−	3.12400i	−0.340054	−	0.866443i	−0.993947	−	0.109856i	\(-0.964961\pi\)
							0.653894	−	0.756586i	\(-0.273134\pi\)
\(17\)	−0.602864	+	2.64132i	−0.146216	+	0.640614i	0.847700	+	0.530475i	\(0.177987\pi\)
							−0.993916	+	0.110138i	\(0.964871\pi\)
\(19\)	−0.0842543			−0.0193293			−0.00966463	−	0.999953i	\(-0.503076\pi\)
							−0.00966463	+	0.999953i	\(0.503076\pi\)
\(23\)	−8.13087	−	2.50804i	−1.69540	−	0.522963i	−0.711658	−	0.702526i	\(-0.752055\pi\)
							−0.983746	+	0.179563i	\(0.942532\pi\)
\(29\)	0.678807	−	0.209384i	0.126051	−	0.0388817i	−0.231088	−	0.972933i	\(-0.574228\pi\)
							0.357139	+	0.934051i	\(0.383752\pi\)
\(31\)	−2.81897	+	4.88260i	−0.506302	+	0.876941i	0.493671	+	0.869649i	\(0.335655\pi\)
							−0.999973	+	0.00729242i	\(0.997679\pi\)
\(37\)	2.30149	−	10.0835i	0.378363	−	1.65772i	−0.324121	−	0.946016i	\(-0.605068\pi\)
							0.702484	−	0.711700i	\(-0.252074\pi\)
\(41\)	5.09579	−	3.47425i	0.795828	−	0.542587i	−0.0957754	−	0.995403i	\(-0.530533\pi\)
							0.891604	+	0.452816i	\(0.149581\pi\)
\(43\)	−5.97774	−	4.07556i	−0.911597	−	0.621516i	0.0140511	−	0.999901i	\(-0.495527\pi\)
							−0.925648	+	0.378385i	\(0.876480\pi\)
\(47\)	6.67939	+	1.00676i	0.974288	+	0.146850i	0.616845	−	0.787084i	\(-0.288410\pi\)
							0.357443	+	0.933935i	\(0.383649\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.337.37	yes	648
9.7	even	3	inner	441.2.ba.a.43.18	✓	648
49.8	even	7	inner	441.2.ba.a.400.18	yes	648
441.106	even	21	inner	441.2.ba.a.106.37	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.43.18	✓	648	9.7	even	3	inner
441.2.ba.a.106.37	yes	648	441.106	even	21	inner
441.2.ba.a.337.37	yes	648	1.1	even	1	trivial
441.2.ba.a.400.18	yes	648	49.8	even	7	inner